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5Positive Output Cascade Boost Converters

Super-lift technique increases the voltage transfer gain in geometric progres-sion. However, these circuits are a bit complex. This chapter introduces anovel approach the positive output cascade boost converter that imple-ments the output voltage increasing in geometric progression, but withsimpler structure. They also effectively enhance the voltage transfer gain inpower-law.

5.1 Introduction

In order to sort these converters differently from existing voltage-lift (VL)

and super-lift (SL) converters, these converters are entitled positive outputcascade boost converters. There are several subseries:

Main series Each circuit of the main series has one switch S, ninductors, n capacitors, and (2n 1) diodes.

Additional series Each circuit of the additional series has oneswitch S, n inductors, (n + 2) capacitors, and (2n + 1) diodes.

Double series Each circuit of the double series has one switch S,

n inductors, 3n capacitors, and (3n 1) diodes. Triple series Each circuit of the triple series has one switch S, n

inductors, 5n capacitors, and (5n 1) diodes.

Multiple series Each multiple series circuit has one switch S anda higher number of capacitors and diodes.

In order to concentrate the super-lift function, these converters work inthe steady state with the condition of continuous conduction mode (CCM).

The conduction duty ratio is k, switching frequency isf, switching period

is T= 1/f, the load is resistive load R. The input voltage and current are Vinand Iin, output voltage and current are VO and IO. Assume no power lossesduring the conversion process, VinIin = VOIO. The voltage transfer gainis G:

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5.2 Main Series

The first three stages of positive output cascade boost converters mainseries are shown in Figure 5.1 to Figure 5.3. For convenience they arecalled elementary boost converter, two-stage circuit, and three-stage circuitrespectively, and numbered as n = 1, 2, and 3.

5.2.1 Elementary Boost Circuit

The elementary boost converter is the fundamental boost converter intro-duced in Chapter 1 (seeFigure 1.24). Its circuit diagram and its equivalentcircuits during switch-on and switch-off are shown in Figure 5.1. The voltageacross capacitor C1 is charged to VO. The current iL1flowing through inductorL1 increases with voltage Vin during switch-on period kTand decreases withvoltage (VOVin) during switch-off period (1 k)T. Therefore, the ripple of

the inductor current iL1 is

(5.1)

(5.2)

The voltage transfer gain is

(5.3)

The inductor average current is

(5.4)

The variation ratio of current iL1 through inductor L1 is

(5.5)

GV

VO

in

=

iV

LkT

V V

Lk TL

in O in1

1 1

1= =

( )

Vk

VO in= 1

1

GV

V kO

in

= =1

1

I kV

RLO

1 1= ( )

11

1 1 1

2

1 2 2= =

=

iI

kTV

k L V R

k R

fLL

L

in

O

/

( ) /

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Usually 1 is small (much lower than unity), which means this converterworks in the continuous mode. The ripple voltage of output voltage vO is

Therefore, the variation ratio of output voltage vO is

(5.6)

Usually R is in k,fin 10 kHz, and C1 in F, the ripple is smaller than 1%.

5.2.2 Two-Stage Boost Circuit

The two-stage boost circuit is derived from elementary boost converter byadding the parts (L2-D2-D3-C2). Its circuit diagram and equivalent circuits

FIGURE 5.1Elementary boost converter.

VIN

+

iIN

S

L1

R

D1

C1 VO

+

iO

(a) Circuit diagram

+

VC1

VIN

+

iIN L

1

RC1 VO

+

iO

(b) Switching-on

VC1

+

VIN

+

iIN L

1

RC1 VO

+

iO

(c) Switching-off

VC1

+

VL1

v

Q

C

I k T

C

k

fC

V

ROO O

= =

=

1 1 1

1 1( )

= =v

V

k

RfCO

O

/ 2 1

2 1


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during switch-on and switch-off are shown in Figure 5.2. The voltage acrosscapacitor C1 is charged to V1. As described in previous section the voltageV1 across capacitor C1 is

The voltage across capacitor C2 is charged to VO. The current flowing throughinductor L2 increases with voltage V1 during switching-on period kT anddecreases with voltage (VOV1) during switch-off period (1 k)T. There-fore, the ripple of the inductor current iL2 is

(5.7)

(5.8)


FIGURE 5.2Two-stage boost circuit.

VIN

+

iIN

L1

C2

(b) Equivalent circuit during switching-on

R VO

+

iO

VC2

+

C

1V

C1

+

L2

V1

VIN

+

iIN

(c) Equivalent circuit during switching-off

R VO

+

iO

L1

C1

VC1

+

VL1

L2

C2

VC2

+

VL2

iIN

S

L1 D1

(a) Circuit diagram

RC2 V

O

+

iO

+

VC2

D3

L2

+

VC1C1VIN

+

D2

V1

Vk

Vin11

1=

iV

LkT

V V

Lk TL

O2

1

2

1

2

1= =

( )

Vk

Vk

VO in= =

1

1

1

112( )


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The voltage across capacitor C3 is charged to VO. The current flowing throughinductor L3 increases with voltage V2 during switching-on period kT anddecreases with voltage (VOV2) during switch-off (1 k)T. Therefore, theripple of the inductor current iL3 is

(5.13)

(5.14)


(5.15)

Analogously,

FIGURE 5.3Three-stage boost circuit.

