# Synchronous Machine Round Rotor (fundamental)

Round-rotor synchronous machine with fundamental parameterization

## Library

Machines / Synchronous Machine (Round Rotor)

## Description

The Synchronous Machine Round Rotor (fundamental) block models a round-rotor 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

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

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

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

and

`${e}_{2}{}_{q}=\frac{1}{{\omega }_{base}}\frac{d{\Psi }_{2q}}{dt}+{R}_{2q}{i}_{2q}=0,$`

where:

• efd is the field voltage.

• e1d, e1q, and e2q are the voltages across the d-axis damper winding 1, q-axis damper winding 1, and q-axis damper winding 2. They are all equal to 0.

• ψfd, ψ1d, ψ1q, and ψ2q are the magnetic fluxes linking the field circuit, d-axis damper winding 1, q-axis damper winding 1, and q-axis damper winding 2.

• Rfd, R1d, R1q, and R2q are the resistances of rotor field circuit, d-axis damper winding 1, q-axis damper winding 1, and q-axis damper winding 2.

• ifd, i1d, i1q, and i2q are the currents flowing in the field circuit, d-axis damper winding 1, q-axis damper winding 1, and q-axis damper winding 2.

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),

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

and

`${L}_{aq}={K}_{s}*{L}_{aqu},$`

where:

• ψad is the d-axis air-gap or mutual flux linkage.

• 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.

• Laqu is the unsaturated mutual inductance of the stator q-axis.

• Laq is the mutual inductance of the stator q-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

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

`$\Psi q=-\left({L}_{aq}+{L}_{l}\right){i}_{q}\text{​}+{L}_{aq}{i}_{1q}+{L}_{aq}{i}_{2q},$`

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

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

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

`${\psi }_{1q}={L}_{11q}{i}_{1q}+{L}_{aq}{i}_{2q}-{L}_{aq}{i}_{q},$`

and

`${\psi }_{2q}={L}_{aq}{i}_{1q}+{L}_{22q}{i}_{2q}-{L}_{aq}{i}_{q},$`

where:

• Lffd, L11d, L11q, and L22q are the self-inductances of the rotor field circuit, d-axis damper winding 1, q-axis damper winding 1 and q-axis damper winding 2. Lf1d is the rotor field circuit and d-axis damper winding 1 mutual inductance. They are defined by the following equations.

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

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

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

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

`${L}_{22q}={L}_{aq}+{L}_{2q}$`

These 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 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 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

## Parameters

### Main Tab

Rated apparent power

Rated apparent power. The default value is `555e6` `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 `1`.

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

Select between

• `Field circuit voltage`

• `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 `1300` `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 `92.95` V.

### Impedances Tab

Stator d-axis mutual inductance (unsaturated), Ladu

Unsaturated stator d-axis mutual inductance. If Magnetic saturation representation is set to `None`, this is equivalent to the stator d-axis mutual inductance. The default value is `1.66` pu.

Stator q-axis mutual inductance (unsaturated), Laqu

Unsaturated stator q-axis mutual inductance. If Magnetic saturation representation is set to `None`, this is equivalent to the stator q-axis mutual inductance. The default value is `1.61` pu.

Stator zero-sequence inductance, L0

Stator zero-sequence inductance. 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 `0.003` pu.

Rotor field circuit inductance, Lfd

Rotor field circuit inductance. The default value is `0.165` 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 `0.1713` pu.

Rotor d-axis damper winding 1 resistance, R1d

Rotor d-axis damper winding 1 resistance. The default value is `0.0284` pu.

Rotor q-axis damper winding 1 inductance, L1q

Rotor q-axis damper winding 1 inductance. The default value is `0.7252` pu.

Rotor q-axis damper winding 1 resistance, R1q

Rotor q-axis damper winding 1 resistance. The default value is `0.00619` pu.

Rotor q-axis damper winding 2 inductance, L2q

Rotor q-axis damper winding 2 inductance. The default value is `0.125` pu.

Rotor q-axis damper winding 2 resistance, R2q

Rotor q-axis damper winding 2 resistance. The default value is `0.02368` 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, 0.80, 1.08, 1.31, 1.40]``` 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 you set Specify initialization by to ```Electrical power and voltage output```. The default value is `500e6` `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.