Synchronous Machine Model 1.0

Synchronous machine with field circuit and no damper

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Simscape / Electrical / Electromechanical / Synchronous

Description

The Synchronous Machine Model 1.0 block uses a simplified parameterization model for synchronous machines. Use the block to model synchronous machines with a field winding and no dampers.

The figure shows the equivalent electrical circuit for the stator and rotor windings.

Motor Construction

The diagram shows the motor construction with a single pole pair on the rotor. For the axes convention, 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

Voltages across the stator windings are defined by

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

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.
$\frac{d ψ_{a}}{d t},$ $\frac{d ψ_{b}}{d t},$ and $\frac{d ψ_{c}}{d t}$ are the rates of change of magnetic flux in each stator winding.

The voltage across the field winding is expressed as

$v_{f} = R_{f} i_{f} + \frac{d ψ_{f}}{d t},$

where:

v_f is the individual phase voltage across the field winding.
R_f is the equivalent resistance of the field winding.
i_f is the current flowing in the field winding.
$\frac{d ψ_{f}}{d t}$ is the rate of change of magnetic flux in the field winding.

The permanent magnet, excitation winding, and the three star-wound stator windings contribute to the flux linking each winding. The total flux is defined by

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

where:

ψ_a, ψ_b, and ψ_c are the total fluxes linking each stator winding.
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.
ψ_am, ψ_bm, and ψ_cm are the magnetization fluxes linking the stator windings.
L_amf, L_bmf, and L_cmf are the mutual inductances of the field winding.

The inductances in the stator windings are functions of rotor electrical angle and are defined by

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

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

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

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

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

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

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

where:

N is the number of rotor pole pairs.

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 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.

The magnetization flux linking winding, a-a’ is a maximum when θ_e = 0° and zero when θ_e = 90°. Therefore:

$L_{m f} = [\begin{matrix} L_{a m f} \\ L_{b m f} \\ L_{c m f} \end{matrix}] = [\begin{matrix} L_{m f} \cos θ_{e} \\ L_{m f} \cos (θ_{e} - 2 π / 3) \\ L_{m f} \cos (θ_{e} + 2 π / 3) \end{matrix}]$

and

$Ψ_{f} = L_{f} i_{f} + L_{m f}^{T} [\begin{matrix} i_{a} \\ i_{b} \\ i_{c} \end{matrix}],$

where:

L_mf is the mutual field armature inductance.
ψ_f is the flux linking the field winding.
L_f is the field winding inductance.
${[L_{m f}]}^{T}$ is the transform of the L_mf vector, that is,

${[L_{m f}]}^{T} = {[\begin{matrix} L_{a m f} \\ L_{b m f} \\ L_{c m f} \end{matrix}]}^{T} = [\begin{matrix} L_{a m f} & L_{b m f} & L_{c m f} \end{matrix}] .$

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 is defined by

$P = 2 / 3 [\begin{matrix} \cos θ_{e} & \cos (θ_{e} - 2 π / 3) & \cos (θ_{e} + 2 π / 3) \\ - \sin θ_{e} & - \sin (θ_{e} - 2 π / 3) & - \sin (θ_{e} + 2 π / 3) \\ 0.5 & 0.5 & 0.5 \end{matrix}]$

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

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

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

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

$v_{f} = R_{f} i_{f} + L_{f} \frac{d i_{f}}{d t} + \frac{3}{2} L_{m f} \frac{d i_{d}}{d t},$

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

and

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

where:

v_d, v_q, and v₀ are the d-axis, q-axis, and zero-sequence voltages. These voltages are defined by

$[\begin{matrix} v_{d} \\ v_{q} \\ v_{0} \end{matrix}] = P [\begin{matrix} v_{a} \\ v_{b} \\ v_{c} \end{matrix}] .$
i_d, i_q, and i₀ are the d-axis, q-axis, and zero-sequence currents, defined by

$[\begin{matrix} i_{d} \\ i_{q} \\ i_{0} \end{matrix}] = P [\begin{matrix} i_{a} \\ i_{b} \\ i_{c} \end{matrix}] .$
L_d is the stator d-axis inductance. L_d = L_s + M_s + 3/2 L_m.
ω is the mechanical rotational speed.
L_q is the stator q-axis inductance. L_q = L_s + M_s − 3/2 L_m.
L₀ is the stator zero-sequence inductance. L₀ = L_s – 2M_s.
T is the rotor torque. For the Synchronous Machine Model 1.0 block, torque flows from the machine case (block conserving port C) to the machine rotor (block conserving port R).
J is the rotor inertia.
T_L is the load torque.
B_m is the rotor damping.

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.

Assumptions

Flux distribution is sinusoidal.

Ports

Conserving

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`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.

`n` — Neutral phase
electrical

Electrical conserving port associated with the neutral phase.

`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.

Parameters

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Main

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

Select the modeling fidelity:

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

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

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

$L_{m f} = f_{3} (i_{d}, i_{q}, i_{f})$

$L_{f} = f_{4} (i_{f})$

Dependencies

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

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

Number of permanent magnet pole pairs on the rotor.

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

Stator parameterization method.

Dependencies

The Stator parameterization setting affects the visibility of other parameters.

`Stator d-axis inductance, Ld` — Inductance
`0.00015` `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, Lq, Lmf and Lf.

`Stator q-axis inductance, Lq` — Inductance
`0.00021` `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, Lq, Lmf and Lf.