Advanced Simulation Techniques for Stability Analysis of Converters

5/27/2018 Advanced Simulation Techniques for Stability Analysis of Converters

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STRATEGIC TECHNOLOGY GROUPExperts in Power Electronics, Motor Drives

and Digital Control of PowerStrategicTechGroup.com

This document contains confidential and privileged information. Any unauthorized review, use,disclosure, or distribution is prohibited.

Disclaimer

The materials are provided "As Is without any warranties orclaims of fitness for any particular purpose. Sale and use forcommercial purposes is prohibited. The user shall always validate

any model, especially if critical engineering decisions are to bemade based on simulation results. The Author is not liable for anydirect or consequential damages resulting from the use of thismaterial. Strategic Technology Group, 2010, all rightsreserved.


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Advanced Simulation

Techniques

Part 6:Stability Analysis


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3

Advanced Simulation Techniques

Table of Contents

Open Loop Transfer Function

The Cross Over Frequency

Stability Criteria

Various RC Networks

Type 1, 2 and 3 Amplifier

K Factor Technique

Example of a Stabilized Buck Converter


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4


Buck Converter Open Loop Transfer Function

Gpwm

Error amplifier

H(s)

D1

SW1

+1

-2 D

3-

+

V1

{Vin}

IN 1 2L1

{L}

C1{C}

R1{R}

Out

Gate_Drv

Gate_Drv

{R_Comp}

C3

{C_Comp}

0

10k

0

+

-

OUT

{Vref}

H( s)

GPWM

Er r or Ampl i f i er


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5


Buck Converter Open Loop Transfer Function

+=

+=

)0(1

)0(1

)0()0(1

)0()0(1

T

T

HG

HG

V

V

ref

out

1/ +Vref

VoutG(o) H(o)-

Error amplifier Power stage


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6


Small Signal Representation

wmH(s)G(s)GpT(s)=Duty_Cycle

G( s)

IN2

V2

{Vin}

1 2L2

{L}

C2{C}

R2{R}

Out2

Duty_Cycle

0.5

a c

d p

SW2

Duty_Cycle

Gpwm

{R_Comp}

C5

{C_Comp}

0

10k

0

+

-

OUT

{Vref}


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7


Voltage Injection

{R_Comp}

C5

{C_Comp}

0

10k

0

+

-

OUT

{Vref}

{Gpwm}

V2

{Vin}

1 2L2

{L}

C2{C}

R2{R}

Duty_Cycle

a c

d p

SW2Out3

Verr

Duty_Cycle

V61Vac

0.5Vdc

Vinjector

Vinjector

10k

)V

V(20.logT(s)

inj

err10=


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8


Voltage Injection

Gain-40db/decade

0

-180

H(s) T(s)

+

+180

0

-20db/decade


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9


Method#1

injectorV

errorV

db{R_Comp}

C6

{C_Comp}

0

10k

0

+

-

OUT

{Vref}

{Gpwm}

V3

{Vin}

1 2L3

{L}

C3{C}

R3{R}

Duty_Cycle

a c

d p

SW3Out3

Duty_Cycle

Verr

V81Vac

0.5Vdc

Vinjector

Vinjector

10k


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10


Method#2

in

out

V

Vdb

Verr

1Vac

Vin

Ac=1

+Duty_cycle

{R_Comp}

C7

{C_Comp}

0

10k

0

+

-

OUT

{Vref}

{Gpwm}

V4

{Vin}

1 2L4

{L}

C4{C}

R4{R}

Duty_Cycle

Out3a c

d p

SW4

10k


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11


The Cross Over Frequency: Example

outCf

1

out

ESRC

fout

C

outI

pV

c

c

2

2

Voltage Undershoot

With RHP ZeroSet fc


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12


Stability Criteria

0

01

+

Vout

Verr

GH

High DC Gain by G(s) Reduce the Static Error Reduce output

Impedance Improve the Input line

Rejection

Phase Margin=45-80

Gain Margin=10-15db Phase Margn0


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13


Stability Criteria

Dc Gain0 Gain

Cross over frequency

0

0 Phase180

When Phase=0 Gain0

Gain=0 Phase0

0

-180


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14


Bode Plot for RC Network

R

c

Vout

Vin

0o

o

w

w1tan

ws1

1log20

V

V10

in

out

=

+= )(.

