CH.8 Feedback

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

CH.8 FeedbackCH.8 Feedback

Prof. Y.Kwon


Feedback AmpFeedback Amp

0xx f β=

1⟩⟩βA

Assumption : 1) open-loop gain, ‘A’ is not affected by loading.2) signal direction specified

: feedback factor

( Aβ : loop gain , : amount of feedback )

When , ( irrespective of A)

When , ( “error” signal )

βA+1

0xx f β= β

βAA

xxA

s

of +

==1

β1

≈fA

1⟩⟩βA si xA

xβ+

=1

1


Properties of FeedbackProperties of Feedback

1. Gain Desensitvity

2. Bandwidth Extension– consider a single pole circuit

AdA

AAdA

f

f

β+=

11

desensitivity factor

H

M

sAsA

ω+=

1)(

)1(1)1()(βω

β

MH

M

M

f

As

AA

sA++

+= )1( βωω MHH A+→

Bandwidth Extended !!



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

2AVV

NS

n

s=∴ββ 21

1

21

21

11 AAAV

AAAAVV nso +

++

=



4. Linear Enhancement

– by reducing gain linearize the amp


Basic Feedback Topologies 1 : Voltage Amp : SeriesBasic Feedback Topologies 1 : Voltage Amp : Series--ShuntShunt

Voltage Amplifiers– Rin = ∞, Rout = 0

– Sample output voltage and mix with input voltageSeries shunt feedback

− Increases input voltageRin

− Increase output currentRout


Voltage Feedback Amp : Series Voltage Feedback Amp : Series –– Shunt FeedbackShunt Feedback

(1) Input Resistance (Rif) (2) Output Resistance (Rof)

( )

)]()(1)[()(

1

ssAsZZor

ARI

VVAI

VVIVR

iif

ii

ii

i

if

i

sif

βω

ββ

+=

+=+

=+

==βA

RII

VI

VR o

fo

t

V

tof

s+

=+

=== 10

Additional voltage Additional current


Basic Feedback Topologies 2 : Basic Feedback Topologies 2 : TransconductanceTransconductance Amp : SeriesAmp : Series--SeriesSeries

Transconductance Amplifier– Rin = ∞, Rout = ∞– Series-Series feedback

)1( βARR iif +=)1( βARR oof +=

):( impedanceβ

•Sample approximate Io by sensing IE•Feed it back to Vs via reducing VBE


Basic Feedback Topologies 3 : Current Amp : ShuntBasic Feedback Topologies 3 : Current Amp : Shunt--SeriesSeries

Current Amplifiers– Rin = 0, Rout = ∞

– Shunt-Series Feedback– Sample output current and mix with input current


for Trans-Impedance Amp – Rin =0, Rout = 0– Sense V, mix with input I

Basic Feedback Topologies 4 : Basic Feedback Topologies 4 : TransimpedanceTransimpedance Amp : Amp : ShuntShunt--ShuntShunt


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

– Practical Situation:

① Rs , RL should be included

② Feedback network has finite Rin and Rout functions as loads to main amp

③ Forward gain of amplifier ≫ Forward gain of feedback network

h21 can be neglected


Transformation to Ideal NetworkTransformation to Ideal Network

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

Step 2: Absorb Rs and RL , Z’s in feedback network into main amplifier

Step 3: Calculate

Step 4: Calculate A from the augmented main amplifier

Step 5: Calculate Rif , Rof , Af from A and β

QuantitySensingOutputQuantityMixingInput

=β

1IV

2VIG =

Consistent !!


Analysis of Practical SeriesAnalysis of Practical Series--Series Feedback AmpSeries Feedback AmpBest for Transconductance Amp : Sense I, mix V


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


Loop Gain DeterminationLoop Gain Determination

Set

When is known

,0=sV ,tr VAV β−=

tZ


Loop Gain DeterminationLoop Gain Determination

When is unknown

From [Rosenstark]

tZ

)11(

1

SCOC TT

A+

−=β


Loop Gain Determination ExampleLoop Gain Determination Example

– Insert Vt where Zt is clearly known: at OP Amp input– Calculate Vr


Example 2Example 2


Equivalence of Circuits Equivalence of Circuits from a Feedbackfrom a Feedback--Loop Point of ViewLoop Point of View

1. Poles of a circuit are independent of the external excitation. (Poles are determined by setting Vs = 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 Vs is injected)Stability is a function of the loop.

