Switch-mode converters as feedback systems A B m e v o v - - PDF document

1

Mor M. Peretz, Switch-Mode Power Supplies

[8-1]

Control of switch-mode converters

Mor M. Peretz, Switch-Mode Power Supplies

[8-2]

Control objectives

Produce control command to

• Regulate the output voltage
• Obtain zero or small steady-state (DC) error
• Quick response to reference changes
• Fast recovery
• Immunity to input and load changes
• Reasonable overshoot

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Mor M. Peretz, Switch-Mode Power Supplies

[8-3]

Switch-mode converters as feedback systems

m



e



• Power stage is a Switching System (non-linear)
• Compensator is an analog or digital controller
• Linear control theory based design  small signal response

A B

 

f v v

• e

Compensator

 

f d vo Power stage

 

f V d

e

Modulator

Mor M. Peretz, Switch-Mode Power Supplies

[8-4]

Control of PWM converters disturbances in voltage mode

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Mor M. Peretz, Switch-Mode Power Supplies

[8-5]

Voltage regulation

( power ) Duty cycle Feedback ( power ) Power stage Vin VO RO CO

Driver k error amp VO PWM modulator Ve Vref

Mor M. Peretz, Switch-Mode Power Supplies

[8-6]

PWM modulator

Ve +

• Vv

comp Vp Ve  

p v t v s

V V t V V T   

 

p v

• n

t e v s

V V t V V V T    

 

e v

• n
• n

s p v

V V t D T V V    

Practical Don max  0.8  0.9

D 1 Vv Vp Ve

Oscillator

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Mor M. Peretz, Switch-Mode Power Supplies

[8-7]

Sawtooth generator

Mor M. Peretz, Switch-Mode Power Supplies

[8-8]

Transfer functions

m



e



t t Zoom Ve D t d D

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Mor M. Peretz, Switch-Mode Power Supplies

[8-9]

Control of PWM converters disturbances in voltage mode

• ut
• ut

vin in d

• ut

Z i A v dA v   

• ut

i in v

• ut

d

d v A

 



• ut

i d in

• ut

vin

v v A

 



d in v

• ut
• ut
• ut

i v Z

 



LG 1 Z i LG 1 A v LG 1 LG K 1 v v

• ut
• ut

vin in t ref

• ut

     

d M t

BA K K LG 

Mor M. Peretz, Switch-Mode Power Supplies

[8-10]

Dynamics of feedback systems

Block diagram division B A ) f ( LG 

A – known (power stage + divider) B – unknown (have to be designed)



 S

in

S

f

S

+

• ut

S

K

P

1

H

A

B

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Mor M. Peretz, Switch-Mode Power Supplies

[8-11]

) s ( LG 1 ) s ( A ACL  

• The system is unstable if {1+LG(s)} has roots in the

right half of the complex plane.

• Nyquist criterion is a test for location of {1+LG(s)}

roots.

• Nyquist criterion is normally translated into the Bode

plane (frequency domain)

LoopGain test

Nyquist Criterion

Mor M. Peretz, Switch-Mode Power Supplies

[8-12]

LoopGain test

f f



 db +180 |LG|

   

f A f  In negative feedback systems At f0 ) 180 ( 180

• 

 

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Mor M. Peretz, Switch-Mode Power Supplies

[8-13]

f f



 db

• 180

A 

m

 180o already substracted 1 A  

• 1

| A |

• 1

| A | m

180 ) 180 (        

   

Bode plot

Mor M. Peretz, Switch-Mode Power Supplies

[8-14]

Graphical representation of BA conventional method

 Tedious – need to re-plot BA  Analysis (not design) oriented  Requires iterations

A

] dB [ B

3

f

B

] dB [ A

2

f

1

f ] Hz [ f ] Hz [ f ] dB [ AB

AB

3

f

2

f

1

f ] Hz [ f

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Mor M. Peretz, Switch-Mode Power Supplies

[8-15]

] dB [ A ] Hz [ fo

A

1  B A

1 BA  1 BA  BA ) f ( LG  B 1

Graphical Representation of BA

20log(BA) B 1 20log A 20log   1 A B B 1 20log 20logA    

Mor M. Peretz, Switch-Mode Power Supplies

[8-16]

Possible compensations

 

dB d Vo Log(f) dec dB 40

• dec

dB 20

• A

1/B 1/B 1/B

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Mor M. Peretz, Switch-Mode Power Supplies

[8-17]

Possible compensations

 1

• m

90  

• m

45  

• m

90  

• m

45  

• m

90  

• m

45  

f

dec db dec db

20 

dec db

40 

dec db

60 

db

A

s s s s u u

db

B 1

dec db

20 

dec db dec db

20 

dec db

40 

Mor M. Peretz, Switch-Mode Power Supplies

[8-18]

