Implement single-phase induction motor drive
Simscape / Electrical / Specialized Power Systems / Electric Drives / AC Drives
The Single-Phase Induction Motor Drive block models a vector-controlled single-phase machine drive. The drive configuration consists of a half-bridge rectifier, a divided DC bus with two filter capacitors, and a two-leg inverter that supplies the motor windings.
The single-phase induction machine (SPIM), without its startup and running capacitors, is treated as an asymmetric two-phase machine. The auxiliary and main windings are accessible and are in quadrature. This configuration provides good performances and operation in regenerating mode.
The single-phase induction motor is asymmetrical due to the unequal resistances and inductances of the main and auxiliary windings. To obtain the mathematical model of the motor with constant parameters (voltage, current, and flux), it is necessary to transform all the variables to the stationary reference frame (d-q) fixed to the stator.
Mathematical Model
This diagram shows the mathematical model of the machine.
N_{a} and N_{m} represent the number of auxiliary and main stator windings, respectively.
The equations that define the voltage for the model (in the stationary reference frame d-q) are:
$${V}_{qs}={R}_{s}{i}_{qs}+\frac{d{\varphi}_{qs}}{dt},$$
$${V}_{ds}={R}_{a}{i}_{ds}+\frac{d{\varphi}_{ds}}{dt},$$
$$0=R{\text{'}}_{r}i{\text{'}}_{qr}+\frac{d{\varphi}_{qr}^{\text{'}}}{dt}-\frac{1}{k}\cdot {\omega}_{r}{\varphi}_{qr}^{\text{'}},$$
$$0={k}^{2}R{\text{'}}_{r}i{\text{'}}_{dr}+\frac{d{\varphi}_{dr}^{\text{'}}}{dt}+k\cdot {\omega}_{r}{\varphi}_{dr}^{\text{'}},$$
and
$$k=\frac{{N}_{a}}{{N}_{m}},$$
where:
V_{qs} is the q-axis stator voltage.
R_{s} is the main stator resistance.
i_{qs} is the q-axis stator current.
ϕ_{qs} is the q-axis stator flux linkage.
V_{ds} is the d-axis stator voltage.
R_{a} is the auxiliary stator resistance.
i_{ds} is the d-axis stator current.
ϕ_{ds} is the d-axis stator flux linkage.
R'_{r} is the rotor winding resistance referred to the main stator winding.
i'_{qr} is the q-axis rotor current referred to the main stator winding.
ϕ'_{qr} is the q-axis rotor flux linkage referred to the main stator winding.
k is the turn ratio of N_{a} to N_{m}.
ω_{r} is the rotor electrical angular velocity.
i'_{dr} is the d-axis rotor current referred to the main stator winding.
ϕ'_{dr} is the d-axis rotor flux linkage referred to the main stator winding.
N_{a} is the number of auxiliary stator windings.
N_{m} is the number of main stator windings.
The equations that define the flux for the model (in the stationary reference frame d-q) are:
$${\varphi}_{qs}={L}_{ls}{i}_{qs}+{L}_{ms}\left({i}_{qs}+{i}_{qr}^{\text{'}}\right)$$
$${\varphi}_{ds}={L}_{la}{i}_{ds}+{k}^{2}{L}_{ms}\left({i}_{ds}+{i}_{dr}^{\text{'}}\right)$$
$${\varphi}_{qr}^{\text{'}}={L}_{lr}^{\text{'}}{i}_{qr}^{\text{'}}+{L}_{ms}\left({i}_{qs}+{i}_{qr}^{\text{'}}\right)$$
$${\varphi}_{dr}^{\text{'}}={k}^{2}{L}_{lr}^{\text{'}}{i}_{dr}^{\text{'}}+{k}^{2}{L}_{ms}\left({i}_{ds}+{i}_{dr}^{\text{'}}\right)$$
where:
L_{ls} is the leakage inductance of the main stator winding.
L_{la} is the leakage inductance of the auxiliary stator winding.
L_{ms} is the magnetizing inductance of the main stator winding.
L'_{lr} is the leakage inductance of the rotor winding referred to the main stator winding.
The electromagnetic torque expressed as a function of the rotor flux linkages and currents is
$${T}_{e}=P(k\cdot {\varphi}_{qr}^{\text{'}}{i}_{dr}^{\text{'}}-\frac{1}{k}\cdot {\varphi}_{dr}^{\text{'}}{i}_{qr}^{\text{'}})$$
where:
T_{e} is the electromagnetic torque.
P is the number of pole pairs.
