harmonic analysis of a three-phase...

IJSS : 6(1), 2012, pp. 7-13

HARMONIC ANALYSIS OF A THREE-PHASE INVERTER WITH SPACEVECTOR MODULATION TECHNIQUE

Md. Forhad Zaman1* and Md. Fayzur Rahman2

1,2Department of Electrical and Electronic Engineering, Rajshahi University of Engineering and Technology2E-mail: [email protected]

Abstract: Harmonics are created in the output of the Thyristor based Converters or Inverters. These harmonics notonly causes excessive heat in the devices or appliances used in the daily life of human being, but also reduces thelife period of the appliances. This reduction of harmonics in such system has become great concern for the engineers.

This paper presents the three phase inverter with space vector modulation technique in R-L loaded condition.Space Vector Pulse Width Modulation (SVPWM) has become the successful techniques to construct three-phasesine wave Voltage Source Inverter (VSI) which can be applied to drive three-phase motor using vector controltechnique. In this paper we presented the harmonic analysis of the VSI with space vector modulation technique.It has been found that the Total Harmonic Distortion (THD) is 0-8% in the proposed space vector modulatedinverter. Whereas 8-11% is the normal value of THD in H-bridge 3 level and 7 level inverters.

Keywords: Three phase Inverter, Space Vector Modulation, Harmonic analysis, MATLAB Simulink, FFT analysis,Total Harmonic Distortion.

* Corresponding Author: [email protected]

1. INTRODUCTION

Three-phase voltage source inverters cover mediumto high power applications. They have been widelyused in adjustable speed drives, active filters, ac powersupply including uninterruptible power supply (UPS)system, dynamic voltage restorer, unified power flowcontrollers in power systems and automatic voltageregulator. In recent years, demands for such sensitiveand critical loads have been increased. These loadsrequire high availability and continuous power supplysystems. Low total harmonic distortion (THD), fastdynamic response, high reliability and high efficiencyare commonly required in sensitive load such ascommunication systems, robots for automation, dataacquisition systems, instrumentation plants andmedical equipment [1, 2].

Sinusoidal PWM has been a very populartechnique used in AC motor control. This relativelyunsophisticated method employs a triangular carrierwave modulated by a sine wave and the points ofintersection determine the switching points of thepower devices in the inverter. However, this methodis unable to make full use of the inverter’s supplyvoltage and the asymmetrical nature of the PWMswitching characteristics produces relatively highharmonic distortion in the supply.

Space Vector PWM (SVPWM) is a more sophisti-cated technique for generating a fundamental sine wavethat provides a higher voltage to the motor and lowertotal harmonic distortion, it is also compatible for usein vector control (Field orientation) of AC motors.

2. SPACE VECTOR MODULATION

One of the most preferred pulse width modulation(PWM) strategies today is space-vector modulation(SVPWM). This kind of scheme in voltage sourceinverter (VSI) drives offers improved bus voltageutilization and less commutation losses. Three-phaseinverter voltage control by space-vector modulationincludes switching between the two active and zerovoltage vectors. During the switching cycle thereference voltage is assumed to be constant as the timeperiod would be very low. By simple digitalcalculation of the switching time one can easilyimplement the SVPWM scheme. However, theswitching sequence may not be unique [4].

Space Vector modulation (SVM) is quite differentfrom the PWM methods. With PWMs, the inverter canbe thought of as three separate push-pull driver stages,which create each phase waveform independently.SVM, however, treats the inverter as a single unit,specially the inverter can be driven to eight uniquestates [3]. The control strategies are implemented indigital systems. SVM is a digital modulating technique

8 Md. Forhad Zaman and Md. Fayzur Rahman

where the objective is to generate PWM load linevoltages that are in average equal to a given (orreference) load line voltages. This is done in eachsampling period by properly selecting the switch statesof the inverter and the calculation of the appropriatetime period of each state. The selection of the statesand their time periods are accomplished by the (SV)transformation [5].

2.1 How the SVPWM Works

For a three-phase voltage source inverter as depictedin Figure 1, each pole voltage may assume one of thetwo values depending upon whether the upper switchor the lower switch is on. Therefore, only eightcombinations of switches are possible; these are shownin Table 1. Of these, two of them have zero states. Zerostates occur when either the upper three or the lowerthree switches are conducting simultaneously.

3. SPACE VECTOR PWM GENERATION

Therefore, space vector PWM can be implemented bythe following steps [7]:

Step 1: Determine Vd, Vq, Vref, and angle ( )

Step 2: Determine time duration Ta, Tb, T0

Step 3: Determine the switching time of eachtransistor (S1 to S6)

3.1 Determine Vd, Vq, Vref, and angle ( )

Figure 3 shows the voltage space vector and its d andq components.

