feedback amplifier stability

Ch. 8 Feedback – Part 5 1ECES 352 Winter 2007

Feedback Amplifier Stability

* Feedback analysis Midband gain with feedback

New low and high 3dB frequencies

Modified input and output resistances, e.g.

* Amplifier’s frequency characteristics

* Feedback amplifier’s gain

* Define Loop Gain as βfA

Magnitude

Phase

)()(1

)()(

)()(1

)()(

A

AA

sAs

sAsA

ff

ff

of

ofo A

AA

1

of

LLfHofHf A

A

11

'1'1 Iofoof

Iof

iif ARR

A

RR

A()

Ao

H HfLf L

Afo

)()()( sFsFAsA LHM

A

A

A

Re

Imtan 1

111

1

sin

180Re

Imtan

1

01

i

ii

iif

jjiiii

ii

iii

ii

A

A

AA

so

eeAA

ce

radiansA

Aand

A

i

Amplifier becomes unstable (oscillates) if at some frequency i we have

i

jeAA


* Amplifier with one pole Phase shift of - 900 is the maximum

Cannot get - 1800 phase shift. No instability problem

* Amplifier with two poles Phase shift of - 1800 the maximum

Can get - 1800 phase shift !* For stability analysis, we define two

important frequencies 1 is where magnitude of βfA goes to

unity (0 dB) 180 is where phase of βfA goes to -180o Instability problem if 1 = 180

Feedback Amplifier Instability

βfA(dB)

0 dB

1

00

- 450

- 900

- 1350

- 1800

180

11

HP

o

j

AA

2111

HPHP

o

jj

AA

1111

11

11801

A

A

AAf

General form of Magnitude plot of βfA

General form of Phase plot of βfA


Gain and Phase Margins

* Gain and phase margins measure how far amplifier is from the instability condition

* Phase margin

* Gain margin

* What are adequate margins? Phase margin = 500 (minimum) Gain margin = 10 dB (minimum)

0 dB

00

- 450

- 900

βfA(dB)

- 1350

- 1800

Phase margin

Gain margin

000

1801

1801

45)180(135

margin Phase

Example

margin Phase

dBdBdB

AA

AA

30)30(0

margin Gain

Example

margin Gain

1801

1801

1801

- 30

1

180

Φ(ω1)

Φ(ω180)


Numerical Example - Gain and Phase Margins

dBdBAA

dBAanddBA

and

sradxandsradx

xj

xj

A

dBdBAA

A

fofo

fofo

fo

fofo

fo

28)28(0)()(

ismargin gain so

28)(0)(

40)180(140

ismargin phase so

180140

/103/106

phase and magnitude of plots From

1031

1081

15)(

5.23)(15

1.0150

1801

1801

0001801

0180

01

7180

61

65

Phase margin

Gain margin

1

180

βfA(dB)

Phase Shift (degrees)

Note that the gain and phase margins depend on the feedback factor βf and so the amount of feedback and feedback resistors. * Directly through the value of βf. * Indirectly since the gain Ao and pole frequencies are influenced by the feedback resistors, e.g. the loading effects analyzed previously.

Pole 1Pole 223.5 dB


Amplifier Design for Adequate Gain and Phase Margins

* How much feedback to use? What βf ?* A change of βf means redoing βf A() plots!* Is there an alternate way to find the gain and

phase margins? YES Plot magnitude and phase of gain A()

instead of βf A() vs frequency Plot horizontal line at 1/ βf on magnitude plot

(since βf is independent of frequency) Why? At intersection of 1/ βf with A(), = 1

So this intersection point gives the value where ω = 1 !

Then we can find the gain and phase margins as before.

* If not adequate margins, pick another βf value, draw another 1/ βf line and repeat process.

* Note: βf is a measure of the amount of feedback. Larger βf means more feedback.

Recall for βf = 0, we have NO feedback

As βf increases, we have more feedback00 ff

o

ff Xso

X

X

11

1

AsoA ff

1

00

- 450

- 900

180

Phase margin

Gain margin 180

1

A(dB)

- 1350

- 1800

1/βf(dB)

0 dB

General form of Magnitude plot of A

General form of Phase plot of A

AA

AA

ff

1

A(ω180)

Φ(ω1)

Φ(ω180)


Example - Gain and Phase MarginsAlternate Method

dBdBA

dBAanddB

and

sradxandsradx

dBdBso

so

xj

xj

A

dBdBAA

of

of

f

ff

o

oo

28)8(20)()(/1

ismargin gain so

8)(20)(/1

40)180(140

ismargin phase so

180140

/103/106


2010log20)(1

101.0

111.0

1031

1081

150)(

5.43)(150

1801

1801

0001801

0180

01

7180

61

65

Phase margin

1

180

1/βf(dB)

A(dB)

These are the same results as before for the gain andphase margins .

