Induction Machine Wound Rotor

Wound-rotor induction machine with per-unit or SI parameterization

Library:
Simscape / Electrical / Electromechanical / Asynchronous

Description

The Induction Machine Wound Rotor block models a wound-rotor asynchronous machine with fundamental parameters expressed in per-unit or in the International System of Units (SI). A wound-rotor asynchronous machine is a type of induction machine. All stator and rotor connections are accessible on the block. Therefore, you can model soft-start regimes using a switch between wye and delta configurations or by increasing rotor resistance. If you do not need access to the rotor windings, use the Induction Machine Squirrel Cage block instead.

Connect port ~1 to a three-phase circuit. To connect the stator in delta configuration, connect a Phase Permute block between ports ~1 and ~2. To connect the stator in wye configuration, connect port ~2 to a Grounded Neutral or a Floating Neutral block. If you do not need to vary rotor resistance, connect rotor port ~1r' to a Floating Neutral block and rotor port ~2r' to a Grounded Neutral block.

The rotor circuit is referred to the stator. Therefore, when you use the block in a circuit, refer any additional circuit parameters to the stator.

Induction Machine Initialization Using Load-Flow Target Values

If the block is in a network that is compatible with the frequency-time simulation mode, you can perform a load-flow analysis on the network. A load-flow analysis provides steady-state values that you can use to initialize the machine.

For more information, see Perform a Load-Flow Analysis Using Simscape Electrical and Frequency and Time Simulation Mode. For an example that shows how initialize an induction machine using data from a load flow analysis, see Induction Motor Initialization with Loadflow.

Equations

For the SI implementation, the block converts the SI values that you enter to per-unit values for simulation. The converted values are based on the machine being connected in a delta-winding configuration.

For the per-unit implementation, you must specify the resistances and inductances in the impedances tab based on the machine being connected in a delta-winding configuration.

For information on the relationship between SI and per-unit machine parameters, see Per-Unit Conversion for Machine Parameters. For information on per-unit parameterization, see Per-Unit System of Units.

The asynchronous machine equations are expressed with respect to a synchronous reference frame, defined by

$θ_{e} (t) = \int_{0}^{t} 2 π f_{r a t e d} d t,$

where f_rated is the value of the Rated electrical frequency parameter.

The Park transformation maps stator equations to a reference frame that is stationary with respect to the rated electrical frequency. The Park transformation is defined by

$P_{s} = \frac{2}{3} [\begin{matrix} \cos θ_{e} & cos (θ_{e} - \frac{2 π}{3}) & cos (θ_{e} + \frac{2 π}{3}) \\ - \sin θ_{e} & - sin (θ_{e} - \frac{2 π}{3}) & - sin (θ_{e} + \frac{2 π}{3}) \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{matrix}],$

where θ_e is the electrical angle.

The rotor equations are mapped to another reference frame, defined by the difference between the electrical angle and the product of rotor angle θ_r and number of pole pairs N:

$P_{r} = \frac{2}{3} [\begin{matrix} \cos (θ_{e} - N θ_{r}) & cos (θ_{e} - N θ_{r} - \frac{2 π}{3}) & cos (θ_{e} - N θ_{r} + \frac{2 π}{3}) \\ - \sin (θ_{e} - N θ_{r}) & - sin (θ_{e} - N θ_{r} - \frac{2 π}{3}) & - sin (θ_{e} - N θ_{r} + \frac{2 π}{3}) \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{matrix}] .$

The Park transformation is used to define the per-unit asynchronous machine equations. The stator voltage equations are defined by

$v_{d s} = \frac{1}{ω_{b a s e}} \frac{d ψ_{d s}}{d t} - ω ψ_{q s} + R_{s} i_{d s},$

$v_{q s} = \frac{1}{ω_{b a s e}} \frac{d ψ_{q s}}{d t} + ω ψ_{d s} + R_{s} i_{q s},$

and

$v_{0 s} = \frac{1}{ω_{b a s e}} \frac{d ψ_{0 s}}{d t} + R_{s} i_{0 s},$

where:

v_ds, v_qs, and v_0s are the d-axis, q-axis, and zero-sequence stator voltages, defined by
$[\begin{matrix} v_{d s} \\ v_{q s} \\ v_{0 s} \end{matrix}] = P_{s} [\begin{matrix} v_{a} \\ v_{b} \\ v_{c} \end{matrix}] .$
v_a, v_b, and v_c are the stator voltages across ports ~1 and ~2.
ω_base is the per-unit base electrical speed.
ψ_ds, ψ_qs, and ψ_0s are the d-axis, q-axis, and zero-sequence stator flux linkages.
R_s is the stator resistance.
i_ds, i_qs, and i_0s are the d-axis, q-axis, and zero-sequence stator currents, defined by
$[\begin{matrix} i_{d s} \\ i_{q s} \\ i_{0 s} \end{matrix}] = P_{s} [\begin{matrix} i_{a} \\ i_{b} \\ i_{c} \end{matrix}] .$
i_a, i_b, and i_c are the stator currents flowing from port ~1 to port ~2.

