*4.2. Selection of the Torque Controller Parameter, Kc*

The impact of the selection of *K*c on the static and dynamic performance is analyzed in this section. The standard deviation is used to evaluate the extent of the torque ripple in the anaylsis process. That is:

$$\sigma\_{\rm T} = \sqrt{\frac{1}{n-1} \sum\_{i=1}^{n} \left( T\_{\rm e}(i) - \overline{T\_{\rm e}} \right)^{2}} \tag{23}$$

where, *T*<sup>e</sup> = <sup>1</sup> *n* "*n i*=1 *T*e(*i*), *n* is the number of samples and *n* = 1000.

In the simulation, the motor is operated stably at 100 r/min with 50 Nm. When *t* = 0.1 s, *n*ref is set to 200 r/min. When *K*<sup>c</sup> is between 0.2 and 2.5, the variation rules of the static torque ripple σ<sup>T</sup> and the regulating time of electromagnetic torque *t*<sup>d</sup> are shown in Figure 8:


**Figure 8.** Steady and transient torque performance of the proposed DTC-SVM scheme under different values of *K*c.

#### *4.3. Dynamic Characteristic of the Motor Control System*

The dynamic simulation waveforms of the improved DTC-SVM scheme and the conventional DTC-SVM scheme are shown in Figure 9. In the simulation, for the conventional DTC-SVM scheme, the parameters of the speed PI controller are consistent with Table 1, the torque PI controller are *<sup>K</sup>*<sup>p</sup> <sup>=</sup> 6.25 <sup>×</sup> <sup>10</sup>−<sup>4</sup> and *T*<sup>i</sup> = 20 *T*s. The motor is operated at a steady-state of 100 r/min with 50 Nm. When *t* = 0.1 s, *n*ref is set to −100 r/min and the motor is rotating in reverse. When *t* = 0.2 s, *n*ref is set to 100 r/min again and the motor rotates normally. When *t* = 0.3 s, the load is suddenly increased to 150 Nm.

**Figure 9.** Speed dynamic simulation waveforms. (**a**) The improved DTC-SVM scheme; (**b**) the conventional DTC-SVM scheme.

As can be seen from Figure 9, the improved DTC-SVM scheme inherits the advantages of the conventional scheme, which has a rapid torque response and excellent stator currents. Meanwhile, because the torque loop of the conventional DTC-SVM is a second-order system, overshoot during torque regulation will inevitably occur. However, the improved DTC-SVM regulates the electromagnetic torque using *k*<sup>T</sup> varied with the current load condition, so overshoots of the electromagnetic torque has been restrained to some extent.

#### **5. Experimental Results**

To verify the feasibility and effectiveness of the improved DTC-SVM scheme, experiments have been carried out on a 6 kW PMSM. The parameters of the experimental setup is consistant with the simulation, which is shown in Table 1. In the experimental setup, which is shown in Figure 10, a TMS320F28335 digital signal processor (DSP) is employed for the control strategy; the stator currents are measured by a LA-50P Hall sensor produced by LEM® (Geneva, Switzerland), and the DC-side voltage is measured by the VSM025A Hall sensor, and the sampling tasks of the stator currents and DC-side voltage are accomplished by the DSP; the angular velocity is obtained from the incremental mode optical shaft angle encoder; the electromagnetic torque is estimated with the mathematical model of the PMSM. Besides, the sampling and control period of the DSP is 200 μs.

**Figure 10.** Photograph of the experimental setup.

#### *5.1. MTPA Operation*

In the simulations and experiments, the motor operated at 100 r/min. At first, the motor is operated at 20 Nm, and the load is added at 20 Nm per time, until the load reaches 200 Nm. The actual electromagnetic torque, amplitude of stator flux linkage, average value of d-/q-axis currents and RMS value of the phase current under each load condition are measured, and the experimental data are plotted in Figure 11.

**Figure 11.** The performance of proposed on-line maximum torque per ampere (MTPA) method. (**a**) *i*<sup>d</sup> – *i*<sup>q</sup> plot; (**b**) *T*<sup>e</sup> − |s| plot; (**c**) *T*<sup>e</sup> *– I*<sup>A</sup> plot.

It can be seen from Figure 11 that the MTPA trajectory obtained by simulation and experiments with the proposed online MTPA method almost coincide with the theoretical MTPA trajectory based on Equation (1).

#### *5.2. Torque Control Performance*

Figures 12 and 13 give the experimental waveforms with the conventional DTC-SVM scheme using two different parameters of torque PI controllers, which are calculated with *k*<sup>T</sup> under no-load and rated load conditions, respectively. Figure 14 gives the experimental waveforms with the improved DTC-SVM scheme. In the experiments, firstly, the motor is operated at 100 r/min with no load and 50 Nm load, respetively. Next, the reference speed is increased to 200 r/min. Besides, to assure the incremental quantities of electromagnetic torque are consistent for different load conditions, the limitation of the speed PI controller is set to 100 Nm when the motor is operating under the no load condition, and 150 Nm for the 50 Nm load, correspondingly.

