Page 290 - 2024-Vol20-Issue2

286 | Abdulabbas & Salih

- C vC2 R +

? (r+R) 1 1 ?
L L L
L r vD L - 0 - (1 - do) do

iL1 -+ r ? 0 - (r+R) 1 do - 1 (1 - do) ?
A=? L L L ?
iD iL2 +
+ iload ? 1 (1 - do ) - 1 do 0 0 ?
L C ?
Vdc ?
-
C1 - 1 do 1 (1 - do) 0 0
vC1 C C
vin
? R (1 - do) 1? ? iL1 ?
L
R - L
L
R B = ? (1 do) 0? , x = ? iL2 ? u= iload ,
- ? ? vC1 ? vin
- 1 (1 - do ) 0 ? ?
? C ? ?

- 1 (1 - do ) 0 vC2
C
(a)
- C vC2 R +

L vD L y= iL1 , C= 1000 , D= 0
vC1 0010 0
r -+ r

iL1 iL2 + The dynamic state variables and inputs can be expressed as
+ iload the sum of the steady state and perturbations of the variables
from the equilibrium point(x = X + x˜ ) as following.
vin C1 Vdc
vC1 -

R
-

d (2)
(X + x˜) = A(X + x˜) + B(U + u)
(b)
dt

Fig. 2. The equivalent circuit of the quasi-Z-source inverter ? IL1 ?
(qZSI) can be represented in two states: (a) the active state
and (b) the shoot-through state. Where X = ? IL2 ? is steady-state state variables of the
? VC1 ?

? ?

VC2

qZSI.

? ˜IL1 ?
1˜ L2
as iin) as the output variables for the analyzed system. In x˜ = ? v˜ C1 ? is perturbations of the state variables from the
order simplification, assume that C = C1 = C2, L = L1 = L2, ? ?
the stray resistances of inductors r = r1 = r2, the Equivalent
Series Resistances (ESR) of capacitors R = R1 = R2. The ? ?
shoot-through interval T0 refers to the duration during which
both switches are turned on simultaneously. The active in- v˜ C2

terval T1, on the other hand, is the duration when only one equilibrium point of the qZSI. U = Iload is steady-state
switch is turned on in one leg. The switching period T is the Vin

sum of T0 and T1. The shoot-through duty ratio, denoted as input of the qZSI. u = 1˜ load is perturbations of the input
D0, is calculated by dividing T0 by T. The state space model v˜ in
of the qZSI in the two intervals (active and shoot-through) can
from the equilibrium point of the qZSI. and d0 = D0 + d˜0 is
be given in Eq. (1).
the shoot-through duty cycle of the qZSI.

The dc-side model of the qZSI can be derived using state

space averaging, as seen in Eq. (3).

? -(r + R) 0 (D0 - 1) D0 ? ? iL1 ?
-(r + R) D0
?0 0 (D0 - 1) ?? iL2 ? +
-Do 0 ?? vC1 ?
? (1 - Do) (1 - Do) 0 ??
? ?

dx -Do 0 vC2
= Ax + Bu
(1) ? R (1 - Do) 1 ?
dt
y = Cx + Du ? R (1 - Do) 0 ? iload =0
? ?
where ? (D0 - 1) 0 ? vin

(D0 - 1) 0

(3)