Purdue University
Purdue e-Pubs
International Compressor Engineering Conference
School of Mechanical Engineering
1992
Induction Motor Modeling Using Coupled
Magnetic Field and Electric Circuit Equations
L. W. Marriott
Tecumseh Products Research Laboratory
G. C. Griner
Tecumseh Products Research Laboratory
Follow this and additional works at: http://docs.lib.purdue.edu/icec
Marriott, L. W. and Griner, G. C., "Induction Motor Modeling Using Coupled Magnetic Field and Electric Circuit Equations" (1992).
International Compressor Engineering Conference. Paper 939.
http://docs.lib.purdue.edu/icec/939
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INDUCTION MOTQR MODELING USING
COUPLED MAGNETIC FIELD AND
ELECTRIC CIRCUIT EQUATIONS
Lee W. Marriott
Glenn C. Griner
Tecums eh Products Research Laboratory
3869 Research Park Drive
Ann Arbor, Michigan 48108
(313) 665·918 2
ABSTR ACT
Traditional design/prototype/test cycles have improved the efficien
cies of small induction motors to high levels. To achieve still higher efficien
cies, motor performance
must be studied in a more detailed and integrated manner. The
authors present a nonlinear, time-dependent comput er simulaton modeling the perform
ance of a permanent-splitcapacito r induction motor. The effects of nonlinear permeability
and time varying terminal
voltages are included in the model. Two-dimensional finite
element magneti c field equations are coupled with the electri~: circuit equations and solved
using the Newton-Raphson
and Crank-Nicholson methods. The simulation predicts main
and aWJ;iliary winding cur·
rents, induced rotor currents, hysteresis losses and motor torque
as functions of time. Due
to the detailed nature of the results a deeper understanding of
motor perfonn ance factors,
normally difficult to determine, can be obtained.
INTRODUCTION
Induction motor design involves the prediction and evaluation
of perform ance parameters such as torque, line current and losses at any given
slip and tennina l conditions.
These parameters can be accurately determined only with knowled
ge of the magnet ic field
distribution within the machine. Approximate analytical method
s [I] have been used in
the past with some success. Continu ed pressure to conserve
energy dictates improve d
motor performance. This requires an even more detailed knowled
ge of the magnetic fields
occurring within motors. Two-dimensional statioruu:y magneti
c field problem s have been
investigated by many authors [2,3]. The nonline ar penneab ility
or saturation characteristics of steel require iterative methods for the solution of these
problems. Tandon , Armor
and Chari [4] were first to combin e the Newton · Rapbson techniqu
e with the CrankNicholson central difference scheme to solve transient field problem
s in electric al machines. The disadvantage of these methods is that they rely on
apriori knowled ge of the
c~ts or current densitie s in the device.
The problem is much more complicated when current densitie
s are unknow n and terminal conditions must be considered. Potter and Cambrell [S]
combin ed a twe>dimensional finite element field solution and circuit equation
s to simulat e an indu~:tion
motor. In their work the stator field was rq>laced with a current
sbeet at slip ftequency.
Strangas and Theis [6] also combin ed the finite element field
solution with circuit equations to model a shaded-pole motor. Their approach used a time-de
pendent grid for rotor
movement. This paper expands the above techniq ues to model
a pennanent-splitA
capacito r induction motor. The two-dimensional magnetic field
equatio ns are combin ed
with electric circuit equations describing end and elCtemal effects.
The partial derivatives
1445
is solved by the
of the combine d system of equations fonn a Jacobian matrix which
step is obtained
time
each
at
um
Equilibri
step.
time
each
for
method
Newton. Raphson
equation s are satiswhen the derivatives of the tirne.dependent nonlinea r magnetic field
s appearin g as
fied. The electric circuit equation s are lincaron iinary differential equation
the
updating
by
time
through
s
advance
solution
The
constant s in the Jacobian matrix.
hand side. The
time depende nt terms of the Jacobian matrix and correspo nding right
R:quircs an inversion
Crank-N icholson finite difference method, being an implicit method,
out by a comto solve for the unknown s in the next time step. This inversion is carried
solver.
