proceedings of the asme 2014 international mechanical

J. Agbormbai Mechanical Engineering Department

University of Maryland, Baltimore County Baltimore, MD 21250 [email protected]

N. Goudarzi Mechanical Engineering Department

University of Maryland, Baltimore County Baltimore, MD21250 [email protected]

ABSTRACT

A modified generator, referred to as the variable

electromotive-force generator, is developed to enhance fuel

efficiency of hybrid vehicles and expand operational range of

wind turbines. Obtaining a numerical model that provides

accurate estimates on the generator output power at different

overlap ratios and rotor speeds, comparable with those from

experimental results, would expand the use of the proposed

modified generator in different applications. The general

behavior of the generated electromotive forces at different

overlaps and rotor speeds is in good agreement with those

from experimental and analytical results at steady-state

conditions. Employing generator losses due to hysteresis and

eddy currents in a three-dimensional model would generate

more realistic and comparable results with those from

experiment. In this work, electromagnetic analysis of a

modified two-pole DC generator with an adjustable overlap

between the rotor and the stator at transient conditions is

performed using finite element simulation in the ANSYS 3D

Low Frequency Electromagnetics package. The model is

meshed with tetrahedral or hexahedral elements, and the

magnetic field at each element is approximated using a

quadratic polynomial. For a fixed number of coils, two cases

are studied; one with constant magnetic properties and the

other with nonlinear demagnetization curves are studied.

NOMENCLATURE

B Magnetic Flux Density - Wb/m2

D Differential Operator

Emax Maximum Electromotive Force - V/m

e Normalized Electromotive Force

f Rotor Speed - Hz

Ke Element Stiffness Matrix

K Global Stiffness Matrix

N Element Shape Function

O Stator Overlap - m

OR Stator Overlap Ratio - %

S Normalized Rotor Speed

µ Relative Permeability Matrix

ϕ΄ Estimated Field Variable

∅ Column Matrix of Nodal Degrees of Freedom

Ф Interpolation Function

P Energy - J

δ Phase Angle - rad

R Resistance - Ohm

L Self-Inductance - Henry

Numerical Study of Nonlinear and Transient Behaviors of a Variable Electromotive-force Generator with an Adjustable overlap between the

Rotor and the Stator Using the Finite Element Method

W. D. Zhu Mechanical Engineering Department

University of Maryland, Baltimore County Baltimore, MD 21250 [email protected]

Proceedings of the ASME 2014 International Mechanical Engineering Congress and Exposition IMECE2014

November 14-20, 2014, Montreal, Quebec, Canada

IMECE2014-38755

1 Copyright © 2014 by ASME

1. INTRODUCTION

The recent climate change is thought to be the result of

greenhouse gasses emitted by manmade sources, such as

transportation systems and power plants, to the atmosphere.

Wind, Sun, Biomass, Biogas, Geothermal Sources, and Ocean

Tides are possible alternatives to conventional energy sources.

In order to reduce the carbon footprint of automobiles,

Hybrid Electric Vehicles(HEV) and Electric Vehicles(EV)

have been developed. HEVs incorporate some variation of the

variable electromotive-force generator(VEG) such as the ones

manufactured by Allison Transmission Inc. [1] and Crosspoint

Kinetics [2]. The systems developed by these manufacturers

have dual functions; they serve as electric motors which work

in conjunction with the vehicle’s Internal Combustion (IC)

engine during acceleration and as generators during

regenerative braking. The electricity generated during

regenerative braking is used to charge the vehicle’s battery

[1,2,3]. Vehicles fitted with these systems make use of

complicated electronics, often have smaller and more efficient

engines than conventional vehicles, consume less fuel and

emit less CO2 than their conventional counterparts[1,2,3].

Alternatively, EVs use electric motors as the main prime

mover. The motors are powered by batteries which may be

Lithium Ion batteries or fuel cells. These batteries often

require charging between drives. Plug-in electric vehicles

(PLEG) are charged by plugging in the vehicles charging lead

into a wall receptacle. Unlike the PLEG,some Commuter

Electric Buses in South Korea are charged (without being

plugged-in) while the vehicle runs [4]. This novel method of

charging the bus’s battery without any magnetic contact is

known as the shaped magnetic field in resonance (SMFIR)

transfer. The SMFIR transfers power via magnetic fields that

are generated by underground power cables. This technology

relies on electromagnetic resonance rather than inductive

coupling. The sending unit and the vehicle’s receiver resonate

at 20,000Hz and power is transferred through an air gap of

about 0.18 m between contacts at a rate of 100 kW [4].

