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Direct vector control strategy of 2-phase induction motor drives based on the conventional rotating transformation matrix and EKF

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Abstract

This research presents a novel mathematical model of 2-phase induction Motor (M2) or star-connected 3-phase induction Motor (M3) under the Open-Phase Fault (FOP). In addition, a Direct Vector Control (DVC) strategy based on the Extended Kalman Filter (EKF) is proposed for M2 drives. It is shown that using the conventional rotating Transformation Matrix (TM), the mathematical model of the M2 can be written as forward and backward quantities. Based on this modeling, a developed DVC strategy according to forcing the backward quantities equal to zero is achieved for M2 drives. In the proposed DVC, an EKF is developed to estimate the rotor flux amplitude and angle. The proposed approach with minor modifications can be employed for both M3 and M2 drives. The experimental results on a star-connected M3 test-bed with a DSP/TMS320F28335 indicate that the proposed EKF-based DVC (DVCEKF) method can precisely and effectively control the faulty M3 drive during different conditions. The results also show that the proposed DVCEKF system has a better performance in comparison to conventional control systems during the FOP.

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Data availability and materials

The data that support the findings of this study are available on request from the corresponding author.

Abbreviations

\(\alpha \beta ,dq\) :

Stationary reference and rotating reference

\(\left[ {TM^{h} } \right]\) :

\(abc \to \alpha \beta\) Transformation during normal mode

\(\left[ {TM^{f} } \right]\) :

\(ab \to \alpha \beta\) Transformation during the fault mode

\(\left[ {TM^{e} } \right]\) :

Conventional rotating TM

\(\left[ {TM_{f}^{e} } \right]\) :

Forward rotating TM

\(\left[ {TM_{b}^{e} } \right]\) :

Backward rotating TM

\(\begin{gathered} V_{\alpha s} ,V_{\beta s} ,V_{\alpha r} ,V_{\beta r} \hfill \\ V_{ds} ,V_{qs} ,V_{dr} ,V_{qr} \hfill \\ \end{gathered}\) :

Stator and rotor \(\alpha \beta\) and \(dq\) voltages

\(\begin{gathered} I_{\alpha s} ,I_{\beta s} ,I_{\alpha r} ,I_{\beta r} \hfill \\ I_{ds} ,I_{qs} ,I_{dr} ,I_{qr} \hfill \\ \end{gathered}\) :

Stator and rotor \(\alpha \beta\) and \(dq\) currents

\(I_{as} ,I_{bs} ,I_{cs}\) :

Stator line currents

\(\Psi_{\alpha r} ,\Psi_{\beta r} ,\Psi_{dr} ,\Psi_{qr}\) :

Rotor \(\alpha \beta\) and \(dq\) fluxes

\(N_{p}\) :

Number of pole pairs

\(\left| {\Psi_{r} } \right|,\delta_{e}\) :

Rotor flux amplitude and angle

\(R_{s} ,R_{r}\) :

Stator and rotor resistances

\(L_{ds} ,L_{qs} ,L_{r} ,L_{d} ,L_{q}\) :

Stator and rotor self and mutual inductances

\(L_{ls} ,L_{ms}\) :

Stator leakage and magnetizing inductances

\(\Omega_{r}\) :

Rotor electrical angular speed

\(T_{e} ,T_{l}\) :

Electromagnetic torque and external load torque

\(J,F_{r}\) :

Moment of inertia and viscous friction coefficient

\(p\) :

Differential operator

\(X,Y,U\) :

State, input, and output vectors

\(W,V\) :

System and measurement noises

\(f,H\) :

Functions of states and outputs

\(A,B,C\) :

System, input, and measurement matrices

\(T_{s}\) :

Sampling time

\(P,Q,R\) :

Covariance matrices of the state estimation error, system noise, and measurement noise

\(K\) :

Kalman filter gain

Superscripts “*”, “^”:

Reference and estimated values

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Funding

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Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Gorgan Branch, Islamic Azad University, Gorgan, Iran

    Amir Mohamad Taghinezhad Vaskeh Mahaleh, Mahmood Ghanbari & Reza Ebrahimi

Authors
  1. Amir Mohamad Taghinezhad Vaskeh Mahaleh
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  2. Mahmood Ghanbari
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  3. Reza Ebrahimi
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Contributions

Conceptualization: AMTVM, MG, RE, Data curation: AMTVM, Formal analysis: AMTVM, Investigation: AMTVM, MG, RE, Methodology: AMTVM, MG, Project administration: AMTVM, MG, RE, Software: AMTVM, MG, RE, Validation: AMTVM, MG, RE, Visualization: AMTVM, MG, Writing: AMTVM, MG.

