Asynchronous Machine
Model dynamics of threephase asynchronous machine, also known as induction machine, in SI or pu units
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Simscape / Electrical / Specialized Power Systems / Electrical Machines
Description
The Asynchronous Machine SI Units and Asynchronous Machine pu Units blocks implement a threephase asynchronous machine (wound rotor, squirrel cage, or double squirrel cage) modeled in a selectable dq reference frame (rotor, stator, or synchronous). Stator and rotor windings are connected in wye to an internal neutral point. The block operates in either generator or motor mode. The mode of operation is dictated by the sign of the mechanical torque:
If Tm is positive, the machine acts as a motor.
If Tm is negative, the machine acts as a generator.
The electrical part of the machine is represented by a fourthorder (or sixthorder for the double squirrelcage machine) statespace model, and the mechanical part by a secondorder system. All electrical variables and parameters are referred to the stator, which is indicated by the prime signs in the following machine equations. All stator and rotor quantities are in the arbitrary twoaxis reference frame (dq frame). The subscripts used are defined in this table.
Subscript 
Definition 

d 
daxis quantity 
q 
qaxis quantity 
r 
Rotor quantity (woundrotor or singlecage) 
r1 
Cage 1 rotor quantity (doublecage) 
r2 
Cage 2 rotor quantity (doublecage) 
s 
Stator quantity 
l 
Leakage inductance 
m 
Magnetizing inductance 
Electrical System of the WoundRotor or SquirrelCage Machine
V_{qs} = R_{s}i_{qs} + dφ_{qs}/dt + ωφ_{ds}
V_{ds} = R_{s}i_{ds} + dφ_{ds}/dt – ωφ_{qs}
V'_{qr} = R'_{r}i'_{qr} + dφ'_{qr}/dt + (ω – ω_{r})φ'_{dr}
V'_{dr} = R'_{r}i'_{dr} + dφ'_{dr}/dt – (ω – ω_{r})φ'_{qr}
T_{e} = 1.5p(φ_{ds}i_{qs} – φ_{qs}i_{ds})
ω — Reference frame angular velocity
ω_{r} — Electrical angular velocity
φ_{qs} = L_{s}i_{qs} + L_{m}i'_{qr}
φ_{ds} = L_{s}i_{ds} + L_{m}i'_{dr}
φ'_{qr} = L'_{r}i'_{qr} + L_{m}i_{qs}
φ'_{dr} = L'_{r}i'_{dr} + L_{m}i_{ds}
L_{s} = L_{ls} + L_{m}
L'_{r} = L'_{lr} + L_{m}
Electrical System of the Double SquirrelCage Machine
V_{qs} = R_{s}i_{qs} + dφ_{qs}/dt + ωφ_{ds}
V_{ds} = R_{s}i_{ds} + dφ_{ds}/dt – ωφ_{qs}
0 = R'_{r1}i'_{qr1} + dφ'_{qr1}/dt + (ω – ω_{r})φ'_{dr1}
0 = R'_{r1}i'_{dr1} + dφ'_{dr1}/dt – (ω – ω_{r})φ'_{qr1}
0 = R'_{r2}i'_{qr2} + dφ'_{qr2}/dt + (ω – ω_{r})φ'_{dr2}
0 = R'_{r2}i'_{dr2} + dφ'_{dr2}/dt – (ω – ω_{r})φ'_{qr2}
T_{e} = 1.5p(φ_{ds}i_{qs} – φ_{qs}i_{ds})
φ_{qs} = L_{s}i_{qs} + L_{m}(i'_{qr1} + i'_{qr2})
φ_{ds} = L_{s}i_{ds} + L_{m}(i'_{dr1} + i'_{dr2})
φ'_{qr1} = L'_{r1}i'_{qr1} + L_{m}i_{qs}
φ'_{dr1} = L'_{r1}i'_{dr1} + L_{m}i_{ds}
φ'_{qr2} = L'_{r2}i'_{qr2} + L_{m}i_{qs}
φ'_{dr2} = L'_{r2}i'_{dr2} + L_{m}i_{ds}
L_{s} = L_{ls} + L_{m}
L'_{r1} = L'_{lr1} + L_{m}
L'_{r2} = L'_{lr2} + L_{m}
Mechanical System
$$\begin{array}{c}\frac{d}{dt}{\omega}_{m}=\frac{1}{2H}\left({T}_{e}F{\omega}_{m}{T}_{m}\right)\\ \frac{d}{dt}{\theta}_{m}={\omega}_{m}\end{array}$$
The Asynchronous Machine block parameters are defined in the table. All quantities are referred to the stator.
Parameters Common to All Models 
Definition 

