z Source Converters

7/30/2019 z Source Converters

1/52

1

Z-SOURCE CONVERTERS

Could be grouped into:o Z-source DC-DC converters,o Z-source inverters and,o Z-source rectifiers

Z-SOURCE DC-DC CONVERTERS

Compared to the conventional DC-DC converters, z-sourceDC-DC converters have the following advantages to offer:

o Lower inrush current,o Lower harmonic injection,o Larger range of output DC voltage,o Improved reliability.

The z-network gives this topology its unique features. Capable of operating in either voltage- or current-fed mode, Capable of buck or boost operation depending on whether it

is voltage or current-fed.


2/52

2

PWM duty ratio control can be employed just like inconventional DC-DC converters to control a z-source DC-DC

converter.

The z-network is symmetrical, i.e., the two inductors areidentical and the same is true of the two capacitors.

Figure 1 shows a voltage fed DC-DC converter. In a voltage-fed mode of operation requiring unidirectional

power flow capability, a diode is connected in series with the

voltage source while a controlled switch is connected in

parallel with the z-network output terminals.

RfC

fL1L

2L

1C

2C

1D

1Ci

2Ci

1Li

2Li

fCi

1Di fL

ioii

oii

fLi

oi

+

+

1Cv

2Cv

+

+

ov

fLv

2T+

inv

Fig. 1: Voltage-fed z-source DC-DC converter

Figure 2 shows a current-fed DC-DC converter.


3/52

3

refV

RfC

fL

1L

2L

1C

2C

d

2T

1T

Fig. 2: Current-fed z-source DC-DC converter

With reference to Fig. 1, the following expressions areobtained

21

21

zz

zz

CC

LL

vv

vv

=

=

(1)

Converter operation can be divided into two distinctinverters. During the first interval, the diode conducts

whereas the controlled switch is turned off.

Let the duty ratio of the controlled switch be . Figure 3presents the equivalent circuit of a voltage-fed z-source DC-DC converter during the interval when the diode conducts,

i.e., 0


4/52

4

RfC

fL1L

2L

1C

2C

1D

1Ci

2Ci

1Li

2Li

fCi

1Di fL

ioii

oii fLi

oi

+

+

1Cv

2Cv

+

+

ov

fLv

+

oiv+

inv

Fig. 3: Voltage-fed z-source converter during the interval when the

diode conducts

During this interval (i.e. 0


5/52

5

fZZ

fZZ

fZ

ZZ

f

Zf

ZZ

Z

ZZ

Z

ZZ

ZZ

Z

Z

ZZ

LLLpkin

LLLpkin

LLC

Z

C

ZL

CL

CactivoiL

LpkinC

zC

zL

CLpkinin

Cin

L

zzLinC

CinL

zLinC

iiii

iiii

iidt

dvC

dt

dvCi

ii

iii

iidt

dvC

dt

dvCi

iiii

vvdt

diLvvv

vvdt

diLvvv

+=

=

=+=

+=

+=

=+=

+==

==

==

21

21

2

1

1

1

1

1

12

1

11

1

11

1

2

1

2

1

12

,

,

,

,11

,

22

1

(2a)


6/52

6

0

12

1

2

2

1

1

11

12

12

2

2

1

2

2

2

22

21

,

,,

,2

,1,

,

,

,2

,

=

+=

==

=

==

+=

=

=

=

+=+=

+=

sw

LLLpkin

pkDpkinin

pkoiCL

z

pkoiCL

zLCpkoi

LLLpkin

LLLpkin

LpkinC

z

LL

C

Z

C

ZLCL

CactivoiL

i

iiii

iii

vvdt

diL

vvdt

diLvvv

iiii

iiii

iidt

dvC

iidt

dvC

dt

dvCiii

iii

fZZ

Z

Z

Z

Z

ZZ

fZZ

fZZ

Z

Z

fZ

Z

Z

fZf

ZZ

(2b)


7/52

7

oinCC

L

fopkoiL

Lactivoioi

LpkinLLactivoiLC

z

LCLactivoioi

LpkinLLactivoiLC

z

LCLactivoioi

LLpkinLCLactivoioi

LLpkinLCLactivoioi

activoiofofCoL

inCC

CinCininpkoioi

CinL

CinL

LLin

LoLLinpkoioi

vvvv

dt

di

Lvvv

iii

iiiiiidt

dvC

iiiii

iiiiiidt

dvC

iiiii

iiiiiiii

iiiiiiii

idtdvC

Rviii

vvv

vvvvvvv

vvv

vvv

but

dt

diL

dt

diLv

vvvvvvv

ZZ

f

f

f

ZfZZ

fzZ

ZfZZ

fzz

fzzzz

fzzz

f

zz

zz

zz

f

+=

==

==

===

===

===

===

=+===

=+===

=+=+=

+=

++==

=

=

=

+===

12

211

2

12

122

1

21

1212

2121

12

12

12

21

21

2

,

,

,,2

,

,,1

,

,,

,,

,

,

21

1,

(3)


8/52

8

For a symmetrical z-network, eqns. (2) and (3) simplify to

pkoiCL

ZLCpkoioi

LLLLLpkin

LLC

zC

CL

CactivoiL

LpkinC

ZC

ZL

CLpkinin

CinL

LinC

vvdt

diLvvvv

iiiiii

iidt

dvCi

ii

iii

iidt

dvC

dt

dvCi

iiii

vv

dt

diLvvv

Z

Z

ZZ

fZfZZ

fZ

Z

Z

Zf

ZZ

Z

ZZ

ZZ

ZZZ

,,

,

,

,

,

2

===

=+=

==

+=

+=

=+=

+==

==

(4)


9/52

9

opkoi

oinC

L

fooiL

Lactivoioi

LLactivoiLoiLC

z

LCLactivoioi

LpkinLactivoioi

LLpkinLCLactivoioi

activoio

fo

fCoL

inCCinCininpkoioi

CinL

LzinLoLinpkoioi

vv

vvv

dt

diLvvv

iii

iiiiiidt

dvC

iiiii

iiiii

iiiiiiii

idt

dvC

R

viii

vvvvvvvvv

vvv

but

dt

diLvvvvvvv

Z

f

f

f

fZZZ

Z

fZZ

fZ

fZZZZ

f

ZZZ

ZZ

Z

fZ

=

===

==

===

===

===

=+===

=+=+=

=++==

=

=+===

,

,

,

,

,,

,,

,

,

,

2

2

2

22

(5)

Figure 4 presents the equivalent circuit of a voltage-fed z-source DC-DC converter during the interval when the

controlled switch conducts.