This study investigates the performance characteristics of

a VEG developed from a synchronous motor using Final

Element Analysis (FEA). The VEG in this study has the

potential for use in HEV, EVs, wind turbines, hydraulic

turbines, telecommunication applications, aircrafts,

spacecrafts, and ships [5,6,7].VEG based wind turbines are

capable of capturing more wind power in the entire

operational range of the wind turbine, including the low wind

speed region with a wind speed lower than the cut-in speed of

conventinal generators[8,9]. For the VEG in this study to be

used as an auxilliary drive on an HEV or a prime mover on an

EV it ought to be self starting, as a consequence of this

requirement, a squirel cage winding ought to be incorporated

into the rotor assembly, hence allowing the synchronous motor

to self start as an induction motor until the synchronous speed

is reached [10].

Presently, wind turbines are designed to operate at

variable speed with constant frequency, capable of adjusting

speed as the wind speed changes, resulting in maximal wind

power capture and smoother output power [11]. As a result of

the increase of wind power levels in the power system,

requirements of wind turbine fault ride-through capabilities

have been introduced in many countries. You et al. [11]

revealed that fault duration varies from 100 ms to 625 ms,

while the voltage drop down varies from 25% to nearly 0%.

Germany specifies a one hundred percent reactive current

injection for better grid recovery.

The unpredictable nature of wind causes fluctuations in

the power generated by wind turbines (resulting in transients);

such that, there is need for utility grid operators to control it.

The reliability of the utility grid depends on real-time

balancing of electricity generation and load. Utility grid

reliability could be achieved by active power control. Aho et

al. [12] recommended that, grid frequency should be kept

within a close tolerance of the desired frequency, in order to

ensure grid reliability. Abrupt imbalances (like transients)

between the generator and the load may cause large

perturbations in grid frequency, known as grid event. A

transmission line fault that causes a branch of the load to

disconnect may cause an above normal frequency event while

a below normal frequency event may occur if a generating

utility trips off line or suddenly shuts down. In order to reduce

these deleterious phenomena, utility operators practice under

frequency load shedding [12]. Sudhoff et al. [7] investigated

the effects of transents caused by closing and opening

switches and thyristors on the stator of a synchronous

generator. they observed that in the design of Direct Current

(DC) power systems, such as those found on ships, spacecraft,

HEVs, EVs and aircrafts, impedance characteristics of both

sources and loads must be compatible to ensure system

stability. While Many researchers have studied the transient

currents at the output side (Stator side) of a generator

[7,11,12], a “cause to effect” approach is employed in this

paper to investigate the effects of transient currents on the

rotor of the VEG at the inception of excitation, during startup.

Wu et al. [6] work on a 200kW, 12000 RPM high power

density aircraft synchronous generator proposes a method for

modelling the air gap between the salient pole rotor through

the inverse of an effective air gap function. In that study, the

ANSYS Maxwell FEA software was used to determine the

inductance of the respective windings of the the synchronous

generator. A close agreement between calculated values and

FEA values was observed[6].

In order to better understand the effects of various

parameters such as the overlap ratio, the rotor speed and

transient currents on the performance of the VEG, the FEA is

implemented as an addition to experimental investigations and

theoretical modelling. The FEA has a potential to serve as a

useful tool in facilitating decision making during the design

phase of a VEG and can also serve as a less expensive

alternative to prototyping and testing.


2. METHODOLOGY

2.1. Overview of Finite Element Analysis

The FEA requires the entity to be analyzed (the domain)

to be discretized into elements either wholly or partially (if it

is symmetrical). Depending on the problem, elements may be

bar elements, triangular elements, quadrilateral elements,

hexehedral elements or tetrahedral elements. Each element

consists of nodes with specific spatial coordinates. Stiffness

matrices for respective elements [Ke] are determined and the

stiffness matrix for the entire domain [K] is assembled. [K] is

usually a sparse matrix with all non-zero entries located at the

vicinity of the leading diagonal while all other entries are zero.

Boundary conditions are appropropriately applied and the

resulting set of simultaneous linear equations are solved for

the unknown [13]. FEA formulations transform partial

differential equations (PDE) into a set of simultaneous linear

equations which after applying boundary conditions, are

solved to yield approximate desired solutions. Various

methods are used to transform these PDEs to linear equations,

namely; variational methods, principle of virtual work, method

of least squares and other residual methods [13,14,15].