The rotor voltage equations are defined by

$v_{d r} = \frac{1}{ω_{b a s e}} \frac{d ψ_{d r}}{d t} - (ω - ω_{r}) ψ_{q r} + R_{r d} i_{d r},$

$v_{q r} = \frac{1}{ω_{b a s e}} \frac{d ψ_{q r}}{d t} + (ω - ω_{r}) ψ_{d r} + R_{r d} i_{q r},$

and

$v_{0 r} = \frac{1}{ω_{b a s e}} \frac{d ψ_{0 r}}{d t} + R_{r d} i_{0 s},$

where:

v_dr, v_qr, and v_0r are the d-axis, q-axis, and zero-sequence rotor voltages, defined by
$[\begin{matrix} v_{d r} \\ v_{q r} \\ v_{0 r} \end{matrix}] = P_{r} [\begin{matrix} v_{a r} \\ v_{b r} \\ v_{c r} \end{matrix}] .$
v_ar, v_br, and v_cr are the rotor voltages across ports ~1r' and ~2r'.
ψ_dr, ψ_qr, and ψ_0r are the d-axis, q-axis, and zero-sequence rotor flux linkages.
ω is the per-unit synchronous speed. For a synchronous reference frame, the value is 1.
ω_r is the per-unit mechanical rotational speed.
R_rd is the rotor resistance referred to the stator.
i_dr, i_qr, and i_0r are the d-axis, q-axis, and zero-sequence rotor currents, defined by
$[\begin{matrix} i_{d r} \\ i_{q r} \\ i_{0 r} \end{matrix}] = P_{r} [\begin{matrix} i_{a r} \\ i_{b r} \\ i_{c r} \end{matrix}] .$
i_ar, i_br, and i_cr are the rotor currents flowing from port ~1r' to port ~2r'.

The stator flux linkage equations are defined by

$ψ_{d s} = L_{s s} i_{d s} + L_{m} i_{d r},$

$ψ_{q s} = L_{s s} i_{q s} + L_{m} i_{q r},$

and

$ψ_{0 s} = L_{s s} i_{0 s},$

where L_ss is the stator self-inductance and L_m is the magnetizing inductance.

The rotor flux linkage equations are defined by

$ψ_{d r} = L_{r r d} i_{d r} + L_{m} i_{d s}$

$ψ_{q r} = L_{r r d} i_{q r} + L_{m} i_{q s},$

and

$ψ_{0 r} = L_{r r d} i_{0 r},$

where L_rrd is the rotor self-inductance referred to the stator.

The rotor torque is defined by

$T = ψ_{d s} i_{q s} - ψ_{q s} i_{d s} .$

The stator self-inductance L_ss, stator leakage inductance L_ls, and magnetizing inductance L_m are related by

$L_{s s} = L_{l s} + L_{m} .$

The rotor self-inductance L_rrd, rotor leakage inductance L_lrd, and magnetizing inductance L_m are related by

$L_{r r d} = L_{l r d} + L_{m} .$

When a saturation curve is provided, the equations to determine the saturated magnetizing inductance as a function of magnetizing flux are:

$L_{m_s a t} = f (ψ_{m})$

$ψ_{m} = \sqrt{ψ_{d m}^{2} + ψ_{q m}^{2}}$

For no saturation, the equation reduces to

$L_{m_s a t} = L_{m}$

Plotting and Display Options

You can perform plotting and display actions using the Electrical menu on the block context menu.

Right-click the block and, from the Electrical menu, select an option:

Display Base Values — Displays the machine per-unit base values in the MATLAB^® Command Window.
Plot Torque Speed (SI) — Plots torque versus speed, both measured in SI units, in a MATLAB figure window using the current machine parameters.
Plot Torque Speed (pu) — Plots torque versus speed, both measured in per-unit, in a MATLAB figure window using the current machine parameters.
Plot Open-Circuit Saturation — Plots terminal voltage versus no-load line current, both in per-unit, in a MATLAB figure window. The plot contains three traces:
- Unsaturated — Stator magnetizing inductance (unsaturated).
- Saturated — Open-circuit lookup table (v versus i) you specify.
- Derived — Open-circuit lookup table derived from the per-unit open-circuit lookup table (v versus i) you specify. This data is used to calculate the saturated magnetizing inductance, L_{m_sat}, and the saturation factor, K_s, versus magnetic flux linkage, ψ_m, characteristics.
Plot Saturation Factor — Plots saturation factor, K_s, versus magnetic flux linkage, ψ_m, in a MATLAB figure window using the machine parameters. This parameter is derived from other parameters that you specify:
- No-load line current saturation data, i
- Terminal voltage saturation data, v
- Leakage inductance, L_ls
Plot Saturated Inductance — Plots magnetizing inductance, L_{m_sat}, versus magnetic flux linkage, ψ_m, in a MATLAB figure window using the machine parameters. This parameter is derived from other parameters that you specify:
- No-load line current saturation data, i
- Terminal voltage saturation data, v
- Leakage inductance, L_ls