VIN

+

iIN

L1

C3


C1 R VO

+

iO

+

V

C1

+

L2

V1

VC2

+

L3

V2

C2 VC3

VIN

+

iIN


L1

C1

VL1

C3 R VO

+

iO

VC3

+

VC1

+

L2

VL2

VC2

+

L3

VL3

C2

V1

V2

iIN

S

L1 D1

(a) Circuit diagram

C3

D4

L2

+

VC1C1

VIN

+

D2

L3

D3

D5

R VO

+

iO

+V

C3

+

VC2

C2

V2

V1

iV

LkT

V V

Lk TL

O3

2

3

2

3

1= =

( )

Vk

Vk

Vk

VO in= =

=

1

1

1

1

1

122

13( ) ( )

GV

V kO

in

= =

( )1

13


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Therefore, the variation ratio of current iL1 through inductor L1 is

(5.16)


(5.17)


(5.18)

and the variation ratio of output voltage vO is

(5.19)

5.2.4 Higher Stage Boost Circuit

Higher stage boost circuit can be designed by just multiple repeating of theparts (L2-D2-D3-C2). For n

th stage boost circuit, the final output voltage acrosscapacitor Cn is


iV

LkTL

in1

1

= II

kLO

1 31=

( )

i VL

kTL21

2

= I IkL

O2 21

=( )

iV

LkTL3

2

3

= II

kLO

3 1=

11

1

3

1

6

1

2 1

2

1

2= = =

i

I

k k TV

L I

k k R

fLL

L

in

O

/ ( ) ( )

22

2

21

2

4

2

2 1

2

1

2= =

=

iI

k k TV

L I

k k R

fLL

L O

/ ( ) ( )

33

3

2

3

2

3

2 1

2

1

2= =

=

i

I

k k TV

L I

k k R

fLL

L O

/ ( ) ( )

= =

v

V

k

RfCO

O

/ 2 1

2 3

Vk

VOn

in= ( )11


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(5.20)

the variation ratio of current iLi through inductor Li (i = 1, 2, 3, n) is

(5.21)


(5.22)

5.3 Additional Series

All circuits of positive output cascade boost converters additional series are derived from the corresponding circuits of the main series by adding a

DEC.The first three stages of this series are shown inFigure 5.4 toFigure 5.6.

For convenience they are called elementary additional circuits, two-stageadditional circuits, and three-stage additional circuits respectively, and num-

bered as n = 1, 2, and 3.

5.3.1 Elementary Boost Additional (Double) Circuit

This elementary boost additional circuit is derived from elementary boost

converter by adding a DEC. Its circuit and switch-on and switch-off equiv-alent circuits are shown in Figure 5.4. The voltage across capacitor C1 andC11 is charged to V1 and voltage across capacitor C12 is charged to VO = 2 V1.The current iL1flowing through inductor L1 increases with voltage Vin duringswitching-on period kTand decreases with voltage (V1Vin) during switch-ing-off (1 k)T. Therefore,

(5.23)

GV

V kO

in

n= =

( )1

1

iLi

Li

n i

i

i

I

k k R

fL= =

+ / ( ) ( )2 12

2 1

= =v

V

k

RfC

O

O n

/ 2 1

2

iV

L

kTV V

L

k TLin in

1

1

1

1

1= =

( )

Vk

Vin11

1=


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The output voltage is

(5.24)


(5.25)

and

(5.26)


FIGURE 5.4Elementary boost additional (double) circuit.

VIN

+

iIN L

1

C1


R VO

+

iO

VC12

+

C12

VC11

C11

VC1

+ +

V1

VIN

+

iIN L

1

C1


VC1

+

VL1

R VO

+

iO

VC12

C12

+

C11

VC11

+

(a) Circuit diagram

VIN

+

iIN

S

L1

R

D1

C1

+

V

C1

VO

+

iO

C11

VC11

+

D11

C12

+

VC12

D12

V Vk

VO in= = 2

2

11

GV

V kO

in

= =2

1

i Ik

Iin L O= = 12

1


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(5.27)

The ripple voltage of output voltage vO is


(5.28)

5.3.2 Two-Stage Boost Additional Circuit

The two-stage additional boost circuit is derived from the two-stage boostcircuit by adding a DEC. Its circuit diagram and switch-on and switch-offequivalent circuits are shown in Figure 5.5. The voltage across capacitor C1is charged to V1. As described in the previous section the voltage V1 acrosscapacitor C

1

is

The voltage across capacitor C2 and capacitor C11 is charged to V2 andvoltage across capacitor C12 is charged to VO. The current flowing throughinductor L2 increases with voltage V1 during switch-on period kT anddecreases with voltage (V2V1) during switch-off period (1 k)T. Therefore,

the ripple of the inductor current iL2 is

(5.29)

(5.30)


(5.31)

11

1 1

2

1

2 1

4

1

8= =

=

iI

k k TV

L I

k k R

fLL

L

in

O

/ ( ) ( )

vQ

C

I k T

C

k

fC

V

ROO O= =

=

12 12 12

1 1( )

= =v

V

k

RfC

O

O

/ 2 1

2 12

Vk

Vin11

1=

iV

LkT

V V

Lk TL2

1

2

2 1

2

1= =

( )

Vk

Vk

Vin2 121

1

1

1=

=

( )

V Vk

Vk

VO in= = =

2

2

12

1

12 12( )


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(5.32)

Analogously,


FIGURE 5.5Two-stage boost additional circuit.

+

VIN

+

iIN


C11L

1

C1 VC1

+

VL1

R VO

+

iO

VC12

C12

+

VC11

+V

L2

L2

V

2

+C

2

V2

V1

VIN

+

iIN

L1

C2


iO

R VO

+V

C12

+

C12

VC11

C11

V2

+ +

V2

C1

VC1

+

L2

V1

VIN

(a) Circuit diagram

iIN

S

L1

R

D3

C1

+

V

C1

VO

+

iO

C11

VC11

+

D11

C12

+

VC12

D12

D1

D2

C2

VC2

+

L2

G

V

V k

O

in

= =2

1

1

2

( )

iV

LkTL

in1

1

= Ik

IL O1 22

1=

( )

iV

L kTL21

2= II

kLO

2

2

1=


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(5.33)

and the variation ratio of current iL2 through inductor L2 is

(5.34)



(5.35)

5.3.3 Three-Stage Boost Additional CircuitThis circuit is derived from the three-stage boost circuit by adding a DEC.Its circuit diagram and equivalent circuits during switch-on and switch-offare shown in Figure 5.6. The voltage across capacitor C1 is charged to V1. Asdescribed previously the voltage V1 across capacitor C1 is ,and voltage V2 across capacitor C2 is .

