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First of all I would like to apologize if I have chosen wrong forum. My question is related to the electrical engineering but basically it is a mathematical problem.

I have the state space model of the three phase squirrel cage induction motor. The state space model contains in the system matrix parameters of the \$T\$ equivalent circuit

$$ \frac{\mathrm{d}}{\mathrm{d}t} \begin{bmatrix} \hat{\mathbf{i}}^{\alpha,\beta}_{sT} \\ \hat{\mathbf{\psi}}^{\alpha,\beta}_{rT} \end{bmatrix} = \begin{bmatrix} -\alpha\cdot\mathbf{E} & \beta\cdot\mathbf{E}-\gamma\cdot p_p\cdot\omega_m\cdot\mathbf{J} \\ R_r\cdot\frac{L_h}{L_r}\cdot\mathbf{E} & -\frac{R_r}{L_r}\cdot\mathbf{E} + \omega_m\cdot p_p\cdot \mathbf{J} \end{bmatrix} \cdot \begin{bmatrix} \hat{\mathbf{i}}^{\alpha,\beta}_{sT} \\ \hat{\mathbf{\psi}}^{\alpha,\beta}_{rT} \end{bmatrix} + \begin{bmatrix} \delta\cdot\mathbf{E} \\ \mathbf{Z} \end{bmatrix} \cdot \hat{\mathbf{u}}^{\alpha,\beta}_{sT} $$

$$ \hat{\mathbf{i}}^{\alpha,\beta}_{sT} = \begin{bmatrix} \mathbf{E} & \mathbf{Z} \end{bmatrix} \cdot \begin{bmatrix} \hat{\mathbf{i}}^{\alpha,\beta}_{sT} \\ \hat{\mathbf{\psi}}^{\alpha,\beta}_{rT} \end{bmatrix} $$ where the auxiliary variables \$\alpha, \beta, \gamma, \delta\$ have following values \$\alpha = \frac{R_s + R_r\cdot\frac{L^2_h}{L^2_r}}{L_{s\sigma} + \frac{L_h}{L_r}\cdot L_{r\sigma}}\$, \$\beta = \frac{\frac{R_r\cdot L_h}{L^2_r}}{L_{s\sigma} + \frac{L_h}{L_r}\cdot L_{r\sigma}}\$, \$\gamma = \frac{\frac{L_h}{L_r}}{L_{s\sigma} + \frac{L_h}{L_r}\cdot L_{r\sigma}}\$, \$\delta = \frac{1}{L_{s\sigma} + \frac{L_h}{L_r}\cdot L_{r\sigma}}\$. The parameter \$R_s\$ is the resistance of the stator winding, \$R_r\$ is the resistance of the rotor winding, \$L_h\$ is the magnetizing inductance, \$L_r\$ is the rotor inductance, \$L_{s\sigma}\$ is the leakage inductance of the stator winding and \$L_{r\sigma}\$ is the leakage inductance of the rotor winding.

T-equivalent circuit

enter image description here

The \$\mathbf{E}, \mathbf{J}, \mathbf{Z}\$ matrices have following meaning

$$ \mathbf{E} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} $$

$$ \mathbf{J} = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} $$

$$ \mathbf{Z} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} $$

My intention is to transform the above given state space model to a different state space model which will contain parameters of the \$\Gamma\$ equivalent circuit of the induction motor.

\$\Gamma\$ equivalent circuit

enter image description here

This transformation basically means that the state variables changes due to the fact that the leakage inductance of the stator winding has been moved onto the rotor side

\begin{equation} \hat{\mathbf{i}}^{\alpha,\beta}_{s\Gamma} = \hat{\mathbf{i}}^{\alpha,\beta}_{sT} \end{equation}

\begin{equation} \hat{\mathbf{\psi}}^{\alpha,\beta}_{s\Gamma} = b\cdot\hat{\mathbf{\psi}}^{\alpha,\beta}_{rT} = \frac{L_s}{L_h}\cdot\hat{\mathbf{\psi}}^{\alpha,\beta}_{rT} \end{equation}

or in matrix form

\begin{equation} \begin{bmatrix} \hat{\mathbf{i}}^{\alpha,\beta}_{s\Gamma} \\ \hat{\mathbf{\psi}}^{\alpha,\beta}_{r\Gamma} \end{bmatrix} = \begin{bmatrix} \mathbf{E} & \mathbf{Z} \\ \mathbf{Z} & b\cdot\mathbf{Z} \end{bmatrix} \cdot \begin{bmatrix} \hat{\mathbf{i}}^{\alpha,\beta}_{sT} \\ \hat{\mathbf{\psi}}^{\alpha,\beta}_{rT} \end{bmatrix} \end{equation}

which means

\begin{equation} \mathbf{x}_T = \begin{bmatrix} \hat{\mathbf{i}}^{\alpha,\beta}_{sT} \\ \hat{\mathbf{\psi}}^{\alpha,\beta}_{rT} \end{bmatrix} = \begin{bmatrix} \mathbf{E} & \mathbf{Z} \\ \mathbf{Z} & \frac{1}{b}\cdot\mathbf{Z} \end{bmatrix} \cdot \begin{bmatrix} \hat{\mathbf{i}}^{\alpha,\beta}_{s\Gamma} \\ \hat{\mathbf{\psi}}^{\alpha,\beta}_{r\Gamma} \end{bmatrix} = \mathbf{T} \cdot \mathbf{x}_{\Gamma} \end{equation}

