j. electromagnetic analysis & applications vol.1 no.4-01-04...2004/04/01 · email:...

J. Electromagnetic Analysis & Applications, 2009 Published Online December 2009 in SciRes (www.SciRP.org/journal/jemaa)

CONTENTS

Volume 1 Number 4 December 2009 Dynamic Dipole-Dipole Magnetic Interaction and Damped Nonlinear Oscillations

H. Sarafian………………………………………………………………………………………………………….195

Power Network Asymmetrical Faults Analysis Using Instantaneous Symmetrical Components

S. Leva……………………………………………………………………………………………………………….205

Shunt Reactor Switching Characteristics and Maintenance Planning in 161 kV System H. C. Hsiao, C. Cheng, C. L. Fan…………………………………………………………………………………...214

Optimal Design and Control of a Torque Motor for Machine Tools Y. P. Yang, S. C. Yang, J. J. Liu………………………………………………………………….............................220

Principle and Characteristic of Lorentz Force Propeller J. Zhu……………………………………………………………………………………………………...................229

Improvement of Cylindrical Cloak by Layered Structure of Homogeneous Isotropic Materials

J. Ding, Y. Liu, C. J. Guo, Q. Xu…………………………………………………………………………………….236

Active Power Filter Based on Adaptive Detecting Approach of Harmonic Currents Y. Zhang, Y. P. Tang……………………………………………………………………………………...................240

Monitor System for Protection Device Based on Embedded RTOS Y. Wang, X. G. Yin, Z Zhang………………………………………………………………………………………245

Effect of Weak Magnetic Intergranular Phase on the Coercivity in the HDDR Nd-Fe-B Magnet M. Liu, G. B. Han, R. W. Gao………………………………………………………………………………………249

Monte Carlo Integration Technique for Method of Moments Solution of EFIE in Scattering Problems

M. Mishra, N. Gupta…………………………………………………………………………………………….….254

Numerical Computation of Resonant Frequency of Shorting Post Loaded Gap-Coupled Circular Microstrip Patch Antennas

P. Kumar, G. Singh, T. Chakravarty………………………………………………………………………………259

A Novel PSO Algorithm for Optimal Production Cost of the Power Producers with Transient Stability Constraints

M. Anitha, S. Subramanian, R. Gnanadass…………………………………………………………………………..265

Radio Wave Propagation Characteristics in FMCW Radar G. M. Sami...................................................................................................................................................................275

Design of a Control Device for Synthetic Making Test Based on Pre-Arcing Current Detection and Phase Control

Z. Y. Zhou, M. F. Liao, X. M. Fan…………………………………………………………………………………...279

Copyright © 2009 SciRes JEMAA

Journal of Electromagnetic Analysis and Applications (JEMAA)

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J. Electromagnetic Analysis & Applications, 2009, 1: 195-204 doi:10.4236/jemaa.2009.14030 Published Online December 2009 (http://www.SciRP.org/journal/jemaa)


195

Dynamic Dipole-Dipole Magnetic Interaction and Damped Nonlinear Oscillations

Haiduke SARAFIAN

University College, The Pennsylvania State University, York, USA. Email: [email protected] Received July 10th, 2009; revised August 4th, 2009; accepted August 12th, 2009.

ABSTRACT

Static dipole-dipole magnetic interaction is a classic topic discussed in electricity and magnetism text books. Its dy-namic version, however, has not been reported in scientific literature. In this article, the author presents a comprehen-sive analysis of the latter. We consider two identical permanent cylindrical magnets. In a practical setting, we place one of the magnets at the bottom of a vertical glass tube and then drop the second magnet in the tube. For a pair of suitable permanent magnets characterized with their mass and magnetic moment we seek oscillations of the mobile magnet re-sulting from the unbalanced forces of the anti-parallel magnetic dipole orientation of the pair. To quantify the observed oscillations we form an equation describing the motion of the bouncing magnet. The strength of the magnet-magnet interaction is in proportion to the inverse fourth order separation distance of the magnets. Consequently, the corre-sponding equation of motion is a highly nonlinear differential equation. We deploy Mathematica and solve the equation numerically resulting in a family of kinematic information. We show our theoretical model with great success matches the measured data. Keywords: Dipole-Dipole Magnetic Interaction, Damped Nonlinear Oscillations, Mathematica 1. Introduction

It is trivial to quantify the electrostatic interaction be-tween two point-like charges; however, in practice, it is challenging to deal with point-like charges. On the con-trary, it is common practice to observe the interaction between two magnets; however, it is not that trivial to quantify their mutual interaction. For instance, the trivi-ality of formulating the mutual interaction force between a pair of electric charge results from the fact that there are electric monopoles. We have not observed similar monopoles for the magnets thus far. The mutual magneto static interaction force between two magnets therefore is elevated beyond monopole-monopole interaction; it is considered as magnetic dipole-dipole interaction. Dipoles are geometrically extended objects. Even for planar di-poles intuitively speaking one speculates the interaction force should depend on the relative orientation of the dipoles, let alone the three dimensional configurations. As a common practice, the planar configuration is trivi-alized further to a one dimensional manageable situation; magnets are aligned along their mutual common axial axis [1]. Even for this configuration to the knowledge of the author there is no report utilizing its practical dy-namic application. We fill in the missing gap, and pro-

pose a practical research project. The problem is posed: Consider two permanent mag-

nets. Position them along their mutual common axial axis and orient their magnetic moments so that are anti-par-allel. Drop one of the magnets vertically on top of the second stationary magnet. Select a set of suitable charac-teristics for the magnets, namely the mass of the falling magnet and their magnetic moments, such that the bal-ance between the weight of the falling magnet and the mutual magnetic force between the two magnets results in oscillations. Model the problem theoretically and con-firm the accuracy of the model vs. data.

The analysis of the proposed project embodies a vari-ety of experimental and theoretical challenges. The paper is organized to address both aspects and is composed of five sections. In Section 2, we brief the theoretical foun-dation evaluating the needed axial magnetic field of a permanent magnet and the magnetic force of two inter-acting magnets. In Section 3, guided by the theoretical insight of Section 2, we utilize two independent experi-mental methods and measure the needed magnetic dipole moment of the magnets. In this section we explain also the actual experiment of the bouncing magnet. In Section 4, we lay the foundation for the theoretical model and compare our model to data. In Section 5, we construct a

Dynamic Dipole-Dipole Magnetic Interaction and Damped Nonlinear Oscillations 196

012 2 1 5

2 2 2

ˆ6 .4

( )

zF k

R z

few useful phase diagrams and in Section 6, we address the energy related issues. We close the paper with con-cluding remarks.

(2)

In summary, the project was stemmed from a hypo-thetical conceptual thought experiment. The paper is written descriptively and navigates the reader through the challenges that faced the author. To transit from a thought experiment to reality, suitable magnets had to be sought, experimental methods had to be explored, data acquisition system had to be utilized, and theoretical models had to be investigated. The proposed problem lends itself as a comprehensive physics research project. The paper provides a road map resolving the issues of interest and proves the usefulness of Mathematica [2] as a valuable research tool. From the view of the author, Mathematica to a theorist is what data acquisition utili-ties are to an experimentalist.

Accordingly, the anti-parallel dipole alignment results in a repulsive force and their parallel orientation provides an attractive force, respectively.

Theoretical modeling of the observed oscillations util-izes Equation (2). As we discuss in Section 4, by includ-ing other relevant forces we form an equation describing the motion of the bouncing magnet. We aim to solve the equation of motion symbolically and apply Mathematica. However, because of the highly nonlinear term of Equa-tion (2) Mathematica provides no symbolic solution; we solve the equation numerically. Furthermore, we have observed the pair of our selected magnets always are subject to R<<<z condition where is the length of the

cylindrical magnet. Implementing this geometrical con-straint Equations (1) and (2) simplify, 2. Axial Magnetic Field of a Permanent

03

2( )

4B z

z

Magnet and Dipole-Dipole Magnetic

(3)

Interaction

Magnetic field at a distance z from the center of a counter clockwise steady current i, looping in a horizon-tal circle of radius R along the symmetry axis z perpen-dicular to the loop according to Biot-Savart law trivially evaluates [1],

B

012 2 1 4

1 ˆ6 .4

F kz

(4)

For the chosen set of parameters describing the mag-nets in use, we then compare the numeric solutions of the associated equations of motion utilizing Equation (2) and Equation (4), separately; within the duration of the ob-served oscillations the solutions are indistinguishable. Henceforth, we utilize the simplified format of Equations (3) and (4) throughout our analysis.

20

32 2 2

2 ˆ( )=4

( )

RB z i k

R z

(1)

where is the unit vector along the z-axis and k3. Experimental Data

70

Tesla.m4 1 0

Amp is the permeability of free space.

It is customary to define and apply Equa-

tion (1) in its entirely to a permanent magnet possessing a magnetic moment

3.1 Magnetic Moment of a Permanent Magnet 2 ˆR ik We acquire a variety of cylindrical magnets [3] and ma-

tching glass tubes [4].The magnets can slide freely within the tubes without touching the walls. One at a time we place the tubes containing one of the paired magnets ver-tically in a drilled wooden base. We run a test experiment, meaning, we drop the second magnet into the tube and see if it meaningfully bounces up and down generating data that can be documented. Among successful cases we

select the pairs supporting <<z condition making Equa-

tions (3) and (4) justifiably applicable. Figure 4 of Sub-section 3.2 is a photograph of the actual experiment.

.

With the given quantified value of the magnetic field it is straight forward to determine the magnetic force be-tween two permanent magnets when their moments align along their common axial axis. Viewing the interaction as being the response of the moment of one magnet to the field of the other one, the energy associated with the pair

is . Its spatial variation is the interaction force, 2 1.U B

112 2

ˆ.B

F kz

, meaning, the force is necessitated by the

inhomogeneity of the field. Utilizing Equation (1) the in-

homogeneity evaluates, 3

2 2 2 22

1[ , ]

( ) (

zD z

R z R z

5

2

3

)

To verify the practical reliability of Equation (3) we measure the field directly. Then we compare the trend of the strength of the measured field vs. distance to its theoretical counterpart, Equation (3). For our measure-ment we utilize Pasco™ [5] Magnetic Field Sen-sor/Gauss meter [6], ScienceWorkshop 750 Interface [7] and DataStudio software [8]. The first element of each

and the force becomes,



Figure 1. The left plot is the display of the strength of the field vs. distance from the tip of the Gauss meter to the center of the cylinder magnet. The right graph is the log-log plot version of the left graph pair of the data set embedded in the accompanied code is the distance from the Gauss meter to the tip of the mag-net in mm units; the second element is the averaged value of the measured fields in Gauss. Linear and loga-rithmic plots of the field vs. distance are depicted in Fig-ure 1. The abscissa of both graphs are the distances from the tip of the Gauss meter to the center of the cylinder magnet.

cylindricalMagnedata=30,110,35,80,40,58.6, 45,45,50,36,60,24,70,17,75,14.6/.p_,q_0.1(p+12.7),q; plotlistdata=ListPlot[cylindricalMagnetdata,AxesLabel "z,cm","B,Gauss",PlotRange3,9,0,120,GridLnesAutomatic,PlotStyleBlack,PointSize[0.02]]; lisplot1=ListLogLogPlot[cylindricalMagnetdata, GridLinesAutomatic,AxesLabel"Logz","LogB", PlotStyleBlack]; listplot1=ListLogLogPlot[cylindricalMagnetdata, GridLinesAutomatic,AxesLabel"Logz","LogB", PlotStyleBlack]; listplot2=ListLogLogPlot[cylindricalMagnetdata, GridLinesAutomatic,AxesLabel"Logz","LogB", JoinedTrue,PlotStyleBlack];

s12=Show[listplot1,listplot2]; Show[GraphicsArray[plotlistdata,s12]]

The linear plot of the raw data is depicted in the left graph of Figure 1.The right graph is the log-log display of the same data. The dimensions of the cylinder magnet are: ,2 R=2.54, 1.27 cm. Notably, the strongest

field is measured at 3.0 cm axial distance away from the tip of the cylinder; as shown in both plots this data point falls along the general trend of the rest of the data. Util-izing the right graph of Figure 1, the slope of the slanted line measures 2.9±0.2. The slope lies within the predicted theoretical exponent of the inverse distance of Equation (3).

Guided by the theoretical format of Equation (3) we fit the data applying B(z)=z-. The plot of Log[B(z)] utiliz-

ing the fitted values of , vs. the Log[z] along with the right graph of Figure 1 is shown Figure 2.

, , ,z z ] fitdata=FindFit[cylindricalMagnetdata,

loglogplotfit=LogLogPlot[

Dashing

[0.02], PlotRange->All,10,120];

/ . , , 4,10,z fitdata z

, [0.3],GratLevelPlotStyle [0.008Thickness

Show[listplot1,loglogplotfit,s12]7299.62,2.8922

According to our setup shown in Figure 4, for a typical run the range of the separation distance of the magnets falls within 4cmz8cm. Hence the small deviation of the fitted line (the tail of the dashed line) vs. data (the tail of the solid line) at distances larger than z=7.5 cm justi-fiably may be ignored. The values of the fitted parame-ters , provide two useful pieces of information. First, the slope of the fitted line is =2.8922; as discussed, this matches the slope of the data shown in Figure 1. In other words, for our selected magnets Equation (3) justifiably is applicable. Secondly, the value of =7299.62; hence by comparing B(z)=z- to Equation (3) we deduce

02 ( )4

. The units of is Gauss.cm3, and when

5. 6. 7. 8.Log z

100

50

20

30

70

Log B

Figure 2. The solid line is the same as the right graph of Figure 1. The dashed line is the plot of the fitted function



converted to MKS units, it yields =3.65 Amp.m2. In short, the measurement of field yields the value of mag-netic moment of the permanent magnet.

3.2. Magnetic Moment of a Permanent Magnet, a Static Method

In this section we consider an alternate method to meas-ure the magnetic dipole moment of a permanent magnet. Contrary to the previous method in reference to equip-ment, and in addition to knowing the weight of one of the magnets the only equipment needed is a straight edge. A version of this method has been discussed in [9]. The method is noble; however, without justification the au-thors [9] suggests to apply the approximated field of a solid disk to a set of nut-shaped magnets. On the contrary, our modified approach, as discussed in Subsection 3.1, relies on the proven consistency of theoretical approxi-mated field given by Equation (3) vs. data.

For a typical run, we place a pair of identical cylindri-cal magnets with their anti-parallel moments aligned along their common axial orientation in a vertical glass tube. The anti-parallel orientation of the moments causes a mutual repulsive force resulting in static equilibrium. We then measure the center-to-center separation distance of the magnets -- this yields the value of the moment. The reason is that at static equilibrium the net force on the floating magnet is zero, giving

Figure 3. Four-magnet assembly setup. Magnet poles are col-ored red and blue

12 0F mg (5)

where F12 is the force of the base magnet (#1) on the suspended magnet (#2). Its value is given by Equation (4), where m is the mass of the suspended magnet. For a pair of identical magnets we solve Equation (5) for ,

220

42

16

4

mg

z

(6)

where z2 is the center-to-center distance of the magnets. We introduced the subscripts to identify various general-ized cases -- more explanations are given in the next paragraph. We repeat of the same procedure with three identical cylindrical magnets, yielding

320

4 42 3 2

1 16 [ ]

4 ( )

mg

z z z

and

330

4 43 3 2

1 16 [ ]

4 (

mg

z z z

(7) Figure 4. The setup of the bouncing magnet experiment

)corresponding static equations for the 2nd and the 3rd magnets, respectively. Note, the first subscript of mu indicates the number of magnets for a given setup, and the second subscript corresponds to the magnet of inter-

Here, z2 and z3 are the center-to-center distances from the base to the 2nd and from the base to the 3rd magnets, respectively, and and are the solutions of the 32 33



est. E.g. the indices of 32 are interpreted as the 2nd

magnet of a three-magnet tower. Note that for the num-ber of conducive equations, the value of ’s are one less than the number of the magnets. In our study we extend the static method from a minimum of a two-magnet tower to a five-magnet assembly. We provide one of the four equations for the five-magnet assembly,

(*Import["C:\\DataFiles\\MagneticDipole_Fall2008\\Cyl-indricalMagnetNov28_2008\\IMG_2594.jpg",ImageSi ze120,200,AlignmentCenter];*)

Import["C:\\DataFiles\\MagneticDipole_Fall2008\\Cy lindricalMagneNov28_2008\\position1_Dec4_2008.bmp ",ImageSize400,AlignmentCenter]

550

4 4 45 5 2 5 3 5 4

1 1 1 16 [ ]

4 ( ) ( ) (

mg

z z z z z z z

4)

The run time of the experiment is about 2.0 s and the magnet on an average oscillates eight times with an ap-proximate period of 0.3 s. The largest amplitude of os-cillation typically is 8 cm. The closest distance of ap-proach of the bouncing magnet from the base magnet on average is greater than 4 cm; this is within the z-3 ap-proximation of magnetic field.

(8)

For each assembly we then measure the needed dis-tances and apply the corresponding equations to evaluate the ’s. For each assembly we then plot the values of the moments vs. the number of the magnets. The graph re-veals the distribution of moments about the overall mean value of the moments is sharp; we then objectively evaluate the mean value of each assembly and then evaluate the mean of the mean. Notice also to measure the mass of the magnet we use a balance with an accu-racy of 0.1 grams. Considering the accuracy of a straight edge is 1 mm, our systematic error yields 12%; Hence, our measurement yields =(3.58±0.43) Amp.m2. Com-paring to the measurement of the previous method, al-though the values of the moments are comparable, it ap-pears that we should prefer the method of the previous section. A photo of the setup for a four-magnet assembly is shown in Figure 3.

4. A Theoretical Model

With the data on hand we devise our model. The basis of our model stems from dynamics, meaning, we deal with active forces. In addition to the relevant forces acting on the mobile magnet namely weight and the dipole-dipole magnetic repulsive force, we introduce a new factor, the speed dependent viscous force. The viscous coefficient for the ceramic magnet against the glass tube and the surrounding air is unknown. Therefore, our model em-bodies an unknown parameter. Guided with data we ad-just the value of this unknown parameter; the procedure follows.

Utilizing DataStudio we export the collected data to Microsoft™ Excel [11], apply ExcelLink for Mathe-matica [12] we import the data to Mathematica. The im-ported data is a list of pairs of numbers; applying Mathematica rules we manipulate the list accordingly. For instance, in accordance with the displayed data in Figure 5, we drop the extraneous irrelevant data points and convert the ordinance into MKS units. Furthermore by knowing the distance of the motion sensor from the base magnet conveniently we replace the ordinance of the raw data with the distance between the magnets. Fig-ure 6 displays the center-to-center distance of the bounc-ing magnet vs. the acquired time.

(*Import["C:\\DataFiles\\MagneticDipole_Fall2008\\Cyl-indricalMagnet Nov28_2008\\IMG_2589.jpg", Image Size120,180];*)

3.3. The Bouncing Magnet

The experiment setup is shown in Figure 4. It consists of a pair of identical cylindrical magnets assembled in a suitable vertical glass tube. The tube is placed in a wooden drilled base. Along the extension of the tube using nonmagnetic rods and clamps we mount the Mo-tion sensor [10] and connect it to the Interface [7]. To run the experiment (shown in Figure 4) we lift the top mag-net with the metallic lip of a measuring tape and position it in the front of the motion sensor. We then activate the motion sensor and with a jolt pull the lip of the measur-ing tape away. The suspended magnet falls and bounces

<<ExcelLink`

Utilizing an active Excel file, we read the data. File:C: |DataFiles|MagneticDipole_Fall2008|CylindricalMagnetNov28_2008|position1_Dec4_2008.xls

data=ExcelRead["a7:b197"]; ListLinePlot[Drop[data,50]/.p_,q_p,100q,PlotR

ange1,3.5,0,30]; off the bottom magnet. While the top magnet oscillates up and down the motion sensor collects data; the sensor measures and stores the coordinates of the magnet along with its corresponding time. Applying DataStudio [8] we store the data set and the displayed image. One such im-age is shown in Figure 5. We then import the image and the data to Mathematica.

plotdata=ListLinePlot[Drop[data,50]/.p_,q_p-data[[50,1]],100(-(q+0.0254)+0.340),PlotRange0,13,PlotStyleDashing[0.01,GrayLevel[0.5],Thickness [0.006];GridLinesAutomatic,AxesLabel“t,s”,”z,cm”]




200

The graph shows the bouncing magnet gets as close as 5.5 cm of the tip of the base magnet, and it settles at 7.5 cm away from it. This justifies the applicability of Equa-tion (3).

We formulate the viscous force as v, where and v are the viscous coefficient and the speed of the mobile magnet, respectively. Applying Newton’s second law,

netF ma

at the instance when the bouncing magnet is

accelerating upward along the z-axis yields, 2..

04

1( ) 6( ) ( ) 0

4 ( )z t z t g

m z t

(9)

where = /m is the viscous coefficient per mass. We utilize the mean value of the measured magnetic dipole moment according to the procedures discussed in Sub-sections 3.1 and 3.2. Therefore, with the exception of all of the coefficients of Equation (9) are known. For a set of appropriately chosen initial conditions, namely the initial height of the freely dropped magnet we apply NDSolve and solve Equation (9) numerically. With trial and error we search for such that the solution of Equa-tion (9) reasonably duplicates the data. Figure 7 displays the solution of Equation (9) for =1.9. Utilizing the magnet mass the coefficient of the viscous force, yields 0.0437 kg.s-1. According to the data shown in Figure 5 we had anticipated a small value for such as the one we deduced from our theoretical model. To run our code we supply the known parameters in MKS units.

Figure 5. The data displays the coordinates of the bouncing magnet measured from the motion sensor vs. the corre-sponding time

0.0 0.5 1.0 1.5 2.0 2.5t,s

2

4

6

8

10

12

z,cm

Figure 6. Display of the center-to-center distance of the bouncing magnet vs. the acquired time

n

0.0 0.5 1.0 1.5 2.0 2.5 3.0t,s

2

4

6

8

10

12

z,cm

Figure 7. Comparison of the theoretical model (the solid black line) vs. the experimental data (the dashed gray line)

values=g->9.8, ,70 4 10 323. 10m , mea

n3.63,1.9; eqz=z''[t]-(6(0/(4))mean2)/(mz[t]4)+z'[t]+g/.valu

es; solz=NDSolve[eqz=0,z[0.0]=12.610-2,z'[0.]=0,z[t],

t,0.,4];positionz,speedz,accz=z[t]/.solz[[1]],D[z[t]/.solz[[1]],t],D[z[t]/.solz[[1]],t,2];

plotz=Plot[102 positionz,t,0.,3,AxesLabel"t,s","z,cm",PlotRangeAll,0.,13,GridLinesAutomatic,PlotStyleThickness[0.008],Black];

tabz=Table[positionz,t,0,4,0.02];Manipulate[Show[plotz,plotdata,Graphics[AbsolutePointSize[6],Black,Point[0,102 tabz[[n]]]],PlotRange->All,0,13],n,1,Length[tabz],1]

As shown in Figure 7 the model (the black solid line) fits quite well vs. the data (the dashed gray line). Figure 7 is a snapshot of the animated oscillations of the magnet -- the black disk represents the mobile magnet. The anima-tion code given in the text allows one to run the program in an interactive mode and watch the bouncing disk along the z-axis. The model's amplitude and the period of the oscillations closely follow the data. Within 2.2 s the magnet oscillates eight times. It appears beyond 1.5 s there are somewhat insignificant differences between the


Figure 8. From left to right the plots are: the oscillation amplitude, velocity, and acceleration of the bouncing magnet vs. time. For clarity purpose only we applied CGS units to the first graph and MKS units to the last two graphs

Figure 9. Implicit two and three-dimensional plots of various combinations of kinematic quantities of the bouncing magnet. For the sake of displaying the graphs clearly axes are scaled by certain factors

0.0 0.5 1.0 1.5 2.0t,s

0.005

0.010

0.015

0.020

0.025

0.030Total Energy ,Joule

0.5 1.0 1.5 2.0t,s

0.020

0.015

0.010

0.005

d dt Energy ,Joule s

Figure 10. The left and the right graphs are the display of total energy and the rate of loss vs. time for the bouncing magnet, respectively



phas2Plotza=ParametricPlot[102 positionz,10-1 accz, model calculation and data. However, beyond 1.5 s the amplitudes are about 1.5 cm, and the precision of the motion sensor for this small range is questionable.

t,10-1,3.999,AxesLabel"z,cm"," ,10-1m/s2",Plo

tRangeAll,PlotStyleBlack,AspectRatio1.2];

( )z t

The damped oscillations of the bouncing magnet shown in Figure 7 deceptively resemble the oscillations of a damped simple harmonic motion; recall that the re-storing force describing the latter is a linear force. To show the impact of the nonlinear force of the former i.e z-4, we plot the velocity and acceleration of the bouncing magnet vs. time. These two graphs along with the oscil-lation amplitudes are depicted in Figure 8. An experi-enced reader would realize that the last two plots of Fig-ure 8 distinctly are different from their counterpart of damped simple harmonic oscillations.

phas2Plotva=ParametricPlot[5speedz, 10-1 accz,t,10-1,3.999,AxesLabel" ,5m/s"," ,10-1m/s2",

PlotRangeAll,PlotStyleBlack];

( )z t ( )z t

phasePlotzva=ParametricPlot3D[102 positionz,5speedz,10-1 accz,t,10-1,3.999,AxesLabel"z,m"," ,5

m/s"," ,10-1m/s2",BoxRatios1,PlotStyleThickn

ess[0.007],Black,PlotPoints120];

( )z t( )z t

Show[GraphicsArray[phas2Plotzv,phas2Plotza,phas2Plotva,phasePlotzva]]

Graphs shown in Figure 9 are useful. For instance, the left upper corner graph shows the impact of the viscous force -- it is the cause of the spiral curvature. Similarly, the bottom left graph displays the relationship between acceleration and its associated velocity.

plotspeedz=Plot[speedz,t,10-2,4,AxesLabel"t,s",",m/s",PlotRangeAll,PlotStyleBlack,AxesOrigi

n0,0]; z

plotaccz=Plot[accz,t,10-2,4,AxesLabel"t,s"," , zm/s2",PlotRangeAll,AxesOrigin0,0,PlotStyleBlack]; 6. Energy Characteristics of the Bouncing

Magnet Show[GraphicsArray[plotz,plotspeedz,plotaccz]]

In practice, the magnet is released from its initial height of 12.6 cm away from the base magnet; its initial velocity and acceleration are zero and -8.2m/s2, respec-tively. These are the ordinances of the curves depicted in the last two plots of Figure 8 at t=0. The initial accelera-tion is less than gravity, indicating the impact of the re-pulsive force exerted by the base magnet. The first graph shows the falling magnet; within 0.164 s, it reaches its closest distance of about 5.5 cm from the base magnet. At that moment, its acceleration is at its greatest, ~27 m/s2. As shown in the first graph, the process accordingly repeats itself until the loss of the oscillating amplitudes due to the viscous force, settles the magnet at its stable, at a static equilibrium height of 7.5 cm.

Knowing the time dependent position of the bouncing magnet, z(t), we write its energy E=KE+U. KE is Kinetic Energy and is KE=1/2 mv2; U is potential energy and is

2 1.U mg B

. Applying Equation (3) the potential

energy for a pair of identical magnets yields,

202(4

U mgz

3

1)

z . Now, we utilize the solution of

Equation (9) and evaluate the time-dependent values of the kinetic and potential energies. A display of the total energy vs. time is shown on the left panel of Figure 10. On the right panel of the same figure we display the rate of loss of energy, d/dt E(t); the loss of energy is associ-ated with the viscous force.

plotTotalEnergy=Plot[Evaluate[(1/2m speedz2+2(0/ 5. Phase Diagrams (4)) mean2 1/positionz3+m g positionz)/.values],

Graphs depicted in Figure 8 display various kinematic quantities of the bouncing magnet vs. time. Utilizing Mathematica we fold the time axis and display a pair of quantities such as, ( , , and

implicitly. We also plot one

three-dimensional set, . With the excep-

tion of the velocity vs. position graph, ( ,

commonly known as the phase diagram, the other three graphs as shown in Figure 9 are innovative and fresh.

t,0,2,PlotRange0,All,AxesLabel"t,s","Total Energy,Joule",PlotStyleGrayLevel[0.2],GridLines Automatic];

), ( )z t z t

(z t z t z ( ),

( ), ( )z t z t

( )

z t

dEnergy=D[Evaluate[(1/2m speedz2+2(0/(4)) mean2 1/positionz3+m g positionz)/.values],t]; ( ), ( )z t z t

), t plotdEnergy=Plot[dEnergy,t,0,2,PlotRangeAll,AxesLbel"t,s","d/dt(Energy),Joule/s",PlotStyleBlack,GridLinesAutomatic];

), ( )z t

Show[GraphicsArray[plotTotalEnergy,plotdEnergy]]

positionz,speedz,accz/.t0.164 The left graph of Figure 10 shows the bouncing mag-

net loses its energy over time. This is attributed to the viscous effect. Because of the smallness of the viscous force the loss over the time-span of the oscillations is small. The graph on the right panel of Figure 10 reveals

0.0552837,0.00654413,26.9876 phas2Plotzv=ParametricPlot[102 positionz,4 speedz, t,10-1,4,AxesLabel"z,cm"," ,4m/s",PlotRange

All,PlotStyleBlack];

( )z t



0 1 2 3 4t,s

0.005

0.010

0.015

0.020

0.025

0.030Potential Energy J　　

th

plotMagneticPE=Plot[(2(0/(4)) mean2 1/positionz3)/.values,t,0,4,PlotStyleGrayLevel[0.4],PlotRangeAll,0,0.03,AxesLabel"t,s","Magnetic PE,Joule",GridLinesAutomatic];

Show[plotMagneticPE,plotGravityPE,plotTotalPE,AxesLabel"t,s","Potential Energy,Joule"]

According to Figure 11 and on a par with one's expec-tation, the gravitational (the dark thin gray line) and the magnetic (the light thin gray line) potential energies are completely out of synch. Meaning, initially when the magnets are farthest apart from each other, the gravity potential with respect to the base magnet is at its maxi-mum and the magnetic potential energy is at its minimum. At the end of the fall when the magnets are at their clos-est distance from each other, the former reaches its minimum potential energy while the latter is at maximum. Furthermore, the time dependent variation of the gravita-tional potential energy by itself vs. time is less interesting; it follows the trend of the z(t). However, the time de-pendent variation of the magnetic potential energy has a peculiar character. Its characteristics are a finger print of the non-linearity of the force. We display collectively the various energies of the bouncing magnet in Figure 12.

Figure 11. Display of the gravitational (the dark thin gray line), magnetic (the light thin gray line) and the total (the dark ick gray line) potential energies of the bouncing magnet, vs. time

1 2 3 4t,s

0.005

0.010

0.015

0.020

0.025

0.030

Energy J ,

z m

　　　　

plotKE=Plot[(1/2 m speedz2/.values),t,0,4,PlotStyleGrayLevel[0.6],Thickness[0.008],PlotRangeAll,0,0.006 ,AxesLabel "t,s","KE,Joule",GridLines　Automatic]; plotEnergy=Show[plotTotalPE,plotKE,plotTotalEnergy,AxesLabel"t,s","Enrgy,Jouels",PlotRangeAll];

plotz1=Plot[0.15positionz,t,0.,3,AxesLabel"t,s","z,cm",PlotRangeAll,All,GridLinesAutomatic,PlotStyleThickness[0.008],Black];

Figure 12. Display of potential energy (the semi-dark thick gray line; the second curve from top), kinetic energy (light thick gray line; the bottom curve) and the total energy (the thin black line; the top curve) of the bouncing magnet vs. time, respectively. The middle thick black line is the display of the vertical position of the bouncing magnet vs. time. For convenience, the latter compared to Figuire 6 is down-sized by a factor of 0.15

Show[plotEnergy,plotz1,AxesLabel"t,s","Energy(J), \n z(m)"]

For a comprehensive understanding in Figure 12 we display the potential, kinetic and the total potential ener-gies along with a scaled-down display of the position of the bouncing magnet. By observing the position z vs. time and its corresponding energies one forms a better understanding about the oscillations of the magnet and its corresponding energies. For instance, the black thin line displays how the total energy of the bouncing magnet due to the viscous force dissipates per bounce. It also shows when the oscillations cease the overall value of the total energy is composed of the combination of the gravitational and the magnetic potential energies.

that the rate of the loss of energy is not uniform. To have a better understanding about the time dependent behavior of energy, we dissect the energy and look at the time variations of its composites. Figure 11 is a display of the time dependent variation of the gravitational and mag-netic potential energies.

plotTotalPE=Plot[(2(0/(4)) mean2 1/positionz3+m gpostionz)/.values,t,0,4,PlotStyleGrayLevel[0.4],Thickness[0.008],PlotRangeAll,0,0.03,AxesLabel"t,s","TotalPE,Joule",GridLinesAutomatic];

7. Conclusions

As we pointed out in the introduction, the practical as-pects of dynamic dipole-dipole magnetic interaction have not been discussed in scientific literature. In this article we propose a research project addressing this outstanding

plotGravityPE=Plot[(m g positionz)/.values,t,0,4,PlotStyleGrayLevel[0.2],PlotRangeAll,0,0.03 ,AxesLabel"t,s","Gravitational PE,Joule",GridLinesAutomatic];




204

issue. We apply two distinct practical methods measuring the magnetic dipole moment of a permanent magnet. The measurement of the magnetic moment in one approach acquires deploying a variety of laboratory equipment; the second method simply deploys a straight edge only. The bouncing magnet experiment itself is a repeatable ex-periment and produces reliable data. The data is inter-preted by proposing a semi-empirical theoretical model. The model embodies the dynamic magnet-magnet inter-action and the viscous force as well. The strength of the latter is determined in accordance with data. The author being aware of the utility of Mathematica had proposed and tackled this laborious project. While exploring an uncharted territory we stumbled upon certain issues. Ad-dressing those issues helped us to devise a few fresh useful phase diagrams, as well as reveal the detailed en-ergy characteristics of the bouncing magnet. The article includes a complete set of the needed Mathematica codes including an animation assisting to visualize the bounc-ing magnet. This project also shows how in practice one may import and export data files amongst software such as, Data Studio, Excel, ExcelLink for Mathematica and Mathematica itself. To guide the interested reader to du-plicate the experiments we include two photos of the setups for reference.

8. Acknowledgement

The author wishes to acknowledge Mrs. Nenette Sarafian Hickey for carefully reading over the article.

REFERENCES [1] E.g. J.D. Jackson, “Classical Electrodynamics,” 3rd Edi-

tion, Wiley, 2005.

[2] S. Wolfram, “The Mathematica book,” 5th Edition, Cam-bridge University Publications, 2003.

[3] www.allelectronics.com.

[4] Chemistry lab stock room.