The voltage across capacitor C3 and capacitor C11 is charged to V3. Thevoltage across capacitor C12 is charged to VO. The current flowing throughinductor L3 increases with voltage V2 during switch-on period kT and

decreases with voltage (V3V2) during switch-off (1 k)T. Therefore,

(5.36)

and

(5.37)


(5.38)

11

1

2

1

4

1

2 1

4

1

8= =

=

iI

k k TV

L I

k k R

fLL

L

in

O

/ ( ) ( )

22

2

1

2

2

2

2 1

4

1

8= =

=

iI

k k TV

L I

k k R

fLL

L O

/ ( ) ( )

vQ

C

I k T

C

k

fC

V

RO

O O= =

=

12 12 12

1 1( )

= =v

V

k

RfCO

O

/ 2 1

2 12

V k Vin1 1 1= ( )V k Vin2

21 1= ( )

iV

LkT

V V

Lk TL3

2

3

3 2

3

1= =

( )

Vk

Vk

Vk

Vin3 22

131

1

1

1

1

1=

=

=

( ) ( )

V Vk

VO in= = 2 2

1

133( )


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(5.39)

Analogously:

FIGURE 5.6Three-stage boost additional circuit.

C1 VC1

D1

+

_

R

Vin

+

_

V

+

_

I in L1IO

D2

O

L2

C3VC3

+

_

SC2 VC2

D3

+

_

D4

D5L3

V1

V2 D11 D12

C12 +

_VC12

C11+

_VC11

V3

C12

VC12+

_ RVin

+

_

+

_

I in

L1 VO

IO

L2

C1 VC1

+

_

V1

L3C2

V2

+

_

V2

C2

V3

+

_

V3

_

+

C2 V2

+

_ R

Vin

+

_

+

_

I in L2

VO

IO

V L1

V1

+

_

L1

V L2

C3

+

_

L3

V L3

V3

V1 V2

+

_

VC12

+_

V3

V3C11

C12

(a) Circuit diagram

(b) Equivalent circuit during switch-on

(b) Equivalent circuit during switch-off

V3

C11

G VV k

O

in

= =

2 11

3( )

iV

LkTL

in1

1

= Ik

IL O1 32

1=

( )

i VL kTL21

2= I k IL O2 221= ( )

iV

LkTL3

2

3

= II

kLO

3

2

1=


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(5.40)


(5.41)


(5.42)



(5.43)

5.3.4 Higher Stage Boost Additional Circuit

Higher stage boost additional circuits can be designed by repeating the parts

(L2-D2-D3-C2) multiple times. For nth stage additional circuit, the final outputvoltage is


(5.44)

Analogously, the variation ratio of current iLi through inductor Li (i = 1, 2, 3,n) is

11

1

3

1

6

1

2 1

4

1

8= =

=

iI

k k TV

L I

k k R

fLL

L

in

O

/ ( ) ( )

22

2

21

2

4

2

2 1

4

1

8= =

=

iI

k k TV

L I

k k R

fLL

L O

/ ( ) ( )

33

3

2

3

2

3

2 14

18

= = = iI

k k TV L I

k k RfL

L

L O

/ ( ) ( )

vQ

C

I k T

C

k

fC

V

ROO O= =

=

12 12 12

1 1( )

= =v

V

k

RfCO

O

/ 2 1

2 12

Vk

VOn

in= 2

1

1( )

GV

V k

O

in

n= =

21

1

( )


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(5.45)


(5.46)

5.4 Double SeriesAll circuits of the positive output cascade boost converter double series are derived from the corresponding circuits of the main series by adding aDEC in each stage circuit. The first three stages of this series are shown inFigures 5.4,5.7, and5.8. For convenience to explain, they are called elemen-tary double circuits, two-stage double circuits, and three-stage double cir-cuits respectively, and numbered as n = 1, 2, and 3.

5.4.1 Elementary Double Boost Circuit

From the construction principle, the elementary double boost circuit isderived from the elementary boost converter by adding a DEC. Its circuitand switch-on and switch-off equivalent circuits are shown in Figure 5.4,which is the same as the elementary boost additional circuit.

5.4.2 Two-Stage Double Boost Circuit

The two-stage double boost circuit is derived from the two-stage boost circuitby adding a DEC in each stage circuit. Its circuit diagram and switch-on andswitch-off equivalent circuits are shown in Figure 5.7. The voltage acrosscapacitor C1 and capacitor C11 is charged to V1. As described in the previoussection, the voltage V1 across capacitor C1 and capacitor C11 is .The voltage across capacitor C12 is charged to 2V1.

The current flowing through inductor L2 increases with voltage 2V1 duringswitch-on period kTand decreases with voltage (V2 2V1) during switch-off period (1 k)T. Therefore, the ripple of the inductor current iL2 is

(5.47)

iLi

Li

n i

i

i

I

k k R

fL= =

+ / ( ) ( )2 18

2 1

= =v

V

k

RfCO

O

/ 2 1

2 12

V k Vin1 1 1= ( )

iV

LkT

V V

Lk TL2

1

2

2 1

2

2 21= =

( )


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(5.48)


(5.49)


(5.50)

FIGURE 5.7Two-stage boost double circuit.