43.5 dB

A(ω180)

Gain margin


Feedback Amplifier with Multiple Poles

* Previous feedback analysis Midband gain

New low and high 3dB frequencies

* Amplifiers typically have multiple high and low frequency poles.

Does feedback change the poles other than the dominant ones?

YES, feedback changes the other poles as well.

This is called pole mixing. This can affect the gain and phase

margin determinations ! The bandwidth is still enlarged as

described previously But the magnitude and phase plots

are changed, so the gain and phase margins are modified.

PnLLPLP

ZnLLZLZL sss

ssssF

....

....)(

21

21

)(1

)()(

)(1

)()(

A

AA

sA

sAsA

ff

ff

of

ofo A

AA

1

of

LLfHofHf A

A

11

A()

Ao

H HfLf L

Afo

)()()( sFsFAsA LHM

PnHHPHP

ZnHHZHZH

sss

sss

sF

1....11

1....11

)(

21

21


0

0

2

2

021

2

210

0

0

21

2

210

0

021

0

21

0

21

0

f

f

11

1

11

11

1

111

1111

1

11

1)(A

get weA(s)for ngSubstituti1

)(A

A

AAwhere

s

Q

s

A

A

s

A

s

A

A

ssA

A

Ass

A

ss

A

ss

A

sA

sAs

sA

sAs

ffo

oo

fo

fPHPHPHPHf

f

PHPHPHPHf

fPHPH

PHPH

f

PHPH

f

f

Feedback Effect on Amplifier with Two Poles

* Gain of a two pole amplifier

* Gain of the feedback amplifier

21

0

11

sA

PHPH

ss

A

New poles of the feedback amplifier occur when

mixing" pole" have weSo

4122

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are amplifier feedback theof poles new The

feedback ofamount the torelated alsobut

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are that sfrequencie pole new twoare thereSo

1

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001

2

21

21

21

21

21

222

2

QQQ

s

AQ

A

where

sQ

sors

Q

s

oo

f

PHPH

PHPH

ofPHPH

PHPH

o

ofPHPHo

oo

oo

Midband gain of feedback amplifier.

Gain of feedback amplifierhaving two new, different high frequency poles.


* Feedback amplifier poles

* For no feedback (βf = 0),

* For some feedback (βf Ao > 0), Q increases and 1- 4Q2 0 so poles move towards each other as βf increases.

* For βf such that Q = 0.5, then 1- 4Q2 = 0 so two poles coincide at

* For larger βf such that Q > 0.5, 1- 4Q2 < 0 so poles become complex frequencies.

Feedback Effect on Amplifier with Two Poles

21

21

21

21

2

1

1

4122

PHPH

ofPHPH

PHPH

o

ofPHPHo

oo

AQ

A

whereQQQ

s

)poles original(, 21

21

21min

PHPH

PHPH

PHPH

s

QQ

2221 PHPHo

Qs

Root-Locus Diagram

s = + jj

j


* Amplifier with some feedback ( βf Ao > 0 ), as βf increases 1 - 4Q2 0 so poles move towards each other. For sufficient feedback, Q = 0.5 and 1 - 4Q2 = 0 so the poles meet at the midpoint.

Frequency Response - Feedback Amplifier with Two Poles

Af(dB)

1PHf

2PH

Af(dB)

221 PHPH

2PHf1PH

Original amplifier

* For Q = 0.707, we get the maximally flat characteristic* For Q > 0.7, get a peak in characteristic at corresponding to oscillation in the amplifier output for a pulsed input (undesirable).

Original amplifier

Increasing feedback, poles meet for Q = 0.5

221 PHPH

Q>0.7

Original poles before feedback


Frequency Response of Feedback on Amplifier with Two Poles

21

21

21

21

2

1

1

4122

PHPH

ofPHPH

PHPH

o

ofPHPHo

oo

AQ

A

whereQQQ

s

* For no feedback (βf = 0), Q is a minimum and we have the original two poles.

* For some feedback (βf Ao > 0), Q > 0 and 1- 4Q2 0 so poles move towards each other.

* For βf such that Q = 0.5, then 1 - 4Q2 = 0 so two poles coincide at - o/2Q.

