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ECE 3204Microelectronics II
Stability Analysis in Feedback Systems
Differentiator Example
Prof. Bitarusers.wpi.edu/~sjbitar
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Stability Analysis in Feedback Systems
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Barkhausen Criteria:
In a feedback system, if the loop gain is greater than or equal to one and the phase shift is zero, the system can sustain oscillations (ie. no input signal required).
1?
In a Negative Feedback System, that means…
A
B
-+
Closed Loop Gain …
1
Unstable when the loop gain AB = -1
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Stability Analysis in Feedback Systems
In a Positive Feedback System, that means…
A
B
+
Closed Loop Gain …
1
Unstable when the loop gain AB = +1
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Stability Analysis in Feedback Systems
+
Stability Analysis - Differentiator
RCVin
Vout
Ideal Differentiator In Time…
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Stability Analysis - Differentiator
RCVin
Vout
Ideal Differentiator In Frequency…
12
1
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| |
∡ 90°
Stability Analysis - DifferentiatorExpected Time Response …
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Stability Analysis - DifferentiatorExpected Frequency Response …
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Stability Analysis - DifferentiatorWhat the … ???
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Stability Analysis - DifferentiatorWhat the … ???
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Stability Analysis - Differentiator
What is going on?
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Stability Analysis - DifferentiatorBack to Control Theory…
A
B
-+
Closed Loop Gain …
1
Unstable when the loop gain AB = -1
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Stability Analysis - Differentiator
Can AB = -1 for the Differentiator ??
What is A ?What is B ?
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Stability Analysis - Differentiator
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RCVin
Vout
1. Redraw and Ground Input
C
R
Vin=0
2. BREAK Loop, Add Ground (if necessary) and Apply Test Signal.
Vtest Vout
R
C
Vfb
To Find Loop Gain AB …
Stability Analysis - Differentiator
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1
1
1
Vtest Vout
R
C
Vfb
12
…and Determine AB.
Two Low-Pass Filter
Responses !
Stability Analysis - Differentiator
Ok, so now determine if AB can = -1 at some frequency…
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Stability Analysis - Differentiator
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1
1
1
Could this = 1 at some frequency??
· 1
YES! At …
A look at Magnitude…
Recall GBWP = = 1 ·
Geometric Mean
Stability Analysis - Differentiator
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How close is this to 180 (or -180 ) at ?Well, , so…
it’s close!
A look at Phase …
Stability Analysis - Differentiator
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Too Many Equations !! Let’s take a look at this graphically…
Stability Analysis - DifferentiatorVin Vout
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∡
| |
Stability Analysis - DifferentiatorR16k
C0.1µF
Vout
Vfb
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| |
∡
Stability Analysis - Differentiator
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Vtest VoutR16k
C0.1µF
Vfb| |
∡
Stability Analysis - Differentiator
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Vtest VoutR16k
C0.1µF
Vfb| |
∡ Trouble!!
Trouble!!
Stability Analysis - Differentiator
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So How Do We Fix This ??
Stability Analysis - Differentiator
R
16k
C
0.1µF
Vin
Vout
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Back to the Closed Loop Response…
fb
ftf0 fGM
Stability Analysis - Differentiator
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Add RS and Limit High Frequency Gain to R/RS≤fT/fGM
R
16k
C
0.1µF
Vout
Rs
160Vin
fb
f0 fGM ft
Stability Analysis - Differentiator
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How does adding RSaffect the loop Gain AB?
Stability Analysis - DifferentiatorVtest Vout
R16k
C0.1µF
Vfb
Rs160
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Phase Margin (PM)
New Loop Gain Phase
New Loop Gain Magnitude
Stability Analysis - Differentiator
R
16k
C
0.1µF
Vin
Vout
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So now the transient response goes from this …
Stability Analysis - Differentiator
R
16k
C
0.1µF
Vout
Rs
160Vin
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… to this! OK, much better ! (and Rs can be increased to dampen the response further…)
Stability Analysis - Differentiator
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And the closed-loop frequency response is back to what we expected.
R
16k
C
0.1µF
Vout
Rs
160Vin
fb
f0 fGM ft
Stability Analysis - Differentiator
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Frequency Response Both Magnitude and Phase with respect to the op-amp open-loop response.
(signal applied to the inverting input)
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