Induction Motor Model for the Analysis of Capacitive and Induced
Shaft Voltages
Petri Mäki-Ontto and Jorma Luomi
Power Electronics Laboratory, Helsinki University of Technology
P.O. Box 3000, FI-02015 TKK, Finland
Abstract — The paper proposes a three-phase motor model
that is suitable for the analysis of bearing voltages and currents
in converter-fed ac motors. Both capacitive and induced shaft
voltages are taken into account. The induced shaft voltage, being
the reason for high-frequency circulating bearing currents, is
important in larger machines. For modeling the induced shaft
voltage with a sufficient accuracy, different stator core models
are compared. The motor model can be connected to threephase circuit simulation models of frequency converters and
cables. Simulation results are validated with experimental results obtained using a 1.4-MW induction motor fed by a frequency converter.
Circumferential flux
Common-mode current
Circulating bearing current
Fig. 1. Paths of common-mode current and circulating bearing
current in the motor. The axial distribution of the circumferential
flux is illustrated with the sizes of the circles.
I. INTRODUCTION
Modern frequency converters have introduced new requirements for the modeling of induction motors since there
is a need to investigate high-frequency phenomena in these
motors and whole drive systems. Increased switching frequencies and short rise times of the PWM voltage pulses require a high bandwidth of the simulation model. A frequency
converter also produces a common-mode voltage, and parasitic capacitances in the motor provide low-impedance paths
for high-frequency common-mode currents. In addition, the
frequency dependence of the resistances and inductances has
an influence on these phenomena.
Common-mode voltages and common-mode currents can
cause destructive currents in motor bearings. On the one
hand, the distribution of the common-mode voltage between
parasitic capacitances in the motor will cause a capacitive
shaft voltage [1]. On the other hand, the high-frequency
common-mode current will generate a circumferential magnetic flux in the stator core, which causes an induced shaft
voltage [2]. These two types of shaft voltages are the reasons
for discharge bearing currents and high-frequency circulating
bearing currents, respectively.
The common-mode impedance of an electric drive depends
on various components included in the system: power grid;
frequency converter; filters; power cables; motor; and
groundings. Therefore, the motor model should be compatible with the models of the other parts of the system. Several
models have been presented for the calculation of the capacitive shaft voltage [3], [4]. For the determination of the model
parameters, impedance measurements in a wide frequency
band may be required, as in [4]. Expressions have also been
derived for the circumferential flux and induced shaft voltage
caused by a harmonic component of the common-mode current [5]. However, a complete model for the evaluation of
0-7803-8987-5/05/$20.00 ©2005 IEEE.
both capacitive and induced shaft voltages has not yet been
presented. Such a model would be very useful in the analysis
of bearing currents and, for example, in designing an output
filter for a frequency converter and motor combination.
This paper presents an induction motor model suitable for
the analysis of bearing currents in both frequency domain and
time domain. Both the capacitive shaft voltage and the induced shaft voltage are taken into account. The motor model
can be included in a three-phase circuit simulation model of
the whole electric drive.
II. MODELING OF INDUCED SHAFT VOLTAGE
Steep edges of the PWM voltage pulses generate a highfrequency common-mode current through the capacitance of
the stator winding insulation. As illustrated in Fig. 1, this
current flows through the stator core and causes a circumferential magnetic flux in the stator yoke [2], [5], [6]. The flux
generates the induced shaft voltage in a loop formed by the
stator, the rotor, and the bearings. The induced shaft voltage,
being the reason for high-frequency circulating bearing currents, is especially important in larger machines.
In the laminated stator core, the high-frequency commonmode current and the magnetic flux flow along the iron sheet
surfaces in a thin layer, corresponding to the skin depth
δ=
2
ωµσ
(1)
where ω is the angular frequency of the harmonic component in question, and σ and µ are the conductivity and the
incremental permeability of the core material, respectively.
The incremental permeability has its maximum value in
the case of zero biasing flux density, and it approaches µ0
1653
Stator frame
N sheets
u sh1
N+2
Zb
Zb
u cm
Stator winding
Z
Fig. 2. Contour plot of magnetic field strength (corresponding to
equal currents between curves) in winding insulation and five
iron laminations.