[5] Pasco Scientific™.

[6] Pasco™ Magnetic Field Sensor/Gauss meter, CI–6520.

[7] Pasco™ ScienceWorkshop 750 Interface.

[8] Pasco™ DataStudio software.

[9] R. H. Enns and G.C. McGuire, “Nonlinear physics with Mathematica for scientists and engineers,” Published by Birkhauser, 2001.

[10] Pasco Motion sensor, 003–06758.

[11] Microsoft Excel™.

[12] Excel Link for Mathematica V3.0, Episoft, Inc., 40258 Highway 41, Suite A, Oakhurst, CA 93644.



205


S. LEVA

Department of Energy, Politecnico di Milano, Via La Masa, Milano, Italy. Email: [email protected] Received October 9th, 2009; revised November 3rd, 2009; accepted November 10th, 2009.

ABSTRACT

Although the application of Symmetrical Components to time-dependent variables was introduced by Lyon in 1954, for many years its application was essentially restricted to electric machines. Recently, thanks to its advantages, the Lyon transformation is also applied to power network calculation. In this paper, time-dependent symmetrical components are used to study the dynamic analysis of asymmetrical faults in a power system. The Lyon approach allows the calculation of the maximum values of overvoltages and overcurrents under transient conditions and to study network under non-sinusoidal conditions. Finally, some examples with longitudinal asymmetrical faults are illustrated. Keywords: Power System Fault Analysis, Asymmetrical Faults, Symmetrical Components, Lyon Transformation 1. Introduction

The general Fortescue Symmetrical Components Transf- ormation (SCT) [1,2] is formalized in phasor terms. It can only be used to study steady-state conditions that follow the fault transient condition. The maximum values of overvoltages and overcurrents can only be calculated in an approximate way by means of corrective factors [3].

Recently, the space-vector transformation – used in machine vector control – has been applied to power sys-tem analysis, too [4,5]. Currently, network theory and complex transformation suggest that the study of asym-metrical faults can be carried out by means of instanta-neous sequence components [6–9].

As a matter of fact, by using the same topological ap-proach of the SCT, it is possible to directly analyze the faulty network by differential equations that represent the faults not only in steady-state conditions but also under transient conditions.

As shown by W. Lyon [10,11], the formal aspects of the procedure can be summarized by the following points:

1) the phasors that represent phase- and sequence- variables, are substituted by time-dependent functions, so that the concept of Fortescue sequences can be general-ized to the concept of instantaneous sequences;

2) the Fortescue matrix [ ]S remains the same, and

hence the method confirms the SCT topological and mo-dal-analysis approach [11,12];

3) the phasor operator j is replaced by the derivative operator /p d dt . Under this assumption, differential

analysis is required and depends on the Cauchy initial conditions; and

4) the sequence impedances are converted form ( )Z j to generalized form z(p), maintaining the same

circuital and topological meaning. This time-domain analysis is characterized by three

fundamental features. The first is an applicative one, which regards the ability to calculate - without the use of corrective coefficients – the maximum values of overvoltages and overcurrents during the transient conditions. This is very important for circuit-breaker sizing and the evaluation of the electro-dynamic force between busbars and in transformer windings. The second characteristic concerns the possibility of studying not only sinusoidal, but also non sinusoidal sources. The last characteristic regards the formal and methodological aspects introduced by using the Lyon approach. By means of the Lyon approach, the procedures of dynamic analysis of the network can be unified. In addition, by substituting the SCT with the Lyon approach, fault analysis can be car-ried out by using the state equations that can be inte-grated by classic procedures based on system analysis and the graph approach. The state-equation solutions, can be expressed in literal form by means of analytical for-mulations if the network is linear and time-invariant.

The relations between real and complex transformati-

Power Network Asymmetrical Faults Analysis Using Instantaneous Symmetrical Components 206

ons, steady-state phasors and well-known sequence net-works are given and illustrated through the use of an example with an asymmetrical fault in [6]. The use of dynamic phasors together with space-vectors – incorpor- ating the frequency information – in power system analy-sis is presented in [7] and [8]. To complete these studies in the following, a systematic analysis of the asymmetri-cal faults is developed and deeper investigated both from the theoretical and applicative points of view, giving some important observations that are very useful to achieve the numerical analysis and to better understand the results obtained by using industrial software pack-ages.

The Lyon approach to study transient and steady-state conditions of transversal and longitudinal faults is de-veloped in terms of the following scheme: in Section 2, the Lyon Transformation is recalled and its link with SCT is investigated. In Section 3, the application of the Lyon method to the study of asymmetric transversal and longitudinal faults is formalized and some remarks con-cerning the connection conditions and the use of state-equation approach are put in evidence; furthermore the equivalent model of each fault is calculated. Finally, in Section 4, some numerical examples emphasize the validity of the proposed approach by comparing the ob-tained results with those derived by the SCT method.

2. The Lyon Transformation

Considering an arbitrary time function three-phase set wa(t), wb(t), wc(t), the Lyon transformation gives the following decomposition (where exp( 2 / 3) j ):

2

2

1 1 11

13

1

a o

b

c

w t w t w t

w t w t S w t

w t w t w t

o

(1)

from which it is possible to observe that the matrix S is formally the same for both SCT and Lyon transformation. On the other hand, the functions sub-jected to the Lyon Transformation assume a generic time trend. Taking into account that

1 TS S

then:

02

2

1 1 11

13

1

a

b

c

w t w t

w t w t

w t w t

(2)

Therefore, it is possible to define, starting from a generic three-phase set in time domain, the instanta-neous symmetric components named, respectively.

zero-, positive-, and negative-sequences. The zero- sequence component 0w t is always real. The nega-

tive-sequence component w t is the complex con-

jugate of the positive sequence component w t .

Analyzing Equations (1) and (2) we can see that the Lyon method suggests, time by time and referring to a generic waveform in time domain, the same topological procedures just used with SCT. Moreover, the Lyon transformation, applied to a generic sinusoidal three-phase set, gives the same results provided by the SCT.

Furthermore, the positive Lyon vector satisfies the following identity:

2 2 j tdqw t w t w t e

(3)

and hence it is linked to both Clarke ( )w t and Park

( )dqw t vectors, except for a trivial proportionality

factor. The Lyon transformation, in the context of the modal analysis procedure of the actual three-phase theory, unifies all transformations normally used for dynamic analysis of power networks. In particular – as

( ) ( )w t w t – the real and complex pair of time

functions and 0 ( )w t ( )w t is totally representative

of the generic three-phase set of real time functions wa(t), wb(t), wc(t). The instantaneous power, in terms of the Lyon com-ponent, is [13]:

0 0

pt

abc abc

t t

t v t i t

v t S S i t

(4)

3. Lyon Approach to the Study of Asymmetrical Faults

Lyon decomposition in instantaneous sequence components allows the use of the SCT topological procedures for studying asymmetrical faults that can occur in a power network, by using the Substitution Theorem and the Su-perposition Principle as in SCT [12-14].

Some fundamental remarks about the application to the fault analysis of the Lyon method rather than the SCT are discussed in the following sections.

3.1 Fault Equivalent Networks

In aggreement with Fortescue SCT, the instantaneous sequence networks connection corresponding to the analysed fault configuration starting from the phase cir-cult fault conditions calculation is necessary.

The Lyon transformed fault conditions show how to handle both real- and complex- time functions, while this



is not possible using Fortescue analysis. Consequently, it is important to verify that the connection conditions ob-tained starting from the real conditions are coherent with respect to the definition of an instantaneous sequence components given by (2).

As an example, in the case of a single-phase-to-ground fault, the following relationa are obtained:

0

0 03 pf

i t i t i t

v t v t v t z i t

(5)

where zf (p) is the fault impedance. The first line of (5) shows that the two current Lyon vectors, which are con-jugates, have to be real in order to obtain the zero-sequence current. Moreover, from the second line of (5):

0 0

0

3

2 Re 3

f

f

v t v t v t z p i t

v t z p i t

(6)

Equation (6) confirms that v0(t) is a real time function, too. Similar observations can be applied to the other fault conditions.

The equivalent sequence networks for each fault type are reported in Table 1. Examination of Table 1 reveals that the instantaneous sequence networks connections for the different fault types are equal to that obtained by us-ing Fortescue SCT. This is in agreement with the fact that the Lyon and Fortescue transformations use the same

transformation matrix S . In the time-differential do-main it is possible to use the same phasor expression only by substituting the j factor with the p operator.

The complex impedance ( )Z j becomes the real imp-

edance [15]. Nevertheless, the Lyon transformati- z p

on is of greater generality than the Fortescue transforma- ation: Lyon acts on the time domain, not only in the phasor domain. The SCT can be considered as a particu-lar case of the more general instantaneous sequence components approach.

The results shown in Table 1 and the listed remarks complete the study presented in [6] and [8] analyzing in a systematic way all the asymmetrical faults and presenting the equivalent models of the faults.

Furthermore, Table 1 data together with the aforementioned remarks are very important not only from the theoretical point of view, but, as a matter of fact, these results can be very useful also to the power system ana-lyst to verify the results obtained by using industrial software packages.

3.2 The State-Matrix Approach

The Lyon dynamic analysis of asymmetrical faults can be performed by using the state-matrix approach. It is divided into three distinct stages. In the first step the

power system is represented by the appropriate equivalent sequence networks. The corresponding Lyon state variables (voltages across the capacitors and current flowing in the inductors) are deduced and collected in the Lyon state-vector [x].

Table 1. Instantaneous sequence networks connection

0 + -

0 + -

0 + -

0v

0i

v+

i+

v-

i-

3 ( )fz p

0v

0i

v+

i+

v-

i-

0v

0i

v+

i+

v-

i-

3 ( )fz p

3 ( )fz p

0 + -0v

0i

v+

i+

v-

i-

0A 0B

+A +B

-A -B

0Av

Av+

Av-

Ai+

Ai-

0Ai

Bi+

Bi-

0Bi

0Bv

Bv+

Bv-

1/3 ( )fz p

0A 0B

+A +B

-A -B

0Av

Av+

Av-

Ai+

Ai-

0Ai

Bi+

Bi-

0Bi

0Bv

Bv+

Bv-

0 + -0v

0i

v+

i+

v-

i-

Faults Instantaneous Sequence Networks Connection

Single-Phase-to-ground

Two-Phase-to-ground

Two-Phase

Two-Phase-to-ground

Three-Phase

Single-phase-opening

Two-phase-opening



In the second step, by means of the system and topo-logical procedures of network theory [14], the mathe-matical model of the dynamics of the fault is deduced. Assuming the constitutive relations linear and time- in-variant, it is a priori formalized as:

dx A x B u

dty C x D u

(7)

where the input [u(t)] and the network variables [y(t)] Lyon vectors are present. The [y(t)] vector can be re-garded as the output of the system.

The solution of (7) is well-known and can be obtained in closed form. In fact, knowing the initial values of state variables (at time t=t0), it is possible to assume the fol-lowing expression [14-16]:

0

0( ) ( ) ( )o

tA t t A t

t

x t e x t e B u d (8)

where the first term represents the solution with zero inputs and the second term represents the zero state solu-tion. This last term is calculated considering the general sources [u(t)] expressed in the time domain. In the par-ticular case of sinusoidal inputs, it corresponds to the results also obtained in the phasor domain with SCT when the transient is finished.

The dynamics of the fault depends on the state fault matrix [A], and its elements depend on the sequence pa-rameters related to the type of fault that occurs in the considered power network, and on the initial conditions [x(t0)] analyzed in the following paragraph. The eigen-values of the fault matrix [A] depend on the type of fault and characterize the dynamic of the power system during the fault.

Finally, in the third stage of the study, the network variables [y(t)] are calculated from the second line of (7). The network variables are usually Lyon voltages [v(t)] and currents [i(t)] expressed in the time domain. Equa-tion (1) allows the derivation of the fault dynamics ex-pressed in phase quantities.

Regarding the role of initial conditions, the zero-state network represents the simpler case for a dynamic analy-sis. In fact, in this case, the inductances and the capaci-tances are in zero-state conditions.

If the fault occurs in a non-zero state network, the state variables assume a non-zero initial state [x(t0)]=[x0]; in this case, the voltage vC(t) across a capacitor C and the current iL(t) flowing in an inductor L result to be:

0

0

0

0

1

1

t

C C C

t

t

L L L

t

v t V i dC

i t I v dL

(9)

0CV

1Cp

0LI

Lp'( )Cv t

( )Ci t

( )Lv t

( )Li t

(a) (b)

1Cp

Lp( )Cv t ( )Lv t

( )Ci t

( ) 00C Cv V- = ( ) 00L Li I- =

( )Li t

( )Cv t

'( )Li t

Figure 1. Method that can be used to include the initial condition of the state variables in the proposed approach; case of the capacitance (a) and of the inductance (b). It is represented in the general case, and it is valid for all the instantaneous sequence (+, − and 0)

The reactive elements can be considered in the dy-

namic analysis with an initial zero state condition simply by linking them with a generator that represents the ini-tial conditions (see Figure 1).

In this way, the capacitor must be connected in series with a voltage generator (equal to VC0) and the inductor must be connected in parallel with a current generator (equal to IL0). The new state variables are represented by the voltage '

Cv t across the capacitor C and the current

'Li t flowing in the inductor L, respectively.

Equation (8) becomes:

0

''( ) ' '( )t

A t

t

x t e B u d (10)

where [A’], [B’] and [u’(t)] are calculated considering the new network.

This method is particularly important and useful in the analysis of power networks where some inductances (capacitors) are connected in series (in parallel) with dif-ferent initial conditions.

4. Applicative Case The Lyon approach to studying power system faults pre-sented in this paper is now applied to investigate the cur-rent fault in two different power systems. The first power system is represented by a classic three-phase line considered by other Author [15]. This example can be considered to validate the Lyon method. The second example regards the fault analysis in a real network used in Italy.

4.1 Transient Fault Analysis of a Basic Power System

The network shown in Figure 2 is composed of a gen-erator A, a transformer T2 to elevate the voltage, a three-



phase line L, and a transformer T1. The system has no load when the fault occurs. No information about the grounding connection of the neutral conductor of the generator and transformers is reported in [15], consequently only the three-phase and two-phase faults are ana-lyzed because they are independent to the grounding connection.

The corresponding positive- and negative- instantane- ous sequence networks are reported in Figure 3; the quantities indicated are deducted from the data reported in Figure 2 [5]. The analysis of the previously indicated fault types does not require the zero-sequence instantaneous network.

The Lyon quantities 0w t , ( ), ( )w t w t are be ex-

pressed as follows:

0

0 01

20

0

1

2 j

w

w S e We

w

(11)

Three-phase fault analysis: in this case the positive- and negative- instantaneous sequence networks are short-circuited at the point of the fault. where 0 represents the a phase angle (respect to the

real axis) in which the fault occurs. The corresponding state equations are:

60 MVA

10.5 kV

12 %

G

nG

d

A

V

x

40 MVA

10.5/63 kV

10 %

0.4 %

T

k

r

A

u

u

60 kV

22 km

0.4 /km

0.255 /km

nL

L

L

V

L

x

r

12 MVA

60/10.5 kV

8 %

0.65 %

T

k

r

A

u

u

A 2T L 1T

Figure 2. Power system under analysis

( )e t+

( )i t+

( )v t+

Al 2Tr 2Tl Lr Ll 1Tr 1Tl

( )e t-

( )i t-

( )v t-

Al 2Tr 2Tl Lr Ll 1Tr 1Tl

Figure 3. Positive- and negative- instantaneous sequence net works for the analysis of three-phase and two-phase faults

2 1 2 1

2 1 2 1

T L T A T L T

T L T A T L T

e t r r r i t l l l l pi t

e t r r r i t l l l l pi t

(12) where r and l are the pu resistances and impedances re-spectively.

The line phase currents calculated during the fault are shown in Figure 4. Figure 4 shows the transient move-ment considering φ0=0 and φ0=π/4 respectively, where φ0 represents the phase angle a in the fault instant.

The steady state (sinusoidal condition) values match those calculated by using STC. The maximum value in the steady state condition of the phase fault currents is equal to 0.5208 pu. Under transient conditions the phase currents calculated by using Lyon or STC are instead different. The maximum value reached by the currents during the entire transient depends on φ0. In fact, during the first period of the transient with φ0 = 0, two phase currents reach the value of 0.8 pu, while with φ0 = π/4 the phase b reaches 0.83 pu.

In Figure 5, the vector i t is depicted in the comp-

lex plane. By using a complex vector and its polar repre-sentation the vector magnitude can easily be depicted [5]. The initial magnitudes correspond to the initial condition equal to zero. During the first fault transient instant the current reaches its instantaneous maximum value. At the end of the transient, under steady-state and symmetric condition, the current i t describe a perfect circle.

Two-phase fault analysis: in this case the instantaneous sequence networks are connected in parallel. The cor-responding state equation is:

2 1

2 1

2

2

T L T

A T L T

e t e t r r r i t

l l l l pi t

(13)

Figure 6 shows the line phase currents during the en-tire transient fault calculated with 0 0 and

0 4 respectively. The maximum value of the currents

at the end of the transient (sinusoidal steady-state condi-tion) is 0.451 pu equal to that calculated by using SCT.

Whit φ0 = 0, the b and c phase fault currents reach, during the first period, a maximum value equal to 0.7317 pu. With φ0 = π/4, the maximum value is 0.64 pu.

In this case the real part of the current vector i t is

approximately zero. In accordance with (2), the vector only moves along the imaginary axis of the complex plane starting from 0.

4.2 Transient Fault Analysis of an Existing Power System

The Lyon approach to study power system faults pre-sented in this paper is now applied to perform transversal fault analysis in an Italian exiting power network (Figure



7). The network under analysis is constituted by a high voltage external grid EG, a transformer T, a line L, and a medium voltage load LD. The faults occur on the me-dium voltage busbars.

(a)

(b)

Figure. 4. Three-phase fault: phase currents transient with (a) φ0=0 and (b) φ0=π/4

-0.6 -0.4 -0.2 0 0.2 0.4

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Re

Im i t

Figure 5. Three-phase fault: Lyon time vector i t with

0 4

(a)

(b)

Figure 1. Two phase fault: phase currents transient with (a) φ0 = 0 and (b) φ0 = π/4

EG T

10 km

15 kV LDL

132 kV

Figure 7. One line diagram

To set up the network circuit, the line L is represented by a “Γ” cell, while the transverse parameters of the tran-sformer T are neglected. Furthermore, the transformer is shell core type, which means that the zero-sequence flux component flows in the low reluctance core. Consquently, the zero-sequence impedance is very high. The load is represented by a simple set of impedances. The neutral condition of the external high voltage grid EG is-grounded, while the medium voltage side is not grounded. The network data are reported in Table 2.

Sequence networks and initial conditions. The instan-taneous sequence networks are shown in Figure 8, where the quantities indicated are the pu parameters



Table 2. Numerical parameters of the network in Figure 7

Elements Data

Nominal voltage 132 kVnV

Short-circuit current (HV side) 3 12 kAkHVI EG

Short-circuit power factor cos 0.1k

Connection 0Y y

Nominal power 25 MVAnA

Nominal voltage (HV side) 132 kVnHVV

Nominal voltage (MV side) 15 kVnMVV

Short-circuit voltage 15 5scv . %

Short-circuit power 35 kWscP

No-load current 0 1%i

Tra

nsfo

rmer

No-load power 0 26 kWP


Length 10 kmL

Resistance 1 0.226 Ω/kmr

Inductance 1 0.35Ω/kmx

Capacitance 1 9.65 nF/kmc

Zero-sequence resistance 0 0.371Ω/kmr

Zero-sequence inductance 0 1.536Ω/kmx

Lin

e

Zero-sequence capacitance 0 4.51 nF/kmc


Active power 9 MWcP Loa

d

Reactive power 3.5 MVArcQ

, ( )e t+-

, ( )i t+-

, ( )v t+-

EGr

EGl

Tr Tl

Lc

,Cv +-

Lr Ll LDr LDl

,LDi +-

,EGi +-

,Ti +-

,Li +-

0v

0i

0c 0Lr 0Ll

Figure 8. Positive-, negative-, and zero-sequence instantane- neous sequence networks for the transversal fault analysis of the network depicted in Figure 7

network is composed only by the line zero-sequence pa-rameters.

The computation of the initial condition is performed

calculated starting from the data reported in Table 2. Based on the hypothesis about the type of load and the neutral point connection, the instantaneous zero-sequence

fault occurs. The state quantities result: considering the network under the sinusoidal condition before the

0.165 0.0921 pu

0.1649 0.0923 pu

EG T

L LD

I I j

I I j

0.5619 0.02685 puCV j

(14)

Single-phase-to-ground fault. The sequeare connected in series. The fault does not change the line current values very much: no more than a very small tra

transient - sinusoidal an

zero-instantaneous sequence network is open

nce networks

nsient in the first instants of the fault transient is pre-sent. The fault current (see Figure 9) instead presents in the first instant high frequency oscillations superimposed to the fundamental network frequency (50 Hz). These oscillations with high amplitude decay very rapidly. Nevertheless, the fault current is very low because the network is not grounded: the unique path to the ground is represented by the line capacitors.

Table 3 shows the maximum value of the line currents and voltages calculate by using Fortescue SCT and Lyon ISCT, which are – at the end of the

d equal. Two-phase fault. In this case, the positive- and nega-

tive-instantaneous sequence networks are connected in parallel, the

Table 3. Comparison between maximum value evaluated by SCT and ISCT

Fortescue [pu] Lyon [pu]

max Li 0.2673 0.2673

max av 0.1236 0.1236

max bv 1.2543 1.2545

max cv 1.3219 1.322

max Fi 2. 9. 648·10-4 827·10-3

Figure. 9. Fault current in the first instant of the fault



as shown in Figure 10. Figure 11 shows the line current movement during the fault transient calculated consider-ing φ0 = 0. A high peak in the considered quantities can be observed. In particular, the phases b and c show a peak in the first instants equal to 1.5141 pu and 1.3826

0v

0i

0c 0Lr 0Ll

( )e t+

( )i t+

( )v t+EGr

EGl

Tr Tl

Lc

Cv +

Lr Ll

LDr

LDl

LD+iEGi +

Ti + Li +

( )e t-

EGrLc

EGl

Tr Tl

Cv -

Lr Ll

EGi -

Ti - Li - ( )i t-

( )v t-

LDr

LDl

LDi -

Figure. 10. Instantaneous sequence networks connection for the analysis of two-phase fault

-1.5 -1 -0.5 0 0.5 1 1.5

-1

-0.5

0

0.5

1

Re

Im

Li t

Li t Figure 12. Two phase fault: Lyon time vector wit

Table 4. Current maximum values as a function of φ0

h

φ0 = 18°

0 max puLa b ci t, , max La b c LMaxi t I, , /

-90 1.2859 4.8109 -80 1.3126 4.9108 -60 1.372 5.1331 -40 1.4323 5.3587 -20 1.4841 5.5525 -10 1.5027 5.6221 0 1.5141 5.6647

10 1.5168 5.6748 20 1.5094 5.6471 40 1.4577 5.4537 60 1.3505 5.0526 80 1.2619 4.7212

Table 5. Current maximum value as a function of φ0

0 max puLa b ci t, , max La b ci t LMaxI, , /

-80 1.6685 1.2429 -60 1.6365 1.2191 -40 1.6515 1.2303 -20 1.6685 1.2429 -10 1.6597 1.2364 0 1.6365 1.2191

10 1.6290 1.2135 30 1.6654 1.2406 50 1.6597 1.2364 70 90 1.6654

1.6290 1.2135 1.2406

pu respectively: th values cann lculated by usi Fortescue .

In t kind of fau ne currents ass e values in the nstant ult. The max alue depends on the phase angle φ . I Table IV the maximum val-

ues

ese peak ot be cang analysishis lt, the li ume larg first i of the fa imum v

0

reached by the line phase currents (

n

max i t, , and the ratio between this value and the m

under the pre-fault sinusoidal condition (

La b c )

aximum value

2Figure 11. Two phase fault: line phase currents transient calculated with φ0=0

LMax LI I )

calculated for different phase angles 0 ar ted. Thee repor




213

max value hed for φ0=18°: the ratio imum is reac

( max La b c t I, , /

Fig 12 shows

LMax ) is i close to 5.70.

vector ure the Lyon Li e com-rst fault transient instant the

ous maximu . At the eady-state but metric

t in thplex plane. Duringcurre s its instantaneend o he transiencondition, the curren

the fint reache m valuef t t, in st

t in asym

L

ha

i t describes an ellipse.

an ungrounded network that

lt case, Table V

The two-phase-to-ground fault analysis leads to very similar results. It depends on

s a very high neutral to ground impedance. Three-phase fault. In this case the instantaneous se-

quence networks are short-circuited at the point where the fault occurs. As for the two-phase fau shows the maximum values reached by the line phase

currents ( max La b ci t, , ) and the ratio between this

value and the maximum value under the pre-fault sinu-

soidal condition ( 2LMax LI I ) calculated for differ-

ent phase angles 0 . In this case, the peak value

achieved in the first instants meis 1.24 ti he maximum value of the current in the final steady-state condition.

5. Conclusions

The use, in the time-doma n analysis, of Lyon transformation of asymmetric transversal faults is shown.

s t

i

st.

na

character

y employed for fault analysis, carticular case of the more gene

[2]

port D’Energie Electrique, Dunod, ” Paris, 1966.

lculations,”

[7] A. M. Stankovi sis of asymmetri-cal faults in p namic phasors,”

anced and

in

cas,

. Williams, J. J. DiStefano, Schaum's,

The proposed approach allows the derivation of the Lyon state model of the faulted network and of the transient and steady state voltages and currents of intere

Thanks to the Lyon approach, the peak values reached in the first instants of the fault by the network voltages and currents can be calculated. Furthermore, the complex vectors allow the use of the state equations approach to perform th mic analysis and provide sim-ple relations to steady-state phasors and their rms values. The Lyon approach can also be used for derivation of equivalent circuits ize the different faults and – thanks to the state-matrix approach – its eigenval-ues. These information can be very useful to the power system analysts before starting their analysis by software package simulations.

e network dy

that

nall pa

The SCT, traditiobe considered as a

n ral

three-wire, three-phase systems,” L’Energia Elettrica, Vol. 81, No. 5–6, pp. 51–56, 2004.

[14] W. Lyon, “Application of the method of symmetrical compo-nents,” Mc Graw-Hill Book Company, New York, 1937.

instantaneous sequence components approach proposed by Lyon.

Finally, the examples here presented confirm that the use of time-dependent symmetrical component in net-work calculations has several advantages with respect to the SCT and simulation software: the Lyon transforma-tion allows transient calculations; the simple relation with their steady-state phasors facilitates the interpreta-tion of the results by the well-known steady-state phasor theory and by using complex plane diagrams.

Finally, it is important to underline that network component data are usually available in these coordinates.

REFERENCES

[1] C. L. Fortescue, “Method of symmetrical coordinates applied to the solution of polyphase networks,” AIEE Trans., pt. II, Vol. 37, pp. 1027–1140, 1918.

C. F. Wagner and R. D. Evans, “Symmetrical components,” Mc-Graw-Hill Book Company, Inc., New York, 1933.

[3] H. Edelman, “ThEorie Et Calcul Des REseaux De Trans

[4] L. O. Chua, C. A. Desoer, and E. S. Ku, “Linear and non linear circuits,” McGraw-Hill Inc, New York, 1994.

[5] S. Leva and A. P. Morando, “Analysis of physically symmetrical lossy three-phase transmission lines in term of space vectors,” IEEE Transactions on Power Delivery, Vol. 21, No. 2, pp. 873–882, 2006.

[6] G. C. Paap, “Symmetrical components in the time domain and their applications to power networks caIEEE Transactions on Power Systems, Vol. 15, No. 2, 522–528, 2000.

ć and T. Aydin, “Analyower systems using dy

IEEE Trans. on Power Systems, Vol. 15, No. 3, pp. 1062–1068, 2000.

[8] S. Huang, R. Song, and X. Zhou, “Analysis of balunbalanced faults in power systems using dynamic phasors,” IEEE Proceedings of International Conference on Power Systems Technology, Vol. 3, pp. 1550–1557. 2002.

[9] J. M. Aller, A. Bueno, M. I. Giménez, V. M. Guzmán, T. Pagà, and J. A. Restrepo, “Space vectors applications power systems,” IEEE proceedings of Third International Conference on Devices, Circuits and Systems, Carapp. 78/1–78/6, 2000.

[10] W. Lyon, “Transient analysis of alternating-current ma-chinery,” J. Wiley & Sons, New York, 1954.

G. J. Retter, “Matrix and space-phas[11] or theory of electrical machines,” Akadémiai Kiadó, Budapest, 1987.

[12] A. Gandelli, S. Leva, and A. P. Morando, “Topological considerations on the symmetrical components transfor-mation,” IEEE Transactions on Circuits and Systems, Vol. 47, No. 8, pp. 1202–1211, 2000.

[13] A. Ferrero, S. Leva, A. P. Morando, and A systematic, “Mathematically and physically sound approach to the energy balance in

[15] S. Leva, A. P. Morando, and D. Zaninelli, “Evaluation of line voltage drop in presence of unbalance, harmonics and interharmonics: Theory and applications,” IEEE Transac-tions on Power Delivery, Vol. 20, No. 1, pp. 390–396, 2005.

[16] A. J. Stubberud and I. J“Outline of feedback and control systems,” Schaum's, McGraw-Hill, New York (USA), 2nd Edition. 1990.

[17] R. Roeper, “Short-circuit currents in three-phase sys-tems,” Berlin and München, Siemens Aktiengesellschaft - John Wiley and Sons, 1985.



Shunt Reactor Switching Characteristics and Maintenance Planning in 161 kV System

Horng-Ching HSIAO1, Chiang CHENG1, Chen-Li FAN2 1Department of Electric Engineering, National Taiwan University of Science and Technology, Taiwan, China; 2High Voltage Re-search Lab of Taipower Research Institute, Taiwan Power Company, Taiwan, China. Email: [email protected] Received June 16th, 2009; revised July 30th, 2009; accepted August 10th, 2009.

ABSTRACT

The high frequency transient recovery voltage caused by usually switching operation of the circuit breakers, used on shunt reactor switching, have become a noticeable problem recently. For extension the service life time and normal operation of the circuit breakers, a well modified maintenance strategy is proposed. The field testing and experimental measurement showed the maintenance strategy proposed had been proved effectively and adopted in Taiwan Power Company. Keywords: GCB, Switching, Shunt Reactors, EMTP 1. Introduction

The increasing leading reactive power caused by the un-derground power cables, used in transmission and distri-bution system from 11 kV to 345 kV, have become a noticeable problem and should be solved for voltage and reactive power flow control. To compensate the leading reactive power and depress the voltage rise at the end terminal of cable, the shunt reactors are equipped in power system. But the circuit breakers used for shunt reactors switching caused the unavoidable high fre-quency transient recovery voltage (TRV) during switch-ing operation. The maintenance policy should be planned based on a fixed period maintenance schedule or condi-tional basic maintenance schedule.

The deterioration of insulation material and wear of arcing contact are related with operation times of circuit breaker, switching current and duration of arcing time. We discuss the maintenance policy and performance evaluation by monitoring the energy I2t accumulation of the circuit breakers that are used in Taiwan Power Com-pany.

In this paper, the Electromagnetic Transient Program (EMTP) has been employed to simulate and analyze the transient phenomenon of gas-insulated circuit breaker (GCB) during shunt reactor switching. To increase the realism, the practical Taipower Shen-Mei EHV substa-tion is considered as the simulation system. Based on the theoretical analysis, the three-phase equivalent model is proposed for the EMTP to simulate the switching. Be-

sides, the analysis of transient characteristics has been conducted as the basis of preventive strategy to avoid abnormal operation of GCB.

We got a lot’s of problems on gas-insulated circuit breaker for shunt reactor switching during the past few years in Taiwan power system. The frequently switching of the GCB for shunt reactors degraded gas insulation level. The melted contactors, in turn, could not clear the current prospectively. Meanwhile, the highly rise rate of transient recovery voltage of inductive current switching caused re-striking phenomenon and incomplete tripping. These two main characteristics make GCB using for shunt reactor un-expectantly damaged.

2. Load Switching Simulation Analysis

The simplified circuit model of series RLC and circuit breaker used for switching analysis can be formulated as differential Equation (1), while the waveform of transient recovery voltage, TRV, between source side and load side contactor after circuit breaker opened can be calcu-lated by solving the Equation (1) and expressed by Equa-tion (2).

1cosm

diE t L IR i

dt C dt

t

(1)

(cos cos )atTRV mV E t e b (2)

where 2

Ra

L , and

2

2

1

4

Rb

LC L

Shunt Reactor Switching Characteristics and Maintenance Planning in 161 kV System 215


The practical inductive load switching circuit is shown as Figure 1.

The chopping over-voltage parameter, ka, is shown as Equations (3) or (4).

2

0 0

1am cha

L

V i Lk

V V

C (3)

231

2a

Nk

Q

(4)

where Q is rated capacity of reactor (VA), λ is chopping factor (AF-0.5), ω is angular velocity of system, and N is the series number of poles in a phase. The three phases shunt reactor switching circuit and the over-voltage phe-nomena can be shown as Figure 2 and Figure 3 respec-tively [1,2].

In Figure 3, the waveform includes four stages that are load side oscillation, re-ignition over-voltage oscillation, second parallel oscillation and main circuit oscillation.

The EMTP model for studying inductive current switching is shown as Figure 4. The basic rated parame-ters of the reactor are shown in Table 1. Let:

Lsr: Source side equivalent inductance of phase r. Csr: Source side equivalent stray capacitance of phase

r. Lbr: Equivalent inductance between CB and reactor. CLr: Stray capacitance of load side. Lr : Inductance of reactor. Rr : Intenal resistance of reactor of phase r. Cr : Interior capacitance of reactor of phase r.

Figure 1. Inductive load switching circuit

Figure 2. Three phases shunt reactor circuit

Cpr: Interior capacitance of CB of phase r. Lpr: Interior inductance of CB of phase r. Vsr: Source voltage of phase r The other parameters in phase r of gas-insulated circuit

breaker and equivalent circuit are Lsr = 4.474 mH, Csr = 0.0028 μF, Lbr = 0.0045 mH, CLr = 0.00185 μF, Cpr = 200 pF, Lpr = 0.001 mH, Vst = 19.05 kV, respectively. Sup-

Figure 3. Over-voltage phenomena of inductive load switch-ing

Table 1. Relative parameters of reactor bank

Items Value Unit Rated voltage 33 kV Rated capacity 40 MVA Inductance L 72.2 mH/Phase

Equivalent resistance 0.1082 Ω/Phase Equivalent capacitance 167 pF/Phase

Neutral inductance ∞ H Neutral resistance ∞ Ω

Figure 4. The equivalent circuit model for EMTP

216 Shunt Reactor Switching Characteristics and Maintenance Planning in 161 kV System


pose the phase s and t have the same parameter as phase r and three phases circuit are balanced. The reactors are Y-connected, neutral independent. There are so many uncertain parameters on both Cassie and Mayr arc mod-els and these parameters are difficult to estimated pre-cisely [3,4] that the arc models were not adopted in this study. The circuit breaker model for EMTP analysis is consisted of switch CB parallel connected with Lp and Cp as shown in Figure 4. The simulation procedure of circuit breaker behavior was executed as follow: phase r is the first pole to clear; phase s and t are cleared 4 ms later after phase r. The measured waveform is shown in Figure 5. The simulated waveform of inductive load switching is shown in Figure 6 which is based on the idealized and simplified model. However the amplitude of measured waveform in Figure 5 will be lessened due to high fre-quency switching and mutual coupling.

