# Synchronous Machine Salient Pole

Salient-pole synchronous machine with fundamental or standard parameterization

• Library:
• Simscape / Electrical / Electromechanical / Synchronous

## Description

The Synchronous Machine Salient Pole block models a salient-pole synchronous machine using fundamental or standard parameters.

### Synchronous Machine Initialization Using Load-Flow Target Values

If the block is in a network that is compatible with the frequency-time simulation mode, you can perform a load-flow analysis on the network. A load-flow analysis provides steady-state values that you can use to initialize the machine.

For more information, see Perform a Load-Flow Analysis Using Simscape Electrical and Frequency and Time Simulation Mode. For an example that shows how initialize an synchronous machine using data from a load flow analysis, see Synchronous Machine Initialization with Loadflow.

### Equations

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

`${\theta }_{e}\left(t\right)=N{\theta }_{r}\left(t\right),$`

where:

• θe is the electrical angle.

• N is the number of pole pairs.

• θr 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

`${P}_{s}=\frac{2}{3}\left[\begin{array}{ccc}\mathrm{cos}{\theta }_{e}& \mathrm{cos}\left({\theta }_{e}-\frac{2\pi }{3}\right)& \mathrm{cos}\left({\theta }_{e}+\frac{2\pi }{3}\right)\\ -\mathrm{sin}{\theta }_{e}& -\mathrm{sin}\left({\theta }_{e}-\frac{2\pi }{3}\right)& -\mathrm{sin}\left({\theta }_{e}+\frac{2\pi }{3}\right)\\ \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\end{array}\right].$`

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

`${e}_{d}=\frac{1}{{\omega }_{base}}\frac{\text{d}{\psi }_{d}}{\text{d}t}-{\Psi }_{q}{\omega }_{r}-{R}_{a}{i}_{d},$`

`${e}_{q}=\frac{1}{{\omega }_{base}}\frac{\text{d}{\psi }_{q}}{\text{d}t}+{\Psi }_{d}{\omega }_{r}-{R}_{a}{i}_{q},$`

and

`${e}_{0}=\frac{1}{{\omega }_{base}}\frac{d{\Psi }_{0}}{dt}-{R}_{a}{i}_{0},$`

where:

• ed, eq, and e0 are the d-axis, q-axis, and zero-sequence stator voltages, defined by

`$\left[\begin{array}{c}{e}_{d}\\ {e}_{q}\\ {e}_{0}\end{array}\right]={P}_{s}\left[\begin{array}{c}{v}_{a}\\ {v}_{b}\\ {v}_{c}\end{array}\right].$`

va, vb, and vc are the stator voltages measured from port ~ to neutral port n.

• ωbase is the per-unit base electrical speed.

• ψd, ψq, and ψ0 are the d-axis, q-axis, and zero-sequence stator flux linkages.

• ωr is the per-unit rotor rotational speed.

• Ra is the stator resistance.

• id, iq, and i0 are the d-axis, q-axis, and zero-sequence stator currents, defined by

`$\left[\begin{array}{c}{i}_{d}\\ {i}_{q}\\ {i}_{0}\end{array}\right]={P}_{s}\left[\begin{array}{c}{i}_{a}\\ {i}_{b}\\ {i}_{c}\end{array}\right].$`

ia, ib, and ic are the stator currents flowing from port ~ to port n.

The rotor voltage equations are defined by

`${e}_{fd}=\frac{1}{{\omega }_{base}}\frac{d{\Psi }_{fd}}{dt}+{R}_{fd}{i}_{fd},$`

`${e}_{1d}=\frac{1}{{\omega }_{base}}\frac{d{\Psi }_{1d}}{dt}+{R}_{1d}{i}_{1d}=0,$`

and

`${e}_{1}{}_{q}=\frac{1}{{\omega }_{base}}\frac{d{\Psi }_{1q}}{dt}+{R}_{1q}{i}_{1q}=0,$`

where:

• efd is the field voltage.

• e1d, and e1q are the voltages across the d-axis damper winding 1 and q-axis damper winding 1. They are equal to 0.

• ψfd, ψ1d, and ψ1q are the magnetic fluxes linking the field circuit, d-axis damper winding 1, and q-axis damper winding 1.

• Rfd, R1d, and R1q are the resistances of rotor field circuit, d-axis damper winding 1, and q-axis damper winding 1.

