Post on 05-Jul-2018
(Analog) Electronics II
Lesson II
Stability of Feedback Amplifiers
Guillermo Carpintero – Universidad Carlos III de Madrid
Ideal Configuration of a Feedback Amplifier
A++
-
si sε= si - sfb soerrorinput
With negative feedback
Guillermo Carpintero – Universidad Carlos III de Madrid
β
-
sfb = β· sofeedback
fa
aA
+=1
What happens with the frequency response?
)()(1
)()(
sfsa
sasA
+=
Guillermo Carpintero – Universidad Carlos III de Madrid
How can we explore the effect of feedback on the
frequency response of the amplifier?
Does feedback affect it anyway?
Why did we use feedback anyway?
At the cost of GAIN REDUCTION, we obtain
An amplifier closer to ideal characteristics:
• Insensitivity against active component parameter dispersion.
Guillermo Carpintero – Universidad Carlos III de Madrid
However, the circuit has tendency toward . . . . .
OSCILLATION.
parameter dispersion.
• Linear Gain
• Input/Output Inpedances
Oscillation ??????????
What frequency?
What amplitude?
Why?
Guillermo Carpintero – Universidad Carlos III de Madrid
Why?
jω
σ
s plane
Impulseinput
outputv(t)
ωσ js ±=�
bass ++21
][)(tjtjt nn eeet
ωωσυ −+= � )cos(2 te n
t ωσ�=
Feedback, Poles and Stability
1. One pole system
2. Two pole system
Guillermo Carpintero – Universidad Carlos III de Madrid
2. Two pole system
3. More than two poles
Feedback, Poles and Stability
asi so
Open Loop Gain
Guillermo Carpintero – Universidad Carlos III de Madrid
●
●
●
Open loop amplifiers are always STABLE systems
Case 1. A(s) has one pole
Gray-Meyer Sedra-Smith
Asa
+=)( 0)(
Ksa
−=
Feedback, Poles and Stability
Frequency Domain
Guillermo Carpintero – Universidad Carlos III de Madrid
P
ssa
ω+=1
)(
11
)(ps
sa−
=
Time Domain
inout
out vAdt
dvv
P 01 =+ ω
Frequency Domain Time Domain
Guillermo Carpintero – Universidad Carlos III de Madrid
Feedback, Poles and Stability
1
1( )
11
(1 )
KA s
sKf
p Kf
=+ −
+
Guillermo Carpintero – Universidad Carlos III de Madrid
●□p1(1+Aβ)p1
unconditionally STABLE
Basic Feedback
GAIN K
BW p1
Feedback, Poles and Stability
Gain x Bandwidth Product
)1(
11
1
1)(
1 fKp
fK
KsA
+−+
=
)1(1 fKp +fK
K
+1
Guillermo Carpintero – Universidad Carlos III de Madrid
Feedback, Poles and Stability
Guillermo Carpintero – Universidad Carlos III de Madrid
Feedback, Poles and Stability
Hints on what happens with higher order systems
Case 2. a(s) has two poles
K□
pp
jω
Guillermo Carpintero – Universidad Carlos III de Madrid
K
●
p1
unconditionally STABLE
●
p2
□
σ
Feedback, Poles and Stability
Case 3. a(s) more than two poles
jω
Guillermo Carpintero – Universidad Carlos III de Madrid
System can become unstable,
depending on f strength.
σ
Origin of the instability
Guillermo Carpintero – Universidad Carlos III de Madrid
Origin of the instability
)()(1
)()(
sfsa
sasA
+=
IF feedback can turn our system unstable . . .
. . . . means that poles of a(s) are not the poles of A(s)
1 + a(s)f = 0 characteristic
equation
Guillermo Carpintero – Universidad Carlos III de Madrid
)()(1 sfsa+
L(s)
fja
jajA
)(1
)()(
ωωω
+= 1 + a(jω)f = 0
1 + | a(jω)f |ejφ(ω) = 0
Origin of the instability
¿What happens when φφφφ = ππππ?
| a(ωπ f | e j π = - | a(ωπ) f |
fja
jajA
)(1
)()(
ωωω
−=
What is significant here?
Guillermo Carpintero – Universidad Carlos III de Madrid
ωπ
fja )(1 ω−
Be careful
with it !!!!
