# How to transform the state space model of the induction motor?

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

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

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

$$$$\hat{\mathbf{i}}^{\alpha,\beta}_{s\Gamma} = \hat{\mathbf{i}}^{\alpha,\beta}_{sT}$$$$

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

or in matrix form

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

which means

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

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

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.

• You could cross-port to the dsp stack exchange maybe ?
– Ben
Oct 7 '21 at 12:36

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