• ifd, i1d, and i1q are the currents flowing in the field circuit, d-axis damper winding 1, and q-axis damper winding 1.

The saturation equations are defined by

`${\psi }_{ad}={\psi }_{d}+{L}_{l}{i}_{d},$`

`${\psi }_{aq}={\psi }_{q}+{L}_{l}{i}_{q},$`

`${\psi }_{at}=\sqrt{{\psi }_{ad}^{2}+{\psi }_{aq}^{2}},$`

${K}_{s}=1$ (if saturation is disabled),

${K}_{s}=f\left({\psi }_{at}\right)$ (if saturation is enabled),

and

`${L}_{ad}={K}_{s}*{L}_{adu},$`

where:

• ψaq is the q-axis air-gap or mutual flux linkage.

• ψat is the air-gap flux linkage.

• Ks is the saturation factor.

• Ladu is the unsaturated mutual inductance of the stator d-axis.

• Lad is the mutual inductance of the stator d-axis.

The saturation factor function, f, is calculated from the per-unit open-circuit lookup table as:

`${L}_{ad}=\frac{d{\psi }_{at}}{d{i}_{fd}},$`

`${V}_{ag}=g\left({i}_{fd}\right),$`

and

`${L}_{ad}=\frac{dg\left({i}_{fd}\right)}{d{i}_{fd}}=\frac{d{V}_{ag}}{d{i}_{fd}},$`

where:

• Vag is the per-unit air-gap voltage.

In per-unit,

`${K}_{s}=\frac{{L}_{ad}}{{L}_{adu}},$`

and

`${\psi }_{at}={V}_{ag}$`

can be rearranged to

`${K}_{s}=f\left({\psi }_{at}\right).$`

The stator flux linkage equations are defined by

`${\Psi }_{d}=-\left({L}_{ad}+{L}_{l}\right){i}_{d}\text{​}+{L}_{ad}{i}_{fd}+{L}_{ad}{i}_{1d},$`

`$\Psi q=-\left({L}_{aq}+{L}_{l}\right){i}_{q}\text{​}+{L}_{aq}{i}_{1q},$`

and

`${\Psi }_{0}=-{L}_{0}{i}_{0},$`

where:

• Ll is the stator leakage inductance.

• Lad and Laq are the mutual inductances of the stator d-axis and q-axis.

The rotor flux linkage equations are defined by

`${\psi }_{fd}={L}_{ffd}{i}_{fd}+{L}_{f1d}{i}_{1d}-{L}_{ad}{i}_{d},$`

`${\psi }_{1d}={L}_{f1d}{i}_{fd}+{L}_{11d}{i}_{1d}-{L}_{ad}{i}_{d},$`

and

`${\psi }_{1q}={L}_{11q}{i}_{1q}-{L}_{aq}{i}_{q},$`

where:

• Lffd is the self-inductance of the rotor field circuit.

• L11d is the self-inductance of the d-axis damper winding 1.

• L11q is the self-inductance of the q-axis damper winding 1.

• Lf1d is the roto field circuit and d-axis damper winding 1 mutual inductance.

The inductances are defined by these equations:

`${L}_{ffd}={L}_{ad}+{L}_{fd}$`

`${L}_{f1d}={L}_{ffd}-{L}_{fd}$`

`${L}_{11d}={L}_{f1d}+{L}_{1d}$`

`${L}_{11q}={L}_{aq}+{L}_{1q}$`

The inductance equations assume that per-unit mutual inductance L12q = Laq, that is, the stator and rotor currents in the q-axis all link a single mutual flux represented by Laq.

The rotor torque is defined by

`${T}_{e}={\Psi }_{d}{i}_{q}-{\Psi }_{q}{i}_{d}.$`

### Plotting and Display Options

You can perform plotting and display actions using the Electrical menu on the block context menu.

Right-click the block and, from the Electrical menu, select one of these options:

• Display Base Values — Displays the machine per-unit base values in the MATLAB® Command Window.

• Display Associated Base Values — Displays associated per-unit base values in the MATLAB Command Window.

• Display Associated Initial Conditions — Displays associated initial conditions in the MATLAB Command Window.

