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:
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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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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:
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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)
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Double Squirrel-Cage
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| Mathblock | ||||
|---|---|---|---|---|
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--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
\; \;\;\;\; \; \; \;\;\;\; \;\;\;\;\;\; \;\;\;\; \; \; \;\;\;\; \;\;\;\;\;
|
| Mathblock | ||||
|---|---|---|---|---|
| ||||
--uriencoded--\begin%7Baligned%7D
I
=
\left[\begin%7Bmatrix%7DI_%7Bsq%7D&I_%7Bsd%7D&I_%7Bs0%7D&I%5e%7B'%7D_%7Brq%7D&I%5e%7B'%7D_%7Brd%7D&I%5e%7B'%7D_%7Br0%7D
\end%7Bmatrix%7D\right] %5e%7Bt%7D
\end%7Baligned%7D
\; \;\;\;\; \; \; \;\;\;\; \;\;\;\;\;\; \;\;\;\; \; \; \;\;\;\; \;\;\;\;\;
|
| Mathblock | ||||
|---|---|---|---|---|
| ||||
--uriencoded--\begin%7Baligned%7D
\psi
=
\left[\begin%7Bmatrix%7D\psi_%7Bsq%7D&\psi_%7Bsd%7D&\psi_%7Bs0%7D&\psi%5e%7B'%7D_%7Brq%7D&\psi%5e%7B'%7D_%7Brd%7D&\psi%5e%7B'%7D_%7Br0%7D
\end%7Bmatrix%7D\right] %5e%7Bt%7D
\end%7Baligned%7D
\; \;\;\;\; \; \; \;\;\;\; \;\;\;\;\;\; \;\;\;\; \; \; \;\;\;\; \;\;\;\;\;
|
| Mathblock | ||||
|---|---|---|---|---|
| ||||
--uriencoded--\begin%7Baligned%7D
R
=
\left[\begin%7Bmatrix%7DR_%7Bsq%7D&R_%7Bsd%7D&R_%7Bs0%7D&R%5e%7B'%7D_%7Brq%7D&R%5e%7B'%7D_%7Brd%7D&R%5e%7B'%7D_%7Br0%7D
\end%7Bmatrix%7D\right] %5e%7Bt%7D
\end%7Baligned%7D
\; \;\;\;\; \; \; \;\;\;\; \;\;\;\;\;\; \;\;\;\; \; \; \;\;\;\; \;\;\;\;\;
|
| Mathblock | ||||
|---|---|---|---|---|
| ||||
--uriencoded--\begin%7Baligned%7D
L
=
\left[\begin%7Bmatrix%7DL_%7Bs%7D&0&0&L_%7Bm%7D&0&0\\
0&L_%7Bs%7D&0&0&L_%7Bm%7D&0\\
0&0&L_%7Bs0%7D&0&0&0\\
L_%7Bm%7D&0&0&L%5e%7B'%7D_%7Br%7D&0&0\\
0&L_%7Bm%7D&0&0&L%5e%7B'%7D_%7Br%7D&0\\
0&0&0&0&0&L%5e%7B'%7D_%7Br0%7D\\
\end%7Bmatrix%7D\right]
\end%7Baligned%7D
\; \;\;\;\; \; \; \;\;\;\; \;\;\;\;\;\; \;\;\;\; \; \; \;\;\;\; \;\;\;\;\;
|
| Mathblock | ||||
|---|---|---|---|---|
| ||||
--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
| Mathblock | ||||
|---|---|---|---|---|
| ||||