Figure 5. The measured waveform of inductive load switch-ing

Figure 6. The simulated waveform of inductive load switch-ing

We compared the measured result and simulated result as in Table 2. f1 and f2 represent the first and second os-cillation frequency, two oscillations take 4ms apart as the interrupting time among phase r and phase s and phase t. The simulated oscillation frequency, f1, is almost the same as measuring data [5].

The waveform, simulated and measured, of source side of circuit breaker are shown in Figure 7 and Figure 8.

Table 2. Comparison between measured and simulated result

Frequency Measured Simulated Error f1 10.57 kHz 10.37 kHz 1.89% f2 14.33 kHz 13.16 kHz 8.18%

Figure 7. The measured waveform of source side of circuit breaker

Figure 8. The simulated waveform of source side of circuit breaker

Table 3. Results of simulation of 20A chopping current compared with theoretical value

k theoretical simulation error ka 6.24 p.u. 6.14 p.u. 1.6 % kc 6.24 p.u. 6.07 p.u. 2.72 % krv 7.24 p.u. 7.07 p.u. 2.35 % fL 11.2 p.u. 10.8 p.u. 3.57 %



The waveforms of both simulation and measurement are very similar also. We knew the over-voltage is much influenced by chopping current of interrupting. We stud-ied the chopping over-voltage, the simulated chopping current from 1 A, 5 A, 10 A and 20 A. The simulated result of 20 A, much more influence than others current, is as shown as Table 3 for simplicity. The theoretical values were calculated from the Equations (3) and (4).

From the Table 3, the over-voltage is very high if the circuit breaker has 20A chopping current. The oscillation frequency of load side oscillation, first parallel oscilla-tion, second parallel oscillation and main oscillation are listed in Table 4 [6].

The above simulation was applied to over-voltage study on 161 kV shunt reactor switching phenomenon also. Normally, the shunt reactors were installed in the overhead transmission system to compensate the line capacitive reactive power, especially when much more 161 kV and 345 kV overhead cables were equipped in power system. However, over voltages produced from flashovers, found at spacers of GIS which is used for switching reactors, and sparks found between support structures and cable termination, may degrade the insula-tion performance due to frequently switching. For sur-veying the amplitude and frequency of TRV, different current chopping over-voltage and re-ignition over- voltage were used during simulations after the simulating circuit model was approved. The circuit model realized is shown in Figure 9, and the relative specifications of shunt reactor are listed in Table 5.

The capacitance of shunt reactor and bushing offered by manufacturer are 10,000 pF and 800 pF respectively. The source impedance parameters are given by power system planning division of Power Company. The simu-lated chopping currents are 20A, 10A and 5A and 1.7A. The simulated over voltage waveforms are shown in Figure 10, Figure 11 and Figure.12. The parameters, ka, kr, krv and fr, were calculated according to formulas pro-posed in IEEE C37.015 [4] and listed in Table 6.

Table 4. Oscillation frequency range of TRV

Oscillation type Freq. range Load side oscillation 1~5 kHz

First parallel oscillation 1~10 kHz Second parallel oscillation 50~1000 kHz

Main oscillation 2~20 kHz

Table 5. Parameters of shunt reactors

Rated voltage (kV) 161 345 Rated capacity (Mvar) 80 100

Frequency (Hz) 60 60 Rated current (A) 287 167

Inductance (H) 0.86 3.16 Capacitance (nF) 10 -

Natural frequency (kHz) 1.5 -

Figure 9. Over-voltage simulation model

(file 161cb-R.pl4; x-var t) v:L

0 5 10 15 20 25 30 35

200

40[ms]-300

-200

-100

0

[kV]

100

Figure 10. The over voltage waveform caused by 20A chop-ping current


5 6 7 8 9 10 11 12

150

13[ms]-200

-150

-100

-50

0

50

[kV]

100



4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5

150

[kV]

12.5[ms]-150

-100

-50

0

100

50


218 Shunt Reactor Switching Characteristics and Maintenance Planning in 161 kV System


We have compared the simulation chopping over voltage parameters, which defined in Figure 3 shown, with theoretical calculated parameters from the Equa-tions (3) and (4). The results are shown in Table 6. The errors for 20 A chopping current among the parameters are about 1 %.

We use the model to simulate re-ignition over voltage also. The waveform of reignition over voltage is as Fig-ure 13 shown.

The over voltage waveform measured at site is as Fig-ure 14 shown.

After we found the switching characteristics of the in-ductive load of circuit breaker, the main purposes of this research will be discussed here also.

Table 6. Simulated chopping over voltage parameters

Chopped I 5 A 10A 20A Calculated

ak

1.04 1.6 1.57 1.59

ck 1.04 1.16 1.57 1.59

rvk 2.04 2.16 2.57 2.59

Lf kHz 1.51 1.51 1.51 1.51

(file 161cb-re.pl4; x-var t) factors:offsets:

10

v:L - 10

c:L - 1000

0 5 10 15 20 25 30[ms]-200

-150

-100

-50

0

50

100

150

*103

Figure 13. The re-ignition over voltage waveform

Figure 14. Field measured chopping over voltage

3. Maintenance Strategy

The life time of inductive load switching circuit breaker are mainly influenced by making angle of circuit breaker, arcing energy during contact opening, re-ignition and over-voltage of the circuit breaker. To increase the sta-bility of power system operation, the following method was proposed.

A 144-kV metal oxide arrester was used for over- voltage protection between shunt reactor and GIS. Ad-justing the system parameters may change the frequency and peak value of the over-voltage.

After series calculating statistically the arcing time and arcing energy of inductive load switching, field testing records excused in Taiwan, we proposed a maintenance strategy that the circuit breaker must be re-flashed after every 500 switching operations. Normally, the circuit breaker should be re-flashed after every 2,000 switching operations recommended by manufacturers. But the in-ductive switching for shunt reactors produces high fre-quency and harmful high re-ignition current. The de-structive high current and energy I2t in turn destroyed contactors of the breaker. In field testing, the destructive damages can be observed after 700 switching operations approximately.

Meanwhile a performance evaluation for the circuit breakers should be executed according to the mainte-nance schedule after a re-ignition phenomenon has been found during monitoring the switching behaviors of the breakers daily.

The maintenance prediction of the circuit breaker can then be executed by monitoring the arc energy I2t by digital protection relay, such as GE SR-760, after the setting parameters had been achieved [7].

4. Conclusions

The unavoidable high frequency transient recovery volt-age (TRV) existed in circuit breakers due to inductive switching can be depressed to reasonable and safely level by equipping suitable arrestor proposed in this paper. The maintenance policy for the breakers should be planned based on a fixed period maintenance schedule or condi-tional basic maintenance schedule. Finally, the circuit breaker should be re-flashed after every 500 switching operations to maintain the power system normally.

REFERENCES [1] A. M. Cassie, “A new theory of rupture and circuit sever-

ity,” CIGRE Report, Vol. 102, 1939.

[2] O. Mayr, “Beitrage zur Theorie des statischen und des dynamischen lichtbogens Elektrotechn,” Vol. 37, pp. 589– 608, 1943.

[3] D. F.Peelo and E. M. Ruoss, “IEEE application guide for shunt reactor switching,” IEEE Transaction on Power De-livery, Vol. 11, No. 2, April 1996.



[4] “IEEE application guide for shunt reactor switching,” IEEE Std C37.015–1993, Page(s): I, 25 July 1994.

[6] P. H. Schavemaker and L. Van der Sluis, “The influence of the topology of test circuits on the interrupting per-formance of circuit breakers,” IEEE Transactions on Power Delivery, Vol. 10, No. 4, pp. 1822–1828, 1995.

[5] Z.Ma, C. A. Bliss, A. R. Penfold, A. F. W. Harris, and S. B. Tennakoon, “An investigation of transient over voltage generation when switching high voltage shunt reactors by

6SF circuit breaker,” IEEE Transactions on Power De-

livery, Vol. 13, No. 2, pp. 472–479, April 1998.

[7] C.-L. Fan, “The study on transient recovery voltage ad-justment for high power test laboratory,” M.S. Thesis, National Taiwan Institute of Technology, Taiwan, 1986.



Optimal Design and Control of a Torque Motor for Machine Tools Yee-Pien YANG, Shih-Chin YANG, Jieng-Jang LIU

Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan, China. Email: ypyang@, r94522816, [email protected] Received July 23rd, 2009; revised August 19th, 2009; accepted August 26th, 2009. ABSTRACT This paper presents a systematic approach of optimal design and control of a surface-mount, permanent-magnet syn-chronous torque motor for the next-generation machine tools. A step-by-step procedure of optimization integrates mul-tiple performance objectives and constraints to help the designer make the best decision on the final motor geometry from both design and control perspectives. In the perspective of design, a torque motor with concentrated windings and similar numbers of slots and poles may achieve the desired performance after optimization for multiple objectives, leading to a sinusoidal flux density for a nearly ripple-free torque distribution. From the control perspective, an optimal current waveform with an ideal shift angle is determined for each phase by aligning the current excitation with the back electromotive force. Both design and control of the surface-mount, permanent magnet machine are verified by the finite element method, and a prototype is fabricated for performance validation. Keywords: Multi-Objective Optimization, Torque Motor, Machine Tool, Optimal Current Waveform 1. Introduction In many industrial applications, torque motors attract much attention by their appealing features of high torque density, high efficiency and low ripples. A particular type of permanent magnet (PM) motor uses advanced electronic commutation to replace traditional mechanical commutation. This simple structure makes it easier to fabricate at low cost in the direct or indirect-drive ar-rangement. The direct-drive arrangement eliminates transmission trains and gearboxes to operate at a lower speed with higher torque in various implementations of machine tools and transportation vehicles. The geometry of assembly of a surface-mount, permanent magnet (SPM) torque motor is illustrated in Figure 1.

To achieve its optimal performance, a torque motor needs a systematic methodology of design and control. From the design perspective, the motor configuration and geometry are determined in an optimal way by maxi-mizing the output torque with minimal weight and least power consumption. First, the configuration of concen-trated windings, instead of distributed windings, is de-termined because of its advantages of short end windings and simple structure. Similar numbers of slots and poles are also selected [1–3], so that the slot pitch is close to the pole pitch, not only to reduce the copper loss but also to produce the largest torque in the concentrated wind-

ings. The symmetric winding layout of similar numbers of slots and poles is very suitable for conventional three-phase drives [4]. Second, the motor geometries, such as the shape of the magnet and stator tooth, are cru-cial parameters for motor performance. Hsieh and Hsu [5] and Islam et al. [6] made several investigations on a mo-tor with sinusoidal current excitations and found that the waveform of magnetic flux was very sensitive to the shape of the magnet. Recently, Yang et al. [7] invented a rim motor of sectional stators and arc magnets; its mag-netic flux distribution in the air gap was close to a sinu-soidal function, thereby producing a near-sinusoidal back

Figure 1. The SPM machine assembly

Optimal Design and Control of a Torque Motor for Machine Tools


221

electromotive force (EMF) with fewer harmonics. From the control perspective, it is usually expected to

find an optimal current waveform to maximize the output torque and efficiency of a motor under prescribed con-straints. Jahns and Soong [8] introduced a control-based technique for minimizing torque ripples by tuning the current waveform with an on-line or off-line controller. Chan et al. [9] and Kim et al. [10] proposed their sen-sorless drives for high bandwidth torque control by both the back EMF estimation with fundamental excitation and the saliency-based method with carrier frequency excitation. As for the driving patterns, the optimal cur-rent waveform to produce a maximum torque is propor-tional to the magnetic flux variation in the air gap, which shares the same waveform with the back EMF [11].

Among the previous researches, few papers have ever integrated both the design and control perspectives into the motor design. This paper initiates a systematic ap-proach for designing a torque motor to improve its torque capacity and to reduce torque ripples from both the de-sign and control points of view. The flowchart of the proposed design process is presented in Table 1.

2. Specifications The proposed torque motor will operate for machine tools over a low speed range with high accuracy and resolution, such as the application to a multi-axis high- precision machine center, computer numerical control milling machine or semiconductor handling equipment. A radial-flux SPM motor is chosen as the target sample to satisfy the major specifications in Table 2.

3. Preliminary Design According to the design specifications, a preliminary model of the SPM motor is proposed in Figure 1, featur-ing a large diameter-to-axial-length ratio, which provides a thin ring with more space for bearings, sensors and other components in the hollow shaft of the rotor. The outer surface of rotor is embedded with permanent mag-nets to face their surrounding stator teeth, and the exte-rior of the stator back iron is enclosed by a water cooling house.

3.1 Basic Motor Configuration 3.1.1 Winding Type Concentrated windings have simpler structure with shorter end windings than distributed windings, thereby yielding less flux leakage and power loss. It is also pos-sible for the motor of concentrated windings to increase torque production by similar numbers of slots and poles, which also result in significantly low torque ripples [12]. Therefore, concentrated winding is preferred and se-lected for the torque motor in this paper.

3.1.2 Determination of the Number of Slots and Poles It has been proved that a motor of fractional slot and pole ratio with concentrated windings may have a high torque density, and a motor of similar numbers of slots and poles will make the torque ripple so low that neither ro-tor nor stator skewing is necessary. Three useful factors will help a designer determine the number of slots and poles:

1) Number of slots per pole per phase

2s

sppph

NNpN

= (1)

where Ns is the number of slots, p is the number of pole pairs and Nph is the number of phases. A motor is called fractional when Nspp has a fractional part; when Nspp is less than one and Ns = 2p±2, the motor is said to have

Table 1. Systematic design and control procedure

Table 2. Specifications of torque motor

Item Specification

Objective Torque motor for machine tools

Rated parameters

(1) Torque 518 Nm continuous (convection cooling) 1011 Nm continuous (water cooling) 1250 Nm peak

(2) Voltage: 300-600 VDC (3) Current: 50-100 Amp (4) Speed: Rated 189 rpm (convection cooling) ,

333 rpm (water cooling)

Geometry Outer diameter : 382 mm, Inner diameter: 240 mm, Motor length: 155 mm

Environment Stator and rotor operation temperature 15-25。C Coil winding operation temperature limited at 100。C



222

Table 3. Fractional Nspp and winding factor kw for 3-phase motors Nspp/kw

2p Ns

2 4 6 8 10 12 14 16 20 22

3 1/2 .866

1/4 .866 - - - - - - - -

6 1/2 .866 - - 1/5

.500 - 1/7 .500

1/8 .866

1/10 .866 -

9 1/2 .866

3/8 .945*

3/10 .945*

1/4 .764

3/14 .473

3/16 .328

3/20 .328

3/22 .617

12 1/2 .866

2/5 .933 - 2/7

.933 1/4 .866

1/5 .500

2/11 .250

15 1/2 .866

1/2 .866

5/14 .951*

5/16 .951*

1/4 .866

5/22 .711

18 Distributed windings

1/2 .866

3/7 .902

3/8 .945

3/10 .945

3/11 .945

21 1/2 .866

7/16 .890

7/20 .953*

7/22 .953*

24 1/2 .866

2/5 .933

4/11 .949

*Not recommended for unbalanced radial force

similar numbers of slots and poles. Both properties ren-der a motor with fewer torque harmonics, hence fewer ripples.

2) Winding factor kw The winding factor of concentrated windings is de-

fined as /w phase ph sk E N N=

r (2)

where phaseEr

is the resultant back EMF phasor of a phase formed by its corresponding winding elements. The winding factor is an indication of what portion of the magnet is covered by the stator windings of a single phase.

3) Index factor CT The index factor CT is defined to evaluate the ampli-

tude of cogging torque for various slot and pole numbers:

CT = 2pNs / Nc (3) where Nc is the least common multiple of the numbers of poles and slots. In general, a small index factor may in-dicate small amplitude of cogging torque; its smallest number is 1 when 2p and Ns are relatively prime.

It is convenient to make a table of Nspp and kw for various combinations of slots and poles, among which the promising ones are selected in regard to the index factor, motor size and the performance of torque and torque ripple. Table 3 shows selected slot and pole com-binations of fractional Nspp. First, those combinations with similar numbers of slots and poles, balanced wind-ings and greater slot numbers are selected and marked in grey. Then, the largest winding factor 0.949 for the ma-chine of 24 slots and 22 poles, designated the 24/22 ma-chine, or other machines of the same slot and pole ratio, such as the 48/44 machine, becomes one of the candi-dates for optimal design. Since the specification of dia-

meter in Table 2 allows a large number of slots and poles, the 48/44 machine is selected.

To validate this choice, the performance test was made by comparing it with other arrangements of slots and poles with winding factors of 0.866 and 0.933, which belong to the classes of the slot and pole ratios of 3:2 and 6:5, respectively. Since the diameter specification allows the inclusion of more slots, the machines of 72/48, 60/50 and 48/44 were selected for the performance test in terms of torque and torque ripple, as shown in Table 4.

In this test, all machines were excited with the same magneto-motive force (MMF) from magnets and the same phase current of 50 A from stator windings. The 48/44 machine produced the highest torque among the three machines. Although the torque ripple of the 48/44 machine was a little higher than the 60/50 machine, be-cause of its flux saturating in the stator teeth, it is still the best choice among the three for its largest torque, wind-ing factor and smallest index factor. Being configured of optimal design and finite element refinement will further improve its performance.

3.1.3 Determination of Magnet Shape The back EMF waveform of SPM motors depends on the shape and pitch of magnet poles and stator teeth. The rectangular magnet is most commonly used because of certain manufacturing and cost advantages. The fan-shaped magnet is an improvement over the rectangular one,

Table 4. Performance test result

Parameter Value Slot/Pole 72/48 60/50 48/44

Winding factor 0.866 0.933 0.949 Index factor 24 10 4

Average torque 1022 Nm 1046 Nm 1095 Nm Torque ripple 12.8% 4.1% 6.6%



223

Table 5. Rated specifications for different magnet shapes Magnet shape

Specification Rectangular Fan Arc

Magnet weight (kg) 6.18 6.32 5.56 Rated torque at 64 A (Nm) 1177 1182 1118 Ripple range (Nm) 57.7 50.6 40.9 Active torque density (Nm/kg) 190 187 201 Magnet properties: Remanence (Br) is 1.23T and relative permeability (µr) is 1.1

(a)

winding

d3 ds

As

wt

d1

wo

db

wtb wss

τs

(b)

Figure 2. (a) Arc magnet and (b) design variables of stator and rotor of SPM machine but it may cause manufacturing complexity. The arc magnet is promising for producing a smooth sinusoidal back EMF so as to match the excitation of the sinusoidal current waveform. An ideal arc shape in Figure 2(a) is expressed as a cosine function of magnet pole pitch τm [5] and the air gap length δr on the rotor side can be ap-proximated by

0

1[ 1] , / 2,cos( / )

, / 2 / 2, 1, 2.3...

r a m mm

r m m m

g g for x n wx

g for w x n n

δ δ τπ τ

δ δ τ τ

′≈ − = − − <′

′= − < − < =

(4)

where δ0 denotes the air gap length between the top of the stator teeth and the bottom of the rotor. The coordi-nates x and x′ are the peripheral coordinate fixed on the stator and rotor, respectively, along the circle of the av-erage radius R = (Rsi+Rro)/2, while Rsi is the inner radius of the stator and Rro is the outer radius of the rotor.

Table 5 compares the motor performance for different magnet shapes. The motor with fan magnets provides the maximum rated torque at 64 A, but has a minimum torque density for extra magnet weight. However, the range of ripples, measuring the difference between the upper and lower values of torque ripple, is minimal for the motor with arc magnets. Its active torque density, defined as the ratio between the rated torque and magnet weight, is also maximal among the three. Besides, the back EMF constant produced by the motor with arc magnets is closest to a sinusoidal function, thus provid-ing the lowest torque ripple. The arc magnets, therefore, become the best choice at the preliminary design stage.

3.2. Analytical Magnetic Circuit Model According to the preliminary motor configuration, a 2D analytical magnetic circuit model based on the theory of electromechanical energy conversion is built to describe the performance of the 3D motor in terms of a set of ob-jective functions, such as motor torque, torque density, speed and efficiency. These objectives are expressed as functions of motor geometries, which are illustrated as design variables in Figure 2(b) and are to be determined through the optimization process.

In addition to the assumptions of material linearity and the collinearity of flux and field densities, it is also nec-essary to make three additional assumptions for the 2D magnetic circuit model:

1) The motor is operated in the linear range of the B-H curve of the magnetic material.

2) The air-gap reluctance of the slotted stator structure is approximated by the effective air-gap length with Carter’s coefficient [13].

3) The flux flows straight across the air gaps between the stator and rotor, ignoring the fringing flux to simplify the analysis.

Therefore, the 3D motor structure in Figure 1 can be approximated by a 2D configuration in Figure 3(a) to facilitate the magnetic circuit analysis. By neglecting the flux leakage and armature reaction between the stator and rotor, the MMF from stator windings Fs is simply a square function of magnitude NtI; while the MMF from rotor magnets Fr is a square function of magnet Hclm, where Hc and lm are the coercivity and length of magnet, respectively. The overall MMF distribution is a linear combination of the MMFs of stator windings and rotor magnets

( , ) ( , ) ( , )s rF x s F x s F x s= + (5)



224

where s denotes the rotor shift, which is defined as the relative angle between the rotor and stator.

The magnetic flux density in the air gap is described as

0( , )( , )g

F x sBx s

µδ

= (6)

where µ0 is the permeability of air, δ (x,s) is the effective air gap as a function of slot opening and slot pitch, as shown in Figure 3(b). By the use of the detailed expres-sion of effective air gap [13], the field coenergy in the air gap is expressed as

2 2

00

( , )( )( , )

R

c gF x sW s FB dA L dx

x s

π

µδ

= =∫ ∫ (7)

where L is the axial length of the motor. The torque pro-duced in the motor is then obtained by calculating the variation of magnetic coenergy in the air gap with re-spect to the rotor shift:

( )( ) c

I constant

W sT s Rs =

∂=

∂ (8)

It may be possible to express the coenergy and torque in analytical forms. However, the numerical analysis is used instead to get torque distribution from such a com-plicated magnetic model. Therefore, the coenergy and torque (7) and (8) are rebuilt in the discrete form as

2

01

( , )( ) ( )( , )

N

c ck

F k hW s W h Lk h

θ θµ θ

δ θ θ=

∆ ∆≈ = ∆

∆ ∆∑ (9)

[ ] constant( ) ( ) (( 1) ) ( ) / | ,c c IT s T h W h W hθ θ θ =≈ = + ∆ − ∆ ∆ h=1,2,3,… (10)

where each electric period is divided into N equally spaced points, separated with mechanical position θ∆ from each other, while x k θ= ∆ and s h θ= ∆ . These discrete equations will be used for the multi-functional optimal design in the next step.

4. Optimal Design The performance of the torque motor is usually evaluated by its maximum torque, torque density and efficiency, which are also known as objective functions or perform-ance indices, describing the mechanical and electrical dynamics in terms of motor geometries, magnetic mate-rials and driving conditions, as follows:

1) Rated torque

maxI rated current

( )average of c

rW sT R

s =

∂=

∂ (11)

2) Torque density

F r (x ,s)

F s (x ,s)

s

M M F

x

12 d rlm

1 2 3

(a)

δ

ow

sτ

x

( )eff xδ

'ow

(b)

Figure 3. (a) 2-D motor configuration and magneto-motive force, and (b) effective air gap width due to slotting effect

max /d rT T W= (12)

3) Rated efficiency

max r rr

r r r c s

TT P P P

ωη

ω=

+ + + (13)

Here, the rated torque Tr is an implicit function of de-sign variables, and is calculated through analytical mag-netic circuit models. The motor weight W, rated speed ωr and rated efficiency ηr are all explicit functions of design variables. Their discrete equations can be easily derived and expressed similarly to (9) and (10).

4.1 Sensitivity Analysis First, sensitivity analysis is required to determine the derivatives of the objective functions with respect to the parameters of interest, then a set of design variables are determined. The purposes of sensitivity analysis are:

1) The designer may want to discard those design variables with the least sensitivity of torque, torque den-sity, torque ripple and/or machine efficiency.

2) The designer may keep those design variables con-stant with sensitivities that are linear, or monotonic func-tions.

3) Only those design variables that are not included in the above two cases are retained for the subsequent op-



225

timal design. Table 6 lists all the variables for the sensitivity analy-

sis, while other motor parameters are predetermined in Table 7 according to physical facts and previous design experiences. It would be a time- and space-consuming process to illustrate all the sensitivity curves, though it is worth defining sensitivity indices as the ratio of the variation of motor performance and the variation of de-sign variable. For example, the sensitivity index of maximum torque is denoted by /rT ϕ∆ ∆ , where ϕ symbolizes for design variables. Similarly, the sensitivity indices of toque density Td and efficiency ηr are ex-pressed, respectively, as /dT ϕ∆ ∆ and /rη ϕ∆ ∆ . These sensitivity indices are easily normalized with their maximum values as denoted on the x-axis of Figure 4.

It is found that the output torque is greatly affected by the variations of air-gap length (δ), tooth width (wtb), slot opening (wo), magnet length (lm) and magnet fraction (am = wm/ τm) as shown in Figure 4(a). However, the back iron thickness (db), number of turns (Nt) and the copper wire diameter (dw) have little influence on the output torque. Figure 4(b) indicates that the torque density is greatly influenced by the magnet fraction, magnet length and slot opening while other variables seem to be unre-lated to torque density. Figure 4(c) shows that the mag-net length, slot opening and air-gap length, and, espe-cially, the magnet fraction, have significant influence on the torque ripple. Other variables, such as tooth width, shoe depth (d1), back iron thickness (db), number of turns and copper diameter, seem to have no influence on the torque ripple. According to the sensitivity index of effi-ciency, it is not surprising that both slot opening and tooth thickness have an indispensable influence on effi-ciency as shown in Figure 4(d). Furthermore, both the air-gap length and the diameter of copper have signifi-cant influence on the motor efficiency.

According to the sensitivity indices, the shoe depth, back iron thickness and copper diameter can be dis-carded because they do not significantly influence the performance of the motor. The air-gap length, which is a compromising factor between motor performance and cost, is set at 1 mm by considering the manufacturing tolerance and accuracy. Finally, five design variables — the magnet fraction, magnet length, slot opening, tooth thickness and number of turns, denoted by stars in Table 6 - are determined for multifunctional optimization.

4.2 Multifunctional Optimization In terms of the design variables chosen from the sensitiv-ity analysis, the performance indices — torque, torque density and efficiency — of the torque motor are im-plicitly or explicitly written as (11) through (13). The compromise programming method in the multifunctional optimization system tool (MOST) [14] is applied to

search for the optimal values of design variables that maximize these performance indices.

The optimizer weights the performance indices to reach a satisfactory compromise among the design vari-ables subject to the prescribed constraints: 1) The motor dimensions must be realized, e.g. Ro>Ri; 2) The slot opening is twice larger than the air gap length, but 0.35 times less than the slot pitch; 3) The shoe depth fraction, defined as the ratio between shoe depth (d1) and tooth length (ds), is confined less than 0.5; 4) The slot current density is less than 9×106 A/m2; 5) The flux density in the electrical steel is less than its saturation value of 1.8 T; 6) The peak value of back EMF per phase should be less than the component of the driving voltage along the back EMF vector.

Different weightings were assigned to the three per-formance indices for which relative importance is ad-dressed for the optimization, but the three best results in terms of motor performances are listed and compared in Table 8. The example with the highest weighting on effi-ciency, column 1:1:6, cannot provide enough torque and torque density, and is eliminated first.

Table 6. Design variables for sensitivity analysis No. Name Variable 1 Magnet fraction* am 2 Magnet length* lm 3 Slot opening* wo 4 Shoe depth ds 5 Tooth width* wtb 6 Stator back iron thickness db 7 Air gap length δ 8 Number of turns* Nt 9 Copper wire diameter dw

*Candidates for optimal design Table 7. Predetermined variables for sensitivity analysis

Specifications Value Stator outer radius 191 mm Rotor inner radius 120 mm Rotor axial length 155 mm

Magnet shape Arc Relative permeability 1.1 H/m

Magnet density 7.45 g/cm3 Magnet remanence 1.23 T

No. of Phase 3 Current waveform Sinusoidal wave

No. of slots 48 No. of poles 44 No. of turns 60

Magnet length 8 mm Slot opening 4 mm Tooth width 11.5 mm

Air gap length 1 mm Magnet fraction 0.8

Length of stator tooth 38.04 mm Stator back iron thickness 10 mm

Shoe depth 5 mm Copper wire diameter 1 mm



226

(a)

(b)

(c)

(d)

Figure 4. Sensitivity index of (a) output torque, (b) torque density, (c) torque ripple, and (d) efficiency

Table 8. Optimized motor variables and performance

Tr : Tr /W : ηr 1:1:1 6:1:1 1:1:6 Stator slot opening(mm) 4.098 4.979 5

Tooth thickness(mm) 11.41 12.01 11.11 Magnet length (mm) 7.98 8 6.56

Magnet fraction 0.823 0.871 0.808 Number of turns 61 65 59

Phase current (Arms) 50 50 50 Motor weight (kg) 58.42 58.62 57.35

Ohmic loss (W) 3089 3937 2073 Core loss (W) 98 98.7 92

Maximum speed (rpm) 132 116 143 Output torque (Nm) 1091 1240 975

Torque density (Nm/kg) 18.34 20.70 16.68 Efficiency (%) 90.02 87.70 92.8

The example of column 6:1:1 with the highest weight-

ing on torque has the highest torque and torque density, but yields the worst performance in ohmic and core loss, maximum speed and efficiency among the three exam-ples. Finally, the optimization result with equally impor-tant weightings on the three objectives, column 1:1:1, presents a balanced and satisfactory performance in torque, torque density and efficiency, and is chosen for the FE verification and refinement.

5. Finite Element Verification and Refinement

The 3D FE analysis is responsible for verifying the effec-tiveness of the 2D optimal design by including nonlinear characteristics. Owing to the symmetric structure, a quarter section of the motor with a half electric period is sufficient to model and analyze the expected performance.

The resulting motor performance by the FE verification is shown in Table 9. It is also expected that the FE refine-ment through the adjustment of motor geometries, such as shoe depth, slot opening and back iron thickness, must further improve the motor performance. Apparently, the torque density is increased because of the reduction of weight, and the torque ripple is also suppressed by re-shaping the geometry of slot opening, tooth depth and back iron thickness.

6. Optimal Current Waveform It has been proved that the maximum torque is produced if the phase current is proportional to the angular rate of change of the field flux of the motor [15]. In other words, the best current waveform is the same as the back EMF wave of the motor in order to produce the maximum torque with minimal current. The reduction of torque



227

ripple in the previous design steps means the back EMF of the motor needs to be made as close as possible to a sinusoidal wave. Besides, the current phase shift may have some influence on the torque production especially for the configuration of similar numbers of slots and poles in the three-phase drive.

Figure 5(a) illustrates the minimum magnetic circuit model with 12 slots and 11 poles for the 48/44 machine, where the coils are arranged as AA’AA’B’BB’BCC’CC’, and coils A and A’ are wound in opposite directions. Figure 5(b) lays out the EMF vectors of windings, where the slots of coils A1, -B1, and C1 are distributed in 120°E (electric degrees) of phase offset and the best shift γ of each phase current must lead them in 22.5°E, thereby remaining a balanced configuration. It was found that not only the torque increased 8.4%, but the ripple also de-creased by changing the current shift from 0 to 22.5°E.

7. Prototype Fabrication and Experiments The axial length of the motor was previously specified at 155 mm. To make the prototyping easier, it was down-sized to 55 mm in view of precise machining and accu-rate assembling. Inevitably, such reduction of axial length must cause the rated torque to decrease by about 1/3 of the optimally designed motor. Figure 6(a) shows the sta-tor core and concentrated windings, and the rotor assem-bly, glued on the outer surface of which are NeFeB 40SH magnets. A complete assembly of the torque motor is shown in Figure 6(b).

The performance of the prototype motor was tested with the voltage supply of 220V. Figure 7 compares the back EMF waveforms from the FE analysis and experi-ment, where both curves are close to the pure sine func-tion, but the error between the experimental back EMF and pure sine function is even less than 5%. It is also interesting to point out that the back EMF constant from the experiment is 4.48 V/rad/s, which is very close to 4.57 V/rad/s from the FE analysis. Figure 8 shows the relationship between the torque and current, where Tmag=31.2Irms is for the magnetic circuit model, TFE=30.7Irms for the FE analysis and Texp=33.7Irms+5.43 for the experiment. The measured torque constant 33.7 Nm/A from the experiment is close to but slightly higher than those from the magnetic circuit model and FE analysis, partially because the measured torque may in-clude additional friction from the bearing of the motor assembly. The offset of 5.434 Nm may account for the friction in bearing or inevitable measurement errors.

8. Summary and Conclusions A systematic approach of design and control for a per-manent magnet synchronous torque motor for machine tools was proposed. The design procedure was illustrated step by step in the order of specification, preliminary

Table 9. FE verification and refinement

Variables FE verification FE refinement Slot opening (mm) 4.098 4.210

Stator back iron (mm) 10 7 Tooth width (mm) 11.41 11.50 Rated torque (Nm) 1081 1122 Motor weight (kg) 60.67 57.96

Torque density (Nm/kg) 18.15 18.93 Torque ripple (%) 3.6 2.29

Copper ohmic loss (W) 3084 3084 Core loss (W) (at rated speed) 96.2 90.0

Efficiency (%) 90.0 90.3

(a)

(b)

Figure 5. (a) Minimum magnetic circuit model of the 48/44 machine, and (b) its balanced winding layout

(a) (b)

Figure 6. (a) Stator, rotor, and (b) their assembly of motor prototype design, optimal design, finite element verification and refinement, and optimal control current waveform. These procedures help an engineer perform a complete design without missing any key issues. The sensitivity analysis and multifunctional optimization provide efficient and essential information for the designer to make a final de-cision for motor geometry. The precise finite element tool not only verifies but also refines the preliminary and



228

-5-4-3-2-1012345

0 50 100 150 200 250 300 350 400

Electric position (degree)

Bac

k EM

F co

nsta

nt (V

/rad/

s)

0

5

10

15

20

25

30

35

Erro

r (%

)

Pure sine Experiment

FEA Error(between experiment and pure sine)

Error(between FEA and pure sine)

Figure 7. Back EMF waves from FE analysis and experi-ment

0.5 1 1.5 2 2.5 3 3.5 40

20

40

60

80

100

120

140

Out

put t

orqu

e (N

-m)

Phase current (A)

AnalyticExperimentalFEA

Figure 8. Torque constant curves

optimal design of the machine by improving its perform-ance. The determination of optimal current waveform and phase shift further allows the machine to operate at its best capacity. Finally, the experiments on a prototype presented satisfactory performance in terms of back EMF wave, back EMF constant and torque constant, thereby validating the effectiveness of the optimal design and control strategy. The proposed design approach must facilitate motor designs in an effective way, especially for an engineer of little experience. The only deficiency of this study is the necessity of creating magnetic circuit models which differ from motor types and are only best approximated in two dimensions under various linear assumptions.