2
with magnetic saturation [7]. The circumferential highfrequency magnetic field in the stator yoke is biased by the
periodic distribution of the rotating fundamental-frequency
magnetic field. The incremental permeability thus varies
along the circumference, and this permeability distribution
rotates at the fundamental frequency. An average value of the
incremental permeability can be used for the analysis of the
circumferential magnetic flux. In the following, it is assumed
that the flux is concentrated in the yoke and the problem is
rotationally symmetric.
The circumferential magnetic field was calculated numerically in [5]. In Fig. 2, the magnetic field and the commonmode current are illustrated in five 0.5 mm thick iron laminations. The other dimensions and the relatively low frequency
of 30 kHz were selected for visual clarity in this example.
The field lines would be nearly indistinguishable from the geometry lines at relevant frequencies (about 1 MHz) since the
skin depth in the core material is of the order of 10 µm.
i cm
⎛r ⎞
1
ln ⎜ 2 ⎟
2πσδ ⎝ r1 ⎠
3
4
N+1
C
1
ib
Fig. 3. Detailed common-mode circuit model.
winding is equal to the common-mode voltage u cm (Node 1),
and the common-mode current i cm is distributed to individual laminations through winding insulation capacitances. The
capacitance between the stator winding and a single lamination of the stator core is C , and each of the N laminations is
modeled by two impedances Z according to (2).
The induced shaft voltage u sh1 is the potential of Node
N + 2 in the circuit of Fig. 3. The circulating bearing current
is denoted by i b . The bearing circuit is modeled by a series
connection of two bearing impedances Z b (which also include the impedances of the end shields and the rotor). For
analysis, the circuit model of Fig. 3 can be formulated in
terms of the ( N + 2 ) × ( N + 2 ) admittance matrix
A. Detailed Common-Mode Circuit Model
Since the current and the corresponding magnetic flux
flow at the surfaces of the laminations, each lamination surface can be modeled by an impedance of a layer whose
thickness equals the skin depth, i.e. by the impedance
Z = (1 + j)
Z
(2)
where r1 and r2 are the inner and outer radii of the stator
yoke, respectively.
A detailed model of the common-mode circuit is depicted
in Fig. 3. The winding resistances and inductances are not yet
taken into account, and the capacitive shaft voltage is also
omitted. The terminals of the winding and the grounding are
at the leftmost end of the core in all examples given in this
paper. It is assumed that the electric potential of the stator
1654
"
− jω C − jω C
− jω C
0 ⎤
⎡ jω NC − jωC
⎢
⎥
3
1
⎢ − jω C
"
+ jω C −
0
0
0 ⎥
⎢
⎥
2Z
2Z
⎢
⎥
1
1
⎢ − jω C − 1
#
0
0 ⎥
+ jω C −
⎢
⎥
Z
2Z
2Z
⎢
⎥
1
⎢ − jω C
⎥
%
%
−
0
0
0
⎥
2Z
Y=⎢
⎢
⎥
1
1
⎢
⎥
C
0
j
0
#
#
%
+
−
ω
⎢
⎥
Z
2Z
⎢
⎥
⎢
1
3
1 ⎥
0
0
−
+ jω C −
"
⎢ − jω C
⎥
Z ⎥
2Z 2Z
⎢
⎢
1
1
1 ⎥
0
0
0
0
−
+
⎢ 0
⎥
2
Z
Z
Z
b⎦
⎥
⎣⎢
m sections
B. Reduced Common-Mode Circuit Model
The distribution of the common-mode current i cm between individual laminations depends on the impedances in
the circuit. At lower frequencies, the voltage drop over the
capacitances is dominating, and the current i cm / N enters
each lamination through the capacitance C . The total current
distribution is linear within the iron core: the current in the
rightmost sheet is i cm / N , and the current in the leftmost
sheet is i cm as can be seen in Fig. 2.
