This document discusses modeling electronic systems using operational amplifiers. It provides operational amplifier basics, examples of calculating transfer functions of operational amplifier circuits, and discusses using operational amplifiers for lead/lag compensation and PID controllers. Specifically, it gives examples of calculating transfer functions, discusses electric network transfer functions, and provides a table of common operational amplifier circuits that can be used as controllers or compensators.

1. Modeling of Electronic System
•Operational Amplifier Basics
•Solved Examples
•Lead, Lag Compensation
•PID Controller
1

2. Operational amplifiers
We now discuss active circuits that can be used to implement controllers and compensators. These are
circuits built around an operational amplifier.
An operational amplifier (or differential amplifier), as pictured in the figure above, is an electronic
amplifier, which has the following characteristics:
1. Differential input, v2(t) −v1(t)
2. High input impedance, Zi = (ideal)
3. Low output impedance, Zo = 0 (ideal)
4. High constant gain amplification, A =  (ideal)
5. The output vo(t) of an operational amplifier is given by vo (t) = A(v2(t) −v1(t))

3. Operational Amplifiers
3
1
2
Z
Z
V
V
in
out


1
2
1
Z
Z
V
V
in
out



4. Example#1
• Find out the transfer function of the following
circuit.
4
1
2
Z
Z
V
V
in
out



5. Example#2
• Find out the transfer function of the following
circuit.
5
v1

6. E(s) Z2(s)
= −
Ei(s) Z1(s)
where
Z1(s) =
R1
R1C1s+ 1
, Z2(s) =
R2C2s+ 1
C2s
Example#3

7. Electric network transferfunctions

8. Operational amplifier circuits used as controllers or compensators
Table below presents a list of operational amplifier circuits that may be used as controllers or
compensators.
.

9. Operational amplifier circuits usedas controllers or compensators