equation
linear
"
"sky-line
plex math
MAGNE TIC FIELD EQUAT IONS
's equation s
The solution of the magnetic field problem is based on three of Maxwell
current
ment
displace
ng
Neglecti
laws.
ive
constitut
and the magnetic and electric field
these equation s are:
(I)
VxH=I
VxE =-
iJB
(2)
dl
V•B =0
H=vB
J =aE
(3)
(4)
(5)
is electric field intenWhere H is magnetic field intensity, B is magnetic flux density, E
vity.
sity, 1 is current density, v is reluctivity and a is an effective conducti
hip is satisA vector potential function A is chosen such that the following relations
fied:
(6)
of the scalar voltCombini ng equation s (I ) through (6) and introducing the gradient
:
age as shown in (12], yields the time-dependent magnetic field equation
· ilA
Vx(vVx A)=-cr ar -crVV
(7)
FINITE ELEMEN T APPROX IMATIO N OF THE MAGNE TIC FIELD
field equaThe finite element method was employe d to approximate ~he magnetic
the earliest magnetic s
tions. Many authors have contributed to this methodology. Some of
of the method is that
work is attributed to Silvester and Chari l2]. The basic assumpti on
a linear combina tion
the vector potential within a finite element can be approxim ated by
:
of shape functions
(8)
element. Example s of
Where N; are shape functions and n is the number of nodes in the
Bathe [8]. Using
or
[7)
icz
Zienkiew
of
books
text
shape functions may be found in the
assemble d finite
the
l,
functiona
energy
the
of
tion
minimiza
the
and
ation
approxim
this
element magnetic field equation s become:
l446
Whe n: 1T 1 is the global cond uctiv ity
matrix, A is the global vecto r of vecto
r potentials,
dA is a vecto r of time deriv ative s of
A, s (A ) is a nonli near global coeff
dt
icien t matrix,
INw J is a matrix of weig hting facto
rs which distri bute and conv ert coil volta
ges into element nodal CUil'Cnt densities, v w is
the appli ed volta ge across a lengt h of
coil lying within the lamination stack , [ Nb] is a matri
x of weig hting factors whic h distri bute
and conv ert
rotor bar volta ges into elem ent noda l
cune nt densi ties and V b is the appli ed
volta
ge
acros s a rotor bar length. The matrices
and vecto rs given abov e are comp rised
of an assemb ly of elem ent matrices, the detai
ls of whic h can be found in refer ence
[ 12].
ELECTRIC CIRCUIT EQU ATIO NS
The electric circu it is chara cteriz ed by
two linea r differential equa tions . The
first
equa tion sums the voltages along a rotor
bar or coil and can be written as:
(10)
Whe re I is the current, V is volta ge
and R is a bar
duced volta ge along a length of a coil
or rotor bar is:
or effective coil resistance. The in·
{II)
Substitution ofthe induced volta ge (II)
into equa tion (10), divid ing throu gh by
the
resistance and introducing t~ as a vecto
r of noda l vecto r potential time deriv ative
s
yield
s
a first order differential equa tion whic
h is coup led to the magnetic field equa
tion (9).
aat
- IN I A + GA.,
L
v -
n I 'ii = 0
(12)
Wher e IN I is a matrix of Newt on-C otes
integration factors, Ac is the area ofa
cond uctor ,
L is the length of the cond uctor s, n
is the numb er of cond uctor s and ii is a
curre nt direc tion vecto r for
the winding.
It shou ld be noted that this equation
is depe nden t on the solution of the magn
etic
field equa tions for the value s of t~
and assum es that the appli ed volta ge
V is unifo rm
acros s a coil or a rotor bar. The secon
d equation is based on Kirch hotfs law
as appli ed to
a winding:
l:nv ii+IR e +Le
~~ + ~""'
Jldt = Vr
(t)
(13)
Wher e n is the numb er of cond uctor s,
V is the individual coil voltage, Re
is the end resistan ce, Le is the end inductance, C~
is the external capacitance, ii is a direc tion
vector,
v rU) is the time- depe nden t tenni nal
voltage and r is a summ ation over the
numb er of
coils contributing to a wind ing. Notic
e that this equa tion is coup led to the first
circu it
equation (12) but does not directly influe
nce the magn etic field equa tions . Note
also that
any other extem al impe danc es in serie
s with the wind ing could be add~ to
equa tion ( 13 ).
