Single Phase Asynchronous Machine - Model the dynamics of single phase asynchronous machine with squirrel-cage rotor

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Machines

Description

This machine has two windings: the main winding and the auxiliary winding. With the model, you can simulate the split-phase, the capacitor-start, the capacitor-start-capacitor-run, and main & auxiliary windings operation modes.

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

Reference frame

The reference frame fixed in the stator is used to convert 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).

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
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

Dialog Box and Parameters

You can choose between two types of unit 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 choose to use, SimPowerSystems automatically converts the parameters that you enter into per unit parameters. The Simulink model of the single phase asynchronous machine block uses per unit parameters.

Configuration Tab

Units

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

Type of machine

Specify one of the four types of single phase asynchronous machine (the split-phase, the capacitor-start, the capacitor-start-capacitor-run, or the main & auxiliary windings).

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 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 having 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.

Parameters Tab

Nominal power, voltage, and frequency: The nominal apparent power Pn (VA), RMS Vn (V), and frequency fn (Hz).
Main winding stator: The stator resistance R_s (Ω or pu) and leakage inductance L_ls (H or 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.
Main winding mutual inductance: The magnetizing inductance L_ms (H or pu).
Auxiliary winding stator: The stator resistance R_S (Ω or pu) and leakage inductance L_lS (H or 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), combined viscous friction coefficient F (N.m.s), number of pole pairs p and ratio of number of auxiliary winding's effective turns, and number of main winding's effective turns. pu units dialog box: the inertia constant H (s), combined viscous friction coefficient F (pu), and number of pole pairs p.
Capacitor-Start: The start capacitance C_s (farad or pu) and capacitor series resistance R_st(Ω or pu).
Capacitor-Run: The run capacitance Crun (farad or pu) and series resistance Rrun (farad or pu).
Disconnection speed: Specifies the speed (%) when the auxiliary winding may be disconnected.
Initial speed: Specifies the initial speed (%).

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.

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 16 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 iq	A or pu	i'_qr
4	Rotor current id	A or pu	i'_dr
5	Rotor flux phir_q	V.s or pu	ϕ'_qr
6	Rotor flux phir_d	V.s or pu	ϕ'_dr
7	Rotor voltage Vr_q	V or pu	v'_qr
8	Rotor voltage Vr_d	V or pu	v'_d
9	Main winding stator current ia	A or pu	i_a
10	Auxiliary winding stator current ib	A or pu	i_b
11	Voltage capacitor Vc	V or pu	V_c
12	Stator flux phis_q(V.s)	V.s or pu	ϕ_qs
13	Stator flux phis_d(V.s)	V.s or pu	ϕ_ds
14	Rotor speed	rad/s	ω_m
15	Electromagnetic torque Te	N.m or pu	T_e
16	Rotor angle thetam	rad	Θ_m

Limitations

The single phase asynchronous machine block does not include a representation of iron losses and saturation.

Example

This example, available in the power_singlephaseASM demo, illustrates the use of the Single Phase Asynchronous Machine block in two modes of operation.

It consists of a single phase asynchronous machine in an open-loop speed control system. The main and auxiliary windings are fed by a single phase power supply. The motor is started at no-load and a step of 1N.m is applied at 2 seconds.

Capacitor-Start Operation Mode

The next series of figures shows the capacitor-start operation where the auxiliary winding is tripped when the machine is accelerated to 75% of the rated speed.

The last figure in the series, Electromagnetic Torque in Capacitor-Start Operation Mode, shows the electromagnetic torque developed by the machine. Because there is a step load of 1 N.m, the average torque is 1 N.m. The torque ripple amplitude is about 2.5 N.m, or 150% of the rated load. The torque pulsation affects the operation of the machine.

Auxiliary Winding Current in Capacitor-Start Operation Mode

The auxiliary winding current is set to zero when the speed reaches 75% of the rated speed. The voltage across the start-capacitor remains at its maximum value, because current and voltage across a capacitor are in quadrature.

Main Winding Current in Capacitor-Start Operation Mode

Capacitor Voltage in Capacitor-Start Operation Mode

Speed in Capacitor-Start Operation Mode

Electromagnetic Torque in Capacitor-Start Operation Mode

Capacitor-Start-Run Operation Mode

An improvement in the operation mode of the single phase asynchronous machine occurs when the auxiliary winding is still connected in series with a capacitor after starting.

The next figures show the simulation wave forms in the capacitor-start-run operation mode single phase asynchronous machine. The magnitude of the torque ripple at steady state is only about 3% of the load torque. It improves the operation of a single phase asynchronous machine by limiting the shaft's vibrations.

Main Winding Current in Capacitor-Start-Run Operation Mode