SimPowerSystems

Asynchronous Machine

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

Library

Machines

Description

The Asynchronous Machine block operates in either generating or motoring mode. The mode of operation is dictated by the sign of the mechanical torque (positive for motoring, negative for generating). 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

Electrical System

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
qs, ds	Stator q and d axis fluxes
'qr, 'dr	Rotor q and d axis fluxes
m	Angular velocity of the rotor
m	Rotor angular position
p	Number of pole pairs
r	Electrical angular velocity (m x p)
r	Electrical rotor angular position (m 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

Parameters and Dialog Boxes

You can choose between two Asynchronous Machine blocks to specify the electrical and mechanical parameters of the model.

Note    Depending on the dialog box you choose to use, Power System Blockset automatically converts the parameters you enter into per unit parameters. The Simulink model of the Asynchronous Machine block uses p.u. parameters.

S.I. Units Dialog Box

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 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, 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 and in each reference frame (_e is the position of the synchronously rotating reference frame).

Reference Frame
Rotor	r	0
Stationary	0	-r
Synchronous	e	e - r

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

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

Stator

The stator resistance Rs ( or p.u.) and leakage inductance Lls (H or p.u.).

Rotor

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

Magnetizing inductance

The magnetizing inductance Lm (H or p.u.).

Mechanical

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.

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

Initial conditions

Specifies the initial slip s, electrical angle e (deg), stator current magnitude (A or p.u.), and phase angles (deg):

[ slip, th, i_as, i_bs, i_cs, phase_as, phase_bs, phase_cs ]

You can also specify optional initial values for the rotor current magnitude (A) or (p.u.), and phase angles (deg):

[ slip, th, i_as, i_bs, i_cs, ph_as, ph_bs, ph_cs, i_ar, i_br, i_cr, phase_ar, phase_br, phase_cr ]

The initial conditions can be computed by the load flow utility in the Powergui block.

Inputs and Outputs

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.

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 only use two sources connected in series.

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

The Simulink output of the block is a vector containing 21 variables. They are, in order (refer to the preceding description section, all currents flowing into machine).

Input	Definition
1 to 3	Rotor currents i'ra, i'rb, and i'rc
4 to 9	i'qr, i'dr, 'qr, 'dr, v'qr, and v'd
10 to 12	Stator currents isa, isb and isc
13 to 18	iqs, ids, qs, ds, vqs, and vds
19 to 21	m, Te, and m

You can demultiplex these variables by using the Machines Measurement Demux block provided in the Machines library.

Limitations

The Asynchronous Machine block does not include a representation of the effects of stator and rotor iron saturation.

Example

The psbpwm.mdl demo illustrates the use of the Asynchronous Machine block in motoring 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 m_f of 33 (60 Hz x 33 = 1980). It is recommended in [2] that m_f 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 p.u. 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 psbpwm 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 p.u.). 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.

References

[1] Krause, P.C., O. Wasynczuk, and S.D. Sudhoff, Analysis of Electric Machinery, IEEE Press, 1995.

[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.

AC Voltage Source

Breaker