**3. Results**

The mathematical model of the full-bridge Buck inverter–DC motor system will be validated here. This validation is carried out in two directions: (1) by circuit simulation through the SimPowerSystems toolbox of Matlab-Simulink and (2) via a built prototype of the system by using Matlab-Simulink and a DS1104 board. The results of the circuit simulation of the full-bridge Buck inverter–DC motor system will be presented first. Later, the corresponding experimental results associated with the system will be presented.

#### *3.1. Circuit Simulation Results*

The connection diagram of the system, built on Matlab-Simulink, along with some simulation results are presented here.

#### 3.1.1. Connection Diagram of the System

The circuit simulation results are obtained through the diagram of the system shown in Figure 4, whose implementation has been executed via the SimPowerSystems toolbox of Matlab-Simulink. The blocks composing the diagram of this figure are detailed below:


$$R = 48 \text{ } \Omega, \text{ } \text{ } \text{ } \text{ } = 4.7 \text{ } \text{ } \text{ } \text{ } L = 4.94 \text{ } \text{mH}, \text{ } E = 32 \text{ V.} \tag{47}$$

The sampling frequency of the four transistors, associated with the full-bridge is 50 kHz. The DC motor was manufactured by ENGEL with a 3.1 gearbox with reduction ratio of 14.5:1. Such a motor is a GNM5440E-G3.1 (24 V, 95 <sup>W</sup>), whose parameters are

$$\begin{aligned} \mathbf{L\_{d}} &= 2.22 \text{ mH}, \qquad k\_{\text{dl}} = 120.1 \times 10^{-3} \frac{\text{N} \cdot \text{m}}{\text{A}},\\ \mathbf{R\_{d}} &= 0.965 \,\Omega, \qquad k\_{\text{c}} = 120.1 \times 10^{-3} \frac{\text{V} \cdot \text{s}}{\text{rad}},\\ \mathbf{J = 118.2 \times 10^{-3} \,\text{kg} \cdot \text{m}^{2}, \qquad b &= 129.6 \times 10^{-3} \frac{\text{N} \cdot \text{m} \cdot \text{s}}{\text{rad}}. \end{aligned}$$

**Figure 4.** Circuit of the "full-bridge Buck inverter–DC motor" system designed via the SimPowerSystems toolbox of Matlab-Simulink. This circuit is used for validating the deduced mathematical model of such a system.

#### 3.1.2. Circuit Simulation Results

With the intention of validating the obtained mathematical model of the full-bridge Buck inverter–DC motor system, this section presents the circuit simulation results for different desired trajectories of the angular velocity.

Circuit Simulation 1

> In this simulation, the desired trajectory *ω*<sup>∗</sup> is generated via the following Bézier polynomial:

$$
\omega^\*(t) = \overline{\omega}\_i(t\_i) + [\overline{\omega}\_f(t\_f) - \overline{\omega}\_i(t\_i)]\varphi(t\_\prime t\_i, t\_f),
\tag{49}
$$

where *ϕ*(*<sup>t</sup>*, *ti*, *tf*) is given by

$$\varphi\left(t, t\_i, t\_f\right) = \begin{cases} 0 & \text{for } t \le t\_i, \\ \left(\frac{t-t\_i}{t\_f-t\_i}\right)^5 \times \left[r\_1 + r\_2\left(\frac{t-t\_i}{t\_f-t\_i}\right) + r\_3\left(\frac{t-t\_i}{t\_f-t\_i}\right)^2\right. & (50) \\ \quad + r\_4\left(\frac{t-t\_i}{t\_f-t\_i}\right)^3 + r\_5\left(\frac{t-t\_i}{t\_f-t\_i}\right)^4 + r\_6\left(\frac{t-t\_i}{t\_f-t\_i}\right)^5 & \text{for } t \in \left(t\_i, t\_f\right), \\ 1 & \text{for } t \ge t\_f. \end{cases} \tag{50}$$

and

$$r\_1 = 252, \quad r\_2 = -1050, \quad r\_3 = 1800, \quad r\_4 = -1575, \quad r\_5 = 700, \quad r\_6 = -126. \tag{51}$$

With proposal (50) and coefficients (51), *ω*<sup>∗</sup> smoothly interpolates between *ωi* = −10rads and *<sup>ω</sup>f* = 10rads over the time interval [*ti*, *tf* ]=[4 s, 6 s]. Note that if (50) were of different order, then coefficients (51) would be also different. The corresponding results are presented in Figure 5.

