**Figure 10.** Voltage signal conditioning circuit.

#### *2.7. Measurement of Rotor Position and Speed*

The measurement of the machine's speed, position and direction of rotation can be accomplished through the encoder. The encoder used, called incremental with quadrature output, is a device capable of converting shaft rotation information into pulse train signals. State-of-the-art encoders mostly operate on potentiometric, capacitive, magnetic, or optical principles [37].

The incremental encoder is composed of a disk with a series of slots, an infrared light source, and a photoelectric sensor that produce the electrical pulses of channel A. The direction of rotation is obtained by the second strip of slots, channel B, positioned such that they produce an electrical 90◦ lag concerning channel A. The frequency of the pulse train of channels A and B are directly proportional to the rotational speed of the machine's rotor. The third channel (index) outputs a high logic level signal with each complete revolution of the encoder disk. Figure 11 shows the encoder conditioning circuit and the pulse train of each output channel.

The output voltage of the channels is 12 *Vdc*, as a function of the encoder supply which is 12 *Vdc*. To connect these signals to the DSP's eQEP (Enhanced Quadrature Encoder Pulse) inputs, voltage divider circuits were used to reduce the voltage of the three output channels. In addition, operational amplifiers were used in the voltage buffer configuration, isolating the input signal from the DSP.

The encoder model used for the development of this work is the E30S4 and has an output with a resolution of 360 pulses per revolution.

**Figure 11.** Encoder signal conditioning circuit.

#### *2.8. Auxiliary Voltage Source*

The auxiliary voltage source has multiple outputs: ±15 *Vdc*, ±12 *Vdc*, 5 *Vdc* and 3.3 *Vdc*. These voltages are required to power the integrated circuits and sensors of the current measurement boards, voltage measurement, encoder, and complementary power driver circuits. It is also important to mention the need to use voltage sources isolated concerning

the sources that power the power driver, to prevent short circuits in the power step damage the inverter measurement and control circuits.

#### *2.9. Induction Motor Parameters*

For the experiments, an induction motor from the manufacturer Voges was used. Table 3 shows the electrical and mechanical parameters of the machine.


**Table 3.** Parameters of the motor used.

Some parameters were collected directly from the motor board data and others were obtained through the blocked-rotor and no-load tests.

### **3. Dynamic Modeling of the Indirect Field Oriented for the MIT**

The field-oriented control strategy is classified according to the method of acquisition of the rotor flux angle. In the direct method (Direct Field Oriented Control—DFOC), sensors installed in the machine's air gap are used to measure the flux. In the indirect method (IFOC), there is no flux measurement, the position and slip are used to obtain the position of the rotor flux angle [5].

The IFOC makes use of the fact that satisfying the relationship between slipping and stator current is a necessary and sufficient condition to produce field orientation [5].

In Field Oriented Control (FOC), one can make a direct analogy with the fieldindependent DC machine control [38]. In DC machines, the magnetic flux established by the armature and field currents is orthogonal to each other, regardless of the rotor's position and mechanical load. Thus, for a constant field current level, the armature current is responsible for the machine's torque production. In asynchronous machine FOC, the same principle applies, where the direct axis current is responsible to establish the machine's magnetic flux and quadrature axis current is responsible for the torque level.

Figure 12 presents the block diagram of the proposed speed control system using the indirect oriented field control strategy for the three-phase induction machine [39]. The system consists of an induction motor with a three-phase inverter, SVPWM modulation block, orientation block in which the rotor flux angle is calculated, block for the referential transformations (ABC to dq0) through Clarke and Park transformations, current control loop, speed control loop and position control loop. In this paper, only the design and tuning of the current and speed control loops will be addressed.

The equation of the dynamic model of the induction machine with indirect orientation presented in this paper is supported on the models and simplifications described in [5,39,40].

In an ideal field oriented of a three-phase induction machine, there is decoupling between the direct and quadrature axes, and the rotor flux is aligned to the direct axis (Figure 13). Thus, the flux and its derivative on the quadrature axis are null.

