ELEC4614 Power Electronics
Lecture 13 DC-DC (Boost) Converter 13-1 F. Rahman
Lecture 13 - Boost DC-DC Converters
Step-Up or Boost converters deliver DC power from a lower voltage DC level (Vd) to a higher load voltage Vo.
(a)
Circuit during ton Circuit during toff
Figure 13.1. (a) The basic boost converter during ton & toff
+
Vd
iL L
+ vL - C RVo
+Vd
iLL
+ vL - C RVo
R (Load)
+
iD
+ vL
iL
Vo CVd
ic
D
Io
id
T
Vd
Vd V0 iL
0
LV
IR 1 D
0
on
s
tD
T
toff = (1-D)Ts
Ts
ton toff
vL
Io
ELEC4614 Power Electronics
Lecture 13 DC-DC (Boost) Converter 13-2 F. Rahman
Analysis of the boost converter with continuous conduction mode of inductor current (CCM)
From
T i( Ts s )
L0 i( 0 )
v dt L di 0 (13.1)
Vd ton + (Vd V0)toff = 0 (13.2)
o
d
V 1
V 1 D
(13.3)
so that Vo > Vd for this converter. Theoretically, Vo when D = 1.
Assuming the converter to be lossless
Pd = VdId = VoIo
0
d
I1 D
I (13.4)
0 0 d
L d 2
I V VI I
1 D R(1 D) R(1 D)
(13.5)
Note that for the boost converter, the average inductor current IL and the average input current Id (not the average output current as in the buck converter) are the same.
ELEC4614 Power Electronics
Lecture 13 DC-DC (Boost) Converter 13-3 F. Rahman
Boundary between continuous/discontinuous conduction
Figure 13.2 Inductor current and voltage waveforms with continuous conduction (CCM) of inductor current.
When conduction is just continuous,
d
LB L max on1 1 V
I i t2 2 L
ds
os
V1D T
2 L
VD( 1 D )T
2L
(13.6)
The inductor current for operation at the boundary of CCM and DCM as a function of D is plotted in figure 13.3.
Vd
d oV V
ton toff t
vL iLmax
iL
sT
ELEC4614 Power Electronics
Lecture 13 DC-DC (Boost) Converter 13-4 F. Rahman
Figure 13.3. Continuous-discontinuous boundary for the boost converter.
From (13.4) IOB = 2o sV T
D(1 D )2L
(13.7)
It can be shown that ILB and IoB become maximum for D = 1
2 and 1
3 . The corresponding maximum values are:
s o
LB maxT V
I8L
(13.8)
and s o
oB maxT V2
I27 L
(13.9)
I L B m a xI o B m a x
0 .5 0 .3 3
D
1 .0
0 .7 5
0 .2 5
ELEC4614 Power Electronics
Lecture 13 DC-DC (Boost) Converter 13-5 F. Rahman
From 13.5
d d sL
L max L 2
V V DTii I
2 2LR( 1 D )
(13.10)
d d sL
L min L 2
V V DTii I
2 2LR( 1 D )
(13.11)
At the boundary of continuous and discontinuous conduction,
Lmini 0
d d s d
2s
V V DT V D
2L 2LfR( 1 D )
(13.12)
2
sD(1 D ) R
Lf2
for operation at the boundary of CCM and DCM operation of inductor current.
ELEC4614 Power Electronics
Lecture 13 DC-DC (Boost) Converter 13-6 F. Rahman
DCM operation with constant Vd
Figure 13.4 Inductor current and voltage waveforms with
discontinuous conduction.
From T i( Ts s )
L0 i( 0 )
v dt L di 0
d s d o 1 sV DT V V T 0
o 1
d 1
V D
V
(13.13)
Because Pd = Po,
0 1
d 1
I
I D
(13.14)
The average inductor current IL is also the average input current Id, thus
Vd iL
Ts ton 1 sT
vL
Vd Vo
ELEC4614 Power Electronics
Lecture 13 DC-DC (Boost) Converter 13-7 F. Rahman
d
d s 1V
I DT ( D )2L
(13.15)
From (13.14), d s
0 1V T
I D2L
(13.15a)
Using (13.13), (13.15a) and (13.9)
1
2o o 0
d d oB max
V V I4D 1
27 V V I
(13.16)
The range of required variation in D to keep Vo constant at a specified level when the inductor current is discontinuous is indicated in figure 13.5.
Figure 13.5 Vo – D characteristic with load; for constant Vo and variable Vd.
