# Synchronous Machine Model 1.0

Synchronous machine with field circuit and no damper

**Libraries:**

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.

You can also model the stator windings either in a delta-wound or in an open-end
configuration by setting **Winding type** to
`Delta-wound`

or `Open-end`

,
respectively.

### 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 rotor 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

$$\left[\begin{array}{c}{v}_{a}\\ {v}_{b}\\ {v}_{c}\end{array}\right]=\left[\begin{array}{ccc}{R}_{s}& 0& 0\\ 0& {R}_{s}& 0\\ 0& 0& {R}_{s}\end{array}\right]\left[\begin{array}{c}{i}_{a}\\ {i}_{b}\\ {i}_{c}\end{array}\right]+\left[\begin{array}{c}\frac{d{\psi}_{a}}{dt}\\ \frac{d{\psi}_{b}}{dt}\\ \frac{d{\psi}_{c}}{dt}\end{array}\right],$$

where:

*v*,_{a}*v*, and_{b}*v*are the individual phase voltages across the stator windings._{c}*R*is the equivalent resistance of each stator winding._{s}*i*,_{a}*i*, and_{b}*i*are the currents flowing in the stator windings._{c}$$\frac{d{\psi}_{a}}{dt},$$$$\frac{d{\psi}_{b}}{dt},$$ and $$\frac{d{\psi}_{c}}{dt}$$ are the rates of change of magnetic flux in each stator winding.

The voltage across the field winding is expressed as

$${v}_{f}={R}_{f}{i}_{f}+\frac{d{\psi}_{f}}{dt},$$

where:

*v*is the individual phase voltage across the field winding._{f}*R*is the equivalent resistance of the field winding._{f}*i*is the current flowing in the field winding._{f}$$\frac{d{\psi}_{f}}{dt}$$ is the rate of change of magnetic flux in the field winding.

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

$$\left[\begin{array}{c}{\psi}_{a}\\ {\psi}_{b}\\ {\psi}_{c}\end{array}\right]=\left[\begin{array}{ccc}{L}_{aa}& {L}_{ab}& {L}_{ac}\\ {L}_{ba}& {L}_{bb}& {L}_{bc}\\ {L}_{ca}& {L}_{cb}& {L}_{cc}\end{array}\right]\left[\begin{array}{c}{i}_{a}\\ {i}_{b}\\ {i}_{c}\end{array}\right]+\left[\begin{array}{c}{L}_{amf}\\ {L}_{bmf}\\ {L}_{cmf}\end{array}\right]{i}_{f},$$

where:

*ψ*,_{a}*ψ*, and_{b}*ψ*are the total fluxes linking each stator winding._{c}*L*,_{aa}*L*, and_{bb}*L*are the self-inductances of the stator windings._{cc}*L*,_{ab}*L*,_{ac}*L*,_{ba}*L*,_{bc}*L*, and_{ca}*L*are the mutual inductances of the stator windings._{cb}*L*,_{amf}*L*, and_{bmf}*L*are the mutual inductances of the field winding._{cmf}

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

${\theta}_{e}=N{\theta}_{r}+rotor\text{\hspace{0.17em}}offset$

$${L}_{aa}={L}_{s}+{L}_{m}\text{cos}(2{\theta}_{e}),$$

${L}_{bb}={L}_{s}+{L}_{m}\text{cos}(2\left({\theta}_{e}-2\pi /3\right)),$

$${L}_{cc}={L}_{s}+{L}_{m}\text{cos}(2\left({\theta}_{e}+2\pi /3\right)),$$

$${L}_{ab}={L}_{ba}=-{M}_{s}-{L}_{m}\mathrm{cos}\left(2\left({\theta}_{e}+\pi /6\right)\right),$$

${L}_{bc}={L}_{cb}=-{M}_{s}-{L}_{m}\mathrm{cos}\left(2\left({\theta}_{e}+\pi /6-2\pi /3\right)\right),$

${L}_{ca}={L}_{ac}=-{M}_{s}-{L}_{m}\mathrm{cos}\left(2\left({\theta}_{e}+\pi /6+2\pi /3\right)\right),$

where:

*N*is the number of rotor pole pairs.

*θ*is the rotor mechanical angle._{r}

*θ*is the rotor electrical angle._{e}*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*is the stator self-inductance per phase. This value is the average self-inductance of each of the stator windings._{s}*L*is the stator inductance fluctuation. This value is the fluctuation in self-inductance and mutual inductance with changing rotor angle._{m}*M*is the stator mutual inductance. This value is the average mutual inductance between the stator windings._{s}

The magnetization flux linking winding, *a-a’* is a maximum when
*θ _{e}* = 0° and zero when

*θ*= 90°. Therefore:

_{e}$${L}_{mf}=\left[\begin{array}{c}{L}_{amf}\\ {L}_{bmf}\\ {L}_{cmf}\end{array}\right]=\left[\begin{array}{c}{L}_{mf}\mathrm{cos}{\theta}_{e}\\ {L}_{mf}\mathrm{cos}\left({\theta}_{e}-2\pi /3\right)\\ {L}_{mf}\mathrm{cos}\left({\theta}_{e}+2\pi /3\right)\end{array}\right]$$

and

$${\Psi}_{f}={L}_{f}{i}_{f}+{L}_{mf}^{T}\left[\begin{array}{c}{i}_{a}\\ {i}_{b}\\ {i}_{c}\end{array}\right],$$

where:

