IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 02 Issue: 08 | Aug-2013, Available http:www.ijret.org 36
ON-state, the equivalent circuit shows that the diode is short circuit and Q is open circuit as presented in Fig -3. At this
state, inductor L1 and L2 are in discharge phase. Energy in L1 and L2 are discharged to capacitor C1 and output part,
respectively. As a result, inductor current iL1 and iL2 is decreasing linearly.
To ensure the inductor current iL1 and iL2 increases and decreases linearly on respective state, the converter must
operate in Continuous Conduction Mode CCM. CCM means the current flows in inductors remains positive for the entire
ON-and-OFF states. Fig -4 shows the waveform of iL1 and iL2 in CCM mode. To achieve this, the inductor L1 and L2
must be selected appropriately. According to [3], the formula for selection the inductor values for dynamic model zeta
converter are as follow:
− +
+ −
D R
D r
R r
Df R
D L
C L
1 1
2 1
1 2
2 1
+ −
R r
f R
D L
L2 2
1 2
1
Fig -1: Dynamic model of zeta converter
Fig -2: Equivalent zeta converter circuit when Q turns on
Fig -3: Equivalent zeta converter circuit when Q turns off
Fig -4: iL1 left and iL2 right waveform in CCM [3]
3.1 State-Space Description of Zeta Converter
ON-state Q turns on Voltage across inductor L1 can be written as:
S L
L L
L
v i
r dt
di L
v +
− =
=
1 1
1 1
1
Or
1 1
1 1
1
L v
i L
r dt
di
S L
L L
+ −
= Voltage across inductor L2 can be written as:
Z C
C S
C C
C L
C C
C L
L L
i R
r R
r v
v R
r R
v i
R r
R r
r r
dt di
L v
+ +
+ +
− +
+ +
+ −
= =
2 2
2 2
1 2
2 2
1 2
2 2
2
Or
Z C
C S
C C
C L
C C
C L
L
i R
r L
R r
v L
v R
r L
R v
L i
R r
R r
r r
L dt
di +
+ +
+ −
+
+
+ +
− =
2 2
2 2
2 2
2 1
2 2
2 2
1 2
2 2
1 1
1
Current flows in capacitor C1 can be written as:
2 1
1 1
L C
C
i dt
dv C
i −
= =
Or
2 1
1
1
L C
i C
dt dv
− =
Current flows in capacitor C2 can be written as:
Z C
C C
L C
C C
i R
r R
v R
r i
R r
R dt
dv C
i +
− +
− +
= =
2 2
2 2
2 2
2 2
1 Or
Z C
C C
L C
C
i R
r C
R v
R r
C i
R r
C R
dt dv
+ −
+ −
+ =
2 2
2 2
2 2
2 2
2
1 Output voltage can be written as:
Z C
C C
C L
C C
O
i R
r R
r v
R r
R i
R r
R r
v +
− +
+ +
=
2 2
2 2
2 2
2
Equation 11 to 14 are combined and rewritten in matrix form as:
11
12
13
14
15
IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
__________________________________________________________________________________________
Volume: 02 Issue: 08 | Aug-2013, Available http:www.ijret.org 37
+ −
+ +
+ −
+ −
+ −
+ +
+ −
− =
Z S
C C
C C
C L
L
C C
C C
C C
L L
C C
L L
i v
R r
C R
R r
L R
r L
L v
v i
i
R r
C R
r C
R C
R r
L R
L R
r R
r r
r L
L r
dt dv
dt dv
dt di
dt di
2 2
2 2
2 2
1 2
1 2
1
2 2
2 2
1 2
2 2
2 2
1 2
2 1
1
2 1
2 1
1 1
1 1
1 1
Equation 15 which is the output equation can be written in matrix form as:
+ −
+
+
+ =