At lower frequencies, it is possible to use the commonmode core impedance [5]
⎛r ⎞
N
ln ⎜ 2 ⎟
Z c = (1 + j)
3πσδ ⎝ r1 ⎠
i cm
Cws
m
Zd
Zd
Zd
Zd
m
m
m
ib m
m
1 N µδ ⎛ r2 ⎞
ln ⎜ ⎟
1 + j 2π
⎝ r1 ⎠
and the corresponding induced shaft voltage is
Zc
m
u sh1
Fig 4. Reduced common-mode circuit model.
i cm
ib
Cws
(4)
(5)
−
Zb
(3)
C. Simple Common-Mode Circuit Model
If equal current is assumed to enter each lamination in the
whole core, the common-mode impedance of the core is
given by (3), and the bearing circuit impedance is given by
(4). The common-mode circuit and the bearing circuit can be
modeled by the simple circuit model shown in Fig. 5.
The common-mode circuit is a series connection of the capacitance Cws and the core impedance Z c if the circulating
bearing current i b = 0 . The circumferential flux in the stator
core is [5]
Cws
m
Zc
Zb
where (2) and (3) have been used.
The number of laminations varies from hundreds in small
motors to thousands in large motors, and the modeling of individual sheets would lead to very large circuit models. To
reduce the model size, the circuit shown in Fig. 4 is proposed. In this reduced model, the core is divided into m sections, and equal current is assumed to enter each lamination
within each section. The circuit consists of the contributions
of the winding-to-core capacitance Cws and the impedances
Z c and Z d . The number of nodes can be much lower than
that in the detailed model of Fig. 3.
A large m increases the accuracy and makes it possible to
model the axial distribution of the current and flux at high
frequencies. On the other hand, the implementation of the
circuit model is easier if m is selected as small as possible.
However, a small value of m causes a discretization error in
the induced shaft voltage even at low frequencies (about 6.5
% for m = 5 and 1 % for m = 35 ).
φ = i cm
Cws
m
u cm
in series with the total stator-winding-to-core capacitance
Cws = N C . In accordance to Fig. 3, the core impedance for
the bearing circuit is the series connection of all 2 N impedances of the lamination surfaces, i.e.
Z d = 2 N Z = 3Z c
Cws
m
u cm
+
3
u1
2 −
−
1
u2
2
+
u1
u2
Zc
Common-mode circuit
2Z b
u sh1
Zd
Bearing circuit
Fig 5. Simple common-mode circuit model.
u sh1 = jω φ .
(6)
The relation between the common-mode current and induced
shaft voltage is obtained by combining (3), (5), and (6):
u sh1 =
3
Z i .
2 c cm
(7)
In Fig. 5, this relation is modeled by the voltage-controlled
voltage source 3 u1 / 2 in a separate bearing circuit, which is
also valid when i b ≠ 0 . The effect of the circulating bearing
current on the common-mode circuit is modeled by the controlled voltage source u 2 / 2 in the common-mode circuit.
The presented core models are valid only for frequencydomain analysis. A core model suitable for time-domain
simulations is obtained by modeling the frequency-dependence of core impedances Z c and Z d (or the impedance Z )
by means of a ladder circuit [8]. A four-step ladder circuit is
shown in Fig. 6. A method based on [9] can be applied for
determining suitable resistance and inductance values. The
parameters needed are the low-frequency values of the resistance and inductance, and the maximum frequency. The ratio
between the resistances of subsequent ladder steps is constant, and so is the ratio between the inductances. The model
1655
Lc2
Rc1
Lc3
Rc2
Rc3
Current (mA)
Lc1
Rc4
3
D
2
C
1
A
B
0
Fig. 6. Ladder circuit approximation for core impedance Z c suitable for both frequency-domain and time-domain modeling.
200
Phase (°)
parameters are optimized by a least-squares fit of the ladder
circuit impedance to the impedance Z c given by (3) in the
frequency band considered.
D
0
A
B
-100
III. COMPARISON OF COMMON-MODE CIRCUIT MODELS
C
-200
0
500
1000
1500
Sheet number
2000
Impedance (Ω)
Fig. 7. Amplitude and phase of currents in the capacitances of the
detailed model for different frequencies: 100 kHz (A); 700 kHz
(B); 1.5 MHz (C); 5 MHz (D).