(a) Circuit diagram


(c) Equivalent circuit during switch-off

C1 VC1

D1

+

_

R+

_

V

+

_

I inIO

D2

O

C2 VC2+

_

SVin

L1 L2V1

C21

C22

+

+

_

VC21

VC22

D3 D21 D22

C11

C12

+

+

_

VC11

VC12

D11 D12

__

VC22+

_

R

+

_

+

_

VO

IO

L2C12

2V1

+

_

2V1

L1

Vin

Iin

C22

C21

V2+

_

V2

V2

+

_

C2

C11

V1+

_

V1

+

_

C1

V1

Vin

V2+

_

R+

_

+

_

I in

VO

IO

+

_C1 VC1 C2

L1 L2

VL1 VL2

V1

VC12+

_C22

C21

V2+_

V2

VC12+

_C12

C11

V1+_

2V1

V k V k Vin2 122

1 21

1= = ( )

V Vk

VO in= = 2

2

122( )

GV

V kO

in

= =

( )2

12


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Analogously,


(5.51)


(5.52)



(5.53)

5.4.3 Three-Stage Double Boost Circuit

This circuit is derived from the three-stage boost circuit by adding a DEC ineach stage circuit. Its circuit diagram and equivalent circuits during switch-on and -off are shown in Figure 5.8. The voltage across capacitor C1 andcapacitor C11 is charged to V1. As described earlier the voltage V1 acrosscapacitor C1 and capacitor C11 is , and voltage V2 across capac-itor C2 and capacitor C12 is .

The voltage across capacitor C22 is . The voltage acrosscapacitor C3 and capacitor C31 is charged to V3. The voltage across capacitorC12 is charged to VO. The current flowing through inductor L3 increases with

iV

L

kTLin

1

1

= I

k

IL O122

1

=

( )

iV

LkTL2

1

2

= II

kLO

2

2

1=

11

1

2

1

4

1

2 1

8

1

16

= =

=i

I

k k TV

L I

k k R

fL

L

L

in

O

/ ( ) ( )

22

2

1

2

2

2

2 1

4

1

8= =

=

iI

k k TV

L I

k k R

fLL

L O

/ ( ) ( )

vQ

C

I k T

C

k

fC

V

ROO O= =

=

22 22 22

1 1( )

= =v

V

k

RfCO

O

/ 2 1

2 22

V k Vin1 1 1= ( )V k Vin2

22 1 1= ( )

2 2 12 2V k Vin= ( )


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FIGURE 5.8Three-stage boost double circuit.

C1 VC1

D1

+

_

+

_

D2

C2 VC2+

_

Vin

L1 L2V1

C21

C22

+

+

_

VC21

VC22

D3 D21 D22

C11

C12

+

+

_

VC11

VC12

D11 D12

__

C3S

D5L3 V3

D4

2V1

2V2

V2

+

_

L2C12

2V1

+

_

2V1

L1Vin

Iin

C21

V2+

_

V2

V2

+

_

C2

C11

V1+

_

V1

+

_

C1

V1VC32

+

_

L3C22

2V2

+

_

2V2

C32

C31

V3+

_

V3

V3

+

_

C3

Vin V2+

_

+

_

+

_C1 VC1 C2

L1 L2

VL1 VL2

V1

VC22+

_C22

C21

V2+_

V2

VC12+

_C12

C11

V1+_

2V1

V3+

_C3

L3

VL3

VC32

C31

V3+_

V32V2

(a) Circuit diagram


(c) Equivalent circuit during switch-off

Iin

Iin


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voltage V2 during switch-on period kTand decreases with voltage (V3 2V2)during switch-off (1 k)T. Therefore,

(5.54)

and

(5.55)


(5.56)


(5.57)

Analogously,


(5.58)


(5.59)

iV

L kTV V

L k TL32

3

3 2

3

2 21= =

( )

VV

k kVin3

23

2

1

4

1=

=

( ) ( )

V Vk

VO in= = 2

2

133( )

GV

V kO

in

= =

( )2

13

iV

LkTL

in1

1

= Ik

IL O1 38

1=

( )

iV

LkTL2

1

2

= Ik

IL O2 24

1=

( )

iV

LkTL3

2

3

= II

kLO

3

2

1=

11

1

3

1

6

1

2 1

16

1

128= =

=

iI

k k TV

L I

k k R

fLL

L

in

O

/ ( ) ( )

22

2

21

2

4

2

2 1

8

1

32= =

=

iI

k k TV

L I

k k R

fLL

L O

/ ( ) ( )


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(5.60)



(5.61)

5.4.4 Higher Stage Double Boost Circuit

The higher stage double boost circuits can be derived from the correspondingmain series circuits by adding a DEC in each stage circuit. For nth stageadditional circuit, the final output voltage is


(5.62)

Analogously, the variation ratio of current iLi through inductor Li (i = 1, 2, 3,n) is

(5.63)

The variation ratio of output voltage vO is

(5.64)

33

3

2

3

2

3

2 1

4

1

8

= =

=i

I

k k TV

L I

k k R

fL

L

L O

/ ( ) ( )

vQ

C

I k T

C

k

fC

V

ROO O= =

=

32 32 32

1 1( )

= =v

V

k

RfCO

O

/ 2 1

2 32

Vk

VOn

in= ( )

2

1

GV

V kO

in

n= =

( )2

1

iLi

Li

n i

ni

i

I

k k R

fL= =

+ / ( ) ( )2 12 2

2 1

2

= =v

V

k

RfCO

O n

/ 2 1

2 2


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5.5 Triple Series

All circuits of P/O cascade boost converters triple series are derivedfrom the corresponding circuits of the double series by adding the DEC twicein each stage circuit. The first three stages of this series are shown inFigure 5.9 to Figure 5.11. For convenience they are called elementary triple

boost circuit, two-stage triple boost circuit, and three-stage triple boost circuitrespectively, and numbered as n = 1, 2 and 3.

5.5.1 Elementary Triple Boost Circuit

From the construction principle, the elementary triple boost circuit is derivedfrom the elementary double boost circuit by adding another DEC. Its circuit

FIGURE 5.9Cascade boost re-double circuit.