Equations simil iarto equa tions (12 and
13) can also be written for the rotor. Usin
g the
!44 7
curre nts and
Strangas and Theis [6), rotor end ring
method of rotor representation used by
nts are
curre
bar
r
Roto
nts.
cwre
bar voltages and
voltages are solved for rather than rotor
nts.
curre
ring
end
priate
appro
the
then computed by summation of
CRA NK-N ICHO LSO N METHOD
rically stable implicit central-difference
The Crank-Nicholson method [9] is a nume
is:
scheme in which the basic asswnption
(14)
I)
(14) for the present (n) and future (n +
Combining equations (9), (12), (13) and
ric
elect
and
field
etic
magn
rence
coup led finite diffe
time steps as show n in (12) yields the
circuit system of equations:
~~ ITl]A n] -INw I v,t' -[Nb]vbn
itaf:NJ]TA n- "':: V n +nl n n
[-[ S (Al+
(IS)
£I·
+l+ Vn+ .l!ll i=l
..Le +-A L)tn +vn
-nvn n-(R e+.1
C.,
T
T
2Ca~
6t
1
NEW TON - RAPHSON METHOD
tions
x., the finite difference system of equa
Due to the nonlinearity of the v[S I matri
tonNew
the
as
such
ique
solution techn
must be solved using a nonlinear equation
each time
n method is used to find equilibrium at
Raphson method. The Newton-Raphso
:
tions
wing equa
step. The method is shown in the follo
L~a~r J{AV ar)., {-'1')
(16a)
{Var };+l :{Va d;+{ AVa r}
(l6b)
a:ar J
is the assem bled
ous equation to be solved , [ a
Where I'I' J is the nonlinear homogene
I Var) ;
algebraic manipulation of equation ( 15),
Jacob•an matrix rendered symmetric by
r of soluprevious iteration, { V,u} i+ 1is a vecto
is a vector of solution variables from the
s.
{A v;~r) is a vecto r of cone ction
tion variables at the new iteration and
ian
repeated inversion of the nonlinear Jacob
The Newton-Raphson method requires
r was
solve
ine"
"skyl
math
lex
comp
a
ite,
matrix. Since the Jacobian is not positive defin .11r is solved for and used to update the
the A v
used for inversion. During each iteration
bles.
varia
own
unkn
prediction for the
1448
TO RQ UE
Mo tor torq ue com put atio n
usin g finite elem ent met hod
s com mo nly relies eith er on
the use of the Ma xwe ll stre
ss ten sor or the virtual wo
rk principle. The stre ss ten
od requires sev era l laye rs
sor met hof air gap elem ent s in ord er
to obt ain acc ura te val ues of
gen tial flux density. The virt
the tanual wor k imp lem enta tion
nor mal ly req uire s two fini
elem ent solutions who se mes
te
hes are disp lace d a small rota
tion from eac h other. A mo
eleg ant virtual wo rk met hod
re
was dev elo ped by Cou lom
b and Me ine r [10]. The ir
requires only a sing le lay er
met hod
of elem ent s in the air gap
and
one
finite elem ent solu tion at
eac h tim e step. The mag net
ic ene rgy exp ress ion is diff
eren tiat ed mat hem atic ally
spe ct to rotation bef ore inte
wit h regration ove r the solu tion spa
ce and results in the followi
equation for the torq ue con
ng
tribution per unit dep th of
eac h virtually dist orte d elem
the air gap:
ent in
Te:-va[B-t /e[~, ~Ai]+By /aL~, :]]Ae-(~·
)[f,~ dj~ 1 ]Ae
(17)
Wh ere Va is the reluctivity
of air, Bx . By are the X and
y flux densities, a is ang ula
. , B 2 · fl d
r
rotatiOn
. squ d,
ts ux ens tty
. th
are A e ts
eI
dN;
e eme nt area, dN;
denv
.
