Synchronous Reluctance Machine

Synchronous reluctance machine with sinusoidal flux distribution

Libraries:
Simscape / Electrical / Electromechanical / Reluctance & Stepper

Description

The Synchronous Reluctance Machine block represents a synchronous reluctance machine (SynRM) with sinusoidal flux distribution. The figure shows the equivalent electrical circuit for the stator windings.

You can also model the stator windings either in a delta-wound or in an open-end configuration by setting Winding type to Delta-wound or Open-end, respectively.

Motor Construction

The diagram shows the motor construction with a single pole-pair on the rotor. For the axes convention shown, when rotor mechanical angle θ_r is zero, the a-phase and permanent magnet fluxes are aligned. The block supports a second rotor axis definition for which rotor mechanical angle is defined as the angle between the a-phase magnetic axis and the rotor q-axis.

Equations

The combined voltage across the stator windings is

$[\begin{matrix} v_{a} \\ v_{b} \\ v_{c} \end{matrix}] = [\begin{matrix} R_{s} & 0 & 0 \\ 0 & R_{s} & 0 \\ 0 & 0 & R_{s} \end{matrix}] [\begin{matrix} i_{a} \\ i_{b} \\ i_{c} \end{matrix}] + [\begin{matrix} \frac{d ψ_{a}}{d t} \\ \frac{d ψ_{b}}{d t} \\ \frac{d ψ_{c}}{d t} \end{matrix}],$

where:

v_a, v_b, and v_c are the individual phase voltages across the stator windings.
R_s is the equivalent resistance of each stator winding.
i_a, i_b, and i_c are the currents flowing in the stator windings.
ψ_a, ψ_b, and ψ_c are the magnetic fluxes that link each stator winding.

The permanent magnet, excitation winding, and the three stator windings contribute to the flux that links each winding. The total flux is defined as

$[\begin{matrix} ψ_{a} \\ ψ_{b} \\ ψ_{c} \end{matrix}] = [\begin{matrix} L_{a a} & L_{a b} & L_{a c} \\ L_{b a} & L_{b b} & L_{b c} \\ L_{c a} & L_{c b} & L_{c c} \end{matrix}] [\begin{matrix} i_{a} \\ i_{b} \\ i_{c} \end{matrix}]$

where:

L_aa, L_bb, and L_cc are the self-inductances of the stator windings.
L_ab, L_ac, L_ba, L_bc, L_ca, and L_cb are the mutual inductances of the stator windings.

$θ_{e} = N θ_{r} + r o t o r o f f s e t$

$L_{a a} = L_{s} + L_{m} cos (2 θ_{e})$

$L_{b b} = L_{s} + L_{m} cos (2 (θ_{e} - 2 π / 3))$

$L_{c c} = L_{s} + L_{m} cos (2 (θ_{e} + 2 π / 3))$

$L_{a b} = L_{b a} = - M_{s} - L_{m} \cos (2 (θ_{e} + π / 6))$

$L_{b c} = L_{c b} = - M_{s} - L_{m} \cos (2 (θ_{e} + π / 6 - 2 π / 3))$

and

$L_{c a} = L_{a c} = - M_{s} - L_{m} \cos (2 (θ_{e} + π / 6 + 2 π / 3))$

where:

θ_r is the rotor mechanical angle.
θ_e is the rotor electrical angle.
rotor offset is 0 if you define the rotor electrical angle with respect to the d-axis, or -pi/2 if you define the rotor electrical angle with respect to the q-axis.
L_s is the stator self-inductance per phase. This value is the average self-inductance of each of the stator windings.
L_m is the stator inductance fluctuation. This value is the amplitude of the fluctuation in self-inductance and mutual inductance with changing rotor angle.
M_s is the stator mutual inductance. This value is the average mutual inductance between the stator windings.

Simplified Equations

Applying the Park transformation to the block electrical defining equations produces an expression for torque that is independent of rotor angle.