10/52

10

RfC

fL1L

2L

1C

2C

1Ci

2Ci

1Li

2Li

fCi

ini fLi

oii

oii fLi

oi

+

+

1Cv

2Cv

+

+

ov

fLv

oiv

+

+

inv

Fig. 4: Voltage-fed z-source converter during the interval when the

switch conducts

Similarly, during the interval (1-)Tsw


11/52

11

dtdiLvv

dt

dvC

R

viii

iiiiii

iii

iii

ii

iiii

f

f

f

f

fZZf

f

ZZ

ZZ

ZZ

LfoL

Cf

ofCoL

LLLLshthrooisw

swLshthrooi

CLshthrooi

LL

CLshthrooioi

==

+=+=

+==

+=

=

+=

==

21

12

21

21

,

,

,

,

(6b)


12/52

12

For a symmetrical z-network, eqn. (6) simplifies to

ZZ

Z

ZZ

Z

LCC

z

in

oi

CLL

z

iidt

dvC

i

v

vv

dt

diL

==

=

=

==

0

0(7a)

dt

diLvv

dt

dvC

R

viii

iiiiiiii

iii

iii

iii

iiii

f

f

f

f

fZfZZf

f

ZZ

ZZZ

ZZ

LfoL

C

f

o

fCoL

LLLLLLshthrooisw

swLshthrooi

CLshthrooi

LLL

CLshthrooioi

==

+=+=

=+==

+=

=

=+=

==

2

2

,

,

,

,

(7b)

During steady state operation, average voltage across theinductor must be zero otherwise the inductor saturates. This

observation allows the derivation of the input-output steady

state transfer function as follows:


13/52

13

( ) ( ) ( )

( ) ( )

( )( ) ( )( ) ( )( ) ( )( )( )

oinC

in

C

CCinin

Cin

CCinLL

VVD

DV

or

D

D

V

V

tdDtvVtdDtvV

vv

vvvvv

Z

Z

ZZ

Z

ZZZZ

=

=

=

++=+

=

=+=+

21

1

21

1

~21~

~1~

211

0111

(8)

It is seen from eqn. (8) that by controlling the duty ratio, it ispossible to have either buck or boost operation depending on

the value of the duty ratio.

Further, it is also seen that the output voltage could be inphase or out of phase with the input voltage.

When duty ratio is less that 0.5, the output voltage is alwaysin phase with the input voltage and the converter operates inthe boost mode.

Additionally, when the duty ratio exceeds 0.5, the outputvoltage is negative in polarity relative to the input voltage

(i.e., the gain is negative). Further, mode of operation is

buck-boost.

The peak output voltage of the converter in one switchingcycle is obtained as follows:


14/52

14

inCC

CinCinin

LLinpkoioi

vvv

vvvvv

vvvvv

ZZ

ZZ

ZZ

+=

++=

==

12

12

21,

(9)

For a symmetrical z-network, eqn. (9) simplifies to

D

v

vvv

in

inCpkoi Z

21

2,

=

=

(10)

Average eqns. (2), (3) and (6) to yield expressions that areapplicable over one switching period as

( ) ( ) ( )

( ) ( )

( )

( ) ( ) ( )

( ) ( )

( )

=

+=

+=+

=

+=

+=+

1

11

111

1

11

111

,2

2

22

,1

1

11

2

2

12

2

12

22

1

1

21

1

21

11

pkoiCL

z

CCinL

z

CCinL

zL

z

pkoiCL

z

CCinL

z

CCinL

zL

z

vvdt

diL

also

vvv

dt

diL

vvvdt

diL

dt

diL

vvdt

diL

also

vvvdt

diL

vvvdt

diL

dt

diL

Z

Z

ZZ

Z

ZZ

ZZ

Z

Z

ZZ

Z

ZZ

ZZ

(11)


15/52

15

( ) ( ) ( )

( )( )( )( )( )

( ) ( )

( ) ( )

( ) ( ) ( )

( )

( )( )( )( )

( )( )

( )( )

21

21

2

21

221

2

2

22

2

12

12

1

12

121

1

1

11

1

1

1

1

1

1

111

11

11

1

1

1

111

,2

,2

,2

,1

,1

,1

ZfZ

ZZ

Z

ZfZ

ZfZZ

Z

Z

ZZ

Z

fZZ

ZZ

Z

ZfZ

ZfZZ

Z

Z

ZZ

Z

LLL

LactivoiLC

z

LLL

LLLL

LpkinC

z

LLpkinC

z

LLL

activoiLLC

z

LLL

LLLL

Lpkin

C

z

LLpkinC

z

iii

iiidt

dvC

iii

iiii

iidt

dvC

iiidt

dvC

iii

iiidt

dvC

iii

iiii

iidt

dv

C

iiidt

dvC

=

=

=

+=

=

=+

=

=

=

+=

=

=+

(12)


16/52

16

( )

( )( )( ) ( )

( ) ( )( ) ( )

( )

R

vi

dt

dvC

dt

dvC

R

viii

iiii

iii

iiii

iii

iiiii

iiii

oL

of

C

fo

fCoL

LLLsw

LLL

LLactivoioi

LLL

LLLLL

pkinLLoi

f

f

f

fZZ

ZZf

ZZ

ZZf

fZZZZ

ZZ

=

+=+=

+=

++=

++=

++=

++=

+=

21

21

21

21

2121

21

1

1

1

1

1

,

,

(13)

( )

( )

( )( )( )( )( )( )

( )( )

( )( )( )( )

=

+=

=

+=

+=

+=

=

=

=

1

11

1

1

1

1

1

1

,

,

,

,

21

21

21

21

21

pkD

LLL

pkinin

oinCC

L

f

inCCoi

ooinCC

ooLLin

opkoi

L

f

Lpkoio

Z

fZZ

ZZ

f

ZZ

ZZ

ZZ

f

f

i

iiiii

vvvvdt

diL

vvvv

vvvvv

vvvvv

vvdt

diL

vvv

(14)


17/52

17

For a symmetrical z-network, eqns. (11)-(14) simplify to( ) ( ) ( )

( ) ( )

( ) ( )

( )

=

=

+=

+=+

1

211

11

111

,pkoiCL

z

Cin

CCinL

z

CCinL

zL

z

vvdt

diL

also

vv

vvvdt

diL

vvv

dt

diL

dt

diL

Z

Z

Z

ZZ

Z

ZZ

ZZ

(15)

( ) ( ) ( )

( )

( )( )

( ) ( )( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

=

=

=

=

=

=

+=

=

=+

121

11

121

11

121

11

1

1

111

,

,

,

,

fZ

fZZ

Z

ZZ

Z

fZ

fZZ

ZfZZ

Z

Z

ZZ

Z

LL

LLL

activoiL

activoiLLC

z

LL

LLL

LLLL

LpkinC

z

LLpkinC

z

ii

iii

ii

iiidt

dvC

ii

iii

iiii

iidt

dvC

iiidt

dvC

(16)


18/52

18

( )

( )( )( ) ( )

( )

( ) ( )( )

( ) ( )

( )

( )( )