(a) Circuit Diagram



VIN

iIN

S

L1

D1

C1

C11

D11

C12

D12

C13

D13

C14

D14

RVO

+

iO

V1

2V1

VIN

+

iIN

L1

C1

C12

VC11

C11

VC1

+ +

V1

C13

VC12

+ +

2V1

R VO

+

iO

VC12

+

VC13

C14

C13

+

VIN

iIN L

1

C1

VC1

+

VL1

R VO

+

iO

VC12

C12

+

C11

VC11

+C

14

2V1

V1


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and switch-on and -off equivalent circuits are shown in Figure 5.9. Theoutput voltage of first stage boost circuit is V1, V1 = Vin/(1 k).

The voltage across capacitors C1 and C11 is charged to V1 and voltage acrosscapacitors C12 and C13 is charged to VC13 = 2V1. The current iL1flowing through

inductor L1 increases with voltage Vin during switch-on period kT anddecreases with voltage (V1Vin) during switch-off (1 k)T. Therefore,

(5.65)


(5.66)


(5.67)

5.5.2 Two-Stage Triple Boost Circuit

The two-stage triple boost circuit is derived from the two-stage double boostcircuit by adding another DEC in each stage circuit. Its circuit diagram andswitch-on and -off equivalent circuits are shown inFigure 5.10. As describedin the previous section the voltage V1 across capacitors C1 and C11 is

. The voltage across capacitor C14 is charged to 3V1.The voltage across capacitors C2 and C21 is charged to V2 and voltage across

capacitors C22 and C23 is charged to VC23 = 2V2. The current flowing throughinductor L2 increases with voltage 3V1 during switch-on period kT, anddecreases with voltage (V2 3V1) during switch-off period (1 k)T. There-fore, the ripple of the inductor current iL2 is

(5.68)

(5.69)

iV

LkT

V V

Lk TL

in in1

1

1

1

1= =

( )

Vk

Vin11

1=

V V V V k

VO C C in= + = = 1 13 13

3

1

GV

V kO

in= =

3

1

V k Vin1 1 1= ( )

iV

L

kTV V

L

k TL21

2

2 1

2

3 31= =

( )

Vk

Vk

Vin2 123

13

1

1=

=

( )


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(5.70)


(5.71)

Analogously,

FIGURE 5.10Two-stage boost re-double circuit.

(a) Circuit diagram



VIN

+

iIN

S

L1

D1

C1

L2

R VO

+

iO

C11

D11

C12

D12

C13

D13

C14

D14

D2

C21

D21

C22

D22

C23

D23

C24

D24

D3

C2

V1

2V1

3V1 V2

2V2

L1

C1

C14

L2VC14

+

+

VIN

iIN

R

VO

+

iO

VC24

+

C24

C12

VC11

C11

VC1

+ +

VC13

C13

VC12

+ +

C22

VC21

C21

VC2

+ +

VC23

C23

VC22

+ +

C2

2V1

V1

3V1

V2

2V2

+

VIN

iIN L1

C1

VC1

+

VL1

VC12

C12

+

C11

VC11

+

C13

C14

VC14

+

VC13 +

L2

R VO

+

iO

C2

+

VC2

C21

VC21

+

C22

VC

22

+

C23

C24

VC23

+

VC24

+

V1

3V1

2V1

V2

2V2

V V V V k

VO C C in= + = = 2 23 223

3

1( )

G

V

V kO

in= = ( )3

12


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(5.72)


(5.73)



(5.74)

5.5.3 Three-Stage Triple Boost CircuitThis circuit is derived from the three-stage double boost circuit by addinganother DEC in each stage circuit. Its circuit diagram and equivalent circuitsduring switch-on and -off are shown in Figure 5.11. As described earlier thevoltage V2 across capacitors C2 and C11 is , and voltageacross capacitor C24 is charged to 3V2.

The voltage across capacitors C3 and C31 is charged to V3 and voltage acrosscapacitors C32 and C33 is charged to VC33 = 2V3. The current flowing throughinductor L3 increases with voltage 3V2 during switch-on period kT and

decreases with voltage (V3 3V2) during switch-off (1 k)T. Therefore, theripple of the inductor current iL3 is

(5.75)

iV

LkTL

in1

1

= Ik

IL O122

1=

( )

i VL

kTL21

2

= I IkL

O2

21

=

11

1

2

1

4

1

2 1

8

1

16= =

=

iI

k k TV

L I

k k R

fLL

L

in

O

/ ( ) ( )

22

2

1

2

2

2

2 1

4

1

8= =

=

iI

k k TV

L I

k k R

fLL

L O

/ ( ) ( )

v

Q

C

I k T

C

k

fC

V

ROO O= =

=

22 22 22

1 1( )

= =v

V

k

RfCO

O

/ 2 1

2 22

V V k V in2 13 3 1= = ( )

iV

LkT

V V

Lk TL3

2

3

3 2

3

3 31= =

( )


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FIGURE 5.11Three-stage boost re-double circuit.