The last matrix equation enables to move from the \$T\$ based state space model to the \$\Gamma\$ state space model. Because in case we use it in the \$T\$ based state space model

\begin{eqnarray} \dot{\mathbf{x}}_T &=& \mathbf{A}_T\cdot \mathbf{x}_T + \textbf{B}_T\cdot \mathbf{u} \nonumber \\ \mathbf{y}_T &=& \mathbf{C}_T\cdot\mathbf{x}_T \nonumber \end{eqnarray}

we have

$$ \frac{\mathrm{d}}{\mathrm{d}t}\left(\mathbf{T}\cdot\mathbf{x}_{\Gamma}\right) = \mathbf{A}_T\cdot\left(\mathbf{T}\cdot\mathbf{x}_{\Gamma}\right) + \mathbf{B}_T\cdot \mathbf{u}_T $$

$$ \dot{\mathbf{x}}_{\Gamma} = \mathbf{T}^{-1}\cdot\mathbf{A}_T\cdot\mathbf{T}\cdot\mathbf{x}_{\Gamma} + \mathbf{T}^{-1}\cdot\mathbf{B}_T\cdot\mathbf{u}_T $$

$$ \mathbf{y}_{T} = \mathbf{y}_{\Gamma} = \mathbf{C}_T\cdot\mathbf{T}\cdot\mathbf{x}_{\Gamma} $$

From the last set of the matrix equations we can write formulas for transforming the matrices of the \$T\$ based state space model into the \$\Gamma\$ based state space model

$$ \mathbf{A}_{\Gamma} = \mathbf{T}^{-1}\cdot\mathbf{A}_T\cdot\mathbf{T} $$

$$ \mathbf{B}_{\Gamma} = \mathbf{T}^{-1}\cdot\mathbf{B}_T $$

$$ \mathbf{C}_{\Gamma} = \mathbf{C}_T\cdot\mathbf{T} $$

My problem is how to write the matrices of the \$\Gamma\$ based state space model i.e. \$\mathbf{A}_{\Gamma}, \mathbf{B}_{\Gamma}, \mathbf{C}_{\Gamma}\$ in such a manner that they contain in their elements only the parameters of the \$\Gamma\$ equivalent circuit i.e. stator resistance \$R_S\$, magnetizing inductance \$L_M\$, total leakage inductance \$L_L\$ and rotor resistance \$R_R\$. There are following formulas for conversion of the parameters of the \$T\$ equivalent circuit into the parameters of the \$\Gamma\$ equivalent circuit

$$ R_S = R_s $$

$$ L_M = L_s $$

$$ L_L = L_r\cdot\frac{L^2_s}{L^2_h} - L_s $$

$$ R_R = R_r\cdot\frac{L^2_s}{L^2_h} $$

I have attempted to made some modifications in the elements of the matrices aiming to exploit the formulas for the \$R_S\$,\$L_M\$,\$L_L\$ \$R_R\$ but I always stuck in a situation where let's say \$R_r\$ is multiplied by the desired "transform" coefficient \$b=\frac{L_s}{L_h}\$ but the necessary modifications create very complex expression from the rest of the matrix element which doesn't contain none of the formulas for the \$R_S\$,\$L_M\$,\$L_L\$,\$R_R\$. I have also attempted to use the wxMaxima software for the symbolic calculations but without success.

One observation I have made. If I chose different transform coefficient \$b=\frac{L_h}{L_r}\$ which corresponds to the another equivalent circuit called \$\Gamma^{-1}\$

\$\Gamma^{-1}\$ equivalent circuit

enter image description here

I am able to get correct state space model based on the above mentioned steps and below given formulas for conversion of the parameters of the \$T\$ equivalent circuit into the parameters of the \$\Gamma^{-1}\$ equivalent circuit

$$ R_S = R_s $$

$$ L_M = L_h\cdot\frac{L_h}{L_r} $$

$$ L_L = L_{s\sigma} + \frac{L_h}{L_r}\cdot L_{r\sigma} $$

$$ R_R = R_r\cdot\frac{L^2_h}{L^2_r} $$

This fact proofs in my opinion the correctness of the derived formulas for the transformation of the matrices.

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  • \$\begingroup\$ You could cross-port to the dsp stack exchange maybe ? \$\endgroup\$
    – Ben
    Oct 7, 2021 at 12:36

1 Answer 1

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I hope I am not wrong but I think that the solution is very simple. Due to the fact that the transition between both the state space models has been done via application of the transform matrix \$\mathbf{T}\$ on the system matrix \$\mathbf{A_T}\$ and the input matrix \$\mathbf{B_T}\$ I think that I can simply substitute the following formulas into the elements of the \$\mathbf{A_{\Gamma}}\$ and \$\mathbf{B_{\Gamma}}\$ matrices and get the state space model related to the \$\Gamma\$ equivalent circuit

$$L_{s\sigma} = L_s - L_h$$ $$L_{r\sigma} = L_r - L_h$$ $$L_r = L_L + L_M$$ $$R_s = R_S$$ $$L_h = L_M$$ $$R_r = R_R$$ $$L_s = L_M$$

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