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Skin Effect in Squirrel Cage Rotor Bars and Its Consideration in
Simulation of Non-steady-state Operation of Induction Machines
Marcel Benecke1 , Reinhard Doebbelin1 , Gerd Griepentrog2 , and Andreas Lindemann1
1
Institute of Electric Power Systems, Otto-von-Guericke-University, Magdeburg, Germany
2
Siemens AG, Corporate Technology CT T DE, Germany
Abstract— This paper deals with squirrel cage induction machines and their modeling concerning the occurrence of current displacement in rotor bars caused by skin effect. Considering
non-steady-state operation such as direct on-line start and especially in the case of sustained
ramp-up, e.g., by using Y-D starter or soft starters (thyristor controlled AC voltage regulator),
an accurate calculation of current and torque is required. An appropriate numerical calculation
model is pointed out and an application example by simulating a double-cage induction machine
start-up demonstrates the benefit of this extended machine model.
1. INTRODUCTION
In order to start-up three-phase induction machines very often three-phase AC voltage controllers
based upon SCRs are being used to achieve a smooth transition to nominal speed and avoid high
torque and current values. Especially widely used squirrel-cage motors are intentionally designed
to develop a higher starting torque by utilizing the skin effect in the rotor bars. This, in turn,
affects the stator currents as well.
To analyze the characteristics of these machines in non-steady-state operations with higher rotor
slip, e.g., start with SCR based AC voltage controller or simple Y-D starter, the skin effect in rotor
bars (also known as deep bar effect or current displacement) has to be taken into consideration in
order to obtain valid results for torque and stator currents during ramp-up.
2. PRINCIPLE OF SKIN EFFECT IN ROTOR BARS
The rotor of current-displacement influenced induction machines is a squirrel cage consisting of a
number of bars (N2 ) arranged all-over the rotor perimeter and grouted with short circuit rings.
The energized three-phase stator windings generate a rotary field to which the squirrel cage is
exposed. This magnetic flux of the stator rotary field induces voltages which excite mesh currents
in the adjacent rotor bars. Their values and phase angles are determined by the resistance and the
leakage inductance of rotor bars and ring segments and also by the rotor slip s. The N2 “windings”
constitute an N2 -phase system which generates a rotor field acting like the field of a three-phase
wound rotor [1]. So it is possible to model the rotor as a three-phase equivalent winding and to
model the whole motor using the well-known T-equivalent circuit.
Under nominal operating conditions the current in the rotor bar cross-section is homogeneously
distributed and the leakage flux lines are shaped like illustrated in Fig. 1(left). In non-steady-state
operation, e.g., starting, the rotational speed does not correspond to the number of revolutions of
the stator rotary field. Therefore, high slip values occur. In the case of a low rotational speed,
which is connected with increased values of rotor current frequency, the current in rotor bars is
cumulatively displaced in radial direction to the air gap. This effect is caused by the slot leakage
field in the environment of the bars.
It is assumed that the rotor bar is divided into several elements (Fig. 1). Combined with
the adjacent rotor bars they form partial coils connected by the corresponding short-circuit ring
segments. Thus, the in radial direction internally located coil is exposed to a stronger leakage field
and shows the highest leakage inductance value compared to the leakage inductance values of the
upper coils close to the air gap [1]. At increased frequency f2 of the rotor current i2 (decreased
rotational speed n) the leakage reactance predominates compared to the resistance and the current
concentrates in the upper coils (Fig. 1(right)). Therefore, the effective conducting cross-section
decreases and with it the resistance increases. As a consequence, leakage reactance and resistance
values of the rotor depend on slip [2, 3].
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s
n
f2
s
n
f2
0
nnominal
0
422
1
0
f1
i2
Φσ
R2(k)
L2 σ(k)
Figure 1: Principle of leakage field distribution in slots of a bar-wound rotor, with s — slip, n — rotational
speed, f — electrical frequency (index: 1 — stator, 2 — rotor), i2 rotor current, Φσ — leakage flux [1].
R1
u1
L1 σ
L2 σ
R2
i1
Lh
u2
i2
R 2 ring
i2(0)-i2(1)
R2(0)
u2
L2σ(0) i2(0)
R 2(1)
L2σ(1)
i2(1)
R2(n-1)
L2σ(n-1)
i2(n-1)
Figure 2: Common T-equivalent circuit diagram of an induction machine and its variation with the right
side extended by RL-ladder network.
3. MODELING OF SKIN EFFECT
One method to model this slip-dependent effect is the determination of correction factors for sinusoidal supply and fundamental slot geometry characteristics [1, 4, 5]. This approach is sufficient in
case of sinusoidal supply and for simple rotor bar geometries.
Apart from that, another option is modeling of the above defined partial coils by using lumped
resistance and inductance elements in an RL-ladder network consisting of n stages, each with a
resistance R2(k) and a leakage inductance L2σ(k) (see Fig. 1(right)) and replacing the right side of
the T-equivalent circuit by this network [4, 6, 7], as illustrated in Fig. 2. This approach allows the
numerical calculation of slip-dependent rotor parameters for more complex rotor bar geometries
(e.g., double-cage induction machines) and non-sinusoidal feeding of the machine.
