CH.8 Feedback CH.8 Feedback

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CH.8 Feedback CH.8 Feedback

Prof. Y.Kwon

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Y. Kwon Chap. 8 : Feedback Amplifier Microelectronic Circuit Course Note, SoEE, SNU

Feedback Amp Feedback Amp

x0

xf =β

〉〉 1 β A

‰ Assumption : 1) open-loop gain, ‘A’ is not affected by loading.

2) signal direction specified

‰ Æ : feedback factor

‰ ( Aβ : loop gain , : amount of feedback )

‰ When , ( irrespective of A)

‰ When , ( “error” signal )

β + A 1

x

0

x

f

= β β β A A x

A x

s o

f

= = +

1

β

≈ 1 A

f

〉〉 1 β

A

i

x

s

x A

β

= + 1

1

(3)

Properties of Feedback Properties of Feedback

‰ 1. Gain Desensitvity

‰ 2. Bandwidth Extension

– consider a single pole circuit

A dA A

A dA

f f

β

= + 1

1

desensitivity factor

H M

s s A

A

+ ω

= 1 ) (

) 1

1 (

) 1

) ( (

β ω

β

M H

M M

f

s A

A A

s A

+ +

= + ω H → ω H ( 1 + A M β )

Bandwidth Extended !!

(4)

Y. Kwon Chap. 8 : Feedback Amplifier Microelectronic Circuit Course Note, SoEE, SNU

Properties of Feedback Properties of Feedback

‰ 3. Noise Reduction

– To keep the same gain, precede noise amp with a noiseless amp and apply negative feedback.

– Also, assume that noise from feedback network = ‘0’

– ex) preamplifier in audio power amp

A

2

V V N

S

n

=

s

β ∴ β

11 2

2 1

2 1

1

1 A A

V A A

A A V A

V

o s n

+ +

= +

(5)

Properties of Feedback Properties of Feedback

‰ 4. Linear Enhancement

– by reducing gain Æ linearize the amp

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Y. Kwon Chap. 8 : Feedback Amplifier Microelectronic Circuit Course Note, SoEE, SNU

Basic Feedback Topologies 1 : Voltage Amp : Series

Basic Feedback Topologies 1 : Voltage Amp : Series - - Shunt Shunt

‰ Voltage Amplifiers – R

in

= ∞, R

out

= 0

– Sample output voltage and mix with input voltage Æ Series shunt feedback

− Increases input voltage Æ R

in

Ï

− Increase output current

Æ R

out

Ð

(7)

Voltage Feedback Amp : Series

Voltage Feedback Amp : Series Shunt Feedback Shunt Feedback

(1) Input Resistance (R

if

) (2) Output Resistance (R

of

)

( )

)]

( ) ( 1 )[

( )

(

1

s s A s

Z Z

or

A I R

V V A I

V V

I R V

i if

i i

i i i

i f i

s if

β ω

β β

+

=

+ + =

+ =

=

= A β

R I

I V I

R V

o

f o

t V

t of

s

= +

= +

=

=0

1

Additional voltage Additional current

(8)

Y. Kwon Chap. 8 : Feedback Amplifier Microelectronic Circuit Course Note, SoEE, SNU

Basic Feedback Topologies 2 :

Basic Feedback Topologies 2 : Transconductance Transconductance Amp : Series- Amp : Series - Series

Series

‰ Transconductance Amplifier – R

in

= ∞, R

out

= ∞

– Series-Series feedback

) 1

( A β R

R

if

=

i

+

) 1

( A β R

R

of

=

o

+

) :

( β impedance

•Sample approximate I

o

by sensing I

E

•Feed it back to V

s

via

reducing V

BE

(9)

Basic Feedback Topologies 3 : Current Amp : Shunt

Basic Feedback Topologies 3 : Current Amp : Shunt - - Series Series

‰ Current Amplifiers – R

in

= 0, R

out

= ∞

– Shunt-Series Feedback

– Sample output current and mix with input current

(10)

