Synchronous Machine (Six-Phase)

Six-phase synchronous machine

Since R2020b

Libraries:
Simscape / Electrical / Electromechanical / Synchronous

Description

The Synchronous Machine (Six-Phase) block models a six-phase synchronous machine, also known as a dual-star machine.

A six-phase synchronous machine has two groups of stator windings: the ABC group and the XYZ group. These two groups have a 30 degree phase shift.

The equivalent circuits of the six-phase synchronous machine for the direct axis, the quadrature axis, and the two zero sequence are:

Equations

The synchronous machine equations are expressed with respect to a synchronously rotating reference frame, defined by:

$θ_{e} (t) = N θ_{r} (t) + x_r o t o r_o f f s e t,$

where:

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

Two Park transformations map the synchronous machine equations to the rotating reference frame with respect to the electrical angle. The Park transformation for the first group of stator windings, the ABC group, is defined by:

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

The Park transformation for the second group of stator windings, the XYZ group, is defined by:

$P_{s 2} = \frac{2}{3} [\begin{matrix} \cos (θ_{e} - \frac{π}{6}) & \cos (θ_{e} - \frac{5 π}{6}) & \cos (θ_{e} + \frac{π}{2}) \\ - \sin (θ_{e} - \frac{π}{6}) & - \sin (θ_{e} - \frac{5 π}{6}) & \sin (θ_{e} + \frac{π}{2}) \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{matrix}] .$

The Park transformations are used to define the per-unit synchronous machine equations.

The stator voltage equations for the ABC group are defined by:

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

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

$v_{01} = R_{s} i_{01} + \frac{1}{ω_{b a s e}} \frac{d ψ_{01}}{d t}$

where:

v_d1, v_q1, and v₀₁ are the d-axis, q-axis, and zero-sequence ABC stator voltages, defined by:

$[\begin{matrix} v_{d 1} \\ v_{q 1} \\ v_{01} \end{matrix}] = P_{s 1} [\begin{matrix} v_{a} \\ v_{b} \\ v_{c} \end{matrix}] .$

v_a, v_b, and v_c are the ABC stator voltages measured from port ~ABC to neutral port n1.
ω_base is the per-unit base electrical speed.
ψ_d1, ψ_q1, and ψ₀₁ are the d-axis, q-axis, and zero-sequence stator flux linkages for the ABC group.
ω_r is the per-unit rotor rotational speed.
R_s is the stator resistance.
i_d1, i_q1, and i₀₁ are the d-axis, q-axis, and zero-sequence ABC stator currents, defined by:

$[\begin{matrix} i_{d 1} \\ i_{q 1} \\ i_{01} \end{matrix}] = P_{s 1} [\begin{matrix} i_{A} \\ i_{B} \\ i_{C} \end{matrix}] .$

i_a, i_b, and i_c are the ABC stator currents flowing from port ~ABC to port n1.

The stator voltage equations for the XYZ group are defined by:

$v_{d 2} = R_{s} i_{d 2} - ψ_{q 2} ω_{r} + \frac{1}{ω_{b a s e}} \frac{d ψ_{d 2}}{d t}$

$v_{q 2} = R_{s} i_{q 2} + ψ_{d 2} ω_{r} + \frac{1}{ω_{b a s e}} \frac{d ψ_{q 2}}{d t}$

$v_{02} = R_{s} i_{02} + \frac{1}{ω_{b a s e}} \frac{d ψ_{02}}{d t}$

where:

v_d2, v_q2, and v₀₂ are the d-axis, q-axis, and zero-sequence XYZ stator voltages, defined by:

$[\begin{matrix} v_{d 2} \\ v_{q 2} \\ v_{02} \end{matrix}] = P_{s 2} [\begin{matrix} v_{x} \\ v_{y} \\ v_{z} \end{matrix}] .$

v_x, v_y, and v_z are the XYZ stator voltages measured from port ~XYZ to neutral port n2.
ψ_d2, ψ_q2, and ψ₀₂ are the d-axis, q-axis, and zero-sequence stator flux linkages for the XYZ group.
i_d2, i_q2, and i₀₂ are the d-axis, q-axis, and zero-sequence XYZ stator currents, defined by:

$[\begin{matrix} i_{d 2} \\ i_{q 2} \\ i_{02} \end{matrix}] = P_{s 2} [\begin{matrix} i_{_{X}} \\ i_{Y} \\ i_{_{Z}} \end{matrix}] .$

i_x, i_y, and i_z are the XYZ stator currents flowing from port ~XYZ to port n2.