**Figure 12.** Indirect field oriented control scheme of induction machine.

**Figure 13.** Analysis of the induction motor in the synchronous rotating referential.

Thus, the Equation (4) is used to calculate the rotor flux, given by

$$
\lambda\_{dr} = \frac{L\_m i\_{ds}}{1 + p \frac{L\_r}{R\_r}} \tag{4}
$$

where *Lm*, *Lr*, *Rr* and *ids* represent mutual inductance, rotor inductance, rotor resistance, direct axis current, and *p* represents the derivative in time (*d*/*dt*), respectively.

The equation of the electromagnetic torque (Equation (5)) can be found considering that the electrical time constant of the system is negligible concerning the mechanical constant in the Equation (4), thus, it is obtained

$$T\_{\mathfrak{C}} = \frac{3}{2} \frac{P}{2} \frac{L\_m}{L\_r} i\_{qs} \lambda\_{dr} \tag{5}$$

where *Te* is the electromagnetic torque, *P* are the number of poles, *λdr* the direct axis flux and *iqs* the quadrature axis current that denotes the torque command of the machine. The Equation (6) shows that the direct axis current is directly related to the magnetizing current of the machine, given by

$$i\_{ds} = \frac{\lambda\_{dr}}{L\_m} \tag{6}$$

In the indirect field-oriented method, the frequency needs to be calculated in dq0 coordinates. Thus, the Equation (7), allows obtaining the slip frequency, given by

$$
\omega\_{sl} = \frac{L\_m R\_r i\_{qs}^\*}{L\_r \lambda\_{dr}} = \frac{R\_r i\_{qs}^\*}{L\_r i\_{ds}^\*} \tag{7}
$$

The Equation (8) relates the torque, rotor velocity and angular position, given by

$$\theta\_{\mathcal{I}} = p\omega\_{\mathcal{I}} = \frac{1/\ \mathcal{I}}{p + B/\mathcal{J}} (T\_{\mathcal{C}}(p) - T\_{\mathcal{C}}(p)) \tag{8}$$

where *θr*, *J*, *B* and *Tc* denote the rotor position, moment of inertia, viscous coefficient of friction and load torque, respectively.

#### **4. Space Vector Pulse Width Modulation—SVPWM**

The SVPWM is a modulation technique that presents reduced switching number and allows better utilization of DC link voltage, compared to the SPWM (Sinusoidal Pulse Width Modutalion), widely used in scalar V/F AC drives. In addition, the SVPWM presents reduced current harmonic distortion and relatively simple digital implementation [3,41–43].

Figure 14 presents the waveforms of the eight possible states for a two-level threephase inverter. Furthermore, it is considered that at the instant when the upper switch of one arm of the converter is closed, the lower one is open and reciprocal.

**Figure 14.** Single-phase voltage combinations for a two-level three-phase converter.

Each of the six nonzero states gives rise to a vector in the complex plane *αβ* (Figure 15). The six active vectors have magnitude 2*Vcc*/3 and lagged from each other by an angle of 60◦. The null vectors are represented at the origin of the complex plane.

**Figure 15.** Hexagon of the converter output voltage spatial vectors.

To synthesize a reference voltage level (*V*∗ *<sup>s</sup>* ) during a sampling time interval (*ts*) it is necessary to use adjacent voltage vectors and null vectors. The Equation (9) is used to calculate the reference voltage, given by

$$
\overline{V\_s^\*} \cdot \mathbf{t\_s} = \overline{V\_i} \cdot \mathbf{t\_a} + \overline{V\_{i+1}} \cdot \mathbf{t\_b} + \overline{V\_0} \cdot \mathbf{t\_0} + \overline{V\_7} \cdot \mathbf{t\_7} \tag{9}
$$

where *Vi* and *Vi*+<sup>1</sup> are adjacent vectors, *V*<sup>0</sup> and *V*<sup>7</sup> null vectors, *ta*, *tb* correspond to the length of time adjacent vectors are used, while *t*<sup>0</sup> and *t*<sup>7</sup> represent the duration of null vectors. The Equations (10)–(12) allow you to calculate the duration times of adjacent and null vectors, given by