The Park transformation, P, is defined as

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

where θ_e is the electrical angle. The electrical angle depends on the rotor mechanical angle and the number of pole pairs such that

$θ_{e} = N θ_{r} + r o t o r o f f s e t$

where:

N is the number of pole pairs.
θ_r is the rotor mechanical angle.

Applying the Park transformation to the first two electrical defining equations produces equations that define the behavior of the block:

$v_{d} = R_{s} i_{d} + L_{d} \frac{d i_{d}}{d t} - N ω i_{q} L_{q},$

$v_{q} = R_{s} i_{q} + L_{q} \frac{d i_{q}}{d t} + N ω i_{d} L_{d},$

$v_{0} = R_{s} i_{0} + L_{0} \frac{d i_{0}}{d t},$

$T = \frac{3}{2} N (i_{q} i_{d} L_{d} - i_{d} i_{q} L_{q})$

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

where:

i_d, i_q, and i₀ are the d-axis, q-axis, and zero-sequence currents, defined by

$[\begin{matrix} i_{d} \\ i_{q} \\ i_{0} \end{matrix}] = P [\begin{matrix} i_{a} \\ i_{b} \\ i_{c} \end{matrix}],$
where i_a, i_b, and i_c are the stator currents.
v_d, v_q, and v₀ are the d-axis, q-axis, and zero-sequence currents, defined by

$[\begin{matrix} v_{d} \\ v_{q} \\ v_{0} \end{matrix}] = P [\begin{matrix} v_{a} \\ v_{b} \\ v_{c} \end{matrix}],$
where v_a, v_b, and v_c are the stator currents.
The dq0 inductances are defined, respectively as
- $L_{d} = L_{s} + M_{s} + \frac{3}{2} L_{m}$
- $L_{q} = L_{s} + M_{s} - \frac{3}{2} L_{m}$
- $L_{0} = L_{s} - 2 M_{s}$ .
R_s is the stator resistance per phase.
N is the number of rotor pole pairs.
T is the rotor torque. For the Synchronous 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.
B_m is the rotor damping.
ω is the rotor mechanical rotational speed.
J is the rotor inertia.

Calculating Iron Losses

To calculate iron losses in the Synchronous Reluctance Machine block, choose whether you want to model them by specifying magnetizing resistance, parameterize them based on the Steinmetz equation, or if you want to tabulate the losses as a function of currents and rotor speed.

If you set the Stator parameterization parameter to Specify Ls, Lm, and Ms or Specify Ld, Lq, and L0 and Modeling fidelity to Constant Ld and Lq, then specify the Magnetizing resistance parameter to model iron losses.

In the Iron Losses settings, if you set the Model parameter to Specify tabulated Steinmetz coefficients or Tabulate with current and speed, the iron losses are applied to the mechanical side as a reduction in the electrical torque:

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

where w_min is minimum rotor speed to start taking iron losses into account, and it is equal to 1 rad/s.

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

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

Steinmetz parameterization method

To model the iron losses based on the Steinmetz equation, set the Stator parameterization parameter to Specify Ld, Lq, and L0 and the Modeling fidelity parameter to Tabulated Ld and Lq. The Steinmetz method scales for different motor speeds or electrical frequencies so that iron loss data is only required as function of motor currents.

If, in the Iron losses settings, you set the Model parameter to Specify tabulated Steinmetz coefficients, the iron losses are given by:

$\begin{array}{l} P_{l o s s, r o t o r} (f) = k_{h r} (i_{d}, i_{q}) f + k_{J r} (i_{d}, i_{q}) f^{2} + k_{e r} (i_{d}, i_{q}) f^{1.5} \\ P_{l o s s, s t a t o r} (f) = k_{h s} (i_{d}, i_{q}) f + k_{J s} (i_{d}, i_{q}) f^{2} + k_{e s} (i_{d}, i_{q}) f^{1.5} \end{array}$

where:

k_hr(i_d,i_q) is the Rotor hysteresis loss coefficient, k_hr(id,iq).
k_Jr(i_d,i_q) is the Rotor eddy current loss coefficient, k_Jr(id,iq).
k_er(i_d,i_dq) is the Rotor excess current loss coefficient, k_er(id,iq).
k_hs(i_d,i_q) is the Stator hysteresis loss coefficient, k_hs(id,iq).
k_Js(i_d,i_q) is the Stator eddy current loss coefficient, k_Js(id,iq).
k_es(i_d,i_q) is the Stator excess current loss coefficient, k_es(id,iq).
$f = w \frac{n P o l e P a i r s}{2 π}$ is the electrical frequency, in Hz.

