# Asynchronous Machine

Model the dynamics of three-phase asynchronous machine, also known as induction machine

Machines

## Description

The Asynchronous Machine 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 fourth-order state-space model and the mechanical part by a second-order system. All electrical variables and parameters are referred to the stator. This is indicated by the prime signs in the machine equations given below. All stator and rotor quantities are in the arbitrary two-axis reference frame (dq frame). The subscripts used are defined as follows:

Subscript

Definition

d

d axis quantity

q

q axis quantity

r

Rotor quantity

s

Stator quantity

l

Leakage inductance

m

Magnetizing inductance

### Mechanical System

The Asynchronous Machine block parameters are defined as follows (all quantities are referred to the stator):

Parameter

Definition

Rs, Lls

Stator resistance and leakage inductance

R'r, L'lr

Rotor resistance and leakage inductance

Lm

Magnetizing inductance

Ls, L'r

Total stator and rotor inductances

Vqs, iqs

q axis stator voltage and current

V'qr, i'qr

q axis rotor voltage and current

Vds, ids

d axis stator voltage and current

V'dr, i'dr

d axis rotor voltage and current

ϕqs, ϕds

Stator q and d axis fluxes

ϕ'qr, ϕ'dr

Rotor q and d axis fluxes

ωm

Angular velocity of the rotor

Θm

Rotor angular position

p

Number of pole pairs

ωr

Electrical angular velocity (ωm x p)

Θr

Electrical rotor angular position (Θm x p)

Te

Electromagnetic torque

Tm

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

## Dialog Box and Parameters

You can choose between two Asynchronous Machine blocks to specify the electrical and mechanical parameters of the model, by using the pu Units dialog box or the SI dialog box. Both blocks are modeling the same asynchronous machine model. Depending on the dialog box you choose to use, SimPowerSystems automatically converts the parameters you enter into per unit parameters. The Simulink model of the Asynchronous Machine block uses pu parameters.

Preset model

Provides a set of predetermined electrical and mechanical parameters for various asynchronous machine ratings of power (HP), phase-to-phase voltage (V), frequency (Hz), and rated speed (rpm).

Select one of the preset models to load the corresponding electrical and mechanical parameters in the entries of the dialog box. Select No if you don't want to use a preset model. Note that the preset models do not include predetermined saturation parameters.

Select the Show detailed parameters parameter to view and edit the detailed parameters associated with the preset model.

Mechanical input

Allows you to select either the torque applied to the shaft or the rotor speed as the Simulink signal applied to the block's input.

Select Torque Tm to specify a torque input, in N.m or in pu, and change labeling of the block's input to Tm. The machine speed is determined by the machine Inertia J (or inertia constant H for the pu machine) and by the difference between the applied mechanical torque Tm and the internal electromagnetic torque Te. The sign convention for the mechanical torque is the following: when the speed is positive, a positive torque signal indicates motor mode and a negative signal indicates generator mode.

Select Speed w to specify a speed input, in rad/s or in pu, and change labeling of the block's input to w. The machine speed is imposed and the mechanical part of the model (Inertia J) is ignored. Using the speed as the mechanical input allows modeling a mechanical coupling between two machines and interfacing with SimMechanics and SimDriveline.

The next figure indicates how to model a stiff shaft interconnection in a motor-generator set when friction torque is ignored in machine 2. The speed output of machine 1 (motor) is connected to the speed input of machine 2 (generator), while machine 2 electromagnetic torque output Te is applied to the mechanical torque input Tm of machine 1. The Kw factor takes into account speed units of both machines (pu or rad/s) and gear box ratio w2/w1. The KT factor takes into account torque units of both machines (pu or N.m) and machine ratings. Also, as the inertia J2 is ignored in machine 2, J2 referred to machine 1 speed must be added to machine 1 inertia J1.

Show detailed parameters

If selected, the mask displays the detailed parameters of the Asynchronous Machine block. The detailed parameters can be modified no matter the preset model you selected in the Preset Model list.

