Switched Reluctance Machine

Three-phase switched reluctance machine

Libraries:
Simscape / Electrical / Electromechanical / Reluctance & Stepper

Description

The Switched Reluctance Machine block represents a three-phase switched reluctance machine (SRM). The stator has three pole pairs, carrying the three motor windings, and the rotor has several nonmagnetic poles. The motor produces torque by energizing a stator pole pair, inducing a force on the closest rotor poles and pulling them toward alignment. The diagram shows the motor construction.

Choose this machine in your application to take advantage of these properties:

Low cost
Relatively safe failing currents
Robustness to high temperature operation
High torque-to-inertia ratio

Use this block to model an SRM using easily measurable or estimable parameters. To model an SRM using FEM data, see Switched Reluctance Motor Parameterized with FEM Data.

Equations

Switched Reluctance Machine Block

The rotor stroke angle for a three-phase machine is

$θ_{s t} = \frac{2 π}{3 N_{r}},$

where:

θ_st is the stoke angle.
N_r is the number of rotor poles.

The torque production capability, β, of one rotor pole is

$β = \frac{2 π}{N_{r}} .$

The mathematical model for a switched reluctance machine (SRM) is highly nonlinear due to influence of the magnetic saturation on the flux linkage-to-angle, λ(θ_ph) curve. The phase voltage equation for an SRM is

$v_{p h} = R_{s} i_{p h} + \frac{d λ_{p h} (i_{p h}, θ_{p h})}{d t}$

where:

v_ph is the voltage per phase.
R_s is the stator resistance per phase.
i_ph is the current per phase.
λ_ph is the flux linkage per phase.
θ_ph is the angle per phase.

Rewriting the phase voltage equation in terms of partial derivatives yields this equation:

$v_{p h} = R_{s} i_{p h} + \frac{\partial λ_{p h}}{\partial i_{p h}} \frac{d i_{p h}}{d t} + \frac{\partial λ_{p h}}{\partial θ_{p h}} \frac{d θ_{p h}}{d t} .$

Transient inductance is defined as

$L_{t} (i_{p h}, θ_{p h}) = \frac{\partial λ_{p h} (i_{p h}, θ_{p h})}{\partial i_{p h}},$

or more simply as

$\frac{\partial λ_{p h}}{\partial i_{p h}} .$

Back electromotive force is defined as

$E_{p h} = \frac{\partial λ_{p h}}{\partial θ_{p h}} ω_{r} .$

Substituting these terms into the rewritten voltage equation yields this voltage equation:

$v_{p h} = R_{s} i_{p h} + L_{t} (i_{p h}, θ_{p h}) \frac{d i_{p h}}{d t} + E_{p h} .$

Applying the co-energy formula to equations for torque,

$T_{p h} = \frac{\partial W (θ_{p h})}{\partial θ_{r}},$

and energy,

$W (i_{p h}, θ_{p h}) = \int_{0}^{i_{p h}} λ_{p h} (i_{p h}, θ_{p h}) d i_{p h}$

yields an integral equation that defines the instantaneous torque per phase, that is,

$T_{p h} (i_{p h}, θ_{p h}) = \int_{0}^{i_{p h}} \frac{\partial λ_{p h} (i_{p h}, θ_{p h})}{\partial θ_{p h}} d i_{p h} .$

Integrating over the phases give this equation, which defines the total instantaneous torque for a three-phase SRM:

$T = \sum_{j = 1}^{3} T_{p h} (j) .$

The equation for motion is

$J \frac{d ω}{d t} = T - T_{L} - B_{m} ω$

where:

J is the rotor inertia.
ω is the mechanical rotational speed.
T is the rotor torque. For the Switched Reluctance Machine block, torque flows from the machine case (block conserving port C) to the machine rotor (block conserving port R).
T_L is the load torque.
J is the rotor inertia.
B_m is the rotor damping.