VIN

+

iIN

S

L2

C21

D21

C22

D22

C23

D23

C24

D24

D3

C2

L1 D1

C1

C11

D11

C12

D12

C13

D13

C14

D14

D2

L3

D5

C3

C31

D31

C32

D32

C33

D33

C

D4

V1

2V1

3V1

V2

2V2

3V2

V3

2V3

(a) Circuit diagram

L1

C1

+

VIN

iIN

C22

C21

C23

C2 C34C14

L2

C12C11 C13 C32C3 C31

L3

C24

C33

V1 2V1 3V1 V2 2V2 3V2 V3 2V3


+

VIN

iIN L

1

C1 VC1

+

VL1

VC12C12

+

C11

VC11

+

C13

C14

VC14

+

VC13 +

L2

R VO

+

iO

C2

+

VC2

C21

VC21

+

C22

VC22

+

C23

C24

VC23

+

VC24

+

L3

C3 VC3

+

VC31 +

C31

C32

C33

VC33

+

C34

VC32

+

VC34

+V

1

2V1

3V1

V2

2V2

3V2

V3

2V3



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and

(5.76)


(5.77)


(5.78)

Analogously,


(5.79)


(5.80)


(5.81)

V

k

V

k

Vin3 233

1

91

1

=

=

( )

V V V V k

VO C C in= + = = 3 33 333

3

1( )

G VV k

O

in

= =

( )31

3

iV

LkTL

in1

1

= Ik

IL O1 332

1=

( )

i VL

kTL21

2

= Ik

IL O2 28

1=

( )

iV

LkTL3

2

3

= Ik

IL O32

1=

11

1

3

1

6

31

2 1

64

1

12= = =

i

I

k k TV

L I

k k R

fLL

L

in

O

/ ( ) ( )

22

2

21

2

4

22

2 1

16

1

12= =

=

iI

k k TV

L I

k k R

fLL

L O

/ ( ) ( )

33

3

2

3

2

3

2 1

4

1

12= =

=

i

I

k k TV

L I

k k R

fLL

L O

/ ( ) ( )


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(j) times in each stage circuit. The first three stages of this series are shownin Figure 5.12 to Figure 5.14. For convenience they are called elementarymultiple boost circuits, two-stage multiple boost circuits, and three-stagemultiple boost circuits respectively, and numbered as n = 1, 2, and 3.

5.6.1 Elementary Multiple Boost Circuit

From the construction principle, the elementary multiple boost circuit isderived from the elementary boost converter by adding DEC multiple (j)times in the circuit. Its circuit and switch-on and -off equivalent circuits areshown in Figure 5.12.

The voltage across capacitors C1 and C11 is charged to V1 and voltage acrosscapacitors C12 and C13 is charged to VC13 = 2V1. The voltage across capacitorsC1(2j-2) and C1(2j-1) is charged to VC1(2j-1) = jV1. The current iL1flowing throughinductor L1 increases with voltage Vin during switch-on period kT anddecreases with voltage (V1Vin) during switch-off (1 k)T. Therefore,

FIGURE 5.12Cascade boost multiple-double circuit.

VIN

+

iIN

S

L1

D1

C1

C11

D11

C12

D12

D1(2j1)

V1

2V1

C1(2j1)

C12j

D12j

R VO

+

iO

(1+j)V1

1 j2...

VIN

+

iIN

L1

C1

C12

VC11

C11

VC1

+ +

V1

R VO

+

iO

C12jC13

VC12

+

2V1

jV1

C12(j1)

C1(2j1)

(j+1)V1

+

VIN

iIN L

1

C1

VC1

+

VL1 R V

O

+

iO

VC12

C12

+

C11

VC11

+

C13

C14

C1(2j1)

C12j

V1

2V1

3V1

(1+j)V1

(a) Circuit diagram




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(5.86)

(5.87)


(5.88)


(5.89)

5.6.2 Two-Stage Multiple Boost Circuit

The two-stage multiple boost circuit is derived from the two-stage boostcircuit by adding multiple (j) DECs in each stage circuit. Its circuit diagram

and switch-on and -off equivalent circuits are shown in Figure 5.13. Thevoltage across capacitor C1 and capacitor C11 is charged to .The voltage across capacitor C1(2j) is charged to (1 +j)V1.

The current flowing through inductor L2 increases with voltage (1 + j)V1during switch-on period kT and decreases with voltage [V2 (1 + j)V1]during switch-off period (1 k)T. Therefore, the ripple of the inductor currentiL2 is

(5.90)

(5.91)


(5.92)


(5.93)

iV

LkT

V V

Lk TL

in in1

1

1

1

1= =

( )

Vk

Vin11

1=

V V V j V j

kVO C C j in= + = + =

+1 1 2 1 1

11

1( )( )

GV

V

j

kO

in

= =+

1

1

V k Vin1 1 1= ( )

ij

LkTV

V j V

Lk TL2

2

12 1

2

1 11=

+=

+

( )( )

Vj

kV j

kVin2 1

21

11

1

1=

+

= +

( )( )

V V V j V j

kVO C C j in= + = + =

+1 1 2 1 2

211

1( )( ) ( )

GV

V

j

kO

in

= =+

( )1

12


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(5.94)

FIGURE 5.13Two-stage boost multiple-double circuit.

VIN

+

iIN

S

L1

D1

C1

L2

R VO

+

iO

C11

D11

C12

D12

C1(2j

1)

D1(2j1)

C12j

D12j

D2

C21

D21

C22

D22

C2(2j

1)

D2(2j1)

C24

D22j

D3

C2

1 2... j j2...1

V1

jV1

(1+)jV1 V2

jV2 (1+j)V2

L1

C1

+

VIN

iIN

C11

VC1

+

C12j

L2

C21C2C12(j1) C1(2j1)

VO

iOR +

VC24

+

C22jC22(j1) C2(2j1)

V1

jV1

(1+)jV1

V2

jV2 (1+j)V2

+

VIN

iIN

L1

C1

VC1

+

VL1

VC12

C12

+

C11

VC11

+

C1(2j1)

C1

L2

R VO

+

iO

C2

+

VC2

C21

VC21

+

C22

VC22

+

C2(2j1)

C22j

VC22j

+

V1

(1+)jV1

V2

(1+j)V2

(a) Circuit diagram



vQ

C

I k T

C

k

fC

V

RO j

O

j j

O= =

=

22 22 22

1 1( )

= =v

V

k

RfCO

O j

/ 2 1

2 22


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5.6.3 Three-Stage Multiple Boost Circuit

This circuit is derived from the three-stage boost circuit by adding multiple(j) DECs in each stage circuit. Its circuit diagram and equivalent circuits

during switch-on and -off are shown in Figure 5.14. The voltage across capac-itor C1 and capacitor C11 is charged to . The voltage acrosscapacitor C1(2j) is charged to (1 +j)V1. The voltage V2 across capacitor C2 andcapacitor C2(2j) is charged to (1 +j)V2.

