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

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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.

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

Voltages across the stator windings are defined by

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

*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

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

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

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

*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}*ψ*,_{am}*ψ*, and_{bm}*ψ*are the magnetization fluxes linking the stator windings._{cm}*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

${\theta}_{e}=N{\theta}_{r},$

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

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

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

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

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

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

where:

*N*is the number of rotor pole pairs.

*θ*is the rotor mechanical angle._{r}

*θ*is the rotor electrical angle._{e}*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 amplitude of 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 _{r}* = 0° and zero when

$${\psi}_{m}=\left[\begin{array}{c}{\psi}_{am}\\ {\psi}_{bm}\\ {\psi}_{cm}\end{array}\right]=\left[\begin{array}{c}{\psi}_{m}\mathrm{cos}{\theta}_{r}\\ {\psi}_{m}\mathrm{cos}\left({\theta}_{r}-2\pi /3\right)\\ {\psi}_{m}\mathrm{cos}\left({\theta}_{r}+2\pi /3\right)\end{array}\right],$$

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

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

*ψ*is the linked motor flux._{m}*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}$${\left[{L}_{mf}\right]}^{T}={\left[\begin{array}{c}{L}_{amf}\\ {L}_{bmf}\\ {L}_{cmf}\end{array}\right]}^{T}=\left[\begin{array}{ccc}{L}_{amf}& {L}_{bmf}& {L}_{cmf}\end{array}\right].$$

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

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

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

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

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

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

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

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

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

*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 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.*T*is the load torque._{L}*B*is the rotor damping._{m}

The block assumes that the flux distribution is sinusoidal.

[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.

Brushless DC Motor | Permanent Magnet Synchronous Motor | Switched Reluctance Machine | Synchronous Machine Field Circuit (SI) | Synchronous Machine Field Circuit (pu) | Synchronous Machine Measurement | Synchronous Reluctance Machine

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