9. Acknowledgments This work was supported by National Science Council of Taiwan, China, under Contract NSC95-2221-E-002-132- MY2.

REFERENCES

[1] B. Stumberger, G. Stumberger, M. Hadziselimovic, A. Hamler, M. Trlep, V. Gorican, and M. Jesenik, “High-

performance permanent magnet brushless motors with balanced concentrated windings and similar slot and pole numbers,” Journal of Magnetism and Magnetic Materials, Vol. 304, pp. e829–e831, 2006.

[2] D. Ishak, Z. Q. Zhu, and D. Howe, “Comparison of PM brushless motors, having either all teeth or alternate teeth wound,” IEEE Transactions on Energy Conversion, Vol. 21, pp. 95–103, 2006.

[3] C. C. Hwang, S. P. Cheng, and C. M. Chang, “Design of high-performance spindle motors with concentrated windings,” IEEE Transactions on Magnetics, Vol. 41, pp. 971–973, 2005.

[4] J. Cros and P. Viarouge, “Synthesis of high performance PM motors with concentrated windings,” IEEE Transa- ction on Energy Conversion, Vol. 17, pp. 248–253, 2002.

[5] M. F. Hsieh and Y. S. Hsu, “An investigation on influence of magnet arc shaping upon back electromotive force waveforms for design of permanent-magnet brushless motors,” IEEE Transactions on Magnetics, Vol. 41, pp. 3949–3951, 2005.

[6] M. S. Islam, S. Mir, T. Sebastian, and S. Underwood, “Design considerations of sinusoidally excited permanent-magnet machines for low-torque-ripple applications,” IEEE Transactions on Industry Applications, Vol. 41, pp. 955–962, 2005.

[7] Y. P. Yang, W. C. Huang, and C. W. Lai, “Optimal design of rim motor for electric powered wheelchair,” IET Electric Power Applications, Vol. 1, pp 825–832, 2007.

[8] T. M. Jahns and W. L. Soong, “Pulsating torque minimization techniques for permanent magnet AC motor drives-a review,” IEEE Transactions on Industrial Electronics, Vol. 43, pp. 321–330, 1996.

[9] T. F. Chan, W. Wang, P. Borsje, Y. K. Wong, and S. L. Ho, “Sensorless permanent-magnet synchronous motor drive using a reduced-order rotor flux observer,” IET Electric Power Applications, Vol. 2, pp. 88–98, 2008.

[10] T. Kim, H. W. Lee, and M. Ehsani, “Position sensorless brushless DC motor/generator drives: Review and future trends,” IET Electric Power Applications, Vol. 1, pp. 557–564, 2007.

[11] Y. P. Yang and D. S. Chung, “Optimal design and control of a wheel motor for electric passenger cars,” IEEE Transactions on Magnetics, Vol. 43, 2007, pp. 51–61.

[12] Z. Q. Zhu and D. Howe, “Influence of design parameters on cogging torque in permanent magnet machines,” IEEE Transactions on Energy Conversion, Vol. 15, 2000, pp. 407–412.

[13] V. Ostovic, “Computer-aided analysis of electric machines,” New York: Prentice Hall, 1994.

[14] C. T. Tseng, W. C. Liao, and T. C. Tang, “MOST user's manual,” in Mechanical Engineering, 1.2ed Taiwan, Hsinchu: National Chiao-Tung University, 1993.

[15] Y. P. Yang, Y. P. Luh and C. H. Cheung, “Design and control of axial-flux brushless dc wheel motors for electric vehicles – Part I: multi-objective optimal design and analysis,” IEEE Transactions on Magnetics, Vol. 40, No. 4, July 2004, pp.1873–1882.



229

Principle and Characteristic of Lorentz Force Propeller

Jing ZHU

Northwest Polytechnical University, Xi’an, Shaanxi, China. Email: [email protected] Received August 4th, 2009; revised September 1st, 2009; accepted September 9th, 2009.

ABSTRACT

This paper analyzes two methods that a magnetic field can be generated, and classifies them under two types: 1) Self-field: a magnetic field can be generated by electrically charged particles move, and its characteristic is that it can’t be independent of the electrically charged particles. 2) Radiation field: a magnetic field can be generated by electric field change, and its characteristic is that it independently exists. Lorentz Force Propeller (ab. LFP) utilize the charac-teristic that radiation magnetic field independently exists. The carrier of the moving electrically charged particles and the device generating the changing electric field are fixed together to form a system. When the moving electrically charged particles under the action of the Lorentz force in the radiation magnetic field, the system achieves propulsion. Same as rocket engine, the LFP achieves propulsion in vacuum. LFP can generate propulsive force only by electric energy and no propellant is required. The main disadvantage of LFP is that the ratio of propulsive force to weight is small. Keywords: Electric Field, Magnetic Field, Self-Field, Radiation Field, the Lorentz Force

1. Introduction

The magnetic field generated by a changing electric field is a kind of radiation field and it independently exists. When the moving electrically charged particles are sub-jected to Lorentz force in the magnetic field, the device generating the changing electric field isn’t subjected to any reacting force.

2. Theoretical Basis

As we know, a magnetic field can be generated by two methods: one is electrically charged particles move and the other is electric field change [1]. However, the mag-netic fields generated by the two methods are entirely different in nature.

We discuss the magnetic field generated by moving electrically charged particles at first. It’s well known that the electric quantity carried by the electrically charged particles isn’t affected by the motion state of the particles. This shows that when the electrically charged particles generate a magnetic field due to their movement, the electric quantity owned by the particles doesn’t change with the generation of the magnetic field. Only the dis-tribution of the electric energy, i.e., the distribution of the electric field, is affected by the motion state of the parti-cles. (We will mention below that even this change is

also due to the changes in observation angle.) “If the electric quantity carried by the particles is certain, the magnetic field generated by the particles is entirely de-termined by the motion speed of the particles” [2] and it is a single-valued function of the motion speed. In other words, there must be a certain magnetic field corre-sponding to the electrically charged particles when the motion state of the particles is determined. This suggests that the magnetic field generated by the moving electri-cally charged particles is the state quantity which de-scribes the motion state of the electrically charged parti-cles. The state quantity, which reflects the sate of a physical object, is different in different reference systems, but it can’t be separated from the physical object and can’t be independent of the physical object. Thus, the magnetic field generated by moving electrically charged particles is different in different reference systems. Of cause the magnetic field can’t be independent of the electrically charged particles and it’s the self-field of the particles.

As the magnetic field generated by moving electrically charged particles, kinetic energy also belongs to state quantity. They reflect the motion state of a physical ob-ject in the form of energy. The difference between them is as follows. The motion state of a physical object is described by its mass in kinetic energy, which is central-

Principle and Characteristic of Lorentz Force Propeller 230

ized at the object. The motion state of the electrically charged particles is described by the electric quantity in the magnetic field, which is generated by the moving particles and distributed in the space around the particles.

Then we discuss the magnetic field generated by a changing electric field. Based on the energy conservation principle, the total energy in the electromagnetic field isn’t changed in the course that a magnetic field is gener-ated by a changing electric field. This indicates that the magnetic energy is continuously generated and the elec-tric energy is continuously decreased at the same time. The magnetic energy is transformed from the decreased electric energy. The course that a magnetic field is gen-erated by a changing electric field is a course that electric energy is transformed into magnetic energy. The electric energy and the magnetic energy are localized in the elec-tric field and the magnetic field respectively. According to “the superposition principle of electric fields” [3], the initial electric field can be regarded as the superposition of two mutually independent electric fields in the above-mentioned course at any time. One of the two electric fields is the electric field with the localized elec-tric energy, which has been transformed into magnetic energy. The other is the electric field with the localized electric energy, which hasn’t been transformed into magnetic energy. Obviously, the former electric field generated a magnetic field but the magnetic field has disappeared. The latter electric field still exists but it hasn’t generated any magnetic field. Because the two electric fields are mutually independent, the magnetic field generated by the former electric field is independent of the latter electric field. In other words, the magnetic field and the electric field are mutually independent in the course that the magnetic field is generated by the changing electric field. The former electric field is also a tie between the magnetic field and the device generating the changing electric field and it’s shown as the device generating the changing electric field → the electric field → the magnetic field. With the disappearance of the former electric field, the magnetic field generated by the changing electric field is not only independent of the electric field, but also independent of the device generat-ing the changing electric field at the same time. The changing electric field and the device generating the changing electric field are only the initial condition to generate a magnetic field. The magnetic field generated by the changing electric field is a radiation field and it’s irrelevant to reference system.

The relation between the magnetic field generated by a changing electric field and the device generating the changing electric field is similar to the relation between electromagnetic waves and an antenna. The antenna is only the initial condition to generate electromagnetic waves. When the electromagnetic waves have been gen-erated, they are irrelevant to the antenna. In other words,

the electromagnetic waves are separated from the an-tenna and independent of the antenna. [4]

It’s another issue that a changing magnetic field also generates an electric field, which is known as the issue that magnetic energy is transformed into electric energy. Similarly, the electric field generated by the changing magnetic field is independent of the magnetic field. “A magnetic field is generated by a changing electric field, and an electric field is generated by the changing mag-netic field” [5], so repeatedly. The process is continu-ously repeated. The most common example is electro-magnetic waves. Just because not only the magnetic field generated by a time varying electric field but also the electric field generated by the time varying magnetic field is a radiation field, electromagnetic waves are formed and spread continuously. On the surface, the electric field and the magnetic field in the electromag-netic waves are tightly fastened and linked. But in fact, they are mutually independent and irrelevant to each other.

The law of a time varying electromagnetic field fol-lows the Maxwell equation. “The differential form of the Maxwell equations in free space is given as follows [6]

D

t

BE

t

D 0

B 0 ”

It can be seen from the equations that the size of the magnetic field intensity is determined by the changing rate of the electric field. However, according to the above-mentioned, the magnetic field is transformed from the disappeared electric field, so the size of the magnetic field intensity should be determined by the changing amount of the electric field. They seem to be contradic-tory. This is because that the two courses that the time varying electric field generates the magnetic field and the time varying magnetic field generates the electric field occur at the same time, rather than the magnetic field begins to transform into the electric field after the electric field has thoroughly transformed into the magnetic field. It can be seen more easily from electromagnetic waves.

Finally, we discuss the different changes in the electric field when the electrically charged particles have differ-ent motion speeds or have acceleration. The relations between the changes and electromagnetic radiation are also investigated.

The situation that the electrically charged particles have different motion speeds is firstly discussed. As shown in Figure 1, a is a stationary particle carrying positive charge and its electric field is distributed in the space



Figure 1. The reason for the deflection of the electric field line around it. An arbitrary electric field line, E, is chosen and discussed. It’s supposed that the angle between the elec-tric field line, E, and the Y-axis is α. A certain part of the electric field line, called E0, which starts from the particle, is randomly intercepted from E. The electric field line E0 is decomposed into two mutually perpendicular compo-nents. One, called EX, is parallel to the X-axis. The other, called EY, is parallel to the Y-axis. When the electrically charged particle, a, moves in a straight line with a uni-form speed along the X-axis direction, the length of EX is shortened because it’s parallel to the motion direction, shown as EX1. The phenomenon is known as Lorentz contraction. Nevertheless, the length of EY is unchanged because it’s perpendicular to the motion direction. It can be seen from Fig.1 that the vector sum of EX1 and EY is no longer E0 but E1. The angle between E1 and the Y-axis is β and β<α is satisfied. This means that the electric field line is deflected towards the Y-axis and the particle is the circle center. So the distribution of the electric field changes. The faster the motion speed of the particle is, the larger the Lorentz contraction is. Thus the deflection of the electric field line is more obvious. In practice, an electric field line starts from a positive charge and ends at a negative charge. So we can only observe the deflec-tion of the electric field line, rather than observe the Lorentz contraction. In Figure 1, the length of EX is shortened caused by the Lorentz contraction, but its in-herent length is unchanged. So the “inherent angle” be-tween the electric field line and the Y-axis is unchanged and the “inherent distribution” of the electric field is also unchanged. From the above-mentioned, it’s conduced that the change of the electric field of the electrically charged particles caused by the different motion speeds of the particles is the relative change between electrically charged particles and reference system in essence. Therefore, the magnetic field generated by the moving electrically charged particles aren’t generated by the

change of the electric field. Electrically charged particles moving in a straight line with a uniform speed can’t gen-erate any electromagnetic radiation.

“The electric field of stationary electrically charged particles and the electric field of electrically charged par-ticles moving in a straight line with a uniform speed are both radial. The electric field lines of them are also straight” [7] and the inherent distributions of them are also the same. This state of the electric field of the parti-cles is called a normal state. When the electrically charged particles have acceleration, the electric fields of the particles are no longer radial and a horizontal com-ponent perpendicular to the radial direction will emerge in the electric field. The horizontal component is irrele-vant to the motion speed of the particles and it’s deter-mined by the accelerations of the particles. Therefore, the inherent distribution of the electric field changes. Ac-cording to the superposition principle of electric fields, the horizontal component and the radial component can be regarded as two mutually independent electric fields. And they are known as the horizontal electric field and the radial electric field. Then they are respectively dis-cussed below.

No matter how the motion speed of the electrically charged particles changes, the electric field of the parti-cles is in a normal state. This indicates that the changing electric field caused by the acceleration of the electrically charged particles tends to return to a normal state. In other words, the horizontal electric field tends to disap-pear. The disappeared horizontal field will generate a magnetic field, that is, a changing electric field generates a magnetic field, and the electrically charged particles generate electromagnetic radiation. When the accelera-tion of the electrically charged particles remains un-changed or increases, the horizontal electric field doesn’t decrease or disappear, and it remains unchanged or in-creases on the contrary. At the same time, the electro-magnetic radiation still exists. It seems incompatible with the opinion that a magnetic field is generated while an electric field disappears. The reason is that an external force (because the electrically charged particles have acceleration) and the electromagnetic radiation act si-multaneously. The external force keeps the horizontal electric field increasing continuously and the electric field lines bending continuously at the same time. The electromagnetic radiation keeps the horizontal electric field decreasing constantly and the electric field lines constantly returning to being straight. When the effect of the external force is equal to or in excess of the effect of the electromagnetic radiation, the increased horizontal electric field is equal to or in excess of the decreased horizontal electric field. At this time, the phenomenon that the electromagnetic radiation is continuously gener-ated and the horizontal electric field keeps unchanged or



increases appears. The deflection of the radial electric field is caused by

the change of the motion speed of the particles. As above-mentioned, it’s irrelevant to the generation of the radiation magnetic field, that is, it is irrelevant to the electromagnetic radiation.

We have got two conclusions that analyze the different origins of magnetic fields:

1) A magnetic field that can be generated by electri-cally charged particles move is the self-field of the parti-cles, and it can’t be independent of the electrically charged particles.

2) A magnetic field can be generated by electric field change is radiation field, and it independently exists.

3. Operating Principle

The magnetic field generated by a changing electric field is a kind of radiation field and it independently exists. When the moving electrically charged particles are sub-jected to Lorentz force in the magnetic field, the device generating the changing electric field isn’t subjected to any reacting force. The Newton’s third law is not tenable between the electrically charged particles and the device generating the changing electric field.

The carrier of the moving electrically charged particles and the device generating the changing electric field are fixed together to form a system. The relative position between them makes the moving electrically charged particles is located in the magnetic field generated by the changing electric field and the motion direction of the electrically charged particles is unparallel to the direction of the magnetic field. The magnetic field generated by the changing electric field is independent of the device generating the changing electric field, so it’s independent of the foregoing system. For the foregoing system, the Lorentz force subjected by the moving electrically charged particles in the magnetic field generated by the changing electric field is attributable to a external force of the system. When the carrier of the electrically charged particles moves under the action of the Lorentz force, the device generating the changing electric field is driven and moves together with the carrier. Thus, the system achieves propulsion. The above-mentioned is the design idea of the new novel propulsion system. The new propulsion system utilizes Lorentz force, so it’s called as the Lorentz force propeller, which is LFP for short here-inafter.

Obviously, conductor can be selected as the carrier of the moving electrically charged particles. Yet, what kind of devices can be used to generate a radiation magnetic field?

The most common radiation magnetic field is the magnetic field in electromagnetic waves. That is to say that an antenna is the most common device being able to generate a radiation magnetic field. But there are two

main problems if an antenna is selected as the device generating a radiation magnetic field. One is that it’s difficult to improve the magnetic induction intensity, which is crucial to improve the propulsion force. The other is that it will produce electromagnetic radiation pollution. So we have to find a new way to solve these problems. A coil (solenoid) is selected as the device to generate a changing electric field.

When we talk about a coil, i.e., inductance, we gener-ally consider that “the energy of the inductance is stored in the magnetic field generated by the current.” [8] How is the energy stored in the inductance? For easily dis-cussed, we suppose that the coil is ideal, that is, it hasn’t internal resistance.

As we all know, “when a time varying current flows through a coil, a magnetic field is generated in the coil and an induced electric field is generated in the coil winding” [9]. Generally, the induced electric field is also time varying. At a steady state, the induced electric field can’t generate any current because it’s disturbed by an external electric field. The changing induced electric field will generate a magnetic field. How can we know this? It has been known that the inductive reactance of the coil increases and the current decreases if other con-ditions are unchanged and only the frequency of the time varying current increases. The limit case that the fre-quency approaches infinity is assumed. In this case, the inductance is broken, no current flows through the coil and the induced electric field reaches the maximum. There is no current means that there is no energy input or output. At this time, the energy stored in the inductance should be a definite value. But the induced electric field is time varying. Where is the energy gone when the in-duced electric field decreases? Where is the energy from when the induced electric field increases? The answer is that the time varying induced electric field generates a time varying magnetic field. The energy is transformed backwards and forwards between the electric field and the magnetic field, and the total energy of the electro-magnetic field remains unchanged.

In general, when a time varying current flows through a coil, two magnetic fields are generated in the coil. One is the magnetic field generated by the current, that is, the magnetic field generated by the moving electrically charged particles. The other is the magnetic field gener-ated by the induced electric field, that is, the magnetic field generated by the changing electric field. The two magnetic fields are superposed.

Therefore, it can be deduced from the abovementioned that the energy stored in the inductance should include three parts: the magnetic field generated by the current, the induced electric field and the magnetic field gener-ated by the induced electric field. The first part of the energy will flow into and out of the inductance together with the current. When it reaches a steady state, the other



two parts of the energy is always stored in the coil. If the frequency of the current is zero, that is, the cur-

rent is a direct current, the induced electric field is zero and only the magnetic field generated by the current ex-ists in the coil. With the increase of the frequency of the current, the current gradually decreases and the magnetic field generated by the current also gradually decreases. At the same time, the induced electric field gradually increases and the magnetic field generated by the in-duced electric field also gradually increases. When the frequency of the current approaches infinity, the current disappears and the magnetic field generated by the cur-rent also disappears. At the same time, the induced elec-tric field reaches the maximum and the magnetic field generated by the current also reaches the maximum. The magnetic field generated by the induced electric field is the magnetic field generated by the changing electric field and it is a kind of radiation field. This is the reason that the higher the frequency of a current is, the more powerful the radiation is. The number and magnitude of radiation magnetic fields isn’t the only factor forming electromagnetic waves. The generation of electromag-netic waves is also determined by the characteristics of the antenna.

As we all know, an antenna is not only a transmitting antenna, but also a receiving antenna. In other words, it can not only transmit electromagnetic waves, but also receive electromagnetic waves at the same time. In fact, an antenna shows the characteristics of transmitting and receiving simultaneously when it transmits electromag-netic waves. When it generates a radiation magnetic field, the antenna will receive them back more or less at the same time. The received part forms electromagnetic in-duction and the unreceived part forms electromagnetic waves. [10]

Strictly speaking, the magnetic field generated by the induced electric field shouldn’t be regarded as the energy stored in the coil because it is a kind of radiation field and is independent of the coil. The magnetic field is gen-erated by the coil and received by the coil in turn. From this point of view, the coil can be regarded as an antenna. The phenomenon that a coil receives its own radiation magnetic field is called self-induction of the coil. More-over, the receiving is not necessarily 100%, for example, inducing mutually in the coil or radiating electromagnetic waves.

As shown in Figure 2, the conductor I0 gets through the coil L0, and they are fixed together and form a system. a is the main view and b is the left view. When there are time varying currents flowing through the conductor I0

and the coil L0 respectively, the moving electrically charged particles in the conductor I0 are located in the magnetic field generated by the coil L0 and their direc-tions are unparallel. The conductor I0 will be subjected to two Lorentz forces. One is the acting force that the cur-

Figure 2. The operating principle of Lorentz Force Propel-ler rent-carrying conductor is subjected in the magnetic field generated by the current and the other is the acting force that it is subjected in the magnetic field generated by the induced electric field. For the above-mentioned system, the former acting force is an internal force of the system. Its reacting force is the Lorentz force that the coil wind-ing is subjected in the magnetic field generated by the current-carrying conductor. Since the conductor I0 and the coil L0 are fixed together, the couple of acting force and reacting force can’t make any relative movement between the conductor I0 and the coil L0. So the relative position between them is also unchanged. The latter act-ing force is an external force of the system and it hasn’t any reacting force. When the current-carrying conductor I0 moves under the action of the force, the coil L0 will be driven and move together with the conductor. Therefore, the system achieves propulsion.

Because the direction of the magnetic field generated by the induced electric field is time varying, the current in the conductor I0 should also be time varying in order to achieve propulsion. The frequencies of them must be same, and the phase difference between them is 0 or 1/2 cycle. The direction of the propulsive force when the phase difference is zero is opposite to that of the propul-sive force when the phase difference is 1/2 cycle. In ad-dition, there is no magnetic core (iron core) in the coil L0. Since it’s generated by the moving electrically charged particles, the magnetic field doesn’t contribute to propul-sion.

Like electric motor, LFP also utilizes the Lorentz force. But there are two fundamental differences between them.

1) Electric motor utilizes the magnetic field generated by electrically charged particles, whereas LFP utilizes the magnetic field generated by a changing electric field.

2) In electric motor, the current-carrying conductor



and the device generating magnetic field can relatively move and they are called stator and rotor. In LFP, the current-carrying conductor and the device generating magnetic field are fixed and there is no relative move-ment between them.

Just because the foregoing differences, LFP and elec-tric motor have completely different functions. Electric motor can’t achieve propulsion by itself and a propulsion device must be adopted. For example, cartwheels are used to achieve propulsion by the friction between cart-wheels and the ground. So it can’t be called a propeller. LFP can achieve propulsion by itself and it can propel in vacuum.

4. Design Optimization

In the LFP shown in Figure 2, both of the two ends of the coil L0 are open, and the external of the coil may gener-ate magnetic field. One part of the conductor I0 is in the coil and the other part is out of the coil. The directions of the two magnetic fields that the two parts are located are opposite. That is to say, the directions of the Lorentz forces subjected are also opposite. It’s harmful to propul-sion. In addition, some other problems, such as radiation pollution and electromagnetic induction, may arise at the same time. In order to overcome these disadvantages, the LFP shown in Figure 2 is optimized and shown in Figure 3. In Figure 3, L1 is an end-to-end annular coil. I1~I6 are the six conductors getting through the coil Ll. They are divided into two layers. I1, I2 and I3 are located at the upper layer and I4, I5 and I6 are located at the lower layer. The two parts are fixed with the coil L1 respectively. F is the direction of the propulsion force.

L1 is a closed coil and the magnetic field in the exter-nal of the coil is eliminated. So the problem resulting from this disappears. It should be explained that because the direction of the magnetic field that the conductors I1, I2, I3 are located is opposite to the direction of the mag-netic field that the conductors I4, I5, I6 are located, there is 1/2 cycle phase difference between the current in the conductors I1, I2, I3 and that in the conductors I4, I5, I6. Figure 3 is the schematic drawing. In practical applica-tion, the shape and turn number of coil, the number and position of conductors can be adjusted and determined according to different requirements.

It’s very important to manufacture propellers with large propulsive force in practice. To obtain more pow-erful propulsive force, the most intuitive method is to increase the amplitude and the frequency of the time varying current. However, it’s restricted by many factors, such as materials and manufacture techniques. Thus, the effect is limited. We conceive a new novel method to obtain large propulsive force, which is described as fol-lows.

L2 is a helical closed coil, shown in Figure 4 a. It can also be considered that one end of the eight L1 is cut and

then they are connected end to end. For simplicity, there are two conductors, I7 and I8, are shown in Figure 4 a. They get through the upper and lower layer of the coil respectively. The direction of the propulsive force is perpendicular to the paper. The propulsive device shown in Figure 4 a can be fixed on a circular pedestal. Therefore, a propulsive unit, which looks like an optical disc in appearance, is ob-tained. Then many of these propulsive units are organi-cally assembled, for example, stacked as shown in Figure4 b, to obtain more powerful propulsive force. In Figure 4(b), F is the direction of the propulsive force.

Figure 3. The three-dimensional schematic drawing of de-sign optimization of Lorentz Force Propeller

Figure 4. The schematic drawing of Lorentz Force Propel-ler with large propulsive force


Principle and Characteristic of Lorentz Force Propeller


235

5. Main Characteristics

An integrated Lorentz force propulsion system includes three main parts: energy supply system, control system and LFP. The function of the energy supply system is to supply electric energy for the whole system. It may be batteries, combustion engines, nuclear reactors and so on. The main function of the control system is to control the magnitude and the direction of the propulsive force by controlling the current inputting into the LFP. The func-tion of the LFP is to transform electric energy into pro-pulsive force. The energy supply system and the control system should be determined according to different con-crete situations and requirements. With different selec-tion, the two systems may be quite different. Therefore, only the characteristics of LFP are discussed in the paper and the integrated Lorentz force propulsion system isn’t involved. Moreover, because LFP is a new novel design, the studies on materials, manufacture techniques and so on related to LFP are blank. And these are the important factors determining the performance of LFP. Thus, only qualitative analysis on LFP is made and no quantitative analysis is done.

Like rocket engine, LFP can generate propulsive force without any external medium. So the main characteristics of LFP are described by the comparison with rocket en-gine.

1) LFP can generate propulsive force only by electric energy and no propellant is required.

2) The specific impulse of LFP is large. If solar energy or nuclear energy is used, the specific impulse may be astronomical. (The specific impulse is an important index to describe the performance of rocket engine through the assumption of propellant. LFP doesn’t use any propellant. For comparison, the assumption of propellant is replaced by the assumption of fuel here. In mechanical rocket en-gine, fuel has the same meaning as propellant.)

3) The propulsive force of liquid-propellant rocket en-gine can be adjusted in a certain range by adjusting the throttle and other mechanical components. The ratio of propulsive force can reach 10:1.

LFP can adjust propulsive force directly by controlling the current and no mechanical component is required. The adjusting accuracy and response speed are far supe-rior to those of rocket engine. The adjusting range is be-tween 0 and the maximum propulsive force. The ratio of propulsive force approaches infinity and the direction of propulsive force can be reversed 1800.

4) There is severe mechanical vibration when rocket engine is working. Although the propulsive force of LFP is time varying, there is no mechanical vibration because the frequency is high.

5) When LFP is working, no sound is produced and no exhaust gas is emitted. So it’s beneficial to protect envi-ronment.

6) Compared with rocket engine, the main disadvan-tage of LFP is that the ratio of propulsive force to weight is small.

6. Claims

LFP has been applied for a patent. So please do not use without permission.

REFERENCES

[1] B. S. Guru and H. R. Hiziroglu, “Electromagnetic field theory fundamentals,” (Second Edition) (Chinese Ver-sion), published by Cambridge University Press in 2005, P5, P7. There have expatiated generating methods of magnetic field.

[2] F. T. Ulaby, “Fundamentals of applied electromagnetics,” (Fourth Edition) (Chinese Version), 2004 Media Edition, 013185089X by ULABY, published by Pearson Educa-tion, Inc., P8.

[3] F. T. Ulaby, “Fundamentals of applied electromagnetics,” (Fourth Edition) (Chinese Version), 2004 Media Edition, 013185089X by ULABY, published by Pearson Educa-tion, Inc., P6.

[4] N. N. Rao, “Elements of engineering electromagnetics,” (Sixth Edition) (Chinese Version), published by Pearson Education, Inc., P427, P449, P450. There has expatiated the relation between electromagnetic waves and an an-tenna.

[5] B. S. Guru and H. R. Hiziroglu, “Electromagnetic field theory fundamentals,” (Second Edition) (Chinese Ver-sion), published by Cambridge University Press in 2005, P185, 2005.

[6] J. A. Edminister, “Schaum’s outlines theory and problems of electromagnetics,” Second Edition (Chinese Version), P148, 2002.

[7] B. S. Guru and H. R. Hiziroglu, “Electromagnetic field theory fundamentals,” (Second Edition) (Chinese Ver-sion), published by Cambridge University Press in 2005, P57.

[8] T. L. Floyd, “Principles of electric circuits: Conventional current, seventh edition,” (Chinese Version), published by Pearson Education, Inc., P430

[9] T. L. Floyd, “Principles of electric circuits: Conventional current version,” (Seventh Edition) (Chinese Version), published by Pearson Education, Inc., P429, P431

[10] B. S. Guru and H. R. Hiziroglu, “Electromagnetic field theory fundamentals,” (Second Edition) (Chinese Ver-sion), published by Cambridge University Press in 2005, P366, P386, P387. There has expatiated the course that antenna radiates electromagnetic waves.




Jun DING, Yang LIU, Chenjiang GUO, Qian XU

School of Electronic and Information Engineering, Northwest Polytechnical University, Xi’an, China. Email: [email protected] Received September 8th, 2009; revised September 26th, 2009; accepted September 29th, 2009.

ABSTRACT

The perfect cylindrical cloak requires three spatial variant material parameters, which is very difficult to realize in practice [Science 312, 1780 (2006)]. The approach of realizing the electromagnetic cloaking by concentric layered structures instead of using the metamaterial was presented [Optics Express, Vol. 15, No. 18 (2007)]. We use the con-centric layered structures to realize a simplified cylindrical cloak with improved parameters and an ideal cylindrical cloak with spatially invariant axial material parameters. Numerical simulation results validate that cloaking perform-ance is significantly improved compared with previously proposed multilayered cloak.

Keywords: Cylindrical Cloak, Concentric Layered Structures

1. Introduction

Recently the idea of designing an invisible cloak by means of coordinate transformation approach has drawn extensive attention [1–23]. Pendry et al. [1] first sug-gested an interesting way of using the coordinate trans-formation method to design a cloak of invisibility. Many theoretical studies have shown the possibility of such invisible cloaks [2,3]. The first experimental demonstra-tion of such invisible cloak using metamaterial with sim-plified material parameters was soon realized [4], the simplified cloak inherits some properties of the ideal cloak, but finite scatterings exist. Perfect electromagnetic invisible cloak requires inhomogeneous and anisotropic media, therefore it is difficult to construct by using natu-rally existed materials. A cylindrical cloak structure that does not require metamaterials to realize the anisotropy and inhomogeneity of the material parameters was pro-posed [9], it was realized by a concentric layered struc-ture consisting of alternating homogeneous isotropic materials, which can be treated as an effective medium with the radius dependent anisotropy.

In this paper, we use the concentric layered structures to construct a simplified cylindrical cloak with improved parameters and an ideal cylindrical cloak with spatially invariant axial material parameters. It may be realized by layered structures of composite media.

2. Cylindrical Cloak by Layered Structure of Homogeneous Isotropic Materials

Cylindrical anisotropy can be attained by layered struc-

ture of homogeneous isotropic materials [10,11]. As shown in Figure 1, we assume that each layer can be described by homogeneous and isotropic permittivity and permeability parameters. The layer thickness in each of these structures is much less than the wavelength

d

and when d , we can treat the structured material as a single anisotropic medium with dielectric permittivity

1 2 1 2

2 1

(1 ),

1r

(1)

where 2 1/d d is the thicknesses ratio of the two

layers. As proposed in Reference [9], the cylindrical invisible

with inhomogeneous and anisotropic media can be con

12

1d 2d

Figure 1. Concentric layers structure alternate with dielec-tric layers



237

structed through layered structures of homogeneous isotropic materials. Now we concentrate on the realization of the improved simplified cloak and the ideal cylindrical cloak through multilayered structures of composite ma-terials.

As proposed in Reference [16], we could obtain an-other set of cloak parameters satisfying the requirement, the associate material parameters of the cloak can be expressed as

2( ) , ,r

b r a b b

b a r b a b a

z (2)

We first apply this idea to construct a cylindrical cloak with improved parameters, combining Equations (1) and (2), only r , are considered and z is invariant, we

obtain the expressions:

2 1 2

2 1

1 2

(1 )( )

1

b r a

b a r

b

b a

(3)

There are two ways to achieve the multilayered cylin-drical cloak. One is to keep the thickness ratio a con-

stant value and alternate layers of isotropic media 1

and 2 vary with the radius. The other way is to con-

struct a concentric layered structure that the permittivity of the alternating dielectrics is fixed, while the thickness ratio of the two layers varies to approximately satisfy the Equation (3).

Here, we take the fist approach as an example, assume the thickness of the two layers are equal ( 1 ), the cy-

lindrical cloaking shell with inner radius a= λ and outer radius b = 2λ, then Equation (3) can be written in the form:

21 2 1 24, 4( )

r a

r (4)

Next, we consider a stepwise homogeneous ten dis-crete layer approximation of the ideal, continuous pa-rameters required by Equation (4), which has removed the most challenging issue of the design, the continuous radius-dependent, anisotropic medium could be repre-sented approximately by ten discrete layers of homoge-neous anisotropic medium. Then we use the alternating layers of isotropic dielectric 1 and 2 to mimic each

homogeneous anisotropic layer, the permitivities of them are designed by Equation (4) according to the corre-sponding anisotropic layer. The thinner the layers are, the better the layered structure approaches the ideal ani-sotropic medium. We can conclude the dielectric 1

and 2 in each discrete layer by Equation (4), as shown

in Table 1.

