Introduction

This document presents the formulation implemented on FPGA for squirrel-cage, double squirrel-cage, and doubly-fed (wound rotor) three-phase induction machine (asynchronous machine) with mechanical model. The three-phase to q-d-0 park transform has the 3/2 factor in rotor reference frame. The machine can operate in both motor mode, when the mechanical torque is positive, and generating mode, when the mechanical torque is negative. When the option of Open Stator Winding is unchecked, 0-axis is not computed for the squirrel-cage machine.

State Space Representation

The modeling of the Generic Machines is based on state space equations. The forward Euler discretization of the state space model can be presented as follows:

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The state space equation can be simplified into:

...

where

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Q-d Transformation

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

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

Induction Machine Electrical Model

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

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Depending on the rotor type, the state variables, outputs, and the size of the state space equation might be different as presented as follows:

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Squirrel-Cage and Doubly-fed (Wound)

--uriencoded--\begin%7Baligned%7D
u
=
\left[\begin%7Bmatrix%7DV_%7Bsq%7D&V_%7Bsd%7D&V_%7Bs0%7D&V%5e%7B'%7D_%7Brq%7D&V%5e%7B'%7D_%7Brd%7D&V%5e%7B'%7D_%7Br0%7D
\end%7Bmatrix%7D\right] %5e%7Bt%7D


\end%7Baligned%7D
\; \;\;\;\; \; \; \;\;\;\; \;\;\;\;\;\; \;\;\;\; \; \; \;\;\;\; \;\;\;\;\;

...

--uriencoded--\begin%7Baligned%7D
\Omega
=
\left[\begin%7Bmatrix%7D0&\omega&0&0&0&0\\
-\omega&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&(\omega - \omega_%7Br%7D)&0\\
0&0&0&-(\omega - \omega_%7Br%7D)&0&0\\
0&0&0&0&0&0\\
\end%7Bmatrix%7D\right] 


\end%7Baligned%7D
\; \;\;\;\; \; \; \;\;\;\; \;\;\;\;\;\; \;\;\;\; \; \; \;\;\;\; \;\;\;\;\;

Double Squirrel-Cage

--uriencoded--\begin%7Baligned%7D
u
=
\left[\begin%7Bmatrix%7DV_%7Bsq%7D&V_%7Bsd%7D&0&0&0&0
\end%7Bmatrix%7D\right] %5e%7Bt%7D


\end%7Baligned%7D
\; \;\;\;\; \; \; \;\;\;\; \;\;\;\;\;\; \;\;\;\; \; \; \;\;\;\; \;\;\;\;\;

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The Induction Machine is modeled in rotor reference frame so ω=ωr.The electrical torque can be calculated as follows:

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Mechanical Model

The mechanical model is the same for all machine types to calculate the rotor mechanical speed (ωm) as follows:

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Nomenclature

x: state variables vector

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