.
dY
,
dX
are
attv es o f
the sha pe functions Ni , A;
are nodal vec tor pot enti als
and
n
is
the
num ber of nod es in the
element. Additional details
of the met hod can be found
in [10) and [12).
RE SU LTS
Tra nsf onn er
For thre e reasons a tran sfo
nne r was sele cted as a mo
del to test the com put er sim
lation. First, a tran sfo nne
ur is simi liar to an ind ucti on
mo tor at standstill. The refo
gen erat ed cod e wou ld be dire
re the
ctly app lica ble to a motor.
Sec ond , an isolation tran sfo
of kno wn con truc tion deta
rme r
ils was available for exp erim
enta l verification. Third,
form er lend s itse lf to a sim
the tran sple mes h whi ch can be eas
ily mo difi ed or refi ned as
The tran stbr rne r was a 1.2
required.
KVA, 120 Vac , 60 hertz mo
del wit h a turns ratio of 1.1
(sec ond ary to prim ary ) and
con stru cted of Arn old Tec
hno log ies, Inc. M-2 2 steel.
poi nts for the steel wer e sele
B-H
cted from a pub lish ed cur ve
and a v-v ersu s-B 2 cur ve
usin g cubic spli nes [II] . The
was fit
finite elem ent grid for one
-qu arte r of the tran sfo nne
sist ed of2 16 triangles and
r con 125 nod es and the time step
use d was one -six tiet h of a
App rox ima tion s for the end
sec ond .
leak age reac tanc es wer e obt
aine d from no- load and sho
sec ond ary tests. The tran sfor
rted mer mo del was test ed und
er thre e conditions: at no- load
with sec ond ary win din g ope
(i.e.,
n-c ircu ited ) for com put atio
n
of
the mag net izin g current; wit
sho rted -sec ond ary to pre dict
h
ind uce d vol tage s and curren
ts; and wit h a purely cap acit
sec ond ary load. A ram ped
ive
sinu soid al vol tage was app
lied
to the mo del to min imi ze
startup transient oft he solu
the
tion, pen nitt ing stea dy stat
e to be ach iev ed in abo ut 2-11
trical cycles. Figure I sho
2 elec ws stea dy- stat e com put ed
and exp erim enta l results for
netizing cur ren t of the tran
the mag sformer. Differences in the
cur ren t wav efo rms are due
hysteresis effects whi ch are
to
not modeled. The exp erim
ent al and calc ulat ed RM S
mag net izm g cur ren t are 0.8
valu es of
4 and 0.73 amp s, respectively
. In the sho rted -sec ond ary
vol tage of3 .87 volts RM S
test, a
was app lied . Th e exp erim
enta l prim ary and sec ond ary
are 11.5 and I 0.6 amp s; calc
cur ren ts
ula ted valu es are 11.5 and
l 0.4 amp s. The se resu lts
stra te that the met hod of calc
dem onulat ing induced vol tage s and
cur ren ts is valid.
1449
were obtained using a capacitive secThe most interesting results on the transfonner
3 show the primary and secondary currents
ondary load of25 microfarads. Figures 2 and
nic caused by resonance occurs in the
hanno
d
secon
for the transfonner. A pronounced
isingly, secondary current remains essenprimary current and is properly modeled. Surpr
ary of the experimental and modeled
summ
A
tially sinusoidal and is also modeled well.
is given in Table 1.
results for the transfonner for the three load cases
Table 1. Tran sform er- Thre e Load Cases
Load
Condition
Input
Voltage
(Volts)
RMS Output Voltage (Volts)
RMS Current (Amps)
Predicted
Mcasi,II"ed
Prim
Se~;.
Prim.
Sec.