The relative permittivity of anisotropic medium varies gradually, dielectric 1 increases gradually from 0.008

to 0.268, while the relative permittivity of dielectric 2

decreases gradually from 3.992 to 3.372. Here, we should note that the relative permeability of the layers is 2, which may be realized by composite media. Now, we have completed the material parameters description.

Following the above approach, we can easily realize the ideal cylindrical cloak through multilayered struc-tures of composite materials. Yu Luo et al. [17] have provided a perfect cylindrical invisible cloak with spa-tially invariant axial material parameters:

2 2 2 2

2 2 2 2, ,r z

r a r b

r r a b

2a (5)

Combining Equations (1) and (5), we obtain the ex-pressions:

2 21 2

22 1

21 2

2 2

(1 )

1

r a

r

r

r a

(6)

Plugging in the cloak parameters value, we find that the media properties are given by

1 2 1 ,2

1 2 2 2

2r

r a

(7)

We can also conclude the dielectric 1 and 2 of

each discrete layer in multilayered cloak with perfect

Table 1. The permittivity of ten-layered multilayered cloak with improved simplified parameters

Layer (unit: ) 1.0~1.1 1.1~1.2 1.2~1.3 1.3~1.4 1.4~1.5

1 0.008 0.028 0.054 0.083 0.114

2 3.992 3.972 3.946 3.917 3.886

Layer (unit: ) 1.5~1.6 1.6~1.7 1.7~1.8 1.8~1.9 1.9~2.0

1 0.146 0.177 0.208 0.239 0.268

2 3.854 3.823 3.792 3.761 3.732

Table 2. The permittivity of ten-layered multilayered cloak with perfect parameters

Layer (unit: ) 1.0~1.1 1.1~1.2 1.2~1.3 1.3~1.4 1.4~1.5

1 0.0874 0.1565 0.2134 0.2617 0.3033

2 11.4364 6.3889 4.6851 3.8217 3.2967

Layer (unit: )

1.5~1.6 1.6~1.7 1.7~1.8 1.8~1.9 1.9~2.0

1 0.3399 0.3723 0.4014 0.4276 0.4514

2 2.9422 2.6859 2.4915 2.3387 2.2153



238

material parameters by Equation (7), as shown in Table 2, here, we note that 4 / 3 .

Full-wave simulations with the commercial finite-element solver COMSOL MULTIPHYSICS were performed to verify the performance of the designed electromagnetic cloak. In the simulations, we assume a TM plane wave incidence along the x direction. The cloaking materials are assumed to be lossless and the interior region is per-fect electric conductor (PEC) ensuring that the fields inside it are ideally zero, and also demonstrating the cloaking performance.

Perfect matched layers(PML) are used to absorb the scattered field, which simulate the infinite domain in which the system resides, and 10 sub-layers of alternat-

ing permittivity are used to realize each layer. The simu-lated fields shown in each case were computed with ap-proximately 170,000 elements and 200,000 unknowns.

Figure 2(a) shows the simulation results for the mag-netic field and scattered magnetic field distribution around the multilayered cylindrical cloak with improved parameters, the magnetic fields are smoothly excluded from the interior region with minimal scattering. The scattered fields are caused by the reduced set of pa-rame-ters and small approximation made in the design of the permittivity and the thickness ratio. Figure 2(b) sug-gests that the multilayered cylindrical cloak with perfect mate rial parameters should cause a relatively low scattering at the same cloak thicknesses. The scattered fields

Figure 2. The magnetic-field and scattered magnetic field distribution around the electromagnetic cloak, (a) a concentric multilayered cylindrical cloak with improved simplified parameters, (b) a multilayered cylindrical cloak with perfect pa-ameters, (c) comparison of absolute scattered magnetic field of z component r



239

are caused by the approximations made in the design of the permittivity and the thickness ratio. As mentioned above, the nonideal cylindrical cloak by layered structure of homogeneous isotropic media has been already dis-cussed in [9]. We explained that the difference is that we are constructing a multilayered cylindrical cloak with improved parameters and a multilayered cloak with ideal parameters by the concentric layered structures. We have also compared the scattering characteristics of three types of cylindrical cloak (Figure 2(c)). It shows clearly that both multilayered cloak with improved simplified parameters and ideal material parameters have smaller scattering, compared with previously reported layered cloak [9].

3. Conclusions

In conclusion, we construct a cylindrical cloak with im-proved simplified parameters and an ideal cylindrical cloak by using spatially invariant axial material parameters through layered structures of alternating homogeneous isotropic materials. These two cloaks exhibit better invisibility performances as compared to the previously reported cloak. In our design, the relative permeability of material media is always larger than 1, which may be realized by composite media. Such cloak has no require-ment of any metamaterial, it is possible to be realized by thin layers of composite media. With its relatively low requirement on material parameters, it can potentially be a better candidate for realizing a near-invisible cloak.

4. Acknowledgment

This work was supported by graduate starting seed fund of Northwestern Polytechnical University.

REFERENCES [1] J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling

electromagnetic fields,” Science 312, 1780, 2006.

[2] J. B. Pendry, D. Schurig, and D. R. Smith, “Calculation of material properties and ray tracing in transformation me-dia,” Optics Express 14, 9794, 2006.

[3] S. Cummer, B.-I. Popa, D. Schurig, D. Smith, and J. Pen-dry, “Full wave simulations of electromagnetic cloaking structures,” Physical Review E74, 036621, 2006.

[4] D. Schurig, J. J. Mock, B.-J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electro-magnetic cloak at microwave frequencies,” Science, 2006.

[5] W. S. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nature Photonics, Vol. 1, pp. 224–27, 2007.

[6] W. S. Cai, U. K. Chettiar, A. V. Kildishev, G. W. Milton, and V. M. Shalaev, “Nonmagnetic cloak with minimized scattering,” Appled Physical Letters 91, 111105, 2007.

[7] W. S. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Sha-laev, “Designs for optical cloaking with high-order trans-formations,” Optics Express 5444, Vol. 16, No. 8, 2008.

[8] M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photon Nano- structured: Fundamental Application, Vol. 6, pp. 87–95, 2008.

[9] Y.Huang, Y. J. Feng, T. Jiang, “Electromagnetic cloaking by layered structure of homogeneous isotropic materials,” Optics Express, Vol. 15, pp. 11133–11141, 2007.

[10] Z. Jacob, L. V. Alekseyev, E. Narimanov, “Optical hy-perlens: Far-field imaging beyond the diffraction limit,” Optics Express, Vol. 14, pp. 8247–8256, 2006.

[11] A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations”, Physical Review, B74, 075103, 2006.

[12] H. S. Chen, B.-I. Wu, B. L. Zhang, and J. A. Kong, “Electromagnetic wave interactions with a metamaterial cloak,” PRL99, 063903, 2007.

[13] B. L. Zhang, H. S. Chen, B.-I. Wu, Y. Luo, L. X. Ran, and J. A. Kong, “Response of a cylindrical invisibility cloak to electromagnetic waves,” Physical Review, B76, 121101, 2007.

[14] Z. C. Ruan, M, Yan, C, W. Neff, and M, Qiu, “Ideal cy-lindrical cloak: Perfect but sensitive to tiny perturbations,” Physical Review Letters, 99, 113, 903, 2007.

[15] M. Yan, Z. Ruan, and M. Qiu, “Cylindrical invisibility cloak with simplified material parameters is inherently visible,” Physical Review Letters 99, 233, 901, 2007.

[16] M. Yan, Z. C. Ruan, and M. Qiu, “Scattering characteris-tics of simplified cylindrical invisibility cloaks,” Optics Experss17772, Vol. 15, No. 26, 2007.

[17] Y. Luo, J. J. Zhang, H. S. Chen, a- S. Xi, and B.-I. Wu, “Cylindrical cloak with axial permittivity/permeability spatially invariant,” Applied Physics Letters 93, 033504, 2008.

[18] G. Isí, R. Gajić, B. Novaković, Z. V. Popović, and K. Hingerl, “Radiation and scattering from imperfect cylin-drical electromagnetic cloaks,” Optics Express 1413, Vol. 16, No. 3, 2008

[19] J. J. Zhang, Y. Luo, H. S. Chen, B.-I. Wu, “Cloak of arbi-trary shape”, Journal of the Optical Society of America B 25, pp. 1776–1779, 2008.

[20] J. J. Zhang, J. T. Huang, Y. Luo, H. S. Chen, J. A. Kong, and B.-I. Wu. “Cloak for multilayered and gradually changing media,” Physical Review B 77, 035116, 2008.

[21] Y. Luo, et al., “Design and analytical full-wave validation of the invisibility cloaks, concentrators, and field rotators created with a general class of transformations”, Physical Review B, 77, 125127, 2008.

[22] J. A. Silva-Maćedo, M. A. Romero, and B.-H. V. Borges, “An extended FDTD method for the analysis of electro-magnetic field rotations and cloaking devices,” Progress In Electromagnetics Research, PIER 87, pp. 183–196, 2008.

[23] J. J. Zhang, Y. Luo, H. S. Chen, and B.-I. Wu, “Sensitiv-ity of transformation cloak in engineering,” Progress In Electromagnetics Research, PIER 84, pp. 93–104, 2008.



Active Power Filter Based on Adaptive Detecting Approach of Harmonic Currents Yu ZHANG, Yupeng TANG

School of Electrial Engineering, Beijing Jiaotong University, Beijing, China. Email: [email protected], [email protected] Received September 17th, 2009; revised October 29th, 2009; accepted November 7th, 2009. ABSTRACT The ip-iq detection method based on instantaneous inactive power theory has been applied widely in active power filter because of its good real-time. But it needs large computation, and three-phase currents are processed as integrity, thus calculation accuracy can't be ensured. Based on adaptive interference canceling theory, this paper presents a new adaptive detection method for harmonic current, it is a continuously regulated closed-loop system, and its operating char-acteristics are almost independent of the parameter variations of the elements, thus it performs better than that based on traditional theory. At last this paper provides the simulation of active power filter including the detecting circuit which proved the design is feasible and correct. Keywords: Adaptive Interference Canceling, Adaptive Harmonic Detecting, Active Power Filter 1. Introduction Because of the use of more nonlinear loads, especially more power electronic equipments, a large number of harmonic and reactive currents have been introduced into power grid, resulting in some problems such as voltage flicker, frequency variation, imbalance of three-phase problem, etc [1]. In order to suppress the harmonics, pas-sive filters have been used in the past years [2], while recently Active Power Filter (APF) has been developed rapidly. Widely used in APF, the harmonic detection method is based on three-phase instantaneous inactive power theory. Thus a lot of analog multipliers and calcu-lation are needed, resulting in difficult adjust and poor performance [3-4]. Furthermore, this method is only suitable for a three-phase equilibrium sinusoidal system.

This paper presents a new adaptive closed-loop detec-tion method based on adaptive interference canceling theory, and the simulation results show that the filter based on this new method performs better than that based on the three-phase instantaneous inactive power theory, and with higher accuracy [5].

2. The Basic Principle of an Active Filter An active power filter is a new power electronic device of dynamic harmonic suppression. Figure 1 shows the basic principle. There are four parts in a shunt APF: the main circuit, command current operational circuit, cur-rent tracking control circuit, and the drive circuit. The

command current operation circuit detects the harmonic component iLh, in the load current iL, and takes the oppo-site value as command signal *

ci . The principle can be expressed by the following formula

s L ci i i= +

L Lf Lhi i i= +

c Lhi i= −

s L c Lfi i i i= + =

where iS, iL are currents of the supply and a nonlinear load, respectively, and ic is the compensation current. iLf, iLh are the fundamental active and harmonic reactive components of the load current, respectively.

si

ci

Li

se

*ci

Figure 1. Basic principle block diagram of an active filter

Active Power Filter Based on Adaptive Detecting Approach of Harmonic Currents


241

3. Adaptive Detecting Algorithm 3.1 The Basic Principle of Adaptive Interference

Canceling Theory The adaptive interference canceling technique has been widely used in recent years [6]. By continuously self- studying and self-adjusting, the detecting system can always operate at its best. The basic noise-canceling the-ory can be illustrated in Figure 2. In the detecting system, there are two unrelated input signals: original input s+n0 and reference input n1. And s is unrelated with n0 and n1, while n0 and n1 are related. The reference input signal n1 is filtered by an adaptive filter to produce an output sig-nal *

0n , which is an approximate replica of n0. This output *

0n is subtracted from the original input signal s+n0 to produce *

0 0y s n n= + − , the system output sig-nal.

In the system shown in Figure 2, the reference input is processed by an adaptive filter which automatically ad-justs its own response through a least-squares algorithm. Thus the filter can detect the noise n0 continuously and adjust the system to minimize the error signal e. It can be proved that *

0n is the best least-squares estimate of n0, when the filter is adjusted to make the error signal power

2[ ]E ε minimum.

3.2 Adaptive Harmonic Detection Based on the principle of adaptive noise canceling theory, adaptive harmonic current detecting circuit is shown in Figure 3. The system is composed of an analog adaptive filter, a BPF (Band Pass Filter) and a 900 phase-shifter [7]. The primary input is the load current:

1( ) ( ) ( ) ( ) ( ) ( )h p q hi t i t i t i t i t i t= + = + + , where i1(t) is the fundamental current, ih(t) is the sum of all harmonic components, and ip(t), iq(t) are the active component and the reactive component of i1(t), respectively in Figure 3. u(t) and u1(t) are the AC source voltage and its funda-mental component, respectively. R1(t) and R2(t) are two reference inputs orthogonal to each other, and i0(t) is the system output.

As shown in Figure 3, because both feedback branches are similar, we take the lower feedback branch as an example. Only the fundamental reactive component which has the same frequency with 1( )R t can produce the DC signal after the output current i0(t) is multiplied by 1( ) cos rR t D tω= , while other components produce AC signals after the same procession. The DC compo-nent can be integrated to get the average value of funda-mental reactive current IFp, while the AC component will be zero after the same calculation. Thus, we can get the instantaneous fundamental reactive current ifq(t) by

0n1n

s0ns +

∗0n

Figure 2. Adaptive noise canceling theory

∫

∫

90° Phase-shifter1( )u t )(2 tR)(1 tR

RPF

1M

2M

1M ′

2M ′

1I

2I

1( )f t

)(2 tf

)()(0 titi h=)()()( 1 tititi h+=

+ -

-

)(tu

Figure 3. Adaptive detecting diagram for harmonic currents multiply IFp with R1(t). Similarly, using R2(t), we can get the instantaneous fundamental active current ifp(t). At last, by adding the reverse of ifp(t)+ifq(t) to i(t), the output current i0(t)=ih(t) is produced. If only the current i0(t)=ih(t)+ifq(t) is needed, what we should do is remove the R1(t) branch.

We can also explain the principle in the phase space. Assume the reference inputs which processed by the BPF are:

1( ) cos rR t D tω= ,

2 ( ) sin rR t D tω= .

Then the output of the multiplier M1 can be expressed as:

0 1 0 0( ) ( ) ( ) cos ( )2rDi t R t i t D t i tω⋅ = ⋅ =

[exp( ) exp( )]r rj t j tω ω× + − (1)

Taking the Laplace transform of (1), we have

0 1 0 0[ ( ) ( )] ( ) ( )2 2r rD DL i t R t I s j I s jω ω⋅ = − + + (2)

where, I0(s) is the Laplace transform of i0(t). After proc-

essed by the integrator, whose transform is Gs

(here G

is the integration gain), the transform of the feedback signal can be expressed as:

1 0 0( ) [ ( ) ( )]2 r r

DGW s I s j I s js

ω ω= − + + (3)

The output of the multiplier '1M is simply the feed-

back signal of the lower branch, which mean 1 1 1( ) ( ) ( )f t W t R t= ⋅ . Its transform is:



242

1 1 1( ) [ ( ) ( )]2 r rDF s W s j W s jω ω= − + +

2

0 0[ ( 2 ) ( )]4( ) r

r

D G I s j I ss j

ωω

= − +−

2

0 0[ ( 2 ) ( )]4( ) r

r

D G I s j I ss j

ωω

+ + +−

2

2 2

02 ( )42( )r

D Gs D GI ss ω

= ++

(4)

Similarly, the transform F2(s) of the feedback signal f2(t) for the upper feedback branch can be expressed as:

2 2

2 02 2( ) ( )42( )r

D Gs D GF s I ss ω

= −+

0 0( 2 ) ( 2 )[ ]

( ) ( )r r

r r

I s j I s js j s j

ω ωω ω

− +× +

− + (5)

The total feedback signal is:

1 2( ) ( ) ( )f t f t f t= +

Its transform is: 2

1 2 02 2[ ( )] ( ) ( ) ( ) ( )r

D GsL f t F s F s F s I ss ω

= = + =+

(6)

Thus the feedback coefficient of the whole system is: 2

2 20

( )( )( ) r

F s D GsB sI s s ω

= =+

(7)

Then the transfer function H(s) of the system is: 2 2

02 2 2

( ) 1( )( ) 1 ( )

r

r

I s sH sI s B s s D Gs

ωω

+= = =

+ + + (8)

From (8), when rω ω= , | ( ) | 0H jω = , which means a zero point exists in the system corresponding to the fundamental frequency rω Consequently the funda- metal signal will be greatly attenuated. It is obvious that the system shown in Figure 3 is equivalent to an ideal second-order notch filter. In addition, the center fre-quency of the system depends solely on the frequency signal rω of the reference input. Therefore, the system is independent of parameter of the circuit components, which means that the system is almost stable while the temperature varies or the circuit components ages.

3.3 DC Side Voltage Control Ideally, what an active filter compensates is the non-active power; that is to say, it neither absorbs active power from the power supply nor outputs to it, so the DC side voltage of an active filter is constant. However, due to the loss of the active filter, energy in the capacitor on

trωsin *cipi∆

Figure 4. DC voltage control block diagram

the DC side will reduce, making the voltage on the ca-pacitor drop.

In order to maintain the voltage on the capacitor, the feedback method has usually been adopted, whose pur-pose is to obtain some active power from the source to compensate the corresponding loss.

As shown in Figure 4, Ucr , Ucf are the reference and feedback values of Uc, respectively. The difference be-tween Ucr and Ucf is regulated by PI to get the signal

pi∆ .

Since *ci contains the fundamental active component ,

ic, which comes from *ci , also contains such a compo-

nent. Therefore, when ic is introduced into the power system, APF can exchange the active energy between AC and DC sides, which keeps Uc constant.

4. Simulation Results In this section, computer simulation is carried out to ver-ify the design of the adaptive shunt active filter. A three-phase distribution system is built using Matlab as shown in Figure 5. Simulation parameters are as follow-ing: AC source is 220V/50Hz, supply side inductance Ls is 0.2μH. The nonlinear load parameters for three-phase full-controlled bridge rectifier are R= 20Ω, L=0.1H. In the main circuit of the active filter, IGBT is used as the switch, and the inductance on the AC side La is 5mH, while the capacitance is 2200μF/1000V on the DC side.

Figure 6 shows the AC source voltage, the power sup-ply currents before and after filterd, respectively, and the harmonic and reactive reference currents. From Figure 6(b), we can see that before filtered, the current lags the source voltage and contains a lot of harmonic and reac-tive components. After filtered by the APF, shown in Figure 6(d), the supply current is nearly sinusoidal and in phase with AC source voltage, which means APF cor-rects the power factor of the supply side nearly to unity. There is a variation in the nonlinear current at t=0.1s, From Figure 6 it can be seen the proposed adaptive shunt active filter only needs approximately half a cycle to adapt itself to the change.

Since the APF adopts traditional hysteresis current control method, the tracking ability of APF is limited, resulting in some ripples in the current when it changes suddenly, as shown in Figure 6(d).

The DC capacitor voltage is shown in Figure 7, it only



243

Figure 5. Active filter simulation model

Figure 6. Simulation result of detection for harmonic and reactive currents. (a) ac source voltage for phase A, (b) power sup-ply currents for phase A before filter input, (c) harmonic and reactive reference current from adaptive detection, (d) power supply currents for phase A after filter input

0 0.05 0.1 0.15 0.20

200

400

600

800

1000

1200

Figure 7. DC side capacitor voltage of active filter

Figure 8. FFT analysis of the supply current before APF input



244

Figure 9. FFT analysis of the supply current after APF in-put based on adaptive interference canceling theory

Figure 10. FFT analysis of the supply current after APF input based on instantaneous inactive power theory takes about 0.05s to reach at the desired value of 1000V and stabilize rapidly.

Compared Figure 8 with Figure 9, it shows the har-monic and reactive currents are greatly restrained.

It shows from Figure 9 and Figure 10, under the same conditions, after APF input, the THD of the supply cur-rent based on adaptive interference canceling theory drops to 10.50%, but that based on instantaneous inactive power theory is only 12.30%, moreover, the method based on traditional theory uses 6 analog summer, 4 mul-tipliers and lots of gains, thus the calculation accuracy is more difficult to be assured in practice. The method based on adaptive interference canceling theory uses only 6 multipliers and 3 integrators, which ensures better per-formance in actual operation than that based on instanta-neous inactive power theory.

Overall, it shows that the proposed adaptive shunt ac-tive filter can compensate nonlinear load current, adapt itself to compensate the variations in nonlinear load cur-rents and correct the power factor of the supply side nearly to unity.

5. Conclusions In this paper, a novel adaptive detection method for har-monic and reactive current is proposed. This method is analyzed systematically and verified by Matlab simula-tion. It is a continuously regulated closed-loop system,

and the operating characteristics are nearly independent of the parameter variations of the elements, and band-width behaving as one of a second-order notch filter can be regulated easily by controlling the amplitude of the reference input and the gain of the integrator. Further-more, this paper also introduces DC side voltage control method, which is simple and effective. Finally, simu-lation result is given to conform the feasibility of the design.

REFERENCES

[1] R. Bojoi, G. Griva, F. Profumo, M. Cesano, and L. Natale, “Shunt active filter implementation for induc-tion heating applications,” Applied Power Electronics Conference and Exposion, Vol. 3, pp. 1674–1679, March 2005.

[2] D. Liu, B. D. Zhang, and X. L. Zhang, “Design of adaptive increment controlled hybrid-type active power filter,” Power and Energy Engineering Conference, Asia-Pacific, Vol. 27–31, No. 3, pp. 1–4. 2009.

[3] H. Akagi, E. H. Watanabe, and M. Aredes, “Instanta-neous power theory and applications to power condi-tioning,” Wiley-Interscience a John Wiley & Sons, Inc., Publication, 2007.

[4] H. Akagi, “Generalized theory of instantaneous reac-tive power and its application,” Elec.Eng. in Japanese, April 1983.

[5] L. H. Tey and Y. C. Chu, “Improvement of power quality using adaptive shunt active filter,” IEEE Transactions on Power Delivery, Vol. 20, pp. 1558–1568, April 2005.

[6] B. Widrow and S. P. Sterns, “Adaptive signal proc-essing,” Englewood Cliffs, Prentice-Hall, NJ, inc., 1985.

[7] J. H. Husy and M. S. E. Abadi, “Unified approach to adaptive filters and their performance,” IET Signal Proc-ess, Vol. 2, No. 2, pp. 97–109, 2008.

[8] A. Nakajima, “Development of active filter with se-ries resonant circuit,” IEEE-PESC, Annual Meeting, pp. 1168–1173, 1988.

[9] Q. Wang, N. Wu, and Z. A. Wang, “A neuron adap-tive detecting approach of harmonic current for apf and its realization of analog circuit,” IEEE Transac-tions on instrumentation and measurement,” Vol. 50, pp. 77–84, February 2001.

[10] H. P. To, F. Rahman, and C. Grantham, “An adaptive algorithm for controlling reactive power compensa-tion in active power filters,” Industry Applications Conference, 39th IAS Annual Meeting, Vol. 1, pp. 3– 7, October 2004.

[11] L. P. Ling and A. Azli, “SVM based hysteresis cur-rent controller for a three phase active power filter,” Power and Energy Conference, pp. 132–136, No-vember 2004.



245

Monitor System for Protection Device Based on Embedded RTOS Yang WANG, Xianggen YIN, Zhe ZHANG

Electric Power Security and High Efficiency Lab, Huazhong University of Science and Technology, Wuhan, China. Email: [email protected] Received September 8th, 2009; revised September 26th, 2009; accepted September 29th, 2009. ABSTRACT For the purpose of the monitor system in digital protection, the embedded real-time operating system (RTOS) and the embedded GUI (Graphical User Interface) is introduced to design the monitor system. Combining the necessity and the application value of the operation system, the choice of embedded Linux and Qt/Embedded is completely viable for the monitor system in digital protection for generator-transformer sets. The design with embedded Linux and embedded GUI enriches system information, increases developing efficiency and improve the generality. Keywords: Real-Time, Operating System, Embedded Linux, Protection, Monitor System 1. Introduction After the development of digital protection for over 20 years in our country, The rapid development of DSP (Digital Signal Processor) and high performance embed-ded microprocessor not only make the digital protection function more perfect and reliable, but also can gather and deal with more abundant data [1,2]. The system needs precise and fast inner communicating function for transmission of vast data, and it also needs the network technology based on Ethernet with the development of power distribution automation System. It is difficult to achieve the requirement of the high-speed dealing with the data from DSP, including using parallel communica-tion, realizing the smooth and amicable man-machine interface in traditional monitor system and programming technique, so that the choice of the more intelligent embedded system with RTOS is needed imminently. It is the trend of developing monitor system with high-per-formance embedded system and embedded GUI in future [3].

2. Embedded System of Real-Time Operation System

2.1 Necessity Embedded system is made of embedded processor, cor-relative hardware and embedded software, which can work independently with the hardware and the software. The 32-bit, 64-bit processors are primarily chosen in en-

gineering at present. Restricted by factors like cost, en-ergy consumption and chip size, embedded systems are closely linked with their application and market. It is critical to design and develop embedded system products with the proper cost, function and performance.

Embedded system is used in power system for a long time, including data collection, automatic equipment, detection and control of instrument and so on [4,5]. Monitor System in digital protection widely adopts embedded system of single chip, Rabbit or ARM without operation system because of the slow processor and the deficient of EMS memory for the moment. Lack of op-eration system induces single configuration of function, low efficiency, small memory capacity and few user in-terfaces. The linear programming is the dominating pro-gram configuration in this embedded system. It is so flexible that the monitor system should be locked owing to the abnormity of every part of the application, other-wise it is usual hard to realize Chinese display. With the development of integrated automation system, the con-nection between protection and power dispatching center is based on Ethernet network. It is complicated to add the network protocol stack in circular control of embedded system, on the contrary, the situation in embedded opera-tion system is opposite. Also, it’s convenient for the transplantation of the network protocol stack in several system platforms. Consequently, the embedded system which is marked by operation system and internet is needed for the monitor system in digital protection ur-gently.

Monitor System for Protection Device Based on Embedded RTOS


246

2.2 Feasibility The frequency of 32-bit, 64-bit processors is over 100 MHz, moreover the memory system is bigger and its read-write function is more convenient, but their cost is falling in prices. In one side, high performance system could afford the additional consumption from CPU and memory in this embedded system. In the other side, the high ratio performence&price of them is fit for the de-velopment of digital protection.

2.3 Real-Time Multi-Task Embedded System The man-machine interface and communicating function of the monitor system are not developed independently owing to the restriction in the speed of manage-CPU, the capability of memory and especially the design of linear program. The traditional monitor system brings us the bad configuration of the program, low efficiency on de-veloping, fussy debugging and so on, so that the embed-ded system of RTOS is chosen for the new research and development platform of monitor system. In this plat-form the object-oriented programming which is unlike the ordinary linear frame is available to adopt. Separating and choosing each function as an individual task, while the system should distribute CPU’s resource to every task according to their priority levels, which could ensure the real time response of multi-task. It is better to realize abundant manage functions in the monitor system than traditional one.

3. Embedded Linux and Embedded GUI 3.1 Technical Characteristics of Embedded

Linux There are many embedded operation systems at present, including VxWorks, pSOS, Nucleus, Palm OS, Windows CE, etc. They are all commercial product, so that a lot of small companies can’t afford the cost for the digital pro-tection by reason of their high price. Embedded Linux is a new part of embedded operation system. As a result of inherited from Linux, embedded Linux is advanced and strong, therefore it offers a steady runtime environment for the monitor system. The following are the primary reasons of choosing embedded Linux for the digital pro-tection:

1) Embedded Linux has been used on multiform plat-form, including X86, Alpha., Sparc, MIPS, PPC, ARM, NEC, MOTOROLA, etc, so the work of transplanting should be convenient and fast. Furthermore, it will re-duce the developing cost as Linux is free for following the GNU General Public License (GPL).

2) It is confirmed that embedded Linux is fit for em-bedded system and adding or reducing each part of the system should be easy at any moment due to the high

modularization of Linux. 3) Compared with other embedded RTOS, Linux has

perfect network function, Communication based on Ethernet network is no longer a problem for digital pro-tection.

4) The digital protection needs lots of memory storage to save SOE and fault report ceaselessly recorded by sets. The embedded Linux2.6 can sustain file system (append NAND FLASH) excellently, so that it is easier and more efficient to manage the plentiful messages from DSP.

3.2 Embedded GUI -QT/Embedded 3.2.1 According with Embedded System Development In order to develop high-performance and perfect func-tions monitor system in embedded Linux, an excellent graph support embedded GUI is needed, which is simple, intuitional and less resource used for limited hardware [6].

There are three mainstream embedded GUI, MiniGUI, Nano-X Window (Microwindows) and QT/Embedded, which is selected. QT/Embedded is an edition for em-bedded system created and maintained from QT by a software company named Trolltech in Norway. Com-pared with other embedded GUI, QT/Embedded is fully object-oriented, easily extensible and abundant controls. Although it has bigger size and costs more resource, it becomes more appropriate for embedded system with the fast development of processors and memory now. There are many embedded Linux developers have turn to QT/ Embedded since it is released under the term of GPL.

3.2.2 Supply High-Performance Display Effect It is a reason for choosing QT/Embedded, which can improve display effect of monitor system remarkable. QT/Embedded pays attention to all elements of graphic user interface for developers, it works on FrameBuffer directly, which is an abstract graphics equipment inde-pendent of hardware. It supplies the mapping of display memory and display register from physical memory in process address space. With developing the driver of FrameBuffer device driver for LCD, it provides running base for GUI and improves operation efficiency of pro-gram. Figure 1 shows the configuration of QT/Embedded realization.

Qt/Embedded application

QWSServer

Qt/Embedded

FrameBuffer Input Device

Linux Operation System

Hardware platform

Figure 1. Configuration of QT/Embedded realization



247

3.2.3 Supply Perfect Development Environment 1) Cross Platform: as a monitor system of protection, it is better to fit different platform and has strong transplanta-tion. The Trolltech assured developers that they would maintenance and develop same API on all kinds of op-eration system, reducing the degree of developing diffi-culty.

2) Object-Oriented: QT/Embedded is fully object ori-ented and allows true component programming. It is easy to realize stronger function and more excellent display with plenty of control resource.

3) Internationalization: in order to be accepted by power plant operation personnel, all the information showed on interface should be realized by Chinese. QT/ Embedded tries to make internationalization as pain- less as possible for developers. One of the solutions is using QString for all user visible text. Since QString uses the Unicode encoding internally, every language in the world can be processed transparently using familiar text proc-essing operations, especially Chinese.

4) Powerful Tools: QT/Embedded is supplied with several powerful command line and graphical tools to ease and speed the development process. Furthermore, the reference-documents of QT are extremely detailed through the effort of the Trolltech and all developers of the world.

4. The Development of Monitor System Monitor system is the embodiment of the most intelligent design for digital protection. As more and more interface developments with embedded GUI on embedded RTOS, but this application has not been realized in the field of digital protection. According to the requirements of the monitor function in digital protection for large genera-tor-transformer sets at Three Georges’s right bank plant, this chapter refers to adopt the 32-bit embedded RISC microprocessor based on ARM920T, and develop a real- time, multi-task, high-performance and flexible monitor system with QT/Embedded-3.3.5 on embedded Linux- 2.6.

4.1 Hardware Design The ARM9 processor EP9315 of the CIRRUS LOGIC in U.S. is selected for the manage CPU, and there are many advantages in this hardware platform. Firstly, it has strong computational capacity, whose frequency is as high as 200MHz and it equips the Maverick Crunch co-processor for floating-point operation. Secondly, it has rich memory function, including 64MB-SDRAM, 32MB- NOR FLASH, 512MB-NAND FLASH and it also allows connecting with IDE hard disk, CF-card and U disk. Thirdly, it has perfect communication function, for in-stance, two 232-serial ports, four RS485/422-serial ports, two CAN bus and two Ethernet ports. Last but not least, the hardware platform is so powerful that it is also con-

sist of standard 8 keyboard, DS1286 real-time clock and 6 inches 640×480 256K-colour LCD. Figure 2 shows the functional configuration of the manage plug-in in this digital protection.

4.2 Software Design 4.2.1 Technique and Application of Multithread In comparison with traditional linear software design, monitor system has always been realized by making use of multithread. The task of main thread is dealing with graphics user interface. There are three threads for com-munication: the communication with DSP based on MODBUS protocol and CAN bus, the communication with host computer based on MODBUS protocol and Ethernet and the communication with power dispatching center based on IEC 60870-5-103 protocol and Ethernet. The four parallel threads work at the same time, in other words, any normal threads will not be influenced by the one with errors which are caused by intense disturb. Ob-viously, it is better to improve the stability of the devices for using technique of multithread, and what’s more, it is easier to add other new manage functions for better ex-pansibility.

There are two choices for realizing multithread: 1)The design for multithread in Linux. 2)The design for multithread in QT. In order to reduce the developing difficulty and coop-

erate with the monitor software developed by QT/ Em-bedded, the second one is more suitable and convenience. Qt provides thread support in the form of basic platform- independent threading classes, a thread-safe way of posting events, and a global Qt library lock that allows you to call Qt methods from different threads. The most

Figure 2. Functional configuration of manage unit

Figure 3. Configuration of multithread



248

important thread in QT is QThread, then it provides the means to start a new thread, which begins execution in your reimplementation of QThread::run(). This is similar to the Java thread class. Don’t mix the normal Qt library and the threaded Qt library in your application. This means that if your application uses the threaded Qt li-brary, you should not link with the normal Qt library, dynamically load the normal Qt library or dynamically load another library or plug-in that depends on the nor-mal Qt library. On some systems, doing this can corrupt the static data used in the Qt library.

4.2.2 Improve Display Effect As can be seen from the analysis of the developing envi-ronment, which is fully object-oriented and allows true component programming, the interface developers can devote their mind to graphics design. With high-resolution color LCD, the familiar PC desktop graphic images should be realized in the monitor system of digital pro-tection again, that is to say this new developing tech-nique brings on huge reform of tradition human machine interface.

