## Chapter 3

## Differential Equation of the Converter

### In this chapter the equations that represent the behavior of the converter will be written.

### The equations are written only considering the resistive model of the electronic components.

### Moreover the blocks diagram of the equation will also be illustrate.

### 3.1 Buck Converter

### In order to write the equations, diagram 3.1 is considered. Switch-on and the Switch-off

### V in = V m

### C

1### R c

1### C

^{2}

### R c

2### R ind

### R load

### i in

### L in

### i c

1### i ind L ind

### i d

### i c

2### i m

### R onm

### R _{ond}

### A i load

### v load

### V fd

### B

### V

A### Figure 3.1: restive step-down circuit

### are the states of the circuit, and in the next paragraphs the equations of every state will

### 8

### be written

### 3.1.1 Equations and Block Diagram with Mosfet on

### Diagram 3.2 shows the circuit when the Mosfet is on and the diode is off. By applying

### V in = V m

### C

1### R c

1### C

^{2}

### R c

2### R ind

### R _{load} i in

### L in

### i c

1### i ind L ind

### i c

2### i m

### R onm

### i load

### v _{load} 1 2

### 3

### 4

### Figure 3.2: restive step-down circuit when the mosfet is on

### Kirchhoff’s voltage law (KVL) at the loop 124, at the loop 234 and at the loop 34, and the Kirchhoff’s current law (KCL) at the nodes 3 and 4 respectively, the following systems equation is obtained.

### I in =

### V in − I C1

### s·C ^{1} − I ^{C1} ·R ^{C1}

### · 1

### L in ·s (3.1)

### I ind =

### V 2 − s·L ^{in} ·I ^{in} − I ^{ind} ·R ^{onm} − I ^{ind} ·R ^{ind} − R ^{load} ·I ^{load}

### · 1

### s·L ^{ind} (3.2)

### I load =

### R _{C2} ·I ^{C2} + I _{C2} s·C ^{2}

### (3.3)

### I in −I ^{m} − I ^{C1} = 0 (3.4)

### I m −I ^{C2} − I ^{load} = 0 (3.5)

### The block diagram of the system, is depicted in figure 3.3. When the Mosfet is on, the

### current through it is the same as the current through the main inductor L ind . The sum

### blocks 2 and 3 represent the KCL at the correspondent nodes.

### I

C1### V

2### R

C1### s 1

### C 1

1

### V _{in} L _{in}

### 1 s 1

### 2

### R onm R _{ind}

### 1 s R

C2### 1 s

### C 1

2

### L _{ind} 1

### R load

### 1 3

### I in

### I ind

### I

C2### I load V _{load}

### I _{m} I _{ind}

### Figure 3.3: Blocks diagram of the step-down circuit when the mosfet is on

### 3.1.2 Equations and Block Diagram with Mosfet off

### Picture 3.4 shows the circuit when the Mosfet is off and the diode is on. By applying

### V _{in} = V m

### C

1### R ^{c}

^{1}

### C

^{2}

### R c

2### R ind

### R _{load} i in

### L in

### i c

1### i ind L ind

### i d

### i c

2### R ond

### 1

### i load

### v load

### V fd

### 2

### 3

### 4 5

### Figure 3.4: restive step-down circuit when the mosfet is off

### Kirchhoff’s voltage law (KVL) at the loop 124, at the loop 534 and at the loop 34 and the

### Kirchhoff’s current law (KCL) at the nodes 3 respectively, the following system equation is

### obtained.

### I in =

### V in − I _{C1}

### s·C ^{1} − I ^{C1} ·R ^{C1}

### · 1

### L in ·s (3.6)

### I ind =

### − V ^{f d} − R ^{ond} ·I ^{d} − I ^{ind} ·R ^{ind} − R ^{load} ·I ^{load}

### · 1

### s·L ^{ind} (3.7)

### I load =

### R C2 ·I ^{C2} + I C2

### s·C 2

### (3.8)

### I d −I ^{C2} − I ^{load} = 0 (3.9)

### The block diagram of the system 3.6 is depicted in figure 3.5

### R ind

### 1 s R

C2### 1 s

### C 1

2

### L ind 1

### R load

### 1 3

### I ind

### I

C2### I load V _{load}

### V _{fd} -

### R ond L _{in}

### 1 s 1

### V

2### R

C1### s 1

### C 1

1

### V _{in}

### I

C1### I in

### I d I ind

### Figure 3.5: Block diagram of the step-down circuit when the Mosfet is off

### 3.1.3 Block Diagram of the whole Circuit

### In order to obtain obtain the whole block diagram of the step-down circuit, the above block

### diagrams must be combined. First of all it is possible to realize that the circuit with the

### Mosfet on (fig. 3.2), and the circuit with the Mosfet off (fig. 3.4) have two equal parts,

### consequently some equations (3.1 with 3.6, 3.3 with 3.8, 3.4 with 3.9 and part of the 3.2

### with part of 3.7) are equal, hence parts of the blocks diagrams are too. Figure 3.6 shows the

### complete block diagram. In the diagram, the states of the switches are so that the Mosfet

### is on, and obviously their control is the same as the control signal of the Mosfet.

### R ind

### 1 s R

C2### 1 s

### C 1

2

### L ind 1

### R load

### 1 3

### I ind

### I

C2### I load V load

### I ind

### switch 1

### switch 3 - fd

### V

### I

C1### V

2### R

C1### s 1

### C 1

1

### V _{in} s 1

### 2 I in

### L in 1

### switch 2

### R ond R onm

### Figure 3.6: Block diagram of the complete step-down circuit

### 3.2 Boost Converter

### As the buck converter, the equation of the boost-converter must also be written. Figure 3.7 show the boost-converter circuit.