The rotor voltage equations are defined by:

$v'_{f d} = R'_{f d} i'_{f d} + \frac{1}{ω_{b a s e}} \frac{d ψ'_{f d}}{d t}$

$v'_{k d} = R'_{k d} i'_{k d} + \frac{1}{ω_{b a s e}} \frac{d ψ'_{k d}}{d t} = 0$

$v'_{k q} = R'_{k q} i'_{k q} + \frac{1}{ω_{b a s e}} \frac{d ψ'_{k q}}{d t} = 0$

where:

v'_fd is the field winding voltage referred to the stator side.
v'_kd and v'_kq are the dq-axes damper winding voltages referred to the stator side. They are all equal to 0.
ψ'_fd, ψ'_kd, and ψ'_kq are the magnetic fluxes linking the field circuit, the d-axis damper winding, and the q-axis damper winding.
R'_fd, R'_kd, and R'_kq are the resistances of the rotor field circuit, d-axis damper winding, and q-axis damper winding.
i'_fd, i'_kd, and i'_kq are the field and dq-axes damper winding currents referred to the stator side.

The stator flux linkage equations are defined by:

$ψ_{d 1} = L_{l} i_{d 1} + L_{m d} (i_{d 1} + i_{d 2} + {i^{'}}_{f d} + {i^{'}}_{k d})$

$ψ_{q 1} = L_{l} i_{q 1} + L_{m q} (i_{q 1} + i_{q 2} + {i^{'}}_{k q})$

$ψ_{01} = L_{l} i_{01}$

$ψ_{d 2} = L_{l} i_{d 2} + L_{m d} (i_{d 1} + i_{d 2} + {i^{'}}_{f d} + {i^{'}}_{k d})$

$ψ_{q 2} = L_{l} i_{q 2} + L_{m q} (i_{q 1} + i_{q 2} + {i^{'}}_{k q})$

$ψ_{02} = L_{l} i_{02}$

where:

L_l is the stator leakage inductance.
L_md and L_mq are the mutual inductances of the stator d-axis and q-axis.

The rotor flux linkage equations are defined by:

$ψ'_{f d} = L'_{l f d} i'_{f d} + L_{m d} (i_{d 1} + i_{d 2} + {i^{'}}_{f d} + {i^{'}}_{k d})$

$ψ'_{k d} = L'_{l k d} i'_{k d} + L_{m d} (i_{d 1} + i_{d 2} + {i^{'}}_{f d} + {i^{'}}_{k d})$

$ψ'_{k q} = L'_{l k q} i'_{k q} + L_{m q} (i_{q 1} + i_{q 2} + {i^{'}}_{k q})$

where:

L'_lfd is the rotor field winding inductance.
L'_lkd is the rotor d-axis damper winding inductance.
L'_lkg is the rotor q-axis damper winding inductance.

The rotor torque is defined by:

$T_{e} = ψ_{d 1} i_{q 1} - ψ_{q 1} i_{d 1} + ψ_{d 2} i_{q 2} - ψ_{q 2} i_{d 2} .$

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 contains expanded electrical conserving ports associated with the stator windings, but does not contain thermal ports.
Show thermal port — The block contains expanded electrical conserving ports associated with the stator windings and thermal conserving ports for each of the windings and for the rotor.

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 prior to 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.

For this block, the Initial Targets settings are visible only if, in the Initial Conditions section, you set the Initialization option parameter to Set targets for rotor angle and Park's transform 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. Nominal values can come from different sources, one of which is the Nominal Values section in the block dialog box or Property Inspector. For more information, see System Scaling by Nominal Values.

Ports

Conserving

expand all

fd+ — Field winding positive terminal
electrical

Electrical conserving port associated with the field winding positive terminal.

fd- — Field winding negative terminal
electrical

Electrical conserving port associated with the field winding negative terminal.

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.

~ABC — Stator ABC windings
electrical

Three-phase electrical port associated with the stator ABC windings.

~XYZ — Stator XYZ windings
electrical

Three-phase electrical port associated with the stator XYZ windings.

n1 — ABC winding neutral point
electrical

Electrical conserving port associated with the neutral point of the ABC winding configuration.

Dependencies

To enable this port, set Zero sequence to Include.

n2 — XYZ winding neutral point
electrical

Electrical conserving port associated with the neutral point of the XYZ winding configuration.

Dependencies

To enable this port, set Zero sequence to Include.