The current flowing through inductor L3 increases with voltage (1 + j)V2during switch-on period kT and decreases with voltage [V3 (1 + j)V2]during switch-off (1 k)T. Therefore,

(5.95)

and

(5.96)


(5.97)


(5.98)



(5.99)

V k Vin1 1 1= ( )

ij

LkTV

V j V

Lk TL3

32

3 2

3

1 11=

+=

+

( )( )

Vj V

k

j

kVin3

22

3

1

1

1

1=

+

=+

( )

( )

( )

( )

V V V j V j

kVO C C j in= + = + =

+3 3 2 1 3

311

1( )( ) ( )

GV

V

j

kO

in

= =+

( )1

13

vQ

C

I k T

C

k

fC

V

RO j

O

j j

O= =

=

32 32 32

1 1( )

= =v

V

k

RfCO

O j

/ 2 1

2 32


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FIGURE 5.14Three-stage boost multiple-double circuit.

VIN

+

iIN

S

L2

R

C21

D21

C22

D22

C2(2j-1)

D2(2j-1)

C22j

D22j

D3

C2

-

L1

D1

C1

C11

D11

C12

D12

C1(2j-1)

D1(2j-1)

C12j

D12j

D2

L3

D5

C3

C31

D31

C32

D32

C3(2j-1)

D3(2j-1)

C32j

D32j

D4

1 j2... 1 2... j 1 2... j

V1

jV1

(1+j)V1

V2

jV2

(1+j)V2

V3

jV3

(1+j)V

iIN

L1

C1

+

-

VIN

C22(j-1)C21

C2(2j-1)C2

R VO-

+

iO

C32j

C12j

L2

C12(j-1)C11

C1(2j-1)

C32(j-1)

C3

C31

L3

C22j

C3(2j-1)

V1

jV1

(1+j)V1

V2

jV2 (1+j)V2 V3 jV3 (1+j)V3

+

-VIN

iIN L

1

C1 VC1

+

-

VL1

VC12C12

+

-

C11

VC11

+-

C1(2j-1)

C12j

L2

R VO

-

+

iO

C2

+

-

VC2

C21

VC21

- +

C22

VC22

+

-

C2(2j-1)

C22j

L3

C3 VC3

+

-

VC31- +

C31

C32

C3(2j-1)

C32j

VC32

-

+

V1

(1+j)V1

V2

(1+j)V2

V3

(1+j)V3

(a) Circuit Diagram




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5.6.4 Higher Stage Multiple Boost Circuit

Higher stage multiple boost circuit is derived from the corresponding circuitof the main series by adding multiple (j) DECs in each stage circuit. For nth

stage additional circuit, the final output voltage is


(5.100)

Analogously, the variation ratio of output voltage vO is

(5.101)

5.7 Summary of Positive Output Cascade Boost Converters

All circuits of the positive output cascade boost converters as a family areshown in Figure 5.15 as the family tree. From the analysis of the previoustwo sections we have the common formula to calculate the output voltage:

(5.102)


Vj

kVO

nin=

+

( )1

1

GV

V

j

kO

in

n= =+

( )1

1

= =v

V

k

RfCO

O n j

/ 2 1

2 2

V

kV main series

kV additional series

kV double series

kV triple series

j

k V multiple j series

O

nin

nin

nin

nin

n in

=

+

( ) _

( ) _

( ) _

( ) _

( ) ( ) _

1

1

21

12

13

11

1


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(5.103)

In order to show the advantages of the positive output cascade boost con-verters, we compare their voltage transfer gains to that of the buck converter,

FIGURE 5.15The family of positive output cascade boost converters

5 Stage BoostCircuit

4 Stage Boost

Circuit

3 Stage Boost

Circuit

2 Stage Boost

Circuit

5 Stage AdditionalBoost Circuit

4 Stage Additional

Boost Circuit

3 Stage Additional

Boost Circuit

2 Stage Additional

Boost Circuit

5 Stage DoubleBoost Circuit




5 Stage MultipleBoost Circuit

4 Stage Multiple

Boost Circuit

3 Stage Multiple

Boost Circuit

2 Stage Multiple

Boost Circuit

Elementary Multiple

Boost Circuit

Elementary Triple

Boost Circuit

Elementary

Additional/Double

Boost Circuit

Positive Output Elementary Boost Converter

MainSeries

AdditionalSeries

DoubleSeries

TripleSeries

MultiSeries

5 Stage TripleBoost Circuit




GV

V

kmain series

kadditional series

kdouble series

ktriple series

j

kmultiple j series

O

in

n

n

n

n

n

= =

+

( ) _

( ) _

( ) _

( ) _

( ) ( ) _

1

1

2 112

13

11

1

GV

VkO

in

= =


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forward converter,

Nis the transformer turn ratio

Ck-converter,

fly-back converter,

Nis the transformer turn ratio

boost converter,

and positive output Luo-converters

(5.104)

If we assume that the conduction duty k is 0.2, the output voltage transfergains are listed inTable 5.1; if the conduction duty k is 0.5, the output voltagetransfer gains are listed in Table 5.2; if the conduction duty k is 0.8, the output

voltage transfer gains are listed in Table 5.3.