For high-fidelity modeling and control development, use empirical data and finite element calculation to determine the flux linkage curve in terms of current and angle, that is,

$λ_{p h} (i_{p h}, θ_{p h}) .$

For low-fidelity modeling, you can also approximate the curve using analytical techniques. One such technique [2] uses this exponential function:

$λ_{p h} (i_{p h}, θ_{p h}) = λ_{s a t} (1 - e^{- i_{p h} f (θ_{p h})}),$

where:

λ_sat is the saturated flux linkage.
f(θ_r) is obtained by Fourier expansion.

For the Fourier expansion, use the first two even terms of this equation:

$f (θ_{p h}) = a + b \cos (N_{r} θ_{p h})$

where a > b,

$a = \frac{L_{\min} + L_{\max}}{2 λ_{s a t}},$

and

$b = \frac{L_{m a x} - L_{m i n}}{2 λ_{s a t}} .$

Switched Reluctance Motor Block

The flux linkage curve is approximated based on parametric and geometric data:

$λ_{p h} (i_{p h}, θ_{p h}) = λ_{s a t} (1 - e^{- L_{0} (θ) i_{p h} / λ_{s a t}}),$

where L₀ is the unsaturated inductance.

The effects of saturation are more prominent as the product of current and unsaturated inductance approach the saturated flux linkage value. Specify this value using the Saturated flux linkage parameter.

Differentiating the flux equation then gives the winding inductance:

$L (θ_{p h}) = L_{0} (θ_{p h}) e^{(- L_{0} (θ_{p h}) i_{p h} / λ_{s a t})}$

The unsaturated inductance varies between a minimum and maximum value. The minimum value occurs when a rotor pole is directly between two stator poles. The maximum occurs when the rotor pole is aligned with a stator pole. In between these two points, the block approximates the unsaturated inductance linearly as a function of rotor angle. This figure shows the unsaturated inductance as a rotor pole passes over a stator pole.

In the figure:

θ_R corresponds to the angle subtended by the rotor pole. Set it using the Angle subtended by each rotor pole parameter.
θ_S corresponds to the angle subtended by the stator pole. Set it using the Angle subtended by each stator pole parameter.
θ_C corresponds to the angle subtended by this full cycle, determined by 2π/2n where n is the number of stator pole pairs.

Calculating Iron Losses

The Switched Reluctance Machine (Multi-Phase) block implements the iron losses as a reduction in the electrical torque.

If you set the Stator parameterization parameter to either Specify parametric data or Specify parametric and geometric data, specify the Magnetizing resistance parameter to model iron losses.

If you set the Stator parameterization parameter to Specify tabulated flux data, the block tabulates the losses as the power lost in each stator phase. The losses for each stator only depends on the stator current in that phase, the rotor angle (geometrically shifted to that phase), and the rotor speed. The iron loss power of the rotor is computed as a percentage of the total iron loss power, as specified in the Percentage of total iron losses associated with the rotor parameter.

If you tabulate the losses with current, speed, and angle, the iron loss torque reduces the electrical torque of a generic "A" phase:

$τ_{e l e c A} = τ (i_{A}, θ_{A}) - τ_{l o s s} (i_{A}, w, θ_{A})$

where $τ_{l o s s} (i_{A}, w, θ_{A}) = s i g n (w) \frac{P_{l o s s} (a b s (i_{A}), a b s (w), θ_{A})}{a b s (w) + w_{m i n}}$ is the torque loss computed as the mechanical power loss divided by the rotor speed. w_min is the minimum rotor speed and it is equal to 1 rad/s. This value prevents a division by zero when the rotor speed is equal to 0.

If you tabulate the losses in function of only current and speed, the iron loss torque reduces the electrical torque of a generic "A" phase:

$τ_{l o s s} (i_{A}, w) = s i g n (w) \frac{P_{l o s s} (a b s (i_{A}), a b s (w))}{a b s (w) + w_{m i n}} .$

The iron losses are applied to the mechanical side as a reduction in torque. If you create the iron loss table, P_loss, from FEM results, the FEM tool might express the power lost as electrical power instead of mechanical power.