The Galerkin Method of Weighted Residuals (GMWR) is

used in this work. An approximate solution of a PDE yields a

non-zero result, called the residual R, difined by

'R D (1)

where, D is a differential operator, ' is an estimated field

variable [13,14,15].The choice of an interpolation function is

usually crucial to FEA. In the case of three dimensional (3-D)

analysis, the interpolation function may have the following

form:

1 2 3 4a a x a y a z (2)

The interpolation function is usually expressed as the product

of a row and column matrix as follows:

[ ][ ]x a (3)

where [ ] [1 ]x x y z and 1 2 3[1 ]Ta a a a .

Equation (3) could also be expressed as

[ ]{ }N (4)

where, [ ]N is the element shape function matrix and { } is a

column matrix of respective nodal degrees of freedom.

Applying the GMWR to a typical element results in [13]

0

v

NRdV (5)

where N is the element shape function or weighting factor and

R is the weighted residual. The element equation is expressed

as

[ ]{ } { }eK F (6)

where Ke is the element stiffness matrix, ∅ is a column matrix

of the unknown nodal degrees of freedom to be solved for, and

F is the constraint or boundary condition matrix. In order to

find the element shape function and subsequently the element

stiffness matrix, one needs to define N in terms of nodal

coordinates. For an element with n nodes and nodal

coordinates 1 1 1( ) ( )n n nx y z x y z ,

1 1 1 1

2 2 2 2

1

1

1

1

1

e

n n n n

x y z a

x y z a

x y z a

(7)

Let

1 1 1

2 2 2

1

1[ ]

1 n n n

x y z

x y zA

x y z

and

1

2

1n

a

aa

a

The coefficients of the interpolation function are given by

1

ea A

(8)

and the shape function matrix is, 1[ ] [ ] [ ]N A x ,

where

[ ] 1x x y z .

The element stiffness matrix is given by

[ ] [ ]

[ ] [ ][ ] [ ]T

Te

v v i i

d N d NK b C b dv C dv

dx dx (9)

where

1 2

1 2

1 2

n

n

n

dNdN dN

dx dx dx

dNdN dNb

dy dy dx

dNdN dN

dz dz dx

(10)

and C is a 3x3 diagonal matrix of material properties.

The elements used to mesh the VEG model in ANSYS are

a degenerate 8 node tetrahedral element and a 10 node regular

tetrahedral element whose interpolation function is a quadratic

function given by [16];


1 2 3 4 5 6 7

2 2 28 9 10

( , , )x y z a a x a y a z a xy a yz a xz

a x a y a z

(11)

An 8 node degenerate tetrahedral element has the same

characteristics as an 8 node hexehedral element. The eight

node degenerate tetrahedral element is preferred to the

hexahedral element because it facilitates the meshing of

complicated shapes and shortens the time taken to generate the

element stiffness matrix. The 10 node tetrahedral element has

4 nodes at the vertices of the terahedron with one itermediate

node at the mid span of each side of the tetrahedron. For an

isotropic material the permeability matrix is given by[16];

0 0

0 0

0 0

(12)

The stiffness matrices of the tetrahedral elements used in

meshing the model are derived as follows:

1

22 2 2

10

1

a

ax y z xy yz xz x y z

a

(13)

For an ANSYS, 8 node tetrahedral element with some

coincident nodes, substituting nodal coordinates in Eq. (13)

leads to Eq. (14), where , {a} is the column matrix of the

unknown coefficients of the interpolation function and [A] is

the 10x8 matrix containing numerical values of components of

nodal field variables. Ordinarily the [A] matrix in Eq. (14) is a

singular matrix, since it contains identical rows; this

singularity is removed by the ANSYS software (For details on

how this is done, consult the software developer). Nodes 3 and

4 have the same spatial co-ordinates and nodes 5, 6, 7, and 8

have the same spatial co-ordinates.