R

vi

dt

dvC

dt

dvC

R

viii

ii

iiii

ii

iii

ii

iiii

ii

iii

iiiii

iiii

oL

of

C

fo

fCoL

LL

LLLsw

LL

LLL

Lactivoi

LLactivoioi

LL

LLL

LLLLL

pkinLLoi

f

f

f

fZ

fZZ

Zf

ZZf

Z

ZZ

Zf

ZZf

fZZZZ

ZZ

=

+=+=

=

+=

+=

++=

+=

++=

+=

++=

++=

+=

2

21

1

21

1

21

1

1

1

,

,

,

(17)


19/52

19

( )

( )

( )( )( )( )( )( )( )( )

( )( )( )( )

( )( )( )

( )( )

( )( )( )

=

=

+=

=

=

+=

+=

=

=

+=

=

=

=

1

12

1

1

12

1

1

12

12

1

1

1

1

,

,

,

,

pkD

LL

LLL

pkinin

oinC

oinCC

L

f

inCCoi

oinC

ooinC

ooinCC

ooLLin

opkoi

L

f

Lpkoio

Z

fZ

fZZ

Z

ZZ

f

ZZ

Z

Z

ZZ

ZZ

f

f

i

ii

iii

ii

vvv

vvvvdt

diL

vvvv

vvv

vvvv

vvvvv

vvvvv

vvdt

di

L

vvv

(18)


20/52

20

Perturb eqns. (11)-(14) to yield( )

( )( ) ( )( )

( )( ) ( )( ) ( )( ) ( )( )

( )( ) ( )( ) ( )( )

( )( )( ) ( )( )

( )( ) ( )( ) ( )( ) ( )( )

( )( ) ( )( ) ( )( )tdDtvVtvV

dt

tidL

also

tdDtvVtdDtvV

tdDtvV

dt

tidL

tdDtvVtvVdt

tidL

also

tdDtvVtdDtvV

tdDtvV

dt

tidL

pkoipkoiCCL

z

CCCC

ininL

z

pkoipkoiCCL

z

CCCC

ininL

z

ZZ

Z

ZZZZ

Z

ZZ

Z

ZZZZ

Z

~1~~

~

~1~

~~

~1~

~

~1~~

~

~1~~

~1~

~

,,2

2

,,1

1

22

2

1122

2

11

1

2211

1

++=

++++

+=

++=

++++

+=

(19)


21/52

21

( )( )( ) ( )( ) ( )

( ) ( )( ) ( )( )( )( ) ( )( )

( )( )( ) ( )( )

( )( ) ( )( )

( )( ) ( )( )

( )( ) ( )( )( )( ) ( )( )

( )( ) ( )( )( )

( )( ) ( )( ) ( )

( ) ( )( ) ( )( )( )( ) ( )( )

( )( ) ( )( ) ( )( )

( )( ) ( )( )

( ) ( )( ) ( )( )( )( ) ( )( )tdDtiI

tdDtiItiI

tdDtiI

tdDtiItiIdt

tvdC

tdDtiI

tdDtiItiI

tiItdDtiIdt

tvdC

tdDtiI

tdDtiI

tdDtiI

tdDtiI

tdDtiI

tdDtiIdt

tvdC

tdDtiI

tdDtiItiI

tiItdDtiIdt

tvdC

ZZ

ffZZ

ZZ

ZZ

Z

ZZ

ffZZ

ZZ

Z

ff

ZZ

ZZ

ZZ

ZZ

Z

ZZ

ffZZ

ZZ

Z

LL

LLLL

LL

activoiactivoiLLC

z

LL

LLLL

LLpkinpkinC

z

LL

LL

LL

activoiactivoi

LL

LLC

z

LL

LLLL

LLpkinpkinC

z

~~

~1

~~

~~

~1

~~~

~~

~1

~~

~~1

~~

~1

~

~~

~1~

~1

~

~~

~1

~~

~~

~1

~~

~~1

~~

22

11

22

11

2

22

11

22

2

11

22

11

22

1

11

22

11

1

,,2

,,2

,,

1

,,1

++

+=

++

+=

++

+=

+=

+

++

+=

+

++

+=

++

+=

+=

(20)


22/52

22

( ) ( ) ( )

( )( ) ( )( )

( )( ) ( )( )( ) ( )( ) ( )( )

( ) ( )( ) ( )( )

( ) ( )( ) ( )( )( ) ( ) ( ) ( )( ) ( )( )

( ) ( ) ( ) ( ) ( )

( )( )

( )

R

tvVtiI

dt

tvdC

dt

tvdC

R

tvVtitiItiI

tdDtiItiItiItiI

tdDtiItiI

tdDtiItiI

tdDtiItiI

tdDtiI

tdDtiI

tiItiItiI

ooLL

of

C

foo

fCooLL

LLLLLLswsw

LLLL

activoiactivoioioi

LLLL

LL

pkinpkin

LLLLoioi

ff

f

ff

ffZZZZ

ZZZZ

ZZZZ

ff

ZZZZ

~~

~

~~~~~

~~~~~

~~~

~1

~~

~~~

~

1

~

~1

~

~~~

2211

2211

2211

2211

,,

,,

++=

++

=++=+

++++=+

+++++

+=+

+++++

+=

+

+++=+

(21)

( ) ( )( ) ( )( ) ( )

( ) ( ) ( )( ) ( )( )( )

( ) ( ) ( ) ( )( ) ( )( )

( ) ( )( ) ( )( )