Z S
C C
C C
L L
C C
C O
i v
R r
R r
v v
i i
R r
R R
r R
r v
2 2
2 1
2 1
2 2
2
The state-space matrices of zeta converter for ON-state are therefore:
+ −
+ −
+ −
+ +
+ −
− =
R r
C R
r C
R C
R r
L R
L R
r R
r r
r L
L r
A
C C
C C
C C
L L
2 2
2 2
1 2
2 2
2 2
1 2
2 1
1 1
1 1
1 1
+ −
+ =
R r
C R
R r
L R
r L
L B
C C
C
2 2
2 2
2 2
1 1
1 1
+ +
= R
r R
R r
R r
C
C C
C 2
2 2
1
+ −
= R
r R
r E
C C
2 2
1
OFF-state Q turns off Voltage across inductor L1 can be written as:
1 1
1 1
1 1
1 C
L L
C L
L
v i
r r
dt di
L v
− +
− =
=
Or
1 1
1 1
1 1
1
1 1
C L
L C
L
v L
i r
r L
dt di
− +
− =
Voltage across inductor L2 can be written as:
Z C
C C
C L
C C
L L
L
i R
r R
r v
R r
R i
R r
R r
r dt
di L
v +
+ +
−
+
+ −
= =
2 2
2 2
2 2
2 2
2 2
2
Or
Z C
C C
C L
C C
L L
i R
r L
R r
v R
r L
R i
R r
R r
r L
dt di
+ +
+ −
+ +
− =
2 2
2 2
2 2
2 2
2 2
2 2
1 Current through capacitor C1 can be written as:
1 1
1 1
L C
C
i dt
dv C
i =
= Or
1 1
1
1
L C
i C
dt dv
= Current through capacitor C2 can be written as:
Z C
L C
C C
i R
r R
i R
r R
dt dv
C i
+ −
+ =
=
2 2
2 2
2 2
Or
Z C
C C
L C
C
i R
r C
R v
R r
C i
R r
C R
dt dv
+ −
+ −
+ =
2 2
2 2
2 2
2 2
2
1 Output voltage can be written as:
Z C
C C
C L
C C
O
i R
r R
r v
R r
R i
R r
R r
v +
− +
+ +
=
2 2
2 2
2 2
2
Equation 16 to 19 are combined and rewritten in matrix form as:
+ −
+ +
+ −
+ +
−
+
+ −
− +
− =
Z S
C C
C C
C L
L C
C C
C C
L L
C C
C L
L
i v
R r
C R
R r
L R
r L
L v
v i
i R
r C
R r
C R
C R
r L
R R
r R
r r
L L
r r
L
dt dv
dt dv
dt di
dt di
2 2
2 2
2 2
1 2
1 2
1 2
2 2
2 1
2 2
2 2
2 2
1 1
1 1
2 1
2 1
1 1
1 1
1 1
1
The output equation in Equation 20 can be written in matrix form as:
+ −
+
+
+ =
Z S
C C
C C
L L
C C
C O
i v
R r
R r
v v
i i
R r
R R
r R
r v
2 2
2 1
2 1
2 2
2
The state-space matrices of zeta converter for OFF-state are therefore:
16 17
18
19
20
IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
__________________________________________________________________________________________
Volume: 02 Issue: 08 | Aug-2013, Available http:www.ijret.org 38
+ −
+ +
−
+
+ −
− +
− =
R r
C R
r C
R C
R r
L R
R r
R r
r L
L r
r L
A
C C
C C
C L
L C
2 2
2 2
1 2
2 2
2 2
2 1
1 1
1 2
1 1
1 1
1
+ −
+ =
R r
C R
R r
L R
r B
C C
C
2 2
2 2
2 2
+ +
= R
r R
R r
R r
C
C C
C 2
2 2
2
+ −
= R
r R
r E
C C
2 2
2
Equation 8 is revisited and the state-space matrices derived previously for ON and OFF-state are used, the weighted
average matrices are:
D A
D A
A −
+ =
1
2 1
+ −
+ −
− +
− +
+ +
+ −
− −
+ −
− =
R r
C R
r C
R C
D C
D R
r L
R L
D R
r L
R r
Dr r
R r
L D
L r
D r
A
C C
C C
C C
L C
L C
2 2
2 2
1 1
2 2
2 2
2 2
1 2
2 1
1 1
1
1 1
1 1
+ −
+ =
− +
= R
r C
R R
r L
R r
L D
L D
D B
D B
B
C C
C
2 2
2 2
2 2
1 2
1
1
+ +
= −
+ =
R r
R R
r R
r D
C D
C C
C C
C 2
2 2
2 1
1
+
− =
− +
= R
r R
r D
E D
E E
C C
2 2
2 1
1
3.2 Zeta Converter Steady-state
