Feedback Amplifier Stability

1
Feedback Amplifier Stability

• Feedback analysis
• Midband gain with feedback
• New low and high 3dB frequencies
• Modified input and output resistances, e.g.
• Amplifiers frequency characteristics
• Feedback amplifiers gain
• Define Loop Gain as ßfA
• Magnitude
• Phase

A(?)
Ao
Afo
Amplifier becomes unstable (oscillates) if at
some frequency ?i we have
2
Feedback Amplifier Instability
General form of Magnitude plot of ßfA

• Amplifier with one pole
• Phase shift of - 900 is the maximum
• Cannot get - 1800 phase shift.
• No instability problem
• Amplifier with two poles
• Phase shift of - 1800 the maximum
• Can get - 1800 phase shift !
• For stability analysis, we define two
  important frequencies
• ?1 is where magnitude of ßfA goes to unity
  (0 dB)
• ?180 is where phase of ßfA goes to -180o
• Instability problem if ?1 ?180

ßfA(dB)
0 dB
General form of Phase plot of ßfA
00
- 450
- 900
- 1350
- 1800
3
Gain and Phase Margins
ßfA(dB)

• Gain and phase margins measure how far
  amplifier is from the instability condition
• Phase margin
• Gain margin
• What are adequate margins?
• Phase margin 500 (minimum)
• Gain margin 10 dB (minimum)

0 dB
Gain margin
- 30
00
- 450
- 900
F(?1)
- 1350
Phase margin
F(?180)
- 1800
4
Numerical Example - Gain and Phase Margins
ßfA(dB)
Pole 1
23.5 dB
Pole 2
Gain margin
Phase Shift (degrees)
Note that the gain and phase margins
depend on the feedback factor ßf and so
the amount of feedback and feedback
resistors. Directly through the value of
ßf. Indirectly since the gain Ao and
pole frequencies are influenced by
the feedback resistors, e.g. the
loading effects analyzed previously.
Phase margin
5
Amplifier Design for Adequate Gain and
Phase Margins
General form of Magnitude plot of A

• How much feedback to use? What ßf ?
• A change of ßf means redoing ßf A(?)
  plots!
• Is there an alternate way to find the
  gain and phase margins? YES
• Plot magnitude and phase of gain A(?)
  instead of ßf A(?) vs frequency
• Plot horizontal line at 1/ ßf on magnitude
  plot (since ßf is independent of
  frequency)
• Why? At intersection of 1/ ßf with A(?),
  ? ?1
• So this intersection point gives the value
  where ? ?1 !
• Then we can find the gain and phase
  margins as before.
• If not adequate margins, pick another ßf
  value, draw another 1/ ßf line and repeat
  process.
• Note ßf is a measure of the amount of
  feedback. Larger ßf means more feedback.
• Recall for ßf 0, we have NO feedback
• As ßf increases, we have more feedback

A(dB)
1/ßf(dB)
Gain margin
A(?180)
0 dB
General form of Phase plot of A
00
- 450
- 900
- 1350
F(?1)
Phase margin
- 1800
F(?180)
6
Example - Gain and Phase MarginsAlternate
Method
A(dB)
43.5 dB
1/ßf(dB)
Gain margin
A(?180)
Phase margin
These are the same results as before for
the gain and phase margins .
7
Feedback Amplifier with Multiple Poles
A(?)

• Previous feedback analysis
• Midband gain
• New low and high 3dB frequencies
• Amplifiers typically have multiple high and
  low frequency poles.
• Does feedback change the poles other than
  the dominant ones?
• YES, feedback changes the other poles as
  well.
• This is called pole mixing.
• This can affect the gain and phase margin
  determinations !
• The bandwidth is still enlarged as
  described previously
• But the magnitude and phase plots are
  changed, so the gain and phase margins are
  modified.

Ao
Afo
8
Feedback Effect on Amplifier with Two Poles
New poles of the feedback amplifier occur
when

• Gain of a two pole amplifier
• Gain of the feedback amplifier

Midband gain of feedback amplifier.
Gain of feedback amplifier having two new,
different high frequency poles.
9
Feedback Effect on Amplifier with Two Poles

• Feedback amplifier poles
• For no feedback (ßf 0),
• For some feedback (ßf Ao gt 0), Q increases
  and 1- 4Q2 ? 0 so poles move towards each
  other as ßf increases.
• For ßf such that Q 0.5, then 1- 4Q2 0
  so two poles coincide at
• For larger ßf such that Q gt 0.5, 1- 4Q2 lt 0
  so poles become complex frequencies.

Root-Locus Diagram
j?
s ? j?
?
j?
?
10
Frequency Response - Feedback Amplifier with
Two Poles

• Amplifier with some feedback ( ßf Ao gt 0 ),
  as ßf increases 1 - 4Q2 ? 0 so poles move
  towards each other. For sufficient feedback,
  Q 0.5 and 1 - 4Q2 0 so the poles meet
  at the midpoint.

Increasing feedback, poles meet for Q 0.5
Af(dB)
Original amplifier
Original poles before feedback
For Q 0.707, we get the maximally
flat characteristic For Q gt 0.7, get a
peak in characteristic at
corresponding to oscillation in the
amplifier output for a pulsed input
(undesirable).
Af(dB)
Qgt0.7
Original amplifier
11
Frequency Response of Feedback on Amplifier
with Two Poles
- 3dB
3dB frequency increases as feedback (Q and ßf)
increases!