* For larger βf such that Q > 0.5, 1- 4Q2 < 0 and the poles become complex

For βf such that Q = 0.7, we get the maximally flat response (largest bandwidth).

For larger βf such that Q > 0.7, we get a peak in the frequency response and oscillation in the amplifier’s output transient response (undesirable).

3dB frequency increases

as feedback (Q and βf) increases!

- 3dB


* DC bias analysis

* Configuration: Shunt-Shunt (ARo = Vo/Is)

* Loading: Input R1 = Rf , Output R2 = Rf

* Midband gain analysis

Example - Feedback Amplifier with Two Poles

Rs=10 K

RB1=90 K

RB2=10 K

RC=1.5 K

Rf=8 K

10 V 10 V

=75rx=0C =7 pFC =0.5 pF

KVmAg

rVmAV

mA

V

Ig

mAAIIAK

VV

R

VI

KKKRVVKK

KV

mT

Cm

BCTh

ThB

ThTh

8.0/98

75/98

0256.0

5.2

5.2)33(75339

7.00.17.0

990100.1109010

10

RB= 9 K

KK

I

VA

KKKKKrRRRI

V

VVKKVmA

RRgV

RRVg

V

V

I

V

V

V

I

VA

s

oRo

fBSS

fCmfCmo

s

o

s

oRo

8463.0124

63.08.08910

/12485.1/98

If


Example - Feedback Amplifier with Two Poles

* Determine the feedback factor βf

* Calculate gain with feedback ARfo

* High frequency ac equivalent circuit

VmA

K

RRI

I

V

I

X

X

Fff

f

o

f

o

ff

/125.08

1

1'

'

'

KK

A

AA

KVmAA

Rof

RoRfo

Rof

3.75.101

84

1

5.10)84(/125.0

Is

Rs RB R1 R2


Original High Frequency Poles

Is

Rs RB R1 R2

C Pole

sradxpFKCR

KKKKKrRRRR

xCC

BsxC

/103.2768.0

11

68.08.08910

8

1

C Pole

Rs RB R1 R2

sradxpFKCR

KKKKKKKVmAKK

rRRRRRgRRR

xCC

BsCmCxC

/101.25.086

11

868.0891085.1/98185.1

1

7

122

See how we did the analysis for the C2 high frequency pole for the Series-Shunt feedback amplifier example.


Gain and Phase Margins

dBdBdBA

dBAanddB

and

sradxandsradx

dBdBso

so

xj

xj

KA

dBdBAKA

of

of

f

ff

o

oo

40)22(18)(/1

ismargin gain so

22)(18/1

margingain For the

50)180(130

ismargin phase so

180130

/103.2/101.2


188log20)(1

8125.0

11125.0

101.21

103.21

84)(

5.3884log20)(84

180

180

0001801

0180

01

9180

81

78

Phase margin

Gain margin

1

180

1/βf(dB)

A(dB)

38.5 dB

A(ω180)

Φ(ω1)

Φ(ω180)


High Frequency Poles for Feedback Amplifier

jxxxxQQQ

sradxsradx

Q

xx

sradxAQ

sradxKxxA

whereQQQ

s

oo

o

PHPH

ofPHPH

PHPH

o

ofPHPHo

oo

88882

88

87

8

21

21

21

88721

2

100.2102.141102.1102.14122

-s

at are poles new So

/102.101.12

/104.2

2

01.1101.2103.2

/104.21

/104.2125.0841103.2101.21

4122

* These new poles are complex numbers !* Original poles were at 2.1x107 and 2.3x108 rad/s.* This means the amplifier output will have a tendency to oscillate (undesirable!)* We have too much feedback ! * Q is too large, because βf is too large, because Rf is too small (βf = - 1/Rf)!* How to change Rf so that βf and Q are not too large?

• Need to increase Rf so that βf and Q are smaller.• How much to increase Rf ?

Note that this Q value is bigger than 0.707 so the amplifier will tend to oscillate (undesirable).


Frequency Response with and without Feedback

dBdBA

KK

A

AA

sradxsradx

Rfo

Rof

RoRfo

CC

3.173.7log20)(

3.75.101

84

1

/101.2 ,/103.2 78

Original amplifier(without feedback)

Amplifier with feedback

,/100.2102.1 ,/100.2102.1

/102.12

882

881

8

sradxjxsradxjx

polesNew

sradx

ff

CC


What is the Optimum Feedback and Rf ?