Corresponding author

Correspondence to Mahmood Ghanbari.

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Appendices

Appendix A

Using the conventional rotating TM, equations of the M2 can be written as (A1)-(A3):

Stator voltage equations:

$$ \begin{aligned} \left[ \begin{gathered} V_{ds} \hfill \\ V_{qs} \hfill \\ \end{gathered} \right] & = \left[ {TM^{e} } \right]\left[ {\begin{array}{*{20}c} {R_{s} + L_{ds} p} & 0 \\ 0 & {R_{s} + L_{qs} p} \\ \end{array} } \right]\left[ {TM^{e} } \right]^{{ - 1}} \left[ \begin{gathered} I_{ds} \hfill \\ I_{qs} \hfill \\ \end{gathered} \right] \\ & \quad + \left[ {TM^{e} } \right]\left[ {\begin{array}{*{20}c} {L_{d} p} & 0 \\ 0 & {L_{q} p} \\ \end{array} } \right]\left[ {TM^{e} } \right]^{{ - 1}} \left[ \begin{gathered} I_{dr} \hfill \\ I_{qr} \hfill \\ \end{gathered} \right] \\ & = \left[ {TM^{e} } \right]\left( {\left[ {\begin{array}{*{20}c} {R_{s} } & 0 \\ 0 & {R_{s} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {L_{ds} p} & 0 \\ 0 & {L_{qs} p} \\ \end{array} } \right]} \right)\left[ {TM^{e} } \right]^{{ - 1}} \left[ \begin{gathered} I_{ds} \hfill \\ I_{qs} \hfill \\ \end{gathered} \right] \\ & \quad + \left[ {TM^{e} } \right]\left[ {\begin{array}{*{20}c} {L_{d} } & 0 \\ 0 & {L_{q} } \\ \end{array} } \right]\left( {\left( {p\left[ {TM^{e} } \right]^{{ - 1}} } \right)\left[ \begin{gathered} I_{dr} \hfill \\ I_{qr} \hfill \\ \end{gathered} \right] + \left[ {TM^{e} } \right]^{{ - 1}} \left[ \begin{gathered} pI_{dr} \hfill \\ pI_{qr} \hfill \\ \end{gathered} \right]} \right) \\ & = \left[ {TM^{e} } \right]\left[ {\begin{array}{*{20}c} {R_{s} } & 0 \\ 0 & {R_{s} } \\ \end{array} } \right]\left[ {TM^{e} } \right]^{{ - 1}} \left[ \begin{gathered} I_{ds} \hfill \\ I_{qs} \hfill \\ \end{gathered} \right] \\ & \quad + \left[ {TM^{e} } \right]\left[ {\begin{array}{*{20}c} {L_{ds} } & 0 \\ 0 & {L_{qs} } \\ \end{array} } \right]\left( {\left( {p\left[ {TM^{e} } \right]^{{ - 1}} \left[ \begin{gathered} I_{ds} \hfill \\ I_{qs} \hfill \\ \end{gathered} \right]} \right) + \left[ {TM^{e} } \right]^{{ - 1}} \left[ \begin{gathered} pI_{ds} \hfill \\ pI_{qs} \hfill \\ \end{gathered} \right]} \right) \\ & \quad + \left[ {TM^{e} } \right]\left[ {\begin{array}{*{20}c} {L_{d} } & 0 \\ 0 & {L_{q} } \\ \end{array} } \right]\left( {\left( {p\left[ {TM^{e} } \right]^{{ - 1}} } \right)\left[ \begin{gathered} I_{dr} \hfill \\ I_{qr} \hfill \\ \end{gathered} \right] + \left[ {TM^{e} } \right]^{{ - 1}} \left[ \begin{gathered} pI_{dr} \hfill \\ pI_{qr} \hfill \\ \end{gathered} \right]} \right) \\ & = \underbrace {{\left[ {\begin{array}{*{20}c} {R_{s} + \left( {\frac{{L_{ds} + L_{qs} }}{2}} \right)p} & { - \Omega_{e} \left( {\frac{{L_{ds} + L_{qs} }}{2}} \right)} \\ {\Omega_{e} \left( {\frac{{L_{ds} + L_{qs} }}{2}} \right)} & {R_{s} + \left( {\frac{{L_{ds} + L_{qs} }}{2}} \right)p} \\ \end{array} } \right]\left[ \begin{gathered} I_{ds1} \hfill \\ I_{qs1} \hfill \\ \end{gathered} \right] + \left[ {\begin{array}{*{20}c} {\left( {\frac{{L_{d} + L_{q} }}{2}} \right)p} & { - \Omega_{e} \left( {\frac{{L_{d} + L_{q} }}{2}} \right)} \\ {\Omega_{e} \left( {\frac{{L_{d} + L_{q} }}{2}} \right)} & {\left( {\frac{{L_{d} + L_{q} }}{2}} \right)p} \\ \end{array} } \right]\left[ \begin{gathered} I_{dr1} \hfill \\ I_{qr1} \hfill \\ \end{gathered} \right]}}_{{\left[ \begin{subarray}{l} V_{ds1} \\ V_{qs1} \end{subarray} \right]}} \\ & \quad + \underbrace {{\left[ {\begin{array}{*{20}c} {\left( {\frac{{L_{ds} - L_{qs} }}{2}} \right)p} & {\Omega_{e} \left( {\frac{{L_{ds} - L_{qs} }}{2}} \right)} \\ {\Omega_{e} \left( {\frac{{L_{ds} - L_{qs} }}{2}} \right)} & { - \left( {\frac{{L_{ds} - L_{qs} }}{2}} \right)p} \\ \end{array} } \right]\left[ \begin{gathered} I_{ds2} \hfill \\ I_{qs2} \hfill \\ \end{gathered} \right] + \left[ {\begin{array}{*{20}c} {\left( {\frac{{L_{d} - L_{q} }}{2}} \right)p} & {\Omega_{e} \left( {\frac{{L_{d} - L_{q} }}{2}} \right)} \\ {\Omega_{e} \left( {\frac{{L_{d} - L_{q} }}{2}} \right)} & { - \left( {\frac{{L_{d} - L_{q} }}{2}} \right)p} \\ \end{array} } \right]\left[ \begin{gathered} I_{dr2} \hfill \\ I_{qr2} \hfill \\ \end{gathered} \right]}}_{{\left[ \begin{subarray}{l} V_{ds2} \\ V_{qs2} \end{subarray} \right]}} \\ \end{aligned} $$
(29)