4.2.3 General Design As a result of the flexible configure for digital protection, the monitor system is expected good action in the update of same sets and other similar devices being transplanted. So the generality is very important for a successful monitor system. The embedded Linux and QT/Embedded both can be transplanted expediently while it is better to adopt a general design for monitor application. Most important, similar functions of digital protection are the base for the generality.

At the beginning of designing this monitor system, its general requirement has been considered already. The pivotal technique is that all the information including names and interrelated properties displayed when the device is running, such as settings, samplings, digital input and output , SOE and fault/pickup report, should be saved in a special file named configuration file. When initializing the monitor system, the information read in memory from the configuration file should be displayed on the interface. This file needn’t be compiled and can be configured flexibly as required. This method has the fol-lowing advantages:

1) The method of the text configuration is very simple and flexible. During the developing process, we just modify the text file when adjusting any information as actual required. It can raise the efficiency of development and debugging.

2) It realizes the generality of monitor system in gen-erator protection and transformer protection. And it can

fit for any digital protection system without any modifi-cation of main program.

According to the excellent transplantation of embed-ded system and embedded GUI, the developing platform of the monitor system can fit for the development of most kinds of digital protection.

5. Conclusions The large generator-transformer digital protection has passed type test and dynamic simulation test in Electric Power Research Institute of China. The real-time multi- task operating system shows the excellent performance in interface display, hand operations, data acquisition and communication. These devices have done the test-run in 300MVA hydraulic generator sets in Geheyan hydro-power station and they work well by far.

The monitor system based on embedded Linux, which is developed on embedded GUI, not only offer enrich man-machine exchange of information, provide high quality of display effect, optimize program structure, and supply powerful function of net, but also save the devel-oping and maintenance cost because of the open source. The development platform of embedded system is so excellent in transplantation and expansibility that it is even fit for developing other automated electric power equipments. To summarize, it will be one of the main-stream platforms for developing monitor system in digi-tal protection in future.

REFERENCES

[1] N. L. Tai, Z. J. Hou, X. H. Li, Z. Zhang, and Y. X. Chen, “Microprocessor-based protection system design for large hydro generator unit,” Relay, Vol. 29, No. 8, April 2001.

[2] Q. Zhou, X. J. Yi, and Z. L. Wang, “Technical scheme of generator-transformer relay protection system for three gorges left bank power station,” Automation of Electric Power Systems, Vol. 11, No. 5, March 1999.

[3] Q. Luo, D. H. You, and H. Y. He, “Human-machine iinterface design based on embedded gui for power auto-mation equipment,” Electric Power Automation Equip-ment, Vol. 24, No. 9, May 2004.

[4] T. Q. Xu, D. H. You, and C. Li, “Microprocessor-based realying protection device for generator based on μC/OS-II,” Power System Technology, Vol. 16, No. 15, August 2005.

[5] H. B. Xiao, X. Li, and L. J. Zhang, “Real-time multi-task kernel and its application in power system AER,” Electric Power Automation Equipment, Vol. 8, No. 10, May 2002.

[6] C. Xie, Y. Tao, and Z. Tan, “Study of multi-process graphic user interface (GUI) based on the embedded linux,” Industrial Control Computer, Vol. 5, No. 1, January 2003.



249

Effect of Weak Magnetic Intergranular Phase on the Coercivity in the HDDR Nd-Fe-B Magnet

M. LIU, G. B. HAN, R. W. GAO

School of Physics, Shandong University, Jinan, China. Email: [email protected]

Received July 2nd, 2009; revised August 11th, 2009; accepted August 19th, 2009.

ABSTRACT

Assuming that intergranular phase (IP) existing between adjacent grains is a weak magnetic phase, we study the effect of IP on the coercivity in the HDDR Nd-Fe-B magnet. The results indicate that the coercivity increases with the in-creasing IP’s thickness d, but decreases with increasing its anisotropy constant K1(0). When the structure defect thick-ness r0 =6nm, d=1nm and K1(0)=0.15K1 (K1 is the normal anisotropy constant in the inner part of a grain), our calcu-lated coercivity is in agreement with available experimental data. Keywords: HDDR Nd-Fe-B Magnet, Intergranular Phase, Coercivity 1. Introduction

The HDDR powder particles, prepared by the HDDR (hydrogenation, decomposition, desorption, and recom-bination) process, consist of fine Nd2Fe14B crystalline grains with diameters ranging from 0.2 to 0.3 μm, which is close to the single domain size of Nd2Fe14B phase [1]. Such unique grain microstructure of HDDR magnet is different from not only the grain microstructure of sin-tered magnet, but also that of nanocrystalline magnet. Generally, the sintered magnet consists of the Nd2Fe14B crystalline grains of 5-10 μm in diameter, and nonmag-netic Nd-rich boundary phases [2] which interrupts the intergrain exchange coupling interaction. Thus, the grain- boundary anisotropy (GBA) of the sintered magnet is mainly affected by the grain-boundary structure defect (GBSD). The nanocrystalline magnet is composed of the directly contacted magnetic grains of a few tens of na-nometers [3], and its GBA is principally influenced by the intergrain exchange coupling interaction (IECI). However, for the HDDR magnet, its GBA may be si-multaneously influenced by GBSD and IECI [4], owing to the unique grain microstructure. Some investigators considered that the adjacent grains directly contacted with each other in the same HDDR powder particle [5,6,7]. However, Nakayama et al [8] observed experi-mentally that a thin grain-boundary layer with the thick-ness of 1 nm exists between adjacent HDDR grains. Theoretically, the effect of intergranular phase (IP) on the coercivity is unclear. Thus, this paper tries to theo-retically study the effect of intergranular phase on the

coercivity in HDDR Nd-Fe-B magnet. The component, structure and character of intergranu-

lar phase sensitively depend on the alloy’s composition and processing technique. The intergranular phase is the crystalline phase with Nd2Fe14B-like structure reported by Reference [9]. Reference [10] pointed out that the Nd6Fe13Al1 phase was identified as an intergranular phase. Thus, the intergranular phase is still magnetic phase. Here, assuming that the IP existing between adja-cent grains is a weak magnetic phase, and using cubic- grain anisotropy model, we study the effect of IP on the coercivity of the HDDR Nd-Fe-B magnet. The results indicate that the coercivity increases with the increasing IP’s thickness d, but decreases with increasing its anisot-ropy constant K1(0). Such conclusion could provide a theoretical reference for preparing high coercivity HDDR Nd-Fe-B magnet.

2. Anisotropy Model

Reference [4] pointed out that the GBA is simultaneously influenced by the GBSD and IECI in the HDDR magnet, and proposed a structure model of a cubic grain with edge of D (where the GBSD’s thickness is r0 and the IECI’s length is lex). Here, we assume that the IP is a weak magnetic phase, and distributes homogeneously between grains. Because of the very small size of IP, we presume that half of the thickness, d/2, is shorter than both lex/2 and r0 (as shown in Figure 1a where r0>lex/2 is supposed). IP weakens the IECI, leading to the IECI’s length reduce from lex/2 to (lex–d)/2. Based on different

250 Effect of Weak Magnetic Intergranular Phase on the Coercivity in the HDDR Nd-Fe-B Magnet

ranges influenced by the GBSD and IECI, a grain is di-vided into three parts in the case of D/2>r0>lex/2. For convenience, the center of IP is chosen as the coordinate origin of r. For d/2<r<lex/2, the GBA is simultaneously affected by the GBSD and IECI. For lex/2<r<r0, it is in-fluenced by the GBSD alone. While r>r0, the GBA isn’t influenced by the GBSD or IECI, and is still the common anisotropy constant K1 in the inner part of the grain. The grain-boundary anisotropy K1'(r) was described by dif-ferent formulae for r0 ≤lex/2 and r0>lex/2 in Reference [4]. Here, we assume that K1(0) is a constant in the IP region. Due to the continuous variation of K1'(r), its ex-pression can be rewritten as Equations (1) and (2). Figure 1b shows the variation of K1'(r) in the case of D/2>r0> lex/2. It can be seen that K1'(r) continuously decreases from K1 in the inner part of a grain to K1(0) in the IP region. when r0 ≤ lex/2,

(a)

(b)

Figure 1. (a) Sketch of a grain divided into three parts due to different ranges influenced by GBSD and IECI in the case of D/2＞ r0＞ lex/2; (b) Variation sketch of grain- boundary anisotropy

1

23

' 21 1

0

3

21 0

(0), 0 2

2( )2( )= (1 ) ,

2( )( )2

2( )2(1 ) ,

( )

dK r

dr d

0

2

K r K K r rd

r lex d

dr lex

K K r rlex d

(1)

when r0>lex/2,

1

23

' 21 1

0

3

21 0

0

(0), 0 2

2( )2( )= (1 ) ,

2 2( )( )2

( )2(1 ) ,

2( )2

dK r

dr d lex

K r K K rd

r lex d

dr lex

K K r rd

r

(2)

where r is the distance to the IP’s center, ΔK=K1–K1(0), and K1(0) ≤K1.

3. Coercivity of the HDDR Nd-Fe-B Magnet

The demagnetization process and coercivity mechanism of the HDDR Nd-Fe-B magnet were studied by Refer-ence [4], where the IP didn’t exist, and it was concluded that both the demagnetization nucleation and pinning of domain wall displacement between grains might occur at the grain boundary. If IP exists, it might become the pin-ning center of the domain wall displacement [11]. When the coercivity of magnet is determined by the irreversible domain wall displacement in the IP region, it can be ex-pressed by [12],

'1 0 1

''1

2

3 3c e

s B

K r KAff sH N M

KAM

( ) (3)

where A, A' and K1, K1' denote the integral constants and anisotropy constants in the inner and boundary parts of a grain, respectively. δB' denotes the domain wall thickness. Ms is the saturation magnetization, and Ms in denomina-tor of Equation (3) can be replaced by the saturation po-larization Js in the International System of Units. Neff is the effective demagnetization factor.


Effect of Weak Magnetic Intergranular Phase on the Coercivity in the HDDR Nd-Fe-B Magnet 251


Reference [12] considered that A' is equal to A, and K1' takes the fixed value less than K1. Based on our proposed anisotropy model, r0 should be the thickness of anisot-

ropic inhomogeneous district, and is denoted by 0r , and

K1' varies between 0 and K1. For convenience, K1' in Equa-tion (3) will be replaced by the average anisotropy <K1'>

in 0r region. Thus, Equation (3) can be rewritten as,

'1 0 1

''1

2

3 3c e

s B

K r KAff sH N M

KAM

( ) (4)

where <K1'> can be expressed as follows,

0

0

23 3

2 22 21 1 1 00

20

'1

23

221 1 10

00 0

2( ) 2( )2 2 2( (0) ( (1 ) ) ( (1 ) ) ),( )( )( )

2

2( ) ( )1 2 2( (0) ( (1 ) ) ( (1( )( ) (

2

d lexr

dr

d

d dr r lex

K dr K K dr K K dr rdlex lex dr lex d

Kd d

r rK dr K K dr K K

dr r lex d r

　 2

03

2 20

22

) ) 2)

2

lexr

lexd

lexdr r

d

),　 (5)

Taking the intrinsic magnetic parameters of Nd2Fe14B: K1 = 4.3 MJ/m3, A= 7.7×10-12 J/m, Ms =1280 kA/m [13], Js = 1.61T [14], lex = 4.2 nm [15], δB = 4.2 nm, Neff = 0.6 [16], into Equations (4) and (5), we can calculate the coercivity of magnet for different values of r0, d and K1(0).

4. Results and Discussion

Figure 2 shows the variations of anisotropy K1'(r) for given values of, r0d and K1(0). For different values of r0, d and K1(0), K1'(r) decreases with decreasing r. This is due to that the closer to the grain surface, the smaller the anisotropy is [4]. It can be also seen that, for the fixed r0 and K1(0) shown by the star and circle lines, the variation velocities of K1'(r) increases with increasing d. This is attributed to the decreasing variation range from d/2 to r0 with increasing d for the fixed value of (K1–K1(0)). But for the fixed r0 and d, shown by the upper triangle and lower triangle lines, the variation velocities of K1'(r) de-creases with increasing K1(0), which is owing to that, the variation value (K1–K1(0)) decreases with increasing K1(0) for the fixed variation range from d/2 to r0. While for the fixed d and K1(0) shown by the circle and lower triangle lines, the variation speeds of K1'(r) decreases with in-creasing r0, attributing to the increasing variation range from d/2 to r0 as increasing r0 for the fixed value of (K1– K1(0)).

Figure 3 shows the dependence of average anisotropy, <K1'>, on d for different values of r0 and K1(0). For dif-ferent r0 and K1(0), <K1'> all decreases with increasing d, which is attributed to the variation speeds of K1'(r) in-creases with increasing d (as shown in Figure 2). So, <K1'> computed by Equation (5) decreases. But for the fixed r0 and d shown by the upper triangle and circle lines, <K1'> increases with increasing K1(0), which is owing to the variation velocities of K1'(r) decrease with increasing K1(0) (as shown in Figure 2). Thereby, <K1'> calculated by Equation (5) increases. It can be also seen

that, for the fixed d and K1(0) shown by the upper trian-gle and square lines, <K1'> increases with increasing r0,

Figure 2. Variations of grain-boundary anisotropy, K1'(r), with r for different values of r0, d and K1(0)

Figure 3. Dependences of average anisotropy, <K1'>, on d for different values of r0 and K1(0)

252 Effect of Weak Magnetic Intergranular Phase on the Coercivity in the HDDR Nd-Fe-B Magnet

Figure 4. Dependence of Coercivity, Hc, on d for different values of r0 and K1(0) ascribing to the variation speeds of K1'(r) decrease with increasing r0 (as shown in Figure 2).Thus, <K1'> com-puted by Equation (5) increases.

Figure 4 shows the dependence of coercivity, Hc, on d for different values of r0 and K1(0). For different values of r0 and K1(0), Hc increases with increasing d. On the one hand, this is owing to that the enhancement of d re-sults in the reduction of <K1'> (as shown in Figure 3), and then Hc calculated by Equation (4) increases. On another hand, the domain wall energy is the lowest in the IP region, where the domain walls located are the most stable. And the domain walls are pinned more strongly in the IP region with increasing IP’s thickness d. Thus, a largely external field is needed if the domain walls tend to deviate from the IP region, and then Hc also increases. But for the fixed r0 and d shown by the circle and upper triangle lines, Hc decreases with increasing K1(0). On the one hand, for the fixed r0 and d, <K1'> increases with increasing K1(0) (as shown in Figure 3). Thus Hc com-puted by Equation (4) decreases. On another hand, with increasing K1(0), the pinning force hindering the moving of domain wall becomes smaller, thus the domain wall deviates from the IP region more easily. So, the coerciv-ity decreases. It can be also seen that, for the fixed K1(0) and d shown by the upper triangle and square lines, Hc increases with increasing r0, attributing to the variable

quantities of 0

B

r

is larger than that of

'1 1'

11

A K

KA

( )

with increasing r0. Consequently, Hc computed by Equa-tion (4) increases. When r0 = 6 nm, d =1 nm and K1(0) = 0.15K1, the calculated coercivity is 1068 kA/m, which is consistent well with the experimental data (IP’s thickness of the HDDR Nd-Fe-B magnet is around 1 nm, and its coercivity is 1058 kA/m) reported by Nakayama et al [8].

In summary, the weak magnetic intergranular phase (IP) existing between adjacent grains weakens the IECI.

The increase of both the IP’s thickness d and GBSD’s thickness r0 or the decrease of the IP’s anisotropy con-stant K1(0) all enhance the coercivity of magnet. Yet, if d and r0 are too larger and K1(0) is too smaller, the mag-netization and remanence would badly fall, then it is im-possible to obtain high-energy product. In order to get high-energy product, it needs not only to enhance coer-civity, but also to keep a sufficiently high remanence. Therefore, it is necessary to ensure that the IP’s thickness is around 1 nm, the GBSD’s thickness is around 6 nm, and K1(0) varies between 0.1 K1 and 0.2 K1, by reasona-bly adjusting the alloy’s composition and technical proc-ess. So, this paper possesses a high preference value for experiment preparing high coercivity HDDR Nd-Fe-B magnet with considerable magnetization and remanence.

5. Conclusions

Effects of the IP’s thickness d, its anisotropy constant K1(0), and the GBSD’s thickness r0 on the coercivity in the HDDR Nd-Fe-B magnet are investigated. The results indicate that Hc increases with the increasing d and r0, but decreases with the increasing K1(0). And while r0 = 6 nm, d =1 nm and K1(0) = 0.15K1, the calculated coerciv-ity is consistent well with experimental data.

7. Acknowledgements

The work is supported by the National Natural Science Foundation of China (50671055) and (50801043).

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[2] M. Sagawa, S. Fujimura, H. Yamamoto, Y. Matsuura, and K. Hiraga, “Permanent magnet materials based on the rare earth-iron-boron tetragonal compounds,” IEEE Tran- sactions on Communications, Vol. 20, No. 1, pp. 1584– 1589, January 1984.

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[4] M. Liu, Y. Sun, G. B. Han, Wu. Y, and R. W. Gao, “De-pendence of anisotropy and coercivity on microstructure in HDDR Nd-Fe-B magnet,” Journal of Alloys Com-pounds, Vol. 478, pp. 303–307, April 2009.

[5] T. Takeshita and R. Nakayama, “Magnetic properties and microstructures of the NdFeB magnet powder produced by hydrogen treatment,” Proceedings of the 10th Interna-tional Workshop on Rare-Earth Magnets and Their Ap-plications, Kyoto, Japan, pp. 551, 1989.

[6] T. Takeshita, K. Morimoto, “Anisotropic Nd-Fe-B bonded magnets made from HDDR powders,” Journal


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Mrinal MISHRA, Nisha GUPTA

Department of Electronics and Communication Engineering, Birla Institute of Technology, Mesra, India Email: [email protected], [email protected]

Received May 8th, 2009; revised July 10th, 2009; accepted July 18th, 2009.

ABSTRACT

An integration technique based on use of Monte Carlo Integration is proposed for Method of Moments solution of Elec-tric Field Integral Equation. As an example numerical analysis is carried out for the solution of the integral equation for unknown current distribution on metallic plate structures. The entire domain polynomial basis functions are em-ployed in the MOM formulation which leads to only small number of matrix elements thus saving significant computer time and storage. It is observed that the proposed method not only provides solution of the unknown current distribution on the surface of the metallic plates but is also capable of dealing with the problem of singularity efficiently.

Keywords: Scattering, EFIE, Method of Moments, Monte Carlo Integration

1. Introduction

The Method of Moments (MoM) [1] is one of the widely used numerical techniques employed for the solution of Integral Equations. The MoM is based upon the trans-formation of an integral equation, into a matrix equation. However, the application of the spatial-domain MoM to the solution of integral equation is quite time consuming. The matrix-fill time would be significantly improved if these integrals can be evaluated efficiently. The MoM employs expansion of the unknown function inside the integral in terms of known basis functions with unknown coefficients to be determined. Point matching technique or Galerkin’s technique commonly employed in MoM results in a system of linear equations equal in number to that of unknown coefficients. This leads to a matrix equation for the coefficients. The matrix thus obtained is called the ‘moment’ matrix. The unknown coefficients can then be obtained by matrix inversion.

The MoM method involves two approaches, the sub domain [2,3] and the entire domain [4,5] approaches, essentially based on the two kinds of basis functions em-ployed for the expansion of the unknown function on the metal surface. The entire domain basis functions extend over the whole region occupied by the structure, whereas the sub domain basis functions are defined to exist over a section of the structure and have a zero value over the rest of its portion. The choice of the type of basis func-tion depends upon the size and shape of the metallic structure in the problem. The advantage with the sub

domain basis functions is that due to their flexibility to be defined over small polygonal domains of varying sizes. The whole structure under investigation can be modeled as consisting of large number of such polygonal sub domains, thus making possible the analysis of com-plicated shaped structures. The disadvantage with these basis functions is that they are limited to electrically small and moderately large structures, as the number of sub domains required to model large structures accu-rately becomes very large. This results in the moment matrix of a large size increasing the computation costs in terms of memory and CPU time.

The entire domain basis functions, on the other hand, require a very few number of expansion terms. These are also capable of analyzing electrically large structures and the solution obtained with these functions are more ac-curate than the sub domain basis functions. This results in a faster and more accurate solution, thus reducing the computational cost. One of the requirements of the entire domain basis functions is a prior knowledge of the dis-tribution of the unknown quantity for the kind of the structure under consideration. The effectiveness of a MoM numerical solution depends on a judicious choice of basis functions. The optimal choice of the basis func-tion is one that provides solutions with the fewest num-ber of expansion terms and in shortest computational time. These functions should incorporate as closely as possible the physical conditions of the actual distribution of the unknown quantity on the region of interest. In this paper, entire domain polynomial basis function is utilized

Monte Carlo Integration Technique for Method of Moments Solution of EFIE in Scattering Problems 255

which results in the reduction of computational cost and an increase in the accuracy of the result.

The other aspect of the MoM formulation is the prob-lem of the singularity of the function that is to be inte-grated to obtain the matrix elements, in both the ap-proaches of the MoM formulation. In the point matching MoM approach using sub-domain basis functions, only a few matrix elements, whereas using entire domain ap-proach, all the matrix elements are obtained by integra-tion of singular functions. Various analytical and nu-merical techniques have been adopted to deal with such integrals. The Monte Carlo Integration (MCI) technique [6,7] proposed in this paper is not only capable of solving the scattering problem but also deals efficiently with the problem of singularity.

To demonstrate the capability of the above mentioned technique, the problem is formulated in terms of an inte-gral equation to determine the current distribution on square metallic plate. The MCI technique has been pro-posed to tackle scattering problem from an infinitesi-mally thin square metallic plate structures in the MoM formulation of the problem. The entire domain basis functions are utilized in the MoM matrix solution and hence reduce the computational cost to the great extent. Besides, it is also capable of handling the singularity aspect of the Green’s function easily and more efficiently. The typical simulation demonstrates the application of the proposed technique and also validates the result against the Benchmark solution [8].

2. Mathematical Concept

Monte Carlo methods are useful for obtaining solutions to problems involving integration which are too compli-cated to be solved analytically or by other numerical methods. Standard numerical integration techniques do not work very well on high-dimensional domains, espe-cially when the integrand is not smooth. Although the quadrature rules of integration typically work very well for one-dimensional integrals, problems occur when ex-tending them to higher dimensions.

Monte Carlo methods have advantages over numerical methods in a space of many dimensions. Their efficien-cies relative to other numerical methods increase when the dimension of the problem increases e.g. Quadrature formula becomes very complex while MCI technique remains almost unchanged in more than one dimension. In addition to this, the convergence of the MCI is inde-pendent of dimensonality regardless of the smoothness of the integrand. Monte Carlo integration is simple since only two basic operations are required, namely sampling and point evaluation. It is also suited for large structures and highly complex problems for which definite integral formulation is not obvious and standard analytical tech-niques are ineffective. Sampling can be used even on domains that are not well-suited to numerical quadrature.

The idea of Monte Carlo integration is to evaluate the integral using random sampling as

( )I f x dx

(1)

where f is a function of vector x, is domain of integra-tion. The Monte Carlo integration is popular for complex f and/or . In its basic form, this is done by independ-ently sampling N points x1, …,xN according to some convenient density function p, and then computing the estimate

1

( )

( )

Ni

Ni i

f xF

N p x

(2)

where p(xi) is the probability density function or pdf. Here the notation NF is used rather than I to emphasize

that the result is approximate, and that its properties de-pend on how many sample points are chosen. If p(xi) is the uniform probability density, then the integral is sim-ply

1

( )N

ii

I f xN

(3)

The MCI methods are better suited than quadrature methods for integrands with singularities. It is particu-larly helpful for integrand that have large values on a relatively small part of the domain due to singularities. It can be applied to handle such integrands effectively, even in situations where there is no analytic transforma-tion available to remove the singularity. The simple way to handle the singularity using MCI is to ignore a region around the singularity and let this region become smaller as N increases [9–13]. In this approach the total region is split into , where the numerical integration is per-

form, and a small region min

minrr r r which is apparently left

out as long as it has a small or negligible contribution for large values of N. The event generators use therefore a cutoff to avoid this region of singularity i.e., the

random points generated in MCI are restricted to fall within this excluded region. This does not require any extra effort to handle the singularity problem as the re-quired condition can be embedded directly in the MCI technique itself in a single statement of the MATLAB code employed for the purpose in simulation.

mir n

3. Numerical Example

As an example, the electric field integral equation is solved by method of moments for the unknown surface current density on a square metallic plate. The plate is an infinitesimally thin λ x λ square in free space, with limits -1.0m to 1.0m along both the x and y axes. The scatterer is excited normally by an incident plane wave with the electric field having a magnitude 1.0 Vm-1 and po-larized along a scatterer edge, in this case the y- axis. The

iE




256

field as:

(5)

where

])()[/1( 200 AA kjs E

| '|'0( ) ( ) '

4 conduc | ' |

jk

tingsurface

ed

r r

A r J r rr r

(6)

is

inte uation (EFIE) for the unknown current density

e inte

current on the conducting surface can be expanded as:

(8)

the vector potential. Applying the boundary condition 0 tEn on the

scatterer, we get the electric field rag l eq: )'(rJFigure 1. Geometry of the λ x λ square P. E. C. scatterer

incident upon by a plane wave with Ei polarized along the y- axis, λ = 2m ])()[/1( 2

00 AAEE kjsi (7)

For method of moments (MoM) solution of th -gral equation, taking some known basis function )'(rnf ,

the unknown

geometry is shown in Figure 1. From Maxwell’s equations we can obtain vector ex-

pression for the total electric field

s (4) it EEE )'()'(

1

rrJ n

M

nn fa

where is the incident field and tE sE is the scattered

(a) (b)

(c) (d)

Figure 2. (a) Solution for the Jy-current component (real part) along the X axis of the scatterer obtained for different number

(imaginary part) along the Y axis of the scatterer obtained for different nu ber of random points generations

of random points generations; (b) Solution for the Jy-current component (real part) along the Y axis of the scatterer obtained for different number of random points generations; (c) Solution for the Jy-current component (imaginary part) along the X axis of the scatterer obtained for different number of random points generations; (d) Solution for the Jy-current component

m

Monte Carlo Integration Technique for Method of Moments Solution of EFIE in Scattering Problems 257

where na ; n = 1, 2, . . ., M are unknown amplitudes of

is fthe bas unctions and are to be determined. This expan-sion is plied over the same number of field points as the number of expansion terms, which transforms the integral equation into a set of simultaneous algebraic equation in the unknown coefficients, which can be writ-ten in the matrix form as:

ap

inmn aZ nE (9)

or

inmnn EZa 1 (10)

The matrix elements of the matriusing numerical integration (such as MCI th

tire domain basis functions for the un

x equation are formed technique) of

e singular function that results after differentiation of the Green’s function.

The problem under investigation is an entire domain MoM problem. The en

known current density are the modified polynomial functions, with edge correction and symmetry considera-tions as stated in the benchmark solution [8] and pre-sented as follows:

( 2) ( 2)

21 3

( , ) ( )1

yx xxn n i

jx xij

i j

yJ x y a x x

y

(11)

( 2) ( 2)

20 3

( , ) ( 1)1

xy yyn n ij

y yiji j

xJ x y a y

x

n = n = 8 and n = n = 7. The su

(12)

where xx yy xy yx bscript (2) below the summation sign means that the indices i and j

are to be increased with a step size 2. Thus the total number of expansion terms for x-current is 12 and for y-current is 16, thus making the total number of unknown coefficients = 28. This leads to the formation of the Z-matrix of order 28 X 28, far less than the number of coefficients required in sub domain analysis, which in case of such a large scatterer becomes extremely large for accurate analysis. Though, the Galerkin’s method in MoM is a suitable choice for the problem under investi-gation, the present formulation employs the point match-ing technique in MoM specifically to demonstrate the singularity aspect of integrand. The point matching tech-nique makes it essential that all the matrix elements that are evaluated, involve integration of singular integrands (singular kernels of the integral equation). Thus it is nec-essary to adopt means that can take care of the singular-ity inside the integral and give a good approximation of the actual result. The technique adopted here is the Monte Carlo Integration (MCI) technique that overcomes the singularity, making integration much simpler and justified in case of two dimensional and three dimension-

(a)

(b)

(c)

Figure 3. (a) Solution for the Jx current component (Real part) over the scatterer; (b) Solution for the Jx current component (Imaginary part) over the Scatterer; (c) Solutionfor the Jx current compone Magnitude) over the Scat-terer

nt (




258

as to prevent the points fall inside a certain region

the M t two things. First, the re

in polynomial basis h Monte Carlo Integration techniquef the problem under investigation. In

h (CSIR), Pusa, New Delhi,

[1] R. F. Harrington, “Field computation by moment meth-ods,” New Yo

975.

ol.

rlo Methods, pp. 403–418, 2004.

entific and Industrial Researcal scattering problems. The method that has dealt with the singularity here is named as the ‘local correction technique’. Since MCI is based on random generation of points inside the entire domain of integration, care must taken

India–110012.

REFERENCES

min

The Figures 2(a) and 2(b) show the real part of Jy as seen along the x-axis while the Figures 2(c) and 2(d) show the real and imaginary parts of Jy as seen along the y-axis, for different numbers N of random points taken for

CI. The figures make eviden

r r , the region of singularity. rk, Macmillan, 1968.

[2] C. A. Balanis, “Antenna theory: Analysis and design,” Harper & Row, New York, pp. 283–321. 1982.

[3] C. M. Bulter and D. R. Wilton, “Analysis of various nu-merical techniques applied to thin-wire scatterers,” IEEE Transactions, Vol. AP–23, No. 4, pp. 524–540, 1

sults for all the values of N show a good agreement with the results obtained in the benchmark solution [8] using MoM for the solution. Second, the solution with N = 10000 points converge to that obtained for N = 45000 and N = 50000 points, stating that N = 10000 points are suffi-cient enough in the MCI for the problem under in vestiga-tion, thereby decreasing the computational burden to a great extent. Figures 3(a), 3(b) and 3(c) show the results for the 3-D current distribution for the real part, imaginary part and magnitude of Jx. Total number of random points for the MCI to evaluate the elements in the impedance matrix for MoM solution has been taken to be 10,000. It is found that the results show good agreement with those obtained in benchmark solution [8].

4. Conclusions

The Monte Carlo integration technique in the MoM solu-tion of the integral equations has been proposed. The technique employs the entire-doma

[4] B. M. Notaros and B. D. Popovic, “General entire-domain method for analysis of dielectric scatterers,” IEE Pro-ceedings - Microwaves, Antennas and Propagation, V143, No. 6, pp. 498–504, 1996.

[5] M. Djordjevic and B. M. Notaros, “Double higher order method of moments for surface integral equation model-ing of metallic and dielectric antennas and scatterers,” IEEE Transactions on Antennas and Propagation, Vol. 52, No. 8, pp. 2118–2129, 2004.

[6] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, “Numerical recipes, second edition,” Cam-bridge University Press, 1992.

[7] M. N. O. Sadiku, “Numerical techniques in electromag-netics,” CRC Press, New York.

act

[8] B. M. Kolundžija, “Accurate solution of square scatterer as benchmark for validation of electromagnetic modeling of plate structures,” IEEE Trans ions on Antennas and Propagation, Vol. 46, No. 7, pp. 1009–1014, 1998.

[9] T. Pillards, “Quasi-Monte Carlo integration over a sim-plex and the entire space,” Ph. D. Thesis, Katholieke Uni-versiteit Leuven, Belgium, ISBN 90–5682–741–3, 2

functions along witin MOM solution o 006.

[10] J. Hartinger, R. F. Kainhofer, and R. F. Tichy, “Quasi- Monte Carlo algorithms for unbounded, weighted inte-gration problems,” Journal of Complexity, Vol. 5, No. 20,

the Monte Carlo Integration technique, a local correction technique is employed to deal with the singularity aspect of the kernel very efficiently and easily. For the demon-stration of the present method, the current distribution is investigated on a two dimensional square plate scatterer. Where comparisons are available, it is found that this technique yields results which compare very closely to those of other methods. In addition to the rapid conver-gence of the Monte Carlo integral with respect to total number of random points, the small number of basis functions (< 10) is needed to provide accurate results. Further research is being carried out to extend the method to three dimensional structures.

5. Acknowledgements

This research work was supported by the Council of Sci-

pp. 654–668, 2004,

[11] A. B. Owen, “Quasi-Monte Carlo for integrands with point singularities at unknown locations,” Monte Carlo and Quasi-Monte Ca

[12] M. Mishra and N. Gupta, “Singularity treatment for inte-gral equations in electromagnetic scattering using Monte Carlo integration technique,” Microwave and OpticalTechnology Letters, Vol. 50, No. 6, pp. 1619–1623, June 2008.

[13] M. Mishra and N. Gupta, “Monte Carlo integration tech-nique for the analysis of electromagnetic scattering from conducting surfaces,” Progress In Electromagnetics Re-search, PIER 79, pp. 91–106, 2008.



259

Numerical Computation of Resonant Frequency of Shorting Post Loaded Gap-Coupled Circular Microstrip Patch Antennas

Pradeep KUMAR1, G. SINGH1, T. CHAKRAVARTY2

1Department of Electronics and communication Engineering, Jaypee University of Information Technology, Solan, India; 2Embedded Systems Innovation Laboratory, Tata Consultancy Services, Bangalore, India Email: [email protected]

Received April 17th, 2009; revised May 30th, 2009; accepted June 5th, 2009.

ABSTRACT

In this paper, the numerical computation of resonant frequency of the two gap-coupled circular microstrip patch an-tenna loaded with shorting post by using cavity model is presented. The numerically computed results are compared with simulated results. The two gap-coupled circular microstrip patch antenna loaded with shorting post miniaturize the cross-sectional dimension of the radiating patch at the microwave frequency, which is useful for short range com-munications or contactless identification systems. The simulation has been performed using method-of-moments based commercially available simulator IE3D.

Keywords: Gap-Coupling, Microstrip Antennas, Shorting Post, Resonant Frequency

1. Introduction

An explosive growth of the wireless radio frequency identification market such as electronic toll collection and more generally wireless road-to-vehicle communica-tion systems is currently observed in the microwave band. In the short range communications or contactless identi-fication systems, antennas are key components, which must be small, low profile, and with minimal processing costs [1,2]. The microstrip patch antennas are of great interest for aforementioned mentioned applications due to their thin and compact structures. The flexibility af-forded by microstrip antenna technology has led to a wide variety of design and techniques. The main limita-tions of the microstrip antennas are low efficiency and narrow impedance bandwidth. The bandwidth of the mi-crostrip antenna can be increased using various tech-niques such as by loading a patch, by using a thicker substrate, by reducing the dielectric constant, by using gap-coupled multi-resonator etc [3–5]. However, using a thicker substrate causes generation of spurious radiation and there are some practical problems in decreasing the dielectric constant. The spurious radiation degrades the antenna parameters. Among various antenna bandwidth enhancement configurations, the two gap-coupled circu-lar microstrip patch antenna is most elegant one. So,

gap-coupling is the suitable method for enhancing the impedance bandwidth of the antennas [6,7]. In the cofi-gration of gap-coupled microstrip antennas method, two patches are placed close to each other. The gap-coupled microstrip antennas generate two resonant frequencies and the bandwidth of the microstrip antennas can be in-creased [6].

