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ELEC4614 Power Electronics
Lecture 12 - DC-DC Buck Converter 12-1 F. Rahman
Lecture 12 - Non-isolated DC-DC Buck Converter
Step-Down or Buck converters deliver DC power from a higher voltage DC level (Vd) to a lower load voltage Vo.
Figure 12.1 The basic buck converter
The switch T connects the output terminals of the converter to Vd and 0V for time Ton in each switching period which remains constant (i.e., T is switched at a constant switching frequency, fs). The output DC voltage Vo is normally proportional to Ton. For DC power supplies with constant output voltage Vo, the turn-on time Ton is normally controlled in closed-loop in order to maintain Vo at a fixed level. For applications where Vo may be variable, Ton is varied accordingley.
vc
Vd
Vref
Controller
+
Vo
Vsense
Vo
ELEC4614 Power Electronics
Lecture 12 - DC-DC Buck Converter 12-2 F. Rahman
Duty cycle, on c
s ST
t vD ;
ˆT V s
s
1f
T
Figure 12.2 Pulse-width modulated (PWM) switching and output voltage waveforms.
fs 3fs2fs0
Vo
4fs 5fs
Figure 12.3 Frequency spectrum of vo
tonto f f to ff
T s
v c
V d
v o
STV̂
v o
ton
0
ELEC4614 Power Electronics
Lecture 12 - DC-DC Buck Converter 12-3 F. Rahman
Figure 12.4 Implementation of the buck converter circuit
The impedance of the capacitor C for fs must be small compared to the impedance of the load.
Load
L
VoVd
T
Controller
C
ELEC4614 Power Electronics
Lecture 12 - DC-DC Buck Converter 12-4 F. Rahman
The Power Transistor BJTs and Darlingtons
hfe < 10 for the BJT
VCEsat 1-2 volts
Turn off time a few hundred nsec to about 60 sec.
VBD up to 1400 V
n+ = 1019/cm3
n+ = 1019/cm3
n- = 1014/cm3
p = 1016/cm3
10 m
5-20 m
50-200 m
250 m
B E
C Figure 12.5 Vertical cross section of a npn power BJT
ELEC4614 Power Electronics
Lecture 12 - DC-DC Buck Converter 12-5 F. Rahman
Figure 12.6 vertical cross section of a npn BJT. Courtesy:
N. Mohan, Undeland & Robins
Figure 12.7 BJT (a) Symbol, (b) v-i characteristic, (c) idealized characteristic
ELEC4614 Power Electronics
Lecture 12 - DC-DC Buck Converter 12-6 F. Rahman
MOSFETs Very fast, toff 50 nsec - 500 nsec.
Rdson increases with 2.6
BDV ; typically, Rds 40 m for
a 500V, 15A device Turned-on and -off by VGS 5-20V
These devices are easily connected in parallel.
n+
n
p (body) p (body)n+ n+ n+n+
S G
D
Figure 12.8 Vertical section of an n-channel Power MOSFET
ELEC4614 Power Electronics
Lecture 12 - DC-DC Buck Converter 12-7 F. Rahman
Figure 12.9 N-Channel MOSFET (a) Symbol, (b) v-i characteristic, (c) Idealised characteristic.
IGBTs (Insulated Gate Bipolar Transistor)
Vsat 2-3 V
tq 1 sec
Ratings up to 3,000V, 3000A
Figure 12.10 Vertical cross section of an IGBT and IGBT
symbols
A
G
K
ELEC4614 Power Electronics
Lecture 12 - DC-DC Buck Converter 12-8 F. Rahman
Figure 12.10 Static characteristic of an IGBT. VGS < 20V max.
ELEC4614 Power Electronics
Lecture 12 - DC-DC Buck Converter 12-9 F. Rahman
Average voltage and current of capacitors In some power electronic circuits, capacitors act as reservoirs of energy. Consider the circuit of figure 12.11 in which a constant amplitude of current pulse is assumed to charge the capacitor for time t.