• Plot Open-Circuit Saturation (pu) — Plots air-gap voltage, Vag, versus field current, ifd, both measured in per-unit, in a MATLAB figure window. The plot contains three traces:

• Unsaturated — Stator d-axis mutual inductance (unsaturated), Ladu you specify

• Saturated — Per-unit open-circuit lookup table (Vag versus ifd) you specify

• Derived — Open-circuit lookup table (per-unit) derived from the Per-unit open-circuit lookup table (Vag versus ifd) you specify. This data is used to calculate the saturation factor,Ks, versus magnetic flux linkage, ψat, characteristic.

• Plot Saturation Factor (pu) — Plots saturation factor,Ks, versus magnetic flux linkage, ψat, both measured in per-unit, in a MATLAB figure window using the present machine parameters. This parameter is derived from other parameters that you specify:

• Stator d-axis mutual inductance (unsaturated), Ladu

• Per-unit field current saturation data, ifd

• Per-unit air-gap voltage saturation data, Vag

### Model Thermal Effects

You can expose thermal ports to model the effects of generated heat and machine temperature. 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

The Variables settings allow you to specify the priority and initial target values for block variables before simulation. For more information, see Set Priority and Initial Target for Block Variables.

For this block, the Variables settings are visible only if, in the Initial Conditions settings, the Initialization option parameter is set to ```Set targets for rotor angle and Park's transform variables```.

## Ports

### Output

expand all

Physical signal vector port associated with the machine per-unit measurements. The vector elements are:

• Field voltage (field circuit base, Efd), pu_fd_Efd

• Field current (field circuit base, Ifd), pu_fd_Ifd

• Electrical torque, pu_torque

• Rotor velocity, pu_velocity

• Stator d-axis voltage, pu_ed

• Stator q-axis voltage, pu_eq

• 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

expand all

Electrical conserving port associated with the field winding positive terminal.

Electrical conserving port associated with the field winding negative terminal.

Mechanical rotational conserving port associated with the machine rotor.

Mechanical rotational conserving port associated with the machine case.

Expandable three-phase port associated with the stator windings.

#### Dependencies

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

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

#### Dependencies

To enable this port, set Zero sequence to `Include`.

Electrical conserving port associated with a-phase.

#### Dependencies

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

Electrical conserving port associated with b-phase.

#### Dependencies

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

Electrical conserving port associated with c-phase.

#### Dependencies

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

Thermal conserving port associated with stator winding a.

#### Dependencies

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

Thermal conserving port associated with stator winding b.

#### Dependencies

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

Thermal conserving port associated with stator winding c.

#### Dependencies

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

Thermal conserving port associated with the rotor.

#### Dependencies

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

## Parameters

expand all

Whether to enable the thermal ports of the block and model the effects of generated heat and machine temperature.

### Main

Whether to have composite or expanded three-phase ports.

Rated apparent power.

RMS rated line-line voltage.

Nominal electrical frequency at which rated apparent power is quoted.

Number of machine pole pairs.

Block parameterization method. Options are:

• `Fundamental parameters` — Specify impedance using fundamental parameters.

• `Standard parameters` — Specify impedance using fundamental parameters and specify d-axis and q-axis time constants.

#### Dependencies

If this parameter is set to ```Fundamental parameters```:

• Fundamental parameters in the Impedances settings are visible.

• Time Constant settings are visible.

If this parameter is set to ```Standard parameters```:

• Standard parameters in the Impedances settings are visible.

• Time Constant settings are not visible.

Field circuit parameterization method. Options are:

• `Field circuit voltage` — Specify the field circuit in terms of voltage.

• `Field circuit current` — Specify the field circuit in terms of current.

This parameter affects the visibility of the Field circuit voltage and Field circuit current parameters.

#### Dependencies

If this parameter is set to ```Field circuit voltage```.

• The Field circuit voltage parameter is visible.

• The Field circuit current parameter is not visible.

If this parameter is set to ```Field circuit current```:

• The Field circuit current parameter is visible.

• The Field circuit voltage parameter is not visible.

Voltage across field circuit which produces rated voltage at machine terminals.

#### Dependencies

This parameter is visible only if the Specify field circuit input required to produce rated terminal voltage at no load by parameter is set to ```Field circuit voltage```.

Current in field circuit which produces rated voltage at machine terminals.

#### Dependencies

This parameter is visible only if the Specify field circuit input required to produce rated terminal voltage at no load by parameter is set to ```Field circuit current```.