10
1
C
10
0
A
B
10
4
10
5
Frequency (Hz)
10
6
10
7
Fig. 8. Common-mode impedance of winding insulation and core,
obtained using different core models: detailed model (A); reduced
model (B); simple model (C).
C
Voltage (V)
The detailed model can be used for studying the commonmode current distribution in the stator winding insulation.
The parameters given in Table I were used for the following
examples. The distribution of the common-mode current
( i cm = 1 + j0 A ) in the winding insulation is shown in Fig. 7
for four frequencies. At 100 kHz, the amplitude and phase of
the current are equal in each capacitance C of the model. At
700 kHz, there is some variation in the amplitude and phase
curves. At 1.5 MHz, the amplitude of the current is clearly
larger at the terminal end, and the phase of the current
changes from a positive value at the terminal end to a negative value at the other end of the core. At 5 MHz, this effect
is even more pronounced, and most of the current flows
through the capacitances near the terminal end of the core.
The detailed model, the reduced model with m = 10 , and
the simple model were compared by calculating the commonmode impedance and the induced shaft voltage as a function
of the frequency. The common-mode impedance of the insulation and core is shown in Fig. 8. At frequencies below 400
kHz, all three models predict equal common-mode impedances. The simple model has one distinct critical frequency,
after which the impedance increases, whereas the impedances
of the other models decrease again after 1.7 MHz. The reduced model gives good results up to frequencies of a few
megahertz.
The induced shaft voltage is shown in Fig. 9. The reduced
model gives again good results up to a few megahertz, but
the accuracy of the simple model decreases rapidly above 1
MHz. A constant discretization error of about 3 % can be
seen in the results obtained by using the reduced model. A
larger value of m = 100 in the reduced model provides results that are identical to the results of the detailed model in
the whole frequency band considered.
100
10
0
A
B
TABLE I
CORE DATA
Conductivity of core material
Insulation capacitance
Incremental permeability
Lamination thickness
Number of laminations
Inner radius of the yoke
Outer radius of the yoke
σ
Cws
µ
N
r1
r2
10 ⋅106 (Ωm) −1
370 nF
700µ0
0.5 mm
2200
362 mm
430 mm
10
-1
10
4
10
5
Frequency (Hz)
10
6
10
7
Fig. 9. Induced shaft voltage obtained using different core models: detailed model (A); reduced model (B); simple model (C).
1656
Lσ s1
i cm
Cws
6
Cwr
u cm
Ls
Rs
Cwr
6
Cwr
6
Cws
6
Ms
Cws
Csr
Zb
Zb
u sh2
Fig. 10. Capacitive shaft voltage model.
i cm
Cws
u cm
−
1
u2
2 +
u1 Z c
−
1
u
2 2+
Cwr
Zb
3 +
u
4 1
−
Csr
u2
1
Z
2 d
Zb
u1 Z c
3 −
u
4 1
+
Zb
3 +
u
4 1
−
Csr
u2
Zb
3 −
u
4 1+
1
1
Z
Z
2 d
2 d
Stator core, rotor, and bearings
1
Z
2 d
Fig. 12. Three-phase motor model including the combined shaft
voltage model.
Stator core, rotor, and bearings
and the core impedance Z d are split in two parts in the bearing circuit. It should be noted that the capacitive shaft voltage
can also be added to the reduced model presented in Fig. 4.
Fig. 11. Combined shaft voltage model including both capacitive
and induced shaft voltages.
IV. MOTOR MODEL
A. Combined Shaft Voltage Model
The capacitive shaft voltage is caused by capacitive couplings between the stator winding, the rotor core, and the
stator core [1], [3]. It can be modeled by the equivalent circuit shown in Fig. 10. The circuit consists of the statorwinding-to-core capacitance Cws , the stator-winding-to-rotor
capacitance Cwr , the stator-core-to-rotor capacitance Csr ,
and the bearing impedances Z b . The capacitive shaft voltage
is denoted by u sh2 . The capacitances can be calculated from
the dimensions of the motor.
