# Hybrid Excitation Synchronous Machine

Hybrid excitation synchronous machine with three-phase wye-wound stator

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## Description

The Hybrid Excitation Synchronous Machine block represents a hybrid excitation synchronous machine with a three-phase wye-wound stator. Permanent magnets and excitation windings provide the machine excitation. 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

`${\theta }_{e}=N{\theta }_{r},$`
`${L}_{aa}={L}_{s}+{L}_{m}\text{cos}\left(2{\theta }_{e}\right),$`
`${L}_{bb}={L}_{s}+{L}_{m}\text{cos}\left(2\left({\theta }_{e}-2\pi /3\right)\right),$`
`${L}_{cc}={L}_{s}+{L}_{m}\text{cos}\left(2\left({\theta }_{e}+2\pi /3\right)\right),$`
`${L}_{ab}={L}_{ba}=-{M}_{s}-{L}_{m}\mathrm{cos}\left(2\left({\theta }_{e}+\pi /6\right)\right),$`
`${L}_{bc}={L}_{cb}=-{M}_{s}-{L}_{m}\mathrm{cos}\left(2\left({\theta }_{e}+\pi /6-2\pi /3\right)\right),$`
`${L}_{ca}={L}_{ac}=-{M}_{s}-{L}_{m}\mathrm{cos}\left(2\left({\theta }_{r}+\pi /6+2\pi /3\right)\right),$`

where:

• N is the number of rotor pole pairs.

• θr is the rotor mechanical angle.

• θe is the rotor electrical angle.

• 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 amplitude of 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 θr = 0° and zero when θr = 90°. Therefore:

`${\psi }_{m}=\left[\begin{array}{c}{\psi }_{am}\\ {\psi }_{bm}\\ {\psi }_{cm}\end{array}\right]=\left[\begin{array}{c}{\psi }_{m}\mathrm{cos}{\theta }_{r}\\ {\psi }_{m}\mathrm{cos}\left({\theta }_{r}-2\pi /3\right)\\ {\psi }_{m}\mathrm{cos}\left({\theta }_{r}+2\pi /3\right)\end{array}\right],$`
`${L}_{mf}=\left[\begin{array}{c}{L}_{amf}\\ {L}_{bmf}\\ {L}_{cmf}\end{array}\right]=\left[\begin{array}{c}{L}_{mf}\mathrm{cos}{\theta }_{r}\\ {L}_{mf}\mathrm{cos}\left({\theta }_{r}-2\pi /3\right)\\ {L}_{mf}\mathrm{cos}\left({\theta }_{r}+2\pi /3\right)\end{array}\right],$`
and
`${\Psi }_{f}={L}_{f}{i}_{f}+{L}_{mf}^{T}\left[\begin{array}{c}{i}_{a}\\ {i}_{b}\\ {i}_{c}\end{array}\right],$`
where:

• ψm is the linked motor flux.

• 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

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

The inverse of the Park transformation is defined by

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

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

`${v}_{d}={R}_{s}{i}_{d}+{L}_{d}\frac{d{i}_{d}}{dt}+{L}_{mf}\frac{d{i}_{f}}{dt}-N\omega {i}_{q}{L}_{q},$`
`${v}_{q}={R}_{s}{i}_{q}+{L}_{q}\frac{d{i}_{q}}{dt}+N\omega \left({i}_{d}{L}_{d}+{\psi }_{m}+{i}_{f}{L}_{mf}\right),$`
`${v}_{0}={R}_{s}{i}_{0}+{L}_{0}\frac{d{i}_{0}}{dt},$`
`${v}_{f}={R}_{f}{i}_{f}+{L}_{f}\frac{d{i}_{f}}{dt}+\frac{3}{2}{L}_{mf}\frac{d{i}_{d}}{dt},$`
`$T=\frac{3}{2}N\left({i}_{q}\left({i}_{d}{L}_{d}+{\psi }_{m}+{i}_{f}{L}_{mf}\right)-{i}_{d}{i}_{q}{L}_{q}\right),$`
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 Hybrid Excitation Synchronous Machine 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.

### Assumptions

The block assumes that the 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.

Electrical conserving port associated with the field winding positive terminal.

Electrical conserving port associated with the field winding negative terminal.

## Parameters

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### Main

Number of permanent magnet pole pairs on the rotor.

Peak permanent magnet flux linkage for any of the stator windings.

Method for parameterizing the stator.

#### Dependencies

Selecting `Specify Ld, Lq and L0` enables these parameters:

• Stator d-axis inductance, Ld

• Stator q-axis inductance, Lq

• Stator zero-sequence inductance, L0

Selecting `Specify Ls, Lm, and Ms` enables these parameters:

• Stator self-inductance per phase, Ls

• Stator inductance fluctuation, Lm

• Stator mutual inductance, Ms

Direct-axis inductance of the machine stator.

#### Dependencies

To enable this parameter, set Stator parameterization to ```Specify Ld, Lq and L0```.

Quadrature-axis inductance of the machine stator.

#### Dependencies

To enable this parameter, set Stator parameterization to ```Specify Ld, Lq and L0```.

Zero-axis inductance for the machine stator.

#### Dependencies

To enable this parameter, set Stator parameterization to ```Specify Ld, Lq and L0```.

Average self-inductance of the three stator windings.

#### Dependencies

To enable this parameter, set Stator parameterization to ```Specify Ls, Lm, and Ms```.

Amplitude of the fluctuation in self-inductance and mutual inductance with the rotor angle.

#### Dependencies

To enable this parameter, set Stator parameterization to ```Specify Ls, Lm, and Ms```.

Average mutual inductance between the stator windings.

#### Dependencies

To enable this parameter, set Stator parameterization to ```Specify Ls, Lm, and Ms```.

Inductance of the field winding.

Armature-field mutual inductance.

Resistance of each of the stator windings.

Resistance of the field winding.

### Mechanical

Inertia of the rotor.

Damping of the rotor.

### Initial Conditions

Initial d-, q-, 0-sequence and field winding currents.

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.

Rotor angle at simulation start time.

Rotor speed at simulation start time. If the rotor inertia, J, is zero, the initial speed of the rotor is zero rpm and the initial rotor speed is ignored.

## References

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

[2] Mbayed, R. Analysis of Faulted Power Systems. Hoboken, NJ: Wiley-IEEE Press, 1995.

[3] Anderson, P. M. Contribution to the Control of the Hybrid Excitation Synchronous Machine for Embedded Applications. Universite de Cergy Pontoise, 2012.

[4] Luo, X. and T. A. Lipo. “A Synchronous/Permanent Magnet Hybrid AC Machine.” IEEE Transactions of Energy Conversion. Vol. 15, No 2 (2000), pp. 203–210.