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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Open

The state space equation can be simplified into:

Open

where

Q-d Transformation

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

θ required for the q-d transformation depends on the chosen reference frame as follows:

• 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.

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:

where the coefficient matrices are the same for all electric machine types as follows:

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

Squirrel-Cage and Doubly-fed (Wound)

Double Squirrel-Cage

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

Mechanical Model

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

Nomenclature

x: state variables vector

y: outputs vector

u: inputs vector

A,B,C: state space coefficient matrices

Ts: simulation time step

n: time step number

Va: stator phase a voltage

Vb: stator phase b voltage

Vc: stator phase c voltage

Vsq: stator q-axis voltage

Vsd: stator d-axis voltage

Vs0: stator 0-axis voltage

V'rq: rotor q-axis voltage (referred to the stator)

V'rd: rotor d-axis voltage (referred to the stator)

I: current vector

Isq: stator q-axis current

Isd: stator d-axis current

Is0: stator 0-axis current

I'rq: rotor q-axis current (referred to the stator)

I'rd: rotor d-axis current (referred to the stator)

I'c1q: cage 1 q-axis current (referred to the stator)

I'c1d: cage 1 d-axis current (referred to the stator)

I'c2q: cage 2 q-axis current (referred to the stator)

I'c2d: cage 2 d-axis current (referred to the stator)

Ψ: magnetic flux vector

ψsq: stator q-axis flux

ψsd: stator d-axis flux

ψs0: stator 0-axis flux

ψ'rq: rotor q-axis flux (referred to the stator)

ψ'rd: rotor d-axis flux (referred to the stator)

ψ'c1q: cage 1 q-axis flux (referred to the stator)

ψ'c1d: cage 1 d-axis flux (referred to the stator)

ψ'c2q: cage 2 q-axis flux (referred to the stator)

ψ'c2d: cage 2 d-axis flux (referred to the stator)

R: resistance matrix

Rs: stator winding resistance

R0: 0-axis stator winding resistance

R'r: rotor winding resistance (referred to the stator)

R'c1: cage 1 resistance (referred to the stator)

R'c2: cage 2 resistance (referred to the stator)

L: inductance matrix

Ls: stator self inductance (mutual (Lm) +leakage (Lls) )

L0: 0-axis stator inductance

L'r: rotor self inductance (referred to the stator) (mutual (Lm) +leakage (Llr) )

L'c1: cage 1 self inductance (referred to the stator) (mutual (Lm) +leakage (Llc1) )

L'c2: cage 2 self inductance (referred to the stator) (mutual (Lm) +leakage (Llc2) )

Lm: mutual inductance

Ω: speed matrix

ω, θ: rotation speed, and position of the reference frame

ωe, θe: rotation speed, and position of the synchronous frame

ωr, θr: rotation speed, and position of the rotor (electrical) frame

ωm, θm: rotation speed, and position of the rotor (mechanical) frame

Te: electromagnetic torque

Tm: mechanical torque

pp: number of pole pairs

J: rotor inertia

Fv: viscous friction coefficient

Fs: static friction torque

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