In the field oriented control technique, 3 phases of current, voltage and flux data of the motor are transferred into two planes with a 90° phase difference. Thus, the speed control of the induction was made similar to separately excited DC motors, which creates two planes independent of each other to provide speed control.

Stator and rotor voltage equations of the motor that were reduced to the two planes may be expressed in the d-q synchronous axis plane;

$$\begin{array}{}{V}_{qs\phantom{\rule{thinmathspace}{0ex}}}^{e}={R}_{s\phantom{\rule{thinmathspace}{0ex}}}{i}_{qs}^{e}+\phantom{\rule{thinmathspace}{0ex}}p\phantom{\rule{thinmathspace}{0ex}}{\lambda}_{qs}^{e}+\phantom{\rule{thinmathspace}{0ex}}{\omega}_{e}\phantom{\rule{thinmathspace}{0ex}}{\lambda}_{ds}^{e}\end{array}$$(1)

$$\begin{array}{}{V}_{\phantom{\rule{thinmathspace}{0ex}}ds\phantom{\rule{thinmathspace}{0ex}}}^{e}={R}_{s\phantom{\rule{thinmathspace}{0ex}}}{i}_{ds}^{e}+\phantom{\rule{thinmathspace}{0ex}}p\phantom{\rule{thinmathspace}{0ex}}{\lambda}_{ds}^{e}-\phantom{\rule{thinmathspace}{0ex}}{\omega}_{e}\phantom{\rule{thinmathspace}{0ex}}{\lambda}_{ds}^{e}\end{array}$$(2)

$$\begin{array}{}0={R}_{r\phantom{\rule{thinmathspace}{0ex}}}{i}_{qr}^{e}+\phantom{\rule{thinmathspace}{0ex}}p\phantom{\rule{thinmathspace}{0ex}}{\lambda}_{qr}^{e}+\left(\phantom{\rule{thinmathspace}{0ex}}{\omega}_{e}-{\omega}_{r}\phantom{\rule{thinmathspace}{0ex}}\right){\lambda}_{dr}^{e}\end{array}$$(3)

$$\begin{array}{}0={R}_{r\phantom{\rule{thinmathspace}{0ex}}}{i}_{dr}^{e}+\phantom{\rule{thinmathspace}{0ex}}p\phantom{\rule{thinmathspace}{0ex}}{\lambda}_{dr}^{e}-(\phantom{\rule{thinmathspace}{0ex}}{\omega}_{e}-{\omega}_{r}\phantom{\rule{thinmathspace}{0ex}}){\lambda}_{qr}^{e}\end{array}$$(4)

In the equation above, R_{s}: stator phase resistance, R_{r}: rotor phase resistance, *ω*_{e}: synchronous speed, *ω*_{r}: rotor speed and p: derivative operator. Flux and torque equation may be expressed as;

$$\begin{array}{}{\lambda}_{qs}^{e}={L}_{s\phantom{\rule{thinmathspace}{0ex}}}{i}_{qs}^{e}+\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{L}_{m\phantom{\rule{thinmathspace}{0ex}}}{i}_{qr}^{e}\end{array}$$(5)

$$\begin{array}{}{\lambda}_{ds}^{e}={L}_{s\phantom{\rule{thinmathspace}{0ex}}}{i}_{ds}^{e}+\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{L}_{m\phantom{\rule{thinmathspace}{0ex}}}{i}_{dr}^{e}\end{array}$$(6)

$$\begin{array}{}{\lambda}_{qr}^{e}={L}_{m\phantom{\rule{thinmathspace}{0ex}}}{i}_{qs}^{e}+\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{L}_{r\phantom{\rule{thinmathspace}{0ex}}}{i}_{qr}^{e}\end{array}$$(7)

$$\begin{array}{}{\lambda}_{dr}^{e}={L}_{m\phantom{\rule{thinmathspace}{0ex}}}{i}_{ds}^{e}+\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{L}_{r\phantom{\rule{thinmathspace}{0ex}}}{i}_{dr}^{e}\end{array}$$(8)

$$\begin{array}{}{\displaystyle {T}_{e}=\frac{3}{2}{\frac{P}{2}\phantom{\rule{thinmathspace}{0ex}}\frac{{L}_{m}}{{L}_{r}}\phantom{\rule{thinmathspace}{0ex}}({\lambda}_{dr}^{e}}_{\phantom{\rule{thinmathspace}{0ex}}}{i}_{qs}^{e}-\phantom{\rule{thinmathspace}{0ex}}{\lambda}_{qr}^{e}\phantom{\rule{thinmathspace}{0ex}}{i}_{ds}^{e}\phantom{\rule{thinmathspace}{0ex}})}\end{array}$$(9)

In the equation above, L_{s}: stator phase inductance, L_{r}: rotor phase inductance and L_{m}: common inductance. Using the electromagnetic torque equation, change of rotor speed may be expressed in terms of electrical speed as;
$$\begin{array}{}{\displaystyle \frac{d\phantom{\rule{thinmathspace}{0ex}}\omega r}{dt}=\left(\phantom{\rule{thinmathspace}{0ex}}{T}_{e}-\phantom{\rule{thinmathspace}{0ex}}B\phantom{\rule{thinmathspace}{0ex}}\frac{2}{P}\phantom{\rule{thinmathspace}{0ex}}{\omega}_{r}-\phantom{\rule{thinmathspace}{0ex}}{T}_{1}\phantom{\rule{thinmathspace}{0ex}}\right)\frac{P}{2j}}\end{array}$$(10)

In the equation above, T_{e}: electromagnetic moments, B: coefficient of friction, T_{1}: load moments, j: moment of inertia, P: number of motor poles. Induction motor speed control is achieved using the equation above. Speed error may be found by calculating the difference between the actual speed and the desired speed. To compensate the speed error, it is necessary to change the conduction time and ranking of the semiconductor switches [11, 12]. Figure 1 illustrates the block diagram of field oriented control of the induction motor.

Figure 1 Block diagram of field oriented control of induction motor

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