There exist a wide range of basic microstrip antenna shapes such as rectangular, circular and triangular patch shapes which are commonly used patches. For these patches, operating at their fundamental mode resonant frequency, are of the dimension of the patch is about half wavelength in dielectric. At lower frequencies the size of the microstrip antennas becomes large. In modern com-munication systems the compact microstrip patch anten-nas are desirable. The size of the microstrip antenna can be reduced by shorting the patch. Changing the basic patch shape can give rise to substantial size reduction. Further decrease in size can be obtained by loading the basic shapes by shorting post or slots [8–10].

In [11,12], circular microstrip patch antenna with dual frequency operation is designed by shorting the patch and the results are compared with the conventional cir-cular microstrip antenna (without a shorting post) which shows that the size of the circular microstrip antenna can be reduced for the same frequency application. It is also

260 Numerical Computation of Resonant Frequency of Shorting Post Loaded Gap-Coupled Circular Microstrip Patch Antennas

observed that the resonant frequency of the circular mi-crostrip antenna with shorting post can be varied by varying its location. In [13,14], it is shown that the loaded circular microstrip antenna has two modes that is TM01 and TM11. Recently, we have also designed two gap-coupled circular microstrip patch antenna without shorting post which generates two modes that is TM11 and TM21 mode [15].

In the present paper, the two gap-coupled circular mi-crostrip patch antenna is loaded with a shorting post to minimize the size of the antenna structure and it is seen that it generates the three modes that is TM01, TM11 and TM21. The resonant frequency of TM01 mode of the pro-posed antenna is lower as compared to TM11 and TM21 mode of the two gap-coupled circular microstrip anten-nas as reported in [15]. The numerical computation of the resonant frequency of the shorting post loaded two gap- coupled circular microstrip antenna is performed and the results are compared with the simulated results. The simulation is performed by using the IE3D simulator which is based on the method-of-moment. This paper is structured as follows. In Section 2, the geometrical con-figuration of the two gap-coupled circular microstrip patch antenna is discussed. In Section 3, the proposed antenna is analyzed theoretically. The Section 4 dis-cusses the analytical and simulated results of the pro-posed antenna. Finally, Section 5 concludes the work.

2. Antenna Configuration

The geometrical configuration of two gap-coupled circu-lar microstrip patch antennas loaded with shorting post is shown in Figure 1. The patch of radius mm is the feed patch and other patch of radius mm is the parasitic patch. The parasitic patch is excited by the gap-coupling whereas the feed patch is excited by the probe feeding technique. The parasitic patch introduces another resonance near the main resonance and proper adjustment of the structure parameters, bandwidth can be enhanced. The feed patch is shorted by shorting post of diameter . The height and permittivity of the substrate

is mm and

15a 15d

p

591.h 2.2r , respectively. The gap

distance between the adjacent edges of the feed patch and parasitic patch is s .

3. Theory

Now we divide the structure in two concentric regions; region I and region II as shown in Figure.2.The inner

radii of region I is 2

p and outer radii of region I is ‘a’.

The inner radius of region II is ‘b’ and outer radius of region II is ‘c’.

Now, we consider the field expressions for given TMnp mode in these two regions [16].

In region I (2

pr a ):

The solution of the wave equation in cylindrical coor-dinates for region I gives the field expressions as:

(1) [ ( ) ( )]cos1 2E j C J kr C N kr nz n n

(1)

(1) ' '[ ( ) ( )]cos1 2H k C J kr C N kr n

n n (2)

The expression for is omitted for brevity. The

and

(1)Hr(1)Ez

(1)H are the axial electric field and azi-

muthal magnetic field, respectively for region I, and k are the angular frequency and propagation constant for TMnp mode, respectively. ( )J xn

Nn

is the Bessel function

of first kind of order and is the Bessel func- n ( )x

Figure 1. Geometrical configuration of the two gap-coupled circular microstrip patch antenna loaded with shorting post

Figure 2. Analytical configuration of the two gap-coupled circular microstrip patch antenna loaded with shorting post


Numerical Computation of Resonant Frequency of Shorting Post Loaded Gap-Coupled Circular Microstrip Patch Antennas 261

tion of second kind of order . C1 and C2 are the am-plitude constants for region I.

n

In region II ( b r ): c The solution of the wave equation in cylindrical coor-

dinates for region II gives the field expressions as:

(2) [ ( ) ( )]cos3 4E j C J kr C N kr nz n n (3)

(2) [ ' ( ) ' ( )]cos3 4H k C J kr C N kr nn n (4)

where C3 and C4 are the amplitude constants for region II. Considering the parasitic patch in isolation the boundary

condition of vanishing (2)H can be applied as:

(2) 0H for and b r c 1 2 (5)

Thus,

'( )4 2

'( )3 1

cJ kr rdrnC Ib

cC IN kr rdrnb

(6)

Therefore the field expression in region II can be rewrit-ten as:

(2) (2) (2) ( )cosE j C F krz n n n (7)

(2) (2) (2) '( ) cosH kC F kr nn n (8)

where is a constant dependant on given mode n

and

(2)Cn

(2) ( ) ( ) ( )1 2F kr J kr I I N krn n n (9)

Similarly, (1) 0Ez for / 2r p

( / 2) 31( / 2)2 4

IC N kpnC J kp In

Thus the field expression in region I is:

(1) (1) (1) ( ) cosE j C F krz n n n (10)

(1) (1) (1) '( ) cosH kC F kr nn n (11)

where is a constant and (1)Cn

(1) ( ) ( ) ( )3 4

F kr J kr I N kr In n n (12)

We now consider the gap between two regions at the point of coupling between two patches as a -type net-work [15] as shown in Figure 3.

Figure 3. An equivalent circuit diagram of the two gap-gap coupled circular microstrip patch antenna

In Figure 3, are the wall admittances of

individual patches and is the mutual admit-

tance between the two patches.

( / )wy a bn

mn

( , )y a b

Here,

1myn j M

where M denotes magnetic coupling and is given by

01 240 0

h h qM dl dl

s

where is the substrate height and is a correction

factor. The formula for magnetic coupling is derived for infinitely thin cylindrical lines. However, to take account for substantial area of each cylinder an empirical correc-tion factor is utilized. From the condition of discontinuity of current in these two regions, we can write:

h q

(1) (2) (1) (2)( ) ( ) ( ) ( )1 2

aH a bH b E a Y E b Yz z (13)

using this expression, we obtain:

(1) (1)(2) '( ) ( )1

(1) (2) (2)'( ) ( )

2

jaF ka F ka YC n nn k

jC bF kb F kb Yn n nk

(14)

For small gap normal component of electric field is

continuous at 'r where '2

b ar

.

Therefore at . (1) (2)( ) ( )E r E rz z

'r r

From this

(1)(2) ( ')

(1) (2)( ')

F krCn n

C F krn n

(15)

Equating Expressions (14) and (15) we get:



(1) (1)(1) '( ) ( )( ') 10

(2) (2) (2)( ') '( ) ( )2

jaF ka F ka YF kr n nn k

jF kr bF kb F kb Y

n n nk

(16)

The above transcendental equation gives the resonant frequency of a given TMnp mode.

4. Results and Discussion

The proposed simulation model of two gap-coupled cir-cular microstrip patch antenna is as shown in Figure 1. It is clearly seen that, for the short-circuited patch antenna, the input impedance become very sensitive to the feed position and strongly depends on the distance between the shorting post and the feed position [2–10]. The de-termination of all relevant parameters for the shorting post microstrip patch antenna is straight forward once the resonant frequency has been determined. It has been shown that the resonant frequency depends critically on dimension of the shorting post and gap distance between adjacent edges of the feed patch and parasitic patch.

In Figure 4, the variation of resonant frequency of dif-ferent modes with radius of shorting post in feed patch is shown. It is important to note that at a fixed frequency, the patch size can be increased or decreased, depending on the radius of the shorting post. As the radius of the shorting post increases the resonance frequency of the proposed antenna is increases for TM01 TM11 and TM21 modes as shown in Figure 4 In Figure 5, the variation of resonant frequency of different modes with gap distance between adjacent edges of the feed patch and parasitic patch is shown. The resonant frequency decreases with increasing the gap distance between adjacent edges of the feed patch and parasitic patch for TM01 TM11 and TM21 modes as shown in Figure 5. The proposed antenna is also simulated using IE3D simulator. The numerically computed and simulated results are compared which shows good agreement as shown in Figure 4 and Figure 5.

The diameter of the shorting post and the gap distance between the adjacent edges of feed patch and parasitic patch also play an important role in the overall size of the patch conductor. Basically, the shorting post is modeled as an inductance parallel to the resonant LC circuit de-scribing a reference resonant mode of the unloaded (without shorting post) patch. In an equivalent circuit, new resonance mode (with shorting post) can be viewed as resulting from the inductance due to shorting post. Using the resonant frequency of TM01 mode, the size of the gap-coupled circular microstrip antenna can be re-duced for same frequency applications, because the resonant frequency of this mode is much less than the resonant frequencies of the conventional gap-coupled circular microstrip antenna presented in [15]. By varying the various parameters such as diameter of shorting post and the gap distance between adjacent edges, the reso-

(a)

(b)

(c)

Figure 4. Variation of the resonant frequency with radius of the shorting post of the proposed antenna for, (a) TM01 mode with s = 0.5 mm; (b) TM11 mode with s = 0.5 mm and (c) TM21 mode with s = 0.5 mm

nant frequencies of the presented antenna can be con-trolled.

To validate the presentation of mode numbers the


Numerical Computation of Resonant Frequency of Shorting Post Loaded Gap-Coupled Circular Microstrip Patch Antennas 263

simulated current density on the patches are presented in Figure 6, 7 and 8. It is seen that the mode numbers are

(a)

(b)

(c)

Figure 5. Variation of the resonant frequency with gap dis-tance between adjacent edges of the feed patch and para-sitic patch for (a) TM01 mode with radius of shorting post = 0.5 mm, (b) TM11 mode with radius of shorting post = 0.5 mm, and (c) TM21 mode with radius of shorting post = 0.5 mm

predicted accurately. In this case, the current density to-wards ‘Red’ colour is maximum and current density to-wards ‘Blue’ colour is minimum. The current density represented by ‘Yellow’ colour is between the ‘Red’ and ‘Blue’. The left hand side patch is the feed patch and the right side patch is parasitic patch. In Figure 6, the yellow region is at the center of the feed patch and blue region at the surroundings of the green region conveys the TM01

mode. In Figure 7, it is seen that on the feed patch a yel-low region and a blue region exists. This accounts for TM11 mode. Two blue regions in Figure 8 convey the existence of TM21 mode.

5. Conclusions

In this paper, a numerical model for shorting post loaded two gap-coupled circular microstrip patch antenna is de-veloped. The comparison between the numerically com-puted and simulated results shows good agreement in resonant frequency for TM01, TM11, and TM21 modes. The proposed gap-coupled microstrip antennas loaded

Figure 6． The current density distribution on the proposed two gap-coupled circular microstrip patch antennas loaded with shorting post for TM01 mode at frequency 2.149 GHz for gap distance between adjacent edges of feed patch and parasitic patch 0.1 mm and radius of shorting post 1.5 mm

Figure 7. The current density distribution on the proposed two gap-coupled circular microstrip patch antennas loaded with shorting post for TM11 mode at frequency 3.6912 GHz for gap distance between adjacent edges of feed patch and parasitic patch 0.1 mm and radius of shorting post 1.5 mm




Figure 8. The current density distribution on the proposed two gap-coupled circular microstrip patch antennas loaded with shorting post for TM21 mode at frequency 3.8474 GHz for gap distance between adjacent edges of feed patch and parasitic patch 0.1 mm and radius of shorting post 1.5 mm

with shorting post can be used for multi frequency applications. Also the size of the proposed antenna can be controlled by varying either gap distance between adja-cent edges of the feed patch and parasitic patch or radius of the shorting post. The proposed model can be ex-tended to multiple resonators as well as analysis with different patch sizes.

6. Acknowledgement

Authors are sincerely thankful to the reviewers for criti-cal comments and suggestions to improve the quality of the manuscript.

REFERENCES

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[2] A Sharma and G. Singh, “Design of single pin shorted three-dielectric-layered substrates rectangular patch mi-crostrip antenna for communication systems,” Progress In Electromagnetic Research Letters, Vol. 2, pp. 157–165, 2008.

[3] R. Garg, P. Bhartia, I. Bahl, and A. Ittipiboon, “Microstrip Antenna Design Handbook,” Artech House: London, 2001.

[4] T. Chakravarty, S. Biswas, A. Majumdar, and A. De, “Computation of resonant frequency of annularing-loaded circular patch”, Microwave and Optical Technology Let-

ter, Vol. 48, No. 3, pp. 622–626, 2006.

[5] D. M. Pozar, “Microstrip antennas,” Proceedings of the IEEE, Vol. 80, No. 1, pp. 79–91, January 1992.

[6] P Kumar, G. Singh, and S. Bhooshan, “Gap-coupled mi-crostrip antennas,” Proceedings of the International Con-ference on Computational Intelligence and Multimedia Application, India, pp. 434–437, 2007.

[7] K. P Ray, S. Ghosh, and K. Nirmala, “Compact broadband gap-coupled microstrip antennas,” Proceedings of Inter-national Symposium on IEEE Antennas and Propagation Society, pp. 3719–3722, July 2006.

[8] R. B. Waterhouse, S. D. Targonski, and D. M. Kokoto, “Design and performance of small printed antennas,” IEEE Transactions on Antennas Propagation, Vol. 46, pp. 1629–1633, 1998.

[9] T. K. Lo, C.-O. Ho, Y. Hwang, E. K. W. Lam, and B. Lee, “Miniature aperture-coupled microstrip antenna of very high permittivity,” Electronics Letter, Vol. 33, pp. 9–10, January 1997.

[10] R. Porath, “Theory of miniaturized shorting post micro-strip antennas,” IEEE Trans. Antennas Propagation, Vol. 48, No. 1, pp. 41–47, January 2000.

[11] C.-L. Tang, H.-T. Chen, and K.-L. Wong, “Small circular microstrip antenna with dual frequency operation,” Elec-tronics Letter, Vol. 33, No. 73, pp. 1112–1113, 1997.

[12] K.-L. Wong and W.-S. Chen, “Compact microstrip an-tenna with dual-frequency operation,” Electronics Letter, Vol. 33, No. 8, pp. 646–647, April 1997.

[13] T. Chakravarty and A. De, “Investigation of modes tun-able circular patch radiator with arbitrarily located short-ing posts,” IETE Technical Review, Vol. 16, No. 1, pp. 109–111, January 1999.

[14] T. Chakravarty and A. De, “Design of tunable modes and dual-band circular patch antenna using shorting posts,” IEE Proceeding Microwaves, Antennas Propagation, Vol. 146, No. 3, pp. 224–228, 1999.

[15] P. Kumar, T. Chakravarty, G. Singh, S. Bhooshan, and S. K. Khah “Numerical computation of resonant frequency of gap coupled circular microstrip antennas,” Journal of Electromagnetic Waves and Applications, Vol. 21, No. 10, pp. 1303–1311, 2007.

[16] T. Chakravarty, S. M. Roy, S. K. Sanyal, and A. De, “A novel microstrip patch antenna with large impedance bandwidth. In VHF/UHF range,” Progress in Electro-magnetics Research, Vol. 54, pp. 83–93, 2005.



265


M. ANITHA1, S. SUBRAMANIAN1, R. GNANADASS2

1Department of Electrical Engineering, Annamalai University, Annamalainagar, Tamil Nadu, India, 2Department of Electrical & Electronics Engineering, Pondicherry Engineering College, Puducherry, India. Email: [email protected] Received April 29th, 2009; revised July 10th, 2009; accepted July 23rd, 2009.

ABSTRACT

The application of a novel Particle Swarm Optimization (PSO) method called Fitness Distance Ratio PSO (FDR PSO) algorithm is described in this paper to determine the optimal power dispatch of the Independent Power Producers (IPP) with linear ramp model and transient stability constraints of the power producers. Generally the power producers must respond quickly to the changes in load and wheeling transactions. Moreover, it becomes necessary for the power pro-ducers to reschedule their power generation beyond their power limits to meet vulnerable situations like credible con-tingency and increase in load conditions. During this process, the ramping cost is incurred if they violate their permis-sible elastic limits. In this paper, optimal production costs of the power producers are computed with stepwise and piecewise linear ramp rate limits. Transient stability limits of the power producers are also considered as additional rotor angle inequality constraints while solving the Optimal Power Flow (OPF) problem. The proposed algorithm is demonstrated on practical 10 bus and 26 bus systems and the results are compared with other optimization methods. Keywords: Optimal Power Flow, Production Cost, Ramping Cost, Fitness Distance Ratio Particle Swarm Optimization,

Transient Stability Limit, Piecewise Linear Ramp Rate Limit 1. Introduction

The electric power industry is restructured around the world to meet the growing load demand. In the restruc-tured power market, most of the power transfer is carried out through wheeling transactions [1]. Power system planners and operators often use Optimal Power Flow (OPF) as a powerful assistant tool in both planning and operating stage. OPF problem is aimed to optimize the operating cost of the power system while satisfying various system operating constraints. Due to the nonlinear nature of the problem, many researchers explored the artificial intelligence techniques to have the minimum solution [1–4].

Though OPF problem minimizes the cost while satis-fying the practical constraints like line flow, voltage lim-its etc., it has to satisfy the stability limits also. Due to the competitive expansion in recent years, power system operation has become highly stressed, unpredictable and vulnerable [5]. A stressed system has more contingencies and become more vulnerable. Due to the occurrence of large blackouts, stability constraints are inevitable while

solving the optimal power flow problem [6]. Many re-searchers have solved the transient stability constrained OPF problem using various methods [7–12].

In the real time power market, the critical contingen-cies and sudden load changes forced the operation of the Independent Power Producers (IPPs) beyond their oper-ating power limits. But the operations are restricted by their ramp rate limits. When the operating range of the generator is within the elastic range of the strength of the shaft, the corresponding ramping process will not shorten the life of the rotor and no ramping costs are incurred. When it violates the elastic range, the economic impact due to rotor fatigue is expressed in terms of the ramping cost [13] and the authors argued that the ramping costs should be incorporated into the system operation costs.

Shrestha et al. derived the operating cost of the power producers by considering ramping rate and time with conventional quadratic cost function [14]. Tanaka de-scribed an extended form of real time pricing that achieves the optimal rate of change in demanded quantity by considering the ramping costs into account. The steepness of the load curve is controlled by the proposed

A Novel PSO Algorithm for Optimal Production Cost of the Power Producers with Transient Stability Constraints 266

optimal pricing policy, which reduces the ramping cost and the possibility of large scale blackout [15].

In the above literature, fixed ramping rate limits are used to incur the cost with respect to the contingencies and system fluctuations. But the strict ramping rate of the power producers will increase the production cost and hence the stepwise ramping rate [16] has been proposed. But in reality the dispatch schedules of the independent system operators have not been executed by the power producers due to the restricted ramping period [17,18]. Hence the piecewise linear ramping model was proposed by Shahidehpour et al. [19] to solve unit commitment problem and find their applications in the operation of Independent System Operators [17,18].

Conventional methods like lambda iteration method offer good results but when the search space is non linear and has discontinuities like piecewise linear ramping, these methods become difficult to solve with a slow con-vergence ratio and not always seeking to the global op-timal solution. Hence, Optimal Power Flow problem considered in this paper is solved using a novel Fitness Distance Ratio based swarm algorithm with transient stability limit. The power producers adjust their operat-ing points when the system is subjected to credible con-tingencies and sudden load variations. If the change in power output of the power producers violates their per-missible limits, the ramping cost is computed. A linear ramping model is validated to compute the ramping cost of the power producers with a utility test system.

2. Problem Formulation

Optimization of production cost function F of generation has been formulated based on classical OPF problem with line flow constraints. For a given power system network, the optimization of production cost of genera-tion is given by the following equation.

F = Min 1

( ( ) )n

i i ii

f P RC

$ / h (1)

where F is the total operating cost of power producers, fi (Pi) is the fuel cost of the ith power producer. RCi is the ramping cost of the power producer and n is the total number of power producers connected in the network.

The fuel cost function of the ith generator is given by

fi (Pi) = a0 + a1 Pi + a2 Pi2 $ / h (2)

where Pi is the real power output of an ith power producer and a0, a1, a2 are the fuel cost coefficients of the ith power producer.

Ramping cost is considered to be negligible when the power producers operate within the limits. However, strict ramping limits restrict their operation. If they are permitted to extend their limits, the life of the rotor will be affected. Hence the operation beyond the safe limits is charged as ramping cost. This ramping cost is cumulated

with the fuel cost which is termed as production cost of the power producers.

Even though, the ramping actions of the power pro-ducers are governed by limits, they change their operat-ing states with respect to time. Figure 1 shows the varia-tion of output during the ramping process. It consists of piecewise linear ramping period [0, RTm] and a constant output period [RTm, 1 h].

The power output of the power producer during the first interval between (0, RT1) is given by

Pi = 2 1

1

( ) (i iP P RT

RT

) + Pi1 (3)

where RT is the total ramping time of the power pro-ducer.

Similarly, the power output of the power producer during the second interval between (RT1 , RT2) is given by

Pi = i3 i2 1

2 1

(P P ) (RT RT )

(RT RT )

+ Pi2

In general, the power output of the power producer in any interval among the m segments during the linear ramping period (0, RTm) is given as follows:

Pi =1mm

im1im

RTRT

PP

(RT – RT (m -1))+ Pim , 0<RT<RTm, (4)

m is segment index

The power output of the power producer during the constant output period between the interval (RTm ,1) is given by

Pi = Pi + RR * RT, RTm< RT <1 (5)

where RR is the ramping (up/down) rate and RT is the ramping time of the ith power producer.

The above formulation includes the m linear segment regions before the power producers obtain its ramping limit in the specified time interval. This formulation is

Figure 1. Power delivery during a time interval

RT1

RTm RT2

Ramping process

0

Pim

Pi2

Pi3

Pi1

1 Hour



used to compute the ramping cost of the power producers at their corresponding operating point. This information is followed by the ISOs [17,18] during the rescheduling process of the power producers at the insecure conditions.

When the ramping process is included in the power dispatch, the effective operating cost of a unit consider-ing the change of the power output is given by

F = (6) )t(c)t(c 2

1

RTt1

RT

0t m

m

where c1(t) is the optimal production cost between the interval 0 and RTm and is given by

c1(t) = , t (0, RTm) 20 1 2( ) ( )i ia a P t a P t

where

Pi(t) = t*RRPRTRT

)RTRT()PP(im

1mm

)1m(im1im

,

(7) (0, )mt RT

c2(t) is the optimal production cost between the interval RTm and 1 and is given by

c2(t) = a0 + a1Pi(t) + a2Pi(t)2, t (RTm , 1)

where Pi(t) = Pi + RR* RT (8)

In the above expressions, the term RR represents ei-ther increase ramp rate (UR) or decrease ramp rate (DR). When the power producers are operating beyond their permissible limits, ramping cost is calculated using the above formulation.

The operating cost function of the power producer given in the equation 1 is subjected to the following con-straints.

* – PL – PD = 0 (9)

n

1iiP

where PD is the total load of the system and PL is the transmission losses of the system.

* The inequality constraint on real power generation Pi of each power producer i is given by

Pimin Pi Pimax (10)

* Power limit on transmission line is given by

MVAfp,q MVAfp,qmax (11)

where MVAfp,qmax is the maximum rating of transmission

line connecting buses p and q. * Transient stability constraint is expressed in terms of

generator rotor angles which are given as follows:

min i max (12)

where i is the relative rotor angle of the ith power pro-ducer with respect to the reference.

3. Overview of Particle Swarm Optimization

Kennedy and Eberhart first introduced the PSO method which is motivated by social behavior of organisms such as fish schooling and birds flocking [20]. In a PSO sys-tem, particles fly around a ‘d’ dimensional problem space. During flight, each particle adjusts its position according to its own experience as well as by the best experiences of other neighboring particles. Let us consider Xi = (Xi1, Xi2 … Xid) and Vi = (Vi1, Vi2 … Vid) be the position and velocity of the ith particle. Velocity Vid is bounded be-tween its lower and upper limits. The best previous posi-tion of the ith particle is recorded and is given by Pbesti = (Pi1, Pi2, … Pid ). Let gbesti = (Pg1, Pg2, … Pgid) be the best position among all individual best positions achieved so far. Each particle’s velocity and position is updated using the following two equations.

Vidk+1=W*Vid

k+C1*rand1*(Pid–Xid)+C2*rand2*(Pgid –Xid)

(13)

Xidk+1 = Xid

k + Vidk+1 (14)

where C1 and C2 are the acceleration constants, which represent the weighting of stochastic acceleration terms that pull each particle towards Pbest and gbest positions. K represents the current iteration and rand1 and rand2 are two random numbers in the range zero to one. Inertia weight W is a control parameter that is used to control the impact of the previous velocities on the current one. Hence, it influences the trade-off between the global and local exploration abilities of the particles. The search process will terminate if the number of iterations reaches the maximum allowable number.

4. FDR PSO Algorithm

In the literature, it has been reported that the particle po-sitions in PSO oscillate in damped sinusoidal waves until they converge to points in between their previous Pbest and gbest positions [21,22]. During this oscillation, if a particle reaches a point, which has better fitness than its previous best position, then the particle continues to move towards the convergence of the global best position discovered so far. All the particles follow the same be-havior to converge quickly to a good local optimum. Suppose, if the global optimum of the problem does not lie on a path between original particle positions and such a local optimum, then the particle is prevented from ef-fective search for the global value. In such case, many of the particles are wasting their computational effort in seeking to move towards the local optimum already dis-covered. Better results may be obtained if various parti-cles explore other possible search directions.

In the FDR PSO algorithm, in addition to the Socio- cognitive learning processes, each particle also learns from the experience of neighboring particles that have a better fitness than itself [23]. The implementation of this




268

5. Simulation Results and Discussion idea is simple based on computing and maximizing the relative fitness distance ratio. The proposed algorithm does not introduce any complex computations in the original PSO algorithm.

The operation of the power producers are limited by their real and reactive power limits. But in the real time opera-tion, they have to be committed beyond their limits for a given period when subjected to credible contingencies and sudden increase in load conditions. This type of op-eration will affect the life of the rotor. But keeping in the view of power system reliability, this operation is inevi-table and the power producers are reasonably compen-sated by the system operators. But the change in state of their operation is also limited by their ramp rate limits.

This approach results in change in the velocity update equation, although the position update equation remains unchanged. It selects only one other particle at a time when updating each velocity dimension and that particle is chosen to satisfy the following two criteria.

1) It must be near the current particle. 2) It should have visited a position of higher fitness. The simplest way to select a nearby particle which

satisfies the above mentioned two criteria is that maxi-mizes the ratio of the fitness difference to the one-di-mensional distance. In other words, the dth dimension of the ith particle’s velocity is updated using a particle called the nbest , with prior best position Pj. It is necessary to maximize the following Fitness Distance Ratio which is given by

The ramping costs are incurred with the fuel cost of the power producers when they violate the elastic power limits for maintaining the system security. The step by step algorithm for computing ramping cost is given in Figure 2.

The optimal power dispatch and minimum production cost of the IPPs were obtained using swarm intelligence algorithms by satisfying the transient stability limits and transmission line constraints. The simulation parameters of the swarm intelligence methods which decide the execution time of the algorithms are given in Appendix A. A linearly decreasing inertia weight W, which varies from 0.9 to 0.2, was used for the convergence character-istics. The line flows were computed using Newton Raphson method and their thermal power limits were satisfied. The simulation studies were carried out on P IV, 3 GHz system in MATLAB environment. The effective-ness of the proposed method has been demonstrated by considering 10 bus and 26 bus test systems.

( ) ( )

| |i

jd id

Cost p Cost X

P X

i (15)

In the FDR PSO algorithm, the particle’s velocity up-date is influenced by the following three factors:

1) Previous best experience i.e. Pbest of the particle 2) Best global experience i.e. gbest , considering the

best P best of all particles. 3) Previous best experience of the “ best nearest”

neighbor i.e. nbest. Hence, the new velocity update equation becomes:

Vi dk+1=W*Vid

k + C1*rand1*(Pid–Xid)+C2*rand2* (Pgid – Xid) +C3 *rand3 (Pnd - Xid) (16) 5.1 10 Bus System

The swarm intelligence algorithms were tested on ten bus system having 5 power producers and 13 transmission lines. The fuel cost coefficients, unit limits of the test system are taken from [24].

where Pnd is the nearby particle that have better fitness. The position update equation remains the same as in

Equation (13). The overall evolution of the PSO and FDR PSO popu-

lation resembles that of other evolutionary algorithms. The main difference is that algorithms in the PSO family retain historical information regarding points in the search space already visited by the various particles, this is a feature not shared by other evolutionary algorithms.

5.1.1 Case 1: Base Load Condition The base load of the 10 bus system is 2.25 p.u. The op-timal power flow solution with transient stability limits has been obtained using the FDR PSO algorithm. The optimal settings of the power producers and the obtained minimum fuel cost values are compared with other opti-mization methods which are given in Table 1.

PSO algorithm performs well in initial iterations but fails to make further progress in later iterations as the population diversity is rapidly lost. Hence PSO algorithm suffers from premature convergence in many applica-tions. But in FDR PSO algorithm, the average and best fitness continue to differ for many more iterations than PSO algorithm [23]. Hence the proposed method is less susceptible to premature convergence and less likely to get into local minimum of the function being optimized. Avoiding premature convergence allows FDR PSO to continue search for global optima in difficult optimiza-tion problems, reaching better solutions than PSO. Thus it outperforms the standard PSO.

Table 1. Comparison among different methods

Generator Power (p.u) Optimi-zation

methods P1 P2 P3 P4 P5

Fuel Cost ($/hr)

LP [24] 0.414 0.050 1.224 0.050 0.059 164.177

EP [25] 0.285 0.052 1.183 0.058 0.727 164.019

PSO 0.417 0.129 0.911 0.196 0.597 164.321

FDR PSO 0.352 0.077 1.079 0.060 0.682 163.850


Start

Preparation of line, bus, load and power producer data base

Perform the increase in load and credible contingencies

Obtain FDR PSO based optimal generation dispatch

Figure 2. Step-by-step algorithm

From this table, it is inferred that the value of the fuel cost obtained through the proposed methods is better than the results obtained through other optimization methods. It was also observed that there was no vast variation in the five generator powers after several runs.

The convergence characteristics of PSO and FDR PSO methods are shown in Figure 3 for illustration. It is found that during initial iterations, FDR PSO concen-trates mainly on finding feasible solutions to the problem. Then the value gradually settles down to the optimum value with most of the particles in the population reach-ing that point.

In the above OPF solution, the transient stability limit has to be incorporated by the following procedure:

A three phase to ground fault was assumed in the transmission line connected between buses 8 and 9. The above fault was cleared by opening the contacts of the circuit breakers by 0.1 Section. The obtained OPF solution satisfies transient stability limits and the cor-responding relative rotor angles of the generators are shown in Figure 4.

From this figure, it is observed that, while obtaining the solution for OPF problem, the relative rotor angles are also within safer limits and hence the system security is ensured.

5.1.2 Case 2: Production Cost with Ramping The power producers have to change their generation

End

Yes

No

Check the power output of power producers lie within elastic limit

Compute the total operating cost

Compute the ramping cost using linear model



Figure 3. Convergence characteristics

Figure 4. Relative rotor angle curves

schedule when the load changes. The operation of their settings depends upon their ramp rate limits. Initially, a stepwise ramping up/down limits is considered. The power producers are rescheduled to obtain their best set-ting point with various stepwise ramping limits. The ob-tained results are given in Table 2. From the table, it is inferred that the production cost is maximum when the producers operate with strict ramp rate limit and the minimum production cost is obtained when the ramp rate limit is around 20%. However, further stricter ramp rate constraint will restrict them from the essential diversity search, and result in more production cost or in less cost saving. Table 2. Ramping cost results -stepwise RR limits

PSO FDR PSO %RR limit FC RC PC FC RC PC

10 164.63 4.12 168.75 164.71 3.35 168.0620 164.84 3.25 168.09 164.64 3.14 167.7830 165.26 4.52 169.70 164.94 3.99 168.9340 165.30 4.35 169.60 164.64 4.54 169.1850 165.40 4.08 169.50 164.78 4.89 169.6775 165.07 5.09 170.17 164.42 5.83 170.25

100 165.41 8.35 173.76 164.74 7.10 171.84FC = Fuel Cost ($/hr) RC = Ramping Cost ($/hr) PC = Production Cost ($/hr)

In reality, the power producers are operated with dis-tinct ramping limits with respect to different operating conditions. Strict ramping limits increase the production cost while liberal ramping limits hurt the life of the rotor. Hence in this paper, a piecewise linear function is used to determine the ramping cost of the power producers. The piecewise linear ramp rate limits of the power producers are given in the Appendix B. The obtained optimal pro-duction cost of the power producers with piecewise lin-ear ramping rate and transient stability limits using swarm intelligence methods is given in Table 3. Hence the proposed method is able to solve the real time ISO problem and able to handle non linear functions.

5.1.3 Case 3: Production Cost with Wheeling Transactions

In the restructured power market, most of the power transfers have been carried out through either bilateral or multilateral wheeling transactions. In this case, two si-multaneous bilateral wheeling transactions and one mul-tilateral transaction were carried out on the test system. The details of the wheeling transactions and magnitude of power transfer are given in Tables 4 and 5. PSO and FDR PSO methods are used to obtain the optimal pro-duction cost with linear ramp model for the practical power system and the results are given in Table 6. Whilesolving the above problem generator bus voltage limits, voltage angles, line flow limits and transformer tap positions are taken into account. The wheeling trans-actions are carried out in the system with transmission line MVA limits. Since the wheeling transactions are satisfying the transmission line limits, they are feasible.

5.2 26 Bus System

The swarm intelligence algorithms were used to obtain optimal production cost with linear ramp model for 26 bus utility system. It consists of six power producers and forty six transmission lines. The fuel cost coefficients, unit limits of the test system are taken from [26]. The piecewise linear ramp rate limits of the system are given Appendix C. The transient stability based OPF solution for the practical utility system was demonstrated with line and generator contingency case studies are illus-trated in the following subsections.