( ) ( ) ( )( ) ( )( )( )( ) ( )( )tdDtiI

tdDtiItiItiI

tdDtiItiI

tdDtvVtvVtvVtvV

tvV

tdDtvVtvVtvV

tvVtdDtvVdt

tidL

pkDpkD

LLLLLL

pkinpkininin

ininCCCCoioi

oo

ininCCCC

oopkoipkoiL

f

ZZ

ffZZZZ

ZZZZ

ZZZZ

f

~1

~

~1

~~~

~1

~~

~1~~~~

~

~1~~~

~~1~~

,,

,,

,,

2211

2211

2211

+=

+++=

+=+

+++=+

+++=

+=

(22)


23/52

23

Obtain the DC-terms from eqns. (19)-(22) to yield( ) ( )

( )

( )

( ) ( )

( )

( )

( ) ( )

( )

( )

( )( )

( )

( )D

DVV

DDVV

DV

DVDV

DVDVDV

VV

DVV

DVV

also

DVDVDV

DVV

DVValso

DVDVDV

inC

inC

C

CC

CCin

CC

pkoiC

pkoiC

CCin

pkoiC

pkoiC

CCin

Z

Z

Z

ZZ

ZZ

ZZ

Z

Z

ZZ

Z

Z

ZZ

21

1

211

21

1

11

1

10

110

1

10

110

2

1

2

22

12

21

2

2

12

1

1

21

,

,

,

,

=

=

=

=

=

=

=

=

+=

=

=

+=

(23)


24/52

24

( )

( )( )( ) ( )

( ) ( )

( )

( )( )( )( )

( )( )( ) ( )

( )

( )

( )

( )

( )

( )D

DI

D

DII

DI

DIDI

DIDIDI

DIDII

DIDII

DIDII

II

IDI

DIDIDI

DIDIDI

DIDII

IDI

ZZf

Z

ZZ

ZZf

ZfZ

ZZ

ZfZ

ZZ

Z

fZZ

ZZ

ZfZ

Z

LLL

L

LL

LLL

LLL

LactivoiL

LLL

LL

Lpkin

LLL

activoiLL

LLL

Lpkin

=

=

=

=

=

=

=

=

=

=

=

=

=

=

1

21

1

21

21

1

11

1

10

1

10

11

110

1

10

21

1

11

21

21

21

21

21

2

12

12

12

1

,

,

,

,

(24)


25/52

25

( )

( )( )( )

( )( )( )

( )

( )

( )

( )

( ) ( )( )

( )( )

( )

( )( ) ( )

( )

( )

( )

( ) ( )( )

R

VI

R

VII

DIIII

DIIDII

D

DII

D

DDDDI

D

DDDI

D

DDIDI

DIDI

I

DIDII

D

DI

D

DII

DIIDI

DIIIII

DIIII

oL

ooL

LLLsw

LLactivoioi

Loi

LL

LL

LL

L

LLoi

LLL

LLL

LLLLL

pkinLLoi

f

f

fZZ

ZZ

f

ff

ff

Zf

Z

ZZ

ZZf

ZZf

fZZZZ

ZZ

=

==

+=

++=

=

+=

+=

+=

+=

=

+=

=

=

++=

++=

+=

0

1

21

1

21

12211

21

121

21

121

21

221

1

21

1

21

1

1

1

21

21

1

1

11

21

21

2121

21

,

,

(25)


26/52

26

( )

( )

( )( )( )( )( )( )

( )

( )( )( )DI

DIII

DII

DVVVV

DVVVV

VDVVV

DVV

VDV

pkD

LLL

pkinin

inCCoi

inCCo

oinCC

pkoio

opkoi

Z

fZZ

ZZ

ZZ

ZZ

=

+=

=

+=

+=

+=

=

=

1

1

1

1

1

10

1

10

,

,

,

,

21

21

21

21

(26)


27/52

27

Obtain the AC-terms from eqn. (19-22) to yield( )

( )( ) ( ) ( ) ( )

( )( ) ( )

( )( ) ( )( ) ( )

( )( )( ) ( ) ( ) ( )

( )( ) ( )

( )( ) ( )( ) ( )tdVDtvtv

dt

tidL

also

tdVDtv

tdVDtvtdVDtv

dt

tidL

tdVDtvtvdt

tidL

also

tdVDtv

tdVDtvtdVDtv

dt

tidL

pkoipkoiCL

z

CC

CCininL

z

pkoipkoiCL

z

CC

CCininL

z

Z

Z

ZZ

ZZ

Z

Z

Z

ZZ

ZZ

Z

~1~~

~

~1~

~~~1~~

~1~~

~

~1~

~~1~~

,,2

2

,,1

1

2

2

11

22

2

1

1

22

11

1

+=

+

++=

+=

+

++=

(27)


28/52

28

( )( ) ( ) ( )

( )

( )( ) ( )( ) ( ) ( )( ) ( )

( )( )( ) ( ) ( ) ( )

( )( ) ( )

( )

( )( ) ( ) ( ) ( )( )( ) ( )

( )( )( ) ( ) ( )

( )( ) ( )( )( ) ( ) ( )

( ) ( )

( )( ) ( )( )( ) ( ) ( )

( ) ( )

( )( ) ( )( )( ) ( ) ( )

( ) ( )tdIDti

tdIIDtitidt

tvdC

tdIDti

tdIIDtitidt

tvdC

tdIDti

tdIIDtitidt

tvdC

titdIDtidt

tvdC

tdIDti

tdIDtitdIDtidt

tvd

C

tdIDti

tdIDtitdIDtidt

tvdC

tdIDti

tdIIDtiDtidt

tvdC

titdItidt

tvdC

ZZ

fZfZ

Z

ZZ

ZZ

Z

ZZ

fZfZ

Z

Z

Z

ff

ZZZZ

Z

ZZZZ

Z

ZZ

fZfZ

Z

Z

Z

LL

LLLLC

z

LL

activoiLactivoiLC

z

LL

LLLLC

z

LpkinpkinC

z

LL

LLLL

C

z

activoiactivoi

LLLLC

z

LL

LLLL

C

z

LpkinpkinC

z

~~

~1

~~~

~~

~1

~~~

~~

~1

~~~

~~1

~~

~1

~

~~~

1

~~

~1

~

~~~1

~~

~~

~

1

~

1

~~

~~~~

22

11

2

22

11

2

22

11

2

2

2

1122

1

1122

1

11

22

1

1

1

2

,,2

2

,,2

1

,,

1

1

,,1

=

=

=

=

+

=

+

=

=

=

(28)


29/52

29

( ) ( ) ( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( ) ( )( ) ( )( ) ( )

( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )( ) ( )

( )( )

( )

R

tvti

dt

tvdC

dt

tvdC

R

tvtititi

tdIIIDtitititi

tdIIDtiti

tdIDtiti

tdIIDtiti

tdIDtiti

tdIDtitititi

oL

of

C

fo

fCoL

LLLLLLsw

LLLL

activoiactivoioi

LLLL

LLoi

pkinpkinLLoi

f

f

f

fZZfZZ

ZZZZ

ZZZZ

ff

ZZ

~~

~

~~~~~

~~~~~

~~~

~1

~~

~~~

~1

~~

~1

~~~~

2121

2121

2121

21

,,

,,

=

+=+=

+++=

++++

=

++++

=

++=

(29)

( )( )( ) ( ) ( )

( )( ) ( ) ( )( )( )

( ) ( ) ( )

( ) ( ) ( ) ( )( )( ) ( ) ( )

( ) ( )( ) ( )

( ) ( ) ( ) ( )( ) ( )( )( ) ( )( ) ( )tdIDtiti

tdDtiItiItiIti

tdIDtiti

tdVVVDtvtvtvtv

tvtdVVV

Dtvtvtvdt

tidL

tvtdVDtv

dt

tidL

pkDpkDin

LLLLLLin

pkinpkinin

inCCinCCoi

oinCC

inCC

L

f

opkoipkoi

L

f

ZZ

ffZZZZ

ZZZZ

ZZ

ZZ

f

f

~1

~~

~1~~~~

~1

~~

~1~~~~

~~

1~~~~

~~1~~

,,

,,

,,

2211

2121

21

21

=

+++=

=

++=

+

+=

=

(30)


30/52

30

Laplace transform eqns. (27)-(30) into the s-domain to yield( ) ( )( ) ( ) ( ) ( )

( )( ) ( )

( ) ( ) ( )( ) ( )

( ) ( )( ) ( ) ( ) ( )

( )( ) ( )

( ) ( ) ( )( ) ( )sdVDsvsvsisLalso

sdVDsv

sdVDsvsdVDsvsisL

sdVDsvsvsisL

also

sdVDsv

sdVDsvsdVDsvsisL

pkoipkoiCLz

CC

CCininLz

pkoipkoiCLz

CC

CCininLz

ZZ

ZZ

ZZZ

ZZ

ZZ

ZZZ

~1~~

~

~1~

