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 Copyright © 2014 by ASME
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];
3 Copyright © 2014 by ASME
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
x y z x y y z x z x y z
x y z x y y z x z x y z
x y z x y y z x z x y z
x y z x y y z x z x y z
x y z x y y z x z x y z
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
x y z x y y z x z x y z
x y z x y y z x z x y z
x y z x y y z x z x y z
x y z x y y z x z x y z
x y z x y y z x z x y z
x y z x y y z x z x y z
x y z x y y z x z x y z
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
ax y z x y y z x z x y z
ax y z x y y z x z x y 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
4 Copyright © 2014 by ASME
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
5 Copyright © 2014 by ASME
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
6 Copyright © 2014 by ASME
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].
7 Copyright © 2014 by ASME
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
8 Copyright © 2014 by ASME
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
9 Copyright © 2014 by ASME
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
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Exposition, Baltimore, MD, USA, Jul. 28-31, 2014.
10 Copyright © 2014 by ASME