### C

^{2}

### R ind

### i ind L ind

### R onm

### i load

### +

### -

### v _{load} i d

### i m

### R ond

### V in

### V fd

### +

### - R c

2### R load

### i c

2### i i in L in

### c

1### C

1### R c

1### Figure 3.7: resistive step-up circuit

### 3.2.1 Equations and Block Diagram with Mosfet on

### Picture 3.8 shows the circuit when the Mosfet is on and the diode is off. By applying

### C

^{2}

### R ind

### i ind L ind

### R onm

### +

### -

### v _{load}

### i m

### V in

### +

### - R c

2### R _{load} i c

2
### i L in

### c

1### C

1### R c

1### i in

### 1 2 3

### 4

### 5 i load 6

### Figure 3.8: resistive step-up circuit when the Mosfet is on

### Kirchhoff’s voltage law (KVL) at the loop 124, at the loop 234 and at the loop 564, and the Kirchhoff’s current law (KCL) at the node 3, the follow equation systems is obtained.

### I in =

### V in − I C1

### s·C 1 − I ^{C1} ·R ^{C1}

### · 1

### L in ·s (3.10)

### I ind =

### V _{2} − I ^{ind} ·R ^{onm} − I ^{ind} ·R ^{ind}

### · 1

### s·L ^{ind} (3.11)

### I load =

### R C2 ·I ^{C2} + I C2

### s·C ^{2}

### · 1 R load

### (3.12)

### I in −I ^{ind} − I ^{C1} = 0 (3.13)

### I m = I ind (3.14)

### The block diagram of the system, is depicted in figure 3.9. When the Mosfet is on, the

### current through it is the same as the current through the main inductor L ind (Eq.3.5). The

### sum blocks 2 represent the KCL at the corresponding node.

### R

C2### 1 s

### C 1

2

### R load 1 V _{load}

### I

C2### I load

### I

C1### V

2### R

C1### s 1

### C 1

1

### V _{in} L _{in}

### 1 s 1

### 2 I in

### R ind

### 1 s L _{ind}

### 1

### I m I ind R onm

### Figure 3.9: Block diagram of the step-up circuit when the Mosfet is on

### 3.2.2 Equations and Blocks Diagram with Mosfet off

### Picture 3.10 shows the circuit when the Mosfet is off and the diode is on. By applying

### C

2### R ind

### i ind L ind

### i load

### +

### -

### v _{load} i d

### R _{ond}

### V in

### V fd

### +

### - R c

2### R _{load} i c

2
### i i in L in

### c

1### C

1### R c

1### 1 2 3

### 4

### 5 6

### Figure 3.10: Resistive step-up circuit when the Mosfet is off

### Kirchhoff’s voltage law (KVL) at the loop 124, at loop 23564 and at loop 564, and the

### Kirchhoff’s current law (KCL) at nodes 2 and 5 respectively, the follow system equation is

### obtained.

### I in =

### V in − I C1

### s·C ^{1} − I C1 ·R C1

### · 1

### L in ·s (3.15)

### I ind =

### V 2 − I ^{ind} ·R ^{ind} − R ^{ond} ·I ^{ind} − V ^{f d} − V ^{load}

### · 1

### s·L ^{ind} (3.16)

### I load =

### R C2 ·I ^{C2} + I C2

### s·C ^{2}

### · 1 R load

### (3.17)

### I in −I ^{ind} − I ^{C1} = 0 (3.18)

### I ind −I ^{load} − I ^{C2} = 0 (3.19)

### I ind = I d (3.20)

### The block diagram of the system is depicted in figure 3.11

### R

C2### 1 s

### C 1

2

### R load 1

### V _{load} I

C2
### I load I

C1### V

2### R

C1### s 1

### C 1

1

### V _{in} L _{in}

### 1 s 1

### 2 I in

### R ind

### s 1 L ind

### 1 V _{fd}

### I ind I _{d}

### I _{ind}

### R ond 5

### Figure 3.11: Block diagram of the step-up circuit when the Mosfet is off

### 3.2.3 Blocks Diagram of the whole Circuit

### As the step-down circuit, both the diagrams are combined. The circuits with the Mosfet on

### (fig. 3.8), and the circuit with the Mosfet off (fig. 3.10) have two equal parts, consequently

### some equations (3.10 with 3.15, 3.12 with 3.17, 3.13 with 3.18 and part of the 3.11 with part of 3.16) are equal, hence parts of the blocks diagrams are too. Figure 3.12 shows the

### 2

### I

C1### V

2### R

C1### s 1 C 1

1

### R

ind### R

ond### R

onm### V

_{fd}

### R

C2### 1 s

### C 1

2

### R

load### 1 V

load### I

C2### I

load### 1 s L

ind### 1 I

ind### I

ind### 5 s 1

### I

in### L

_{in}

### 1 V

_{in}