This block implements a squirrel-cage induction machine (SCIM)

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The SCIM block implements a three-phase induction machine (asynchronous machine) with a squirrel-cage rotor model with resolvers and encoders. The machine can operate in both motoring mode, when the mechanical torque is positive, and generating mode when the mechanical torque is negative.

Model Formulation

Q-d Transformation

The 3-phase to q-d transformation and the inverse used for the model are:

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This block implements a squirrel-cage induction machine (SCIM)

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The SCIM block implements a three-phase induction machine (asynchronous machine) with a squirrel-cage rotor model with resolvers and encoders. The machine can operate in both motoring mode, when the mechanical torque is positive, and generating mode when the mechanical torque is negative.

Model Formulation

Q-d Transformation

The 3-phase to q-d transformation and the inverse used for the model are:

\left[\begin{array}{l}
{V_{q}} \\
{V_{d}} \\
{V_{0}}
\end{array}\right]=\frac{2}{3}\left[\begin{array}{ccc}
{\cos (\theta)} & {\cos \left(\theta-\frac{2 \pi}{3}\right)} & {\cos \left(\theta+\frac{2 \pi}{3}\right)} \\
{\sin (\theta)} & {\sin \left(\theta-\frac{2 \pi}{3}\right)} & {\sin \left(\theta+\frac{2 \pi}{3}\right)} \\
{\frac{1}{2}} & {\frac{1}{2}} & {\frac{1}{2}}
\end{array}\right]\left[\begin{array}{cccl}
{\cos (\theta)} & {\cos \left(\theta-\frac{2 \pi}{3}\right)} & {\cos \left(\theta+\frac{2 \pi}{3}\right)V_{a}} \\
{V_{b}} \\
{V_{c}}
\end{array}\right]

\left[\begin{array}{c}
{V_{a}} \\
{\sin (\theta)} V_{b}} \\
{V_{c}}
\end{array}\right]=\left[\begin{array}{ccc}
{\cos (\theta)} & {\sin (\theta)} & {1} \\
{\cos \left(\theta-\frac{2 \pi}{3}\right)} & {\sin \left(\theta+-\frac{2 \pi}{3}\right)} & {1} \\
{\frac{1}{2}cos \left(\theta+\frac{2 \pi}{3}\right)} & {\sin \left(\theta+\frac{12 \pi}{23}\right)} & {\frac{1}{2}}
\end{array}\right]\left[\begin{array}{l}
{V_{aq}} \\
{V_{bd}} \\
{V_{c0}}
\end{array}\right]

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• Rotor reference frame: θ=θr,

• Stationary reference frame: θ=0,

• Synchronous reference frame: θ=θe.

Since the Induction Machine is modeled in rotor reference frame, the θ required for the q-d transformation is the rotor electrical angle (θr).

Induction Machine Electrical Model

Induction machine models in state space framework are based on magnetic fluxes as the state variables and winding currents as the outputs, and can be represented as follows:

\begin{aligned}
\psi(n+1) &=A_{d}(n) \psi(n)+B_{d}(n) u(n) \\
I(n+1) &=C(n+1) \psi(n+1)
\end{aligned}

where the coefficient matrices are as follows:

A_{d}=T_{s}\left(-R L^{-1}-\Omega\right)+ I

B_{d}=T s \times I

\boldsymbol{c}=\boldsymbol{L}^{-1}

with 

and

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\left[\begin{array}{c}
{V_{a}} \\
{V_{b}} \\
{V_{c}}
\end{array}\right]=\left[\begin{array}{ccc}
{\cos (\theta)} & {\sin (\theta)} & {1} \\
{\cos \left(\theta-\frac{2 \pi}{3}\right)} & {\sin \left(\theta-\frac{2 \pi}{3}\right)} & {1} \\
{\cos \left(\theta+\frac{2 \pi}{3}\right)} & {\sin \left(\theta+\frac{2 \pi}{3}\right)} & {1}
\end{array}\right]\left[\begin{array}{l}
{V_{q}} \\
{V_{d}} \\
{V_{0}}
\end{array}\right]

• Rotor reference frame: θ=θr,

• Stationary reference frame: θ=0,

• Synchronous reference frame: θ=θe.

Since the Induction Machine is modeled in rotor reference frame, the θ required for the q-d transformation is the rotor electrical angle (θr).

Induction Machine Electrical Model

Induction machine models in state space framework are based on magnetic fluxes as the state variables and winding currents as the outputs, and can be represented as follows:

\begin{aligned}
\psi(n+1) &=A_{d}(n) \psi(n)+B_{d}(n) u(n) \\
I(n+1) &=C(n+1) \psi(n+1)
\end{aligned}

where the coefficient matrices are as follows:

A_{d}=T_{s}\left(-R L^{-1}-\Omega\right)+ I

B_{d}=T s \times I

\boldsymbol{c}=\boldsymbol{L}^{-1}

with 

and

\boldsymbol{u}=\left[\begin{array}{llll}
{V_{s q}} & {V_{s d}} & {V_{r q}^{\prime}} & {V_{r d}^{\prime}}
\end{array}\right]^{t}

