Model dynamics of single phase asynchronous machine with squirrel-cage rotor
This machine has two windings: main and auxiliary. With the model, you can simulate the split-phase, the capacitor-start, the capacitor-start-capacitor-run, and main & auxiliary windings operation modes.
For the split-phase mode, the main and auxiliary windings are internally connected as follows:
For the capacitor-start mode, the main and auxiliary windings are internally connected as follows:
For the capacitor-start-capacitor-run mode, the main and auxiliary windings are internally connected as follows:
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, indicated by the following prime signs in the machine equations. All stator and rotor quantities are in the stator reference frame (dq frame). The subscripts are defined in the following table.
d axis quantity
q axis quantity
Referred to the main winding rotor quantity
Referred to the auxiliary winding rotor quantity
Main winding stator quantity
Auxiliary winding stator quantity
|Vqs = Rsiqs + dφqs/dt||φqs = Lssiqs + Lmsi'qr|
|Vds = RSids + dφds/dt||φds = LSSids + LmSi'dr|
|V'qr = R'ri'qr + dφ'qr/dt – (Ns/NS)ωrφ'dr||φ'qr = L'ri'qr + Lmsiqs|
|V'dr = R'Ri'dr + dφ'dr/dt + (NS/Ns)ωrφ'qr||where||φ'dr = L'RRi'dr + LmSids|
|Te = p[(NS/Ns)φ'qri'dr – (Ns/NS)φ'dri'qr]||Lss = Lls + Lms|
|LSS = LlS + LmS|
|L'rr = L'lr + Lms|
|L'RR = L'lR + LmS|
The reference frame fixed in the stator converts voltages and currents to the dq frame.
The following relationships describe the ab-to-dq frame transformations applied to the single phase asynchronous machine.
The variable f can represent either voltage, currents, or flux linkage.
The single phase asynchronous machine block parameters are defined as follows (all quantities are referred to the stator).
Main winding stator resistance and leakage inductance
Auxiliary winding stator resistance and leakage inductance
Main winding rotor resistance and leakage inductance
Auxiliary winding rotor resistance and leakage inductance. The two values are equal to the main winding rotor resistance and leakage inductances values, respectively.
Main winding magnetizing inductance
Auxiliary winding magnetizing inductance
Total main winding stator and rotor inductances
Total auxiliary winding stator and rotor inductances
Main winding stator voltage and current
Auxiliary winding stator voltage and current
q axis stator voltage and current
q axis rotor voltage and current
d axis stator voltage and current
d axis rotor voltage and current
Stator q and d axis fluxes
Rotor q and d axis fluxes
Angular velocity of the rotor
Rotor angular position
Number of pole pairs
Electrical angular velocity (ωm x p)
Electrical rotor angular position (Θm x p)
Shaft mechanical torque
Combined rotor and load inertia coefficient in (kg.m2). Set to infinite to simulate locked rotor.
Combined rotor and load viscous friction coefficient.
Combined rotor and load inertia constant in (s). Set to infinite to simulate locked rotor.
Number of main winding's effective turns.
Number of auxiliary winding's effective turns.
Ratio of number of auxiliary winding's effective turns and number of main winding's effective turns.
You can choose between two types of units to specify the electrical and mechanical parameters of the model, the per unit dialog box, and the SI dialog box. Both blocks are modeling the same machine. Depending on the dialog box that you use, Simscape™ Power Systems™ automatically converts the parameters that you specify into per unit parameters. The Simulink® model of the Single Phase Asynchronous Machine block uses per unit parameters.
Select the torque applied to the shaft as a Simulink input of the block, or to represent the machine shaft by a Simscape rotational mechanical port.
Torque Tm (default) to specify
a torque input, in N.m or in pu, and change labeling of the block
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 when the
speed is positive, a positive torque signal indicates motor mode and
a negative signal indicates generator mode.
Mechanical rotational port to
add to the block a Simscape mechanical rotational port that allows
connection of the machine shaft with other Simscape blocks that
have mechanical rotational ports. The Simulink input representing
the mechanical torque Tm of the machine is then removed from the block.
The next figure indicates how to connect an Ideal Torque Source block from the Simscape library to the machine shaft to represent the machine in motor mode, or in generator mode, when the rotor speed is positive.
Specify the per unit dialog box or the SI dialog box. Default
Specify one of the four types of single phase asynchronous machines:
Main & auxiliary windings.
When this check box is selected, the measurement output uses the signal names to identify the bus labels. Select this option for applications that require bus signal labels to have only alphanumeric characters.
When this check box is cleared (default), the measurement output uses the signal definition to identify the bus labels. The labels contain nonalphanumeric characters that are incompatible with some Simulink applications.
The nominal apparent power Pn (VA), RMS Vn (V), and frequency
fn (Hz). Default is
[ .25*746 110 60].
The stator resistance Rs (Ω or
pu) and leakage inductance Lls (H or pu). Default
[2.02 7.4e-3] (SI) and
[0.031135 0.042999] (pu).
The rotor resistance Rr' (Ω or
pu) and leakage inductance Llr' (H or pu),
both referred to the stator. Default is
[4.12 5.6e-3] (SI)
[0.063502 0.03254] (pu).
The magnetizing inductance Lms (H or
pu). Default is
0.1772 (SI) and
The stator resistance RS (Ω or
pu) and leakage inductance LlS (H or pu). Note
that the Auxiliary winding rotor parameters are assumed to be equal
to the main winding rotor resistance and leakage inductances values.
Therefore it is not required to specify them in the dialog box. Default
[7.14 8.5e-3] (SI) and
[0.11005 0.049391] (pu).
For the SI units dialog box:
the combined machine and load inertia coefficient J (kg.m2),
the combined viscous friction coefficient F (N.m.s), the number of
pole pairs p and ratio of number of auxiliary winding's effective
turns, and the number of main winding's effective turns. pu units dialog box: the inertia constant H
(s), the combined viscous friction coefficient F (pu), and the number
of pole pairs p. Default is
[0.0146 0 2 1.18] (SI)
[1.3907 0 2 1.18] (pu).
The start capacitance Cs (farad or pu)
and capacitor series resistance Rst(Ω
or pu). Default is
[2 254.7e-6] (SI) and
The run capacitance Crun (farad or pu) and series resistance
Rrun (farad or pu). Default is
[18 21.1e-6] (SI)
[0.27744 0.51608] (pu).
Specifies the speed (%) when the auxiliary winding may be disconnected.
Specifies the initial speed (%). Default is
Specifies the sample time used by the block. To inherit the
sample time specified in the Powergui block, set this parameter to
The Simulink input of the block is the mechanical torque at the machine shaft. When you use the SI parameters mask, the input is a signal in N.m; otherwise it is in pu.
The Simulink output of the block is a vector containing measurement 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.
Rotor current ir_a
A or pu
Rotor current ir_b
A or pu
Rotor current iq
A or pu
Rotor current id
A or pu
Rotor flux phir_q
V.s or pu
Rotor flux phir_d
V.s or pu
Main winding stator current ia
A or pu
Auxiliary winding stator current ib
A or pu
Stator flux phis_q(V.s)
V.s or pu
Stator flux phis_d(V.s)
V.s or pu
Voltage capacitor Vc
V or pu
Electromagnetic torque Te
N.m or pu
Rotor angle thetam
The Single Phase Asynchronous Machine block does not include a representation of iron losses and saturation.
shows the use of the Single Phase Asynchronous Machine block
in two modes of operation.
 Krause, P.C., O. Wasynczuk, and S.D. Sudhoff, Analysis of Electric Machinery, IEEE Press, 1995.