Using the stator currents and rotor flux linkages as state-space variables for the SPIM model, the electromagnetic torque equation is
$${T}_{e}=P\cdot \frac{{L}_{ms}}{{L}_{r}^{\text{'}}}(\frac{1}{k}\cdot {i}_{qs}{\varphi}_{dr}^{\text{'}}-k\cdot {i}_{ds}{\varphi}_{qr}^{\text{'}}).$$
Using the next change of variable,
$${i}_{qs}={k}^{2}\cdot {i}_{qs1}$$
and
$${i}_{ds}={i}_{ds1}.$$
Therefore, the electromagnetic torque equation can be rewritten as
$${T}_{e}=P\cdot \frac{{L}_{ms}}{{L}_{r}^{\text{'}}}k({\varphi}_{dr}^{\text{'}}{i}_{qs1}-{\varphi}_{qr}^{\text{'}}{i}_{ds1}).$$
In the indirect rotor flux-oriented control, the d-axis of the reference frame is oriented along the rotor flux linkage vector ϕ'_{r}, then
$${\varphi}_{dr}^{\text{'}}={\varphi}_{r}^{\text{'}}$$
and
$${\varphi}_{qr}^{\text{'}}=0.$$
The electromagnetic torque results in
$${T}_{e}=P\cdot \frac{{L}_{ms}}{{L}_{r}^{\text{'}}}k\cdot {\varphi}_{r}^{\text{'}}{i}_{qs1}^{e}.$$
From here, the q-axis current component is
$${i}_{qs1}^{e}=\cdot \frac{{T}_{e}\cdot {L}_{r}^{\text{'}}}{k\cdot {L}_{ms}\cdot P\cdot {\varphi}_{r}^{\text{'}}}.$$
The resulting slip speed, ω_{s}, is
$${\omega}_{s}=\frac{{k}^{3}\cdot {L}_{ms}\cdot {i}_{qs1}^{e}}{\left(\frac{{L}_{r}^{\text{'}}}{{R}_{r}^{\text{'}}}\right)\text{\hspace{0.05em}}\text{\hspace{0.05em}}\cdot {\varphi}_{r}^{\text{'}}}.$$
From here, the d-axis current component is
$${i}_{ds1}^{e}=\frac{{\varphi}_{r}^{\text{'}}}{{k}^{2}\cdot {L}_{ms}\text{\hspace{0.05em}}}.$$
where the e superscript indicates that the variable is referred to the synchronous reference frame.
This block diagram shows field-oriented control.
This type of control selects the voltage vector from a switching table to control the power switches in the inverter to obtain the required stator flux and corresponding motor torque. From the motor equations in the stationary reference frame d-q, estimate the stator flux and the torque:
$${\varphi}_{qs}={\displaystyle \int \left({V}_{qs}-{R}_{s}{i}_{qs}\right)}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}dt$$
and
$${\varphi}_{ds}={\displaystyle \int \left({V}_{ds}-{R}_{a}{i}_{ds}\right)}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}dt.$$
Assuming the approximation
$${L}_{la}={k}^{2}\cdot {L}_{ls},$$
and using the stator variables (flux linkages and currents) as state-space variables of the SPIM model, the electromagnetic torque is given by
$${T}_{e}=P(\frac{1}{k}\cdot {i}_{qs}{\varphi}_{ds}-k\cdot {i}_{ds}{\varphi}_{qs}).$$
Using the cross-product, the torque is
$${T}_{e}=P({\varphi}_{s}\times {i}_{s}),$$
that is
$${T}_{e}=P\left|{\varphi}_{s}\right|\cdot \left|{\varphi}_{s}\right|\mathrm{sin}\delta ,$$
where:
|ϕ_{s}| and |ϕ_{r}| are the magnitudes of the stator and rotor flux linkage space vectors, respectively.
δ is the angle between the space vectors.
A change in the relative movement of ϕ_{s} and ϕ_{r} (defined by the angle, δ) affects the motor instantaneous torque. If the voltage drop on the stator resistance is omitted, the stator flux linkage directly depends on the inverter output voltage.
The next diagram shows the available voltage vectors, which correspond to possible inverter states, and the four distinct sectors in the d-q plane for a two-leg inverter.
Selecting the appropriate inverter voltage vectors can directly change the magnitude of ϕ_{s} (flux control) and the rotating speed of ϕ_{s} (torque control) as shown in this diagram of sector 1.
This block diagram shows direct-torque control.
The estimated flux and torque are compared with the references using hysteresis control. The digitalized output variables and the stator flux position sector are used to select the appropriated voltage vector from the switching table. This table shows the appropriate voltage vector for the inverter where H_{ϕ} and H_{Te} are the output of the flux and torque hysteresis blocks
H_{ϕ} | H_{Te} | Sector 1 | Sector 2 | Sector 3 | Sector 4 |
---|---|---|---|---|---|
1 (Flux is up) | 1 (Torque is up) | V_{1} | V_{2} | V_{3} | V_{4} |
0 (Torque is down) | V_{4} | V_{1} | V_{2} | V_{3} | |
0 (Flux is down) | 1 (Torque is up) | V_{2} | V_{3} | V_{4} | V_{1} |
0 (Torque is down) | V_{3} | V_{4} | V_{1} | V_{2} |