Lecture 12 - DC-DC Buck Converter 12 - DC-DC Buck...ELEC4614 Power Electronics Lecture 12 - DC-DC Buck Converter 12-3 F. Rahman Figure 12.4 Implementation of the buck converter circuit

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



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



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



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



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



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



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



Figure 12.10 Static characteristic of an IGBT. VGS < 20V max.



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 .



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



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



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)



+ 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



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.



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



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



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.



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



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.



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



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



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



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.



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



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%.

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Lecture 12 - DC-DC Buck Converter 12 - DC-DC Buck...ELEC4614 Power Electronics Lecture 12 - DC-DC Buck Converter 12-3 F. Rahman Figure 12.4 Implementation of the buck converter circuit