• For no feedback (ßf 0), Q is a minimum
  and we have the original two poles.
• For some feedback (ßf Ao gt 0), Q gt 0 and
  1- 4Q2 ? 0 so poles move towards each
  other.
• For ßf such that Q 0.5, then 1 - 4Q2 0
  so two poles coincide at - ?o/2Q.
• For larger ßf such that Q gt 0.5, 1- 4Q2 lt 0
  and the poles become complex
• For ßf such that Q 0.7, we get the
  maximally flat response (largest bandwidth).
• For larger ßf such that Q gt 0.7, we get a
  peak in the frequency response and
  oscillation in the amplifiers output
  transient response (undesirable).

12
Example - Feedback Amplifier with Two Poles

• DC bias analysis
• Configuration Shunt-Shunt (ARo Vo/Is)
• Loading Input R1 Rf , Output R2 Rf
• Midband gain analysis

10 V
10 V
RC1.5 K
RB190 K
Rf8 K
Rs10 K
?75 rx0 C? 7 pF C? 0.5 pF
If
RB210 K
RB 9 K
13
Example - Feedback Amplifier with Two Poles

• Determine the feedback factor ßf
• Calculate gain with feedback ARfo
• High frequency ac equivalent circuit

Rs
Is
RB
R1
R2
14
Original High Frequency Poles
Rs
Is
RB
R1
R2
C? Pole
C? Pole
Rs
RB
R1
R2
See how we did the analysis for the C?2
high frequency pole for the Series-Shunt
feedback amplifier example.
15
Gain and Phase Margins
A(dB)
38.5 dB
1/ßf(dB)
Gain margin
A(?180)
F(?1)
Phase margin
F(?180)
16
High Frequency Poles for Feedback Amplifier
Note that this Q value is bigger than
0.707 so the amplifier will tend to
oscillate (undesirable).

• These new poles are complex numbers !
• Original poles were at 2.1x107 and
  2.3x108 rad/s.
• This means the amplifier output will
  have a tendency to oscillate (undesirable!)
• We have too much feedback !
• Q is too large, because ßf is too
  large, because Rf is too small (ßf -
  1/Rf)!
• How to change Rf so that ßf and Q are
  not too large?
• Need to increase Rf so that ßf and Q
  are smaller.
• How much to increase Rf ?

17
Frequency Response with and without Feedback

Original amplifier (without feedback)
Amplifier with feedback
18
What is the Optimum Feedback and Rf ?
Select Q 0.707 and work backward to
find ßf and then Rf Q 0.707 gives
maximally flat response !
New poles are still complex numbers, but
Q 0.707 so okay.
19
Gain and Phase Margins for Optimal Bandwidth
A(dB)
38.5 dB
1/ßf(dB)
Gain margin
A(?180)
Phase margin
20
Frequency Response for Optimum Feedback
Original amplifier (without feedback)
Amplifier with feedback
Note Better midband gain than before
Improved bandwidth and No
oscillation tendency !
21
Design for Minimum Phase Shift (50o)
A(dB)

• Construct plots of A(?) in dB and phase
  shift versus frequency as before.
• Determine frequency ?180 for -180o phase shift
  
• For minimum 50o phase shift,
• Measure up from -180o to -130o on phase
  shift plot, draw horizontal line and
  determine frequency where this is reached.
  This gives ?1.
• Find corresponding magnitude of A(?) in
  dB at this frequency (?1) on the magnitude
  plot. Draw a horizontal line. This gives
  the magnitude of 1/ßf in dB.
• Convert 1/ßf in dB to a decimal
• Calculate Rf from ßf

1/ßf(dB)
Gain margin
A(?180)
Phase margin

• Check gain margin to see if it is okay.
  YES, 40 dB !
• But Q 0.94 gt 0.707 so some tendency to
  oscillate .

22
Design for Minimum Gain Margin (10 dB)
A(dB)

• Construct plots of A(?) in dB and phase
  shift versus frequency
• Determine frequency ?180 for -180o phase shift
  
• For minimum 10dB gain margin,
• Draw vertical line on magnitude plot at
  ?180 frequency. Where it intersects the
  magnitude plot, draw a horizontal line.
• Measure upward from this horizontal line by
  10 dB and draw a second horizontal line.
  This gives the magnitude of 1/ßf in dB
  (-12dB here).
• Convert 1/ßf in dB to a decimal
• Calculate Rf from ßf

1/ßf(dB)
Gain margin
Phase margin

• Check phase margin to see if it is okay.
  NO, only 12o !
• Q is also very large 5.1 gtgt 0.707 so
  strong oscillation tendency !

23
Summary of Feedback Amplifier Stability

• Feedback amplifiers are potentially unstable
• Can break into oscillation at a particular
  frequency ?i where
• Need to design feedback amplifier with
  adequate safety margin
• Minimum gain margin of 10 dB
• Minimum phase margin of 50o.
• Adjust ßf by varying size of Rf .
• Smaller Rf is, the larger is the amount of
  feedback.

A(?)
Ao
Afo
Oscillation instability is a design problem
for any amplifier with two or more high
frequency poles. This is the case for ALL
amplifiers since each bipolar or MOSFET
transistor has two capacitors and each
capacitor gives rise to a high frequency
pole!