Rotor type

Specifies the branching for the rotor windings.

Reference frame

Specifies the reference frame that is used to convert input voltages (abc reference frame) to the dq reference frame, and output currents (dq reference frame) to the abc reference frame. You can choose among the following reference frame transformations:

• Rotor (Park transformation)

• Stationary (Clarke or αβ transformation)

• Synchronous

The following relationships describe the abc-to-dq reference frame transformations applied to the Asynchronous Machine phase-to-phase voltages.

In the preceding equations, Θ is the angular position of the reference frame, while is the difference between the position of the reference frame and the position (electrical) of the rotor. Because the machine windings are connected in a three-wire Y configuration, there is no homopolar (0) component. This also justifies the fact that two line-to-line input voltages are used inside the model instead of three line-to-neutral voltages. The following relationships describe the dq-to-abc reference frame transformations applied to the Asynchronous Machine phase currents.

The following table shows the values taken by Θ and β in each reference frame (Θe is the position of the synchronously rotating reference frame).

Reference Frame

Θ

β

Rotor

Θr

0

Stationary

0

r

Synchronous

Θe

Θe - Θr

The choice of reference frame affects the waveforms of all dq variables. It also affects the simulation speed and in certain cases the accuracy of the results. The following guidelines are suggested in [1]:

• Use the stationary reference frame if the stator voltages are either unbalanced or discontinuous and the rotor voltages are balanced (or 0).

• Use the rotor reference frame if the rotor voltages are either unbalanced or discontinuous and the stator voltages are balanced.

• Use either the stationary or synchronous reference frames if all voltages are balanced and continuous.

Nominal power, L-L volt, and freq.

The nominal apparent power Pn (VA), RMS line-to-line voltage Vn (V), and frequency fn (Hz).

Stator

The stator resistance Rs (Ω or pu) and leakage inductance Lls (H or pu).

Rotor

The rotor resistance Rr' (Ω or pu) and leakage inductance Llr' (H or pu), both referred to the stator.

Mutual inductance

The magnetizing inductance Lm (H or pu).

Inertia, friction factor, and pairs of poles

For the SI units dialog box: the combined machine and load inertia coefficient J (kg.m2), combined viscous friction coefficient F (N.m.s), and pole pairs p. The friction torque Tf is proportional to the rotor speed ω (Tf = F.w).

For the pu units dialog box: the inertia constant H (s), combined viscous friction coefficient F (pu), and pole pairs p.

Initial conditions

Specifies the initial slip s, electrical angle Θe (degrees), stator current magnitude (A or pu), and phase angles (degrees):

[slip, th, ias, ibs, ics, phaseas, phasebs, phasecs]

For the wound-rotor machine, you can also specify optional initial values for the rotor current magnitude (A or pu), and phase angles (degrees):

``` [slip, th, ias, ibs, ics, phaseas, phasebs, phasecs, iar, ibr, icr,
phasear, phasebr, phasecr]
```

For the squirrel cage machine, the initial conditions can be computed by the load flow utility in the Powergui block.

Simulate saturation

Specifies whether magnetic saturation of rotor and stator iron is simulated or not.

Saturation parameters

Specifies the no-load saturation curve parameters. Magnetic saturation of stator and rotor iron (saturation of the mutual flux) is modeled by a nonlinear function (in this case a polynomial) using points of the no-load saturation curve. You must enter a 2-by-n matrix, where n is the number of points taken from the saturation curve. The first row of this matrix contains the values of stator currents, while the second row contains values of corresponding terminal voltages (stator voltages). The first point (first column of the matrix) must correspond to the point where the effect of saturation begins.

You must select the Simulate saturation check box to simulate saturation. If the Simulate saturation is not selected, the relationship between the stator current and the stator voltage is linear.

## Inputs and Outputs

Tm

The Simulink input of the block is the mechanical torque at the machine's shaft. When the input is a positive Simulink signal, the asynchronous machine behaves as a motor. When the input is a negative signal, the asynchronous machine behaves as a generator.