2 2 21 1 1 1 1 1 1 1 1 1 1 1

2 2 22 2 2 2 2 2 2 2 2 2 2 2

2 2 23 3 3 3 3 3 3 3 3 3 3 3

2 2 23 3 3 3 3 3 3 3 3 3 3 3

1 2 2 25 5 5 5 5 5 5 5 5 5 5 5

2 2 25 5 5 5 5 5 5 5 5 5 5 5

2 2 25 5 5 5 5 5 5 5 5 5 5 5

1

1

1

1

1

1

1

e

x y z x y y z x z x y z







1

2

3

4

5

6

7

8

92 2 25 5 5 5 5 5 5 5 5 5 5 5

10

1

a

a

a

a

a

a

a

a

ax y z x y y z x z x y z

a

(14)

Similarly for an ANSYS, 10 node tetrahedral element

substituting nodal coordinates in Eq. (13) yields;

2 2 21 1 1 1 1 1 1 1 1 1 1 1

2 2 22 2 2 2 2 2 2 2 2 2 2 2

2 2 23 3 3 3 3 3 3 3 3 3 3 3

2 2 24 4 4 4 4 4 4 4 4 4 4 4

2 2 25 5 5 5 5 5 5 5 5 5 5 5

2 2 2 26 6 6 6 6 6 6 6 6 6 6 6

2 2 27 7 7 7 7 7 7 7 7 6 7 7

1

1

1

1

1

1

1

e








1

2

3

4

5

6

7

2 2 2 88 8 88 8 8 8 8 8 8 8 8

92 2 29 9 9 9 9 9 9 9 9 9 9 9

102 2 210 10 10 10 10 10 10 10 10 10 10 10

1

1

1

a

a

a

a

a

a

a

ax y zx y z x y y z x z



(15)

The element stiffness matrix for each tetrahedral element is

given by;

[ ][ ]T

ev

K b b dv (16)

where b is given in Eq. (10) above, 1[ ] [ ] [ ]N A x and

2 2 21x x y z xy yz xz x y z

. Element

stiffness matrices for all the elements in the domain are

determined and appropriately assembled into a global stiffness

matrix [K]. ANSYS performs all these tasks automatically.

The following system of equations results

[ ]{ } { }K B H (17)

Where, B is column matrix of unknown nodal magnetic flux

densities and H is the constraint or boundary condition column

matrix. Equation (17) is solved with a sparse solver or any

other solver.

Each solver, has a fundamental defining equation which

provides a residual for the solved field. In the case of

magnetostatic simulations, this defining equation is the no

magnetic monopoles(or magnetic particles with single poles)

maxwell equation [16]:

0B (18)

When the field solution obtained from the analysis is

substituted in Eq. (18), the result is not zero, but a residual is

returned instead, Viz, lnsoB R .

2.2. ANSYS FEA Modelling Steps

The VEG studied is a two-pole synchronous generator with

a round rotor. For the purpose of saving computational time,

half the VEG is modelled and used to study the magnetic flux

density variation in it. The plane of symmetry is the vertical

plane with both halves of the stator poles lying on the vertical

plane in the N-S direction. The upper pole is modelled to


curve concave upwards with a cylindrical arc spanning from

15° to 90° in the counterclockwise direction and the lower

pole is modelled to curve concave downwards with a

cylindrical arc spanning from -15° to -90° in the clockwise

direction. The rotor is modelled as a semicircular cylinder

spanning from -90° to 90° in the counter clockwise direction.

Each pole piece is modelled to be attached to the stator

housing via yokes of rectangular cross section, and the stator

housing is modelled as a thin hollow semi circular cylinder

spaning from -90° to 90° in the counter clockwise direction.

The annular space between the two-pole pieces and the rotor

represent the air gap; it is modelled as thin cylindrical arcs

with a length corresponding to the length of the respective

pole pieces in the studied VEG ( as shown in Fig. 1).

Having modelled the VEG (for each overlap ratio), the

analysis preference was set to electromagnetic analysis-

magnetic nodal. Two element types were chosen (each at a

time) in the modelling menu namely; a 10 node scalar

tetrahedral solid element (solid 98) and a solid hexahedral

element (solid 96). Solid 96 is a magnetostatic scalar 8-node

hexahedral element which could be authomatically changed by

ANSYS to a degenerate tetrahedral element or a pyramidal

element, based on the shape of the domain to be meshed while

solid 98 is a regular 10 node tetrahedral element. After

selecting the element type, material properties of the laminated

iron core rotor, the carbon-steel stator, and the airgap between

the rotor and the stator were entered in the material modelling

tool. A magnetization curve (Fig. 2) was plotted with B-H

values estimated from reference[17] and a coercive force of

4000 N were used for the rotor, a magnetization curve (Fig. 3)

was plotted with B-H values estimated from reference[17] for

the pole pieces and carbon-steel yokes (stator) and a relative

permeability of 1 was used for the air gap. After assigning

material properties, each component is meshed accordingly

using mesh size 4 (fine mesh), as shown in Figs. 4 and 5.