Tabulate with current and speed

To model the iron losses by tabulating them independently for rotor and stator, set the Stator parameterization parameter to Specify Ld, Lq, and L0 and the Modeling fidelity parameter to Tabulated Ld and Lq.

Then, if in the Iron losses settings, you set the Model parameter to Tabulate with current and speed, the iron losses are given by:

$P_{l o s s} (i_{d}, i_{q}, w) = t a b l e l o o k u p (P_{t a b l e}, i_{d}, i_{q}, w) .$

where w is the Rotor speed vector, w parameter.

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 stator and in the rotor:

$\begin{array}{l} P_{d i s s A} = R_{a} i_{a}^{2} + \frac{P_{l o s s, s t a t o r}}{3} \\ P_{d i s s B} = R_{b} i_{b}^{2} + \frac{P_{l o s s, s t a t o r}}{3} \\ P_{d i s s C} = R_{c} i_{c}^{2} + \frac{P_{l o s s, s t a t o r}}{3} \\ P_{d i s s R} = P_{l o s s, r o t o r} \end{array}$

If you specify the magnetizing resistance, the iron loss in the rotor is equal to a percentage of the total iron losses. This percentage remains constant and does not depend on other states:

$P_{l o s s, r o t o r} = \frac{p e r c e n t_{r o t o r}}{100} P_{i r o n l o s s, t o t a l} .$

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

$P_{l o s s, s t a t o r} = (1 - \frac{p e r c e n t_{r o t o r}}{100}) P_{i r o n l o s s, t o t a l} .$

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.

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

Synchronous Reluctance Machine Torque Control

Control the torque in a synchronous reluctance machine (SynRM) based electrical drive. A high-voltage battery feeds the SynRM through a controlled three-phase converter. An ideal angular velocity source provides the load. The Control subsystem uses an open-loop approach to control the torque and a closed-loop approach to control the current. At each sample instant, the torque request is converted to relevant current references using the maximum torque per Ampere strategy. The current control is PI-based. The simulation uses torque steps in both the motor and generator modes. The Visualization subsystem contains scopes that allow you to see the simulation results.

Open Model

Synchronous Reluctance Machine Velocity Control

Control the rotor angular velocity in a synchronous reluctance machine (SynRM) based electrical drive. A high-voltage battery feeds the SynRM through a controlled three-phase converter. An ideal torque source provides the load. The Control subsystem includes a multi-rate PI-based cascade control structure. The control structure has an outer angular-velocity-control loop and two inner current-control loops. The Visualization subsystem contains scopes that allow you to see the simulation results.

Open Model

Assumptions

The flux distribution is sinusoidal.

Ports

Conserving

expand all

R — Machine rotor
mechanical rotational

Mechanical rotational conserving port associated with the machine rotor.

C — Machine case
mechanical rotational

Mechanical rotational conserving port associated with the machine case.

~ — Three-phase composite
electrical

Expandable three-phase port associated with the stator windings.

Dependencies

To enable this port, set Electrical connection to Composite three-phase ports and Winding type to Wye-wound or Delta-wound.

a — a-phase
electrical

Electrical conserving port associated with a-phase.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports and Winding type to Wye-wound or Delta-wound.

b — b-phase
electrical

Electrical conserving port associated with b-phase.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports and Winding type to Wye-wound or Delta-wound.

c — c-phase
electrical

Electrical conserving port associated with c-phase.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports and Winding type to Wye-wound or Delta-wound.

