lecture 2 feedback amplifier -...

Ref:080130HKN EE3110 Feedback Amplifiers 1

Lecture 2 Feedback Amplifier

• Introduction of Two-Port Network

• Negative Feedback (Uni-lateral Case)

• Feedback Topology

• Analysis of feedback applications

– Close-Loop Gain

– Input/Output resistances


Two-Port Network (z-parameters)(Open-Circuit Impedance)

2121111 IzIzV +=

2221212 IzIzV +=

021

111 ==

II

Vz

012

1

12 ==

II

Vz

Open-circuit

input impedance

At port 1

Open-circuit reverse

transimpedance 012

2

22 ==

II

Vz

021

221 ==

II

Vz

At port 2

Open-circuit forward

transimpedance

Open-circuit

output impedance

V1

+

−

I1

V2

+

−

I2

z11

z22

z12I2 z21I1+−

+−

=

2

1

2221

1211

2

1

I

I

zz

zz

V

V


Two-Port Network (y-parameters)(Short-Circuit Admittance)

2121111 VyVyI +=

2221212 VyVyI +=

021

111 ==

VV

Iy

012

1

12 ==

VV

Iy

Short-circuit

input admittance

At port 1

Short-circuit reverse

transadmittance 012

2

22 ==

VV

Iy

021

221 ==

VV

Iy

At port 2

Short-circuit forward

transadmittance

Short-circuit

output admittance

V1

+

−

V2

+

−

I1 I2

1/y11 1/y22

y12V2 y21V1

=

2

1

2221

1211

2

1

V

V

yy

yy

I

I


Two-Port Network (h-parameters)(hybrid)

2121111 VhIhV +=

2221212 VhIhI +=

021

111 ==

VI

Vh

012

1

12 ==

IV

Vh

Short-circuit

input impedance

At port 1

Open-circuit reverse

voltage gain 012

2

22 ==

IV

Ih

021

221 ==

VI

Ih

At port 2

Short-circuit forward

current gain

Open-circuit

output admittance

V2

+

−

I2

1/h22

h21I1

V1

+

−

I1

h11

h12V2 +−

=

2

1

2221

1211

2

1

V

I

hh

hh

I

V


Two-Port Network (g-parameters)(inverse-hybrid)

2121111 IgVgI +=

2221212 IgVgV +=

021

111 ==

IV

Ig

012

1

12 ==

VI

Ig

Open-circuit

input admittance

At port 1

Short-circuit reverse

current gain 012

2

22 ==

VI

Vg

021

221 ==

IV

Vg

At port 2

Open-circuit forward

current gain

Short-circuit

output impedance

V2

+

−

I2

g22

g21V

1

V1

+

−

I1

1/g11

g12I

2 +−

=

2

1

2221

1211

2

1

I

V

gg

gg

V

I


z-parameter exampleI1 I2

V1

+

−V2

+

−6Ω

I1 I2

V2

+

−V1

+

−

12Ω

3ΩV2

+

−V1

+

−

I1 I23Ω12Ω

6Ω

[ ]

=

Ω==

=

Ω==

=

Ω=Ω=

30

012

00

00

312

21

221

12

112

2211

Z

II

VZ

II

VZ

ZZ

[ ]

=

Ω==

=

Ω==

=

Ω=Ω=

66

66

60

60

66

21

221

12

112

2211

Z

II

VZ

II

VZ

ZZ

[ ]

=

Ω===

=

Ω===

=

Ω=Ω=

96

618

66

0

66

0

918

1

1

21

221

2

2

12

112

2211

Z

I

I

II

VZ

I

I

II

VZ

ZZ

Note: (1) z-matrix in the last circuit = sum of two former z-matrices

(2) z-parameters is normally used in analysis of series-series circuits

(3) Z12 = Z21 (reciprocal circuit)

(4) Z12 = Z21 and Z11 = Z22 (symmetrical and reciprocal circuit)


y-parameter exampleI1 I2

V1

+

−V2

+

−

0.05S

V2

+

−V1

+

−

I1 I20.2S0.1S

0.025S

[ ]

−

−=

−=−

==

=

−=−

==

=

==

05.005.0

05.005.0

05.005.0

0

05.005.0

0

05.005.0

1

1

21

221

2

2

12

112

2211

y

V

V

VV

Iy

V

V

VV

Iy

ySy

S

S

S

[ ]