5.2.1 Case 1: Optimal Production Cost with Line Contingency

In this case study, the optimal production solution of the test system is obtained through the FDR PSO algorithm when subjected to different types of line contingency (single line outage, during fault and double line outage) studies. 10% increase in base load condition is assumed. The transient stability limit of the power producers is also satisfied during the contingency based optimal solu-tion procedure. The obtained OPF results for the various




271

contingency case studies are consolidated in Table 7. It includes the minimum fuel and linear ramping costs in-curred by the power producers and their corresponding settings. When the power limits of the power producers are violated during the contingency based rescheduling

operation, then ramping cost is incurred, otherwise no ramping cost is levied. In Case (a), single line contin-gency is illustrated by making the line connected be-tween buses 20 and 21 as out of service.

In Case (b), contingency based OPF solution is obtained

Table 3. Production cost -linear ramping

PSO FDR PSO Gen No. Gen set FC RC PC Gen set FC RC PC

G1 0.64 32.08 0.81 32.90 0.54 31.41 0.81 32.22

G2 0.26 33.87 1.10 34.97 0.31 33.60 1.09 34.69

G3 0.63 32.08 0.71 32.80 0.93 34.13 0.00 34.13

G4 0.11 35.67 1.31 36.99 0.09 35.91 1.28 37.19

G5 0.58 31.18 0.85 32.03 0.37 29.39 0.82 30.21

Total 164.90 4.81 169.70 164.44 4.02 168.45

Table 4. Details of bilateral wheeling transactions

Bus No. Real Power (p.u) Transaction

From To

TB1 10 4 0.20

TB2 8 5 0.10

Table 6. Optimal production cost with wheeling transactions

Method FC($/hr) RC($/hr) PC($/hr) PSO 165.08 3.88 168.96

FDR PSO 164.04 3.17 167.21

Table 5. Details of multilateral wheeling transactions

Bus No. Bus No. Transaction

From Real Power

(p.u ) To

Real Power (p.u )

9 0.20 6 0.10

7 0.15 3 0.15 TM1

2

Total 0.35 0.35

Table 7. OPF solution – line contingency case studies

G1 G2 G3 G4 G5 G6 Total

Gen set 478.83 198.94 271.81 124.49 199.18 116.01 1389.26

FC 5197.49 2565.38 3195.31 1708.87 2628.77 1683.05 16978.87

RC 0.00 96.13 0.00 0.00 95.40 45.17 236.17

Case a

PC 5197.49 2661.51 3195.31 1708.87 2724.17 1728.22 17215.04

Gen set 482.05 199.55 298.12 100.27 199.16 110.14 1389.29

FC 5240.95 2573.79 3553.89 1394.74 2628.49 1602.66 16994.52

RC 0.00 95.88 161.17 0.00 0.00 45.14 302.19 Case b

PC 5240.95 2669.67 3715.06 1394.74 2628.49 1647.80 17296.71

Gen set 498.36 195.50 258.05 149.7 179.61 108.05 1389.27

FC 5467.06 2538.09 3023.17 2048.39 2367.99 1574.16 17018.86

RC 264.98 95.67 0.00 76.33 0.00 0.00 436.98 Case c

PC 5732.04 2633.76 3023.17 2124.72 2367.99 1574.16 17455.84


Figure 5. Relative rotor angles curve - Case b Figure 6. Power Flow in transmission lines – Case c during the fault condition. Along with the single line outage in the Case (a), a three phase to ground fault is also assumed in transmission line connected between buses 1 and 18. The fault is cleared by isolating the transmission line by 0.1 Sec. The corresponding relative rotor angle diagram with respect to the reference is given in Figure 5. It is inferred that the rotor angles of the power producers are within acceptable limits which en-sures the system security.

In Case (c), two line contingencies are considered while obtaining the solution of the transient stability constrained OPF problem. The lines connected between buses (20-21) and (9-10) are made to be out of service to clear the fault. The power flow in the lines during the pre and post contingency conditions are shown in Figure 6. The obtained OPF solution satisfies the thermal power limits of the transmission lines during the credible con-tingencies.

5.2.2 Case 2: Generator Contingency Condition A severe generator outage contingency based power flow solution is illustrated to demonstrate the effectiveness of the proposed FDR PSO approach. A generator connected at bus 5 is made out of service for maintenance purpose. The minimum fuel, ramping and production costs for the above critical condition are obtained as 15604.31 $/hr, 639.16 $/hr 16243.47 $/hr respectively. The simulation results are shown in Figures 7 and 8 for rotor angles and angular frequencies respectively. From the waveforms shown in these figures, it can be concluded that the system remains stable since the waveforms do not diverge dur-ing the simulation interval.

6. Conclusions

FDR PSO algorithm for solving the OPF problem with credible contingencies considering transient stability

constraints and piecewise linear ramping model are pre-sented in this paper. The feasibility of the proposed

Figure 7. Relative rotor angle curves - Generator contin-gency condition

Figure 8. Angular frequency diagram - Generator contin-gency condition



method for solving the transient stability constrained OPF problem was demonstrated with two test systems considering various contingencies and nonlinearities like piecewise linear ramping model. The computation of linear ramping cost illustrates the increase in production cost of the power producers when subjected to load changes in the deregulated environment. The obtained results give feasible economic and security signals to the power utilities when subjected to vulnerable conditions in the deregulated power industry.

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systems: An overview,” International Journal of Emerging Electric Power Systems, Vol. 2, No. 1, Article 1021, 2005.

[2] J. G. Vlachogiannis and K. Y. Lee, “A comparative study on particle swarm optimization for optimal steady-state performance of power systems,” IEEE Transactions on Power Systems, Vol. 21, No. 4, pp. 1718–1728, Novem-ber 2006.

[3] I. Selvakumar and K. Thanushkodi, “A new particle swarm optimization solution to non convex economic dispatch problems,” IEEE Transactions on Power Sys-tems, Vol. 22, No. 1, pp. 42–51, February 2007.

[4] P. E. Onate Yumbla, J. M. Ramirez, and C. A. Coello, “Optimal power flow subject to security constraints solved with a Particle swarm optimizer,” IEEE Transac-tions on Power Systems, Vol. 23, No. 1, pp. 33–40, Feb-ruary 2008,.

[5] M. Ni, J. D. McCalley, V. Vittal, T. Tayyib, “Online Risk-based security assessment,” IEEE Transactions on Power Systems, Vol. 18, No. 1, pp. 258–265, February 2003.

[6] US-Canada Power System Outage Task Force, Final Re-port on the August 14, 2003 Blackout in the United States and Canada: Causes and Recommendations, Issued April 2004.

[7] D. Gan, R. J. Thomas, R. D. Zimmerman, “Stability con-strained optimal power flow,” IEEE Transactions on Power Systems, Vol. 15, No. 2, pp.535–540, 2000.

[8] D.-H. Kuo and A. Bose, “A generation rescheduling method to increase the dynamic security of power sys-tems,” IEEE Transactions on Power Systems, Vol. 10, No. 1, pp. 68–76, February 1995.

[9] N. Mo, Z. Y. Zou, K. W. Pong, and T. Y. G. Pong, “Transient stability constrained optimal power flow using particle swarm optimization”, IET. Generation, Trans-mission and Distribution, Vol. 1, No. 3, pp. 476–483, 2007.

[10] X. J. Lin, C. W. Yu, and A. K. David, “Optimum tran-sient security constrained dispatch in competitive de-regulated power systems,” Electric Power Systems Re-search, Vol. 76, pp. 209–216, 2006.

[11] W. Gu, F. Milano, P. Jiang, and J. Y. Zheng, “Improving large-disturbance stability through optimal bifurcation control and time domain simulations,” Electric Power Systems Research, Vol. 78, pp. 337–345, 2008.

[12] S. V. Ravikumar and S. S. Nagaraju, “Transient stability improvement using UPFC and SVC,” ARPN Journal of Engineering and Applied Sciences, Vol. 2, No. 3, pp. 38–45, June 2007.

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[23] T. Peram, K. Veeramacheneni, and C. K. Mohan, “Fit-ness distance ratio based particle swarm optimization,” Proceedings of the IEEE Swarm intelligence Symposium, 24–26, pp. 174–181, April 2003.

[24] A. Farag, S. Al-Baiyat, and T. C. Cheng, “Economic load dispatch multiobjective optimization procedures using linear programming techniques,” IEEE Transactions on Power Systems, Vol. 10, No. 2, pp. 731–737, May 1995.

[25] R. Gnanadass, K. Manivannan, and T. G. Palanivelu, “Application of evolutionary programming approach to economic load dispatch problem,” National Power Sys-tem Conference, IIT, Kharagpur, 2002.

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Appendix

A. Simulation Parameters

AI Methods C1 C2 C3 Population

size Maximum No. of generations

PSO 1.0 1.0 -- 50 5000

FDR PSO 1.0 1.0 2.0 50 5000

B. Piecewise Linear Rr Limit – 10 Bus System

Pi (p.u) RR (MW/hr)

Min Max DRi URi

RT

(Min)

0.05 0.50 0.45 0.25 0.333

0.51 0.75 0.40 0.20 0.250

0.76 1.50 0.60 0.30 0.417




275

Radio Wave Propagation Characteristics in FMCW Radar

Ghada M. SAMI

Mathematics Department, Faculty of Science, Ain Shams University, Cairo, Egypt Email: [email protected]

Received July 3rd, 2009; revised August 11th, 2009; accepted August 20th, 2009.

ABSTRACT

FMCW Radar (Frequency Modulated Continuous Wave Radar) is used for various purposes, such as atmospheric Re-mote Sensing, inter-vehicle ranging, etc. FMCW radar systems are usually very compact, relatively cheap in purchase as well as in daily use, and consume little power. In this paper, FMCW radar determines a target range by measuring the beat frequency between a transmitted signal and the received signal from the target, and Combines between PO and radar single. The approach based on frequency domain physical optics for the scattering estimation and the linear sys-tem modeling for the estimation of time domain response, and FMCW Radar signal processing.

Keywords: Radio Wave, FMCW Radar, Cloud Profiling Radar

1. Introduction

The FMCW radar has to adjust the range of frequencies of operation to suit the material and targets under inves-tigation. The transceiver generates a signal of linearly increasing frequency for the frequency-sweep period. The signal propagates from the antenna to a static target and back. The value of the received-signal frequency compared to the transmitted-signal frequency is propor-tional to the propagation range. The main advantages of the FMCW radar are the wider dynamic range, lower noise figure and higher mean powers that can be radiated. In addition a much wider class of antenna is available for use by the designer.

The further advantage of FMCW radar is its ability to adjust the range of frequencies of operation to suit the material and targets under investigation if the antenna has an adequate pass-band of frequencies. This radar system mixes the wave reflected by a target object and part of the radiated wave to obtain a beat signal that con-tains distance and speed components. For large scatterer, the physical-optics approximation is an efficient method in the frequency domain [1,2]. This physical optics (PO) approximation is initially applied in the frequency-domain with the inverse Fourier transform [3], [4], [5], [6]. With FMCW, the high-frequency circuitry for beat signal detection is relatively simple and distance can be directly obtained. By mixing the received FMCW and transmitted FMCW signals, the system obtains a beat signal having a frequency f b.

2. The Principles of the FMCW Radar

The principle of the FMCW radar is shown in Figure 1. Transmitted signal from one of the antennas is reflected, and is received by the other antenna with delay time To relative to the original transmitted signal. Mixing the received and transmitted frequencies, the beat frequen-cies bf are observed in the spectra.

The time takes for the signal to travel the two-way dis-tance between the target and the radar is To, hence [7], [8]:

Frequency

fb

Transmitted signal

Received Signal

Tm

To

B

Timefo

Figure 1. Principle of the FMCW radar

Radio Wave Propagation Characteristics in FMCW Radar 276

2o

RT

c (1)

From the geometry of transmit and receive waveforms we can derive a relationship between the beat frequency f b, the range R, and c is the velocity of light. From Figure 1, we can see,

o

m

T f

T B b (2)

Substituting (1) in (2), we get

2

bm

RBf

cT (3)

3. Scattered Wave from Radar Target

3.1 Frequency Domain Physical Optics

For a perfectly conducting body, the frequency-domain PO-induced current distribution over the illuminated surface is [9,10,11]:

ˆ( , ) 2 ( , )po incsj R n H R

(4)

where is the unit vector normal to the surface n R

and ( , )incH R is the incident magnetic field with an-

gular frequency The frequency-domain scattered field is obtained by

calculating the integral over the illuminated surface using

the free space Green’s function:

` `

`

`

( , ) ( , ) ( , ) ds4`

inc PO POo st

jk R Roe

H R E R j RR R

s

(5) where the vector \R locates the integration point on the

scatterer surface, is the wave number, ok2

o

f

kc

is the velocity of the light and o is the intrinsic free

space impedance, and `( , )postj R

is the surface-current

distribution. The frequency transfer function ( )H is defined as

( )( )

( )o

i

VH

V

(6)

where ( )iV is the input waveform in frequency do-

main physical optics, this is just a magnitude of the source.

The output Voltage ( )oV is calculated from

( , )poE R by considering the receiver antennas as

[12,13],

( , )( )

( )

po

oc

E RV

F

(7)

`inc `

k`

1ˆ( ) 2 n H ( , ) s

4 ( ) ( )

kRo

kkc i k

jk RoH R

F V R R

e

(8)

where ( )cF is the Complex antenna factor.

3.2 Treatment of FMCW Signal

Instantaneous frequency fi(t), is given as

( )i om

tf t f B

T

(0 )mt T

and the instantaneous phase ( )i t , is defined as

( )i t =0

2 ( ) dt

if = 22 om

Bf t

T t (9)

Using Equation (9), the FMCW signal waveform is defined as

( ) i iv t V Cos 2(2 )om

Bf t t

T (10)

The output of mixer VFMCW1(t) is expressed as

iFMCW1v ( ) (t) ot v v t ( )

)

(11)

The output waveform vo(t) is

vo(t) = vi(t )* h(t)

when h(t) is a sample delay of To, i.e.

h(t)= ( ot T (12)

Therefore, mixer output signal is

FMCW1v t( ) = vi (t) vi (t-To) (13)

where

vi (t-To) iV Cos 2[2 ( ) ( )o o om

Bf t T t T

T ] (14)

From Equations (10) and (11) we can calculate: 2

2 21 0( ) (4 2 (2 2 ))

2i

FMCW o om

V BV t Cos f t f T t T t T

T 0 0

2(2 (2 ))o o o om

BCos f T T t T

T (15)

The first Cos term describes a linearly increasing FM signal (chirp) at about twice the carrier frequency with a phase shift that is proportional to the delay time To. This term is generally filtered out.

The second Cos term:

2 ( )FMCWV t 2

2iV

Cos 2(2 (2 ))o o o om

Bf T T t T

T (16)


Radio Wave Propagation Characteristics in FMCW Radar 277

describes a beat signal at a fixed frequency b om

Bf T

T .

3.3 Combination of PO and Radar Single Processing

As h(t) given in section 3.2 is just an idealized model, more realistic h(t) obtained by PO in section 3.1, Equa-tion (12) shall be used.

However, Equation (6) is given in frequency do-main ( )H , and it shall be Inverse Fourier transformed

j t1( ) ( ) e d

2h t H

(17)

It is sufficient that ( )H is computed only within

the source frequency range, i.e. .o of f f B Out-

side the band, ( )H can be assumed zero, (iV ) is

also zero in this region. In reality, Fourier transform shall be executed nu-

merically. Let us assume the sampling interval T which shall satisfy the following relation

1

2( )o

Tf B

(18)

1sf T

(19)

o Lf N f s

)B s

(20)

(o Lf B N N f (21)

where sf is a sampling frequency, LN , and BN are

some certain integers. Now is denoted as , and ( )iv t (iv l T ) )(h n T is

given as,

jm 2 s

-jm 2 *

( ) ( ( 2 f ) e4( )

(m 2 )e )

L B

s

L

s

N Nf n Ts

m NL B

f n Ts

fh n T H m

N N

H f

(22)

Convolution (11) is now implemented as,

2( )

0

( ) (( ) ) h(nL BN N

o in

v l T T v l n T T

) (23)

Substituting from (23) and (10) in (13), we can get

21( ) (2 )fmcw i o

m

BV t V Cos f t t

T 2

2( )2

0

(2 (( ) ) (( ) ) ) h(n )L BN N

on m

BT Cos f l n T l n T

T

T

(24)

Figure 2. Frequency domain representation of FMCW sen-sor output

where ( )L L BN m N N ,

Now, we use Fourier Transformation, we get 2

21( ) (2 )

2i

FMCW c fb fbm

V BV f Cos f

T

2 -j(m2 n T)

0

(2 (( ) ) (( ) ) )h(n ) e fo

m

BT Cos f l n T l n T T T

T

(25)

And also we can get 2

22 ( ) (2 )

2i

FMCW c fb fbm

V BV f Cos f

T (26)

describes a beat signal at a fixed frequency bf

Bf

T .

It can be seen that the signal frequency is directly pro-portional to the time delay time , and hence is directly proportional to the round trip time to the target.

4. Conclusions

This paper presents the time domain linear system analy-sis for FMCW radar response by performing the inverse Fourier transform over the frequency-domain scattered field which obtained by calculating the integral over the illuminated surface using the free space Green’s function. Then we got the received FMCW signal and transmitted FMCW signal, the product detection is implemented to get the beat signal. The Fourier transform is used to find the beat frequency.

REFERENCES

[1] W. V. T. Rusch and P. D. Potter, “Analysis of reflector antenna,” Academic, New York, pp. 46–49, 1970.

[2] R. F. Harrington, “Time-harmonic electromagnetic fields,” McGraw-Hill, New York, pp. 127, 1961.

[3] E. M. Kennaugh and R. L. Cosgriff, “The use of impulse


Radio Wave Propagation Characteristics in FMCW Radar


278

response in electromagnetic scattering problems,” IRE Natl. Conv. Rec., Part 1, pp. 72–77, 1958.

[4] S. Hatamzadeh-Varmazyar and M. Naser-Moghadasi, “An integral equation modeling of electromagnetic scat-tering from the surfaces of arbitrary resistance distribu-tion,” Progress In Electromagnetics Research B, Vol. 3, pp. 157–172, 2008.

[5] C. A. Valagiannopoulos, “Electromagnetic scattering from two eccentric Metamaterial cylinders with frequency-dependent permittivities differing slightly each other,” Progress In Electromagnetics Research B, Vol. 3, pp. 23–34, 2008.

[6] E. M. Kennaugh and D. L. Moffatt, “Transient and im-pulse response approximation,” Proceeding IEEE, pp. 893–901, August 1965.

[7] S. Hoshi, Y. Suga, Y. Kawamura, T. Takano, and S. Shi-makura, “Development of an FMCW radar at 94 Ghz for observations of cloud particles-antenna section,” Pro-ceeding of the Society of Atmospheric Electricity of Ja-pan, No. 58, pp. 116, 2001.

[8] T. Takano, Y. Suga, K. Takei, Y. Kawamura, T. Taka-mura, and T. Nakajima, “Development of an cloud pro-

filing FMCW radar at 94 Ghz, International Union of Ra-dio Science,” General Assembly, Session FP, No. 1786, 2002.

[9] E. Yuan and W. V. T. Rusch, “Time-domain physical optics,” IEEE Transactions on Antenna and Propagation, Vol. 42, No. 1, January 1994.

[10] L.-X. Yang, D.-B. Ge, and B. Wei, “FDTD/TDPO hybrid approach for analysis of the EM scattering of combinative objects,” Progress In Electromagnetics Research, PIER 76, pp. 275–284, 2007.

[11] G. M. Sami, “Time-domain analysis of a rectangular re-flector,” Submitted.

[12] S. Ishigami, H. Iida, and T. Iwasaki, “Measurements of complex antenna factor by the Near 3-antenna method,” IEEE Transaction on Electromagnetic Compatibility, Vol. 38, No. 3, August, 1996.

[13] L. Li and C.- H. Liang, “Analysis of resonance and qual-ity factor of antenna and scattering systems using com-plex frequency method combined with model-based pa-rameter estimation,” Progress In Electromagnetics Re-search, PIER 46, pp. 165–188, 2004.



279


Zhengyang ZHOU1, Minfu LIAO1, Xingming FAN2

1Department of Electrical and Electronics Engineering, Dalian University of Technology, Dalian, China; 2School of Mechanical of Electrical Engineering, Guilin University of Electronic Technology, Guilin, China. Email: [email protected]

Received July 22nd, 2009; revised August 12th, 2009; accepted August 20th, 2009.

ABSTRACT

The synthetic making test has been widely used in evaluating the break ability of high-voltage circuit breaker. However, the test research and application are still inadequate, especially in the condition of rated voltage. According to the re-alistic conditions of test stations in China, a control device based on pre-arcing current detection and phase control is proposed in this paper. A sample of the control device made up of DSP TMS320LF2407A is fabricated, in which the CPLD MAX7064 is used to transmit signals for EMC design. It can be applied in full voltage synthetic making test at a level of 126kV/63kA. The test results show that, it is accurate to control the making phase of the applied voltage, whether the closing is demanded at voltage peak or zero.

Keywords: Synthetic Making Test, Pre-Arcing Current, Phase Control

1. Introduction

It is well recognized that the verification of the making ability of circuit breakers is an integral part of the certi-fication procedure. A circuit breaker rated 145kV/40kA can be tested three-phase at full voltage and current, which is the highest rating that can be carried out with direct three-phase short-circuit power at KEMA high- power laboratory in Holland [1]. However, in many cases direct making test exceeds the available power of the test laboratories. So the synthetic test circuit has been put into service and it is widely accepted at present. The test basic rules and applied conditions have been discussed at CIGRE in 1960’s and 1970’s [2,3]. According to the lat-est IEC60427 and China National Standard GB1984- 2003, for the circuit breaker whose pre-arcing is obvi-ous, it is necessary to test not only the rated short circuit close ability at reduced applied voltage, but also the close and open ability with maximum pre-arcing at standard applied voltage, which can prove the loading ability of circuit breaker when the pre-arcing occurs at applied voltage peak, leading to a symmetrical closing current [4,5].

Synthetic making test of high-voltage circuit breakers at full voltage is a close test at rated applied voltage. It is reported that the KEMA laboratory can do three-phase

synthetic making test at 245kV/63kA [1], and the ABB Company at 362kV [6]. On the basis of realistic condi-tions of high power laboratories in China, a control de-vice for the synthetic making test at full voltage based on pre-arcing current detection and phase control is pro-posed in this paper. A test circuit and its control system made up of DSP and CPLD are set up. The whole system has been applied in synthetic making test at a full voltage of 126kV/63kA.

2. Experiment Setup

The experimental circuit of synthetic making test is shown in Figure 1, with one circuit supplying the full rated voltage to initiate pre-strike and the second circuit to supply the short circuit current succeeding the pre- strike at reduced voltage.

In Figure 1 U1 and U2 mean the current and voltage source respectively. QFt is the switching device under test. The unit of Sa and CH constitutes the fast making switch, which is made up of triggered vacuum switch (TVS) and fast vacuum circuit breaker (FVCB) with permanent magnetic actuator [7]. Parallel-connected TVS and FVCB make up the basic optic-controlled mod-

Design of a Control Device for Synthetic Making Test Based on Pre-Arcing Current Detection and Phase Control 280

T1

QFt

QFb

Controller

PhaseDetection

L1 L2Sa

Rogowski Coil

i i'1

2 3

T2

U2U1

CH

Figure 1. Circuit for synthetic making test

ule. And the arranging of basic modules in series can increase the withstand voltage. The closing commands of Sa and CH are sent out simultaneously. QFb is connected to the current source rapidly (in several microseconds) and accurately through Sa when pre-strike occurs in QFt. CH (FVCB) turns on subsequently and the making cur-rent will shift to this branch, to reduce and protect the wear of Sa (TVS). The Rogowski Coil is used to detect the pre-arcing current.

The aim of design is to control the making phase of the applied voltage, whether the closing is demanded at voltage peak or zero. From the Controller in Figure 1, we can see that the control strategy for synthetic making test lies on three aspects, i.e., the fast making device, pre-arcing current detection and phase control. Firstly, the detected information of pre-arcing current and volt-age/current phase are input to the controller. Secondly, the controller calculates the delay time. Lastly, controller outputs a signal to the fast making switch, making it to turn on at the voltage peak or zero. The fast making switch has been discussed in the previous paragraph, and the other two aspects will be expanded in Section 3 and Section 4.

3. Pre-Arcing Current Detection

3.1 Entire Structure

The frame of pre-arcing current detection is shown as Figure 2. It is the time when the pre-arcing appears to close the fast making device. So it is important to detect pre-arcing current fleetly and accurately.

3.2 Rogowski Coil

There are three ways to detect the pulse high current: divider, optics device and coil [8]. Because the pulse current can reach a level of several ten thousands to even millions amperes, it is difficult for a divider to work ef-fectively, and the optics device is complex. The coil is suited to detect the pulse high current, which is measured through the voltage in the coil induced by the aim current [8]. The detected current doesn’t flow through the coil, so it is isolated between the detection system and the tested main loop, which is helpful for safe request and EMC design. The principle of Rogowski coil shows that

the last result of aim current is in direct ratio to the turn number of coil (N), and in inverse ratio to the signal re-sistant (R) which is connected to the output of coil [9]. It is shown as:

Ni U

R (1)

U is the voltage of tested signal. In this paper, the N adopted is 1000, and R is 0.019Ω. So the current detected can be measured as:

410005.263 10 52.63 (kA)

0.019i U U U (2)

In order to get the properties of coil, a standard divider is used to check the output of coil. The resistant value of standard divider is 600μΩ. The tested current is re-strained from 8kA to 15kA, due to the measurement range of divider. A typical wave of tested results is shown as Figure 3, where CH1 is the signal wave of Rogowski Coil, and the CH2 is the signal wave of di-vider. The multiple results are stored in Table 1. The compare of these results shows that, the Rogowski Coil is credible.

Pre-arcingCurrent

RogowskiCoil

WindowComparator

Commutating &filter circuit

FiberTransmission

DSP

Figure 2. Frame of pre-arcing current detection

Figure 3. Tested result of Rogowski coil

Table 1. Multiple tested results of Rogowski coil

U(mV) i(kA) U’(V) I’(kA) 160 8.4208 5 8.3333 170 8.9471 5.5 9.1667 230 12.1049 7.2 12.0000 300 15.7890 9 15.0000

U: the signal voltage of coil; U’: the signal voltage of divider



3.3 Precise Commutating and Filter Circuit

The high-speed operational amplifier LM318 is adopted in the circuit, with typical signal bandwidth of 15MHz, and inversion frequency of 70-150V/μs. It will be of benefit to reduce the hardware delay of detecting the pre-arcing current signal and leave enough time for DSP processing. The precise commutating and filter circuit is shown in Figure 4.

The output of filter circuit makes the input signal of hyper speed window comparator, which is made up of hyper speed comparator TL3016 of TI Company. The threshold value of window comparator is the set range of pre-arcing current. When the transient value of signal meets the set range, the optical fiber transmission system will send an interrupt to the DSP. That is the total proc-ess of pre-arcing current detection.

4. Phase Control

4.1 Control Method

The closing command of device is random in power sys-tem. To simulate the actual operation of circuit breaker, distinction is made in the standards between two extreme cases: one condition is making at the voltage peak, lead-ing to a symmetrical short-circuit current and the longest pre-strike; the other condition is making at the voltage zero, without pre-arcing, leading to a fully asymmetrical short-circuit current [4,5].

Figure 5 takes the condition of making at voltage peak for example to illustrate the control method. The proces-sor samples the applied voltage in real time, and once the zero of voltage is detected the timer begins to count until the random closing command comes (th). If

, the timer resets and counts up from 0

again, where the f denotes the frequency of applied volt-age. Or else, the phase control closing command will be sent after delay of tc+td, where tc is operation and calcula-tion time of DSP, and td is the waiting time to sent the phase control closing command. And tmaking later, pre- strike occurs between the approaching electrodes of switch to be tested. At this time the fast making device acts, so the short circuit current flows through the tested switch. tc+td can be calculated by the DSP from the fol-lowing Equations [10]:

h zero 1 /t t f

c d making h zero

1( ) mod (

4 2

Tt t nT t t t

f ) (3)

d making h zero c

1( ) mod ( )

4 2

Tt T t t t t

f (4)

making closing arcingt t t (5)

Because the timer will reset if , the value

of n can be testified as 1 in Equation (3). Mod is the re-

mainder. tclosing and tarcing are the closing and pre-arcing time of the tested switch respectively, which can be known before the test.

h zero 1 /t t f

4.2 Hardware Configuration

The hardware configuration of the system is shown in Figure 6. The DSP TMS320Lf2407A is adopted as the CPU, which offers up to 16 channels 10-bit ADC, CAN 2.0B module, SCI and Watchdog Timer Module, etc. The DSP takes the pre-strike current and random closing command as its input signals, meanwhile detects the phases of current and voltage in real time.

The data and signals transmitted in the system are all in the optical fibers. The structure of control signals transmission system with fiber is shown in Figure 7,

n

V_INR1

R2

R3

R4

R5

R6 R7D1

D2

C1 D3LM318

LM318

Figure 4. Precise commutating and filter circuit

AppliedVoltage

1/4T

tarcing

tclosing

tmaking

tc

td

th tmtzero tprk tmkd

MakingCurrent

RadomClose

PhaseControl

ElectrodePosition

Close

Figure 5. Phase control closing time sequence

pre-arcing current

parameter setdata process

display

random close

maincontroller

DSP current phasedetection

voltage phasedetection

fast makingdevice

phase controlclose

Fiber1 Fiber2

Fiber3

Fiber4RS485

Figure 6. Diagram of the control system

DSPMAX7064

encodefiber

transmitterfiber

receptor

fiber controlsignal

fiber transmitdriver element

MAX7064decode

close

open

fiber

signals to device

Figure 7. Structure of control signals transmission



where the MAX7064 is the CPLD made by Altera Com-pany. Two channels of control signals are sent by DSP, i.e. closing signal and opening signal for Sa and CH. When the closing signal jumps to 1 while opening signal keeps 0, MAX7064 will send closing control code 1010 to the fiber. Otherwise the opening signal jumps to 1 while closing signal keeps 0, MAX7064 will send open-ing control code 1001 to the fiber. Thus the EMC design of the system can be improved distinctly especially in the high power laboratory environment.

4.3 Software Design

The software flow chart of main program in DSP is shown in Figure 8.

The random close command and the phase control close command are the close instructions for the switch-ing device to be tested.

4.4 Discussions

It is necessary to control the making phase of the applied voltage accurately, whether the closing is demanded at voltage peak or zero. There are many factors that influ-ence the test accuracy and success rate, such as the varia-tions of operating times and pre-strike characters of the switching device to be tested. According to the standards [4,5], the time from the point of pre-strike to the voltage peak should be limited in 1.5ms, i.e. the pre-strike occurs at the phase of 90˚±27˚ for power current.

The system has been applied in the full voltage syn-thetic making test at a level of 126kV/63kA.The test re-sult is shown as Figure 9 and Figure 10. The waveforms presented in Figure 9 are the results of making at voltage peak. It is evident that the percent of DC component of the tested making current is low, i.e., symmetrical short-circuit current. The waveforms of Figure 10 are the results of making at voltage zero, and the percent of DC component is high, i.e., asymmetry short-circuit current.

The phase control is accurate at the voltage peak, be-cause the pre-strike characters of tested switching device are stable and no pre-strike occurs before the voltage peak. From abundant experiments, it is proved that the typical deviation of time from the point of pre-strike to the voltage peak is about 800μs, and it is limited in 1.5ms. Another way, the fast making switch of current source is controlled not to turn on if the deviation is much large, which can reduce the failure of making (see the phase sat-isfied pre-strike current of software flow chart in Figure 8).

5. Conclusions

1) A control device for synthetic making test at full volt-age based on pre-arcing current detection and phase con-trol is proposed in this paper.

2) Rogowski coil is adopted to detect the pre-arcing current of synthetic making test, and DSP is used as main controller to complete the phase control and data proce-

Start

Parameters preset

Random closecommand

tc+td delay

Phase controlclose command

Phase satisfiedPre-strike current

Trigger fast making deviceShort-circuit current flows

End

No

Yes

No

Yes

Figure 8. Software flow chart of main program

U2/

kV 300

200

100

1000 40 100 120 20 60 80 t/ms140

I/kA

Figure 9. Test result of voltage peak

U2/

kVI/

kA

-100

100-200

300

4.6 20 40 60 80 100 120 t/ms

Figure 10. Test result of voltage zero

ssing. 3) The system has been applied in the full voltage syn-




283

thetic making test at a level of 126kV/63kA. It is accu-rate to control the making phase of applied voltage, whether the closing is demanded at voltage peak or zero.

6. Acknowledgment

This work is supported partly by National Natural Sci-ence Foundation of China (No. 50507001).

REFERENCES

[1] R. P. P. Smeets and W. A. van der Linden, “Verification of the short-circuit current making capability of high- voltage switching devices,” IEEE Transactions on Power delivery, Vol. 16, No. 4, pp. 611–618, 2001.

[2] Discussion Group 13, “Synthetic circuit for making tests,” CIGRE Conference, pp. 444–445, 1966.

[3] CIGRE WG 13.04, “Requirements for testing making performance of high-voltage circuit breakers at reduced applied voltage,” Electra, Vol. 163, No. 6, pp. 27–33. 1979.

[4] IEC60427, “Synthetic testing of high-voltage alternating- current circuit breakers,” 2000

[5] GB 1984–2003, “High-voltage alternating-current circuit-

breakers.”

[6] B L Sheng, “Design consideration of weil-dobke syn-thetic testing circuit for the interrupting testing of HV AC breakers,” IEEE Power Engineering Society Winter Meeting, Vol. 1, No. 28, pp. 295–299, 2001.

[7] X. M. Fan, J. Y. Zou, and J. Y. Cong, “Research on the measurement and controlling system for synthetic making test of high-voltage circuit breakers at full voltage,” 6th International Symposium on Test and Measurement, Vol. 3, pp. 2709–2712, 2005.

[8] X. P. Li, C. Zhao, L. Y. Li, J. Y. Zhou and T. Meng, “The test system of pulse high current in the reconnection elec-tromagnetic gun,” 6th International Symposium on Test and Measurement, Vol. 3, pp. 2847–2850, 2005.

[9] J. Y. Zou, X. Y. Duan, and T. Zhang, “The simulating calculation and experimental research of Rogowski coil for current measurement,” Transactions of China Electro-technical Society, Vol. 16, No. 1, pp. 82–84, Feburay 2001.

[10] X. M. Fan, J. Y. Zou, E. Y. Dong, and J. Y. Cong, “Con-trol strategy of synthetic making test for high voltage cir-cuit breakers at full voltage and its realization,” Power system technology, Vol. 29, No. 17, pp. 8–13, 2005.

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