Figure 2- Magnetization Curve for the Rotor

Figure 3- Magnetization Curve for the Stator

Figure 1- 3-D schematic model of the VEG in ANSYS

Figure 4 - Meshed Model with 100% overlap


After meshing the model, boundary conditions and other

constraints including the torque on the rotor are applied as

follows: The node at the origin of the rotor is picked (to

prevent singularity) and the magnetic potential is set to zero.

The rotor is flagged. A Maxwell surface constraint is applied

on all the nodes in the air gap. The flux density and the

magnetic potential on the plane of symmetry going through

the N-S direction are respectively set to zero. Finally, the flux

parallel boundary condition is applied on the whole model.

After applying the boundary conditions to the model, the

analysis type is set to non-linear static, a sparse solver is

selected, the number of iterations is set to 25 and the

convergence criterion is set to 0.1%. The ANSYS solver is run

for each overlap ratio (ranging from 0-100%); the total

computational time per run is 8.04 Sec and the maximum

resultant nodal magnetic flux density is estimated to be 1.20T

for all cases (for both solid 96 and solid 98). Figure 6 is a

contour plot of the resultant magnetic flux density of the

model at 70% overlap ratio. The magnetic flux density varies

from a minimum of zero to a maximum of 0.60001 T.

Since it is a half-symmetry model of the VEG, the maximum

resultant magnetic flux density for the actual model should be

doubled.

2.3. Transient Analysis

The rotor of the VEG is modelled as a resistance-

inductance (R-L) circuit carrying an excitation current of 1

Amp. Prior to the rotation, the steady unit excitation current

on the rotor results in a transient current of [18, 19]

( / )( ) 1 Rt LI t e (19)

where R is the resistance in ohms and L is the self-inductance

in Henry of the rotor coil, L/R is the time constant and t is

time. The self-inductance causes the current to rise slowly to

its maximum value. The current reaches 63% of the maximum

value after a time equal to the time constant [18].

The 1 Amp excitation current on the rotor when rotation and

excitation commence simultaneously results in a transient

current of the form

( / )( ) cos( )Rt LI t e t (20)

where δ is the phase angle and ω is the rotor speed in

radians/sec. The phase angle is defined as

1tan ( / )L R (21)

2 f (22)

The resistance of the rotor coil was measured in the laboratory

using an Ampere-Volt-Ohm meter to be 106 Ohms. The self-

inductance of the rotor was calculated using procedures

outlined in the next section.

3. RESULTS

3.1 Electromotive Force Calculations

The resultant flux density obtained from FEA is used to

calculate the induced electromotive force, E(O) per unit width

per turn for each stator overlap (O) and is given by

( )E O O B f (23)

where B is the magnetic flux density in Tesla and f is the rotor

speed in Hz. Equation (23) was written in Engineering

Equation Solver (EES) for various overlaps between the rotor

and the stator and the respective rotor speed; the results are

plotted on the graphs shown below. The FEA results are valid

for rotor speeds ranging from 75RPM (1.25Hz) to 300 RPM

(5Hz). The normalized rotor speed S was also calculated in

EES using the expression below

Figure 5 - Meshed Model with 80% overlap

Figure 6 - Contour plot of Nodal Flux density at 70% overlap


5

fS (24)

and the normalized emf was also calculated in EES as follows:

max

( )( )

fe O O B

E O (25)

Where E (O) max is the maximum induced emf per turn per unit

length at 100% overlap and a rotor speed of 5Hz (300RPM).

The results of the respective parameters expressed in Eqns.

(23), (24), and (25) are plotted on the graphs in Figs. 7, 8, 9

and 10 below.

3.2 Self-Inductance Calculations and Related Expressions

The self-inductance of the rotor coil was estimated using

a unit excitation current and the flux density obtained from

FEA, using the following expression [19]:

2 201 2 1 2B dV LI (26)

Where μ0 is the permeability of free space

(μ0=4πx107kgm/C

2), I is the excitation current and V is the

volume of the rotor coil. The estimated value of the rotor coil’s

self-inductance is 5.24 H. This value of L was substituted in

Eqs. (19), (20), and (21). The resulting graphs which were

plotted in EES are shown in Figs. 13, 14 and 15 below. The

energy stored in the magnetic field within the air gap is

expressed as a function of the stator overlap as follows:

4 2

01.466 10 2P O B (27)

Where, P is the energy stored in the magnetic field in J.