--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
\; \;\;\;\; \; \; \;\;\;\; \;\;\;\;\;\; \;\;\;\; \; \; \;\;\;\; \;\;\;\;\;
|
| Mathblock | ||||
|---|---|---|---|---|
| ||||
--uriencoded--\begin%7Baligned%7D
I
=
\left[\begin%7Bmatrix%7DI_%7Bsq%7D&I_%7Bsd%7D&I%5e%7B'%7D_%7Bc1q%7D&I%5e%7B'%7D_%7Bc1d%7D&I%5e%7B'%7D_%7Bc2q%7D&I%5e%7B'%7D_%7Bc2d%7D
\end%7Bmatrix%7D\right] %5e%7Bt%7D
\end%7Baligned%7D
\; \;\;\;\; \; \; \;\;\;\; \;\;\;\;\;\; \;\;\;\; \; \; \;\;\;\; \;\;\;\;\;
|
| Mathblock | ||||
|---|---|---|---|---|
| ||||
--uriencoded--\begin%7Baligned%7D
\psi
=
\left[\begin%7Bmatrix%7D\psi_%7Bsq%7D&\psi_%7Bsd%7D&\psi%5e%7B'%7D_%7Bc1q%7D&\psi%5e%7B'%7D_%7Bc1d%7D&\psi%5e%7B'%7D_%7Bc2q%7D%7D&\psi%5e%7B'%7D_%7Bc2d%7D
\end%7Bmatrix%7D\right] %5e%7Bt%7D
\end%7Baligned%7D
\; \;\;\;\; \; \; \;\;\;\; \;\;\;\;\;\; \;\;\;\; \; \; \;\;\;\; \;\;\;\;\;
|
| Mathblock | ||||
|---|---|---|---|---|
| ||||
--uriencoded--\begin%7Baligned%7D
R
=
\left[\begin%7Bmatrix%7DR_%7Bs%7D&R_%7Bs%7D&R%5e%7B'%7D_%7Bc1%7D&R%5e%7B'%7D_%7Bc1%7D&R%5e%7B'%7D_%7Bc2%7D&R%5e%7B'%7D_%7Bc2%7D
\end%7Bmatrix%7D\right] %5e%7Bt%7D
\end%7Baligned%7D
\; \;\;\;\; \; \; \;\;\;\; \;\;\;\;\;\; \;\;\;\; \; \; \;\;\;\; \;\;\;\;\;
|
| Mathblock | ||||
|---|---|---|---|---|
| ||||
--uriencoded--\begin%7Baligned%7D
L
=
\left[\begin%7Bmatrix%7DL_%7Bs%7D&0&0&L_%7Bm%7D&0&L_%7Bm%7D&0\\
0&L_%7Bs%7D&0&0&L_%7Bm%7D&0&L_%7Bm%7D\\
0&0&L_%7Bs0%7D&0&0&0&0\\
L_%7Bm%7D&0&0&L%5e%7B'%7D_%7Bc1%7D&0&L_%7Bm%7D&0\\
0&L_%7Bm%7D&0&0&L%5e%7B'%7D_%7Bc1%7D&0&L_%7Bm%7D\\
L_%7Bm%7D&0&0&L_%7Bm%7D&0&L%5e%7B'%7D_%7Bc2%7D&0\\
0&L_%7Bm%7D&0&0&L_%7Bm%7D&0&L%5e%7B'%7D_%7Bc2%7D\\
\end%7Bmatrix%7D\right]
\end%7Baligned%7D
\; \;\;\;\; \; \; \;\;\;\; \;\;\;\;\;\; \;\;\;\; \; \; \;\;\;\; \;\;\;\;\;
|
| Mathblock | ||||
|---|---|---|---|---|
| ||||
--uriencoded--\begin%7Baligned%7D
\Omega
=
\left[\begin%7Bmatrix%7D0&\omega&0&0&0&0&0\\
-\omega&0&0&0&0&0&0\\
0&0&0&0&0&0&0\\
0&0&0&0&(\omega - \omega_%7Br%7D)&0&0\\
0&0&0&-(\omega - \omega_%7Br%7D)&0&0&0\\
0&0&0&0&0&0&(\omega - \omega_%7Br%7D)\\
0&0&0&0&0&-(\omega - \omega_%7Br%7D)&0\\
\end%7Bmatrix%7D\right]
\end%7Baligned%7D
\; \;\;\;\; \; \; \;\;\;\; \;\;\;\;\;\; \;\;\;\; \; \; \;\;\;\; \;\;\;\;\;
|
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
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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