# Synchronous Machine Model 1.0

Synchronous machine with field circuit and no damper

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• 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.

### Motor Construction

The diagram shows the motor construction with a single pole pair on the rotor. For the axes convention, when rotor mechanical angle θr is zero, the a-phase and permanent 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:

• va, vb, and vc are the individual phase voltages across the stator windings.

• Rs is the equivalent resistance of each stator winding.

• ia, ib, and ic are the currents flowing in the stator windings.

• $\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:

• vf is the individual phase voltage across the field winding.

• Rf is the equivalent resistance of the field winding.

• if is the current flowing in the field winding.

• $\frac{d{\psi }_{f}}{dt}$ is the rate of change of magnetic flux in the field winding.

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

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

where:

• ψa, ψb, and ψc are the total fluxes linking each stator winding.

• Laa, Lbb, and Lcc are the self-inductances of the stator windings.

• Lab, Lac, Lba, Lbc, Lca, and Lcb are the mutual inductances of the stator windings.

• ψam, ψbm, and ψcm are the magnetization fluxes linking the stator windings.

• Lamf, Lbmf, and Lcmf are the mutual inductances of the field winding.

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

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

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

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

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

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

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

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

where:

• N is the number of rotor pole pairs.

• θr is the rotor mechanical angle.

• θe is the rotor electrical angle.

• 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.

• Ls is the stator self-inductance per phase. This value is the average self-inductance of each of the stator windings.

• Lm is the stator inductance fluctuation. This value is the fluctuation in self-inductance and mutual inductance with changing rotor angle.

• Ms is the stator mutual inductance. This value is the average mutual inductance between the stator windings.

The magnetization flux linking winding, a-a’ is a maximum when θe = 0° and zero when θe = 90°. Therefore:

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

and

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

where:

• Lmf is the mutual field armature inductance.

• ψf is the flux linking the field winding.

• Lf is the field winding inductance.

• ${\left[{L}_{mf}\right]}^{T}$ is the transform of the Lmf vector, that is,

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

### 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:

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

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

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

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

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

and

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

where:

• vd, vq, and v0 are the d-axis, q-axis, and zero-sequence voltages. These voltages are defined by

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

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

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

• Ld is the stator d-axis inductance. Ld = Ls + Ms + 3/2 Lm.

• ω is the mechanical rotational speed.

• Lq is the stator q-axis inductance. Lq = Ls + Ms − 3/2 Lm.

• L0 is the stator zero-sequence inductance. L0 = Ls – 2Ms.

• 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.

• TL is the load torque.

• Bm is the rotor damping.

### Thermal Ports

The block has four optional thermal ports, one for each of the three stator windings and one for the rotor. These ports are hidden by default. To expose the thermal ports, right-click the block in your model, select Simscape > Block choices, and then select the desired block variant with thermal ports: Composite three-phase ports | Show thermal port or Expanded three-phase ports | Show thermal port. This action displays the thermal ports on the block icon, and exposes the Thermal parameters. These parameters are described further on this reference page.

Use the thermal ports to simulate the effects of copper resistance and iron losses that convert electrical power to heat. For more information on using thermal ports in actuator blocks, see Simulating Thermal Effects in Rotational and Translational Actuators.

### Variables

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

## Assumptions

Flux distribution is sinusoidal.

## Ports

### Conserving

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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.

Electrical conserving port associated with the neutral phase.

#### Dependencies

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

Electrical conserving port associated with the field winding positive terminal.

Electrical conserving port associated with the field winding negative terminal.

Thermal conserving port associated with winding A. To expose the thermal ports, right-click the block in your model, select Simscape > Block choices, and then select the desired block variant with thermal ports. For more information, see Thermal Ports.

Thermal conserving port associated with winding B. To expose the thermal ports, right-click the block in your model, select Simscape > Block choices, and then select the desired block variant with thermal ports. For more information, see Thermal Ports.

Thermal conserving port associated with winding C. To expose the thermal ports, right-click the block in your model, select Simscape > Block choices, and then select the desired block variant with thermal ports. For more information, see Thermal Ports.

Thermal conserving port associated with the rotor. To expose the thermal ports, right-click the block in your model, select Simscape > Block choices, and then select the desired block variant with thermal ports. For more information, see Thermal Ports.

## Parameters

expand all

### Main

Select the modeling fidelity:

• `Constant Ld, Lq, Lmf and Lf`Ld, Lq, Lmf, Lf, and PM values are constant and defined by their respective parameters.

• `Tabulated Ld, Lq, Lmf and Lf`Ld, Lq, Lmf, Lf, and PM values are computed online from DQ and field currents look-up tables as follows:

`${L}_{d}={f}_{1}\left({i}_{d},{i}_{q},{i}_{f}\right)$`

`${L}_{q}={f}_{2}\left({i}_{d},{i}_{q},{i}_{f}\right)$`

`${L}_{mf}={f}_{3}\left({i}_{d},{i}_{q},{i}_{f}\right)$`

`${L}_{f}={f}_{4}\left({i}_{f}\right)$`

#### Dependencies

This parameter is visible only when you set the Stator parameterization parameter to ```Specify Ld, Lq, and L0```.

