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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.
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 |
V_{qs} = R_{s}i_{qs} + dφ_{qs}/dt | φ_{qs} = L_{ss}i_{qs} + L_{ms}i'_{qr} | |
V_{ds} = R_{S}i_{ds} + dφ_{ds}/dt | φ_{ds} = L_{SS}i_{ds} + L_{mS}i'_{dr} | |
V'_{qr} = R'_{r}i'_{qr} + dφ'_{qr}/dt – (N_{s}/N_{S})ω_{r}φ'_{dr} | φ'_{qr} = L'_{r}i'_{qr} + L_{ms}i_{qs} | |
V'_{dr} = R'_{R}i'_{dr} + dφ'_{dr}/dt + (N_{S}/N_{s})ω_{r}φ'_{qr} | where | φ'_{dr} = L'_{RR}i'_{dr} + L_{mS}i_{ds} |
T_{e} = p[(N_{S}/N_{s})φ'_{qr}i'_{dr} – (N_{s}/N_{S})φ'_{dr}i'_{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} |
$$\begin{array}{c}\frac{d}{dt}{\omega}_{m}=\frac{{T}_{e}-F{\omega}_{m}-{T}_{m}}{2H}\\ \frac{d}{dt}{\theta}_{m}={\omega}_{m}.\end{array}$$
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.
$$\begin{array}{c}\left[\begin{array}{c}{f}_{qs}\\ {f}_{ds}\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{c}{f}_{as}\\ {f}_{bs}\end{array}\right]\\ \left[\begin{array}{c}{f}_{qr}\\ {f}_{dr}\end{array}\right]=\left[\begin{array}{cc}\mathrm{cos}(\theta r)& -\mathrm{sin}\left(\theta r)\right)\\ -\mathrm{sin}(\theta r)& -\mathrm{cos}(\theta r)\end{array}\right]\left[\begin{array}{c}{f}_{ar}\\ {f}_{br}\end{array}\right].\end{array}$$
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.m^{2}). 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. |
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, SimPowerSystems™ 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.
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 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.
Specify one of the four types of single phase asynchronous machines (the split-phase, the capacitor-start, the capacitor-start-capacitor-run, or the 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, 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).
The stator resistance R_{s} (Ω or pu) and leakage inductance L_{ls} (H or pu).
The rotor resistance R_{r}' (Ω or pu) and leakage inductance L_{lr}' (H or pu), both referred to the stator.
The magnetizing inductance L_{ms} (H or pu).
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.
For the SI units dialog box: the combined machine and load inertia coefficient J (kg.m^{2}), 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.
The start capacitance C_{s} (farad or pu) and capacitor series resistance R_{st}(Ω or pu).
The run capacitance Crun (farad or pu) and series resistance Rrun (farad or pu).
Specifies the speed (%) when the auxiliary winding may be disconnected.
Specifies the initial speed (%).
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.
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 |
The Single Phase Asynchronous Machine block does not include a representation of iron losses and saturation.
The power_singlephaseASMpower_singlephaseASM example shows 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.
In capacitor-start operation mode 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
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
Auxiliary Winding Current in Capacitor-Start-Run Operation Mode
Capacitor Voltage in Capacitor-Start-Run Operation Mode
Electromagnetic Torque in Capacitor-Start-Run Operation Mode
Speed in Capacitor-Start-Run Operation Mode