DC to DC Converter (chopper)

1
DC to DC Converter (chopper)
VG
Buck dc to dc converter
output voltage input voltage
Is
IL
t
2T
TKT
0
T
KT
R
VG
Pulse Width Modulation PWM
L
VL
Vs
E
ID
K is the duty cycle 0 K 1 T is the period
2
(No Transcript)
3
For KT ?t T iL - E/R B exp-(t-KT)/?) iL
Imax _at_ ?t KT Imax - E/R B B Imax
E/R iL - E/R (Imax E/R) exp-(t-KT)/?) iL
Imin _at_ ?t T Imin - E/R (Imax E/R)
exp-(T-KT)/?.. (2) Substituting from (1) into
(2) Imin - E/R (Vs E) / R
Imin (Vs-E) / R exp-(KT/?) E/R
exp-(T-KT)/?) Imin1 exp-(T /?) Vs/R
exp-(T-KT)/ ? -exp-(T /?) E/R
1exp-(T-KT)/ ?- exp-(T /?) -exp (T-KT)/?)
Assuming continuous load current
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Imin1 exp-(T /?) Vs/R exp-(T-KT)/ ?
exp-(T /?) E/R 1-exp-(T-KT)/ ?- exp-(T
/?) - exp
(T-KT)/?) Imin vs exp-(T-KT)/ ? -
exp-(T /?) - E R 1
exp-(T/ ?) R Imin Vs
exp(KT/ ?) -1 - E R exp (T/?)
1 R Substituting for Imin in (1) Imax(Vs
E) Vs exp(KT/ ?) -1 - E Vs-E
exp-KT/?) R R exp(T/?)
-1 R R Imax Vs 1 1 -
exp-(KT/?) - exp-(KT/?) - E R
exp(T/?) - 1
R Imax Vs exp(T/?) - 1 1 - exp-(KT/ ?) -
exp(T-KT)/? exp-(KT/ ?) - E R
exp(T/?) - 1
R Imax Vs
exp (T/?) - exp (T-KT/ ?) - E R
exp (T/?) - 1 R
Imax Vs 1 exp-(KT/ ?) - E R
1 exp- (T/?) R
VLav K Vs
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For discontinuous load current conduction
iL
For 0 t KT i (Vs-E) / R A exp-(t/?) i
0 _at_ t 0 0 (Vs-E) / R A A - (Vs-E)
/ R i (Vs-E) / R 1 exp-(t/?) i Imax
_at_ t KT Imax (Vs-E) / R 1 exp-(KT/?)
For KT t tx i - (E/R) B
exp-(t-KT)/? i Imax _at_ t KT Imax (E/R)
B B Imax E/R
t
tx
Ttx
0
KT
T
TKT
vL
Vs
E
tx
Ttx
VLav K Vs (T-tx)/T E
B (Vs-E) / R 1 exp-(KT/?) E/R Vs/R
Vs/R exp-(KT/?) E/R exp-(KT/?) i - E/R
Vs/R Vs/R exp-(KT/?) E/R exp-(KT/?)
exp-(t-KT)/? i - E/R Vs/R exp-(t-KT)/ ?
Vs/R exp-(t/?) E/R exp-(t/?)..(1) i 0 _at_
t txx 0 - E exp-(tx/?) Vs exp(KT/?)
Vs E
E exp-(tx/?)
Vs exp(KT/?) Vs E
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exp (tx/?) Vs/E exp(KT/?) Vs/E 1
tx ? ln Vs/E exp(KT/?) 1 1
To calculate the critical duty cycle ratio Kcri,
corresponding to critical conduction by
substituting into (1) by K Kcri , tx T and
i 0 0 - E/R Vs/R exp-(T-KcrT)/ ?
Vs/R exp-(T/?) E/R exp-(T/?) Vs
exp-(T-KcriT)/ ? E Vs exp-(T/?) - E
exp-(T/?) Vs exp(KcriT/?) E exp(T/?) Vs -
E exp(KcriT/?) E/Vs exp(T/?) 1 - E/Vs Kcri
?/T ln 1 E/Vs (exp(T/?) -1)
iL
Imax
t
0
KcriT
T
TKcriT
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Load characteristics of a dc to dc converter
VL
K1
K0.8
K0.6
Continuous conduction
Discontinuous conduction
K0.4
K0.2
IL
8
vG1
2-quadrant dc to dc converter
Is
IL
D2
R
VG1
t
T1
T2
L
Vs
VL
D1
E
VG2
vG2

• No discontinuous load current since there is a
  pathe
• for negative load current through T2 if
  conducting.
• 3 cases are present
• ILmax ILmin are both ve
• ILmax is ve ILmin is ve
• ILmax ILmin are both ve

t
Complementary control for T1 T2
IL
VL
9
iL
Imax
1st case ILmax ILmin are both ve
Imin
Is
IL
D2
vL
R
VG1
Vs
T1
T2
L
Vs
VL
D1
E
VG2
In this case the circuit acts exactly like a
single quadrant dc to dc converter, no current
flows through T2 or D2 . During the interval
from 0 to KT, power is transferred from Vs to the
load. During the interval from KT to T, power
stored in L is dissipated in the R and the
forward resistance of D1 if considered.
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iL
1st case Imax Imin are both ve
Imax
i2
i2
i1
i1
Imin
i1
i1
D2
R
VG1
T1
T2
L
Vs
VL
D1
E
VG2
i1
i1
i2
D2
R
VG1
T1
T2
L
VL
Vs
D1
E
i2
VG2
i2
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i3
iL
i3
i3
D2
Imax
i4
i4
i1
i1
t
R
VG1
i3
i3
Imin
T1
i2
T2
i2
L
Vs
VL
0
KT
T
TKT
D1
E
VG2
i3
i1
i1
D2
i3
R
VG1
T2
T1
L
Vs
VL
D1
E
VG2
i1
i1
i2
D2
R
VG1
T1
T2
L
Vs
D1
E
i2
VG2
i2
2nd case Imax is ve Imin is -ve
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3rd case Imax Imin are both -ve
i1
i1
i1
D2
R
VG1
T1
T2
L
Vs
VL
D1
E
0
VG2
TKT
KT
T
i1
t
Imax
i2
i2
i1
i1
i1
Imin
Note that the load must be active such that it
can support negative load current with a positive
supply voltage Vs.
13
4-quadrant dc to dc converter
IL
T1
T3
D2
D4
VG1
VG3
Vs
VL
T2
T4
D1
D3
VG2
VG4
Operation in the 1st and 4th quadrants
Operation in the 2nd and 3rd quadrants
vG1
vG3
vG1
vG3
IL
t
t
t
t
T
TKT
T
TKT
0
KT
KT
0
KT
T
TKT
0
T
TKT
vG3
0
KT
vG2
vG2
vG4
t
t
t
KT
T
TKT
t
0
0
KT
T
TKT
T
TKT
KT
0
0
KT
T
TKT