*L*is the mutual field armature inductance._{mf}*ψ*is the flux linking the field winding._{f}*L*is the field winding inductance._{f}$${\left[{L}_{mf}\right]}^{T}$$ is the transform of the

*L*vector, that is,_{mf}$${\left[{L}_{mf}\right]}^{T}={\left[\begin{array}{c}{L}_{amf}\\ {L}_{bmf}\\ {L}_{cmf}\end{array}\right]}^{T}=\left[\begin{array}{ccc}{L}_{amf}& {L}_{bmf}& {L}_{cmf}\end{array}\right].$$

### 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

$P=2/3\left[\begin{array}{ccc}\mathrm{cos}{\theta}_{e}& \mathrm{cos}\left({\theta}_{e}-2\pi /3\right)& \mathrm{cos}\left({\theta}_{e}+2\pi /3\right)\\ -\mathrm{sin}{\theta}_{e}& -\mathrm{sin}\left({\theta}_{e}-2\pi /3\right)& -\mathrm{sin}\left({\theta}_{e}+2\pi /3\right)\\ 0.5& 0.5& 0.5\end{array}\right]$

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

${v}_{d}={R}_{s}{i}_{d}+{L}_{d}\frac{d{i}_{d}}{dt}+{L}_{mf}\frac{d{i}_{f}}{dt}-N\omega {i}_{q}{L}_{q},$

${v}_{q}={R}_{s}{i}_{q}+{L}_{q}\frac{d{i}_{q}}{dt}+N\omega ({i}_{d}{L}_{d}+{i}_{f}{L}_{mf}),$

${v}_{0}={R}_{s}{i}_{0}+{L}_{0}\frac{d{i}_{0}}{dt}$

$${v}_{f}={R}_{f}{i}_{f}+{L}_{f}\frac{d{i}_{f}}{dt}+\frac{3}{2}{L}_{mf}\frac{d{i}_{d}}{dt},$$

$T=\frac{3}{2}N\left({i}_{q}\left({i}_{d}{L}_{d}+{i}_{f}{L}_{mf}\right)-{i}_{d}{i}_{q}{L}_{q}\right),$

and

$J\frac{d\omega}{dt}=T={T}_{L}-{B}_{m}\omega .$

where:

*v*,_{d}*v*, and_{q}*v*are the_{0}*d*-axis,*q*-axis, and zero-sequence voltages. These voltages are defined by$\left[\begin{array}{c}{v}_{d}\\ {v}_{q}\\ {v}_{0}\end{array}\right]=P\left[\begin{array}{c}{v}_{a}\\ {v}_{b}\\ {v}_{c}\end{array}\right].$

*i*,_{d}*i*, and_{q}*i*are the_{0}*d*-axis,*q*-axis, and zero-sequence currents, defined by$\left[\begin{array}{c}{i}_{d}\\ {i}_{q}\\ {i}_{0}\end{array}\right]=P\left[\begin{array}{c}{i}_{a}\\ {i}_{b}\\ {i}_{c}\end{array}\right].$

*L*is the stator_{d}*d*-axis inductance.*L*=_{d}*L*+_{s}*M*+ 3/2_{s}*L*._{m}*ω*is the mechanical rotational speed.*L*is the stator_{q}*q*-axis inductance.*L*=_{q}*L*+_{s}*M*− 3/2_{s}*L*._{m}*L*is the stator zero-sequence inductance._{0}*L*=_{0}*L*– 2_{s}*M*._{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*is the load torque._{L}*B*is the rotor damping._{m}

### Model Thermal Effects

You can expose thermal ports to model the effects of losses that convert power to heat. To
expose the thermal ports, set the **Modeling option** parameter to either:

`No thermal port`

— The block contains expanded electrical conserving ports associated with the stator windings, but does not contain thermal ports.`Show thermal port`

— The block contains expanded electrical conserving ports associated with the stator windings and thermal conserving ports for each of the windings and for the rotor.

For more information about using thermal ports in actuator blocks, see Simulating Thermal Effects in Rotational and Translational Actuators.

### Variables

To set the priority and initial target values for the block variables before simulation,
use the **Initial Targets** section in the block dialog box or Property
Inspector. For more information, see Set Priority and Initial Target for Block Variables.

Nominal values provide a way to specify the expected magnitude of a variable in a model.
Using system scaling based on nominal values increases the simulation robustness. You can
specify nominal values using different sources, including the **Nominal
Values** section in the block dialog box or Property Inspector. For more
information, see System Scaling by Nominal Values.

## Examples

## Assumptions

Flux distribution is sinusoidal.

## Ports

### Conserving

## Parameters

## References

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

[2] Anderson, P. M. *Analysis of Faulted Power Systems.* IEEE
Press, Power Systems Engineering, 1995.

[3] Retif, J. M., X. Lin-Shi, A. M. Llor, and F. Morand “New hybrid
direct-torque control for a winding rotor synchronous machine.” *2004
IEEE 35th Annual Power Electronics Specialists Conference.* Vol. 2
(2004): 1438–1442.

[4] IEEE Power Engineering Society. IEEE Std 1110-2002. *IEEE Guide for
Synchronous Generator Modeling Practices and Applications in Power System Stability
Analyses.* Piscataway, NJ: IEEE, 2002.

## Extended Capabilities

## Version History

**Introduced in R2018a**