~~~1~~

~1~~

~

~1~

~~1~~

,,2

2

,,1

1

22

11

222

11

22

111

+=

+

++=

+=

+

++=

(31)


31/52

31

( ) ( ) ( ) ( )

( ) ( )( ) ( )( ) ( ) ( )

( ) ( )( ) ( )( ) ( ) ( ) ( )

( )( ) ( )

( ) ( )( ) ( ) ( ) ( )

( )( ) ( )

( ) ( )( ) ( ) ( )( ) ( ) ( )( )( ) ( ) ( )

( ) ( )

( ) ( ) ( )( )( ) ( ) ( )

( ) ( )

( ) ( ) ( )( )( ) ( ) ( )( ) ( )sdIDsi

sdIIDsisisvsC

sdIDsi

sdIIDsisisvsC

sdIDsi

sdIIDsisisvsC

sisdIDsisvsC

sdIDsi

sdIDsisdIDsisvsC

sdIDsi

sdIDsisdIDsisvsC

sdIDsi

sdIIDsiDsisvsC

sisdIsisvsC

ZZ

fZfZZ

ZZ

ZZZ

ZZ

fZfZZ

ZZ

ff

ZZZZZ

ZZZZZ

ZZ

fZfZZ

ZZ

LL

LLLLCz

LL

activoiLactivoiLCz

LL

LLLLCz

LpkinpkinCz

LL

LLLLCz

activoiactivoi

LLLLCz

LL

LLLLCz

LpkinpkinCz

~~

~1~~~

~~

~1

~~~

~~

~1

~~~

~~

1~~

~1

~

~~~1

~~

~1

~

~~~1

~~

~~

~1

~1

~~

~~~~

22

112

22

112

22

112

22

11221

11221

11

221

11

2

,,2

2

,,2

1

,,

1

1

,,1

=

=

=

=

+

=

+

=

=

=

(32)


32/52

32

( ) ( ) ( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( ) ( )( ) ( )( ) ( )

( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )( )

( )

( ) ( )( )

R

svsisvsC

svsC

R

svsisisi

sdIIIDsisisisi

sdIIDsisi

sdIDsisi

sdIIDsisi

sdIDsisi

sdIDsisisisi

oLof

Cfo

fCoL

LLLLLLsw

LLLL

activoiactivoioi

LLLL

LLoi

pkinpkinLLoi

f

ff

fZZfZZ

ZZZZ

ZZZZ

ff

ZZ

~~~

~~

~~~

~~~~~

~~~

~1

~~

~~~

~1

~~

~1

~~~~

2121

2121

2121

21

,,

,,

=

+=+=

+++=

++++

=

++++

=

++=

(33)

( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( )( )( )( ) ( ) ( )

( ) ( ) ( ) ( )( )( ) ( ) ( )

( ) ( )( ) ( )

( ) ( ) ( ) ( )( )( )

( ) ( )( ) ( )( ) ( )sdIDsisi

sdIII

Dsisisisi

sdIDsisi

sdVVVDsvsvsvsv

svsdVVV

DsvsvsvsisL

svsdVDsvsisL

pkDpkDin

LLL

LLLin

pkinpkinin

inCCinCCoi

oinCC

inCCLf

opkoipkoiLf

ZZ

fZZ

fZZ

ZZZZ

ZZ

ZZf

f

~1

~~

~

1~~~~

~1

~~

~1~~~~

~~1

~~~~

~~1~~

,,

,,

,,

21

21

2121

21

21

=

+

+=

=

++=

+

+=

=

(34)


33/52

33

Re-arrange eqns. (31)-(32) to obtain expressions relatingcapacitor voltage to duty ratio and input current as follows

( ) ( )( ) ( )

( ) ( ) ( )( ) ( )

( )( )

( ) ( ) ( )( ) ( )

( )( ) ( )( ) ( )

1

,,

1

1

1

1

~1~~~

~

1

~~

1~

~1~

~~

~

1

~

~

1

1

1221

2211

1

z

pkoipkoiCL

z

CCinCC

z

in

z

CCCC

z

ininL

sL

sdVDsvsvsi

also

sLDsvDsvsdVVV

sL

Dsv

sL

sdVDsvsdVDsv

sLsdVDsvsi

Z

Z

ZZZZ

zzzz

Z

+

=

++

+

=

++

+

=

(35a)

( )( )( ) ( )

( ) ( ) ( )( ) ( )

( )( )

( ) ( ) ( )( ) ( )

( )( ) ( )( ) ( )

2

,,

2

2

2

2

~1~~~

~1~~

1~

~1~

~~

~1~~

2

2

2121

1122

2

z

pkoipkoiCL

z

CCinCC

z

in

z

CCCC

z

ininL

sL

sdVDsvsvsi

also

sL

DsvDsvsdVVV

sL

Dsv

sL

sdVDsvsdVDsv

sL

sdVDsv

si

Z

Z

ZZZZ

ZZZZ

Z

+

=

++

+

=

++

+

=

(35b)


34/52

34

( ) ( )( )

( )

( ) ( ) ( )( )DsisdIII

Dsi

DsisvsC

activoiactivoiLL

L

LCz

ZZ

Z

ZZ

+

=

1

~~

~

1~~

,,

1

21

1

21

(36a)

( )

( )( )

( ) ( ) ( )( ) ( )( )

( )( )

( ) ( ) ( )( ) ( )

( ) ( ) ( )( )

( )( ) ( )( ) ( )

( ) ( ) ( )( )

( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( )DsisdIIIsL

DDsvDsv

sL

sdDVVV

sL

DDsv

sL

DsvDDsv

sL

sdDVVV

sL

Dsv

DsisdIII

D

sL

DsvDsvsdVVV

sLDsv

D

sL

DsvDsvsdVVV

sL

Dsv

svsC

activoiactivoiLL

z

CC

z

inCC

z

in

z

CC

z

inCC

z

in

activoiactivoiLL

z

CCinCC

z

in

z

CCinCC

z

in

Cz

ZZ

ZZ

ZZ

ZZ

ZZ

ZZ

ZZZZ

ZZZZ

Z

+

+

+

+

+

+

+

=

+

++

+

++

+

=

1~~

1~~

~1~

1~1~

~11~

1~~

~1~~

1~

1~1~~

1~

~

,,

1

2

11

2

2

22

2

,,

1

1

2

2

1

21

21

21

12

21

21

1221

2121

1

(36b)


35/52

35

( )

( )( ) ( )( )

( )( ) ( )

( ) ( )

( ) ( ) ( )( )

( ) ( ) ( )