R_{s}, L_{ls} 
Stator resistance and leakage inductance 
L_{m} 
Magnetizing inductance 
L_{s} 
Total stator inductance 
V_{qs}, i_{qs} 
qaxis stator voltage and current 
V_{ds}, i_{ds} 
daxis stator voltage and current 
ϕ_{qs}, ϕϕ_{ds} 
Stator qaxis and daxis fluxes 
ω_{m} 
Angular velocity of the rotor 
Θ_{m} 
Rotor angular position 
p 
Number of pole pairs 
ω_{r} 
Electrical angular velocity (ω_{m} × p) 
Θ_{r} 
Electrical rotor angular position (Θ_{m} × p) 
T_{e} 
Electromagnetic torque 
T_{m} 
Shaft mechanical torque 
J 
Combined rotor and load inertia coefficient. Set to infinite to simulate locked rotor. 
H 
Combined rotor and load inertia constant. Set to infinite to simulate locked rotor. 
F 
Combined rotor and load viscous friction coefficient 
Parameters Specific to SingleCage or Wound Rotor 
Definition 

L'_{r} 
Total rotor inductance 
R'_{r}, L'_{lr} 
Rotor resistance and leakage inductance 
V'_{qr}, i'_{qr} 
qaxis rotor voltage and current 
V'_{dr}, i'_{dr} 
daxis rotor voltage and current 
ϕ'_{qr}, ϕ'_{dr} 
Rotor qaxis and d axis fluxes 
Parameters Specific to DoubleCage Rotor 
Definition 

R'_{r1}, L'_{lr1} 
Rotor resistance and leakage inductance of cage 1 
R'_{r2}, L'_{lr2} 
Rotor resistance and leakage inductance of cage 2 
L'_{r1}, L'_{r2} 
Total rotor inductances of cage 1 and 2 
i'_{qr1}, i'_{qr2} 
qaxis rotor current of cage 1 and 2 
i'_{dr1}, i'_{dr2} 
daxis rotor current of cage 1 and 2 
ϕ'_{qr1}, ϕ'_{dr1} 
qaxis and daxis rotor fluxes of cage 1 
ϕ'_{qr2}, ϕ'_{dr2} 
qaxis and daxis rotor fluxes of cage 2 
Assumptions and Limitations