IoBmax
0.5
D
1.0
0.75
0.25
Io, Amps
Vo = constantd
o
V0.25
V
d
o
V0.5
V
d
o
V0.75
V
ELEC4614 Power Electronics
Lecture 13 DC-DC (Boost) Converter 13-8 F. Rahman
Note that during discontinuous conduction, the energy
stored in the inductor at the end of each ton, i.e., 2L max
1Li
2 ,
is transferred to the load circuit which includes the capacitor C. If the load is not able to absorb this energy, the capacitor voltage must rise, leading possibly to too high capacitor voltage. This can be avoided if Vo is controlled (i.e., regulated) during every switching cycle (Ts). Output Voltage Ripple
During ton, the diode current is zero. The load current is maintained by the capacitor. Assuming continuous conduction of load current and also assuming that the average load current is Io, which is also the average diode current, the change in the load voltage, Vo, during ton is given by
o s o s
oI DT V DTQ
VC C RC
(13.17)
o s
o
V DT
V RC
(13.18)
The above analysis is valid only for continuous conduction mode and when D min oi I or ID
ELEC4614 Power Electronics
Lecture 13 DC-DC (Boost) Converter 13-9 F. Rahman
iDID = Io
DTs (1-D)Ts
Vo
Vo
t
ton toff
Ts
Q
Q
0
Figure 13.6 Waveforms of output voltage, capacitor current and diode current ripple.
ELEC4614 Power Electronics
Lecture 13 DC-DC (Boost) Converter 13-10 F. Rahman
Example: design of a boost converter Given, Vd = 12 V; Vo = 30 V; R = 50 ; fs = 25 kHz
(i) Find L so that operation is in CCM for the load specified. What L would you choose for DCM operation up to the maximum (specified) load?
(ii) Find iL
(iii) Find C for Vo/Vo < 10%
(iv) Sketch the voltage and current waveforms of the switch and the diode.
on
s
tD
T toff = (1-D)Ts
+
iD
T
+ vL
iL
Vo CVd R
Load
ic
D
Io
Vd
Vd V0 iL0
0V
IR
0
Ts
ton toff
ELEC4614 Power Electronics
Lecture 13 DC-DC (Boost) Converter 13-11 F. Rahman
Solution:
o
d
V 1
V 1 D
; D = 0.6
6
ss 1040f1T s.
At the continuous /discontinuous boundary,
2
s min
D( 1 D ) RLf
2
Lmax = 96 H
Alternatively, the DC output current IoB = 30/50 = 0.6A.
s2o
oB TD1DL2
VI ; from 13.7.
L
106.57 6 A
For operation in CCM with Io ≥ IoB = 0.6A, L = 96µH
Normally, we choose L < 96 H, say, 85 µH, to ensure DCM for the maximum (50Ω in this example) load. With L = 96µH,
(ii) A3DTL
VI s
dL
(iii) s
3o o
so o
DT D 0.6C 48 F
V V 25 10 50 0.01R f R
V V
ELEC4614 Power Electronics
Lecture 13 DC-DC (Boost) Converter 13-12 F. Rahman
(vi)
A5.1D1R
V
D1R
V
D1
II
2doo
L
d s
L maxV DT
I 1.5 3 A2L
d s
LminV DT
& I 1.5 0 A2L
Sketch of iT , vT, iD and vD are left for you as an exercise. Boost Converter Gain
Note that for a boost converter,
o
d
V 1
V 1 D
With continuous conduction, Vo approaches as D → 1. This does not happen in practice because parasitic elements, such as the switch, diode and inductor resistances and the capacitor ESR, prevent the rise of Vo when D approaches unity. Note also that as D → 1, iL approaches .
ELEC4614 Power Electronics
Lecture 13 DC-DC (Boost) Converter 13-13 F. Rahman
o
d
V
V
D 1.0
1
1 D
0
With cont. conduction
With disc. conduction
Figure 13.7 Boost converter gain characteristic. The sign of practical boost converter gain becomes negative as D approaches unity. Because of this and the other problems mentioned above, the boost converter is normally operated with duty cycle less than 0.85 or so. Also, the small-signal transfer function of the boost converter in CCM exhibits a zero in the right-half s-plane. This makes it difficult to obtain good closed-loop performance and stability. Because of these reasons, the boost converter is often operated in DCM to ensure stable operation and fast dynamics of control.
ELEC4614 Power Electronics
Lecture 13 DC-DC (Boost) Converter 13-14 F. Rahman
Buck-Boost Converter
The output DC voltage of a buck-boost converter can be lower or higher than the input DC voltage, with reversed polarity with respect to the input DC voltage.