( ) ( ) ( )( )DsisdIIIsL

DDsvDsv

sL

DsvDDsv

sL

sdDVVV

sL

sdDVVV

sL

DDsv

sL

Dsv

svsC

activoiactivoiLL

z

CC

z

CC

z

inCC

z

inCC

z

in

z

in

Cz

ZZ

ZZ

ZZ

ZZ

ZZ

Z

+

+

+

+

+

=

1~~

1~~

1~1~

~

~1

1~1~

~

,,

1

2

2

2

1

2

12

2

1

21

21

12

21

21

1

(36c)

( )( )

( ) ( ) ( ) ( ) ( ) ( )

( )( ) ( )( )

( ) ( ) ( )( )DsisdIII

sdsL

DVVV

sL

DVVV

svsL

DD

sL

DDsv

sL

DD

sL

D

svsL

D

sL

DsC

activoiactivoiLL

z

inCC

z

inCC

Czz

inzz

Czz

z

ZZ

ZZZZ

Z

+

+

+

+

+

+

=

+

+

1~~

~1

~11~11

~1

,,

12

1212

2

1

2

2

2

1

21

2121

2

1

(36d)


36/52

36

( ) ( )

( ) ( )( ) ( )( ) ( )( ) ( )

( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( )siDLsLsdLLIIIs

sdDVVVLDLVVV

svDDLDDL

svDDLDL

svDLDLCLLs

activoizzzzactivoiLL

inCCinCC

Czz

inzz

Czzz

ZZ

ZZZZ

Z

Z

,2121,

21

21

22

1

22

21112

2

~1

~

~1

~11

~11

~1

21

2121

2

1

+

+++

++

=

++

(37a)

( )

( )

( )( )

( )

( )( ) 22

21112

2

,21

22211122

21

0~

0~

0~

1

1

1

~

~

21

21

,

2

1

DLDLCLLs

IIILsL

DLDLCLLs

DLDLVVV

sd

sv

zzzzz

activoiLLzz

zzzzz

zzinCC

i

v

vC

ZZ

ZZ

activoi

ZCin

Z

++

+

++

+

=

=

=

=

(37b)

( )

( )

( ) ( )

( ) 222

11122

22

1

0~

0~0

~

1

11

~

~

,

2

1

DLDLCLLs

DDLDL

sv

sv

zzzzz

zz

i

vin

C

activoi

ZC

Z

++

=

=

=

= (37c)

( )

( )

( ) ( )

( ) 222

11122

21

0

~0~

0~

1

11~

~

,

2

1

DLDLCLLs

DDLDDL

sv

sv

zzzzz

zz

i

vC

C

activoi

inZ

Z

++

+=

=

=

=(37d)

( )

( )

( )

( ) 222

11122

21

0~

0~0~

, 1

1~

~

2

1

DLDLCLLs

DLsL

si

sv

zzzzz

zz

v

vactivoi

C

ZC

in

Z

++

=

=

=

=

(37e)


37/52

37

( )( ) ( )( ) ( )

( )

( ) ( )( ) ( )

( ) ( ) ( )( )DsisdIII

DsL

sdVDsvsv

DsL

sdVDsvsvsvsC

activoiactivoiLL

z

pkoipkoiC

z

pkoipkoiCCz

ZZ

Z

Z

Z

+

+

+=

1~~

~1~~

1

~1~~~

,,

1

,,

2

,,1

21

1

2

1

(38a)

( )( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )( )DsisdIII

sL

sdDVDDsvsvD

sL

sdDVDsvsvD

svsC

activoiactivoiLL

z

pkoipkoiC

z

pkoipkoiCCz

ZZ

Z

ZZ

+

+

+=

1~~

~1~~

~11~~1

~

,,

1

,,

2

,2

,1

21

1

21

(38b)

( )

( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )

( )( )Dsi

sdIIIsdsL

DVsL

DV

svsL

D

sL

DD

sL

svD

svsL

DsC

activoi

activoiLLz

pkoi

z

pkoi

pkoizzz

C

Cz

z

ZZ

Z

Z

+

+

+

=

+

1~

~~1

~11~1

~

,

,1

,

1

,

,2

2

12

11

21

2

1

(38c)


38/52

38

( )

( )

( ) ( )

( )( ) ( )DLCLLs

IIILsLDLDLV

sL

DsC

IIIsL

DV

sL

DV

sd

sv

zzzz

activoiLLzzzzpkoi

zz

activoiLLz

pkoi

z

pkoi

i

v

C

ZZ

ZZ

activoi

pkoi

Z

21212

,2121,

11

,1

,

2

,

0

0

21

21

,

,

1

1

1

~

~

+

+

=

+

+

=

=

=

(39a)

( )

( )

( ) ( )

DLCLsL

DLDDL

sv

sv

zzzz

zz

i

vpkoi

C

acticoi

pkoi

Z

2121

212

0

0,

11~

~

,

,

1

+

=

=

=

(39b)

( ) ( )

( ) ( )( ) ( )

( )

( ) ( )

( )

( )( ) 1

1~

~

1

~

~

~1~~1

~1~

20

~,

2

,

0~

,,2

,

++

=

++

=

++

+=

=

=

R

sLCLs

D

sv

svR

sLCLs

V

s

sv

sVDsvsvR

sLCLs

svR

sCsi

fffs

pkoi

o

fff

pkoi

sv

o

pkoipkoiof

ff

ofL

pkoi

f

(40)


39/52

39

( ) ( ) ( ) ( )( )( )

( ) ( )( )

( )( )

( )

( )

( )

( )

( )( )

( )( )

( )

( )( )

( )

( )

( )

( )( )

( )( )

1

1~

~

1

1~

~

1

1~

~

1

~

~

~

1~~~~1

20~0~

0~

2

0~

0~

0~

2

0~0

~

0~

2

0~

0~

0~

2

2

1

12

21

21

2

1

21

21

++

=

++

=

++

=

++

+

=

+

+=

++

=

=

=

=

=

=

=

=

=

=

=

=

R

sLCLs

D

sv

sv

R

sLCLs

D

sv

sv

R

sL

CLs

D

sv

sv

R

sLCLs

VVV

s

sv

sVVV

DsvsvsvsvR

sLCLs

fff

svsv

s

in

o

fff

sv

sv

sC

o

f

ffsvsv

sC

o

fff

inCC

sv

sv

svo

inCC

inCCof

ff

C

C

in

C

in

C

in

C

C

(41)


40/52

40

For a symmetrical z-network, perturb eqns. (15)-(18) to yield( )

( )( ) ( )( )

( )( ) ( )( )( )

( )( ) ( )( ) ( )( )tDtvVtvV

dt

tidL

also

tDtvV

tDtvV

dt

tidL

pkoipkoiCCL

Z

CC

ininL

z

ZZ

Z

ZZ

Z

~1~~

~

~21~

~1~

~

,, ++=

++

+=

(42)

( )( )( ) ( )( ) ( )

( )( ) ( )( ) ( )( )

( )

