Transactions on Electrical Engineering, Vol. 3 (2014), No. 1
23
Field-Circuit Modelling
of Self-Excited Induction Generators
Miksiewicz Roman
Silesian University of Technology, Gliwice, Poland, roman.miksiewicz@polsl.pl
Abstract — In the paper computational models of an
induction self-excited generator are presented. Circuit
models taking into consideration nonlinearity of the
magnetic circuit enable calculations of the generator static
characteristic at an autonomic operation. The field-circuit
model using Maxwell 2D software allows determination of
time curves of any electrical variable in different conditions
of the generator operation. There are presented basic
determined static characteristics using both models and
time curves of currents, voltages, torque during self
excitation and under symmetrical load and one-phase short
circuit, in cooperation with a rectifier system and during
connection of the excited generator to network.
Keywords — induction generator; squirrel cage motor;
cuircuit-field modelling.
magnetizing reactance Xm; it depends on magnetizing
current Im. This equivalent circuit diagram does not take in
consideration iron losses. For the considered induction
machine the parameters of the equivalent circuit diagram
were determined using the RMxprt Maxwell software.
The data of the induction motor are the following: rated
power 7,5 kW; rated voltage 380 VAC; rated rotational
speed 965 rpm; winding connection delta. On the basis of
the circuit calculations characteristics Xm(Im) and values of
other parameters of the equivalent circuit were
determined; they are :
Rs = 1,215 ; X σ s = 2,72 ; Rr' = 2,29 ; X σ' r = 4,42
Io
I. INTRODUCTION
Both squirrel-cage and slip-ring machines are used as
wind generators. Squirrel-cage induction machines are
used especially in smaller wind power plants mainly due
their greater reliability nevertheless their worse control
characteristics. They require to use condensers (Fig. 1)
and existence of residual magnetism for their excitation.
At autonomic operation the generated voltage frequency
depends on rotational speed and load what is the essential
disadvantage of the induction squirrel-cage generators. In
order to assure constant frequency additional powerelectronics equipment with conversion are used, for
example AC-DC-AC. Many publications [1-7; 9] discuss
problems related to features of the induction generators at
autonomic operation and cooperation with a network.
n
G
3
Ro
C
Fig. 1. Circuit diagram of an induction generator at autonomic
operation.
II. CIRCUIT MODEL
The equivalent circuit diagram of the circuit model is
presented in Fig. 2. The circuit diagram takes into
consideration non-linearity of the magnetic circuit as
fr X o
Rs
Is
fr X σs
'
Rr
fr
f r - nr
'
fr Xσ r
Uo
XC
fr
fr X m
Ro
Fig. 2. Equivalent circuit diagram of an induction generator for one
phase.
In the equivalent circuit diagram fr means relative
frequency - frequency of the stator voltage in comparison
with the rated frequency of the motor.
The prepared algorithm enables by use of Mathcad
software for determination of static characteristics taking
into consideration non-linearity of magnetization
characteristics determination of characteristics at a
constant rotational speed or constant frequency of voltage
at the load [5].
The relative frequency may be determined by
comparison of the zero real part of the equivalent
impedance seen from terminals of the magnetizing
reactance to Re(Z z ) = 0 . The stationary working point is
calculated by comparison of the imaginary part of the
equivalent impedance to the magnetizing reactance:
Im(Z z ) = f r X m .
'
Ir
Transactions on Electrical Engineering, Vol. 3 (2014), No. 1
A. Self-excitation of the generator
In Fig. 3 there are presented results of calculations of
line voltage and phase current time curves at constant
rotational speed n = 1040 rpm, during self-excitation of
the loaded generator. In order to make possible the
generator self-excitation during digital calculations nonzero initial conditions (current) in one of phase windings
were set.
400
0.05
a)
0.07
0.09
0.11
0.13
0.15
0.11
0.13
0.15
Torque (Nm)
− 30
− 60
− 90
− 120
− 150
Time (s)
t [s]
30
b)
15
Current (A)
i(t) [A]
III. FIELD-CIRCUIT CALCULATIONS
In the field-circuit calculations the Maxwell software
with Transient solver was used. Using this software it is
possible to determine various transient states of the
generator at autonomic operation at symmetric and nonsymmetric loads and also at cooperation with rectifiers. It
is also possible to make calculations for various nonsymmetric states. In 2-dimensional calculations there were
taken into consideration the resistance of the stator and
leakage inductances of the stator and rotor, determined
using RMxprt, the same as in the circuit calculations. The
software does not take iron losses into consideration.
24
0.05
0.07
0.09
− 15
a)
200
Voltage (V)
− 30
0
0.05
0.1
0.15
0.2
Time (s)
400
c)
− 200
Voltage (V)
200
− 400
Time (s)
30
0.05
0.07
0.09
0.11
0.13
b)
− 200
Current (A)
20
10
− 400
0
0.05
0.1
0.15
Time (s)
0.2
t [s]
Fig. 4. Torque, phase current and line voltage time curves at rotational
speed change (n1 = 1040 rpm) (n2= 1140 rpm).
− 10
− 20
.
− 30
Time (s)
t [s]
Fig. 3. Line voltage and phase current time curves during self-excitation
of the generator.
B. Transient state at rotational speed change.
In Fig. 4 curves for transient state (a generator with
resistance load and separate operation) are presented at
discrete change of the rotational speed from n = 1040 rpm
to n = 1140 rpm. There were presented the torque, phase
current and line voltage time curves. In the stationary state
preceding voltage frequency change was of 49.74 Hz and
after the speed change it increased to f = 54.42 Hz.
C. Transient state at single-phase short circuit.
For the presented field-circuit model it may be
performed calculations for an unsymmetrical load. In Fig.
