Simplified Synchronous Machine

Simplified synchronous machine with electromotive force

Library:
Simscape / Electrical / Electromechanical / Synchronous

Description

The Simplified Synchronous Machine block models a simplified synchronous machine with a voltage source that represents electromotive force (EMF). You can specify the internal resistance and inductance with per-unit or SI parameters.

The equivalent circuits of the simplified synchronous machine for the direct axis, the quadrature axis, and the zero sequence are:

Equations

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

$θ_{e} (t) = N θ_{r} (t),$

where:

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

The Park transformation maps the synchronous machine equations to the rotating reference frame with respect to the electrical angle. The Park transformation is defined by:

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

The Park transformation is used to define the per-unit simplified synchronous machine equations. The voltage equations are defined by:

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

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

$e_{0} = \frac{1}{ω_{b a s e}} \frac{d Ψ_{0}}{d t} + R i_{0} + v_{0}$

where:

e_d, e_q, and e₀ are the d-axis, q-axis, and zero-sequence voltages, defined by:

$[\begin{matrix} e_{d} \\ e_{q} \\ e_{0} \end{matrix}] = P_{s} [\begin{matrix} e_{a} \\ e_{b} \\ e_{c} \end{matrix}] .$

e_a, e_b, and e_c are the per-unit internal voltage sources, defined by:

$\begin{array}{l} e_{a} = E_{p u} s i n θ_{e} \\ e_{b} = E_{p u} s i n (θ_{e} - 120 °) \\ e_{c} = E_{p u} s i n (θ_{e} + 120 °) \end{array}$

e_pu is the per-unit amplitude of the internal generated voltage.
v_d, v_q, and v₀ are defined by:

$[\begin{matrix} v_{d} \\ v_{q} \\ v_{0} \end{matrix}] = P_{s} [\begin{matrix} v_{a} \\ v_{b} \\ v_{c} \end{matrix}] .$

v_a, v_b, and v_c are the stator voltages measured from port ~ to neutral port n.
ω_base is the per-unit base electrical speed.
ψ_d, ψ_q, and ψ₀ are the d-axis, q-axis, and zero-sequence flux linkages.
ω_r is the per-unit rotor rotational speed.
R is the stator resistance.
i_d, i_q, and i₀ are the d-axis, q-axis, and zero-sequence stator currents, defined by:

$[\begin{matrix} i_{d} \\ i_{q} \\ i_{0} \end{matrix}] = P_{s} [\begin{matrix} i_{a} \\ i_{b} \\ i_{c} \end{matrix}] .$

i_a, i_b, and i_c are the stator currents flowing out of port ~.

The stator flux linkage equations are defined by

$\begin{array}{l} ψ_{d} = L \cdot i_{d} \\ ψ_{q} = L \cdot i_{q} \\ ψ_{0} = L \cdot i_{0} \end{array}$

where L is the stator leakage inductance.

The power equation of the simplified synchronous machine in per-unit is defined by:

$P = e_{d} i_{d} + e_{q} i_{q} .$

Ports

Input

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`E_si` — Amplitude of internal generated voltage (phase-to-neutral)
vector of physical signals

Physical signal port associated with the amplitude of the internal generated voltage (phase-to-neutral), specified as a vector of physical signals.

Dependencies

To enable this port, set Parameterization unit to SI.

`E_pu` — Amplitude of internal generated voltage, in per-unit
vector of physical signals

Physical signal port associated with the per-unit amplitude of the internal generated voltage, specified as a vector of physical signals.

Dependencies

To enable this port, set Parameterization unit to Per unit.

Output

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`o` — Machine per-unit measurements
vector of physical signals

Physical signal port associated with the machine per-unit measurements, returned as a vector of physical signals. The vector elements are:

Electromotive force associated with phase a, pu_EMF(1)
Electromotive force associated with phase b, pu_EMF(2)
Electromotive force associated with phase c, pu_EMF(3)
Electrical torque, pu_torque
Rotor velocity, pu_velocity
Stator d-axis voltage, pu_vd
Stator q-axis voltage, pu_vq
Stator zero-sequence voltage, pu_e0
Stator d-axis current, pu_id
Stator q-axis current, pu_iq
Stator zero-sequence current, pu_i0
Rotor electrical angle, electrical_angle_out

To connect to this port, use the Synchronous Machine Measurement block.

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.

`~` — Stator windings
electrical

Expandable three-phase electrical port associated with the stator windings.

`n` — Neutral phase
electrical

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

Parameters

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Main

`Rated apparent power` — Rated apparent power
`555e6` `V*A` (default) | scalar

Rated apparent power.

`Rated voltage` — RMS rated line-to-line voltage
`24e3` `V` (default) | scalar

RMS rated line-to-line voltage.

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

Nominal electrical frequency at which rated apparent power is quoted.

`Number of pole pairs` — Number of pole pairs
`2` (default) | scalar

Number of machine pole pairs.