+
vc
C
icvc
tt = 0
ic
Figure 12.11
The capacitor voltage vc is given by
t
c C00
1v ( t ) idt v
C 12.1
The capacitor voltage will rise linearly with time.
In figure 12.14, a capacitor is connected between two circuits. Circuit 1 charges the capacitor with a constant amplitude of current during Ton while Circuit 2 discharges the capacitor, also with constant amplitude of current during time toff. The average voltage across the capacitor remains constant over the switching period
offons ttT .
ELEC4614 Power Electronics
Lecture 12 - DC-DC Buck Converter 12-10 F. Rahman
vo
Circuit 2
Circuit 1
ic
B
A
Vdc
ic
ton toff
Fig. 12.12
If vC0 at the end of a period is the same as at the beginning, the average current through the capacitor must be zero. In other words, Area A = Area B in figure 12.12. Average voltage and currents in inductors
In some circuits, inductors act as reservoirs of energy. Consider the circuit of figure 12.13 in which a constant amplitude voltage V is applied across the inductor L.
L
ViL
t t = 0
iL+
vL
V
Figure 12.13
The inductor voltage is given by
Ldi
v Ldt
so that, t
L00
1i Vdt i
L 12.2
ELEC4614 Power Electronics
Lecture 12 - DC-DC Buck Converter 12-11 F. Rahman
In figure 12.14, an inductor is connected between two circuits. Circuit 1 applies a constant amplitude voltage to L; iL increases linearly with time during ton. During toff, a negative but constant amplitude voltage is applied across the inductor, so that iL decreases linearly with time. The average current through the inductor remains constant around a mean (DC) value over the switching period.
Figure 12.14 If iL0 at the end of a period is the same as at the beginning, the average voltage across the inductor must be zero. In other words, Area A = Area B in figure 12.14.
iL V iIdc
Ckt 1
Ckt 2
L
B
A
ELEC4614 Power Electronics
Lecture 12 - DC-DC Buck Converter 12-12 F. Rahman
Analysis of the Step-Down (Buck) Converter in CCM
+ vL Vd
R (Load)
io
VoC
iLL
D
id
+ voi
vo
T
Figure 12.15. The basic buck converter topology
During 0 < t < ton, voltage across the inductor L is Vd - Vo; iL rises to ILmax. During ton < t < Ts, voltage across the inductor L is -Vo, and iL falls to ILmin. In the steady-state, the inductor current must return to ILmin at the end of the switching period Ts, and the integral of the inductor voltage (i.e., the DC voltage supported across the inductor) must be zero. In the following we assume that the output voltage ripple is negligible. We also assume that the inductor current is continuous throughout the switching period Ts. This is the so-called continuous conduction mode (CCM) of operation.
The voltage across the inductor L is
L
Ldiv L
dt (12.3)
Over one switching period Ts,
s L s
L
T i ( T )
L L0 i ( 0 )
v dt L di 0 (12.4)
ELEC4614 Power Electronics
Lecture 12 - DC-DC Buck Converter 12-13 F. Rahman
+ vL Vd
R (Load)
io
VoC
iLL
D
id
+ voi
vo
T
Vd
iL IL = Io
Vo
T
voi
t
t
Id id
0
0 toff ton
vL Vd - Vo
Figure 12.16 Buck converter waveforms
ELEC4614 Power Electronics
Lecture 12 - DC-DC Buck Converter 12-14 F. Rahman
on s
on
T To
d o0 t
V1V V dt dt 0
L L
on s
on
T T
d o o0 t
V V dt V dt 0 (12.5)
d 0 on 0 s onV V t V T t (12.6)
0 on
d s
V tD
V T (12.7)
Also, d 0P P or d d 0 0V I V I (12.8)
0 d
d 0
V ID
V I (12.9)
Note that the DC inductor current equals the DC output or load current for a buck converter. This follows from the assumption of constant Vo. Note also IL = Io is not the case with other DC-DC converters to be studied later.