5.8 Simulation and Experimental Results

5.8.1 Simulation Results of a Three-Stage Boost CircuitTo verify the design and calculation results, the PSpice simulation packagewas applied to a three-stage boost converter. Choosing Vin = 20 V, L1 = L2 =L3 = 10 mH, all C1 to C8 = 2 F, and R = 30 k, k = 0.7 andf= 100 kHz. The

GV

V

kNO

in

= =

GV

V

k

kO

in

= =1

GV

V

k

kNO

in

= =1

GV

V kO

in

= =1

1

GV

V

n

kO

in

= =1


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obtained voltage values V1, V2, and VO of a triple-lift circuit are 66 V, 194 V,and 659 V respectively and inductor current waveforms iL1 (its average valueIL1 = 618 mA), iL2, and iL3. The simulation results are shown in Figure 5.16.The voltage values match the calculated results.

TABLE 5.1

Voltage Transfer Gains of Converters in the Condition k = 0.2

Stage No. (n) 1 2 3 4 5 n

Buck converter 0.2Forward converter 0.2N(Nis the transformer turn ratio)Ck-converter 0.25Fly-back converter 0.25N(Nis the transformer turn ratio)Boost converter 1.25Positive output Luo-converters 1.25 2.5 3.75 5 6.25 1.25nPositive output cascade boostconverters main series

1.25 1.563 1.953 2.441 3.052 1.25n

Positive output cascade boostconverters additional series

2.5 3.125 3.906 4.882 6.104 21.25n

Positive output cascade boost

converters double series

2.5 6.25 15.625 39.063 97.66 (21.25)n

Positive output cascade boostconverters triple series

3.75 14.06 52.73 197.75 741.58 (31.25)n

Positive output cascade boost (j = 3)converters multiple series

5 25 125 625 3125 (41.25)n

TABLE 5.2Voltage Transfer Gains of Converters in the Condition k = 0.5

Stage No. (n) 1 2 3 4 5 n

Buck converter 0.5Forward converter 0.5N(Nis the transformer turn ratio)Ck-converter 1Fly-back converter N(Nis the transformer turn ratio)Boost converter 2Positive output Luo-converters 2 4 6 8 10 2nPositive output cascade boost converters

main series

2 4 8 16 32 2n

Positive output cascade boost converters additional series

4 8 16 32 64 22n

Positive output cascade boost converters double series

4 16 64 256 1024 (22)n

Positive output cascade boost converters triple series

6 36 216 1296 7776 (32)n

Positive output cascade boost (j = 3)converters multiple series

8 64 512 4096 32,768 (42)n


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5.8.2 Experimental Results of a Three-Stage Boost Circuit

A test rig was constructed to verify the design and calculation results, and

compared with PSpice simulation results. The test conditions are still Vin =20 V, L1 = L2 = L3 = 10 mH, all C1 to C8 = 2 F and R = 30 k, k = 0.7, andf= 100kHz. The component of the switch is a MOSFET device IRF950 withthe rate 950 V/5 A/2 MHz. The measured values of the output voltage and

TABLE 5.3

Voltage Transfer Gains of Converters in the Condition k = 0.8

Stage No. (n) 1 2 3 4 5 n

Buck converter 0.8Forward converter 0.8N(Nis the transformer turn ratio)Ck-converter 4Fly-back converter 4N(Nis the transformer turn ratio)Boost converter 5Positive output Luo-converters 5 10 15 20 25 5nPositive output cascade boostconverters main series

5 25 125 625 3125 5n

Positive output cascade boostconverters additional series

10 50 250 1250 6250 25n

Positive output cascade boost

converters double series

10 100 1000 10,000 100,000 (25)n

Positive output cascade boostconverters triple series

15 225 3375 50,625 759,375 (35)n

Positive output cascade boost (j=3)converters multiple series

20 400 8000 160,000 32105 (45)n

FIGURE 5.16The simulation results of a three-stage boost circuit at condition k = 0.7 andf= 100 kHz.

9.980ms 9.984ms 9.988ms 9.992ms 9.996ms 10.000ms

V(R:2) V(D2:2) V(D5:2)

0V

0.5KV

1.0KV

SEL>>

I(L1) I(L2) I(L3)0A

0.5A

1.0A

Time

(9.988m, 111m)

(9.988m, 659m)

(9.988m, 194m)

(9.988m, 66m)

(9.988m, 218m)

(9.988m, 618m)


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first inductor current in a three-stage boost converter. After careful measure-ment, we obtained the current value of IL1 = 0.62 A (shown in channel 1 with1 A/Div) and voltage value of VO = 660 V (shown in channel 2 with 200 V/Div). The experimental results (current and voltage values) in Figure 5.17match the calculated and simulation results, which are IL1 = 0.618 A and VO= 659 V shown in Figure 5.16.

5.8.3 Efficiency Comparison of Simulation and Experimental Results

These circuits enhanced the voltage transfer gain successfully, and efficiently.Particularly, the efficiency of the tested circuits is 78%, which is good forhigh voltage output equipment. Comparison of the simulation and experi-mental results is shown in Table 5.4.

5.8.4 Transient Process

Usually, there is high inrush current during the first power-on. Therefore,the voltage across capacitors is quickly changed to certain values. The tran-sient process is very quick taking only a few milliseconds.

FIGURE 5.17

The experimental results of a three-stage boost circuit at condition k = 0.7 andf= 100 kHz

TABLE 5.4

Comparison of Simulation and Experimental Results of a Triple-Lift Circuit

Stage No. (n) IL1 (A) Iin (A) Vin (V) Pin (W) VO (V) PO (W) (%)

Simulation results 0.618 0.927 20 18.54 659 14.47 78Experimental results 0.62 0.93 20 18.6 660 14.52 78


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Addison-Wesley, New York, 1991.Khan, I.A., DC-to-DC converters for electric and hybrid vehicles, in Proceedings of

Power Electronics in Transportation, 1994, p. 113.Luo, F.L. and Ye, H., Positive output cascade boost converters, in Proceedings of IEEE-

IPEMC2003, Xian, China, 866, 2002.Luo, F.L. and Ye, H., Fundamentals of positive output cascade boost converters,

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