From Figure 3, the values of Vd, Vq, Vref, andangle ( ) can be determined as follows [7]:

Figure 1: Six-switch Voltage Source Inverter

The inverter has six active states when a voltageis applied to the motor and two states (0 and 7) whenthe motor is shorted through the upper or lowertransistors resulting in zero volts being applied to themotor. The six vectors including the zero voltagevectors can be expressed geometrically as shown inFigure 2. SVPWM seeks to average out the adjacentvectors for each sector. Using the appropriate PWMsignals a vector is produced that moves smoothlybetween sectors and thus provide sinusoidal line toline voltages to the motor. In order to generate thePWM signals that produce the rotating vector,formulae must be derived to determine the PWM timeintervals for each sector [6].

2.2 Space-Vector Sequence

The sequence to be used should ensure load linevoltages that feature quarter-wave symmetry in orderto reduce unwanted harmonics in their spectra (evenharmonics) [8]. To reduce the switching frequency, itis also necessary to arrange the switching sequence insuch a way that the transition from one to next sectoris performed by switching only one inverter leg at atime. Table 1 shows the SV sequence.

Table 1Switching Vectors, Phase Voltages and Output Line to

Line Voltages

State On Devices Van Vbn Vcn Space VoltageVector

0 T2,T4,T6 0 0 0 V0(000)

1 T1,T2,T6 2/3 Vdc –1/3 Vdc –1/3 Vdc V0(100)

2 T1,T3,T2 1/3 Vdc 1/3 Vdc –2/3 Vdc V0(110)

3 T2,T3,T4 –1/3 Vdc 2/3 Vdc –1/3 Vdc V0(010)

4 T3,T4,T5 –2/3 Vdc 1/3 Vdc 1/3 Vdc V0(011)

5 T4,T5,T6 –1/3 Vdc –1/3 Vdc 2/3 Vdc V0(001)

6 T5,T6,T1 1/3 Vdc –2/3 Vdc 1/3 Vdc V0(101)

7 T1,T3,T5 0 0 0 V0(111)

Figure 2: Space Vector Diagram – Line to Neutral Voltages

Harmonic Analysis of a Three-Phase Inverter with Space Vector Modulation Technique 9

1 11

2 2 2

3 3 30

2 2

and

bnq

cn

VV

VV

V

2 2 1and tan 2q

ref d qd

VV V V t ft

V

Where f is the fundamental frequency.

3.2 Determine Time Duration Ta, Tb, T0

This divides the plane into six equal regions within aregular hexagon. These voltage vectors are of equalmagnitude and mutually phase-displaced by 60°.

Consider sector 1 in figure 2, bounded by Vectors100, 110 and the null vectors 000 and 111. The vectorVx within this sector can be resolved and vector Va andVb become as [6]

2sin

33a xV V

and2

sin3

b xV V

Where Va and Vb are the components of Vxaligned in the directions of V100 and V110 respectively.And the corresponding time is

0 0

100 100

,a ba b

V Vt T t T

V V

and t0 = T

0(1 – ta – tb)

Substituting Va and Vb in above equation theresultant time for respective vectors become [6]

1cos sin

3at U

2sin

3bt U

Where U is the ration of 110,100

xVV for the period T

0

in segment- 1. And sometimes referred to as themodulation index.

And switching time duration for any sector canbe found as [7]

03 sin3

a

nt U T

0

13 sin

3b

nt U T

t0 = T

0 – ta – tb

Where, n = 1 through 6 (i.e. Sector 1 to 6) and 0 60°

3.3 Determine the Switching Time of EachTransistor (Switching Signals)

Figure 3: Voltage Space Vector and Its Components in (d, q)

Figure 4: Switching Signals for the SVPWM

As has been mentioned previously, the switchingsequence is not unique. Although there is not asystematic approach to generate a SV sequence, theseconditions meet by the sequence as Vz, Vn, Vn + 1

, Vz(where Vz is alternately chosen among V

7 and V

8). If

for example, the reference vector falls in section 1, theswitching sequence is V

0, V

1, V

2, V

7, V

7, V

2, V

1, V

0. The

time interval Tz (= T0 = T

7) can be split and distributed

at the beginning and at the end of the sampling periodTs. Additionally, the zero SV selection should be donein order to reduce the switching frequency. It provideshigh performance in terms of minimizing unwantedharmonics and reducing the switching frequency.Figure 4 clearly demonstrated the switching in Sector1. Here, the cycle begins in State 0, i.e., [000], with eachinverter pole being successively toggled until State 8,[111], is obtained. The pattern is then reversed to


complete the modulation cycle. Figure 1 show thetimes from the start of each modulation cycle at whichthe inverter poles are toggled, TAon, TBon, and TCon,respectively. Taking the variations from one sector toanother into consideration, it is possible to tabulatethese times as functions of both the active and zerostate times. It can be easily seen that from one state tothe other only one inverter pole is toggled [7].