The waveforms of figure 12.16 are for continuous conduction of current in the inductor, the so called CCM (continuous conduction mode) of operation. If the inductor current iL becomes discontinuous during toff, equations 12.7-12.9 do not hold, leading to a few problems.
ELEC4614 Power Electronics
Lecture 12 - DC-DC Buck Converter 12-15 F. Rahman
Boundary between Continuous-Discontinuous Conduction
Figure 12.17 Inductor voltage and current waveforms; with just continuous operation.
From figure 12.17,
d 0
LB Lmax on oBV V1 1
I i t I2 2 L
(12.10)
s ds sd 0 d d
T V D 1 DDT DT(V V ) V DV
2L 2L 2L
(12.11)
ILB = IoB
Vd - V0
-V0
Ts
(1-D)Ts
iLmax
vL
iL
ton = DTs
0
ELEC4614 Power Electronics
Lecture 12 - DC-DC Buck Converter 12-16 F. Rahman
ILB becomes maximum when 5.0D (this is found by differentiating ILB with respect to D and equating the derivative to zero). For D = 0.5,
s d
LB maxT V
I8L
(12.12)
And LB LBmaxI 4I D(1 D) (12.13)
Figure 12.20 Converter characteristics with duty-cycle and load
During normal operation, ILB should be smaller than the lowest load current, so that the converter operates in continuous conduction mode (i.e., in the linear mode with Vo = DVd). The minimum inductance L and the switching frequency fs for this condition of operation are obtained from the following consideration:
D
ILBmax Io
1.0
0.25
0.75
0.5
ILB locus
ELEC4614 Power Electronics
Lecture 12 - DC-DC Buck Converter 12-17 F. Rahman
From T i( T )s s
L
0 i( 0 )
vdt di 0
L (12.14)
d o o
s sV V V
DT ( 1 D )T 0L L
(12.15)
The first term in (12.15) is iL (rise) and the second term is iL (fall).
For a given load resistance R,
o o oLL smax
V V Vii 1 D T
R 2 R 2L
(12.16)
and o o oLL smin
V V Vii 1 D T
R 2 R 2L
(12.17)
At the boundary of continuous-discontinuous conduction,
iLmin = 0, so that
s min
1 D RLf
2
(12.18)
for operation at the boundary of CCM and DCM operation of inductor current.
s
1 D RLf
2
(12.19)
for operation with CCM.
ELEC4614 Power Electronics
Lecture 12 - DC-DC Buck Converter 12-18 F. Rahman
DCM Operation with constant Vd
In some applications, the output DC voltage is variable while the input DC voltage is maintained constant. If
o LBI I , then Li is discontinuous.
Figure 12.21 vL and iL waveforms with discontinuous conduction.
d 0 s 0 1 s(V V )DT V T 0 (12.20)
o
d 1
V D
V D
(12.21)
where 1D 1
Now 0
L max 1 sV
i TL (12.22)
and s 1 s
0 L max L max sDT T
I i i / T2 2
1Ts
Vd V0 iL
DTs2Ts
Ts
A
B V0
vL
ELEC4614 Power Electronics
Lecture 12 - DC-DC Buck Converter 12-19 F. Rahman
1
L max( D )
i2
0 1
1 sV D
TL 2
(using 22) (12.23)
d 1
1 s1
V DDT
L D 2
(using 21)
d
s 1V
T D2L
(12.24)
LB max 14I D (using 12) (12.25)
DI4
I
maxLB
01 (12.26)
20
2d0 LB max
V D1V D ( I / I )4
(using 21) (12.27)
Equation 12.27 shows that with DCM, the converter output Vo has a non linear relationship with D.
ELEC4614 Power Electronics
Lecture 12 - DC-DC Buck Converter 12-20 F. Rahman
Figure 12.22 Converter characteristics with discontinuous
conduction. Note that with DCM operation, V0 falls sharply with load when the inductor current is discontinuous. Note also that with discontinuous conduction, Vo/Vd ratio becomes higher than D, implying loss of voltage gain of the converter. Converter gain with continuous and discontinuous conduction (Vd = constant).