Rotor voltage equations:

$$ \begin{aligned} \left[ \begin{gathered} V_{\alpha r} \hfill \\ V_{\beta r} \hfill \\ \end{gathered} \right] & = \left[ {TM^{e} } \right]\left[ {\begin{array}{*{20}c} {L_{d} p} & {\Omega_{r} L_{q} } \\ { - \Omega_{r} L_{d} } & {L_{q} p} \\ \end{array} } \right]\left[ {TM^{e} } \right]^{ - 1} \left[ \begin{gathered} I_{ds} \hfill \\ I_{qs} \\ \end{gathered} \right] + \left[ {TM^{e} } \right]\left[ {\begin{array}{*{20}c} {R_{r} + L_{r} p} & {\Omega_{r} L_{r} } \\ { - \Omega_{r} L_{r} } & {R_{r} + L_{r} p} \\ \end{array} } \right]\left[ {TM^{e} } \right]^{ - 1} \left[ \begin{gathered} I_{dr} \hfill \\ I_{qr} \hfill \\ \end{gathered} \right] \\ & = \left[ {TM^{e} } \right]\left( {\left[ {\begin{array}{*{20}c} {L_{d} p} & 0 \\ 0 & {L_{q} p} \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} 0 & {\Omega_{r} L_{q} } \\ { - \Omega_{r} L_{d} } & 0 \\ \end{array} } \right]} \right)\left[ {TM^{e} } \right]^{ - 1} \left[ \begin{gathered} I_{ds} \hfill \\ I_{qs} \hfill \\ \end{gathered} \right] + \left[ {TM^{e} } \right]\left( {\left[ {\begin{array}{*{20}c} {R_{r} } & {\Omega_{r} L_{r} } \\ { - \Omega_{r} L_{r} } & {R_{r} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {L_{r} p} & 0 \\ 0 & {L_{r} p} \\ \end{array} } \right]} \right)\left[ {TM^{e} } \right]^{ - 1} \left[ \begin{gathered} I_{dr} \hfill \\ I_{qr} \hfill \\ \end{gathered} \right] \\ & = \left[ {TM^{e} } \right]\left[ {\begin{array}{*{20}c} {L_{d} } & 0 \\ 0 & {L_{q} } \\ \end{array} } \right]\left( {\left( {p\left[ {TM^{e} } \right]^{ - 1} } \right)\left[ \begin{gathered} I_{ds} \hfill \\ I_{qs} \hfill \\ \end{gathered} \right] + \left[ {TM^{e} } \right]^{ - 1} \left[ \begin{gathered} pI_{ds} \hfill \\ pI_{qs} \hfill \\ \end{gathered} \right]} \right) + \left[ {TM^{e} } \right]\left[ {\begin{array}{*{20}c} 0 & {\Omega_{r} L_{q} } \\ { - \Omega_{r} L_{d} } & 0 \\ \end{array} } \right]\left[ {TM^{e} } \right]^{ - 1} \left[ \begin{gathered} I_{ds} \hfill \\ I_{qs} \hfill \\ \end{gathered} \right] + \left[ {TM^{e} } \right]\left[ {\begin{array}{*{20}c} {R_{r} } & {\Omega_{r} L_{r} } \\ { - \Omega_{r} L_{r} } & {R_{r} } \\ \end{array} } \right]\left[ {TM^{e} } \right]^{ - 1} \left[ \begin{gathered} I_{dr} \hfill \\ I_{qr} \hfill \\ \end{gathered} \right] \\ & \quad + \left[ {TM^{e} } \right]\left[ {\begin{array}{*{20}c} {L_{r} } & 0 \\ 0 & {L_{r} } \\ \end{array} } \right]\left( {\left( {p\left[ {TM^{e} } \right]^{ - 1} } \right)\left[ \begin{gathered} I_{dr} \hfill \\ I_{qr} \hfill \\ \end{gathered} \right] + \left[ {TM^{e} } \right]^{ - 1} \left[ \begin{gathered} pI_{dr} \hfill \\ pI_{qr} \hfill \\ \end{gathered} \right]} \right) \\ & = \underbrace {{\left[ {\begin{array}{*{20}c} {\left( {\frac{{L_{d} + L_{q} }}{2}} \right)p} & { - \left( {\Omega_{e} - \Omega_{r} } \right)\left( {\frac{{L_{d} + L_{q} }}{2}} \right)} \\ {\left( {\Omega_{e} - \Omega_{r} } \right)\left( {\frac{{L_{d} + L_{q} }}{2}} \right)} & {\left( {\frac{{L_{d} + L_{q} }}{2}} \right)p} \\ \end{array} } \right]\left[ \begin{gathered} I_{ds1} \hfill \\ I_{qs1} \hfill \\ \end{gathered} \right] + \left[ {\begin{array}{*{20}c} {R_{r} + L_{r} p} & { - \left( {\Omega_{e} - \Omega_{r} } \right)L_{r} } \\ {\left( {\Omega_{e} - \Omega_{r} } \right)L_{r} } & {R_{r} + L_{r} p} \\ \end{array} } \right]\left[ \begin{gathered} I_{dr1} \hfill \\ I_{qr1} \hfill \\ \end{gathered} \right]}}_{{\left[ \begin{subarray}{l} V_{dr1} \\ V_{qr1} \end{subarray} \right]}} \\ & \quad + \underbrace {{\left[ {\begin{array}{*{20}c} {\left( {\frac{{L_{d} - L_{q} }}{2}} \right)p} & {\left( {\Omega_{e} - \Omega_{r} } \right)\left( {\frac{{L_{d} - L_{q} }}{2}} \right)} \\ {\left( {\Omega_{e} - \Omega_{r} } \right)\left( {\frac{{L_{d} - L_{q} }}{2}} \right)} & { - \left( {\frac{{L_{d} - L_{q} }}{2}} \right)p} \\ \end{array} } \right]\left[ \begin{gathered} I_{ds2} \hfill \\ I_{qs2} \hfill \\ \end{gathered} \right]}}_{{\left[ \begin{subarray}{l} V_{dr2} \\ V_{qr2} \end{subarray} \right]}} \\ \end{aligned} $$
(30)