A combined model including both the induced and capacitive shaft voltage models is shown in Fig. 11. The simple
common-mode circuit model presented in Fig. 5 is augmented with the capacitive shaft voltage model presented in
Fig. 10. The polarity of the capacitive voltage is the same in
both bearings, whereas the induced shaft voltage causes contributions that have opposite polarities at the opposite ends of
the machine. Therefore, the controlled voltage source 3 u1 / 2
B. Three-Phase Model and Parameter Evaluation
In Fig. 12, the combined shaft voltage model is included in
a three-phase motor model. The model of stator winding consists of resistances Rs , self-inductances Ls , and mutual inductances M s . Ls includes most of the leakage inductance.
In practice, it was found that a small part Lσ s1 of the leakage
inductance should be placed in series with each winding
phase model. In the investigated drive, Lσ s1 was 1.5 % of the
total leakage inductance. The capacitances Cws and Cwr are
divided into contributions of the individual phases (corresponding to π-equivalent circuits for each phase).
The model in Fig. 12 corresponds to the no-load operation
of the motor. In addition, the model could be augmented with
a d-q model of the rotor winding and the equation of motion
in a fashion described in [10]. In practical time-domain
simulations, however, the required minimum time step is
very small due to the rapid voltage changes of the PWM inverter, and the long time constants associated with the rotor
winding and mechanics would lead to very long computation
times. Therefore, the model in Fig. 12 is used for the simulations described in this paper.
1657
V. RESULTS
1658
Drive end
Insulations
Non-drive end
Fig. 13. Experimental setup. The motor is equipped with insulations and copper brushes with adjustable connections.
TABLE II
PARAMETER VALUES USED IN SIMULATIONS
Winding Parameters
Ls
1707 µH *
Ms
–833 µH
Lσ s1
0.6 µH *
Rs
1.5 Ω *
Cws
370 nF
Csr
3.70 nF
Cwr
0.228 nF
Core Model Parameters
Rc1
8.68 Ω
Rc2
1.86 Ω
Rc3
0.40 Ω
Rc4
0.086 Ω
Lc1
24.5 nH
Lc2
186 nH
Lc3
1415 nH
* high-frequency values
3
Magnitude (Ω)
A. Measurement of Capacitive Shaft Voltage
In the capacitive shaft voltage measurement, a voltage
probe was connected between the shaft and the end shield at
the drive end of the motor. The measured common-mode current is shown in Fig. 15. The waveform is a typical example
of an LC circuit response to steep voltage pulses. A high-frequency oscillation occurs in the current at the beginning of
the voltage pulse, and the amplitude and length of the lowerfrequency pulses are determined by the slope and rise time of
the voltage pulses ( i = Cws ducm dt ). The simultaneously
measured common-mode voltage shown in Fig. 16 is in
agreement with this reasoning. The current in Fig. 15 was
used as the input for the simulation. The simulation result is
shown in Fig. 17. A good correlation is found between the
simulated and measured common-mode voltages.
V
2
1
0 3
10
10
4
10
Frequency (Hz)
10
4
10
Frequency (Hz)
90
Phase (°)
A six-pole 1.4-MW induction motor (690 V, 1500 rpm)
fed by 1160-kVA frequency converter was used for laboratory experiments. The common-mode voltage at the motor
terminals was calculated from the phase voltages measured
using three voltage probes, and a Rogowski coil was used for
measuring the common-mode current. The motor was prepared for measuring the capacitive and induced shaft voltages. Thick insulating rings and the connections depicted in
Fig. 13 were placed in both end shields. Both shaft ends were
equipped with slip rings and copper brushes.
The capacitive shaft voltage was measured between the
shaft and the end shield when both short-circuit connections
were open (as in Fig. 13). The induced shaft voltage was distinguished from the capacitive voltage by closing one of the
short circuits and measuring the shaft voltage at the opposite
end.
The parameter values used in the simulations are given in
Table II. In high-bandwidth simulations, the frequency-dependence of the stator resistance and leakage inductance
should be modeled. In the simulations presented in this paper,
constant high-frequency values were used for these parameters. The oscillation frequency of the measured highfrequency component of the common-mode current was 700
kHz. The reduction in the slot leakage inductance and the
increase in the winding resistance at this frequency were calculated using the finite element method. The resistance was
increased by a factor of 1500, and the slot leakage inductance
was reduced by 40 % as compared with the values at the
nominal frequency.