D
C R L
T
Vd iL Vo
+ Io id
iD
Figure 13.8 Buck-boost converter circuit
Operation with continuous inductor current
during ton during toff
Figure 13.8 Buck-boost converter operating modes
RC L Vd
iL
+ vL _
io
Vo
+
RC LVd
iL
+vL_
io
Vo
+
Vd Li
0V
sT
ton = DTs toff
t
vL IL= Id + Io
0
ELEC4614 Power Electronics
Lecture 13 DC-DC (Boost) Converter 13-15 F. Rahman
d s 0 sV DT V (1 D )T 0
0
d
V D
V 1 D
(13.19)
From power balance:
d d 0 0V I V I (13.20)
0 d
d 0
I V 1 D
I V D
(13.21)
Boundary between cont/disc conduction
Figure 13.9. Inductor voltage and current waveforms
s d
LB L maxDT V1
I i2 2L
(13.22)
IL,max
t ton= DTs
iL,
Ts
(1-D)Ts
Vd
vL,
Vo
IL = ILB
0
ELEC4614 Power Electronics
Lecture 13 DC-DC (Boost) Converter 13-16 F. Rahman
Because the average capacitor current is zero
0 L dI I I (13.23)
Using 13.19 and 13.22
s o
LBT V ( 1 D )
I2L
(13.24)
Using 13.20 and 13.23 and 13.24
2
s ooB
T V 1 DI
2L
(13.25)
From (24) and (25), ILB,max and IoB,max occur for D = 0, hence,
s o
LB,maxT V
I2L
(13.26)
and s o
oB,maxT V
I2L
(13.27)
Also from (24) and (25),
LB LB,maxI I 1 D (13.28)
2oB oB,maxI I 1 D (13.29)
ELEC4614 Power Electronics
Lecture 13 DC-DC (Boost) Converter 13-17 F. Rahman
Figure 13.10 Continuous-discontinuous boundaries for Io and IL for constant Vo.
Now d oD
I I1 D
(13.30)
L 0 d d d d1 D
I I I I I I / DD
(13.31)
Also (from 31) 2
0d d d L
VV I V DI
R
From 13.19 2
0 dL 2
d
V V DI
V DR R( 1 D )
(13.32)
Now L
L max Li
i I2
D
ILB,/ILB,max
IoB/IoB,max 1.00.5
0.5
1.0
ILB,/ILB,max
IoB/IoB,max
Vo = constant
ELEC4614 Power Electronics
Lecture 13 DC-DC (Boost) Converter 13-18 F. Rahman
d d s
2
V D V DT
2LR 1 D
(13.33)
d d s
L min 2
V D V DTi
2LR 1 D
(13.34)
For Lmini 0
d d
2s
V D V D
2LfR( 1 D )
(13.35)
2
s min(1 D ) R
Lf2
(13.36)
for operation at the boundary of CCM and DCM.
ELEC4614 Power Electronics
Lecture 13 DC-DC (Boost) Converter 13-19 F. Rahman
Operation with discontinuous conduction
Figure 13.11. Inductor current and voltage waveforms with discontinuous conduction.
From Ts
L0
v dt 0,
d s o 1 sV DT V T 0
so that o
d 1
V D
V (13.37)
and from Pd = Po
o 1
d
I
I D
(13.38)
From the above figure
dL s 1
VI DT D
2L (13.39)
Vd iL
ton = DTs s1T
vL
Votoff = (1-D)Ts
0
Ts
ELEC4614 Power Electronics
Lecture 13 DC-DC (Boost) Converter 13-20 F. Rahman
Using 13.23, 13.37, 13.38 and 13.39 it can be shown that, with discontinuous mode of operation, D as a function of Vo and Io is given by
o o
d oB,max
V ID
V I
(13.40)
where IoBmax is given by 13.27 and oo
VI
R .
When Vo is kept constant, the required range of variation of D when the inductor current is discontinuous is indicated in the figure 13.12.
13.12 Buck-boost converter characteristic for constant Vo.
D
IoIoB,max½ IoB,max
0.5
1.0
d
o
V0.33
V
d
o
V1.0
V
d
o
V3.0
V
Vo = constant
ELEC4614 Power Electronics
Lecture 13 DC-DC (Boost) Converter 13-21 F. Rahman
Buck-Boost Converter Gain
As for the boost converter, the voltage-gain of this converter remains constant for low D and it increases sharply as D 1. Too high a gain is prevented by the parasitic elements in the switch and the inductor.
Figure 13.13.
o
d
V
V
D1.0
D
1 D
0
ELEC4614 Power Electronics
Lecture 13 DC-DC (Boost) Converter 13-22 F. Rahman
Output Voltage Ripple
Figure 13.14. Output voltage and diode current ripples and capacitor current
Assuming continuous conduction and considering that the ripple current in iD flows through the capacitor, the output voltage ripple is found from
o s o s
oI DT V DTQ
VC C R C
(13. 34)
o s s
o
V DT TD
V RC
(13.35)
where = RC = time-constant of the capacitor-load circuit.
iDID = Io
DTs (1-D)Ts
Vo Vo
t
ton toff
Ts
Q Q
0