Y. Kwon Chap. 8 : Feedback Amplifier Microelectronic Circuit Course Note, SoEE, SNU

‰ for Trans-Impedance Amp – R

in

=0, R

out

= 0

– Sense V, mix with input I

Basic Feedback Topologies 4 :

Basic Feedback Topologies 4 : Transimpedance Transimpedance Amp : Amp : Shunt

Shunt - - Shunt Shunt

(11)

Practical Network (Loading Effect) Practical Network (Loading Effect) Î Î Transformation into Ideal Network Transformation into Ideal Network

– Practical Situation:

① R

s

, R

L

should be included

② Feedback network has finite R

in

and R

out

Æ functions as loads to main amp

③ Forward gain of amplifier ≫ Forward gain of feedback network

Æ h

21

can be neglected

(12)

Y. Kwon Chap. 8 : Feedback Amplifier Microelectronic Circuit Course Note, SoEE, SNU

(13)

Transformation to Ideal Network Transformation to Ideal Network

Step 1: Represent feedback network with appropriate network parameters (Input: Series R , Output: Shunt R)

Step 2: Absorb R

s

and R

L

, Z’s in feedback network into main amplifier

Step 3: Calculate

Step 4: Calculate A from the augmented main amplifier

Step 5: Calculate R

if

, R

of

, A

f

from A and β

Quantity Sensing

Output

Quantity Mixing

Input β =

I

1

V

V

2

G = I

Consistent !!

(14)

Y. Kwon Chap. 8 : Feedback Amplifier Microelectronic Circuit Course Note, SoEE, SNU

Analysis of Practical Series

Analysis of Practical Series - - Series Feedback Amp Series Feedback Amp

‰ Best for Transconductance Amp : Sense I, mix V

(15)

Practical Series

Practical Series - - Series FA : Derivation of A and Series FA : Derivation of A and β

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Y. Kwon Chap. 8 : Feedback Amplifier Microelectronic Circuit Course Note, SoEE, SNU

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Y. Kwon Chap. 8 : Feedback Amplifier Microelectronic Circuit Course Note, SoEE, SNU

(19)

Loop Gain Determination Loop Gain Determination

‰ Set

‰ When is known

,

= 0

V

s

V

r

= − A β V

t

,

Z

t

(20)

Y. Kwon Chap. 8 : Feedback Amplifier Microelectronic Circuit Course Note, SoEE, SNU

Loop Gain Determination Loop Gain Determination

‰ When is unknown

‰ From [Rosenstark]

Z

t

1 ) ( 1

1

SC

OC

T

T A

+

β =

(21)

Loop Gain Determination Example Loop Gain Determination Example

– Insert V

t

where Z

t

is clearly known: at OP Amp input

– Calculate V

r

(22)

Y. Kwon Chap. 8 : Feedback Amplifier Microelectronic Circuit Course Note, SoEE, SNU

Example 2

Example 2

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(24)

Y. Kwon Chap. 8 : Feedback Amplifier Microelectronic Circuit Course Note, SoEE, SNU

Equivalence of Circuits Equivalence of Circuits from a Feedback

from a Feedback- -Loop Point of View Loop Point of View

1. Poles of a circuit are independent of the external excitation. (Poles are determined by setting V

s

= 0)

2. Poles of a FA depend only on the feedback loop.

3. Characteristic equation (whose roots are poles) is completely determined by the loop gain.

4. A given feedback loop may be used to generate different circuits with same poles and different zeros. (depending on where V

s

is injected)

Æ Stability is a function of the loop.