The total electrical torque is equal to the sum of the electrical torque of each phase:

$τ_{e l e c} = τ_{e l e c A} + τ_{e l e c B} + τ_{e l e c C} .$

The total power dissipated is equal to the sum of the iron losses and the copper losses:

$P_{d i s s i p a t e d} = (τ_{l o s s A} + τ_{l o s s B} + τ_{l o s s C}) w + R_{a} i_{a}^{2} + R_{b} i_{b}^{2} + R_{c} i_{c}^{2} .$

Iron losses with thermal ports

If you expose the thermal ports of the block, the power dissipated in each phase act as heat sources in the motor and in the rotor:

$\begin{array}{l} P_{d i s s A} = R_{a} i_{a}^{2} + P_{i r o n A} \\ P_{d i s s B} = R_{b} i_{b}^{2} + P_{i r o n B} \\ P_{d i s s C} = R_{c} i_{c}^{2} + P_{i r o n C} \\ P_{d i s s R} = P_{i r o n R} \end{array}$

The iron loss in the rotor is equal to a percentage of the total iron losses. This loss remains constant and does not depend on other states:

$P_{i r o n R} = \frac{p e r c e n t_{r o t o r}}{100} (τ_{l o s s A} + τ_{l o s s B} + τ_{l o s s C}) w .$

Then the remaining iron loss is distributed among the different stator phases:

$\begin{array}{l} P_{i r o n A} = (1 - \frac{p e r c e n t_{r o t o r}}{100}) τ_{l o s s A} w \\ P_{i r o n B} = (1 - \frac{p e r c e n t_{r o t o r}}{100}) τ_{l o s s B} w \\ P_{i r o n C} = (1 - \frac{p e r c e n t_{r o t o r}}{100}) τ_{l o s s C} w \end{array}$

Model Thermal Effects

You can expose thermal ports to model the effects of losses that convert power to heat. To expose the thermal ports, set the Modeling option parameter to either:

No thermal port — The block does not contain thermal ports.
Show thermal port — The block contains multiple thermal conserving ports.

For more information about using thermal ports in actuator blocks, see Simulating Thermal Effects in Rotational and Translational Actuators.

Numerical Smoothing

In practice, magnetic edge effects prevent the inductance from taking a trapezoidal shape as a rotor pole passes over a stator pole. To model these effects, and to avoid gradient discontinuities that hinder solver convergence, the block smooths the gradient ∂L₀/∂θ at inflection points. To change the angle over which this smoothing is applied, use the Angle over which flux gradient changes are smoothed parameter.

Assumptions

The block assumes that a zero rotor angle corresponds to a rotor pole that is aligned perfectly with the a-phase.

Variables

To set the priority and initial target values for the block variables before simulation, use the Initial Targets section in the block dialog box or Property Inspector. For more information, see Set Priority and Initial Target for Block Variables.

Nominal values provide a way to specify the expected magnitude of a variable in a model. Using system scaling based on nominal values increases the simulation robustness. You can specify nominal values using different sources, including the Nominal Values section in the block dialog box or Property Inspector. For more information, see System Scaling by Nominal Values.

Examples

Switched Reluctance Machine Speed Control

Control the rotor speed in a switched reluctance machine (SRM) based electrical drive. A DC voltage source feeds the SRM through a controlled three-arm bridge. The converter turn-on and turn-off angles are maintained constant.

Open Model

Switched Reluctance Motor Parameterized with FEM Data

Build a model of a Switched Reluctance Motor (SRM). A custom Simscape™ composite component is used to construct the three stator pole pairs using the Simscape Electrical™ FEM-Parameterized Rotary Actuator as a base component. The states of the input pins (reverse, on/off) control the torque applied to the motor shaft.

Open Model

Ports

Conserving

expand all

~1 — Positive three-phase composite terminal
electrical

Electrical conserving three-phase port associated with the positive terminals of the stator windings.

Dependencies

To enable this port, set Electrical connection to Composite three-phase ports.

~2 — Negative three-phase composite terminal
electrical

Electrical conserving three-phase port associated with the negative terminals of the stator windings.

Dependencies

To enable this port, set Electrical connection to Composite three-phase ports.