CH.8 Feedback CH.8 Feedback
Prof. Y.Kwon
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
0x
f= β β β A A x
A x
s o
f
= = +
1
β
≈ 1 A
f〉〉 1 β
A
ix
sx A
β
= + 1
1
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 !!
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
2V V N
S
n
=
sβ ∴ β
11 22 1
2 1
1
1 A A
V A A
A A V A
V
o s n+ +
= +
Properties of Feedback Properties of Feedback
4. Linear Enhancement
– by reducing gain Æ linearize the amp
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
inÏ
− Increase output current
Æ R
outÐ
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
of o
t V
t of
s
= +
= +
=
=0
1
Additional voltage Additional current
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
oby sensing I
E•Feed it back to V
svia
reducing V
BEBasic 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
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
Practical Network (Loading Effect) Practical Network (Loading Effect) Î Î Transformation into Ideal Network Transformation into Ideal Network
– Practical Situation:
① R
s, R
Lshould be included
② Feedback network has finite R
inand R
outÆ functions as loads to main amp
③ Forward gain of amplifier ≫ Forward gain of feedback network
Æ h
21can be neglected
Y. Kwon Chap. 8 : Feedback Amplifier Microelectronic Circuit Course Note, SoEE, SNU
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
sand 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
ffrom A and β
Quantity Sensing
Output
Quantity Mixing
Input β =
I
1V
V
2G = I
Consistent !!
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
Practical Series
Practical Series - - Series FA : Derivation of A and Series FA : Derivation of A and β
Y. Kwon Chap. 8 : Feedback Amplifier Microelectronic Circuit Course Note, SoEE, SNU
Y. Kwon Chap. 8 : Feedback Amplifier Microelectronic Circuit Course Note, SoEE, SNU
Loop Gain Determination Loop Gain Determination
Set
When is known
,
= 0
V
sV
r= − A β V
t,
Z
tY. Kwon Chap. 8 : Feedback Amplifier Microelectronic Circuit Course Note, SoEE, SNU
Loop Gain Determination Loop Gain Determination
When is unknown
From [Rosenstark]
Z
t1 ) ( 1
1
SC
OC
T
T A
+
−
β =
Loop Gain Determination Example Loop Gain Determination Example
– Insert V
twhere Z
tis clearly known: at OP Amp input
– Calculate V
rY. Kwon Chap. 8 : Feedback Amplifier Microelectronic Circuit Course Note, SoEE, SNU
Example 2
Example 2
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
sis injected)
Æ Stability is a function of the loop.
Ex)
Inverting or
Non-inverting
depending on
V
slocation
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
) ( )]
(
[
0L
f(t) F(s)
δ(t) 1
u(t) e
-atsin 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
Y. Kwon Chap. 8 : Feedback Amplifier Microelectronic Circuit Course Note, SoEE, SNU
Basic on Transfer Function Basic on Transfer Function
2. Transfer Function
①
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 ti
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
= ωω
j
s
sH H
H
) ( )
( circuit
stable for
))
(
(
Function
Network
Transfer Function of Feedback Amplifier Transfer Function of Feedback Amplifier
–
– Loop gain :
Æ determines stability ( φ ( ω
180) = 180º) Linear System
e
jωtF(j ω ) e
jωtTransfer 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 ωL = =
. 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
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 =
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
nv ( t ) = 2 e
σotcos ω
nt
t
e
jωY. Kwon Chap. 8 : Feedback Amplifier Microelectronic Circuit Course Note, SoEE, SNU
A A 『 『 Single Single - - pole pole 』 』 Case Case
Assume
① Open-loop amp(A) has real poles and no finite zero’s
② β is frequency independent and negative feedback
) 90 )
( max (
stable
) 1
( at
) pole 1
( 1
) 1
) ( (
at 1 pole
)
(
0Q ∠ = −
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
A A 『 『 Two Two - - pole』 pole 』 Case Case
Æ
How does pole change with loop gain A
0β ?
— 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
o0 )
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
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
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
Two Two - - pole Response pole Response
Frequency Dependence
① Q=0.707, maximally flat
② Can have peaks
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
180ω = A β < 1
∴ β
Å no Zero’s
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
1phase = −
−ω a
∠
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
180ω
1) ( A β
ω
180ω
1Stability 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
1A
fβ ω ω ω
) (
1
) ) (
(
1 1
1
A j
j j A
A
f= + A
f( j ω
1) = 1 + e
−jθβ
θω
jf
j e
A
−= +
1 1 ) 1
(
1LF cloosed –loop gain overshoot effect