In Figure 12, the parameters of the torque controller of the conventional DTC-SVM scheme are calculated by the constant *k*<sup>T</sup> corresponding to the no load condition. Then, it can be obtained that *K*<sup>p</sup> = 6.25 <sup>×</sup> 10−<sup>4</sup> and *T*<sup>i</sup> = 20 *T*s. It can be seen from Figure 12 that the motor has favorable torque performance when operating under the no load condition. However, the dampening of the torque loop will be decreased on account of the decreased *k*<sup>T</sup> when the motor operates with a 50 Nm load, and this will cause na oscillation process during the transiente torque.

In Figure 13, the parameters of the torque controller of the conventional DTC-SVM scheme are calculated by the constant *k*<sup>T</sup> corresponding to the rated load condition. Then, it can be obtained *K*<sup>p</sup> = 2.0 <sup>×</sup> 10−<sup>4</sup> and *T*<sup>i</sup> = 20 *T*s. It can be seen from Figure 13 that the torque response is favorable when the motor is operated with a 50 Nm load. However, the damping of the torque loop will be increased because of the increment of *k*<sup>T</sup> when the motor is operating under the no load condition, resulting in a longer regualtion time of the eletromagnetic torque than the no load condition.

It can be seen from Figure 14 that the motor has a rapid and consistent torque response when operated under different load conditions for the proposed DTC-SVM scheme.

**Figure 12.** Experimental waveforms of the conventional DTC-SVM scheme during transient operation at *<sup>K</sup>*<sup>p</sup> <sup>=</sup> 6.25 <sup>×</sup> <sup>10</sup>−<sup>4</sup>*; Ti* <sup>=</sup> <sup>20</sup> *<sup>T</sup>*s. (**a**) No load; (**b**) 50 Nm load.

**Figure 13.** Experimental waveforms of the conventional DTC-SVM scheme during transient operation at *<sup>K</sup>*<sup>p</sup> <sup>=</sup> 2.0 <sup>×</sup> <sup>10</sup><sup>−</sup>4; *<sup>T</sup>*<sup>i</sup> <sup>=</sup> <sup>20</sup> *Ts*. (**a**) No load; (**b**) 50 Nm load.

**Figure 14.** Experimental waveforms of the improved and conventional DTC-SVM scheme during transient operation. (**a**) No load; (**b**) 50 Nm load.

#### *5.3. Characteristics of the Torque Controller*

Figure 15 gives experimental waveforms with/without the compensation algorithm mentioned in Section 3.3. In the experiment, the motor is operated at 200 r/min with no load steadily. As shown in Figure 15, with the compensation algorithm, the electromagnetic torque could track its reference value without static error. While without the compensation algorithm, there are deviations between the electromagnetic torque and reference torque.

**Figure 15.** Experimental waveforms with/without compensation for the proposed DTC-SVM. (**a**) Without compensation; (**b**) with compensation.

Figure 16 gives the experimental waveforms of torque response for the value of *K*<sup>c</sup> equal to 0.2, 1.0 and 2.0, respectively. In the experiments, firstly, the motor is operated at 100 r/min with no load. Next, the load is increased to 50 Nm. As can be seen from Figure 16, with the increase of *K*c, the ripple amplitude of the static torque is also increased, but the motor has a faster torque dynamic response; with the decrease of *K*c, although the ripple amplitude of the static torque is decreased, the dynamic performance of the motor deteriorated. Obviously, the above experimental results are consistent with the simulation.

**Figure 16.** Transient and steady torque performance of the proposed DTC-SVM scheme with different values of *K*c. (**a**) *K*c = 0.2; (**b**) *K*c = 1.0; (**c**) *K*c = 2.0.

#### **6. Conclusions**

The constant torque change rate is used to regulate the electromagnetic torque in the conventional DTC-SVM, and the variation of the torque change rate has not been taken into consideration. Therefore, this kind of control mode will lead to a phenomenon where dampening of the torque control changes with the variation of the output torques. So, the dynamic performances of the electromagnetic torque are different under different output torque conditions. In order to solve this problem, this paper puts forward an improved DTC-SVM scheme. Compared with the conventional scheme, the proposed scheme adopts a torque controller with torque angle estimation (TC-LAE). With this torque controller, the torque change rate is adjusted in real-time according to the variation of the output electromagnetic torques. The dynamic performance of the torque control is improved. Meanwhile, for the determination of the reference flux amplitude, the MTPA criterion expressed by the flux linkage is established and the Lagrange interpolation fitting method is used to realize the on-line MTPA operation of the PMSM. With the proposed online MTPA method, we can determine the reference flux amplitude directly that could ensure the PMSM-MTPA operation, instead of utilizing the traditional two-dimensional look-up table. The simulation and experimental results of the improved and conventional scheme were researched using a 6 kW PMSM. The conclusions are as follows:


**Author Contributions:** Conceptualization, Z.Z. and Z.W.; methodology, Z.Z.; software, Z.Z.; validation, X.G. and G.Z.; formal analysis, Z.Z. and Q.G.; writing—original draft preparation, Z.Z.; writing—review and editing, Z.Z. and G.Z., and; funding acquisition, Z.Z.

**Funding:** This research was funded by "The Science & Technology Development Fund of Tianjin Education Commission for Higher Education, grant number 2018KJ207".

**Conflicts of Interest:** The authors declare no conflict of interest.