$$t\_a = \frac{\sqrt{3V\_s^\*}}{V\_{cc}} \cdot t\_s \cdot \text{sen}(\frac{\pi}{3} - \theta) \tag{10}$$

$$t\_b = \frac{\sqrt{3V\_s^\*}}{V\_{cc}} \cdot t\_s \cdot \text{sen}(\theta) \tag{11}$$

$$t\_0 = \frac{t\_a + t\_b - t\_s}{2} \tag{12}$$

To avoid over-modulation in SVPWM modulation, the amplitude of the reference vector cannot be larger than the magnitude of the largest circumscribed radius of the hexagon. In SPWM the use of the DC link voltage is restricted to *Vcc*/ <sup>√</sup>3. In, SVPWM the maximum magnitude of the voltage reference is *Vcc*/ <sup>√</sup>2, i.e., 15% increase in the utilization of the voltage available on the DC link [44,45].

#### **5. Controller's Tuning**

Numerous types of speed controllers are available for induction motors. The Proportional Integral Derivative (PID) controller, which is widely utilized in industrial applications because of its simple design and structure [3].

To perform the tuning of the current and speed control loops, simplified linear representations of the system were used, which are based on the analysis and simplifications

described in [5,46]. The values of the gains of the PI controllers were initially determined by the root locus method in conformity with performance specifications and subsequently adjusted in the real system by an iterative approach to meet the required performance.

#### *5.1. Current Control*

In the design of the current loop controller the switching is considered perfect, so the reference voltage produced by the controller is exactly equal to the voltage applied to the machine terminals. The Equations (13) and (14) relate the reference voltages on the *d* and *q* axes of the controller with the currents that circulate through the windings of the machine, given by

$$\upsilon\_{ds} = \underbrace{R\_{s}i\_{ds} + \sigma L\_{s}\frac{d}{dt}i\_{ds}}\_{\upsilon\_{ds}^{'}} + \underbrace{\frac{L\_{\text{m}}}{L\_{r}}\frac{d}{dt}\lambda\_{dr} - \omega\_{\text{l}}\sigma L\_{\text{s}}i\_{\text{qs}}}\_{\upsilon\_{ds\,rfd}}\tag{13}$$

$$w\_{qs} = \underbrace{R\_s i\_{qs} + \sigma L\_s \frac{d}{dt} i\_{qs}}\_{v\_{qs}'} + \underbrace{\omega\_c \frac{L\_m}{L\_r} \lambda\_{dr} + \omega\_c \sigma L\_s i\_{ds}}\_{v\_{qs, ffd}}\tag{14}$$

where *Rs*, *Ls*, *σ*, *ids*, *iqs*, *ωe*, *vds*, *ffd*, *vqs*, *ffq* represent stator resistance, stator inductance, dispersion coefficient, direct axis current, squaring axis current, synchronous rotary reference system speed, direct axis feedforward compensation, and feedforward compensation of the squaring axis, respectively. Figure 16 shows the current control loop.

*ds*

**Figure 16.** Current control loop diagram.

The voltage equations have coupling terms between them, in which the direct axis voltage depends on the quadrature axis current, and the quadrature axis voltage depends on the direct axis current. For control purposes, the compensation terms are considered disturbances and will therefore be disregarded for current controller design purposes. The reduced equations of the direct and quadrature axis voltages are given by Equations (15) and (16), assuming that the controller will be dominant enough to reject this error.

$$
\sigma\_{ds}^{'} = R\_s i\_{ds} + \sigma L\_s \frac{d}{dt} i\_{ds} \tag{15}
$$

$$
\sigma\_{q\text{s}}^{'} = R\_{\text{s}} i\_{\text{q\text{s}}} + \sigma L\_{\text{s}} \frac{d}{dt} i\_{\text{q\text{s}}} \tag{16}
$$

For the current loop controller design, the criteria presented in Table 4 were defined.

