z source converters
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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.
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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.
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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
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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
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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)
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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)
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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)
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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)
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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.
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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
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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)
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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:
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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:
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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)
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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)
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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)
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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)
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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)
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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)
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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)
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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)
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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)
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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)
-
7/30/2019 z Source Converters
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)
-
7/30/2019 z Source Converters
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)
-
7/30/2019 z Source Converters
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)
-
7/30/2019 z Source Converters
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)
-
7/30/2019 z Source Converters
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)
-
7/30/2019 z Source Converters
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)
-
7/30/2019 z Source Converters
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)
-
7/30/2019 z Source Converters
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)
-
7/30/2019 z Source Converters
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)
-
7/30/2019 z Source Converters
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)
-
7/30/2019 z Source Converters
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)
-
7/30/2019 z Source Converters
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)
-
7/30/2019 z Source Converters
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)
-
7/30/2019 z Source Converters
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)
-
7/30/2019 z Source Converters
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)
-
7/30/2019 z Source Converters
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)
-
7/30/2019 z Source Converters
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)
-
7/30/2019 z Source Converters
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
-
7/30/2019 z Source Converters
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)
-
7/30/2019 z Source Converters
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)
-
7/30/2019 z Source Converters
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)
-
7/30/2019 z Source Converters
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)
-
7/30/2019 z Source Converters
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)
-
7/30/2019 z Source Converters
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)
-
7/30/2019 z Source Converters
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)
-
7/30/2019 z Source Converters
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)
-
7/30/2019 z Source Converters
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)
-
7/30/2019 z Source Converters
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)
-
7/30/2019 z Source Converters
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)
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