* Select Q = 0.707 and work backward to find βf and then Rf

* Q = 0.707 gives maximally flat response !

feedback.much toous gave which ,resistancefeedback a small tooas which w8K, R had we,Previously

1.18/055.0

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using for expressionan find tosolvefirst we,R desired ingcorrespond thefind To

102.1102.121102.1102.14122

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at are poles new So

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f

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28

21

2

21

f

88882

88

88721

21

KVmA

R

so

VmAKxx

x

KA

soA

jxxxxQQQ

sradxsradx

Q

sradxxxQsoQ

ff

PHPH

o

of

ofPHPHo

f

oo

o

PHPHoPHPH

o

New poles are still complex numbers, but Q = 0.707 so okay.


Gain and Phase Margins for Optimal Bandwidth

dBdBdBA

dBAanddB

and

sradxandsradx

dBdBso

soNew

xj

xj

KA

dBdBAKA

of

of

f

ff

o

oo

2.48)23(2.25)(/1

ismargin gain so

23)(2.25/1

margingain For the

80)180(100

ismargin phase so

180100

/103.2/105.9


2.252.18log20)(1

2.18055.0

11055.0

101.21

103.21

84)(

5.3884log20)(84

180

180

0001801

0180

01

9180

71

78

Phase margin

Gain margin

1

180

1/βf(dB)

A(dB)

38.5 dB

A(ω180)


Frequency Response for Optimum Feedback

dBdBAKK

KVmA

K

A

AA

VmA

sradxxx

sradx

sradx

RfoRof

RoRfo

f

CCC

C

5.2315log20)( 156.41

84

)84)(/055.0(1

84

1

/055.0

/102.12

101.2103.2

2 /101.2

/103.2

878

7

8

Note: * Better midband gain than before * Improved bandwidth and * No oscillation tendency !

Original amplifier(without feedback)

Amplifier with feedback


Design for Minimum Phase Shift (50o)

* Construct plots of A() in dB and phase shift versus frequency as before.

* Determine frequency 180 for -180o phase shift

* For minimum 50o phase shift, Measure up from -180o to -130o on

phase shift plot, draw horizontal line and determine frequency where this is reached. This gives 1.

Find corresponding magnitude of A() in dB at this frequency (1) on the magnitude plot. Draw a horizontal line. This gives the magnitude of 1/βf in dB.

Convert 1/βf in dB to a decimal

Calculate Rf from βf

125.01010get weHere

101011

log20)(1

20

18

20

2020

x

f

x

f

x

fff

xdB

KVmA

Rf

f 8/125.0

11

A(dB)

Phase margin

Gain margin

1

180

1/βf(dB)

*Check gain margin to see if it is okay. YES, 40 dB !*But Q = 0.94 > 0.707 so some tendency to oscillate .

A(ω180)


Design for Minimum Gain Margin (10 dB)

* Construct plots of A() in dB and phase shift versus frequency

* Determine frequency 180 for -180o phase shift

* For minimum 10dB gain margin, Draw vertical line on magnitude plot at

180 frequency. Where it intersects the magnitude plot, draw a horizontal line.

Measure upward from this horizontal line by 10 dB and draw a second horizontal line. This gives the magnitude of 1/βf in dB (-12dB here).

Convert 1/βf in dB to a decimal

Calculate Rf from βf

0.41010get weHere

101011

log20)(1

20

12

20

2020

x

f

x

f

x

fff

xdB

KVmA

Rf

f 25.0/4

11

Phase margin

Gain margin

1

180

1/βf(dB)

A(dB)

*Check phase margin to see if it is okay. NO, only 12o !*Q is also very large 5.1 >> 0.707 so strong oscillation tendency !


Summary of Feedback Amplifier Stability

* Feedback amplifiers are potentially unstable

Can break into oscillation at a particular frequency i where

* Need to design feedback amplifier with adequate safety margin

Minimum gain margin of 10 dB Minimum phase margin of 50o. Adjust βf by varying size of Rf .

Smaller Rf is, the larger is the amount of feedback.

111

1

sin

180Re

Imtan

1

01

i

ii

iif

jjiiii

ii

iii

ii

A

A

AA

so

eeAA

ce

radiansA

Aand

A

i

A()

Ao

H HfLf L

Afo

i

Oscillation instability is a design problem for any amplifier with two or more high frequency poles. This is the case for ALL amplifiers since each bipolar or MOSFET transistor has two capacitors and each capacitor gives rise to a high frequency pole!

feedback amplifier stability

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