Ex) Inverting

orNon-inverting

depending on Vs location


Time Domain

Solution of algebraicequations

Basics on Transfer FunctionBasics on Transfer Function

1. Laplace Transform

dsesFj

tf

sdtetftf

stj

j

st

∫

∫∞+

∞−

∞ −

=

=−

α

απ)(

21)(

.complexfor)()]([0

L

f(t) F(s)δ(t) 1u(t)e-at

sinωt

s1

)(1 as +

)( 222 ωω +s

Frequency Domain

Solution of algebraicequations

Circuit describedin the time domain

by differential equations

Circuit described in thefrequency domain

by algebraic equations

Solution expressedin the time domain

Solution expressedin the frequency domain

Transformation intothe frequency domain

Transformation intothe time domain


Basic on Transfer FunctionBasic on Transfer Function

2. Transfer Function

①

② When Vi(t) = δ(t), Vi(s) = 1.H(s) = Vo(s) = L[impulse response]

3. StabilityWith given Vi(t)=δ(t), Vo(t) = h(t)

Poles have negative real part

H(s)Vi(s) Vo(s) = H(s)·Vi(s)

)()()(

sVsVsH

i

o=

)()()()(

ttVoi

tVthδ=

==

0)(lim =∞→

tht

stablep

tueApspsps

sNth

i

N

i

tpi

N

i

,0]Re[

)()())((

)()(121

1

<∴

=⎥⎦

⎤⎢⎣

⎡−⋅⋅⋅−−

= ∑=

−L

⎢⎢⎣

⎡=⇒

= ωω

ω

jssHH

H)()(circuitstablefor

))((FunctionNetwork


Transfer Function of Feedback AmplifierTransfer Function of Feedback Amplifier

–

– Loop gain :

determines stability (φ(ω180) = 180º)

Linear Systemejωt F(jω) ejωt

Transfer function

)()(1)()(

ωβωωω

jjAjAjAf +

=

)()()()()()( ωφωβωωβωω jejjAjjAjL ==

.tiesnonlineariby1)(togose:1)(:3

sustainedbecan,0with1)(:2:1)(:1

180180

180

180

−=−<⇒

=→−=

<

ωω

ωω

LLcasenoscillatio

VVLcasestableLcase

os

Usually indep. of frequency


NyquistNyquist PlotPlot

polar plot of for

,

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

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

)()( ωβωjAβAβA∠

∞<<∞− ω

realvvg

i

o →= )()( * ωω jGjG =)()( ωω jGjG =−

)()( ωω jGjG −∠=−∠

,180ωω =


Stability and Pole LocationStability and Pole Location

If Re(a) > 0, pole exists in the left half plane StableIf Re(a) = 0, : sinusoidal oscillationIf Re(a) < 0, pole exists in right half plane unstable

ex) tetv nto ωσ cos2)( =no js ωσ ±=

tje ω


AA『『SingleSingle--polepole』』CaseCase

Assume① Open-loop amp(A) has real poles and no finite zero’s② β is frequency independent and negative feedback

)90)(max (stable

)1(at pole)1(1

)1()(

at pole1

)( 0

oQ −=∠→

+−=⇒++

+=

−=⇒+

=

sA

AsAs

AAsA

ssAsA

pp

of

pp

βωβω

β

ωω


AA『『TwoTwo--polepole』』CaseCase

How does pole change with loop gain A0β ?— Characteristic eq.:

— Solution:

— Root-Locus Diagram:

locus of poles as loop gain is increased

)1)(1()(

21

0

pp ssAsA

ωω ++=

⎟⎟⎠

⎞⎜⎜⎝

⎛ ∞==∠stable

sA ωat happensbut 180)(max o

0)1()(0)(1

210212 =++++

=+

pppp AsssA

ωωβωω

β

2102

2121 )1(4)(21)(

21

pppppp As ωωβωωωω +−+±+−=


TwoTwo--pole Responsepole Response

For general second-order response, characteristic equation becomes

polescomplex : 0.5 poles, real:5.0

with changecan factor pole :

frequency pole :

0

00

202

><⇒⎭⎬⎫

⎜⎜⎝

⎛

=++

QQ

AQQ

Qss

βω

ωω

axis frompart real of distance :center thefrom distance :0

ωω

jQ


TwoTwo--pole Responsepole Response

Frequency Dependence

① Q=0.707, maximally flat② Can have peaks


Amp with 3 or more polesAmp with 3 or more poles

One pole moves outwardTwo poles move inward and then become complex poles

can reach up to enters right half planeHow to guarantee stability

– at , ( Nyquist plot dose not encircle (-1,0) )For given A, max and min exist beyond which the amplifier becomes unstable

fA

°− 270fA∠

180ωω = 1<βA∴ β

no Zero’s


Basics of Bode PlotsBasics of Bode Plots

Bode plot : approximate plot of magnitude and phase of a transfer function driven by poles and zerosMagnitude ( zero : )

Phase ( pole : )

as+1

210 )(1log20 aω+

as+1

1

)(tan 1

aphase ω−−=∠


Stability using Bode Plots of Loop GainStability 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 : (@ )

βA

180ω1ω)( βA

180ω

1ω


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

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

)( βA

)( 1ωjAf

βωωω

)(1)()(1

11 jA

jAjAf += θω j

f ejA −+= 1)( 1

θβω

jfe

jA−+

=1

11)( 1

LF cloosed –loop gain overshoot effect

Phase margin

1.3 1.931Overshoot factor 0.707

°90 °60 °45 °30 °0∞

CH.8 Feedback

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