What is significant here?
Origin of the instability
+
Phase change in the loop gain Aβ has influence in the type of feedback of the amplifier :
• β < 1
• Aβ is always positive
Guillermo Carpintero – Universidad Carlos III de Madrid
A
β
++
-180º
sε and sfb are in phase. . . . . thus the comparator SUBSTRACTS
Origin of the instability
-A++
At ωπ :
• It is no longer true that Aβ es positive
Guillermo Carpintero – Universidad Carlos III de Madrid
POSITIVE feedback
-A-
β
360º
Origin of the instability
• We have clear out the OSCILATION FREQUENCY.
• How about the conditions for oscillation, . . . Why does it depend on f ?
Guillermo Carpintero – Universidad Carlos III de Madrid
. . . Why does it depend on f ?
Origin of the instability
-a β
a(ωπ) f < 1
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-a β
a(ωπ) f > 1
Tools to analyze the stability of feedback amplifiers
Guillermo Carpintero – Universidad Carlos III de Madrid
Tools to analyze the stability . . .
a(jω) f
| a(jω) f | ejφ(ω)
The secret is within the LOOP GAIN, a(s) f
Guillermo Carpintero – Universidad Carlos III de Madrid
| a(jω) f | ejφ(ω)
ωπωπ a(ωπ) f > 1
Tools to analyze the stability . . .
Starting point. . . . The frequency response of the BASIC AMPLIFIER
Guillermo Carpintero – Universidad Carlos III de Madrid
Tools to analyze the stability . . .
Nyquist diagram
Polar representation of a(jω)f
Guillermo Carpintero – Universidad Carlos III de Madrid
Stable? . . .or
not?
Tools to analyze the stability . . .
PM|Aβ|=1
Criteria for stability in the Nyquist diagram
Guillermo Carpintero – Universidad Carlos III de Madrid
PM180 – Φ0
GM
Tools to analyze the stability . . .
20 log10(a(jω)f ) = 20 log10 [a(jω)] + 20 log10 [ f ] =
= 20 log10 [a(jω)] - 20 log10 [ 1/ f ] =
a(jω)f in the Bode plot
Guillermo Carpintero – Universidad Carlos III de Madrid
= 20 log10 [a(jω)] - 20 log10 [ 1/ f ] =
= 20 log10 [a(jω)] - 20 log10( A )
(feedback network is resistive and presents a flat frequency response)
Tools to analyze the stability . . .
f0 fπ
a(f) a0 aπ
Φ(f) Φ0 Φπ
Two basic reference frequencies,
Guillermo Carpintero – Universidad Carlos III de Madrid
donde, a0 = A ( =1/f )
Φπ = -180
PM = 180º + Φ0GM = G - aπ
Tools to analyze the stability . . .
Guillermo Carpintero – Universidad Carlos III de Madrid
Tools to analyze the stability . . .
For ω0
fa
aAf
)(1
)()( 00
0jωjω
jω+
=
0)/1(θj
ef−
=
The importance of the Phase Margin,
Guillermo Carpintero – Universidad Carlos III de Madrid
en las que
θ0 = 180 - PM
0
0
1
)/1(θj
e
ef−+
=
01
)/1()( 0 θ
ωjf
e
fjA
−+=
Compensation
Guillermo Carpintero – Universidad Carlos III de Madrid
Compensation
Tools to control the gain/phase margins
Modify the BASIC AMPLIFIER freq. response a(s), by:
• DOMINANT POLE:
Place a new pole at LF
Guillermo Carpintero – Universidad Carlos III de Madrid
• POLE-ZERO:
Place a new pole at LF and cancel lowest frequency pole of basic amplifier with a new zero.
AIM: Keep the phase change in the loop below -180º at all frequencies where the loop gain is greater that unity (UNITY GAIN COMPENSATION)
Compensation
Modify the frequency response
of the Basic Amplifier
Guillermo Carpintero – Universidad Carlos III de Madrid
Compensation
OP AMP 741 before . . . .
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ADC = 120 dB
Compensation
. . . . and after compensation
Guillermo Carpintero – Universidad Carlos III de Madrid
Compensation
Guillermo Carpintero – Universidad Carlos III de Madrid
Compensation
Guillermo Carpintero – Universidad Carlos III de Madrid