Single Phase Asynchronous Machine

Model dynamics of single phase asynchronous machine with squirrel-cage rotor

Library

Simscape / Electrical / Specialized Power Systems / Electrical Machines

Description

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.

Subscript	Definition
d	d axis quantity
q	q axis quantity
r	Referred to the main winding rotor quantity
R	Referred to the auxiliary winding rotor quantity
s	Main winding stator quantity
S	Auxiliary winding stator quantity
l	Leakage inductance
m	Magnetizing inductance

Electrical System

V_qs = R_si_qs + dφ_qs/dt		φ_qs = L_ssi_qs + L_msi'_qr
V_ds = R_Si_ds + dφ_ds/dt		φ_ds = L_SSi_ds + L_mSi'_dr
V'_qr = R'_ri'_qr + dφ'_qr/dt – (N_s/N_S)ω_rφ'_dr		φ'_qr = L'_ri'_qr + L_msi_qs
V'_dr = R'_Ri'_dr + dφ'_dr/dt + (N_S/N_s)ω_rφ'_qr	where	φ'_dr = L'_RRi'_dr + L_mSi_ds
T_e = p[(N_S/N_s)φ'_qri'_dr – (N_s/N_S)φ'_dri'_qr]		L_ss = L_ls + L_ms
		L_SS = L_lS + L_mS
		L'_rr = L'_lr + L_ms
		L'_RR = L'_lR + L_mS

Mechanical System

$\begin{matrix} \frac{d}{d t} ω_{m} = \frac{T_{e} - F ω_{m} - T_{m}}{2 H} \\ \frac{d}{d t} θ_{m} = ω_{m} . \end{matrix}$

Reference frame

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.

$\begin{matrix} [\begin{matrix} f_{q s} \\ f_{d s} \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}] [\begin{matrix} f_{a s} \\ f_{b s} \end{matrix}] \\ [\begin{matrix} f_{q r} \\ f_{d r} \end{matrix}] = [\begin{matrix} \cos (θ r) & - \sin (θ r)) \\ - \sin (θ r) & - \cos (θ r) \end{matrix}] [\begin{matrix} f_{a r} \\ f_{b r} \end{matrix}] . \end{matrix}$

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).

Parameter	Definition
R_s, L_ls	Main winding stator resistance and leakage inductance
R_S, L_lS	Auxiliary winding stator resistance and leakage inductance
R′_r, L′_lr	Main winding rotor resistance and leakage inductance
R′_R, L′_lR	Auxiliary winding rotor resistance and leakage inductance. The two values are equal to the main winding rotor resistance and leakage inductances values, respectively.
L_ms	Main winding magnetizing inductance
L_mS	Auxiliary winding magnetizing inductance
L_ss, L′_rr	Total main winding stator and rotor inductances
L_SS, L′_RR	Total auxiliary winding stator and rotor inductances
V_{as ,}i_as V_{bs ,}i_bs V_qs, i_qs	Main winding stator voltage and current Auxiliary winding stator voltage and current q axis stator voltage and current
V′_qr, i′_qr	q axis rotor voltage and current
V_ds, i_ds	d axis stator voltage and current
V′_dr, i′_dr	d axis rotor voltage and current
ϕ_qs, ϕ_ds	Stator q and d axis fluxes
ϕ′_qr, ϕ′_dr	Rotor q and d axis 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 x p)
Θ_r	Electrical rotor angular position (Θ_m x p)
T_e	Electromagnetic torque
T_m	Shaft mechanical torque
J F	Combined rotor and load inertia coefficient in (kg.m²). Set to infinite to simulate locked rotor. Combined rotor and load viscous friction coefficient.
H	Combined rotor and load inertia constant in (s). Set to infinite to simulate locked rotor.
N_s N_S R_st C_s R_run C_run	Number of main winding's effective turns. Number of auxiliary winding's effective turns. Capacitor-Start resistance Capacitor-Start Capacitor-Run resistance Capacitor-Run
N	Ratio of number of auxiliary winding's effective turns and number of main winding's effective turns.

Parameters

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™ Electrical™ Specialized 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.

Configuration Tab

Mechanical input

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.

Select 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.

Select 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.

Units

Specify the per unit dialog box or the SI dialog box. Default is SI.

Type of machine

Specify one of the four types of single phase asynchronous machines: Split Phase (default), Capacitor-Start, Capacitor-Start-Run, or Main & auxiliary windings.

Use signal names to identify bus labels

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.

Parameters Tab

Nominal power, voltage, and frequency: The nominal apparent power Pn (VA), RMS Vn (V), and frequency fn (Hz). Default is [ .25*746 110 60].
Main winding stator: The stator resistance R_s (Ω or pu) and leakage inductance L_ls (H or pu). Default is [2.02 7.4e-3] (SI) and [0.031135 0.042999] (pu).
Main winding rotor: The rotor resistance R_r' (Ω or pu) and leakage inductance L_lr' (H or pu), both referred to the stator. Default is [4.12 5.6e-3] (SI) and [0.063502 0.03254] (pu).
Main winding mutual inductance: The magnetizing inductance L_ms (H or pu). Default is 0.1772 (SI) and 1.0296 (pu).
Auxiliary winding stator: The stator resistance R_S (Ω or pu) and leakage inductance L_lS (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 is [7.14 8.5e-3] (SI) and [0.11005 0.049391] (pu).
Inertia, friction factor, pole pairs, turn ratio (aux/main): 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) and [1.3907 0 2 1.18] (pu).
Capacitor-Start: The start capacitance C_s (farad or pu) and capacitor series resistance R_st(Ω or pu). Default is [2 254.7e-6] (SI) and [0.030826 6.2297] (pu).
Capacitor-Run: The run capacitance Crun (farad or pu) and series resistance Rrun (farad or pu). Default is [18 21.1e-6] (SI) and [0.27744 0.51608] (pu).
Disconnection speed: Specifies the speed (%) when the auxiliary winding may be disconnected. Default is 75.
Initial speed: Specifies the initial speed (%). Default is 0.

Advanced Tab

Sample time (−1 for inherited): Specifies the sample time used by the block. To inherit the sample time specified in the Powergui block, set this parameter to −1 (default).

Inputs and Outputs

Tm: 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.

m

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.

Name	Definition	Units
iar	Rotor current ir_a	A or pu
ibr	Rotor current ir_b	A or pu
iqr	Rotor current iq	A or pu
idr	Rotor current id	A or pu
phiqr	Rotor flux phir_q	V.s or pu
phidr	Rotor flux phir_d	V.s or pu
ias	Main winding stator current ia	A or pu
ibs	Auxiliary winding stator current ib	A or pu
phiqs	Stator flux phis_q(V.s)	V.s or pu
phids	Stator flux phis_d(V.s)	V.s or pu
vc	Voltage capacitor Vc	V or pu
w	Rotor speed	rad/s
Te	Electromagnetic torque Te	N.m or pu
theta	Rotor angle thetam	rad

Limitations

The Single Phase Asynchronous Machine block does not include a representation of iron losses and saturation.

Examples

The Single-Phase Asynchronous Machine example shows the use of the Single Phase Asynchronous Machine block in two modes of operation.

References

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