Zero-sequence model:

• `Include` — Prioritize model fidelity. This model is the default zero-sequence model. 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 Specify parameterization by is set to Fundamental parameters — The Stator zero-sequence inductance, L0 parameter in the Impedances settings is visible.

• `Include` and Specify parameterization by is set to Standard parameters — The zero-sequence reactance, X0 parameter in the Impedances settings is visible.

• `Exclude` — The zero-sequence parameter in the Impedances settings is not visible.

Reference point for the rotor angle measurement.

When you select the default value, the rotor d-axis and stator a-phase magnetic axis are aligned when the rotor angle is zero.

The other value you can choose for this parameter is `Angle between the a-phase magnetic axis and the q-axis`. When you select this value, the rotor q-axis and stator a-phase magnetic axis are aligned when the rotor angle is zero.

### Impedances

Unsaturated stator d-axis mutual inductance. If Magnetic saturation representation is set to `None`, this is equivalent to the stator d-axis mutual inductance. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify parameterization by parameter is set to `Fundamental parameters`.

Stator q-axis mutual inductance. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify parameterization by parameter is set to `Fundamental parameters`.

Stator zero-sequence inductance. This parameter must be greater than 0.

#### Dependencies

This parameter is visible if the Specify parameterization by parameter is set to `Fundamental parameters` and the Zero Sequence parameter is set to `Include`.

Stator leakage inductance. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify parameterization by parameter is set to `Fundamental parameters`.

Stator resistance. This parameter must be greater than 0.

Rotor field circuit inductance. This parameter must be greater than 0.

Rotor field circuit resistance. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify parameterization by parameter is set to `Fundamental parameters`.

Rotor d-axis damper winding 1 inductance. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify parameterization by parameter to `Fundamental parameters`.

Rotor d-axis damper winding 1 resistance. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify parameterization by parameter is set to `Fundamental parameters`.

Rotor q-axis damper winding 1 inductance. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify parameterization by parameter is set to `Fundamental parameters`.

Rotor q-axis damper winding 1 resistance. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify parameterization by parameter is set to `Fundamental parameters`.

Stator leakage reactance. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify parameterization by parameter to `Standard parameters`.

d-axis synchronous reactance. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify parameterization by parameter to `Standard parameters`.

q-axis synchronous reactance. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify parameterization by parameter to `Standard parameters`.

Zero-sequence reactance. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify parameterization by parameter to `Standard parameters` and the Zero Sequence parameter is set to `Include`.

d-axis transient reactance. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify parameterization by parameter is set to `Standard parameters`.

d-axis subtransient reactance. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify parameterization by parameter is set to `Standard parameters`.

q-axis subtransient reactance. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify parameterization by parameter is set to `Standard parameters`.

### Time Constants

Select between `Open circuit` and `Short circuit`.

The setting for this parameter affects the visibility of the d-axis time constant parameters.

d-axis transient open-circuit time constant. This parameter must be:

• Greater than 0.

• Greater than d-axis subtransient open-circuit, Td0''.

#### Dependencies

This parameter is visible only if the Specify d-axis transient time constant parameter is set to `Open circuit`.

d-axis transient short-circuit time constant. This parameter must be:

• Greater than 0.

• Greater than d-axis subtransient short-circuit, Td''.

#### Dependencies

This parameter is visible only if the Specify d-axis transient time constant parameter is set to `Short circuit`.

d-axis subtransient open-circuit time constant. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify d-axis transient time constant parameter is set to `Open circuit`.

d-axis subtransient short-circuit time constant. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify d-axis transient time constant parameter is set to `Short circuit`.

Select between `Open circuit` and `Short circuit`.

The setting for this parameter affects the visibility of the q-axis time constant parameters.

q-axis subtransient open-circuit time constant. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify q-axis transient time constant parameter is set to `Open circuit`.

q-axis subtransient short-circuit time constant. This parameter must be greater than 0.

#### Dependencies

This parameter is visible only if the Specify q-axis transient time constant parameter is set to `Short circuit`.

### Saturation

Block magnetic saturation model:

• `None`

• ```Open-circuit lookup table (v versus i)```

#### Dependencies

If you set this parameter to ```Open-circuit lookup table (v versus i)```, related parameters are visible.

Field current, ifd, data that populates the air-gap voltage, Vag, versus field current, ifd, lookup table. This parameter must contain a vector with a number of elements between `5` and `10`.