For the SI implementation, v is in V (phase-phase RMS) and i is in A (rms).

Variables

Use the Variables settings to specify the priority and initial target values for the block variables before simulation. For more information, see Set Priority and Initial Target for Block Variables.

The type of variables that are visible in the Variables settings depends on the initialization method that you select, in the Main settings, for the Initialization option parameter. To specify target values using:

Flux variables — Set the Initialization option parameter to Set targets for flux variables.
Data from a load-flow analysis — Set the Initialization option parameter to Set targets for load flow variables.

If you select Set targets for load flow variables, to fully specify the initial condition, you must include an initialization constraint in the form of a high-priority target value. For example, if your induction machine is connected to an Inertia block, the initial condition for the induction machine is completely specified if, in the Variables settings of the Inertia block, the Priority for Rotational velocity is set to High. Alternatively, you could set the Priority to None for the Inertia block Rotational velocity, and instead set the Priority for the induction machine block Slip, Real power generated, or Mechanical power consumed to High.

Ports

Output

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`pu` — Per-unit measurements output port
physical

Physical signal vector port associated with the machine per-unit measurements. The vector elements are:

pu_torque
pu_velocity
pu_vds
pu_vqs
pu_v0s
pu_ids
pu_iqs
pu_i0s

Conserving

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`R` — Machine rotor
mechanical

Mechanical rotational conserving port associated with the machine rotor.

`C` — Machine case
mechanical

Mechanical rotational conserving port associated with the machine case.

`~1` — Stator positive-end connections
electrical

Expandable three-phase port associated with the stator positive-end connections.

`~2` — Stator negative-end connections
electrical

Expandable three-phase port associated with the stator negative-end connections.

`~1r'` — Rotor positive-end connections
electrical

Expandable three-phase port associated with the rotor positive-end connections.

`~2r'` — Rotor negative-end connections
electrical

Expandable three-phase port associated with the rotor negative-end connections.

Parameters

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All default parameter values are based on a machine delta-winding configuration.

Main

`Rated apparent power` — Rated apparent power
`15e3` `V*A` (default)

Rated apparent power of the induction machine.

`Rated voltage` — RMS voltage
`220` `V` (default)

RMS line-line voltage.

`Rated electrical frequency` — Nominal electrical frequency
`60` `Hz` (default)

Nominal electrical frequency corresponding to the rated apparent power.

`Number of pole pairs` — Machine pole pairs
`1` (default)

Number of machine pole pairs.

`Parameterization unit` — Unit system for block parameterization
`SI` (default) | `Per unit`

Unit system for block parameterization. Choose between SI, the international system of units, and Per unit, the per-unit system.

Dependencies

Selecting:

SI exposes SI parameters in the Impedances and Saturation settings.
Per unit exposes per-unit parameters in the Impedances and Saturation settings.

`Zero sequence` — Zero sequence
`Include` (default) | `Exclude`

Zero-sequence model:

Include — Prioritize model fidelity. An error occurs if you Include zero-sequence terms for simulations that use the Partitioning solver. For more information, see Increase Simulation Speed Using the Partitioning Solver.
Exclude — Prioritize simulation speed for desktop simulation or real-time deployment.

Dependencies

If this parameter is set to:

Include and Parameterization unit is set to SI — The Stator zero-sequence reactance, X0 parameter in the Impedances settings is visible.
Include and Parameterization unit is set to Per unit — The Stator zero-sequence inductance, L0 (pu) parameter in the Impedances settings is visible.
Exclude — The stator zero-sequence parameter in the Impedances settings is not visible.

`Initialization option` — Initialization method
`Set targets for flux variables` (default) | `Set targets for load flow variables`

Initialization method. You can initialize a machine for steady-state simulation using either flux data or data from a load-flow analysis.

Dependencies

Flux variables — Set the Initialization option parameter to Set targets for flux variables.
Data from a load-flow analysis — Set the Initialization option parameter to Set targets for load flow variables.

Impedances

For the Parameterization unit parameter in the Main settings, select SI to expose SI parameters or Per unit to expose per-unit parameters.