# Simulation Study of Two Torque Optimization Methods for Direct Torque-Controlled Induction Motors
## 2 IM Torque Response Optimization
### Torque Optimization Method (1)  
Considering Figure 2, the IM flux vectors, stator (\(\mathbf{\lambda_{s}}\)) and rotor (\(\mathbf{\lambda_{r}}\)), are indicated. The applied stator voltage vector (\(\mathbf{V_{k}}\)) is defined as follows:  
\[\mathbf{V_{k}}=V<\theta_{k} \tag{1}\]  
where (\(\theta_{k}\)) is the angle of the voltage vector, (\(k\)) is the order of the voltage vector, and (\(V\)) is the inverter DC link voltage. Typically, the rotor time constant of IM is relatively large compared to the stator time constant [31]. Hence, the rotor flux vector is supposed to revolve with constant amplitude and speed through the changes of the stator variables [8]. The equation of the flux vector of the rotor may be represented by:  
\[\overline{\mathbf{\lambda}}_{r}=\lambda_{r}\angle(\omega t+\theta_{ro}) \tag{2}\]  
where (\(\omega\)) is the rotor flux angular speed. Suppose that the stator flux vector initial value (\(\overline{\mathbf{\lambda}}_{o}\)) is given by:  
\[\overline{\mathbf{\lambda}}_{s}\big{|}_{t=0}=\overline{\mathbf{\lambda}}_{o}=\lambda_ {o}\angle\theta_{o} \tag{3}\]  
Figure 1: The effect of voltage vector selection on the flux vector trajectory in the hysteresis band region. Figure 2: The trajectory of the rotor and stator flux vectors. Assuming that the stator resistance is neglected, the stator voltage is related to the stator flux by:  
\[V_{k}=\frac{d\overline{\lambda}_{s}}{dt} \tag{4}\]  
Solving Equation (4) for the stator flux, we get:  
\[\overline{\lambda}_{s}=V_{k}t+\overline{\lambda}_{o} \tag{5}\]  
According to Equation (5), the stator flux vector tip moves in a direction parallel to the selected voltage vector. Hence, the trajectory of the stator flux can be directed by the selected voltage vector, as shown in Figure 1. The torque equation of the IM is specified by the following:  
\[T=k_{c}\cdot Im\left\{\overline{\lambda}_{s}\overline{\lambda}_{r}\right\} \tag{6}\]  
where (\(k_{c}\)) is the torque constant. The time rate differentiation of Equation (6) is:  
\[\frac{dT}{dt}=k_{c}\cdot Im\left\{\frac{d\overline{\lambda}_{s}}{dt}\overline{ \lambda}_{r}+\overline{\lambda}_{s}\frac{d\overline{\lambda}_{r}}{dt}\right\} \tag{7}\]  
Incorporating Equations (2) and (5) provides the following:  
\[\left\{ \begin{array}{c}\frac{d\overline{\lambda}_{r}}{dt}=-j\omega \overline{\lambda}_{r}\\ \frac{d\overline{\lambda}_{r}}{dt}=V_{k}\end{array} \right. \tag{8}\]  
Therefore, substituting Equation (8) into Equation (7), the equation is as follows:  
\[\frac{dT}{dt}=k_{c}\cdot Im\left\{V_{k}\overline{\lambda}_{r}^{*}-j\omega \overline{\lambda}_{s}\overline{\lambda}_{r}\right\} \tag{9}\]  
when substituting Equation (5) into Equation (9), the equation is as follows:  
\[\frac{dT}{dt}=k_{c}\cdot Im\left\{(V_{k}-j\omega\left(V_{k}t+\overline{ \lambda}_{o}\right))\overline{\lambda}_{r}\right\} \tag{10}\]  
Solving for the imaginary part and substitute with \(t=0\) (the time of checking for the optimum voltage vector) to obtain the following:  
\[\frac{dT}{dt}=k_{c}\lambda_{r}[Vsin(\theta_{k}-\theta_{ro})-\omega\lambda_{o} cos(\theta_{k}-\theta_{ro})] \tag{11}\]  
Using math operations and assuming that \(V\), \(\lambda_{o}\), and \(\omega\) are considered constants during the optimization process, Equation (11) can be reduced to:  
\[\frac{dT}{dt}=k^{*}sin(\theta_{k}-\theta_{ro}-\theta_{x}) \tag{12}\]  
where \(k^{*}\) and \(\theta_{x}\) are constants. Finally, the voltage vectors selection will be governed by Equation (12), which will optimize the selection to the vector that gives quick torque response. The optimization function (\(f\)) may be written as:  
\[f=\max_{k}[sin(\theta_{k}-\theta_{ro}-\theta_{x})] \tag{13}\]  
The main disadvantage of this method is the requirement to determine the rotor flux angle (\(\theta_{ro}\)). A rotor flux observer must be adapted to determine \(\theta_{ro}\). Unfortunately, this requirement utilizes some parameters of the IM that may change with the operating conditions.