5 there are presented results of calculations
(electromagnetic torque, phase currents, terminal voltages)
for the following conditions of the generator operation:
constant rotational speed n = 1040 rpm; symmetrical load
by receiver resistance Ro=30 Ω and next single-phase
short circuit of the receiver. As a result of the
unsymmetrical short circuit large pulsating component
arises in the torque time curve.
0.15
Transactions on Electrical Engineering, Vol. 3 (2014), No. 1
25
600
0.06
0.08
0.1
0.12
0.14
0.16
a)
− 40
Voltage (V)
Torque (Nm)
a)
− 80
− 120
400
200
− 160
0.06
− 200
0.07
Current (A)
Current (A)
15
0.1
0.12
0.08
0.09
t [s]
b)
0.08
0.09
40
b)
30
0.06
0.08
Time (s)
Time (s)
0.14
20
0.06
0.07
0.16
− 20
− 15
− 40
Time (s)
− 30
c)
Time (s)
Current (A)
t [s]
600
c)
300
Voltage (V)
20
10
0.06
0.07
0.08
0.09
− 10
0.06
0.08
0.1
0.12
0.14
0.16
− 20
Time (s)
− 300
0.06
0.07
0.08
− 600
Time (s)
Fig. 5. Torque, phase current and line voltage time curves before and
after the single phase short circuit.
Torque (Nm)
d)
− 50
− 100
− 150
Time (s)
Fig. 7. Steady state of: a) line voltage, b) phase currents, c) line currents,
d) electromagnetic torque time curves.
Fig. 6. Circuit diagram of a generator with rectifiers of the Schematic
Capture module.
D. Cooperation of the generator with rectifiers
The circuit diagram of the Schematic Capture module
of the Maxwell software is presented in Fig. 6.
Calculations were performed for the resistance load
Ro=30 Ω, capacitors C=90 µF, at the constant rotational
speed n = 1040 rpm. Exemplary voltage at the receiver,
phase current of the winding and line current time curves
0.09
Transactions on Electrical Engineering, Vol. 3 (2014), No. 1
E. Switching on of excited generator to a network
The Maxwell software enables calculations of various
transient states. It was considered the case of the nonexcited generator connection to a network and setting a
chosen torque. The circuit diagram of the circuit system is
presented in Fig. 8.
200
a)
0.05
Torque (Nm)
before rectifiers and electromagnetic torque are presented
in Fig. 7. Knowledge of these curves allows accurate
determining of their RMS values and permissible load.
26
0.1
0.15
0.2
0.25
− 200
− 400
− 600
− 800
− 1 × 10
3
Time (s)
Fig. 8. The circuit diagram of the generator in the Schematic Capture
module at connection to a network.
In Fig. 9 there are presented the torque, rotational
speed and current time curves of the excited generator
(initial rotational speed of 1040 rpm) during its switching
on to a network and loading by torque T = 100 Nm. As
can be seen from presented time curves no problems
occur with synchronization while large switching currents
can be expected.
Speed (rpm)
b)
1.1 × 10
3
1.05 × 10
3
1 × 10
3
950
900
0.05
0.1
0.15
0.2
0.25
Timet [s]
(s)
300
c)
200
0.05
0.1
0.15
0.2
0.25
0.2
0.25
− 100
− 200
− 300
Time (s)
400
d)
Current (A)
IV. STATIC CHARACTERISTICS OF THE GENERATOR
Static characteristics of the generator may be
determined on the basis of above calculated parameters of
the equivalent circuit diagram and on the basis of the
field-circuit calculations for stationary operational
conditions. In Fig. 10 there are presented external
characteristics U = f(I) determined using both above
mentioned methods for two values of capacitors at the
constant rotational speed n = 1040 rpm and resistance
load.
On the basis of the circuit calculations it is easy to
determine static characteristics at a constant frequency
what is more troublesome for the field-circuit calculations
and it requires repeated time-consuming recalculations.
Comparing the determined static characteristics using
both methods it may be noticed slight differences
between them. It influences accuracy of the equivalent
circuit diagram parameter determination using the circuit
method and stability of these parameters, however from
practical point of view it should be used for initial
determination of capacitors and generator windings circuit
calculations. The circuit calculations give greater
variability of voltage at generator terminals.
Current (A)
100
200
0.05
0.1
0.15
− 200
− 400
Time (s)
Fig. 9. Time curves : a) electromagnetic torque, b) speed, c) phase
currents, d) line currents at switching on the excited generator to a
network.
Transactions on Electrical Engineering, Vol. 3 (2014), No. 1
V. SUMMARY
The circuit and circuit-field calculations show good
conformity of calculation results. Therefore it is possible
to use the circuit model in the initial stage of designing.
The prepared field-circuit model allows for calculations
of various transient states types and analysis of results
both from point of view of the generator and for various
systems.
500
a)
U [V]
Voltage (V)
400
300
200
100
REFERENCES
C=90 uF
C=70 uF
0
5
10
Current (A)
15
500
b)
400
Voltage (V)
300
200
100
C=90 uF
C=70 uF
0
2 × 10
3
3
4 × 10
6 × 10
3
8 × 10
3
1 × 10
4
Power (W)
P [W]
500
c)
U [V]
Voltage (V)
400
300
200
100
C=90 uF
C=70 uF
0
5
10
15
Current (A)
I [A]
500
d)
400
Voltage (V)
27
300
200
100
C=90 uF
C=70 uF
0
2 ×10
3
4 × 10
3
6 ×10
3
8 ×10
3
1 ×10
PPower
[W] (W)
Fig. 10. Static characteristics calculated using the field-circuit model
a) U = f(I), b) U = f(P), and using the circuit model c) U = f(I),
d) U = f(P).
4
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