### Healthy motor model

The dynamic model of the WRIM can be built under healthy condition in the ABC frame as functions of rotor position by its voltage differential and mechanical equations as follows:

$$\left[ {\begin{array}{*{20}c} {V_{\rm s} } \\ {V_{\rm r} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {Z_{\rm ss} } &\quad {Z_{\rm sr} } \\ {Z_{\rm rs} } &\quad {Z_{\rm rr} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {I_{\rm s} } \\ {I_{\rm r} } \\ \end{array} } \right]$$

(1)

Where

$$\left[ {V_{\rm s} } \right] = \left[ {\begin{array}{*{20}c} {v_{\rm as} } & {v_{\rm bs} } & {v_{\rm cs} } \\ \end{array} } \right]^{\text{T}} ,\quad\left[ {V_{\rm r} } \right] = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 \\ \end{array} } \right]^{\text{T}}$$

(2)

$$\left[ {I_{\rm s} } \right] = \left[ {\begin{array}{*{20}c} {i_{\rm as} } & {i_{\rm bs} } & {i_{\rm cs} } \\ \end{array} } \right]^{\text{T}}, \quad\left[ {I_{\rm r} } \right] = \left[ {\begin{array}{*{20}c} {i_{\rm ar} } & {i_{\rm br} } & {i_{\rm cr} } \\ \end{array} } \right]^{\text{T}}$$

(3)

$$\left[ {Z_{\rm ss} } \right] = \left[ {\begin{array}{*{20}c} {R_{\rm s} + L_{\rm s} p} &\quad { - \,0.5L_{\rm o} p} &\quad { - \,0.5L_{\rm o} p} \\ { - \,0.5L_{\rm o} p} &\quad {R_{\rm s} + L_{\rm s} p} &\quad { - \,0.5L_{\rm o} p} \\ { - \,0.5L_{\rm o} p} &\quad { - 0.5L_{\rm o} p} &\quad {R_{\rm s} + L_{\rm s} p} \\ \end{array} } \right]$$

(4)

$$\left[ {Z_{\rm rr} } \right] = \left[ {\begin{array}{*{20}c} {R_{\rm r} + L_{\rm r} p} &\quad { - \,0.5L_{\rm o} p} &\quad { - \,0.5L_{\rm o} p} \\ { - \,0.5L_{\rm o} p} &\quad {R_{\rm r} + L_{\rm r} p} &\quad { - \,0.5L_{\rm o} p} \\ { - \,0.5L_{\rm o} p} &\quad { - 0.5L_{\rm o} p} &\quad {R_{\rm r} + L_{\rm r} p} \\ \end{array} } \right]$$

(5)

$$\left[ {Z_{\rm sr} } \right] = L_{\rm o} p\left[ {\begin{array}{*{20}c} {\cos (\theta_{\rm r} )} &\quad {\cos (\theta_{\rm r} - 2\pi /3)} &\quad {\cos (\theta_{\rm r} + 2\pi /3)} \\ {\cos (\theta_{\rm r} + 2\pi /3)} &\quad {\cos (\theta_{\rm r} )} &\quad {\cos (\theta_{\rm r} - 2\pi /3)} \\ {\cos (\theta_{\rm r} - 2\pi /3)} &\quad {\cos (\theta_{\rm r} + 2\pi /3)} &\quad {\cos (\theta_{\rm r} )} \\ \end{array} } \right]$$

(6)

$$\left[ {Z_{\rm rs} } \right] = \left[ {Z_{\rm sr} } \right]^{\text{T}}$$

(7)

$$L_{\rm s} = L_{\rm ls} + L_{\rm o}, \quad L_{\rm r} = L_{\rm lr} + L_{\rm o}$$

(8)

$$T_{\text{em}} = \frac{P}{2}\left[ {\begin{array}{*{20}c} {I_{\rm s} } \\ {I_{\rm r} } \\ \end{array} } \right]^{\text{T}} \frac{{{\text{d}}L}}{{{\text{d}}\theta_{\rm r} }}\left[ {\begin{array}{*{20}c} {I_{\rm s} } \\ {I_{\rm r} } \\ \end{array} } \right]$$

(9)

$$T_{\text{em}} - T_{\text{load}} = J\frac{{{\text{d}}\omega_{m} }}{{\text{d}}t} + B\omega + T_{\rm f}$$

(10)

$$p\theta_{\rm r} = \omega_{\rm r} = P\omega_{m}$$

(11)

$$p = \frac{\text{d}}{{{\text{d}}t}}$$

(12)

### Open-phase motor models

The dynamic model of the WRIM is modified to represent the open rotor phase condition. The stator windings of the studied motor are connected as star connection, while the rotor windings could be star or delta connection. The rotor windings could be connected in different four connections, which are isolated neutral star connection, standard delta connection with internal open circuit rotor phase, standard delta connection with external open circuit rotor phase and delta connection with reversed one phase and external open circuit. These connections are shown in Fig. 1.