( )( )( ) ( )( )( )

( )( ) ( )( )( )

( )( ) ( )( )( )

( )( ) ( )( )tDtiI

tDtiIdt

tvdC

tDtiI

tDtiIdt

tvdC

tiI

tDtiItiIdt

tvdC

tiItDtiIdt

tvdC

ff

ZZ

Z

ZZ

Z

ZZ

ffZZ

Z

ZZ

Z

LL

LLC

Z

activoiactivoi

LL

C

Z

LL

LLLLC

Z

LLpkinpkinC

Z

~1

~

~21

~~

~1

~

~21

~~

~

~1

~~22

~

~~1

~~

,,

,,

+

++=

+

++=

+=

+=

(43)


41/52

41

( ) ( ) ( )( ) ( )

( ) ( )( ) ( )( ) ( )( ) ( )( )

( ) ( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( )

( ) ( )( ) ( )( ) ( )( )

( ) ( ) ( )( ) ( )

( )( )

( )

R

tvVtiIdt

tvdC

dt

tvdC

R

tvVtitiItiI

tdDtiItiItiI

tdDtiItdDtiI

tdDtiItdDtiItiI

tiItdDtdDtiItiI

tdDtiItiItiI

ooLLof

C

foo

fCooLL

LLLLswsw

LLLL

LLactivoiactivoioioi

LLLLoioi

pkinpkinLLoioi

ff

f

ff

ffZZ

ZZff

ZZ

ZZff

ZZ

~~

~

~~~~~

~~~2

~

~~2

~1

~

~~2

~1

~~

~~2

~1

~~

~1

~~2

~

,,

,,

+

+=

++

=++=+

++=+

++++=

++++=+

++++=+

++=+

(44)

( )( )( ) ( )( ) ( )

( )( ) ( )( ) ( )( ) ( )

( ) ( )( ) ( )( ) ( )( )

( ) ( )( ) ( )( )

( )( ) ( )( ) ( )( )( )( ) ( )( )tdDtiI

tdDtiItiI

tdDtiItiI

tdDtvVtvVtvV

tvVtdDtvVtvV

tvVtdDtvVdt

tidL

pkDpkD

LLLL

pkinpkininin

ininCCoioi

ooininCC

oopkoipkoi

L

f

ZZ

ffZZ

ZZ

ZZ

f

~1

~

~1

~~2

~1

~~

~1~~2~

~~1~~2

~~1~~

,,

,,

,,

+=

+=

+=+

+=+

+=

+=

(45)

For a symmetrical z-network, obtain the DC-terms fromeqns. (42)-(45) as


42/52

42

( ) ( )

( )

( )

( )DVV

D

DVV

DVDV

pkoiC

inC

inC

=

=

=

1

21

1

121

,

(46)

( )

( )

( )( )( ) ( )

( ) ( )

( )

( )

( )

( )

( )( )

( )

( )

( )DII

DII

D

DIDI

D

DII

D

DII

DIDI

DIDI

IDII

DII

IDI

pkinactivoi

activoipkin

activoipkin

Lactivoi

activoiL

activoiL

activoiL

LLL

pkinL

Lpkin

Z

Z

Z

Z

ZfZ

Z

Z

21

21

1

21

11

1

21

21

1

121

1210

120

1

10

,,

,,

,,

,

,

,

,

,

,

=

=

=

=

=

=

=

=

=

=

(47)


43/52

43

( )

( )

( )

( )

( )( )

( )

( )( )

( )( )( ) ( )

( )

( )

( )

R

VI

R

VII

D

DI

D

DDDDI

D

D

DIDII

D

DII

DIII

DIDI

DIDII

DIDII

DIII

oL

ooL

LL

LLoi

LL

LLsw

LL

Lactivoioi

LLoi

pkinLoi

f

f

ff

ff

fZ

fZ

Zf

Z

Zf

Z

=

==

=

+=

+=

=

=

+=

+=

+=

=

0

21

1

21

1221121

121

21

1

2

21

21

21

12

,

,

(48)

( )

( )

( )( )( )( )( )( )

( )

( )( )( )DI

DII

DII

DVVV

DVVV

VDVV

DVV

VDV

pkD

LL

pkinin

inCoi

inCo

oinC

pkoio

opkoi

Z

fZ

Z

Z

Z

=

=

=

=

=

=

=

=

1

12

1

12

12

120

1

10

,

,

,

,

(49)


44/52

44

For a symmetrical z-network, obtain AC-terms from eqns.(42)-(45) as

( )( )( ) ( ) ( )( ) ( )

( )( ) ( )( ) ( )tVDtvtv

dt

tidL

also

tVDtvtVDtvdt

tidL

pkoipkoiCL

CCininL

~1~~

~

~221~

~1~

~

,, +=

+=

(50)

( )( )( ) ( ) ( )

( )( )( ) ( )

( )( ) ( )tIDti

tIDtidt

tvdC

titIDtidt

tvd

C

activoiactivoi

LLC

z

zLpkinpkinC

z

ZZ

Z

Z

~1

~

~221

~~

~~

1~

~

,,

,,

+

=

=

(51)

( ) ( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )

( ) ( )( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( )( )

( )

R

tvti

dt

tvdC

dt

tvdC

R

tvtititi

tdIIDtititi

tdItiDtdIDti

tdItiDtdIDtiti

tdIDtitdIDtiti

tdIDtititi

oL

of

C

fo

fCoL

LLLLsw

LLLL

LLactivoiactivoioi

LLLLoi

pkinpkinLoi

f

f

f

fZfZ

ZZff

ZZ

ZZff

Z

~~

~

~~~~~

~2

~~2

~

~2

~2

~1

~

~2

~2

~1

~~

~2

~2

~1

~~

~

1

~~

2

~

,,

,,

=

+=+=

+=

++=

++=

++=

+=

(52)


45/52

45

( )( )( ) ( ) ( )

( )

( ) ( )( )( ) ( ) ( ) ( )( ) ( ) ( )( )( ) ( ) ( )

( ) ( )( ) ( )

( ) ( )( )( ) ( ) ( )( )( ) ( )tdIDti

tdIIDtiti

tdIDtiti

tdVVDtvtvtv

tvtdVVDtvtvdt

tid

L

tvtdVDtvdt

tidL

pkDpkD

LLLL

pkinpkinin

inCinCoi

oinCinC

L

f

opkoipkoi

L

f

ZZ

fZfZ

ZZ

ZZ

f

f

~1

~

~21

~~2

~1

~~

~21~~2~

~~

21~~2

~

~~1~~

,,

,,

,,

=

=

=

=

=

=

(53)


46/52

46

For a symmetrical z-network, transform the AC-terms ofeqns. (50)-(53) to the s-domain to yield

( ) ( )( ) ( ) ( )( ) ( )

( ) ( ) ( )( ) ( )sVDsvsvsisL

alsosVDsvsVDsvsisL

pkoipkoiCL

CCininL

~1~~

~

~

221

~~

1

~~

,, +=

+=

(54)

( ) ( )( ) ( ) ( )

( ) ( )( ) ( )

( )( ) ( )sIDsi

sIDsisvsC

sisIDsisvsC

activoiactivoi

LLCz

zLpkinpkinCz

ZZZ

Z

~1

~

~221

~~

~~1

~~

,,

,,

+

=

=

(55)

( ) ( ) ( )( ) ( )

( ) ( )( ) ( ) ( ) ( )

( ) ( )( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )( )

( )

( ) ( )( )

R

svtisvsC

svsCR

svsisisi

sdIIDsisisi

sdIsiDsdIDsi

sdIsiDsdIDsisi

sdIDsisdIDsisi

sdIDsisisi

oLof

Cfo

fCoL

LLLLsw

LLLL

LLactivoiactivoioi

LLLLoi

pkinpkinLoi

f

ff

fZfZ

ZZff

ZZ

ZZff

Z