3 3

3 3 3 1

3 3 9 3

0

1[ ]

0

if

B I u VL

Note that load line to line voltage VL, inverteroutput current Ii, and the load current IL are the statevariables of the system, and the inverter output line-to-line voltage Vi is the control input (u).

4. SIMULATION 0F SVPWM

For the figure 1, an example of use of SV PWM hasbeen created a program that SV PWM is used togetherwith reference voltage is assumed 400 V andfundamental frequency 50 Hz and switching frequency3 kHz to control an AC induction motor. The programis divided into three main parts. First part is aninitialization process that is responsible forinitialization of filter parameters and motor ratings.The second part of the program is represented ofproposed VSI in simulink. And finally the output ofthe simulink is simulated in MATLAB.

4.1 Simulation Results

The SVPWM algorithm was simulated in MATLABSIMULINK and it is done by 3 steps:

(a) Model for “Space Vector PWM Generator”.

(b) Model for “Making Switching time”.

(c) Model for “Overall System”.

Then the result has been shown in differentfigures.

Table 2Switching Time Calculation at Each Sector

Sector Upper Switches(S1, S3, S5) Lower Switches (S4, S6, S2)

1 T1 = ta + tb + t

0/2 T

4 = t

0/2

T3 = tb + t

0/2 T

6 = ta + t

0/2

T5 = t

0/2 T

2 = ta + tb + t

0/2

2 T1 = ta + t

0/2 T

4 = tb + t

0/2

T3 = ta + tb + to/2 T

6 = to/2

T5 = T

0/2 T

2 = ta + tb + t

0/2

3 T1 = t

0/2 T

4 = ta + tb + t

0/2

T3 = ta + tb + to/2 T

6 = to/2

T5 = tb + t

0/2 T

2 = ta + t

0/2

4 T1 = t

0/2 T

4 = ta + tb + t

0/2

T3 = ta + to/2 T

6 = tb + to/2

T5 = ta + tb + t

0/2 T

2 = t

0/2

5 T1 = tb + t

0/2 T

4 = ta + t

0/2

T3 = to/2 T

6 = ta + tb + to/2

T5 = ta + tb + t

0/2 T

2 = t

0/2

6 T1 = ta + tb + t

0/2 T

4 = t

0/2

T3 = to/2 T

6 = ta + tb + to/2

T5 = ta + t

0/2 T

2 = tb + t

0/2

3.4 State Space Model

From the figure 1, the state-space mode can beexpressed as the following continuous-time state spaceequation [7]

( ) ( ) ( )X t AX t Bu t

Where,

3 3 3 3 3 3

3 3 3 3 3 3 3 3

3 3 3 3 3 3

9 9

1 10

3 3

1, 0 0 0

10

f fL

if

L

load

load load

I IC C

VX I A I

LI

RI I

L L

Figure 5: Switching Signals to Control the VSI

Figure 5 shows the switching signal of the inverter.In 180 conduction schedule three devices are always ON.There are three signals to control the three phase of theinverter. When the switch voltage is high then it meanthat upper switches of Figure 1 is on and vice-versa.


In figure 5 it is seen that, each state, there are twoswitches of upper/ lower and another switch will onlower/upper state. For example if conserving state 1,then it is seen that T

1, T

3 T

4 switches will be on and so on.

4.2 Harmonic Analysis

Since the simulation process is done with adaptiveRunge-kutta method of ode45 so that here samplingtime is not constant rather it is changedcontinuously. So that normal classic method likeFourier series here is not applied to analysefrequency spectra. Here another method has beenused called Lomb’s method.

Figure 6: Simulation Result of Inverter Output Voltages

Figure 7: Inverter Output Currents

The output of the inverter is not sine wave ratherits combination of sine waves. In figure 5, it is clear that,the inverter output line to line voltages are looks like arectangular wave and three waves are same. It is notedthat, the filter and load to be assumed balanced.

In figure 7, it is seen that, the inverter line to linecurrent contains some harmonics and it would createnoise and affect to apparatus. A filter must beconnected to the inverter before supplying to load.

To run the three phase motor it is mandatory that,the applied voltage of motor must be sine wave and threevoltages are same frequency and amplitude but phase shiftmust be 1200 displaced from each other. In figure 8, it isclear that, the requirement of motor is completely fulfilledand here no other signal except fundamental signal. Sothe filter’s job is done satisfactory.