−

−=

−==

−=

−=−=⇒

+=

−

==

==

=

+

+=

=

+

+=

−

−

0769.00615.0

0615.00692.0

S0615.0 ,reciprocalBy

S0615.0

0615.08.0

025.01.0

0769.0But

0

S0769.0025.01.0

1

2.0

1

S0692.0025.02.0

1

1.0

1

1221

12

221

121

22222

12

112

1

22

1

11

y

yy

y

VII

III

VVyI

VV

Iy

y

y


y-parameter example (Cont’)

V2

+

−V1

+

−

I1 I20.2S0.1S

0.025S

0.05S

[ ]

−

−=

=−=⇒

−=−−=

=+

=

=−=

==

==

===

=+=

=+=

1269.01115.0

1115.01192.0

1115.0

1115.005.00615.0

0615.00769.0025.01.0

1.0

0769.0

05.01269.0

05.0

1269.0

0

1269.0769.005.0

1192.0692.005.0

2112

2221

221.0

205.022.0

2205.0

22222

12

11221

22

11

Y

yy

VVVI

VVI

VIII

VII

VVyI

VV

Iyy

y

y

S

SS

S

Note: the y-matrix is equal to

the sum of two former ones.

Therefore, y-parameters is

normally used in analysis of

shunt-shunt circuits

What connection should be for

h- or g- parameters?


General Feedback Structure

∑ A

β

Source Load+

-

Vs

Vf

Vε Vο A : Open Loop Gain

A = Vo / Vε

β : feedback factor

β = Vf / Vo

ε

ε

ε

β

β

VAV

VVV

VV

VVV

o

oS

of

fs

⋅=

⋅−=

⋅=

−=

β

ββ

ββ

1 :Note

1 :feedback ofAmount

:Gain Loop

)1(

1

1 :gain loop Close

=

⋅+

⋅=

+=

+==

∞→ACL

s

oCL

A

A

AT

T

T

A

A

V

VA


Negative Feedback Properties• Negative feedback takes a sample of the output signal and applies

it to the input to get several desirable properties. In amplifiers,

negative feedback can be applied to get the following properties

– Desensitized gain : gain less sensitive to circuit component

variations

– Reduce nonlinear distortion : output proportional to input

(constant gain independent of signal level)

– Reduce effect of noise

– Control input and output impedances by applying appropriate

feedback topologies

– Extend bandwidth of amplifier

• All of these properties can be achieved by trading off gain


Gain De-sensitivity• Feedback can be used to desensitize the closed-loop gain to variations in the

basic amplifiler.

• Assume β is constant. Take differentials of the closed loop gain equation

gives,

• Divided by Av, the close loop gain sensitivity is equal to,

• This result shows the effects of variations in A on ACL is mitigated by the

feedback amount.

• (1+Aβ) is also called the desensitivity amount.

22 )1(or

)1(

1

1 βββ A

dAdA

AdA

dA

A

AA CL

CLCL +

=+

=+

= Differential respected with A

A

dA

AA

A

A

dA

A

dA

CL

CL

ββ

β +=

++

=1

1)1(

)1( 2


Basic Feedback Topologies

Depending on the input signal (voltage or current) to be amplified

and form of the output (voltage or current), amplifiers can be

classified into four categories. Depending on the amplifier

category, one of four types of feedback structures should be used.

(Type of Feedback) (Type of Sensing)

(1) Series (Voltage) Shunt (Voltage)

(2) Series (Voltage) Series (Current)

(3) Shunt (Current) Shunt (Voltage)

(4) Shunt (Current) Series (Current)


Feedback Structure (Series-Shunt)

Voltage amplifier voltage-controlled

voltage source

Requires high input impedance, low

output impedance

Voltage-voltage feedback

Voltage Gain Calculation:

)1(

1

get weAnd,

where

)1(

1

Gain) Voltage Loop (Close

ββ

ββ

β

β

ε

ε

ε

⋅+=

⋅+⋅

=

=

+==⇒

⋅+=+=

⋅=

⋅=

AVV

A

AVV

AT

T

T

V

VA

VA

VVVV

VV

VAV

i

io

i

oCL

oo

fi

of

o

+−

+−

+−

Basic amplifier

Feedback network

Ii +

−

+

−Vori

ro

AVε

Vf=βVo

Vi

Vε


Input/Output Resistance (Series-Shunt)