The value of the incremental permeability was determined
from the measurements of the common-mode current and the
induced shaft voltage. For the motor studied, µ = 700 µ0 was
selected and the ladder circuit parameters were determined
accordingly. The magnitude and the phase of the ladder circuit impedance and the impedance Z c given by (3) are
shown in Fig. 14. The ladder circuit parameters were fitted in
the frequency band of 10 kHz – 10 MHz.
5
10
6
10
7
5
10
6
10
7
45
0 3
10
Fig. 14. Stator core common-mode impedance Z c , given by (3)
(solid) and by ladder circuit approximation (dashed).
50
30
25
Voltage (V)
Current (A)
15
0
-15
0
-25
-30
0
0.1
0.2
0.3
Time (ms)
0.4
-50
0
0.5
800
60
400
30
0
-400
-800
0
0.1
0.2
0.3
Time (ms)
0.4
0.5
0
-60
0
0.5
0.1
0.2
0.3
Time (ms)
0.4
0.5
Fig. 19. Simulated capacitive shaft voltage.
pacitive shaft voltages at the drive and non-drive ends were
identical, whereas the polarity of the induced shaft voltage
component at one end of the machine was opposite to the
corresponding component at the other end. The simulation
result is shown in Fig. 19. The waveform is similar to the
measured result in Fig. 18, and the maximum and minimum
high-frequency spikes are approximately equal.
1000
500
Voltage (V)
0.4
-30
Fig. 16. Measured common-mode voltage.
0
-500
-1000
0
0.2
0.3
Time (ms)
Fig. 18. Measured capacitive shaft voltage.
Voltage (V)
Voltage (V)
Fig. 15. Measured common-mode current.
0.1
0.1
0.2
0.3
Time (ms)
0.4
0.5
Fig. 17. Simulated common-mode voltage.
Fig. 18 shows the measured shaft voltage. In addition to
the capacitive shaft voltage, contributions of the induced
shaft voltage can be seen as high-frequency spikes. The ca-
B. Measurement of Induced Shaft Voltage
The induced shaft voltage was measured at the drive end
when the short-circuit connection at the non-drive end was
closed. The common-mode current was re-measured in order
to get a source current for the simulation. The measured
common-mode current and induced shaft voltage are shown
in Figs. 20 and 21, respectively. It can be seen that the highfrequency spikes in the induced shaft voltage correspond to
those in the common-mode current, but the lower-frequency
pulses induce a shaft voltage that is less pronounced. The
simulated induced shaft voltage is shown in Fig. 22. A good
correlation is found between the simulation and measurement
results.
1659
VI. CONCLUSION
30
Current (A)
15
0
-15
-30
0
0.1
0.2
0.3
Time (ms)
0.4
0.5
Fig. 20. Measured common-mode current.
The induced shaft voltage can be modeled with a special
common-mode circuit model, which represents the stator core
laminations and the winding insulation. In the investigated
drive, the complexity of this model can be further decreased,
and a simple model is obtained without compromising the
accuracy of the simulation results. Augmenting this model
with a conventional capacitive shaft voltage model results in
a combined model, which makes it possible to simulate both
types of shaft voltages and the corresponding bearing currents simultaneously.
The combined shaft voltage model can be added to a threephase model of an induction motor, which can be included in
a three-phase circuit simulation model of the whole electric
drive [11]. The comparison of measured and simulated shaft
voltages shows that the model is able to predict both capacitive and induced shaft voltage phenomena with a good accuracy.
40
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20
0
-20
-40
0
0.1
0.2
0.3
Time (ms)
0.4
0.5
Fig. 21. Measured induced shaft voltage.
40
Voltage (V)
20
0
-20
-40
0
0.1
0.2
0.3
Time (ms)
0.4
0.5
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[11] P. Mäki-Ontto, J. Luomi, and H. Kinnunen, “Three-phase model for the
circuit simulation of common-mode phenomena and shaft voltages in
AC motor drive systems,” in IEEE International Electric Machines and
Drives Conference, IEMDC’05, San Antonio, TX, 2005
Fig. 22. Simulated induced shaft voltage.
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