0

-20db/Decade


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15


Bode Plot for CR Network

C

R

VinVout

o

o

ws

ws

Vin

Vout

+=1

-

+20db/Decade

+90o

0o


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16

Advanced Simulation Technique

Right Half Plane Zero

Occurs in Indirect Energy Transfer Topologies

Flyback Boost

Buck Boost

Position Changes with the Duty Cycle

Select a Cross Over Frequency to be of 1/3rd of RHPZ


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STRATEGIC TECHNOLOGY GROUPExperts in Power Electronics, Motor Drivesand Digital Control of PowerStrategicTechGroup.com

17


Right Half Plane Zero

)ww(120.log

)w

s(1T(s)

o10

o

=

=

0o

-90o


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18


When to Use Type 1 Amplifier

Power Stage Phase Shift is Small

Roll of the Gain far away from the ResonantFrequency

Brings the Largest Overshoot in the Presenceof Sudden Load

Widely Used in PFC Application


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19

Advanced Simulation TechniquesType 1 Amplifier, No Phase Boost Just dcGain

C

+

-

OUT

U6

OPAMPR

Vin

VVref

2.5

V1Vac

0VdcR

+Vout

11CsR1G(s) =


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20


Type 1 Amplifier, The K Factor

0

G

fc

1

db

K=1Log(f)

1c

20

Gf

GRf2

1C

10Gc

=

=

Type 1 Derivation


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21



Power Stage Phase Shift Lags Down of -90

Phase Boost of ESR Must be Canceled

Use for Current Mode (CCM) and VoltageMode(DCM)


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22


Type 2 Amplifier Can Boost the Phase

+

+

+=

=

+++

+=

10

1

0

1

1

1

)(

pp

z

211P

12z

21

1

2211

12

ss

s

sG

)C(CR

1

CR1

)CC

CCsR)(1C(CsR

CsR1G(s)

2

C

C

R

+

-

OUT

U6

OPAMPR

Vin

VVref

2.5

V1Vac

0VdcR

+Vout


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23


Type 2 Amplifier Can Boost the Phase

+

=

21

212

11

CC

CCR

p

But if C2


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0

G

k

k

Log(f)1

0

Type 1 Reference

Pole 1Zero 1

1

=

+

+==

=+

00

0

0

arctanarctan

1

1

arg))(arg(

arctan)arg(

pz

p

z

ff

ff

ff

ff

boostfT

a

bjba

Let us assume we place a zero at frequencyf/k and a pole at a frequency kf

Type 2 Derivation

k

fc kfccf


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+=

+=

+=

=

+

=

452

tan

452

)arctan(

90)arctan(2

)arctan(90)arctan(

90

1

arctan)arctan(

1arctan)arctan(

boostk

boostk

k

kkBoost

xx

kkBoost

Type 2 Derivation


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26



Type 2 Derivation

12

2

21

12

2

)1(

2

190

18090

Cf

kR

kCC

GkRfC

PSPMBoost

PMBoostPS

c

c

=

=

=

=

+=+


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Power Stage Phase Shift Lags Down to-180 CCM Voltage mode Buck Boost Type


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28


Type 3 Amplifier Associated with anIntegrator

22p

33p

11p

31z

12z

PPP

ZZ

CR1

CR

1

CR1

CR

1

CR

1

)s)(1s(1s

)s)(1s(1

G(s)

2

1

0

2

1

210

21

=

=

=

=

=

++

++

C

C

R

+

-

OUT

U6

OPAMP

R

R Vin

VVref

2.5

V1Vac

0VdcR

+

C

Vout

AssumingC2


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0

GDoubleZero

k

k

1

1Log(f)

+1

Type 1 reference

Double Pole

2

454

tan

1arctan)arctan(2

1arctan)arctan(

+=

=

=

boostk

kkBoost

kkBoost

Type 3 Derivation

k

fc cf

k

fc


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Type 3 Derivation

3

3

13

12

21

12

2

1

1

2

)1(

2

1

RkfC

k

R

R

Cf

kR

kCC

GRfC

c

c

c

=

=

=

=

=


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31


Stabilizing a Voltage-Mode Buck Converterwith k Factor

Generate an Open-Loop Bode Plot

Select a crossover Frequency and a Phase Margin

Read the Bode Plot at the Crossover Frequency

Select the Amplifier type

Apply Formulas

Sweep the Open-Loop Gain with the Above Values

Vary the output Capacitor(s) ESRs Step Load the Output


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Stability Analysis: Demonstration

Demonstration#24: Buck Converter Compensated

Documents

Advanced Simulation Techniques for Stability Analysis of Converters