**Figure 5.** Circuit simulation results for a Bézier polynomial-type trajectory. The results related to the mathematical model are denoted by *ω*, *ia*, *υ*, *i*, and the results associated with the reference variables are labeled as *<sup>ω</sup>*∗, *i*∗*a*, *<sup>u</sup>*∗*av*, *<sup>υ</sup>*∗, *i*<sup>∗</sup>.

Circuit Simulation 2

> Here, *ω*<sup>∗</sup> is defined by the following sinusoidal function:

$$
\omega^\*(t) = 10\sin(0.8\pi t). \tag{52}
$$

Figure 6 depicts the corresponding simulation results.

**Figure 6.** Circuit simulation results for a sinusoidal trajectory. The results of the mathematical model correspond to variables *ω*, *ia*, *υ*, *i*, whereas the results related to the reference variables are *<sup>ω</sup>*∗, *i*∗*a* , *<sup>u</sup>*∗*av*, *<sup>υ</sup>*∗, *i*<sup>∗</sup>.

#### Circuit Simulation 3

In this case, the trajectory to be tracked has been proposed as

$$
\omega^\*(t) = 10 \left( 1 - e^{-2t^2} \right) \sin(0.8\pi t),
\tag{53}
$$

and the results are shown in Figure 7.

**Figure 7.** Circuit simulation results for a sinusoidal trajectory with exponential amplitude. The results associated with the mathematical model are *ω*, *ia*, *υ*, *i*, and the results of the reference variables correspond to *<sup>ω</sup>*∗, *i*∗*a* , *<sup>u</sup>*∗*av*, *<sup>υ</sup>*∗, *i*<sup>∗</sup>.

## Circuit Simulation 4

0 2 4 6 8 10

Lastly, the trajectory to be tracked in this simulation is given by Equation (54) and the corresponding results are presented in Figure 8.

*ω*<sup>∗</sup>(*t*) = 10 sin 0.125*π<sup>t</sup>* 32 . (54) -20-1001020-4 -2 0 2 4 -1 -0.5 0 0.5 1 -30 -15 0 15 30 -4 -2 0 2 4

0 2 4 6 8 10

0 2 4 6 8 10

**Figure 8.** Circuit simulation results for a sinusoidal trajectory with time-varying frequency. The results related to the mathematical model correspond to the signals denoted by *ω*, *ia*, *υ*, *i*, and the results associated with the reference variables are labeled as *<sup>ω</sup>*∗, *i*∗*a*, *<sup>u</sup>*∗*av*, *<sup>υ</sup>*∗, *i*<sup>∗</sup>.

0 2 4 6 8 10

#### *3.2. Results from the Experimental Prototype*

0 2 4 6 8 10

This section describes the connection diagram that allows the implementation of the input signal *<sup>u</sup>*<sup>∗</sup>*av* on the built prototype of the full-bridge Buck inverter–DC motor system shown in Figure 9. Also, the corresponding experimental results are presented.

**Figure 9.** Experimental prototype of the full-bridge Buck inverter–DC motor system.

#### 3.2.1. Experimental Diagram of the System

The experimental results were obtained by using the connection diagram depicted in Figure 10. The blocks composing this figure are described below.


**Figure 10.** Experimental diagram of the "full-bridge Buck inverter–DC motor" system.

#### 3.2.2. Experimental Results

With the purpose of validating the obtained mathematical model (21)–(24), this section presents the experimental results for the system. In these experiments, and with the purpose of making a fair comparison with the simulation results, the desired trajectories for *ω* are the same as those considered in the simulation results.
