*3.1.3. Closed-loop buck converter*

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

*3.1.2. Open-loop buck converter* 

vm F Control

Constant

2 F 30

The PWM control block is illustrated in figure 6b.

vi F io

F

vi -vo

double

**/F**

vo

sign(iL)

vo

iL

Buck

**Figure 6.** Buck converter described in Simulink

NAND

iL to zero.

0.5 Constant

> 1 vi

*CLo dv i t it it C*

() () () *<sup>o</sup>*

1 1 () () () () *o C Lo v t i t dt i t i t dt*

Simulink model of the open-loop buck converter is shown in figure 6a. The Buck block is illustrated in figure 6c. Equation (*12*) is modelled by blocks addition, multiplication and logic. The structure of the converter requires a current iL necessarily positive or zero. Also, the inductance current is modelled by an integrator block that limits the minimum value of

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

Terminator

vL


vo io Load

a) Global view b) PWM Control blockl

iL

1 s

2 iL

3 io

c) Buck block

*dt* (10)

Repeating Sequence

1 vm

1 F

>=

vt

Relational Operator

s


iC

1

1 vo

*C C* (11)

( ) ( ) ( ) \* ( )\* \* ( ) *L io o L v t v t v t F v t F sign i* (12)

A closed-loop buck converter circuit is illustrated in figure 7a. The measurement of the output voltage is realized by 2 resistances R1 and R2. The regulation is achieved by a PID controller. Simulink model of the closed loop converter is shown in figure 7b. Simulink PID control block is illustrated in figure 7c .

The parameters used for the closed-loop simulation are :


The voltage reference was fixed to 2.5 V. The simulation of the closed-loop buck converter is illustrated in figure 7d. The list of configuration parameters used for is:

Start time : 0 Stop time : 0.5 e-3

a) Closed-loop buck converter circuit b) Buck Simulink diagram

c) Simulink PI regulator

Type : Variable-step Solver : ode15s (stiff/NDF) Max step size : 1e-6 Relative tolerance : 1e-3 Min step size : auto absolute tolerance : auto

d) Output voltage of the buck converter

**Figure 7.** Modeling a closed loop DC / DC converter

In steady-state, Vref = Vmes = 2.5 V. From figure 7a, we deduce the theoretical value of Vo:

$$\left.V\_o\right|\_{steady-state} = \frac{R\_1 + R\_2}{R\_2}\left.V\_{ref} = 5\text{ V}\tag{13}$$

Simulation is in good agreement with theoretical value. From figure 7d, we deduce that the transient state last roughly 0.2 ms.

#### **3.2. Boost converter**

#### *3.2.1. Operating phases*

The boost converter circuit is illustrated in figure 8a. The principle of the switch control is described in figure 5b Three operating phases are counted (figure 8c) :


The variation of the voltage across the inductance L (equation *14*) and the current through the capacity (equation *15*) depend on the operating phase.

$$\upsilon\_L(t) = \upsilon\_i(t) \ast F + \left(\upsilon\_i(t) - \upsilon\_o(t)\right) \ast \overline{F} \ast \operatorname{sign}(i\_L) \tag{14}$$

$$\dot{i}\_{\mathcal{C}}(t) = -\dot{i}\_o(t) \, ^\*F + \dot{i}\_L(t) \, ^\*\overline{F} \, ^\*\text{sign}(\dot{i}\_L) = \mathbb{C} \, \frac{dv\_o}{dt} \tag{15}$$

$$\left\|\boldsymbol{v}\_{o} = \frac{1}{C} \int \dot{\boldsymbol{i}}\_{\mathcal{C}}(t) \, dt = \frac{1}{C} \int \left( -\dot{\boldsymbol{i}}\_{o}(t) \, . \, \boldsymbol{F} + \dot{\boldsymbol{i}}\_{L} \cdot \overline{\boldsymbol{F}} \, . \, \text{sign}(\dot{\boldsymbol{i}}\_{L}) \right) dt \tag{16}$$

#### *3.2.2. Open-loop operation*

Simulink model of a open-loop boost converter is shown in figure 9a. The Boost block is illustrated in figure 9b. Equation (14), (15) and (16) are modeled by addition blocks, multiplication blocks and logic blocks. The structure of the converter requires a current iL necessarily positive or zero. Also, the inductance current is modeled by an integrator block that limits the minimum value of iL to zero.

The PWM control block is illustrated in figure 6b.

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

Simulation example:

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

$$\begin{array}{ccccc} \text{V}\_{\text{l}} = 12 \text{ V} & \text{L} = 200 \,\mu\text{H} & \text{C} = 50 \,\mu\text{F} & \text{R} = 5 \,\Omega & \text{f}\_{\text{l}} = 50 \,\text{kHz} \\ \text{Control block:} & V\_{t \,\text{max}} = 1 \text{ V} & V\_{t \,\text{min}} = -1 \text{ V} & V\_{\text{m}} = 0 \\ \end{array}$$

The simulation of the open-loop boost converter is illustrated in figure 9c. The list of configuration parameters used is:


Knowing that vt varies from – 1 V to + 1 V and vm = 0, we deduce that the duty cycle is equal to 0.5. In steady-state, we deduce theoretical value of Vo :

$$\left.V\_o\right|\_{\text{steady}-\text{state}} = \frac{V\_i}{\alpha} = \text{24 V} \tag{17}$$

Simulation is in good agreement with theoretical value. From figure 9c, we deduce that the transient state last roughly 2.5 ms.

b) Operating phases

**Figure 8.** Boost converter

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

transient state last roughly 0.2 ms.

**3.2. Boost converter** 

*3.2.1. Operating phases* 



*3.2.2. Open-loop operation* 

Simulation example:

that limits the minimum value of iL to zero.

The PWM control block is illustrated in figure 6b.

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

In steady-state, Vref = Vmes = 2.5 V. From figure 7a, we deduce the theoretical value of Vo:

1 2 2 <sup>5</sup> *<sup>o</sup> ref steady state R R <sup>V</sup> V V <sup>R</sup>*

Simulation is in good agreement with theoretical value. From figure 7d, we deduce that the

The boost converter circuit is illustrated in figure 8a. The principle of the switch control is

The variation of the voltage across the inductance L (equation *14*) and the current through

( ) ( )\* ( )\* \* ( ) *<sup>o</sup>*

1 1 ( ) ( ). . . ( ) *oC o L L v i t dt i t F i F sign i dt C C*

Simulink model of a open-loop boost converter is shown in figure 9a. The Boost block is illustrated in figure 9b. Equation (14), (15) and (16) are modeled by addition blocks, multiplication blocks and logic blocks. The structure of the converter requires a current iL necessarily positive or zero. Also, the inductance current is modeled by an integrator block

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

Vi = 12 V L = 200 H C = 50 F R = 5 ft = 50 kHz Control blok: *Vt* max = 1 V *Vt* min = - 1 V Vm = 0

*Co L L dv i t i t F i t F sign i C dt*

( ) ( )\* ( ) ( ) \* \* ( ) *Li io <sup>L</sup> v t v t F v t v t F sign i* (14)

(16)

*(*15*)*

described in figure 5b Three operating phases are counted (figure 8c) :

the capacity (equation *15*) depend on the operating phase.

(13)

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

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