**Table 4.** Performance criteria for the current loops controllers (*id* and *iq*).


With the current loop model and performance criteria, the projected gains of the PI controller used in simulation environments and experimental tests are *Kp* = 9.7158 and *Ki* = 4395.

## *5.2. Speed Control*

The mechanical model of the induction machine is linked to the electrical torque (*Te*), load torque (*Tc*), the moment of inertia (*J*), mechanical velocity (*ωr*) and viscous coefficient of friction, as shown in Equation (17)

$$\frac{d\omega\_{\tau}}{dt} = \frac{1}{J}(T\_{\text{f}} - B\omega\_{\tau} - T\_{\text{c}}) \tag{17}$$

Applying the Laplace transform in the Equation (17) and considering the load torque as an external disturbance, Equation (18) is obtained

$$p\Omega\_r = \frac{1}{I}(T\_\varepsilon(p) - B\Omega\_r(p))\tag{18}$$

The quadrature axis current reference is provided from the PI speed controller, considering that the current control is perfect, and the reference current is reproduced in the machine windings, the speed controller model is presented in Figure 17, in which the relationship between the quadrature axis current and the electric torque is given by the machine's torque equation.

**Figure 17.** Speed control loop diagram.

For the speed loop controller design, the criteria presented in Table 5 were defined.

**Table 5.** Performance criteria for the speed loop controller.


The projected gains of the speed loop controller used in simulation environments and experimental tests were *Kp* = 0.0676 and *Ki* = 0.6939.

#### **6. Results**

To analyze the behavior of control loops and vector control strategy, simulations were performed in the MATLAB/Simulink® software. Subsequently, the same tests were performed on the commissioned converter prototype. In this way, it is possible to examine the behavior of the converter both with regard to hardware operation and the practical implementation of the field-oriented control strategy on a real machine.

Initially, reference profiles were applied only to the direct axis current loop to evaluate the magnetization of the machine. Figure 18 presents the behavior of the direct axis loop for references of *i* ∗ *<sup>d</sup>* = 0.2 A and *i* ∗ *<sup>d</sup>* = 0.4 A. In addition, the bottom part shows the rotor speed of the machine.

**Figure 18.** Current control loop behavior for *i* ∗ *<sup>d</sup>* =0.2 A and *i* ∗ *<sup>d</sup>* = 0.4 A.

It is verified in Figure 18 that the controller can keep the currents at the reference values. Furthermore, it is possible to analyze the behavior of the rotor speed, which due to the presence of only magnetizing current, the rotor stands still. Table 6 presents the performance results of the direct axis current controller.

**Table 6.** Direct axis current controller performance.


The overshoot present in the experimental *id* signal is related to simplifications adopted for the current controller design. The closed-loop transfer function of the current plant presents a zero. This zero was disregarded to calculate the PI controller gains. Thus, in the real system, this zero will cause a proportional overshoot for fast responses, but the settling time will be preserved.

To verify the behavior of the quadrature axis current controller, reference profiles were applied in positive step of 1.0 A and negative step of −1.0 A, as shown in Figures 19 and 20, respectively.

Analyzing the Figures 19 and 20, the controller can maintain the imposed references. Still, it is possible to highlight that for positive current values in the quadrature axis, the machine rotor rotates clockwise. However, for negative values of current in the quadrature axis, the machine rotation is reversed, characterized by negative speed values.

**Figure 19.** Current control loop behavior for *i* ∗ *<sup>q</sup>* = 1.0 A.

**Figure 20.** Current control loop behavior *i* ∗ *<sup>q</sup>* = −1.0 A.

For both situations, the speed of the machine increases wildly, because there is no speed control loop. It is also noted that the speed of the machine in simulation assumes higher values than those collected in the experiments. The explanation for this is since in simulation many electrical and mechanical elements are considered ideal or even not present in the machine model. Thus, they do not contribute to the speed behavior of the machine. In the same way as clarified for the current *id*, the overshoot of the current *iq* control loop is justified.