`Parameterization unit` — Block parameterization
`SI` (default) | `Per unit`

Block parameterization method. Options are:

SI
Per unit

`Internal resistance (pu)` — Internal per-unit resistance
`0.05` (default) | scalar

Internal per-unit resistance.

Dependencies

To enable this parameter, set Parameterization unit to Per unit.

`Internal inductance (pu)` — Internal per-unit inductance
`0.3` (default) | scalar

Internal per-unit inductance in per-unit.

Dependencies

To enable this parameter, set Parameterization unit to Per unit.

`Internal resistance` — Internal resistance
`0.0519` `Ohm` (default) | scalar

Internal resistance.

Dependencies

To enable this parameter, set Parameterization unit to SI.

`Internal inductance` — Internal inductance
`8.2589e-04` `H` (default) | scalar

Internal inductance.

Dependencies

To enable this parameter, set Parameterization unit to SI.

Documentation

Simplified Synchronous Machine

Description

Equations

Ports

Input

`E_si` — Amplitude of internal generated voltage (phase-to-neutral)
vector of physical signals

Dependencies

`E_pu` — Amplitude of internal generated voltage, in per-unit
vector of physical signals

Dependencies

Output

`o` — Machine per-unit measurements
vector of physical signals

Conserving

`R` — Machine rotor
mechanical rotational

`C` — Machine case
mechanical rotational

`~` — Stator windings
electrical

`n` — Neutral phase
electrical

Parameters

Main

`Rated apparent power` — Rated apparent power
`555e6` `V*A` (default) | scalar

`Rated voltage` — RMS rated line-to-line voltage
`24e3` `V` (default) | scalar

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

`Number of pole pairs` — Number of pole pairs
`2` (default) | scalar

`Parameterization unit` — Block parameterization
`SI` (default) | `Per unit`

`Internal resistance (pu)` — Internal per-unit resistance
`0.05` (default) | scalar

Dependencies

`Internal inductance (pu)` — Internal per-unit inductance
`0.3` (default) | scalar

Dependencies

`Internal resistance` — Internal resistance
`0.0519` `Ohm` (default) | scalar

Dependencies

`Internal inductance` — Internal inductance
`8.2589e-04` `H` (default) | scalar

Dependencies

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

See Also

Simscape Electrical Documentation

Support

Documentation

Simplified Synchronous Machine

Description

Equations

Ports

Input

E_si — Amplitude of internal generated voltage (phase-to-neutral) vector of physical signals

Dependencies

E_pu — Amplitude of internal generated voltage, in per-unit vector of physical signals

Dependencies

Output

o — Machine per-unit measurements vector of physical signals

Conserving

R — Machine rotor mechanical rotational

C — Machine case mechanical rotational

~ — Stator windings electrical

n — Neutral phase electrical

Parameters

Main

Rated apparent power — Rated apparent power 555e6 V*A (default) | scalar

Rated voltage — RMS rated line-to-line voltage 24e3 V (default) | scalar

Rated electrical frequency — Nominal electrical frequency 60 Hz (default) | scalar

Number of pole pairs — Number of pole pairs 2 (default) | scalar

Parameterization unit — Block parameterization SI (default) | Per unit

Internal resistance (pu) — Internal per-unit resistance 0.05 (default) | scalar

Dependencies

Internal inductance (pu) — Internal per-unit inductance 0.3 (default) | scalar

Dependencies

Internal resistance — Internal resistance 0.0519 Ohm (default) | scalar

Dependencies

Internal inductance — Internal inductance 8.2589e-04 H (default) | scalar

Dependencies

Extended Capabilities

C/C++ Code Generation Generate C and C++ code using Simulink® Coder™.

See Also

Simscape Electrical Documentation

Support

`E_si` — Amplitude of internal generated voltage (phase-to-neutral)
vector of physical signals

`E_pu` — Amplitude of internal generated voltage, in per-unit
vector of physical signals

`o` — Machine per-unit measurements
vector of physical signals

`R` — Machine rotor
mechanical rotational

`C` — Machine case
mechanical rotational

`~` — Stator windings
electrical

`n` — Neutral phase
electrical

`Rated apparent power` — Rated apparent power
`555e6` `V*A` (default) | scalar

`Rated voltage` — RMS rated line-to-line voltage
`24e3` `V` (default) | scalar

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

`Number of pole pairs` — Number of pole pairs
`2` (default) | scalar

`Parameterization unit` — Block parameterization
`SI` (default) | `Per unit`

`Internal resistance (pu)` — Internal per-unit resistance
`0.05` (default) | scalar

`Internal inductance (pu)` — Internal per-unit inductance
`0.3` (default) | scalar

`Internal resistance` — Internal resistance
`0.0519` `Ohm` (default) | scalar

`Internal inductance` — Internal inductance
`8.2589e-04` `H` (default) | scalar

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.