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ECE 100
Lab 4
Design of a Differentiator
Winter 2011
The goal of this lab is to design a signal processing circuit that will output the time derivative of the input scaled
by the appropriate factor. The specification for this
differentiator
is that it should convert a triangle wave of 1v
peak to peak at 1 KHz into a square wave of
±
1 v (at 1 KHz of course). The polarity of the square wave is not
important, but the 10% to 90% rise time must be less than 20
μ
s and the output voltage must not overshoot more
than 15%, i.e. 0.30 v.
The circuit can be based on the operational amplifier inverter topology. If the op amp is ideal this simply requires
an input capacitor and a feedback resistor. Then the ideal transfer function is H
D
(s) = -sRC, which (apart from a
phase shift of 180deg) is what we need. Unfortunately the simple design will be either marginally stable or
completely unstable, and some
compensation
will be required. In this case “unstable” means that the circuit will
oscillate with no input present, and thus is useless as a differentiator. The nature of the compensation is very much
like the way the follower was modified to drive a large capacitive load. However in this assignment we will study
how to do this compensation theoretically.
The transfer function H(s) for the inverter, including the opamp gain A(s) is:
H(s) = V
0
(s)/V
I
(s) = -(Z
F
(s)/Z
I
(s))*{T(s)/(1 + T(s))} where T(s) = A(s) * Z
I
(s)/(Z
I
(s) + Z
F
(s)) is the loop gain.
When |T| >> 1 H(s) -> -Z
F
(s)/Z
I
(s) which we can call the “ideal” transfer function, because it would apply if the
opamp were ideal. The factor {T/(1+T)} gives the deviation from ideal behavior. All feedback circuits have a
factor like this in their transfer function. It is the possibility of a zero in the term (1+T) that causes instability.
1. System Level Design:
Here Z
I
(s)=1/sC and Z
F
(s)=R, so H
IDEAL
(s) = -sRC and V
0
(t) = -RC dV
IN
/dt. We must
choose the RC product necessary to meet the spec. We need V
0
(t) =
±
1v for the given triangle input. All we need
is dV
IN
/dt for the triangle wave and we can find RC. A reasonable choice is R = 100K, so we can then find C.

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