~1 — Stator positive-end connections
electrical

Since R2024a

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

Dependencies

To enable this port, set Electrical connection to Composite three-phase ports and Winding type to Open-end.

~2 — Stator negative-end connections
electrical

Since R2024a

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

Dependencies

To enable this port, set Electrical connection to Composite three-phase ports and Winding type to Open-end.

a1 — a positive-end connection
electrical

Since R2024a

Electrical conserving port associated with the a positive-end connection.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports and Winding type to Open-end.

b1 — b positive-end connection
electrical

Since R2024a

Electrical conserving port associated with the b positive-end connection.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports and Winding type to Open-end.

c1 — c positive-end connection
electrical

Since R2024a

Electrical conserving port associated with the c positive-end connection.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports and Winding type to Open-end.

a2 — a negative-end connection
electrical

Since R2024a

Electrical conserving port associated with the a negative-end connection.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports and Winding type to Open-end.

b2 — b negative-end connection
electrical

Since R2024a

Electrical conserving port associated with the b negative-end connection.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports and Winding type to Open-end.

c2 — c negative-end connection
electrical

Since R2024a

Electrical conserving port associated with the c negative-end connection.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports and Winding type to Open-end.

n — Neutral phase
electrical

Electrical conserving port associated with the neutral phase.

Dependencies

To enable this port, set Winding type to Wye-wound and Zero sequence to Include.

HA — Winding A thermal port
thermal

Thermal conserving port associated with winding A.

Dependencies

To enable this port, set Modeling option to Show thermal port.

HB — Winding B thermal port
thermal

Thermal conserving port associated with winding B.

Dependencies

To enable this port, set Modeling option to Show thermal port.

HC — Winding C thermal port
thermal

Thermal conserving port associated with winding C.

Dependencies

To enable this port, set Modeling option to Show thermal port.

HR — Rotor thermal port
thermal

Thermal conserving port associated with the rotor.

Dependencies

To enable this port, set Modeling option to Show thermal port.

Parameters

expand all

Modeling option — Whether to enable thermal ports
`No thermal port` (default) | `Show thermal port`

Whether to enable the thermal ports of the block and model the effects of losses that convert power to heat.

Main

Electrical connection — Electrical connection
`Composite three-phase ports` (default) | `Expanded three-phase ports`

Whether to have composite or expanded three-phase ports.

Winding type — Windings configuration
`Wye-wound` (default) | `Delta-wound` | `Open-end`

Since R2024a

Select the configuration for the windings:

Wye-wound — The windings are wye-wound.
Delta-wound — The windings are delta-wound. The a-phase is connected between ports a and b, the b-phase between ports b and c and the c-phase between ports c and a.
Open-end — The windings are in an open-end configuration. The block exposes each end of the windings as electrical connections.

Number of pole pairs — Rotor pole pairs
`6` (default) | integer

Number of permanent magnet pole pairs on the rotor.

Stator parameterization — Parameterization model
`Specify Ld, Lq and L0` (default) | `Specify Ls, Lm, and Ms`

Stator parameterization model.

Dependencies

The Stator parameterization setting affects the visibility of other parameters.

Modeling fidelity — Modeling fidelity
`Constant Ld and Lq` (default) | `Tabulated Ld and Lq`

Select the modeling fidelity:

Constant Ld and Lq — Ld and Lq values are constant and defined by their respective parameters.
Tabulated Ld and Lq — Ld and Lq values are computed online from DQ currents look-up tables as follows:

$L_{d} = f_{1} (i_{d}, i_{q})$

$L_{d} = f_{2} (i_{d}, i_{q})$

Dependencies

This parameter is visible only when you set the Stator parameterization parameter to Specify Ld, Lq, and L0.

Stator d-axis inductance, Ld — Inductance
`0.0031` `H` (default)

Direct-axis inductance of the machine stator.

Dependencies

This parameter is visible only when you set the Stator parameterization parameter to Specify Ld, Lq, and L0 and the Modeling fidelity parameter to Constant Ld and Lq.

