Transcript

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


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