#### Isolated neutral star connection

When one rotor phase is open-circuited in case of isolated neutral star connection, the other two phases become connected in series as shown in Fig. 1a. So the voltage differential equations can be rewritten as in healthy case with the following changes:

$$\left[ {V_{\rm r} } \right] = \left[ 0 \right]$$

(13)

$$\left[ {I_{\rm r} } \right] = \left[ {i_{\rm r} } \right], \quad i_{\rm ar} = - i_{\rm br} = i_{\rm r}$$

(14)

$$\left[ {Z_{\rm rr} } \right] = \left[ {2R_{\rm r} + (2L_{\rm lr} + 3L_{\rm o} )p} \right]$$

(15)

$$\left[ {Z_{\rm sr} } \right] = L_{\rm o} p\left[ {\begin{array}{*{20}c} {\cos (\theta_{\rm r} ) - \cos (\theta_{\rm r} - 2\pi /3)} \\ {\cos (\theta_{\rm r} + 2\pi /3) - \cos (\theta_{\rm r} )} \\ {\cos (\theta_{\rm r} - 2\pi /3) - \cos (\theta_{\rm r} + 2\pi /3)} \\ \end{array} } \right]$$

(16)

#### Standard delta connection with internal open circuit rotor phase

For standard delta with internal open circuit, when one rotor phase is internally open-circuited, each phase of the other two phases will be short-circuited upon itself as shown in Fig. 1b, the voltage differential equations can be rewritten with the following changes:

$$\left[ {V_{\rm r} } \right] = \left[ {\begin{array}{*{20}c} 0 & 0 \\ \end{array} } \right]^{\text{T}}$$

(17)

$$\left[ {I_{\rm r} } \right] = \left[ {\begin{array}{*{20}c} {i_{\rm ar} } & {i_{\rm br} } \\ \end{array} } \right]^{\text{T}}$$

(18)

$$\left[ {Z_{\rm rr} } \right] = \left[ {\begin{array}{*{20}c} {R_{\rm r} + L_{\rm r} p} &\quad { - 0.5L_{\rm o} p} \\ { - 0.5L_{\rm o} p} &\quad {R_{\rm r} + L_{\rm r} p} \\ \end{array} } \right]$$

(19)

$$\left[ {Z_{\rm sr} } \right] = L_{\rm o} p\left[ {\begin{array}{l} {\begin{array}{l} {\cos (\theta_{\rm r} )} \\ {\cos (\theta_{\rm r} + 2\pi /3)} \\ {\cos (\theta_{\rm r} - 2\pi /3)} \\ \end{array} } {\begin{array}{l} {\cos (\theta_{\rm r} - 2\pi /3)} \\ {\cos (\theta_{\rm r} )} \\ {\cos (\theta_{\rm r} + 2\pi /3)} \\ \end{array} } \\ \end{array} } \right]$$

(20)

#### Standard delta connection with external open circuit phase

In case of standard delta connection with external open circuit, when one rotor phase is externally open-circuited, two of rotor phases are connected in series and the combination of them is connected in parallel with the third phase, the voltage differential equations of the this configuration can be rewritten as in healthy case with the following changes:

$$\left[ {V_{\rm r} } \right]\,\, = \,\,\left[ {\begin{array}{*{20}l} 0 & 0 \\ \end{array} } \right]^{\rm T}$$

(21)

$$\left[ {I_{\rm r} } \right] = \left[ {\begin{array}{*{20}l} {i_{\rm ar} } & {i_{\rm br} } \\ \end{array} } \right]^{\text{T}}$$

(22)

$$\left[ {Z_{\rm rr} } \right] = \left[ {\begin{array}{*{20}l} {R_{\rm r} + L_{\rm r} p} &\quad { - L_{\rm o} p} \\ { - L_{\rm o} p} &\quad {2R_{\rm r} + (2L_{\rm r} + L_{\rm o} )p} \\ \end{array} } \right]$$

(23)

$$\left[ {Z_{\rm sr} } \right] = L_{\rm o} p\left[ {\begin{array}{l} {\begin{array}{l} {\cos (\theta_{\rm r} )} \\ {\cos (\theta_{\rm r} + 2\pi /3)} \\ {\cos (\theta_{\rm r} - 2\pi /3)} \\ \end{array} } {\begin{array}{l} {\cos (\theta_{\rm r} + 2\pi /3) + \cos (\theta_{\rm r} - 2\pi /3)} \\ {\cos (\theta_{\rm r} ) + \cos (\theta_{\rm r} + 2\pi /3)} \\ {\cos (\theta_{\rm r} - 2\pi /3) + \cos (\theta_{\rm r} )} \\ \end{array} } \\ \end{array} } \right]$$

(24)

#### Delta connection with reversed one phase and external open circuit rotor phase

For delta connection with reversed one, when one rotor phase is externally open-circuited, two of rotor phases are connected in series and the combination of them is connected in parallel with the third phase, the voltage differential equations of the this configuration can be rewritten as in healthy case with the following changes:

$$\left[ {V_{\rm r} } \right]\,\, = \,\,\left[ {\begin{array}{*{20}c} 0 &\quad 0 \\ \end{array} } \right]^{\rm T}$$

(25)

$$\left[ {I_{\rm r} } \right] = \left[ {\begin{array}{l} {i_{\rm ar} } \quad {i_{\rm br} } \\ \end{array} } \right]^{\text{T}}$$

(26)

$$\left[ {Z_{\rm rr} } \right] = \left[ {\begin{array}{ll} {R_{\rm r} + L_{\rm r} p} &\quad 0 \\ 0& \quad {2R_{\rm r} + (2L_{\rm r} + 3L_{\rm o} )p} \\ \end{array} } \right]$$

(27)

$$\left[ {Z_{\rm sr} } \right] = L_{\rm o} p\left[ {\begin{array}{l} {\begin{array}{l} {\cos (\theta_{\rm r} )} \\ {\cos (\theta_{\rm r} + 2\pi /3)} \\ {\cos (\theta_{\rm r} - 2\pi /3)} \\ \end{array} } {\begin{array}{l} {\cos (\theta_{\rm r} + 2\pi /3) - \cos (\theta_{\rm r} - 2\pi /3)} \\ {\cos (\theta_{\rm r} ) - \cos (\theta_{\rm r} + 2\pi /3)} \\ {\cos (\theta_{\rm r} - 2\pi /3) - \cos (\theta_{\rm r} )} \\ \end{array} } \\ \end{array} } \right]$$

(28)