The Asynchronous Machine blocks do not include a representation of the saturation of leakage fluxes. Be careful when you connect ideal sources to the stator of the machine. If you choose to supply the stator via a threephase, Yconnected infinite voltage source, you must use three sources connected in Y. However, if you choose to simulate a delta source connection, you must use only two sources connected in series.
When you use Asynchronous Machine blocks in discrete systems, you might have to connect a small parasitic resistive load at the machine terminals to avoid numerical oscillations. Large sample times require larger loads. The optimum resistive load is proportional to the sample time. With a 25 μs time step on a 60 Hz system, the minimum load is approximately 2.5% of the machine nominal power. For example, a 200 MVA asynchronous machine in a power system discretized with a 50 μs sample time requires approximately 5% of resistive load or 10 MW. If the sample time is reduced to 20 μs, a resistive load of 4 MW is sufficient.
Ports
The stator terminals of the Asynchronous Machine blocks are identified by the letters A, B, and C. The rotor terminals are identified by the letters a, b, and c. The neutral connections of the stator and rotor windings are not available. Threewire Y connections are assumed.
Input
Output
Conserving
Parameters
Examples
Example 1: Use of the Asynchronous Machine Block in Motor Mode
The power_pwm example uses a Asynchronous Machine block in motor mode. The example consists of an asynchronous machine in an openloop speed control system.
The machine rotor is shortcircuited, and the stator is fed by a PWM inverter built with Simulink blocks and interfaced to the Asynchronous Machine block through the Controlled Voltage Source block. The inverter uses sinusoidal pulsewidth modulation. The base frequency of the sinusoidal reference wave is set at 60 Hz and the triangular carrier wave frequency is set at 1980 Hz. This frequency corresponds to a frequency modulation factor m_{f} of 33 (60 Hz x 33 = 1980).
The 3 HP machine is connected to a constant load of nominal value (11.9 N.m). It is started and reaches the set point speed of 1.0 pu at t = 0.9 seconds.
The parameters of the machine are the same as the Asynchronous Machine SI Units block, except for the stator leakage inductance, which is set to twice the normal value to simulate a smoothing inductor placed between the inverter and the machine. Also, the stationary reference frame was used to obtain the results shown.
Open the power_pwm
example. In the simulation parameters, a small relative tolerance is required
because of the high switching rate of the inverter.
Run the simulation and observe the machine's speed and torque.
The first graph shows the machine's speed going from 0 to 1725 rpm (1.0 pu). The second graph shows the electromagnetic torque developed by the machine. Because the stator is fed by a PWM inverter, a noisy torque is observed.
However, this noise is not visible in the speed because it is filtered out by the machine's inertia, but it can be seen in the stator and rotor currents.
Look at the output of the PWM inverter. Because nothing of interest can be seen at the simulation time scale, the graph concentrates on the last moments of the simulation.
Example 2: Effect of Saturation of the Asynchronous Machine Block
The power_asm_sat
example illustrates the effect of saturation of
the Asynchronous Machine block.
Two identical threephase motors (50 HP, 460 V, and 1800 rpm) are simulated, with and without saturation, to observe the saturation effects on the stator currents. Two different simulations are realized in the example.
The first simulation is the noload steadystate test. This table contains the values of the saturation parameters and the measurements obtained by simulating different operating points on the saturated motor (noload and in steadystate).
Saturation Parameters 
Measurements 


Vsat (Vrms LL) 
Isat (peak A) 
Vrms LL 
Is_A (peak A) 
 
 
120 
7.322 
230 
14.04 
230 
14.03 
 
 
250 
16.86 
 
 
300 
24.04 
322 
27.81 
322 
28.39 
 
 
351 
35.22 
 
 
382 
43.83 
414 
53.79 
414 
54.21 
 
 
426 
58.58 
 
 
449 
67.94 
460 
72.69 
460 
73.01 
 
 
472 
79.12 
 
 
488 
88.43 
506 
97.98 
506 
100.9 
 
 
519 
111.6 
 
 
535 
126.9 
 
 
546 
139.1 
552 
148.68 
552 
146.3 
 
 
569 
169.1 
 
 
581 
187.4 
598 
215.74 
598 
216.5 
 
 
620 
259.6 
 
 
633 
287.8 
644 
302.98 
644 
313.2 
 
 
659 
350 
 
 
672 
383.7 
 
 
681 
407.9 
690 
428.78 
690 
432.9 
The next graph illustrates these results and shows the accuracy of the saturation model. The measured operating points fit well the curve that is plotted from the saturation parameters data.
You can observe the other effects of saturation on the stator currents by running the simulation with a blocked rotor or with many different values of load torque.
References
[1] Krause, P.C., O. Wasynczuk, and S.D. Sudhoff, Analysis of Electric Machinery, IEEE^{®} Press, 2002.
[2] Mohan, N., T.M. Undeland, and W.P. Robbins, Power Electronics: Converters, Applications, and Design, John Wiley & Sons, Inc., New York, 1995, Section 8.4.1.