Fundamentals of PWM Dc-to-Dc Power Conversion Dynamic Performance of PWM Dc-to-Dc Converters

Fun

dam

enta

ls o

f PW

M D

c-to

-Dc

Pow

er C

onve

rsio

n

Dynamic Performance ofPWM Dc-to-Dc Converters

2

Performance of PWM Dc-to-Dc Converters

Power

stage

Control

Load

Dc-

to-D

c C

on

vert

er P

erf

orm

anc

e

● Static performance:

● Dynamics performance

Stability Frequency-domain response:

Time-domain response:

3

Buck Converter ExampleS

tab

ility

PWM rampV

ref 4.0VV

20 s0V

3.8V

470 F

0.0540 H

16V

0.1

1

4

Stability of Buck ConverterS

tab

ility 1.5 2.0 2.5 3.0 3.5 4.0 4.5

0

2

4

6

8

2

3

4

5

6

i L(t

) [A

]

Time [ms]

v O(t

) [V

]

Unstable operation

Stable operation

5

Loop GainS

tab

ility

v ( s )s

i ( s )o

d ( s )

v ( s )o

conv ( s )Fm - Fv(s)

Gvd(s)

Zp(s)

Gvs(s)

Tm

mT ( s )

6

Stability of Buck ConverterS

tab

ility

0.1 1 10

-180

-150

-120

-90

-60

-30

0 -40

-20

0

20

40

Pha

se [

deg]

Frequency [kHz]

Mag

nitu

de [

dB]

Stable compensation

Stable compensation

-3 -2 -1 0 1 2 3

3

2

1

-1

-2

-3

Stable design

Unit circle

Re

Im

Bode plot of loop gain Polar plot of loop gain

7

Input-to-Output Frequency Response:

Power

stage

Control

Sv

Input

Ov

Output

● Input-to-output transfer function:

: Laplace transformation of ac component of

: Laplace transformation of ac component of

(s)ov ( )ov t

(s)sv ( )sv t

Fre

que

ncy-

Do

ma

in P

erfo

rma

nce

Cri

teri

a

8

Audio-SusceptibilityT

ime

-Do

ma

in P

erfo

rma

nce

Cri

teri

a

v ( s )s

i ( s )o

d ( s )

v ( s )o

conv ( s )Fm - Fv(s)

Gvd(s)

Zp(s)

Gvs(s)

Tm

0.01 0.1 1 10 100-70

-60

-50

-40

-30

-20

Mag

nitu

de [

dB]

Frequency [kHz]

Audio-susceptibility

9

Input-to-Output Frequency Response

-60

-40

-20

0

20

Ma

gn

itud

e[d

B]

Buck converter

with 0.25D Sv

ˆsv

20 Vov

Sv

20

5

ov

Fre

que

ncy-

Do

ma

in P

erfo

rma

nce

Cri

teri

a

10

Load Current-to-Output Transfer Function:

Power

stage

Control

oi

Loadcurrent

oV

● Load current-to-output transfer function: :

: Laplace transformation of ac component of

: Laplace transformation of ac component of

(s)ov ( )ov t

(s)oi ( )oi t

Fre

que

ncy-

Do

ma

in P

erfo

rma

nce

Cri

teri

a

11

Output ImpedanceT

ime

-Do

ma

in P

erfo

rma

nce

Cri

teri

a

v ( s )s

i ( s )o

d ( s )

v ( s )o

cv ( s )Fm - Fv(s)

Gvd(s)

Zp(s)

Gvs(s)

Tm

0.01 0.1 1 10 100

-80

-60

-40

-20

0

Mag

nitu

de [

dB]

Frequency [kHz]

Output impedance

12

Load Current-to-Output Frequency Response

-60

-40

-20

0

20

Buck converter

with 0.5D 10 V

5

ov

1 ˆoi

Tim

e-D

om

ain

Per

form

an

ce C

rite

ria

Output impedance

13

Step Load Response

● Transient response of the output voltage due to step change in the input voltage

Tim

e-D

om

ain

Per

form

an

ce C

rite

ria

Power

stage

Control

10 A

5 A

14

Output Impedance and Step Load ResponseT

ime

-Do

ma

in P

erfo

rma

nce

Cri

teri

a

0.0 0.5 1.0 1.5 2.03.6

4.0

4.4

v O(t

) [V

]

Time [ms]

0.01 0.1 1 10 100

-80

-60

-40

-20

0

Mag

nitu

de [

dB]

Frequency [kHz]

Output impedance Step load response

15

Step Input ResponseT

ime

-Do

ma

in P

erfo

rma

nce

Cri

teri

a

● Transient response of the output voltage due to step change in load current

Power

stage

Control

24 V

20 V

Control

16

Audio-Susceptibility and Step Input ResponseT

ime

-Do

ma

in P

erfo

rma

nce

Cri

teri

a

0 1 2 3 4 53.5

4.0

4.5

Time [ms]

v O(t

) [V

]

0.01 0.1 1 10 100-70

-60

-50

-40

-30

-20

Mag

nitu

de [

dB]

Frequency [kHz]

Audio-susceptibility Step input response

17

Definition of Stability

● Stability of transfer function T(s) or stability of an LTI system having T(s) as its transfer function

2 10 1 1 1

20 1 2

2 2 21 2 1 1 1

( )T(s)=

( )

( ) ( )( ) 2 ( ) ( )( )

( ) 0 :

Roots of (s)=0 :

nn

nn

b b s b s b s N s

D sa a s a s a s

D s s a s a s s

D s

D

● T(s) or LTI system is stable if and only if all the roots of the characteristic equation are located in the left-half plane(LHP) of the s-plane