Overshoot and Q in Closed Loop in Response to step in Sin

t f Overshoot Q Overshoot

m

• 50

Excitation ACL

• m

m m

50 for sin cos Q     

• m

45 target Design  

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Mor M. Peretz, Switch-Mode Power Supplies

[8-19]

Extracting the power stage control-to-output transfer function

L

• C
• R

in

V

• V

L

I

in

E

b

G S L

• C
• R

in

V D

• V

in

• L

E V V  

in in

• n

E V D  

b

• n

L

G I D  

Mor M. Peretz, Switch-Mode Power Supplies

[8-20]

Linearization

R V(in) I(3)

• ut

( ) ( ) (3) V out V in I  

( ( )) ( ( )) ( ( )) ( ) (3) ( ( )) ( (3)) V out V out d V out v in i V in I       ( ) ( ) ( ) ( ) (3) ( ) (3) V out V out V out v in i V in I      

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Mor M. Peretz, Switch-Mode Power Supplies

[8-21]

SPICE Linearization (AC Analysis)

(3) ( ) F I V in   

R V(in) I(3)

• ut

R

• ut

) 3 ( i ) 3 ( I F

• 

       

) in ( V ) in ( V F

• 

       

( ) (3) F V in I   

Mor M. Peretz, Switch-Mode Power Supplies

[8-22]

Buck linearization

L

• C
• ut

in

V

L

I

b

G

in

E

in

• R

D I G

L b 

D V E

in in 

• ut
• in

D E        

• in

in

V E         L Vin ) in ( v ) d ( V

0 

) d ( v ) in ( V

0 

in

• b

D G        

• L

b

I G         ) d ( v ) L ( I

0 

) L ( i ) d ( V

0  L

I Co Ro

D

V VAC d R

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Mor M. Peretz, Switch-Mode Power Supplies

[8-23] f f p

R C 2 1 f  

in f

• R

R A 

Possible phase compensation schemes Lag network

Mor M. Peretz, Switch-Mode Power Supplies

[8-24]

Design example

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Mor M. Peretz, Switch-Mode Power Supplies

[8-25]

Lag network

Frequency 100Hz 10KHz 1.0MHz 10Hz p(-V(out1))

• 100d
• 50d

0d SEL>> db(V(out1))

• 40

40 E1 V(%IN+, %IN-)*1E6 EVALUE OUT+ OUT- IN+ IN- R1 1k 0V C1 10n R2 100k V1 1Vac 0Vdc 0V 0V

• ut1

Mor M. Peretz, Switch-Mode Power Supplies

[8-26]

2

( .) 1 2

• OL

L f f f in

A A ampl f C R R A R    

f  1 f

dec db

20  f1 f2 A0 A2

Lag – Lead network

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Mor M. Peretz, Switch-Mode Power Supplies

[8-27]

Lag-Lead network

Frequency 100Hz 10KHz 1.0MHz 10Hz p(-V(out2))

• 100d
• 50d

0d SEL>> db(V(out2)) 50 100 E3 V(%IN+, %IN-)*1E6 EVALUE OUT+ OUT- IN+ IN- C2 10n R9 1g 0V

• ut2

0V R3 10k 0V R4 1k V2 1Vac 0Vdc

Mor M. Peretz, Switch-Mode Power Supplies

[8-28]

Double zero compensation scheme

2 1 3 2

R R C C  

β β 1

dec dB 20                                         

3 3 1 1 1 2 2 3

C R 1 C R 1 C R 1 C R 1        π π π π 2 2 2 2

3 2C

R f 2 1  π OL

A

2 3

R R

3 2C

R f 2 1  π

1 3

R R

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Mor M. Peretz, Switch-Mode Power Supplies

[8-29]

Double Zero

Frequency 100Hz 10KHz 1.0MHz 10Hz p(-V(out3))

• 100d

0d 100d SEL>> db(V(out3)) 20 40 C5 10n C3 10n 0V R7 100k V3 1Vac 0Vdc 0V R8 1g

• ut3

R5 1k E2 V(%IN+, %IN-)*1E6 EVALUE OUT+ OUT- IN+ IN- R6 100k C4 100p 0V

Mor M. Peretz, Switch-Mode Power Supplies

[8-30]

The relationship to PID compensators

 

dB V V

e c

Log(f) 1

f

2

f B

   

s ω s ω s K K K s s K K v v

z2 z1 I d d I p e c

       

 

dB V V

e c

Log(f) 1

f

2

f 1 / B

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Mor M. Peretz, Switch-Mode Power Supplies

[8-31]

The relationship to PID compensators

 

dB d Vo Log(f) dec dB 40

• dec

dB 20

• A

1/B 1/B 1/B

Mor M. Peretz, Switch-Mode Power Supplies

[8-32]

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Mor M. Peretz, Switch-Mode Power Supplies

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Mor M. Peretz, Switch-Mode Power Supplies

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