PE_ENEL37_4

7/31/2019 PE_ENEL37_4

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1

ENEL371S2

Power Electronics 1

Lecture 4 – Buck ConverterDiscontinuous Conduction

Department of ECE

University of Canterbury2012

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Continuous and DiscontinuousConduction

Read Chapter 7, Mohan etc.

CONCEPT

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Continuous and Discontinuous

i L

t

i L(min)

i L(max)

i L(min)

Continuous Conduction

i L

t

Average

Value

i L(min)

i L(max)

i L(min)

Current Conduction State

of Inductor (EnergyStorage Device) in

switching cell play animportant role in DC-DC

converter circuit analysis

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Buck RegulatorDiscontinuous Conduction Mode (DCM)

Read Chapter 7, Mohan etc.

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Basic Requirements

Things we need to calculate

1. Output/Input voltage ratio

2. Boundary between CCM and DCM Things we need to know

1. Draw the inductor current waveform

2. Draw the inductor voltage waveform3. Draw three circuit states (ON/OFF/IDLE)

For the steady-state analysis of BuckRegulator in DCM

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Assumptions

voR C

L

vC

i L

v L

iC io

To simplify the analysis, for steady-state buckvoltage regulator, it is reasonable to assume

1. vC is reasonably ripple free (~1% ripple) and io is

ripple-free (io= I o) too

2. Since average capacitor current ic is zero, averagevalue I L of inductor current i L and average value I o

of resistor current io are equal

0 0

1 10, 0s s

o o L

T T

L C

s s

i I I

v dt i dt T T

C ov v

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Buck Converter (Regulators)

Two Tasks:

1. Boundary betweenCCM and DCM

2. Step-down Ratio inDCM

:on on

s on off

duty ratiot t

k T t t

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Boundary btw CCM and DCM

I L: average inductor current

I o: average output/load current

When inductor current i Lgoes to zero at the end ofswitch off period, circuitreaches the boundarybetween CCM and DCM.

Average inductor current

at this boundary I LB :

(max) (min) (max)

12

1 1

2 2

LB L

L L L

I i

i i i

CCM

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(max)

1 1

1 121 1

2 2

d o

L

d s s LB OB

s

LB L L

v v I k k k

v f L f L I I k k f L

I i I

With given T s, v d , v o, L and k,if average inductor current I L(output current I o) becomes

less than I LB, then i L willbecome discontinuous.

(max)

10 0.5,

8

d LB

LB

s

vdI k I

dk f L

CCM

Inductor current ripple

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1

1

2

d

LB OB

s

v I I k k

f L

min

11 1 1

2 2 2

d o

s o o s s

v v R L k k k k

f I I f f

o d v kv

With given T s, v d , v o, I O and k, if the inductorvalue L is less than Lmin, i L will become

discontinuous.

o ov I R Resistive Load

Can be used for the design of inductor value L

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Circuit States in DCM

charging

discharging

0 idle

d o

L o

v v

v v

Steady State

0 0

( ) 0, ( ) 0

T T

L C v t dt i t dt

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Step-Down Ratio in DCM

1

0

0

1

1

( ) 0

(

0 1

) 0

,

ss

s

k k T kT

d o o

k

o

d

T

L

T

v v dt v t dt

v k

v t d

k k DCM v k

t

k

1, 1o

d

v

k k k CCM v

Discharging duration k 1T s is still unknown, how to get it ?

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o

d

v

v

Increased gain due to discontinuous conduction;

but variable gain makes control difficult.

TypicalGainResistorLoadGain

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1 1

(max)

Discharging inductor :1 sk k T

o s

L okT

v k T I v dt

L L

(max) 1

1

2o L

I I k k

1 1

2

o

o

s

v k k k I

f L

2

1 1

20s o

o

f LI k kk

v

Quadratic equation can be

easily solved for k 1.

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Quadratic Equation

http://en.wikipedia.org/wiki/Quadratic_equation

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1

2

1 1

2

1 1

2 2

1

0

1

20

20

8 81 1

2 2

o

o s

s o

o

s sk

v R

I f Lk kk

f LI Rk kk

v

f L f Lk k k k k k

R R

Scenario 1: With Constant Resistance Load R

1 2

2

8

o

d s

v k k

v k k f Lk k

R

Step-down

Ratio in DCM

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1 0

21 1

2

1

20

81

2

s o

k

o

s o

o

f LI k kk v

f LI

k k k v

Scenario 2: If vo is fixed and I o is variable

1 2

28

o

d s o

o

v k k v k k f LI

k k v

Step-downRatio in DCM

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Scenario 2k

DCM

CCM

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1 1

1 1 1

1

1

2 2

2 2

oo

os d s o

o

s s d o

d

v k k k I v k k k f L v k k f LI

I k f L f L v k v k

v k k

Scenario 3: If vd is fixed and vo is variable

12

o

s od

d

v k k f LI v k k

k v k

Step-downRatio in DCM

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k =1.0

The duty-ratio of 0.5 has the highest value of the critical current

Scenario 3

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Lecture 4 - Summary

Boundary between CCM and DCM

Step-down Ratio in DCM – It willchange with the load ( I o or R)

0

1) ( ) 0

T

Lv t dt

2) L Li i

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Homework #4

Download homework #4 questions for

Lecture 4 fromLearn/ENEL371/Term3/Lecture4

How to calculate the capacitor/outputvoltage ripples of Buck Converter in

DCM ? Do homework #4

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Contact Information:

Dr. Keliang Zhou

Office: Electrical A512

Email: [email protected]

Phone: 3642240 Ext.6240

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PE_ENEL37_4