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
Ref:080130HKN EE3110 Feedback Amplifiers 2
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
Ref:080130HKN EE3110 Feedback Amplifiers 3
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
Ref:080130HKN EE3110 Feedback Amplifiers 4
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
Ref:080130HKN EE3110 Feedback Amplifiers 5
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
Ref:080130HKN EE3110 Feedback Amplifiers 6
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)
Ref:080130HKN EE3110 Feedback Amplifiers 7
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
Ref:080130HKN EE3110 Feedback Amplifiers 8
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?
Ref:080130HKN EE3110 Feedback Amplifiers 9
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
Ref:080130HKN EE3110 Feedback Amplifiers 10
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
Ref:080130HKN EE3110 Feedback Amplifiers 11
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
Ref:080130HKN EE3110 Feedback Amplifiers 12
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)
Ref:080130HKN EE3110 Feedback Amplifiers 13
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
Ref:080130HKN EE3110 Feedback Amplifiers 14
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
Ref:080130HKN EE3110 Feedback Amplifiers 15
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
Ref:080130HKN EE3110 Feedback Amplifiers 16
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
+−
+−
Ref:080130HKN EE3110 Feedback Amplifiers 17
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
Ref:080130HKN EE3110 Feedback Amplifiers 18
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
Ref:080130HKN EE3110 Feedback Amplifiers 19
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
Ref:080130HKN EE3110 Feedback Amplifiers 20
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
Ref:080130HKN EE3110 Feedback Amplifiers 21
Input/Output Resistance (Series-Series)
Input Resistance:
i
i
i
i
rT
I
VT
I
VR
⋅+=
⋅+=
=
)1(
)1(
in
ε
Output Resistance (Closed loop output resistance with zero input voltage)
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
Ref:080130HKN EE3110 Feedback Amplifiers 22
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
Ref:080130HKN EE3110 Feedback Amplifiers 23
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
Ref:080130HKN EE3110 Feedback Amplifiers 24
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
+=
++=
+++=++=
++=
++
+=
+=
+==
=
+=
Ref:080130HKN EE3110 Feedback Amplifiers 25
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+=
=
Ref:080130HKN EE3110 Feedback Amplifiers 26
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+−
Ref:080130HKN EE3110 Feedback Amplifiers 27
Input/Output Resistance (Shunt-Shunt)
Input Resistance:
)1(
)1(
in
T
r
TI
rI
I
VR
i
i
i
i
+=
+⋅
=
=
ε
ε
Output Resistance (Closed loop output resistance with zero input voltage)
)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
Ref:080130HKN EE3110 Feedback Amplifiers 28
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
Ref:080130HKN EE3110 Feedback Amplifiers 29
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
Ref:080130HKN EE3110 Feedback Amplifiers 30
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
Ref:080130HKN EE3110 Feedback Amplifiers 31
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
Ref:080130HKN EE3110 Feedback Amplifiers 32
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
Output Resistance (Closed loop output resistance with zero input voltage)
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
Ref:080130HKN EE3110 Feedback Amplifiers 33
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
Ref:080130HKN EE3110 Feedback Amplifiers 34
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
Ref:080130HKN EE3110 Feedback Amplifiers 35
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
Ref:080130HKN EE3110 Feedback Amplifiers 36
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
Ref:080130HKN EE3110 Feedback Amplifiers 37
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
Top Related