*2.2. MTPA Equations*

As it could be found from the torque Equation (4), the total torque produced by machine includes magnetic component proportional to the rotor magnetic flux and reluctance component proportional to the difference between direct and quadrature inductances. The magnetic and reluctance torques depend on the phase of the stator current as sine and sine of doubled angle, respectively, therefore the resulting torque of a machine also has a maximum which has to be defined and used for efficient control.

In order to do this, the magnitude of the stator current vector is fixed at *Is* and the relation between the current components, which corresponds to the maximum can be found. The quadrature component of the stator current is:

$$i\_q = \sqrt{l\_s^2 - i\_d^2} \tag{5}$$

Combining (5) and (4) with the following differentiation, with respect to *id*, results in the expression, which is used for the calculation of the maximum of torque [16]:

$$i\_d = -\frac{\Psi\_m}{4\left(L\_d - L\_q\right)} - \sqrt{\frac{\Psi\_m^2}{16\left(L\_d - L\_q\right)^2} + \frac{I\_s^2}{2}}\tag{6}$$

Equation (6) can be rewritten in terms of the current components:

$$i\_d = -\frac{\Psi\_m}{2\left(L\_d - L\_q\right)} - \sqrt{\frac{\Psi\_m^2}{4\left(L\_d - L\_q\right)^2} + i\_q^2} \tag{7}$$

The last two equations define the MTPA condition of synchronous machines with constant parameters. They are used for the implementation of the conventional MTPA control and move stator current vector along the trajectory, the typical shape is depicted in Figure 2. Good examples of MTPA implementation according to this approach can be found in [40–42].

**Figure 2.** MTPA trajectory in *idiq* plane.

Since Equations (6) and (7) are calculation intensive, the MTPA trajectory is frequently written in terms of the stator current and its phase, which is a function of the stator current amplitude. In this case, the MTPA trajectory is described by the following equations:

$$\begin{aligned} \dot{a}\_d &= -I\_s \sin(\gamma), \\ \dot{a}\_q &= I\_s \cos(\gamma); \end{aligned} \quad \begin{aligned} \text{g} &= \\ \text{g}, \end{aligned} \tag{8}$$

$$\gamma(I\_s) = \arccos\left(\frac{-\Psi\_m + \sqrt{\Psi\_m^2 + 8\left(L\_d - L\_q\right)^2 I\_s^2}}{4\left(L\_d - L\_q\right)I\_s}\right). \tag{9}$$

Despite the MTPA angle being hard for calculation, it is a smooth function, which can be easily approximated with a first- or second-order polynomial, which can be easily seen from Figure 3. Good examples of MTPA implementation according to this approach can be found in [43–45].

**Figure 3.** Dependence of the MTPA angle γ on the stator current amplitude.

#### **3. Impact of the Motor Parameters Variation on the MTPA Trajectories**

In order to design constraints properly, the impact of motor parameters on the MTPA trajectory must be analyzed. The analysis was performed for the motor used in the experimental verification. The rated parameters are demonstrated in Table 1.

**Table 1.** Motor rated parameters.