Table 7 presents the performance parameters of the quadrature axis current controller.


**Table 7.** Controller performance for reference *i* ∗ *<sup>q</sup>* = 1.0 A and *i* ∗ *<sup>q</sup>* = −1.0 A.

After validating the correct operation of the current control loop, the speed control loop controller was simulated and implemented in the digital signal processor. The direct axis current reference was set at a value of 0.4 A, which represents the magnetization of the machine, and experimentally was the value that presented the best speed and torque results. Tests were performed with step, trapezoidal, and sinusoidal references. For a positive step reference, initially with a reference of 300 RPM and then increasing to 500 RPM and returning to 300 RPM, as shown in Figure 21.

**Figure 21.** Behavior of the speed control loop with step reference.

Figure 22 shows the machine speed behavior for a trapezoidal reference, with a lower reference of 300 RPM and higher than 400 RPM.

Figure 23 shows the behavior of the machine speed for a sine reference with a frequency of 0.1 Hz and amplitude of 200 RPM.

Analyzing the results for the different speed references, it is evident the correct functioning of the implemented speed control loop. It is important to highlight that due to the accuracy of the encoder available during the tests, the speed signal has low amplitude oscillations. Table 8 presents the speed controller performance results for the three reference profiles.

**Figure 22.** Behavior of the speed control loop with trapezoidal reference.

**Figure 23.** Behavior of the speed control loop with 0.1 Hz sinusoidal reference.



The overshoot signal for the speed loop with different speed profiles shows little difference from the performance specifications. The closed loop of the speed controller has a zero that was disregarded for the calculation of the controller gains. In the experimental environment, this zero contributes to the divergence of the peak speed from the simulated speed signal.

Due to the accuracy of the electrical parameters obtained through routine tests to make up the machine model, the dynamic performance of the current and speed controllers both in simulation and in the experimental tests did not rigidly follow the design specifications. In addition, simplifications in the current loop were adopted concerning the coupling terms that influence the actual behavior of the machine. However, the divergences present in the compliance with the performance criteria did not compromise the operation of the indirect field-oriented control strategy. The results presented validate the correct operation of the hardware, control loops and the implemented strategy.

#### **7. Conclusions**

The built AC drive presented results within the expected range. Thus, the methodology adopted for the commissioning of the power circuit and acquisition system met the intended goals. The use of the FNA41560 power module made the construction of the converter more versatile and economically viable, and it can be used for different drive applications for small induction machines.

The results show that both current and speed controllers presented similar results to the results obtained during the simulation step. To solve the divergences in performance, settling time and overshoot, verified in the tests, it is proposed to implement feedforward compensation in the current control loop. Furthermore, as the machine used is of small size, it is difficult to obtain electrical parameters with good accuracy only with conventional blocked-rotor and no-load tests.

Vector control allows you to control the torque and flux of the machine independently. However, this technique is dependent on the angle of the rotor flux and the fidelity of the machine's electrical and mechanical parameters. Thus, lines of research focus on approaches that use flux estimators or the use of controllers that are robust to parametric variations or adaptive or predictive type controllers.

SVPMW modulation proved to be an effective technique for driving the semiconductor switches of the IGBT bridge and relatively simplicity of implementation in the DSP. In addition, the use of this technique allows greater use of the voltage available on the DC link, reduced switching numbers and lower harmonic distortion in the machine line currents.

The developed test bench allows work focusing on other control strategies, activation of induction machines and PWM modulation techniques to be studied and validated.

**Author Contributions:** Conceptualization, R.R.G. and L.F.P.; methodology, R.R.G.; validation, R.R.G. and W.W.A.G.S.; writing—original draft preparation, R.R.G.; writing—review and editing, R.R.G., L.F.P. and W.W.A.G.S.; visualization, C.V.S., G.M.R. and F.F.R.; supervision, C.V.S., G.M.R. and F.F.R.; project administration, L.F.P. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work has financial support from Federal University of Itajubá.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.

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