When you use the SI parameters mask, the input is a signal in N.m, otherwise it is in pu.

m

The Simulink output of the block is a vector containing 21 signals. You can demultiplex these signals by using the Bus Selector block provided in the Simulink library. Depending on the type of mask you use, the units are in SI, or in pu.

Signal

Definition

Units

Symbol

1

Rotor current ir_a

A or pu

i'ra

2

Rotor current ir_b

A or pu

i'rb

3

Rotor current ir_c

A or pu

i'rc

4

Rotor current iq

A or pu

i'qr

5

Rotor current id

A or pu

i'dr

6

Rotor flux phir_q

V.s or pu

ϕ'qr

7

Rotor flux phir_d

V.s or pu

ϕ'dr

8

Rotor voltage Vr_q

V or pu

v'qr

9

Rotor voltage Vr_d

V or pu

v'd

10

Stator current is_a

A or pu

isa

11

Stator current is_b

A or pu

isb

12

Stator current is_c

A or pu

isc

13

Stator current is_q

A or pu

iqs

14

Stator current is_d

A or pu

ids

15

Stator flux phis_q

V.s or pu

ϕqs

16

Stator flux phis_d

V.s or pu

ϕds

17

Stator voltage vs_q

V or pu

vqs

18

Stator voltage vs_d

V or pu

vds

19

Rotor speed

ωm

20

Electromagnetic torque Te

N.m or pu

Te

21

Rotor angle thetam

Θm

The stator terminals of the Asynchronous Machine block are identified by the A, B, and C letters. The rotor terminals are identified by the a, b, and c letters. Note that the neutral connections of the stator and rotor windings are not available; three-wire Y connections are assumed.

## Limitations

1. The Asynchronous Machine block does not include a representation of the saturation of leakage fluxes. You must be careful when you connect ideal sources to the machine's stator. If you choose to supply the stator via a three-phase Y-connected 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.

2. When you use Asynchronous Machine blocks in discrete systems, you might have to use a small parasitic resistive load, connected at the machine terminals, in order to avoid numerical oscillations. Large sample times require larger loads. The minimum resistive load is proportional to the sample time. As a rule of thumb, remember that 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 should be sufficient.

## Examples

### Example 1: Use of the Asynchronous Machine Block in Motor Mode

The power_pwm demo illustrates the use of the Asynchronous Machine block in motor mode. It consists of an asynchronous machine in an open-loop speed control system.

The machine's rotor is short-circuited, 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 pulse-width modulation, which is described in [2]. The base frequency of the sinusoidal reference wave is set at 60 Hz and the triangular carrier wave's frequency is set at 1980 Hz. This corresponds to a frequency modulation factor mf of 33 (60 Hz x 33 = 1980). It is recommended in [2] that mf be an odd multiple of three and that the value be as high as possible.

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 second.

The parameters of the machine are those found in the SI Units dialog box above, except for the stator leakage inductance, which is set to twice its normal value. This is done 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 demo. Note in the simulation parameters that 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 also be seen in the stator and rotor currents, which are observed next.

Finally, 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 demo illustrates the effect of saturation of the Asynchronous Machine block.

Two identical three-phase motors (50 HP, 460 V, 1800 rpm) are simulated with and without saturation, to observe the saturation effects on the stator currents. Two different simulations are realized in the demo.

The first simulation, is the no-load steady-state test. The table below contains the values of the Saturation Parameters and the measurements obtained by simulating different operating points on the saturated motor (no-load and in steady-state).

Saturation Parameters

Measurements

Vsat (Vrms L-L)

Isat (peak A)

Vrms L-L

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 graph below illustrates these results and shows the accuracy of the saturation model. As you can see, the measured operating points fit well the curve that is plotted from the Saturation Parameters data.

Running the simulation with a blocked rotor or with many different values of load torque will allow the observation of other effects of saturation on the stator currents.

## 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, Section8.4.1.