Predicted
Measured
No Load
120
0.84
.
0.73
.
131.9
132
Shorted Sec.
3.87
11.5
10.6
11.5
10.4
-
.
25 MfdCap.
120
0.83
1.24
0.83
1.25
130
132
AC Induction Moto r- Locked Rotor
was also modeled. The stator has 24
A two-pole pennanent-split-capacitor motor
osed of34 64 triangles and 1761
comp
grid
slots and the rotor 34 bars. A finite element
was again a ramped sine wave
l
mode
the
for
input
The
l.
nodes was used for the mode
outline of the motor lamination
An
d.
voltage with time steps of one-sixtieth of a secon
n in Figure 4. For reference, the
show
is
lines)
(flux
urs
and resulting vecto r potential conto
ary winding axis along the
auxili
and the
main winding axis is located along the horizontal
in time for the locked rot
instan
one
at
n
show
vertical. The vector potential contours are
ical cycle woul d show the contours rotattor condition. A series of figures over one electr
revolves within the motor. The·
field
ing and changing in magnitude as the magnetic
currents and currents induced in
ing
wind
both
of
ant
result
the
plotted vector potential is
the rotor.
AC Induction Moto r-
Movjn~
Rotor
on of a time-dependent mesh to
The final step·in code development was the creati
nected to the stator through the airrecon
ly
atical
autom
be
permit the rotor to rotate and to
ng in their own coordinate system, it is ungap mesh. Because the rotor nodes are movi
tion (8) with this technique. The solution
Equa
to
necessary to add the speed voltage tenn
much bookkeeping. At each time step the
procedure is conceptually simple but involves
angle, the airgap elements between the
small
a
nodes in the rotor are incremented through
idth minimized, electric circuit equations .
stator and rotor reconnected, the model bandw
iteration perfonned. Much smaller time
added to the matrices and a Newton-Raphson
motor at locked rotor conditions. Steps
or
onner
steps were required than for the transf
es were required both for computational stacorresponding to one or two mechanical degre
The mesh, nodal results (i.e., vector
bility and for sufficient definition in plotted results.
s windings and winding and roacros
es
voltag
),
atives
potential and vector potential deriv
state is reached are saved for post-processing.
torbar currents at each time step after steady
1450
It is again noted that the solution of equation
(15) is time-dependent in nature and
several electrical cycles must be solved to
allow any startup trans-ients to die out. Our
procedure was to set rotor rotation at the desir
ed speed and then apply a ramped sinusoidal
·
voltage to the motor windings for the first
two or three electrical line cycles. Seven
to ten .
additional line cycles were then required to
reach a quasi-steady state solution; that is,
where each succeeding cycle was approxima
tely the same as the previous cycle. Cycl
es
will not repeat exactly due to motor slip when
running at other than synchronous speed.
Post-processing for each time step consists
of calculation of the magnetic field intensity, eddy currents in rotor bars, moto r torqu
e and an instantaneous powe r balance for
the
motor. The power balance includes the rate
of change of magnetic energy the finite
elements, the rates of change of energy in
circuit inductances and capacitor, winding
1squared-R losses and eddy current losses
in rotorbars, the sum of which must equal
the instantaneous input line power. Powe r balan
ces within a few percent were achieved.
Motor torque at each time step is computed
using the virtual work principle described previously. In contrast to Strangas
and Theis [6], who used an airgap mesh composed of elements which could stretch elasti
cally, the authors found it necessary to
"lockstep" the rotor to prevent element disto
rtion during rotation. Element distortion drastically affects the magnetic energy in the
element as the element area changes and prevents accurnte torque calculation. The "lock
stepping" technique consists of meshing the
airgap with uniform elements and rotating
the mesh an integral numb er of airgap elem
ents
at each time step.
in
Calculated results from the model are instan
taneous values of flux density, rotor bar
currents, torques, line, main and auxiliary
currents and voltages, and many others. Air
gap flux densities can be plotted to check
winding distributions, torques examined for
alternating components, eddy current losses
checked in rotor bars, etc. A plot of input
power and motor output torque is shown in Figur
e 5 which clearly shows the alternating
torque characteristic of single-phase moto
rs. Table 2 shows representative time-avera
ged
results together with dynamometer data wher
e available.
