6.302 Feedback Systems
Recitation 5: Op-amp Circuits and Analog Computers
Prof. Joel L. Dawson
In lecture, Prof. Roberge spoke of building elaborate systems using op-amp circuits. When we do this, we
habitually make statements like, “...and we’ll choose the dynamics such that the poles contributed by the
op-amps themselves are negligible.” Let’s discuss.
Consider a two-pole system whose behavior interests us only in the range from DC to ω0. A pole-zero
diagram:
jω
φL
x
ωLARGE
x
ω0
φs
σ
ωSMALL
-ω0
The angle φL is small compared to φS. => pole @ ωLARGE contributes very little phase shift.
For the frequency response evaluated at s=jωo:
1
( ωs +1) ( sω +1)
L
S
ω0
<<1
ωL
1
( jωω0 +1) ( jωω 0 +1)
=
s=jω0
L
ω0
ωS
,
S
is on the order of unity (at least)!
So this transfer function is well-approximated by the single-pole transfer function
frequencies from DC to ωo.
1
( ωs +1)
for
S
Page Cite as: Joel Dawson, course materials for 6.302 Feedback Systems, Spring 2007.
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6.302 Feedback Systems
Recitation 5: Op-amp Circuits and Analog Computers
Prof. Joel L. Dawson
But what determines ωo? Depends on the context, but for feedback systems we often look at the loop trans­
mission.
X(s)
-
Σ
Y(s)
G
H
When |L(s)| = |GH| <<1, we arguably have an open-loop system:
Y(s)
X(s)
=
G
1+GH
≈
G
So when we’re looking for dynamics to ignore, we will often discard poles that are large compared to the
loop crossover frequency, or the frequency at which |L(S)| = .
Op-amp circuits for modeling our systems
We need an integrator, a summer, an inversion, and a gain.
+
∑
=>
v2
R
f
…
+
R2
-
v1
R1
V0 = -
+
vO
( )
Rƒ
R
V1 + - ƒ V2 + …
R1
R2
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6.302 Feedback Systems
Recitation 5: Op-amp Circuits and Analog Computers
Prof. Joel L. Dawson
Rf
Ri
k1
+
=>
k1 =
Rf
Ri
(Note that this also covers us for an inversion.)
=>
∫
v0 =
vi
C
Ri
+-
vo
1
sRiC
And, a block that we do not use:
R
vi
d
dt
v0
vi
C
+-
vo
= -sCR
Difficult to manage in a noisy world:
High frequency noise is empha­
sized in the output
|H(s)|
20dB/dec
ω
Page 3
Cite as: Joel Dawson, course materials for 6.302 Feedback Systems, Spring 2007.
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6.302 Feedback Systems
Recitation 5: Op-amp Circuits and Analog Computers
Prof. Joel L. Dawson
Also against the differentiator: it’s hard enough to get high gain at DC. High gain at high frequencies?
Forget it.
Now, on to building analog computers. Suppose we have an all-pole system. It begins as a differential
equation:
an
dny
dn-1y
dn-2y
+
a
+
a
+ .......+a0y = x
n-1 dtn-1
n-2 dtn-2
dtn
Completely general procedure starts with taking the Laplace transform:
(ansn + an-1sn-1 + an-2sn-2 + .....+ a0)Y(s) = x(s)
The system function, BTW, is: Y(s)
x(s)
=
1
ansn + an-1sn-1 + an-2sn-2 + .....+ a0
Solve for the highest order derivatve:
ansnY(s) = X(s) - (an-1sn-1 + an-2sn-2 + .....+ a0)Y(s)
snY(s) =
1
X(s) ­
an
an-1 n-1
a
a
s + n-2 sn-2 + .......+ 0
an
an
an
Y(s)
Put down a big summing junction:
1
X(s)
an
Σ
snY(s)
Page Cite as: Joel Dawson, course materials for 6.302 Feedback Systems, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.302 Feedback Systems
Recitation 5: Op-amp Circuits and Analog Computers
Prof. Joel L. Dawson
Generate the derivatives that you need:
x(s)
1
an
Σ
snY(s)
1
s
sn-1Y(s)
1
s
1
s
Y(s)
Complete the mapping:
x(s)
1
an
Σ
snY(s)
1
s
sn-1Y(s)
1
s
1
s
Y(s)
an-1
an
an-2
an
a0
an
EXAMPLE: First order system
Y(s)
=
x(s)
1
τs+1
(τs+1)Y(s) = x(s)
sY(s) = 1 x(s) - 1 Y(s)
τ
τ
=>
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6.302 Feedback Systems
Recitation 5: Op-amp Circuits and Analog Computers
Prof. Joel L. Dawson
x(s)
1
τ
sY(s)
Σ
-
1
s
Y(s)
1
τ
2nd order system from class:
v0
vi
=
s
ωn2
2
+ 2ζs + 1
ωn
With a little bit of manipulation, we can write the block diagram as
s2 v
sv0
0
2
ω
vi
ωn
ωn
n
Σ
s
- -
ωn
s
v0
2ζ
If we built this as an electronic circuit, it would be analogous to our mechanical system consisting of a mass,
a viscous fluid, and a forcing mechanism:
βv
F
M
x
xD
Σ
- -
av
M
s2X
1
s
sX
1
s
Is the
po
where sition
I want
it?
x
β
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6.302 Feedback Systems
Recitation 5: Op-amp Circuits and Analog Computers
Prof. Joel L. Dawson
An analog computer might look something like this:
vi
+
+
+
+
vO
+
Make sure that ωn is small compared to the parasitic poles of the op-amps. Then, we get a very good analog.
Page Cite as: Joel Dawson, course materials for 6.302 Feedback Systems, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].