**7. Proposed Design**

*7.1. Dynamic Model*

There is a very deep relationship between the stator and the rotor of an induction motor. If one is to extract parameters of the rotor, there is a need to know the relationship of the currents and voltage between these two elements of an induction machine. A d–q axis model of an induction machine is presented in Figure 7a. A q-axis equivalent circuit for an adaptable design of an induction machine is presented in Figure 7b. The stator can then be represented by (1) to (10), and the rotor's d–q transformation is represented by (11) to (18). The torque is then represented by the stator and rotor parameters (19) [31].

**Figure 7.** (**a**) The d-axis equivalent circuit of an induction motor. (**b**) The q-axis equivalent circuit of an induction motor.

Stator voltage modelling:

$$V\_{sd} = \sqrt{\frac{2}{3}} \left[ \cos(\theta\_{ds} \times v\_{ds}) + \left( \cos(\theta\_{ds} \times \frac{2\pi}{3} \times v\_{b}) \right) + \left( \cos(\theta\_{ds} \times \frac{4\pi}{3} \times v\_{d}) \right) \right] \tag{1}$$

$$V\_{sq} = -\sqrt{\frac{2}{3}} \left[ \sin(\theta\_{ds} \times v\_{ds}) + \left( \sin \theta\_{ds} \times \frac{2\pi}{3} \times v\_{b} \right) + \left( \sin \theta\_{ds} \times \frac{4\pi}{3} \times v\_{a} \right) \right] \tag{2}$$

*Vsd* and *Vsq* may now be simplified to:

$$V\_{sd} = R\_S \times i\_{sd} + \frac{d}{dt}(\lambda\_{sd}) - \omega\_d \times \lambda\_{sq} \tag{3}$$

$$V\_{s\eta} = R\_s \times i\_{s\eta} + \frac{d}{dt}(\lambda\_{sd}) - \omega\_d \times \lambda\_{sd} \tag{4}$$

Stator fluxes:

$$
\lambda\_{sd} = L\_S \times i\_{sd} + L\_{m} \times i\_{rd} \tag{5}
$$

$$
\lambda\_{sq} = L\_S \times i\_{sq} + L\_m \times i\_{rd} \tag{6}
$$

Stator currents:

$$\dot{a}\_{ds} = \frac{1}{Xl\_S}(\lambda\_{ds} - \lambda\_{\text{md}}) \tag{7}$$

$$i\_{\rm qs} = \frac{1}{Xl\_S} \left(\lambda\_{\rm qs} - \lambda\_{\rm mq}\right) \tag{8}$$

Stator voltages:

$$V\_{sd} = R\_S \times i\_{sd} + \frac{d}{dt}(\lambda\_{sd}) - \omega\_d \times \lambda\_{sq} \tag{9}$$

$$V\_{sq} = R\_S \times i\_{sq} + \frac{d}{dt} \left(\lambda\_{sq}\right) - \omega\_d \times \lambda\_{sd} \tag{10}$$

Mathematical model of the rotor:

$$V\_{rd} = R\_r \times i\_{rd} + \frac{d}{dt}(\lambda\_{rd}) - \omega\_{dA} \times \lambda\_{rq} \tag{11}$$

$$V\_{r\eta} = R\_r \times i\_{rd} + \frac{d}{dt}(\lambda\_{rd}) - \omega\_{dA} \times \lambda\_{rd} \tag{12}$$

Rotor flux equations:

$$
\lambda\_{rd} = L\_r \times i\_{rd} + L\_m \times i\_{sd} \tag{13}
$$

$$
\lambda\_{r\eta} = L\_r \times i\_{r\eta} + L\_m \times i\_{sd} \tag{14}
$$

Rotor currents:

$$i\_{dr} = \frac{1}{Xl\_S}(\lambda\_{dr} - \lambda\_{md})\tag{15}$$

$$i\_{q\mathcal{V}} = \frac{1}{Xl\_S} \left(\lambda\_{q\mathcal{V}} - \lambda\_{mq}\right) \tag{16}$$

Rotor voltages:

$$V\_{rd} = R\_r \times i\_{rd} + \frac{d}{dt}(\lambda\_{rd}) - \omega\_{dA} \times \lambda\_{rq} \tag{17}$$

$$V\_{r\eta} = R\_r \times i\_{r\eta} + \frac{d}{dt}(\lambda\_{r\eta}) - \omega\_{dA} \times \lambda\_{rd} \tag{18}$$

Electromagnetic torque:

$$T\_{\rm em} = \frac{P}{2} \times L\_{\rm m} \left( i\_{sq} \times i\_{rd} - i\_{sd} \times i\_{rq} \right) \tag{19}$$

where


Figure 8a,b present the implantation of the dynamic model of the induction motor. The stator supply voltages and currents are presented in Figure 8a,b. The rotor currents

are presented in Figure 8b. The torque and speed are presented in Figure 9a,b [32], where Figure 10 illustrates the complete model of the dynamic induction motor.

**Figure 8.** (**a**) Simulation of a dynamic model of an induction motor; (**b**) current equation of the stator.

**Figure 9.** (**a**) Torque equation; (**b**) speed equation.

**Figure 10.** A complete model of the dynamic induction motor.