~~~

~~

~~~

~2

~~2

~

~2~2~1~

~2

~2

~1

~~

~2

~2

~1

~~

~1

~~2

~

,,

,,

=

+=+=

+=

++=

++=

++=

+=

(56)


47/52

47

( ) ( )( ) ( ) ( )

( ) ( ) ( )( )( ) ( ) ( ) ( )

( ) ( ) ( )( )( ) ( ) ( )( ) ( )( ) ( )

( ) ( )( )( ) ( ) ( )( )( ) ( )sdIDsi

sdIIDsisi

sdIDsisi

sdVVDsvsvsv

svsdVVDsvsvsisL

svsdVDsvsisL

pkDpkD

LLLL

pkinpkinin

inCinCoi

oinCinCLf

opkoipkoiLf

ZZ

fZfZ

ZZ

ZZf

f

~1

~

~21

~~2

~1

~~

~21~~2~

~~21~~2~

~~1~~

,,

,,

,,

=

=

=

=

=

=

(57)


48/52

48

For a symmetrical z-network, obtain transfer functionsrelating capacitor voltage to duty ratio, input voltage and

current as follows

( )( )( ) ( ) ( )( ) ( )

( )( ) ( )( ) ( )

sL

sVDsvsvsi

also

sL

sVDsvsVDsvsi

pkoipkoiCL

CCininL

~1~~~

~221~~1~~

,, +=

+=

(58)

( )( )( ) ( ) ( )( ) ( )

( )

( )

( )( ) ( )sIDsi

sI

DsL

sVDsvsVDsvsvsC

activoiactivoi

L

CCininCz

Z

Z

~1

~

~2

21~

221~~

1~~

,, +

+=

(59a)

( )( )( )( ) ( )( )

( ) ( )( ) ( )

( ) ( ) ( )( )DsisIIsL

sDVDV

sL

Dsv

sL

DDsvsvsC

activoiactivoiL

inC

CinCz

Z

Z

+

=

1~~

2

~21212

21~211~~

,,

2

(59b)


49/52

49

( )( )

( )( )( )

( )( ) ( )

( ) ( )

( )( )Dsi

sII

sL

sVVD

sL

DDsvsv

sL

DsC

activoi

activoiL

inC

inCz

Z

Z

+

=

+

1~

~2

~221

211~~21

,

,

2

(59c)

( ) ( ) ( )( ) ( )

( )( ) ( )

( ) ( )

( ) ( )

( )

( )

( )( ) ( )( )22

,

0~

0~

,

,

22

21

2221~

~

~1

~2

~221

~211~21

,

DLCs

IIsLVVD

s

sv

siDsL

sIIsL

sVVD

svDDsvDLCs

z

activoiLinC

i

v

C

activoi

activoiL

inC

inCz

Z

activoi

in

Z

Z

Z

+

=

+

=+

=

=

(59d)

( )

( )

( )( )

( )220

~0

~21

211~

~

,

DLCs

DD

sv

sv

zi

in

C

activoi

Z

+

=

=

=(59e)


50/52

50

( )( )

( )( )

( )

( )

( )( )

( )( ) ( )( )( )( )

( ) [ ]( ) ( )

( ) ( )

( )( ) ( )

( )

( ) [ ]( )( )

( )

( ) ( )( )

( )

( )

( )

( ) ( )( )

( )( )

( )DLCs

DD

sv

sv

DLCs

DsL

si

sv

DLCsIIsLDV

ssv

siDsL

sIIsLDV

DDsv

svDLCs

siD

D

sD

IIsv

D

sCsi

si

spkoi

C

sv

soi

C

activoiLpkoi

sv

siC

activoi

activoiLpkoi

pkoi

C

activoi

activoiLCL

oi

pkoi

pkoi

oi

21

211

~

~

21

1~

~

21221~

~

~1

~221

211~

~21

~

21

1

~

21

2~21

~

2

0~

0~,

2

0~

0~

2,,

0~

0~

,

,,

,

2

,

,

,

,

=

=

=

+

+

+

=

=

=

=

=

=

=

(60)


51/52

51

( ) ( ) ( )( ) ( )

( ) ( )( ) ( ) ( ) ( )

( ) ( )

( )( ) ( )

( )( )

( )

( ) ( ) ( )( )11~

~

11

~~

~

~1~

~2

~2

~1

~~

~1

~~2

~

0~

,,

+

=

+

+

=

+=

++=

+=

=RsCD

RI

s

sv

RsCD

sRIsiRsv

svR

sCsi

sdIsiDsIDsisi

sdIDsisisi

f

L

si

o

f

Loio

ofL

LLLLoi

pkinpkinLoi

f

oi

f

f

ZZff

Z

(61)

( ) ( )

( )( )( ) ( )

( )

( ) ( )

( )

( )( )

( )

++

=

++

=

++

=

+=

=

=

1

1~

~

1

~

~

1

~1~~

~1~

20~,

2

,

0~

2

,,

,

R

sLCLs

D

sv

sv

R

sLCLs

V

s

sv

R

sLCLs

sVDsvsv

svR

sCsi

fff

spkoi

o

fff

pkoi

sv

o

fff

pkoipkoio

ofL

pkoi

f

(62)


52/52

( )( ) ( )( )( ) ( ) ( )

( )( ) ( )

( )

( )

( )

( ) ( )( )

( )

( )( ) ( )

( )

( )

( ) ( ) ( )( )( ) ( ) ( )sVVDsvsvsv

R

sLCLs

D

sv

sv

R

sLCLs

D

sv

sv

R

sLCLs

VV

s

sv

R

sLCLs

sVVDsvsvsv

inCinCoi

fffs

svC

o

fffs

svin

o

fff

inC

sv

sv

o

fff

inCinCo

in

C

in

C

~21~~2~

1

12~~

1

1~

~

1

2~~

1

~21~~2~

20

~0~

20

~0~

20~

0~

2

=

++

=

++

=

++

=

++

=

=

=

=

=

=

=

(63)

Documents

z Source Converters