Equation (27) was plotted in EES for various stator overlap

ratios as shown in Fig. 16 below.

3.3 Discussion of Analysis Results

The graphs of induced emf per turn per unit width of coil

versus stator overlap ratio in Fig.7 and normalized emf versus

stator overlap ratio (Fig. 8) at respective rotor speeds are linear

and pass through the origin. One can infer from Figs. 7, 8 and

11 that, there is a linear relation between the change in the

overlap between the rotor and the stator and the induced emf;

and hence the output voltage. Both the induced emf and the

normalized emf decrease with rotor speed. At a given speed,

the induced emf increases with the overlap between the rotor

and the stator. This trend is in correlation with experimental,

theoretical and previous FEA results [5, 8, 20]. As depicted in

Figs. 9 and 10, both induced and normalized emf vary linearly

with the rotor speed and the normalized rotor speed,

respectively. The induced emf (hence the output voltage)

decreases with the overlap between the rotor and the stator.

Similarly, both the induced and the normalized emf increase

with rotor speed for a given overlap. This result is in

correlation with experimental results (Fig. 12) [5]. The

induced emf (output voltage) is a function of both the overlap

ratio and the rotor speed. Hence,

( , ) ( ) ( )R RE O f p O f (28)

E( )R RO m O (29)

( )R RO O (30)

E( ) ( 1.25)f a f c (31)

( ) ( 1.25)c

f fa

(32)

Where m, a, c, and p are constants that have been determined

from numerical data by averaging out the emf at each overlap

ratio and each rotor speed, respectively. E (OR, f) is a

multivariable second-order polynomial. For an average rotor

speed, the constants have been estimated to be m≅0.0023,

a=0.04, c=0.0511 and p≅0.00074. Substituting the values of

m, a, and c in Eqs. (29) and (31), Eq. (28) becomes;

4 5( , ) (7.4*10 2.22*10 )R RE O f O f (33)

Experimental results revealed that the output voltage of

the studied VEG is a seventh order multi-variable polynomial

function of the overlap and the rotor speed (i.e.

(O ) ( )RV f where (O )R is a third order polynomial

and ( )f is a fourth order polynomial [8]. Another

experimental result revealed that for a given overlap ratio, the

relationship between the normalized output voltage and the

normalized rotor speed is a logarithmic function (Fig. 12) [5].

While experimental results show a non-linear relation between

the normalized emf and the normalized rotor speed, the FEA

results show a linear relation. This happens because the

experimental results account for material imperfections,

energy losses due eddy current, hysteresis and joule losses, but

the current FEA model does not take these imperfections into

account. The previous FEA results [20] which were based on

linear magnetic properties (i.e. relative permeability),

overestimated the magnetic flux density(the induced emf) of

the VEG (Fig.11) whereas, this study which is based on the

nonlinear magnetic properties of the components of the VEG

resulted in a more accurate estimate of the magnetic flux

density. The magnetic flux density estimated in the current

study is in keeping with the research findings of other scholars

[19]. A magnetic flux density value of 1Tesla is typical of

electrical machines, such as; electric motors, generators,

magnetic resonance imaging (MRI) machines [19].


0 20 40 60 80 1000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Overlap Ratio(%)

Induced E

mf

per

Turn

(V

/m) 5.00Hz5.00Hz

4.52 Hz4.52 Hz4.054.053.58Hz3.58Hz3.12Hz3.12Hz

2.65Hz2.65Hz2.18Hz2.18Hz1.72Hz1.72Hz

1.25Hz1.25Hz

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

Overlap Ratio

Norm

alized E

mf

5.00Hz5.00Hz4.52Hz4.52Hz4.05Hz4.05Hz3.58Hz3.58Hz3.12 Hz3.12 Hz2.65Hz2.65Hz2.18Hz2.18Hz1.72Hz1.72Hz1.25Hz1.25Hz

1 1.5 2 2.5 3 3.5 4 4.5 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Rotor Speed (Hz)

Induced E

mf

per

turn

(V

/m)

100%100%90%90%80%80%70%70%

60%60%50%50%

40%40%30%30%20%20%10%10%0%0%

20 30 40 50 60 70 80 90 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalized Rotor Speed