Rotor flux equations:

$$ \begin{aligned} \left[ {\begin{array}{*{20}c} {\Psi_{dr} } \\ {\Psi_{qr} } \\ \end{array} } \right] & = \left[ {TM^{e} } \right]\left[ {\begin{array}{*{20}c} {L_{d} } & 0 \\ 0 & {L_{q} } \\ \end{array} } \right]\left[ {TM^{e} } \right]^{{ - 1}} \left[ \begin{gathered} I_{ds} \hfill \\ I_{qs} \hfill \\ \end{gathered} \right] \\ & \quad + \left[ {TM^{e} } \right]\left[ {\begin{array}{*{20}c} {L_{r} } & 0 \\ 0 & {L_{r} } \\ \end{array} } \right]\left[ {TM^{e} } \right]^{ - 1} \left[ \begin{gathered} I_{dr} \hfill \\ I_{qr} \hfill \\ \end{gathered} \right] \\ & = \underbrace {{\left[ {\begin{array}{*{20}c} {\left( {\frac{{L_{d} + L_{q} }}{2}} \right)} & 0 \\ 0 & {\left( {\frac{{L_{d} + L_{q} }}{2}} \right)} \\ \end{array} } \right]\left[ \begin{gathered} I_{ds1} \hfill \\ I_{qs1} \hfill \\ \end{gathered} \right] + \left[ {\begin{array}{*{20}c} {L_{r} } & 0 \\ 0 & {L_{r} } \\ \end{array} } \right]\left[ \begin{gathered} I_{dr1} \hfill \\ I_{qr1} \hfill \\ \end{gathered} \right]}}_{{\left[ {\begin{array}{*{20}c} {\Psi_{dr1} } \\ {\Psi_{qr1} } \\ \end{array} } \right]}} \\ & \quad + \underbrace {{\left[ {\begin{array}{*{20}c} {\left( {\frac{{L_{d} - L_{q} }}{2}} \right)} & 0 \\ 0 & { - \left( {\frac{{L_{d} - L_{q} }}{2}} \right)} \\ \end{array} } \right]\left[ \begin{gathered} I_{ds2} \hfill \\ I_{qs2} \hfill \\ \end{gathered} \right]}}_{{\left[ {\begin{array}{*{20}c} {\Psi_{dr2} } \\ {\Psi_{qr2} } \\ \end{array} } \right]}} \\ \end{aligned} $$
(31)

Appendix B

The recursive EKF equations are as (32–36):

$$ \hat{X}_{k + 1/k} = A_{k} \hat{X}_{k/k} + B_{k} U_{k} $$
(32)
$$ P_{k + 1/k} = F_{k} P_{k/k} F_{k} + Q $$
(33)
$$ K_{k + 1} = HP_{k + 1/k} \left( {HP_{k + 1/k} H^{T} + R} \right)^{ - 1} $$
(34)
$$ \hat{X}_{k + 1/k + 1} = \hat{X}_{k + 1/k} + K_{k + 1} \left( {Y_{k} - HX_{k + 1/k} } \right) $$
(35)
$$ P_{k + 1/k + 1} = \left( {I - K_{k + 1} H} \right)P_{k + 1/k} $$
(36)

Appendix C

The EKF formulation in Fig. 8 is according to (32–36) and the healthy motor equations, where \(I_{\alpha s} ,I_{\beta s} ,\Psi_{\alpha r} ,\Psi_{\beta r} ,\Omega_{r}\) are considered as state variables. Based on the healthy motor equations the following equations can be written:

$$ \begin{aligned} \mathop {\left[ {\begin{array}{*{20}c} {pI_{\alpha s} } \\ {pI_{\beta s} } \\ {p\Psi_{\alpha r} } \\ {p\Psi_{\beta r} } \\ {p\Omega_{r} } \\ \end{array} } \right]}\limits^{pX} & = \mathop {\left[ {\begin{array}{*{20}c} { - \frac{{L_{r} }}{{L_{s} L_{r} - L^{2} }}\left( {R_{s} + \frac{{R_{r} L^{2} }}{{L_{r}^{2} }}} \right)} & 0 & {\frac{{R_{r} L}}{{L_{s} L_{r}^{2} - L^{2} L_{r} }}} & {\frac{{\Omega_{r} L}}{{L_{s} L_{r} - L^{2} }}} & 0 \\ 0 & { - \frac{{L_{r} }}{{L_{s} L_{r} - L^{2} }}\left( {R_{s} + \frac{{R_{r} L^{2} }}{{L_{r}^{2} }}} \right)} & { - \frac{{\Omega_{r} L}}{{L_{s} L_{r} - L^{2} }}} & {\frac{{R_{r} L}}{{L_{s} L_{r}^{2} - L^{2} L_{r} }}} & 0 \\ {\frac{{R_{r} L}}{{L_{r} }}} & 0 & { - \frac{{R_{r} }}{{L_{r} }}} & { - \Omega_{r} } & 0 \\ 0 & {\frac{{R_{r} L}}{{L_{r} }}} & {\Omega_{r} } & { - \frac{{R_{r} }}{{L_{r} }}} & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]}\limits^{A} \mathop {\left[ {\begin{array}{*{20}c} {I_{\alpha s} } \\ {I_{\beta s} } \\ {\Psi_{\alpha r} } \\ {\Psi_{\beta r} } \\ {\Omega_{r} } \\ \end{array} } \right]}\limits^{X} \\ & \quad + \mathop {\left[ {\begin{array}{*{20}c} {\frac{{L_{r} }}{{L_{s} L_{r} - L^{2} }}} & 0 \\ 0 & {\frac{{L_{r} }}{{L_{s} L_{r} - L^{2} }}} \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ \end{array} } \right]}\limits^{B} \mathop {\left[ {\begin{array}{*{20}c} {V_{\alpha s} } \\ {V_{\beta s} } \\ \end{array} } \right]}\limits^{U} + W\left( t \right) \\ \end{aligned} $$
(37)

In this case, the discrete Jacobian and measurement matrices should be achieved in each sampling time as (38) and (39):

$$ F\left( k \right) = \left[ {\begin{array}{*{20}c} {1 - \frac{{L_{r} }}{{L_{s} L_{r} - L^{2} }}\left( {R_{s} + \frac{{R_{r} L^{2} }}{{L_{r}^{2} }}} \right)T_{s} } & 0 & {\frac{{R_{r} L}}{{L_{s} L_{r}^{2} - L^{2} L_{r} }}T_{s} } & {\frac{{\Omega_{r} L}}{{L_{s} L_{r} - L^{2} }}T_{s} } & 0 \\ 0 & {1 - \frac{{L_{r} }}{{L_{s} L_{r} - L^{2} }}\left( {R_{s} + \frac{{R_{r} L^{2} }}{{L_{r}^{2} }}} \right)T_{s} } & { - \frac{{\Omega_{r} L}}{{L_{s} L_{r} - L^{2} }}T_{s} } & {\frac{{R_{r} L}}{{L_{s} L_{r}^{2} - L^{2} L_{r} }}T_{s} } & 0 \\ {\frac{{R_{r} L}}{{L_{r} }}T_{s} } & 0 & {1 - \frac{{R_{r} }}{{L_{r} }}T_{s} } & { - \Omega_{r} T_{s} } & 0 \\ 0 & {\frac{{R_{r} L}}{{L_{r} }}T_{s} } & {\Omega_{r} T_{s} } & {1 - \frac{{R_{r} }}{{L_{r} }}T_{s} } & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} } \right] $$
(38)
$$ H\left( k \right) = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ \end{array} } \right] $$
(39)

Based on the above equations, the rotor flux amplitude and angle of the healthy motor can be achieved as (27) and (28).

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Taghinezhad Vaskeh Mahaleh, A.M., Ghanbari, M. & Ebrahimi, R. Direct vector control strategy of 2-phase induction motor drives based on the conventional rotating transformation matrix and EKF. Electr Eng (2023). https://doi.org/10.1007/s00202-023-02063-3

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