Differential Equation of the Converter

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

R c

C

R c

R ind

R load

i in

L in

i c

i ind L ind

i d

i c

i m

R onm

R _ond

A i load

v load

V fd

B

V

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

R c

C

R c

R ind

R _load i in

L in

i c

i ind L ind

i c

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

V

R

s 1

C 1

1

V _in L _in

1 s 1

2

R onm R _ind

1 s R

1 s

C 1

2

L _ind 1

R load

1 3

I in

I ind

I

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

R ^c

C

R c

R ind

R _load i in

L in

i c

i ind L ind

i d

i c

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

1 s

C 1

2

L ind 1

R load

1 3

I ind

I

I load V _load

V _fd -

R ond L _in

1 s 1

V

R

s 1

C 1

1

V _in

I

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

1 s

C 1

2

L ind 1

R load

1 3

I ind

I

I load V load

I ind

switch 1

switch 3 - fd

V

I

V

R

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

R ind

i ind L ind

R onm

i load

+

-

v _load i d

i m

R ond

V in

V fd

+

- R c

R load

i c

i i in L in

c

C

R c

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

R ind

i ind L ind

R onm

+

-

v _load

i m

V in

+

- R c

R _load i c

i L in

c

C

R c

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

1 s

C 1

2

R load 1 V _load

I

I load

I

V

R

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

R ind

i ind L ind

i load

+

-

v _load i d

R _ond

V in

V fd

+

- R c

R _load i c

i i in L in

c

C

R c

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

1 s

C 1

2

R load 1

V _load I

I load I

V

R

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

V

R

s 1 C 1

1

R

R

R

V

R

1 s

C 1

2

R

1 V

I

I

1 s L

1 I

I

5 s 1

I

L

1 V

0 switch 1

switch 2

Figure 3.12: Block diagram of the complete step-up circuit

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

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