Stator q-axis inductance, Lq — Inductance
`0.004` `H` (default)

Quadrature-axis inductance of the machine stator.

Dependencies

This parameter is visible only when you set the Stator parameterization parameter to Specify Ld, Lq, and L0 and the Modeling fidelity parameter to Constant Ld and Lq.

Direct-axis current vector, iD — Direct-axis current vector
`[-200, 0, 200]` `A` (default)

Direct-axis current vector, iD.

Dependencies

This parameter is visible only when you set the Stator parameterization parameter to Specify Ld, Lq, and L0 and the Modeling fidelity parameter to Tabulated Ld and Lq.

Quadrature-axis current vector, iQ — Quadrature-axis current vector
`[-200, 0, 200]` `A` (default)

Quadrature-axis current vector, iQ.

Dependencies

This parameter is visible only when you set the Stator parameterization parameter to Specify Ld, Lq, and L0 and the Modeling fidelity parameter to Tabulated Ld and Lq.

Ld matrix, Ld(id,iq) — Ld matrix
`0.0031 * ones(3, 3)` `H` (default)

Ld matrix.

Dependencies

This parameter is visible only when you set the Stator parameterization parameter to Specify Ld, Lq, and L0 and the Modeling fidelity parameter to Tabulated Ld and Lq.

Lq matrix, Lq(id,iq) — Lq matrix
`0.004 * ones(3, 3)` `H` (default)

Lq matrix.

Dependencies

This parameter is visible only when you set the Stator parameterization parameter to Specify Ld, Lq, and L0 and the Modeling fidelity parameter to Tabulated Ld and Lq.

Stator zero-sequence inductance, L0 — Inductance
`0.0005` `H` (default)

Zero-axis inductance for the machine stator.

Dependencies

This parameter is visible only if Stator parameterization is set to Specify Ld, Lq and L0 and Zero sequence is set to Include.

Stator self-inductance per phase, Ls — Inductance
`0.0025` `H` (default)

Average self-inductance of the three stator windings.

Dependencies

This parameter is visible only if Stator parameterization is set to Specify Ls, Lm, and Ms.

Stator inductance fluctuation, Lm — Inductance
`-0.0003` `H` (default)

Amplitude of the fluctuation in self-inductance and mutual inductance with the rotor angle.

Dependencies

This parameter is visible only if Stator parameterization is set to Specify Ls, Lm, and Ms.

Stator mutual inductance, Ms — Inductance
`0.0010` `H` (default)

Average mutual inductance between the stator windings.

Dependencies

This parameter is visible only if Stator parameterization is set to Specify Ls, Lm, and Ms.

Stator resistance per phase, Rs — Resistance
`0.7` `Ohm` (default)

Resistance of each of the stator windings.

Zero sequence — Zero-sequence model
`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 Stator parameterization is set to Specify Ld, Lq, and L0 — The Stator zero-sequence inductance, L0 parameter is visible.
Exclude — The Stator zero-sequence inductance, L0 parameter is not visible.

Iron Losses

For more information on calculating iron losses, see Calculating Iron Losses.

Model — Iron losses modeling option
`Specify magnetizing resistance` (default) | `Specify tabulated Steinmetz coefficients` | `Tabulate with current and speed`

Whether to enable iron losses modeling.

Dependencies

To enable this parameter, set Modeling fidelity to Tabulate Ld and Lq.

Magnetizing resistance — Magnetizing resistance for iron losses
`inf` `Ohm` (default)

Magnetizing resistance. The value must be greater than zero. The default value is inf, which implies that there are no iron losses.

Dependencies

To enable this parameter, either set Model to Specify magnetizing resistance or set Modeling fidelity to Constant Ld and Lq.

Rotor hysteresis loss coefficient, k_hr(id,iq) — Rotor hysteresis loss coefficient
`zeros(3, 3)` `W/Hz` (default)

Hysteresis loss coefficient of the rotor depending on the d-axis and q-axis currents. It is used by the Steinmetz equation.