Number of permanent magnet pole pairs on the rotor.

Stator parameterization method.

#### Dependencies

The Stator parameterization setting affects the visibility of other parameters.

Direct-axis inductance of the machine stator.

#### Dependencies

This parameter is visible only when you set the Stator parameterization parameter to ```Specify Ld, Lq, and L0``` and the Modeling fidelity parameter to ```Constant Ld, Lq, Lmf and Lf```.

Quadrature-axis inductance of the machine stator.

#### Dependencies

This parameter is visible only when you set the Stator parameterization parameter to ```Specify Ld, Lq, and L0``` and the Modeling fidelity parameter to ```Constant Ld, Lq, Lmf and Lf```.

Direct-axis current vector, iD.

#### Dependencies

This parameter is visible only when you set the Stator parameterization parameter to ```Specify Ld, Lq, and L0``` and the Modeling fidelity parameter to ```Tabulated Ld, Lq, Lmf and Lf```.

#### Dependencies

This parameter is visible only when you set the Stator parameterization parameter to ```Specify Ld, Lq, and L0``` and the Modeling fidelity parameter to ```Tabulated Ld, Lq, Lmf and Lf```.

Excitation current vector, iF.

#### Dependencies

This parameter is visible only when you set the Stator parameterization parameter to ```Specify Ld, Lq, and L0``` and the Modeling fidelity parameter to ```Tabulated Ld, Lq, Lmf and Lf```.

Ld matrix.

#### Dependencies

This parameter is visible only when you set the Stator parameterization parameter to ```Specify Ld, Lq, and L0``` and the Modeling fidelity parameter to ```Tabulated Ld, Lq, Lmf and Lf```.

Lq matrix.

#### Dependencies

This parameter is visible only when you set the Stator parameterization parameter to ```Specify Ld, Lq, and L0``` and the Modeling fidelity parameter to ```Tabulated Ld, Lq, Lmf and Lf```.

Mutual field armature inductance.

#### Dependencies

This parameter is visible only when you set the Stator parameterization parameter to ```Specify Ld, Lq, and L0``` and the Modeling fidelity parameter to ```Tabulated Ld, Lq, Lmf and Lf```.

Lf vector.

#### Dependencies

This parameter is visible only when you set the Stator parameterization parameter to ```Specify Ld, Lq, and L0```, the Modeling fidelity parameter to ```Tabulated Ld, Lq, Lmf and Lf```.

Zero-axis inductance for the machine stator.

#### Dependencies

This parameter is visible only if Stator parameterization is set to ```Specify Ld, Lq and L0``` and Zero sequence is set to `Include`.

Average self-inductance of the three stator windings. This parameter must be:

• Greater than 0.

• Greater than the magnitude of the Stator inductance fluctuation, Lm.

• Greater than the magnitude of the Stator mutual inductance, Ms.

#### Dependencies

This parameter is visible only if Stator parameterization is set to ```Specify Ls, Lm, and Ms```.

Fluctuation in self-inductance and mutual inductance with the rotor angle.

#### Dependencies

This parameter is visible only if Stator parameterization is set to ```Specify Ls, Lm, and Ms```.

Average mutual inductance between the stator windings.

#### Dependencies

This parameter is visible only if Stator parameterization is set to ```Specify Ls, Lm, and Ms```.

Inductance of the field winding.

#### Dependencies

This parameter is visible only when you set the Stator parameterization parameter to ```Specify Ld, Lq, and L0``` and the Modeling fidelity parameter to ```Constant Ld, Lq, Lmf and Lf``` or Stator parameterization parameter to ```Specify Ls, Lm, and Ms```

Armature-field mutual inductance.

#### Dependencies

This parameter is visible only when you set the Stator parameterization parameter to ```Specify Ld, Lq, and L0``` and the Modeling fidelity parameter to ```Constant Ld, Lq, Lmf and Lf``` or Stator parameterization parameter to ```Specify Ls, Lm, and Ms```

Resistance of each of the stator windings.

Resistance of the field winding.

Zero-sequence model:

• `Include` — Prioritize model fidelity. 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 Stator parameterization is set to Specify Ld, Lq, and L0 — The Stator zero-sequence inductance, L0 parameter is visible.

• `Exclude` — The Stator zero-sequence inductance, L0 parameter is not visible.

### Mechanical

Inertia of the rotor.

Damping of the rotor.

Reference point for the rotor angle measurement. If you select the default value, the rotor and a-phase fluxes are aligned for a zero-rotor angle. Otherwise, an a-phase current generates the maximum torque value for a zero-rotor angle.

### Thermal

These parameters appear only for blocks with exposed thermal ports. For more information, see Thermal Ports.

The temperature for which motor parameters are quoted.

Coefficient α in the equation relating resistance to temperature, as described in Thermal Model for Actuator Blocks. The default value is for copper.

The thermal mass value for the A, B, and C windings. The thermal mass is the energy required to raise the temperature by one degree.

The thermal mass of the rotor, that 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] 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.

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