Ex)

Inverting or

Non-inverting

depending on

V

s

location

(25)

Time Domain

Solution of algebraic equations

Basics on Transfer Function Basics on Transfer Function

1. Laplace Transform

ds e s j F

t f

s dt

e t f t

f

j st j

st

+

=

=

α

π

α

( )

2 ) 1 (

. complex for

) ( )]

(

[

0

L

f(t) F(s)

δ(t) 1

u(t) e

-at

sin ω t

s 1

) ( 1 s+a

) ( 2 2

2 ω

ω s +

Frequency Domain

Solution of algebraic equations Circuit described

in the time domain by differential equations

Circuit described in the frequency domain by algebraic equations

Solution expressed in the time domain

Solution expressed in the frequency domain Transformation into

the frequency domain

Transformation into the time domain

(26)

Y. Kwon Chap. 8 : Feedback Amplifier Microelectronic Circuit Course Note, SoEE, SNU

Basic on Transfer Function Basic on Transfer Function

2. Transfer Function

② When V

i

(t) = δ(t), V

i

(s) = 1.

H(s) = V

o

(s) = L[impulse response]

3. Stability

With given V

i

(t)=δ(t), V

o

(t) = h(t) Æ

Æ Poles have negative real part H(s)

V

i

(s) V

o

(s) = H(s)·V

i

(s)

) (

) ) (

( V s

s s V

H

i

=

o

) ( )

)

(

( )

(

o V t t

i

t V t

h =

=δ

=

0 )

(

lim =

h t

t

stable p

t u e p A

s p

s p s

s t N

h

i

N

i

t p i N

i

, 0 ] Re[

) ) (

( ) )(

(

) ) (

(

2 1 1

1

<

⎥ =

⎢ ⎤

⋅⋅

= − ∑

=

L

⎢ ⎢

=

⇒ ω

= ω

ω

j

s

s

H H

H

) ( )

( circuit

stable for

))

(

(

Function

Network

(27)

Transfer Function of Feedback Amplifier Transfer Function of Feedback Amplifier

– Loop gain :

Æ determines stability ( φ ( ω

180

) = 180º) Linear System

e

jωt

F(j ω ) e

jωt

Transfer function

) ( ) ( 1

) ) (

( ω β ω

ω ω

j j

A j j A

A

f

= +

)

)

(

( ) ( )

( ) ( )

( j ω A j ω β j ω A j ω β j ω e

jφ ω

L = =

. ties nonlineari by

1 )

( to gose :

1 )

( : 3

sustained be

can ,

0 with

1 )

( : 2

: 1 ) (

: 1

180 180

180 180

=

<

=

=

<

ω ω

ω ω

L L

case

n oscillatio

V V

L case

stable L

case

o s

Usually indep. of frequency

(28)

Y. Kwon Chap. 8 : Feedback Amplifier Microelectronic Circuit Course Note, SoEE, SNU

Nyquist

Nyquist Plot Plot

‰ polar plot of

Æ for

‰ ,

‰ At contour crosses real axis – if G < 1, unstable

‰ Nyquist Criterion : if Nyquist plot encircles the point simplified (-1,0), amplifier is unstable

) ( ) ( j ω β ω A

β A

β

A

<

<

− ω

v real g v

i

o

= G ( j ω ) = G

*

( j ω ) G ( − j ω ) = G ( j ω )

) ( )

( j ω G j ω

G − = −∠

180

,

ω

ω =

(29)

Stability and Pole Location Stability and Pole Location

c If Re(a) > 0, pole exists in the left half plane ÆStable d If Re(a) = 0, : sinusoidal oscillation

e If Re(a) < 0, pole exists in right half plane Æunstable ex) Æ s = σ

o

± j ω

n

v ( t ) = 2 e

σot

cos ω

n

t

t

e

jω

(30)

Y. Kwon Chap. 8 : Feedback Amplifier Microelectronic Circuit Course Note, SoEE, SNU

A A 『 『 Single Single - - pole pole 』 』 Case Case

ƒ Assume

① Open-loop amp(A) has real poles and no finite zero’s

② β is frequency independent and negative feedback

ƒ

ƒ

) 90 )

( max (

stable

) 1

( at

) pole 1

( 1

) 1

) ( (

at 1 pole

)

(

0

Q ∠ = −

o

+

= + ⇒

+

= +

= + ⇒

=

s A

A A s

s

A s A

A

s s s A

A

p p

o f

p p

β β ω

ω

β

ω ω

(31)

A A 『 『 Two Two - - pole』 poleCase Case

ƒ

Æ

ƒ How does pole change with loop gain A

0

β ?