Sta

bili

ty D

efin

itio

n

18

Nyquist (Stability) Criterion

● Nyquist(Stability) Criterion: Graphical method to determine the number of RHP roots in 1+Tm(s)=0

N Z P

Im[T( )]j

mRe[T ( )]j( 1,0)

Nyq

uist

Cri

teri

on

19

Application Example

● 2 1

0 1 1 12

0 1 2

T(s)=

nn

nn

b b s b s b s

a a s a s a s

● Characteristic equation

20 1 2

22

0 1

0

1+ 0

nn

nn

a a s a s a s

a s a s

a a s

Nyq

uist

Cri

teri

on

20

Stability Analysis of PWM Converters

( )Sv s

( )Oi s

( )d s

( )vsG s

( )pZ s

( )vdG s

mF vF

( )Ov s

( )

( )

O

S

v s

v s

Nyq

uist

Cri

teri

on

21

Nyquist Analysis on 1+Tm(s)=0Im[T(j )]

mRe[T (j )]( 1,0)

m

m

: number of encirclements of (-1,0) point

: number of RHP roots in 1+T (s) =0

: number of RHP poles in T (s)

N Z P

N

Z

P

Nyq

uist

Cri

teri

on

22

Absolute Stability

Im

Re

Im

Re( 1,0) ( 1,0)

Im

Re( 1,0)

Im

Re

( 1,0)

Stable Unstable

Nyq

uist

Cri

teri

on

23

Stability Analysis Using Bode Plot

mT mTmT

mTmT

mT

0dB0dB 0dB

180 180 180

Nyq

uist

Cri

teri

on

24

Marginally Stable Buck ConverterM

arg

ina

lly S

tab

le B

uck

Co

nve

rte

r

0.1 1 10

-180

-135

-90

-45

0 -40

-20

0

20

40

Phas

e [d

eg]

Frequency [kHz]

Mag

nitu

de [

dB]

-6

-5

-4

-3

-2

-1

1

2

3

4

5

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Bode plot of loop gainPolar plot of loop gain

25

Marginally Stable Buck ConverterM

arg

ina

lly S

tab

le B

uck

Co

nve

rte

r

0.0 0.5 1.0 1.5 2.0 2.5

3.0

3.5

4.0

4.5

5.0

Time [ms]

Vol

tage

[V

]

3

3

2 2 2 10 1 180 1

1 2 2 10 0

m c m c m

m

T ( j ) T ( j f ) T ( j )

T ( j )

26

Conditionally Stable SystemM

arg

ina

lly S

tab

le B

uck

Co

nve

rte

r Re

Iv

Ov

Region A

Im

I oRegion A: Large slope of v -v curve Largegain Stable

27

Effect of Gain and Phase Delay

m( )

T (s)( )( )

mK

s

Re

lativ

e S

tab

ility

Im

Re

Larger gain

More phase delay

Im

Re

28

Gain Margin

● Gain margin : the amount of gain increase that can be added to before the system becomes unstable,

mT

mT

Re

Im

(-1,0)6dB

mT

0dB

Sta

bili

ty M

arg

ins

29

Phase Margin

mT

Re

Im

mT

0dB

( 1,0)

PM

● Phase margin : the amount of phase delay that can be added to before the system becomes unstable,

mT

Sta

bili

ty M

arg

ins

30

Gain Margin and Phase Margin

( 1,0)

( K,0) Re

Im

-180

| |mT

mT

0 dB

Sta

bili

ty M

arg

ins

31

Stability Margins and Closed-Loop Performance

mPolar plot of T (s) mLocation of roots in 1+T (s) 0

mIm[T ]

mRe[T ]

j

Smallstabilitymargins

Nearness toimaginary axis

Sta

bili

ty M

arg

ins

and

Clo

sed

-Loo

p P

erf

orm

an

ce

● Proximity to (-1,0) point

Small stability margin

Nearness of poles to imaginary axis:

32

Buck Converter ExampleS

tab

ility

Ma

rgin

s an

d C

lose

d-L

oop

Pe

rfo

rma

nce

0.1 1 10 100

0.1 1 10 100

0.1 1 10 100

-180

-135

-90

-45

0

-40

-20

0

20

40

Pha

se [

deg]

Frequency [kHz]

Mag

nitu

de [

dB]

0.1 1 10 100

0.1 1 10 100

°PM = 60

°0

°0

°60

-2 -1 1 20

1

2

-1

-2

°PM = 60°45

°30

°15

°0

Bode plot of loop gain Polar plot of loop gain

33

Buck Converter ExampleS

tab

ility

Ma

rgin

s an

d C

lose

d-L

oop

Pe

rfo

rma

nce

0.1 1 10 100

0.1 1 10 100

0.1 1 10 100

-180

-135

-90

-45

0

-40

-20

0

20

40

Phas

e [d

eg]

Frequency [kHz]

Mag

nitu

de [

dB]

0.1 1 10 100

0.1 1 10 100

°PM = 60

°0

°0

°60

°PM = 60

0

45

30

15

Bode plot of loop gain

0.1 1 10 100-80

-60

-40

-20

0

20

Mag

nitu

de [

dB]

Frequency [kHz]

°PM = 60

°45°35

°15

°0

Output impedance

34

Buck Converter Example: Step Load Response

2.0 2.5 3.0 3.5

3.8

4.0

4.2

3.8

4.0

4.2

3.8

4.0

4.2

3.8

4.0

4.2

3.8

4.0

4.2

v(t)

[V

]

Time [ms]

v(t)

[V

]

v(

t) [

V]

v(t)

[V

]

v(

t)[V

]

Sta

bili

ty M

arg

ins

and

Clo

sed

-Loo

p P

erf

orm

an

ce