From figure, we can say that, motor current iscompletely unique and it has no components ofharmonic. Also it is seen that, each line to line currentare phase displaced 1200 to each other.

Figure 8: Simulation Result of Load Voltages

Figure 9: Simulation Result of Load Currents

Figure 10: Harmonics Analysis of SVPWM


In figure 10, it is seen the inverter voltage andcurrent contains some higher order components offundamental frequency. But in load voltage andcurrent only 50 Hz frequency component is present.

Figure 11 also show the total harmonic distortion(THD) of these signals and it is found that, THD islow in range and it is within acceptable value.

Figure 11: (a) Harmonics Analysis of SVPWM

Figure 11: (b) FFT Analysis of Load Current for ModulationIndex = 0.7

Figure 11: (c) FFT Analysis of Inverter Current forModulation Index = 0.7

Figure 11 shows that, the FFT analysis of theseveral signals like load voltage, load current andinverter current. Here it is seen that, only the fewerorder components persist in tiny magnitude offundamental components.

Figure 12: Harmonic Analysis Corresponding to ModulationIndex

Figure 12 shows that the Total HarmonicDistortion (THD) of different signals against themodulation index of the VSI. The THD is increasedwith increasing the modulation index. It is seen that,THD is go above 10% when the modulation indexexceed 0.7 for the same designed parameter (Vdc =400 V, Lf = 800 H, Cf = 400 F, Lload = 2 mH, Rload= 5 , f = 50 Hz and Switching frequency = 3 kHz).And if the modulation index is made less that 0.7, thenthe output voltage is affected and it may decrease withdecreasing modulation index.

Figure 12 also shows the THD when the inverterin over modulation region. And in this situation theharmonic is very high and the output voltage is notaccepted due non sine-wave.

5. COMPARISON OF DIFFERENT INVERTER

As mentioned above, SPWM only reaches to 78 percentof square-wave operation, but the amplitude ofmaximum possible voltage is 90 percent of square-wave in the case of space vector PWM.

The maximum phase-to-center voltage bysinusoidal and space vector PWM are respectively:

max2dcV

V for Sinusoidal

max3dcV

V for Space - Vector

Where Vdc is the dc link voltage. In this paper theoutput voltage of SVPWM is about 300 V (at m = 0.7)where the dc voltage was 400 V.


In our proposed inverter with space vectormodulation technique the total harmonic THD is veryquit low than any other conventional inverter. Heresome comparison of few Inverters is shown in Table 3.H-bridge inverter data collect from paper [9].

Table 3Comparison of Harmonics in Multilevel Inverter

S.I Types of Inverter %THD

1 H-bridge 3-level inverter 10-11

2 H-bridge 7-level inverter 8-9

3 Space Vector Inverter 0-8

6. CONCLUSION

In this paper, the SVPWM algorithm is developed fora recent technology to construct the three phase sinewave from a dc source. The exact three phase balancedsine wave easily determined in this paper satisfactory.Once that, the harmonic analysis of the output voltagesand currents are described and nothing found exceptfundamental wave. Again the modulator wasevaluated extensively with open loop volts/Hzcontrolled 3-� induction motor drive with referencevoltage is Vdc = 400 V.

In the paper, the voltage source inverter with spacevector technique is analyzed and three phase voltages

and currents are generated. And it has shown that afterusing filter there are very few harmonics present inthe wave. In the paper, it has also shown that, fullmodulation index of the inverter not good hence 0.7is better than others.

REFERENCES

[1] E.G. Carati, C.M. Richter, H.A. Grundling, IEEE-ICCA, 896 (2000).

[2] Y.B. Byun, K.Joe, S.Park, C.Kim, IEEE/INTELEC,195 (1997).

[3] M.H. Rashid, “Power Electronics Circuits, Devices,and Applications”.

[4] J. David Irwin, Auburn University, “The PowerElectronics Handbook, Industrial ElectronicsSeries”.

[5] Lenk R., “Practical Design of Power Supply”.Piscataway, NJ: IEEE Press Inc,. 1998.

[6] Ing. Pavel Gajd ů šek, “Programable LaboratoryInvertor And Space Vector Pwm”.

[7] Jin-Woo Jung, Ph.D Student, “Project #2 SpaceVector Pwm Inverter”.

[8] Muhammad H. Rashid, Power ElectronicsHandbook.

[9] G. Mahesh, Manivanna Kumar and S. Rama Reddy,“Simulation and Experimental Results of 7-LevelInverter System”, Research Journal of Applied Sciences,Engineering and Technology, 3(2), 88-95, 2011.

harmonic analysis of a three-phase...

Documents