**4.1. Electrical circuit**

52 MATLAB – A Fundamental Tool for Scientific Computing and Engineering Applications – Volume 1

vm F Control

vL


Constant

Vi

vm Constant

F

vi -vo

NAND

sign(iL)

Voltage (V)

and Current (A)

vi

/F

1 vi

2 F a) Open-loop Boost block diagram

1 s Buck

Boost

2 iL

vi F io

vo

iL

/F

sign(iL)

3 io

> 24 24.5

> > 9.5 10

> > > 9

23.5

zoom

Terminator

vo io Load

s

1

1 vo

vs

iL

iC

1/C

4.5 4.52 4.54 4.56


b) Boost block

vs

iL

**Figure 9.** Boost converter described in Simulink

**4. DC-AC converter model in Simulink** 

control of the switches do not change the waveform of the inverter.

account the dead times introduced into the control of the switches.

An inverter is a DC – AC power converter. This converter obtains AC voltage from DC voltage. The applications are numerous: power backup for the computer systems, variable speed drive motor, induction heating... In most cases, the dead times introduced into the

c) Open-loop Boost simulation

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 <sup>3</sup> 3.5 <sup>4</sup> 4.5 <sup>5</sup> <sup>0</sup>

Transient state Steady-state

Time (ms)

This paragraph is dedicated to the simulation of a three-phase inverter without taking into

A variable speed drive for AC motor is shown in figure 10. It consists on a continuous voltage source and a three-phase inverter feeding an AC motor.

In order to simplify the modelling, the electrical equivalent circuit of the AC motor is described by an inductance LM in series with a resistance RM. The motor runs with delta connection of the stator.

**Figure 10.** Electrical circuit

There are many strategies for controlling the switches. The most common control strategy is the intersective PWM. Its principle is reminded in figure 11. The switch control signals are generated by comparing three sinusoidal voltages (modulating) which are phase-shifted through 2/3 [rad] with a same triangular voltage waveform (carrier).

**Figure 11.** Three phase PWM control

Knowing the conduction intervals of the switches, it is then possible to determine the waveform of different voltages and currents.

The line to neutral voltage v10, v20 et v30 are dependent on the state of the switches. Examples:


The phase-phase voltage can be deduce from the line to neutral voltage:

$$\mathbf{u}\mathbf{u}\mathbf{u} = \mathbf{v}\mathbf{v}\mathbf{u} - \mathbf{v}\mathbf{v}\mathbf{u} \tag{18}$$

$$\mathbf{u}\mathbf{u}\mathbf{z} = \mathbf{v}\mathbf{v}\mathbf{u} - \mathbf{v}\mathbf{v}\mathbf{o} \tag{19}$$

$$\mathbf{u}\mathbf{u} = \mathbf{v}\mathbf{v}\mathbf{u} - \mathbf{v}\mathbf{u} \tag{20}$$

The input current ii is deduced from the current of switches K1, K3 and K5:

$$\dot{i}\_i = \dot{i}\_{K1} + \dot{i}\_{K3} + \dot{i}\_{K5} = F\_1 \ . \ i\_1 + F\_3 \ . \ i\_2 + F\_5 \ . \ i\_3 \tag{21}$$

#### **4.2. Simulink model**

Simulink model of the three-phase inverter is shown in figure 12a. The control block is illustrated in figure 12b. It models a three phases PWM control. The inverter block is illustrated in figure 12c.

In the case of a resistive load, the load block is constituted by a gain block (value 1/R).

#### **4.3. Simulation example**

The parameters used for of an open-loop simulation are :


The simulation of the open-loop three-phase inverter is illustrated in figure 13. The list of configuration parameters used is:


The relation between the amplitude of the sinusoidal voltage and the triangular voltage determines the maximum value of the fundamental line-line voltage of the inverter:

Simulation of Power Converters Using Matlab-Simulink 55

$$\mathcal{U}\_{\text{max}} = \frac{\sqrt{3}}{2} \frac{V\_{m \text{ max}}}{V\_{t \text{ max}}} \, V\_{DC} = \frac{\sqrt{3}}{2} \, \frac{0.5}{1} \, 400 = 173 \, V \tag{22}$$

Neglecting the current harmonics, the maximum value of the line current is deduced from equation (22) :

$$I\_{1\,\text{max}} = \sqrt{3} \,\text{J}\_{12\,\text{max}} = \frac{\sqrt{3} \,\text{J}\_{\text{max}}}{\sqrt{\text{R}\_{\text{M}}^{2} + \text{(L}\_{\text{M}} \,\text{2}\,\text{\pi} \, f\_{\text{m}})^{2}}} = \frac{\sqrt{3} \,\text{.}173}{\sqrt{4^{2} + \text{(10}^{-2} \,\text{2}\,\text{\pi} \, 50)^{2}}} = 59 \,\text{A} \tag{23}$$

Simulations are in good agreement with theoretical values.

54 MATLAB – A Fundamental Tool for Scientific Computing and Engineering Applications – Volume 1

The phase-phase voltage can be deduce from the line to neutral voltage:

The input current ii is deduced from the current of switches K1, K3 and K5:

waveform of different voltages and currents.

Examples:

**4.2. Simulink model** 

illustrated in figure 12c.

**4.3. Simulation example** 

configuration parameters used is:

The parameters used for of an open-loop simulation are :

fm = 50 Hz Vm max = 0.5

Knowing the conduction intervals of the switches, it is then possible to determine the

The line to neutral voltage v10, v20 et v30 are dependent on the state of the switches.

u12 = v10 – v20 (18)

u23 = v20 – v30 (19)

u31 = v30 – v10 (20)

1 3 5 11 32 53 ... *iK K K*

Simulink model of the three-phase inverter is shown in figure 12a. The control block is illustrated in figure 12b. It models a three phases PWM control. The inverter block is

The simulation of the open-loop three-phase inverter is illustrated in figure 13. The list of

Type: Variable-step Solver: ode15s (stiff/NDF) Max step size: 1e-5 Relative tolerance: 1e-3 Min step size: auto absolute tolerance: auto

The relation between the amplitude of the sinusoidal voltage and the triangular voltage

determines the maximum value of the fundamental line-line voltage of the inverter:

In the case of a resistive load, the load block is constituted by a gain block (value 1/R).

Power Circuit : VDC = 400 V LM = 10 mH RM = 5 Control blok: ft = 20 kHz Vt max = 1 V Vt min = - 1 V

Start time: 0 Stop time: 1.5

*i i i i Fi Fi Fi* (21)

K1 state-on and K2 state-off: v10 = + VDC K1 state-off and K2 state-on: v10 = 0 K3 state-on and K4 state-off: v20 = + VDC K3 state-off and K4 state-on: v20 = 0 K5 state-on and K6 state-off: v30 = + VDC K5 state-off and K6 state-on: v30 = 0

c) Inverter Block

d) Motor block

**Figure 12.** Three phase inverter

**Figure 13.** Simulation example of a three-phase inverter with PWM control