\boldsymbol{I}=\left[\begin{array}{llll}
{I_{s q}} & {I_{s d}} & {I_{r q}^{\prime}} & {I_{r d}^{\prime}}
\end{array}\right]^{t}

\boldsymbol{\psi}=\left[\begin{array}{llll}
{\psi_{s q}} & {\psi_{s d}} & {\psi_{r q}^{\prime}} & {\psi_{r d}^{\prime}}
\end{array}\right]^{t}

\boldsymbol{uR}=\left[\begin{array}{llll}
{VR_{s q}} & {VR_{s d}} & {VR_{r q}^{\prime}} & {VR_{r d}^{\prime}}
\end{array}\right]^{t}

\boldsymbol{I}L=\left[\begin{array}{llllcccc}
{IL_{s q}}}} & {0} & {IL_{s d}m}} & {0} \\
{0} & {IL_{r q}^{\primes}} & {0} & {L_{m}} \\
{L_{m}} & {I0} & {L_{r d}^{\prime}} \end& {array0}\right]^{t}

\boldsymbol{\psi}=\left[\begin{array}{llll}
{\psi_{s q \\
{0} & {L_{m}} & {\psi_{s d}}0} & {\psiL_{r q}^{\prime}}
& {\psi_{r d}^{\prime}}
\end{array}\right]^{t}

\boldsymbol{R}=\left[\begin{array}{llll}
{R_{s}} & {R_{s}} & {R_{r}^{\prime}} & {R_{r}^{\prime}}
\end{array}\right]^{t}

L=\left[\begin{array}{cccc}
{L_{s}} & {0} & {L_{m}} & {0} \\
{0} & {L_{s}} & {0} & {L_{m}} \\
{L_{m}} & {0} & {L_{r}^{\prime}} & {0} \\
{0} & {L_{m}} & {0} & {L_{r}^{\prime}}
\end{array}\right]

\Omega=\left[\begin{array}{cccc}
{0} & {\omega} & {0} & {0} \\
{-\omega} & {0} & {0} & {0} \\
{0} & {0} & {0} & {\omega-\omega_{r}} \\
{0} & {0} & {-\left(\omega-\omega_{r}\right)} & {0}
\end{array}\right]

The Induction Machine is modeled in rotor reference frame so ω=ωr.

In the model, all the rotor parameters and variables are seen from the stator distinguished by a prime sign. Since the squirrel-cage rotor type is not supplied by an external source, then it is always the case that V'rq=V'rd=0.

The electrical torque can be calculated as follows:

T_{e}=\frac{3}{2} p p\left(\psi_{d} i_{q}-\psi_{q} i_{d}\right)

Mechanical Model

The equation of the mechanical model in torque mode can be expressed as follows:

\omega_{m}(t)=a \omega_{m}\left(t-T_{s}\right)+b\left(T_{e}-T_{m}\right)

a=\left(1-T_{s}\right) \frac{F_{v}}{J}

b=60 \frac{T_{s}}{2 \pi J}

where ω_m is the rotor speed, T_e is the electromagnetic torque, T_m is the torque command, F_v is the viscous friction coefficient, J is the inertia and T_s the time step. There is a dead-zone implementation with the static friction torque, if the electromagnetic doesn't exceed the static friction torque, the speed remains zero.

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end{array}\right]

\Omega=\left[\begin{array}{cccc}
{0} & {\omega} & {0} & {0} \\
{-\omega} & {0} & {0} & {0} \\
{0} & {0} & {0} & {\omega-\omega_{r}} \\
{0} & {0} & {-\left(\omega-\omega_{r}\right)} & {0}
\end{array}\right]

The Induction Machine is modeled in rotor reference frame so ω=ωr.

In the model, all the rotor parameters and variables are seen from the stator distinguished by a prime sign. Since the squirrel-cage rotor type is not supplied by an external source, then it is always the case that V'rq=V'rd=0.

The electrical torque can be calculated as follows:

T_{e}=\frac{3}{2} p p\left(\psi_{d} i_{q}-\psi_{q} i_{d}\right)

Resolver Encoder Model

The equations of the resolver encoder can be expressed as follows:

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\text { sine }=\text { Excitation } \times\left(\cos \left(\theta_{\text {res}}\right) * R_{k_{c c}}+\sin \left(\theta_{\text {res}}\right) * R_{k_{s c}}\right)

where θ_res is the resolver angle, θ_mec is the mechanical angle of the machine, θ_offset is the angle offset, R_pp is the Number of pole pairs of the resolver and R_k are the resolver sine cosine gains.

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Electrical Parameters and Measurements

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Mechanical Parameters and Measurements

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OPAL-RT's resolver models are based off of the following sets of equations:

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Where Sin.Sin, Sin.Cos, Cos.Sin, and Cos.Cos represent gains that are applied to simulate a non-ideal resolver.  To simulate an ideal resolver, set the Sin.Sin and Cos.Cos gains to 1, set the Sin.Cos and Cos.Sin gains to 0, set the pp to 1, and set the θ_Offset to 0.  This results in the following equations:

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Resolver Parameters and Measurements

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Encoder Parameters and Measurements

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Visualization of Resolver Encoder Parameters Effects

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