Input Resistance:

i

i

i

i

i

i

i

i

i

i

rTI

VR

rT

V

r

VI

VTV

I

VR

⋅+==

⋅+==

⋅+=

=

)1(

)1(

)1(

in

in

ε

ε

Output Resistance (Closed loop output resistance with zero input voltage)

+−AVε

ro

Vo

Io

+−

T

r

A

r

I

VR

r

VAVI

VV

VVV

r

VAVI

I

VR

oo

o

o

o

ooo

o

io

o

oo

o

oVi

+=

⋅+==⇒

⋅⋅+=

⋅−=

==⋅+

⋅−=

==

11

0

|

out

0out

β

β

ββ

ε

ε

ε


h-parameter Modeling

Only uni-lateral case

will be considered :

(1) NO reverse

dependent signal

found in the

amplifier network.

|h12a| = 0

(2) NO reverse

dependent signal

found in the

feedback network.

|h21f| = 0

+−

+−

V1

+−

zs +

− V2

1/yL1/h22a

h11a

h11f

1/h22f

h12aV2

h12f V2

h21aI1

h21 f I1

I1


Uni-lateral

faoi

a

oi

fa

oi

a

CL

f

Lfaosfai

oi

a

Lfasfa

a

f

Lfa

a

aLfa

fsfa

hhyz

h

yz

hh

yz

h

A

AA

hf

YhhyZhhz

yz

h

YhhZhh

h

V

VA

h

Yhh

IhV

IhVYhh

VhIZhhV

1221

21

1221

21

12

22221111

21

22221111

21

1

2

12

2222

1212

12122222

212111111

11

isGain Loop Closed the,feedback With the

and where

))((

Gain, LoopOpen the,0For

equation,first back to )(

Put

0)( :port output From

)( :port Input From

⋅−

−=

⋅−+

−

=+

=

=

++=++=

−=

++++

−==

=

++−=

=+++

+++=

β

V21/yL1/h22a

1/h22f

h21aI1

V1

zs

h11a

h11f

h12f V2

I1

+−

+−


Series-Shunt Example

+

−Vε

Vi

Vf

R1

R2

Vo

Equivalent circuit

+

−−−− AVε

Vε

Vi

rπ rο+−

Vo

Vf Vo

R1R2

Amplifier

Feedback

It is observed that:

(1) Series connection in input ports

(2) Shunt connection in output ports

⇒ Series-Shunt connection

h-parameter should be used.

Ω=Ω=Ω=Ω== 40 and M10 ,9 ,1 ,10 :Given 21

5

orrkRkRA π


h-parameter analysis

+−

h11f

h22f

h12fV2

R1

R2

V1 V2

I1

kRRRRI

I

IV

Ih

RR

R

RRI

RI

IV

Vh

RR

RRRR

VI

Vh

f

f

f

10

11

)(0

1.0)(0

9.0//0

21212

2

12

222

21

1

212

12

12

112

21

2121

21

111

=+

=+

==

=

=+

=+

==

==

=+

===

=

β

1


10~)1.0)(10(1

10

1A

gain, voltageloop close theAnd

10~)(

A

isgain voltageloopopen theTherefore,

11

get we,1

and puttingby

0

port,output from

~V

port,input thefrom

andcircuit loopopen for 0set Firstly,

5

5

CL

5

210

21

1

2OP

12

21

21

221

2222

1

11

1

12

+=

+=

=++

+==

=

++

+==

=⋅+−

+=

=

β

ε

π

πε

OP

OP

oo

fε

f

o

f

f

A

A

ARRr

RRA

V

V

r

AVV

RRr

RRhVV

Vhr

AVV

Vhr

rV

h

01

)/1//(

))(1(

22

11

→+

=

∞→++=

β

β π

OP

fo

fOP

A

hr

rhA

out

in

R

impedance,output The

R

impedance,input The

V21/h22f

AVε

V1

rπ

h11f

h12f V2

I1

+−

+−

Vε

+

−

+

−

+

−

rο


Feedback Structure (Series-Series)

+−

Basic amplifier

Feedback network

Ii

+

−Vε ri

Vf=βIo

ro

AVε

Iο

Vi

Vο+−

)1(

1

get weAnd,

where

)1

(1

Gain) tanceTransadmit Loop (Close

:nCalculatioGain

ββ

β

β

β

β

ε

ε

ε

⋅+=

⋅+⋅

=

=

+==⇒

⋅+=+=

⋅=

⋅=

AVV

A

AVI

AT

T

T

V

IA

IA

IVVV

IV

VAI

i

io

i

oCL

oo

fi

of

o


Input/Output Resistance (Series-Series)