#### Dependencies

This parameter is visible only if the Magnetic saturation representation parameter is set to ```Open-circuit lookup table (v versus i)```.

Air-gap voltage, Vag, data that populates the air-gap voltage, Vag, versus field current, ifd, lookup table. This parameter must contain a vector with a number of elements between `5` and `10`.

#### Dependencies

This parameter is visible only if the Magnetic saturation representation parameter is set to ```Open-circuit lookup table (v versus i)```.

### Initial Conditions

Model for specifying values for certain parameters and variables at the start of simulation. To:

Set an operating point regardless of the connected network, select ```Set real power, reactive power, terminal voltage and terminal phase```.

• Specify the priority and initial target values for block variables before simulation using the Variables settings, select ```Set targets for rotor angle and Park's transform variables```. For more information, see Set Priority and Initial Target for Block Variables.

• Select a bus type and specify the related parameters for a load-flow analysis in the Initial Conditions settings, select ```Set targets for load flow variables```.

#### Dependencies

If you set this parameter to:

• ```Set targets for rotor angle and Park's transform variables``` — The Variables settings become visible.

• ```Set real power, reactive power, terminal voltage, and terminal phase``` — Related parameters become visible.

• ```Set targets for load flow variables``` — Related parameters become visible.

Type of voltage source that the block models.

#### Dependencies

This parameter is visible only if the Initialization option parameter is set to ```Set targets for load flow variables``` and the Source type parameter is set to ```Swing bus``` or `PV bus`.

The visibility of Terminal voltage magnitude, Terminal voltage angle, Active power generated, Reactive power generated, Minimum terminal voltage magnitude (pu, Phase search range at terminals, and Phase search range at terminals depend on the value that you choose for this parameter.

Terminal voltage magnitude.

#### Dependencies

This parameter is visible only if the Initialization option parameter is set to ```Set real power, reactive power, terminal voltage, and terminal phase``` or if the Initialization option parameter is set to ```Set targets for load flow variables``` and the Source type parameter is set to ```Swing bus``` or `PV bus`.

Terminal voltage angle.

#### Dependencies

This parameter is visible only if the Initialization option parameter is set to ```Set real power, reactive power, terminal voltage, and terminal phase``` or if the Initialization option parameter is set to ```Set targets for load flow variables``` and the Source type parameter is set to ```Swing bus```.

Active power generated.

#### Dependencies

This parameter is visible only if the Initialization option parameter is set to ```Set real power, reactive power, terminal voltage, and terminal phase``` or if the Initialization option parameter is set to ```Set targets for load flow variables``` and the Source type parameter is set to ```PV bus``` or `PQ bus`.

Reactive power generated.

#### Dependencies

This parameter is visible only if the Initialization option parameter is set to ```Set real power, reactive power, terminal voltage, and terminal phase``` or if the Initialization option parameter is set to ```Set targets for load flow variables``` and the Source type parameter is set to ```PQ bus```.

#### Dependencies

This parameter is visible only if the Initialization option parameter is set to ```Set targets for load flow variables``` and the Source type parameter is set to ```PQ bus```.

Vector that defines the phase angle search range.

#### Dependencies

This parameter is visible only if the Initialization option parameter is set to ```Set targets for load flow variables``` and the Source type parameter is set to ```PV bus``` or `PQ bus`.

Parasitic conductance to the electrical reference.

#### Dependencies

This parameter is visible only if the Initialization option parameter is set to ```Set targets for load flow variables```.

### Thermal

To enable these parameters, set Modeling option to `Show thermal port`.

Temperature for which motor parameters are quoted.

Coefficient α in the equation relating resistance to temperature for all three windings, as described in Thermal Model for Actuator Blocks. The default value, `3.93e-3` 1/K, is for copper.

Thermal mass value for each stator winding. The thermal mass is the energy required to raise the temperature by one degree.

Thermal mass of the rotor. The thermal mass is the energy required to raise the temperature of the rotor by one degree.

## References

[1] Kundur, P. Power System Stability and Control. New York, NY: McGraw Hill, 1993.

[2] Lyshevski, S. E. Electromechanical Systems, Electric Machines and Applied Mechatronics. Boca Raton, FL: CRC Press, 1999.

## Version History

Introduced in R2013b

expand all

Behavior changed in R2021b