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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?
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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
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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
+
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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°
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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 …
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Stability Analysis - Differentiator
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1
1
1
Vtest Vout
R
C
Vfb
12
…and Determine AB.
Two Low-Pass Filter
Responses !
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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
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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 …
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Stability Analysis - Differentiator
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Too Many Equations !! Let’s take a look at this graphically…
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Stability Analysis - DifferentiatorVin Vout
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∡
| |
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Stability Analysis - DifferentiatorR16k
C0.1µF
Vout
Vfb
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| |
∡
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Stability Analysis - Differentiator
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Vtest VoutR16k
C0.1µF
Vfb| |
∡
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Stability Analysis - Differentiator
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Vtest VoutR16k
C0.1µF
Vfb| |
∡ Trouble!!
Trouble!!
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Stability Analysis - Differentiator
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So How Do We Fix This ??
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Stability Analysis - Differentiator
R
16k
C
0.1µF
Vin
Vout
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Back to the Closed Loop Response…
fb
ftf0 fGM
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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
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Stability Analysis - Differentiator
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How does adding RSaffect the loop Gain AB?
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Stability Analysis - DifferentiatorVtest Vout
R16k
C0.1µF
Vfb
Rs160
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Phase Margin (PM)
New Loop Gain Phase
New Loop Gain Magnitude
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Stability Analysis - Differentiator
R
16k
C
0.1µF
Vin
Vout
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So now the transient response goes from this …
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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…)
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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
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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)