— Characteristic eq.:

— Solution:

— Root-Locus Diagram:

locus of poles as loop gain is increased )

1 )(

1 ) (

(

2 1

0

p

p

s

s s A

A = + ω + ω

⎟⎟ ⎠

⎜⎜ ⎞

⎛ ∠ = = ∞

stable s

A ( ) 180 but happens at ω

max

o

0 )

1 ( ) (

0 )

( 1

2 1 0

2 1

2

+ + + + =

= +

p p p

p

A

s s

s A

ω ω β ω

ω

β

2 1 0

2 2 1

2

1

( ) 4 ( 1 )

2 ) 1 2 (

1

p p p

p p

p

A

s = − ω + ω ± ω + ω − + β ω ω

(32)

Y. Kwon Chap. 8 : Feedback Amplifier Microelectronic Circuit Course Note, SoEE, SNU

Two Two - - pole Response pole Response

‰ For general second-order response, characteristic equation becomes

ƒ

poles complex

: 0.5

poles, real

: 5 . 0

with change

can factor

pole :

frequency pole

:

0

0 0

0 2 2

>

<

⎭ ⎬

⎜⎜ ⎫

= +

+

Q Q

Q A Q

s Q s

ω β

ω ω

axis from

part real

of distance :

center the

from distance

0

:

ω ω

j

Q

(33)

Two Two - - pole Response pole Response

‰ Frequency Dependence

① Q=0.707, maximally flat

② Can have peaks

(34)

Y. Kwon Chap. 8 : Feedback Amplifier Microelectronic Circuit Course Note, SoEE, SNU

Amp with 3 or more poles Amp with 3 or more poles

‰ One pole moves outward

‰ Two poles move inward and then become complex poles

‰ can reach up to Æenters right half plane

‰ How to guarantee stability

– at , ( Nyquist plot dose not encircle (-1,0) )

For given A, max and min exist beyond which the amplifier becomes unstable

A

f

°

− 270

A

f

ω

180

ω = A β < 1

β

Å no Zero’s

(35)

Basics of Bode Plots Basics of Bode Plots

‰ Bode plot : approximate plot of magnitude and phase of a transfer function driven by poles and zeros

‰ Magnitude ( zero : )

‰ Phase ( pole : )

s a + 1

2

10

1 ( )

log

20 + ω a

s a + 1

1

) ( tan

1

phase =

ω a

(36)

Y. Kwon Chap. 8 : Feedback Amplifier Microelectronic Circuit Course Note, SoEE, SNU

Stability using Bode Plots of Loop Gain

Stability using Bode Plots of Loop Gain ( )

‰ Idea : two break points phase :

gain :

Gain margin : amount by which the loop gain can increase and still guarantee

stability (@ )

phase margin : (@ )

β

A

ω

180

ω

1

) ( A β

ω

180

ω

1

(37)

Stability using Bode Plots of Loop Gain Stability using Bode Plots of Loop Gain

‰ Phase margin and (closed loop gain at unity loop gain frequency)

) ( A β

) ( j ω

1

A

f

β ω ω ω

) (

1

) ) (

(

1 1

1

A j

j j A

A

f

= + A

f

( j ω

1

) = 1 + e

jθ

β

θ

ω

j

f

j e

A

= +

1 1 ) 1

(

1

LF cloosed –loop gain overshoot effect

Phase margin

1.3 1.93

1 Overshoot factor 0.707

°

90 60 ° 45 ° 30 ° 0 °

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