The voltage gain, Gc, of the buck converter is normally expressed as
o
c ddV
G VdD
(12.28)
o
d
V
V
ILBmax Io
1.0
0.25
0.75
0.5
D = 1.0
D = 0.75
D = 0.5
D = 0.25
ILB locus
Vd = constant
ELEC4614 Power Electronics
Lecture 12 - DC-DC Buck Converter 12-21 F. Rahman
Gc remains constant (= Vd) when the inductor current is continuous. It falls as the inductor current becomes more and more discontinuous.
Figure 12.23 Variation of converter gain with cont & disc conduction.
DCM operation with constant Vo
In many applications, such as power supplies, Vo is kept constant (by regulating the duty cycle D), when Vd varies over some range. From (12.11), at the boundary of continuous-discontinuous conduction,
s d s o
LB
T V D 1 D T V 1 DI
2L 2L
(12.29)
The average inductor current at the boundary of continuous-discontinuous conduction varies linearly with D as indicated by the dotted line of figure 12.24. It is maximum for D = 0 and zero for D = 1.
Cont. conduction
Disc. cond.
Io
Gc Vd
IoB
ELEC4614 Power Electronics
Lecture 12 - DC-DC Buck Converter 12-22 F. Rahman
s o
LB max
T VI
2L (12.30)
Figure 12.24 Converter duty-cycle and load characteristic
for constant Vo and variable Vd in cont and disc conduction.
d 0 s 0 1 s(V V )DT V T 0 (12.31)
o
d 1
V D
V D
(12.32)
where 1D 1
Now 0
L max 1 sV
i TL (12.33)
and s 1 s
0 L max L max sDT T
I i i / T2 2
IoILBmax=s oT V
2L
D
0.25
Vd/Vo = 1.25
Vd/Vo = 1.0
Vd/Vo = 2
Vd/Vo = 4
0.5
0.75
1.0
or IL
Vo = constant
ELEC4614 Power Electronics
Lecture 12 - DC-DC Buck Converter 12-23 F. Rahman
1
L max( D )
i2
0 1
1 sV D
TL 2
(using 12.32) (12.34)
o s d
1o
V T DV
2L V (using 12.33)
d
oB max 1o
VI D
V (12.35)
0
1max
o
oB d
I V
I D V (12.36)
Thus, when Vo is kept constant,
o o LB max
d o d
V I / ID
V 1 V / V
(12.37)
Figure 12.24 also indicates the range of variation of D required for maintaining Vo constant for varying Vd and Io, when the inductor current becomes discontinuous.
ELEC4614 Power Electronics
Lecture 12 - DC-DC Buck Converter 12-24 F. Rahman
Output voltage ripple of the buck converter (approximate analysis) The following analysis assumes continuous conduction.
Figure 12.25. Inductor current waveform For constant DC level of Vo, the filter capacitor can not carry any DC current. Thus, Ic = 0, and IL=I0 . iLripple = ic (12.38)
s L sL
0T I TIQ 1 1
VC C 2 2 2 8C
(12.39)
0
L sV
I ( 1 D )TL
(From 12.15) (12.40)
s0s
0 T)D1(L
V
C8
TV (12.41)
0
L maxi
R
VI 0
0 Q
Ts
Ts/2
IL/2
ELEC4614 Power Electronics
Lecture 12 - DC-DC Buck Converter 12-25 F. Rahman
2
0 sV ( 1 D ) T
8 LC
(12.42)
o
o
V
V
2sT( 1 D )
8 LC
(12.43)
22
c
s
f1 D
2 f
(12.44)
where c1
f2 LC
, is the cut-off frequency of the LC
filter.
Therefore, it is desirable to have fs fc ! Design considerations of the buck converter
High fs reduces the sizes of L and C.
The core of inductor L not to saturate for iLmax.
Sufficient L to maintain continuous conduction for the lowest load current.
C to limit 0
0
V
V
, typically, to less than 1%.