Tabl e 2. PSC lndq ction Mot or- 230 Vac
Line Volts
3600R PM
3500 RPM
2900 RPM
Locked Rotor
(Synchronous}
(Full Load}
{Breakdown.)
Measured Predicted Measured Predic
ted Measured Predicted Measured Predic
ted
Torqu e•
·2.1
-2.1
67
67
158
156
13-15
11.7
Line Amps
1.26
1.36
10.2
10.3
41.9
58-62
60.2
Line Watts
172
150
2,363
2,365
8,113
8,521
Main Amps
5.07
5.14
8.1
8.1 .
41.8
62.7
Main Watts
·690
-711
1,575
1,561
7,711
8,466
Awe Amps
5.02
5.24
4.4
4.6
2.87
3.23
333
Aux Watts
860
861
794
804
402
55
Aux Volts
280
257
262
141
23
Cap. Volts
377
397
333
343
214
245
250
*Torq ue m Ounce-Feet ( 11.8 oz-ft = I nt-m}
-
-
1451
-
-
-
-
reactances such as
The modeling technique inherently accounts for motor leakage
must be included in
which
,
leakage
skew
or
end
not
but
[I],
s
leakage
slot, belt and zig-zag
model, but are
the
in
d
include
ely
separat
the circuit equations. Iron losses must also be
quantities.
field
ic
magnet
the
of
dge
knowle
te
comple
more
the
easier to estimate with
ons made to
correcti
and
red
Friction and windage losses must also be separately conside
calculated currents.
CONCLUSIONS
al finite element a~
A method of modeling magnetic devices using a tw~dimension
finite element
The
ed.
develop
been
has
ns
proach coupled with electric circuit equatio
and the circuit
device
the
within
solution
field
ic
magnet
rying
time-va
solution yields the
governing equaThe
ts.
elemen
circuit
l
equations account for both end effects and externa
rying tennitime-va
be
to
ns
functio
forcing
the
allows
that
form
tions have been cast in a
device, but
entire
the
for
s
solution
nal voltages. Results give not only the magnetic field
techsolution
the
of
result
r
Anothe
tors.
conduc
in
s
current
also permit calculation of eddy
element of the mesh.
every
in
ted
calcula
is
n
variatio
density
flux
ic
magnet
that
is
nique
core losses to be esti·
Both major and minor flux reversals are modeled which permits
a pennanent-splitand
orrner
a'transf
for
results
ed
predict
and
ental
mated. Experim
have been presented.
rotor
moving
with
and
capacitor induction motor both at standstill
cases.
all
in
ent
agreem
nt
excelle
in
are
ent
The model and experim
"..,"
CALCUL.O.TEO CURRENT
S:)(P~RIME.NTAL
~
C\JRRENi
3
""
""
Qe
5 0
HME
10 0
(m'!l"e)
20 0
time.
Figure 1. Transformer input voltage and magnetizing current versus
~~o.-o------~s~.o-------,~o~o~----~,~5~0~--~l~O~o
ft!JE (l'l~•d
or).
Figure 2. Transformer primary current versus time (25 MFD Capacit
1452
Q
i
5. 0
TIME
10
a
20. 0
(mn~::)
Figure 3. Transformer secondary current versus time (25
MFD Capacitor).
V!:CT~ ~Q"f'urr:r~
- · 0.0071.114
..
I
!111011~5
0
00~
~.
110:17!!
O.ll.:'l:;;t_,
o.atnl5
I;;
"
...O,Q!JI;I=
-o. 1:111:.=1
-C1 CIO:I7!
....C,0(15
K
-o.IJOU5
llltll~-0(1?'11114
Figure 4. PSC induction motor vector potential contour
plot.
Figure 5. Calculated torque and power input for PSC inducti
on motor_
1453
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1454