Norm

alized E

mf

100%100%90%90%80%80%

70%70%60%60%

50%50%

40%40%30%30%20%20%10%10%0%0%

0 20 40 60 80 1000

10

20

30

40

50

60

Overlap ratio(%)

Ind

uce

d e

mf/

turn

/wid

th(V

/m) 5Hz5Hz

4.52Hz4.52Hz4.05Hz4.05Hz3.58Hz3.58Hz3.12Hz3.12Hz2.65Hz2.65Hz

2.18Hz2.18Hz

1.72Hz1.72Hz

1.25Hz1.25Hz

Once the rotor’s direct current (DC) excitation source is

turned on, the current rises steeply from zero (Fig.13) to its

steady state value. Figure 14 illustrates changes in the

excitation current with time, as the generator rotor spins; a

sinusoidal behavior is observed after a quick jump at the time

of adding the excitation current to the generator. It could be

seen from Fig.14 that, the value of the overshoot current

increases with decreasing rotor speed and the time period of

the steady state current also increases with decreasing rotor

speed. All curves have a common intersection point after the

transient; it shows that for a particular generator, steady state

condition is achieved after a specific time and this time is

independent of rotor speed. The time taken for transients to die

down after startup does not depend on the rotor speed.

As shown in Fig. 15, the phase angle between the rotor

excitation voltage and current drops with increasing rotor

speed. Figure 16 shows the energy stored in the magnetic field

within the air gap increases with an increase in the overlap

between the stator and rotor.

Figure 9 - Induced Emf per turn per unit width vs. Rotor Speed

Figure 11 - Induced Emf per turn per unit width vs. Overlap

Ratio(base on FEA results plotted in [20]

Figure 10 - Normalized Emf vs. Normalized Rotor Speed

Figure 8 - Normalized Emf vs. Overlap Ratio

Figure 7 - Induced Emf per turn per unit width vs. Overlap


0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (s)

Excita

tio

n C

urr

en

t (A

mp

)

0 2 4 6 8 10

-1

-0.5

0

0.5

1

1.5

2

Time (s)

Excita

tio

n C

urr

en

t (A

mp

)

5.00Hz5.00Hz4.52Hz4.52Hz4.05Hz4.05Hz

3.58Hz3.58Hz3.12Hz3.12Hz2.65Hz2.65Hz2.18Hz2.18Hz1.72Hz1.72Hz1.25Hz1.25Hz

1 1.5 2 2.5 3 3.5 4 4.5 5-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

Rotor Speed (Hz)

Ph

ase

An

gle

(ra

d)

0 20 40 60 80 1000

1

2

3

4

5

6

Overlap Ratio(%)

Energ

y S

tore

d (

J)

4. CONCLUSION

The numerical study of a variable electromotive-force

(VEG) generator shows that for a given rotor speed, the

induced emf (or the output voltage) decreases with a decrease

in the overlap between the rotor and the stator, and increases

with an increase in the overlap between the rotor and the

stator. It is in agreement with previous analytical and

experimental studies.

Using nonlinear magnetic properties in the FEA resulted

in a more accurate estimate of the magnetic flux density of the

VEG. No matter whether the magnetic properties used in the

FEA are linear or nonlinear, the magnetic flux density

estimate has a constant value and does not vary with the

overlap between the stator and the rotor.

At startup the excitation current on the rotor’s coil

overshoots to a value much greater than 1.5 times the initial

value, before dropping to the steady state value in 2 seconds.

Similarly the induced current on the stator coil would

overshoot and result in control issues in wind turbines, at the

Figure 16 - Energy Stored vs. Overlap Ratio

Figure 14 - Transient Current for simultaneous excitation and

Rotation of the Rotor at startup

Figure 13 - Transient Current before Rotor Rotation

Figure 12 - Normalized Output Voltage vs. Normalized Rotor Speed

[5,6]

Figure 15 - Phase Angle vs. Rotor Speed


grid or in the case of a HEV or an EV may result in difficulty

to control the vehicle. Based on this observation, it is

advisable to turn the excitation current on while the rotor is

still static until the transients die down, prior to rotating the

rotor.

5. REFERENCES

[1] Allison Transmission Inc., 2011,”Allison Hybrid-

H40EP/H50EP Specifications,” available at:

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2002 “Concepts and Applications of Finite Element

Analysis,” John Wiley and Sons, 3rd

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179-198; Ch.7, pp.263.

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[17] www.electronics-

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http://www.fueleconomy.gov/

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