Input Resistance:

i

i

i

i

rT

I

VT

I

VR

⋅+=

⋅+=

=

)1(

)1(

in

ε


o

o

o

o

oo

o

oo

of

o

oV

rTI

VR

r

VIT

r

VAVI

IVV

I

VR

i

)1(

port,output from

port,input from

|

out

0out

+==⇒

+⋅−=+=

⋅−==

==

ε

ε β


Series-Series Example

CE amplifier with an un-bypassed emitter ac small signal equivalent circuit

R1

R2 RE

RC

vs

vo

+VCC

vs vo

B

E

C

R1//R2 RC

RE

Feedback network

rπ+

−

+

−

rο


Feedback Network with z-parameter

REV1 V2

I1 I2

Ef

Ef

Ef

Rii

vZ

Rii

vZ

Rii

vz

==

=

==

==

==

=

0

0

0

12

222

12

112

21

111

β

Reduce equivalent circuit

+

−rπ vπ

+−z12fio

vo

+

−

rogvπ

io

z11f

z22f

+− vs

ii


Close loop analysis

)]1)([(R

:is impedanceOutput

)(

)(1)()1)((R

:is impedanceInput

11

isgain ttance transadmiloop close The

Therefore,

isgain ttance transadmiloopopen Then

and

22out

11in

11

β

β

β

ππ

π

πππ

ππ

π

π

π

π

π

π

π

ππ

ππ

OLf

EE

E

EEOLf

EE

E

E

E

op

op

CL

Es

oop

os

f

Az

RgrRr

Rr

gRrRrAzr

gRrRr

gr

Rr

gRr

Rr

gr

A

AA

Rr

gr

v

iA

gvivZr

rv

+=

++=

+++=++=

++=

++

+=

+=

+==

=

+=


Final Rin and Rout

+

−rπ vπ

+−z12fio

vo

+

−

ro

gvπ

io

z11f

z22f

+−

vs

ii

R1//R2RC

RinR'in Rout R'out

21

21inin

////])[(

////RR'

RRRgrRr

RR

EE ππ ++=

=

COPf

C

RAz

R

//)]1)([(

//RR'

22

outout

β+=

=


Feedback Structure (Shunt-Shunt)

)1(

1

get weAnd,

where

)1

(1

Gain) anceTransimped Loop (Close

)1(

)(

)(

:nCalculatioGain

ββ

β

β

β

β

ε

ε

⋅+=

⋅+

⋅=

=

+==⇒

+=

=−

⋅=

−=⋅=

AII

A

AIV

AT

T

T

I

VA

VTAI

VVIA

VI

IIAIAV

i

io

i

oCL

oi

ooi

of

fio

Feedback network

Ii ri

If=β Vo

Vi

+

−

Iε

Basic amplifier

+

−Vo

ro

AIε+−


Input/Output Resistance (Shunt-Shunt)

Input Resistance:

)1(

)1(

in

T

r

TI

rI

I

VR

i

i

i

i

+=

+⋅

=

=

ε

ε


)1(

port,output from

port,input from

|

out

0out

T

r

I

VR

r

TVV

r

AIVI

VII

I

VR

o

o

o

o

oo

o

oo

of

o

oVi

+==⇒

+=

−=

−=−=

==

ε

ε β


Shunt-Shunt Example

+−

RS

RL

RC

vS

C1

C2

Vcc

RF

CE amplifier

+−

Rs

rπ+

−

VπgVπ

Rc RLVs Vo

RF

ac small signal equivalent circuit

Shunt-Shunt connection found! ⇒ y-parameter


F

F

F

RVV

Iy

RV

I

VV

Iy

RVV

Iy

VyVyI

VyVyI

1

0

1

0

1

0

12

222

2

2

12

112

21

111

2221212

2121111

==

=

−=−

==

=

==

=

+=

+=

I1 I2

V1 V2RF

V1

I1 I2

RFRF

−1/RFVo

Feedback Network

y-parameter modeling


β

β

ππ

π

π

π

π

ππ

OP

OPCL

F

FLCFOP

S

o

LCFo

LCF

o

F

S

FS

A

AA

R

rRRRRgVA

I

V

RRRgVV

gVRRR

V

rR

VI

rRIV

+=

−=

−=

−=

=+

=⇒

=

1

:gain dance transimpeloop close the

,1

factor feedback With

)//)(////(

:gain ance tranimpedloopOpen

)////(

0////

port,output from And

)//(

)//(

port,input From

)( :Gain Voltage

)1(

)////(

)1(

)1(

)//(

)1(

out

in

inss

o

s

o

OP

LCF

OP

o

OP

F

OP

i

RRI

V

V

V

A

RRR

A

rR

A

rR

A

rR

+=

+=

+=

+=

+=

β

β

β

β

π

Is

RF

−1/RF¡EVo

rπ

+

−

VπgVπ

RF RC//RL

Vo


Feedback Structure (Shunt-Series)

)1(

1

get weAnd,

where

)1

(1

Gain)Current Loop (Close

)1(

)(

)(

:nCalculatioGain

ββ

β

β

β

β

ε

ε

⋅+=⋅+⋅

=

=

+==⇒

+=

=−

⋅=

−=⋅=

AII

A

AII

AT

T

T

I

IA

ITAI

IIIA

II

IIAIAI

i

i

o

i

o

CL

oi

ooi

of

fio

Feedback network

If=β Io

+

−

Iε

AIε

IοBasic amplifier

Ii Vi ri ro


Input/Output Resistance (Shunt-Series)

Input Resistance:

)1(

)1(

in

T

r

I

rT

I

I

rI

I

VR

i

i

ii

i

i

i

i

+=

⋅+

=

== ε


o

o

o

oooo

ooo

ooo

of

o

oV

rTI

VR

rITIV

rAIIV

AIrVI

III

I

VR

i

)1(

)(

)(

/ port,output from

port,input from

|

out

0out

+==⇒

⋅+=

−=

+=

−=−=

==

ε

ε

ε β


SummaryFeedback

Structure

Close loop

gain

Input

impedance

Output

impedance

Parameter

used

Series-

Shunt h-parameter

Series-

Series z-parameter

Shunt-

Shun y-parameter

Shunt-

Series g-parameter

)1(

1

T

T

V

V

i

o

+=β

)1(

1

T

T

V

I

i

o

+=β

)1

(1

T

T

I

V

i

o

+=β

)1

(1

T

T

I

I

i

o

+=β

irTR ⋅+= )1(

in

irTR ⋅+= )1(

in

T

rR i

+=

1in

T

rR i

+=

1in

T

rR o

+=

1out

T

rR o

+=

1out

orTR ⋅+= )1(out

orTR ⋅+= )1(out


Supplementary

R1

RE

RC

vs

vo

+VCC

Ω=

=

Ω=

Ω=

Ω=

kr

kR

kR

R

E

C

10

200

2

1

1001

π

β

Find the input and output resistance from

- Two port network, and

- Circuit theory


Circuit Theory

Ω==∴

=⇒=

Ω++=

++=+=⇒

+=−

=

=′′=

kRR

ivR

RrRR

Rrivriv

iR

v

r

vvi

i

vRRRR

Cout

bsout

Ein

EbEbs

b

E

EEsb

b

sin

1

00 , find To

100~])1(//[

])1([

)1( and but

where//

1

1

β

β

β

π

ππ

π

R1

RE

RC

vs voβ ib

ib

rπvE


Two Port Network

RE

vs voβ ib

ib

rπ

RE ⇒RE

+−

RE i2

RE

Ef

Ef

Ef

Rii

vZ

Rii

vZ

Rii

vz

==

=

==

=

==

=

0

0

0

12

222

12

112

21

111


Ω==′

Ω==′

++=

+++=

+=

+==

=

+=

=

kRRRR

RRRR

RRr

RR

RRr

RrR

RrRri

i

v

iA

ii

Rriv

CCoutout

inin

E

E

Eout

E

E

Ein

EEb

b

s

oOL

bo

Ebs

1~//

100~//

)(1

)(1)(

)()(

)(

0 signalfeedback settingby found isGain tanceTransadmit LoopOpen The

11

π

ππ

ππ

π

β

β

βββ

+−

rπ

RE

RE io

RE

vs vo

β ib

ib

RinR'in Rout R'out

R1 RC

lecture 2 feedback amplifier -...

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