The concrete rotor bar geometry affects the resulting current displacement. Therefore the calculation of the bar-element’s electrical parameters R2(k) and L2σ(k) is necessary. In the case of
unknown rotor-bar geometry this can be done by an approximation assuming a homogenous rectangular rotor-bar divided into n elements. The resistance values R2(k) can be calculated as parallel
connection and the inductance values can be calculated on closer examination of the current fragmentation in the RL-ladder network (see Fig. 2). In the first element flows the total current i2(0)
and in every subsequent element (every k-th element) a current representing a portion of (n − k)/n
of the total current. Based on the law of energy conservation and the known equation for the
n
P
k 2 = n(n+1)(2n+1)
the following calculation formula can be deduced:
partial sum
6
k=1
L2σ(k) = L2 ·
6n
(n + 1) (2n + 1)
(1)
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In the case of known rotor bar geometry (esp. shape of the cross sectional area of the bars)
approximating the real geometry is possible and more accurate. An example concerning modeling a
double-bar geometry is given in Fig. 3. The parameters of the approximated rectangular elements
can be calculated from geometry [8]:
R2(k)
1
=
l
κ · ∆x · bk
L2σ(k)
µ0 · ∆x
=
l
bk
(2)
with κ — electrical conductance, µ0 — magnetic permeability, ∆x — height of each element, bk —
width of one element, l — length of the rotor bars.
By extending the T-equivalent circuit as described, an implementation algorithm for computing
the current i2(0) is convenient. This can be done using a system of state-space equations (Equation (3)) to express the mesh equations of the RL-ladder network:
di2(k)
= A · i2(k) + B · i2(0)
dt
i2(1) = C · i2(k) + D · i2(0)
with the state vector representing the partial currents of the
of coefficient matrices (A, B) consisting of R/L terms.
With this state-space model the relevant partial current
can be calculated for known rotor bar parameters. Using
current displacement is considered in a calculation algorithm
implementation of a specific motor model.
(3)
ladder elements calculated by means
and with it the entire rotor current
this state-space model the effect of
R by
realized in MATLAB- Simulink°
4. APPLICATION EXAMPLE: SIMULATION OF A DOUBLE-CAGE INDUCTION
MACHINE
An industrial induction machine (Table 1) with double-cage rotor profile (Fig. 3) is the basis to
parameterize the proposed model enabling the simulation of the non-steady-state behavior. The
intention of using a double-cage rotor profile in induction motors is utilization of the skin effect by
a special deep bar geometry (Fig. 3), to obtain a powerful starting torque.
To show the impact of modeling of current displacement, comparable simulations with consideration of skin effect (established motor model) and without consideration of skin effect (common
motor model in Simulink) were performed. The simulation results of a direct start-up are illustrated
in Fig. 4 by comparison of mechanical speed, electric torque and stator current.
As expected the ramp-up-time (t = 0 s . . . t = 0.1 s) is shorter if the skin-effect is considered
because of a higher starting torque. The oscillation of the torque is more damped in this case,
bz
”x
rk
ht
hr
hg
ro
ru
Figure 3: Profile section of a rotor double bar with geometry data (see Table 1) and illustrated approximation
with rectangular elements (n = 10).
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Table 1: Parameters of the used industrial induction motor.
rotor bar geometry:
Machine model 1LA5-207-4AA
ht = 30.85 mm
(data for Y-connection)
hg = 29.60 mm
PN =
hr = 16.45 mm
nN = 1465 U/min
rk =
3.05 mm
mN =
196
bz =
1.50 mm
p =
2
ro =
2.75 mm
IN =
55
ru =
1.30 mm
Number of ladder elements:
30
kW
Nm
A
J = 0.24 Nm · s2
T-equivalent circuit parameters:
n = 10
R1 =
98 mΩ
R20
=
62 mΩ
RF e =
50 kΩ
L1σ = 1.235 mH
L02σ = 1.146 mH
Lh3 =
mechanical speed, n / rpm
calculation of current displacement
35.3 mH
no current displacement calculation
1500
1000
500
0
1000
torque, m / Nm
800
600
400
200
0
-200
800
1
current i / A
600
400
200
0
-200
-400
0
0.1
0.2
0.3
time t in s
0.4
0.5
0.6
Figure 4: Comparison of machine models with (calculation of current displacement) and without consideration of skin effect (no current displacement calculation) by means of the example of a direct ramp-up,
simulation of mechanical speed, electric torque and the stator current of one phase.
which yields a faster decline of the stator current amplitude. The simulation results of the skin
effect model are in accordance with calculation results from a reference calculation tool and with
data sheet specifications of inrush current (iRM S = 385 A corresponding to the RMS-value of the
current i1 between t = 0 s and t = 0.1 s in Fig. 4) and average starting torque (M = 510 Nm
corresponding to the average value between t = 0 s and t = 0.1 s in Fig. 4).
5. CONCLUSIONS
The paper presents a method to calculate the slip-dependent rotor current based on an extended
T-equivalent circuit for non-sinusoidal power supply and arbitrary rotor-bar geometry. For complex
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rotor-bar geometries such as double-cage rotor profile these parameters can be obtained by approximation of the rotor bar cross-section using rectangle elements and deriving the R2(k) and L2σ(k)
R by
values for each element. The calculation algorithm has been realized in MATLAB-Simulink°
implementation of a specific motor model. The simulation results demonstrate the advantages of
the proposed model in comparison to existing machine models.
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