# Synchronous Machine Salient Pole (fundamental)

Salient-pole synchronous machine with fundamental parameterization

## Library

Machines / Synchronous Machine (Salient Pole)

## Description

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

## Electrical Defining Equations

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

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

where:

• θe is the electrical angle.

• N is the number of pole pairs.

• θr is the rotor angle.

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

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

Park's 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

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

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

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, L11d, and L11q are the self-inductances of the rotor field circuit, d-axis damper winding 1, and q-axis damper winding 1. Lf1d is the rotor field circuit and d-axis damper winding 1 mutual inductance. They are defined by the following equations.

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

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

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

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

These equations assume that per-unit mutual inductance L12q = Laq, i.e., 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 Power Systems menu on the block context menu.

Right-click the block and, from the Power Systems menu, select an option:

• 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 is derived from parameters you specify:

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

• Per-unit field current saturation data, ifd

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

## Dialog Box and Parameters

### Main Tab

Rated apparent power

Rated apparent power. The default value is `300e6` `V*A`.

Rated voltage

RMS rated line-line voltage. The default value is `24e3` `V`.

Rated electrical frequency

Nominal electrical frequency at which rated apparent power is quoted. The default value is `60` `Hz`.

Number of pole pairs

Number of machine pole pairs. The default value is `10`.

Specify field circuit input required to produce rated terminal voltage at no load by

Choose between `Field circuit voltage` and ```Field circuit current```. The default value is ```Field circuit current```.

Field circuit current

This parameter is visible only when Specify field circuit input required to produce rated terminal voltage at no load by is set to `Field circuit current`. The default value is `1000` `A`.

Field circuit voltage

This parameter is visible only when Specify field circuit input required to produce rated terminal voltage at no load by is set to `Field circuit voltage`. The default value is `216.54` V.

### Impedances Tab

Stator d-axis mutual inductance (unsaturated), Ladu

Unsaturated stator d-axis mutual inductance, Ladu. If Magnetic saturation representation is set to `None`, this is equivalent to the stator d-axis mutual inductance, Lad. The default value is `0.9` pu.

Stator q-axis mutual inductance, Laq

Stator q-axis mutual inductance, Laq. The default value is `0.55` pu.

Stator zero-sequence inductance, L0

Stator zero-sequence inductance, L0. The default value is `0.15` pu.

Stator leakage inductance, Ll

Stator leakage inductance. The default value is `0.15` pu.

Stator resistance, Ra

Stator resistance. The default value is `0.011` pu.

Rotor field circuit inductance, Lfd

Rotor field circuit inductance. The default value is `0.2571` pu.

Rotor field circuit resistance, Rfd

Rotor field circuit resistance. The default value is `0.0006` pu.

Rotor d-axis damper winding 1 inductance, L1d

Rotor d-axis damper winding 1 inductance. The default value is `0.2` pu.

Rotor d-axis damper winding 1 resistance, R1d

Rotor d-axis damper winding 1 resistance. The default value is `0.0354` pu.

Rotor q-axis damper winding 1 inductance, L1q

Rotor q-axis damper winding 1 inductance. The default value is `0.2567` pu.

Rotor q-axis damper winding 1 resistance, R1q

Rotor q-axis damper winding 1 resistance. The default value is `0.0428` pu.

### Saturation Tab

Magnetic saturation representation

Block magnetic saturation representation. Options are:

• `None`

• ```Per-unit open-circuit lookup table (Vag versus ifd)```

The default value is `None`.

Per-unit field current saturation data, ifd

The field current, ifd, data populates the air-gap voltage, Vag, versus field current, ifd, lookup table. This parameter is only visible when you set Magnetic saturation representation to ```Per-unit open-circuit lookup table (Vag versus ifd)```. This parameter must contain a vector with at least five elements. The default value is ```[0.00, 0.48, 0.76, 1.38, 1.79]``` pu.

Per-unit air-gap voltage saturation data, Vag

The air-gap voltage, Vag, data populates the air-gap voltage, Vag, versus field current, ifd, lookup table. This parameter is only visible when you set Magnetic saturation representation to ```Per-unit open-circuit lookup table (Vag versus ifd)```. This parameter must contain a vector with at least five elements. The default value is ```[0.00 0.43 0.59 0.71 0.76]``` pu.

### Initial Conditions Tab

Specify initialization by

Select between `Electrical power and voltage output` and ```Mechanical and magnetic states```. The default value is ```Electrical power and voltage output.```

Terminal voltage magnitude

Initial RMS line-line voltage. This parameter is visible only when you set Specify initialization by to ```Electrical power and voltage output```. The default value is `24e3` `V`.

Terminal voltage angle

Initial voltage angle. This parameter is visible only when you set Specify initialization by to ```Electrical power and voltage output```. The default value is `0` `deg`.

Terminal active power

Initial active power. This parameter is visible only when Specify initialization by is set to ```Electrical power and voltage output```. The default value is `270e6` `V*A`.

Terminal reactive power

Initial reactive power. This parameter is visible only when you set Specify initialization by to ```Electrical power and voltage output```. The default value is `0` `V*A`.

Initial rotor angle

Initial rotor angle. During steady-state operation, set this parameter to the sum of the load angle and required terminal voltage offset. This parameter is visible only when you set Specify initialization by to ```Mechanical and magnetic states```. The default value is `0` `deg`.

Initial stator d-axis magnetic flux linkage

Stator d-axis initial flux linkage. This parameter is visible only when you set Specify initialization by to `Mechanical and magnetic states`. The default value is `0` pu.

Initial stator q-axis magnetic flux linkage

Stator q-axis initial flux linkage. This parameter is visible only when you set Specify initialization by to `Mechanical and magnetic states`. The default value is `0` pu.

Initial stator zero-sequence magnetic flux linkage

Zero-sequence initial flux linkage. This parameter is visible only when you set Specify initialization by to ```Mechanical and magnetic states```. The default value is `0` pu.

Initial field circuit magnetic flux linkage

Field circuit initial flux linkage. This parameter is visible only when you set Specify initialization by to ```Mechanical and magnetic states```. The default value is `0` pu.

Initial d-axis damper winding 1 magnetic flux linkage

The d-axis damper winding 1 initial flux linkage. This parameter is visible only when you set Specify initialization by to ```Mechanical and magnetic states```. The default value is `0` pu.

Initial q-axis damper winding 1 magnetic flux linkage

The q-axis damper winding 1 initial flux linkage. This parameter is visible only when you set Specify initialization by to ```Mechanical and magnetic states```. The default value is `0` pu.

Initial q-axis damper winding 2 magnetic flux linkage

The q-axis damper winding 2 initial flux linkage. This parameter is visible only when you set Specify initialization by to ```Mechanical and magnetic states```. The default value is `0` pu.

## Ports

The block has the following ports:

`fd+`

Electrical conserving port corresponding to the field winding positive terminal

`fd-`

Electrical conserving port corresponding to the field winding negative terminal

`R`

Mechanical rotational conserving port associated with the machine rotor

`C`

Mechanical rotational conserving port associated with the machine case

`pu`

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

• `pu_fd_Efd`

• `pu_fd_Ifd`

• `pu_torque`

• `pu_velocity`

• `pu_ed`

• `pu_eq`

• `pu_e0`

• `pu_id`

• `pu_iq`

• `pu_i0`

`~`

Expandable three-phase port associated with the stator windings

`n`

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

## 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.

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