Systematic methods for labeling
circuits and finding a solvable set of
equations, Operational Amplifiers
Kevin D. Donohue, University of Kentucky
1
R0
i1
R1
Simple circuits with single
loops or node-pairs can
result in one equation with
one unknown, when
properly labeled.
R
Ø
R4
R2
R3
V
Kevin D. Donohue, University of Kentucky
2
For more complex circuits a set of labels and
equations in terms of node voltages can be
developed.
2Vx
R
5
R0
15
Kevin D. Donohue, University of Kentucky
R2
5
R1
10
3A
Is
+
Vx
-
3
For more complex circuits a set of labels
and equations only in terms of loop currents
can be developed.
3I2
R
5
R0
15
R2
5
R1
10
Vs
12V
I2
Kevin D. Donohue, University of Kentucky
4
Ø
Ø
Ø
Identify and label all nodes in the system.
Select one node as a reference node (V=0)
Perform KCL at each non-reference node
expressing each branch current in terms of the
node voltages
Ø
If any branch contains a voltage source
Ø
Ø
One way: Make reference node the negative end of the voltage source
and set node values on the positive end equal to the source values
(reduces number of equations and unknowns by one)
Another way: Create an equation where the difference between the
node voltages on either end to source is equal to the source value and
then use a surface around both nodes for KCL (provides an extra
equation lost from the unknown current in voltage source)
Kevin D. Donohue, University of Kentucky
5
Ø
Perform nodal analysis on circuits with
current sources and resistors.
Ø
Perform nodal analysis on circuits with
voltage sources and and resistors.
Kevin D. Donohue, University of Kentucky
6
Ø
Ø
Create loop current labels that include every
circuit branch where each loop contains a branch
included by no other loop and no loops cross each
other.
Perform KVL around each loop expressing all
voltages in terms of loop currents.
Ø
If any branch contains a current source,
Ø
Ø
One way: Let only only one loop current pass through source so loop
current then equals the source value (reduces number of equations and
unknowns by one)
Another way: Let more than one loop pass through source and set
combination of loop current equal to source value (provides an extra
equation lost from the unknown voltage drop on current source)
Kevin D. Donohue, University of Kentucky
7
Ø
Perform loop analysis on circuits with
voltage sources and resistors.
Ø
Perform loop analysis on circuits with
current sources and and resistors.
Kevin D. Donohue, University of Kentucky
8
Ø
Operational amplifiers (op amps) were originally
developed to amplify DC voltage levels in analog
computers. Today, their applications are many. Apply
the model for the ideal op amp to find the voltage gain
(Vo/Vi) in the given circuit:
Rf
10K
Ri
1K
Ro
1K
Vi
1
+
Vo
Kevin D. Donohue, University of Kentucky
9
Ø
The actual op amp is composed of many transistors,
but can be approximated with a simpler circuit model:
Ro
V+
A(V+ - V-)
V-
Ri
V+
V-
Kevin D. Donohue, University of Kentucky
+
Vo
-
+
Vo
10
Ø
Ø
Ø
Typical values for Ri = 2 MΩ, A=106, and Ro=50Ω.
For circuit in the first example, use op amp model with
dependent source to justify the ideal approximations made
in the first example.
Ideal op amp
approximation:
I-≈0
Vd≈0
+
I+≈0
Kevin D. Donohue, University of Kentucky
11
Ø The
simplifications for the op amp
model suggest that nodal analysis will
often be the best method of analyzing
op amp circuits.
Ø Do
examples of circuits with op amps,
independent sources, and resistors.
Kevin D. Donohue, University of Kentucky
12
Ø
Ø
Ø
http://www.housing.uoguelph.ca/~antoon/gadgets/
741.htm (tutorial)
http://www.williamson-labs.com/480_opam.htm
(tutorial – WARNING: crude language and humor
used at this site. Not recommended for more
sensitive or unstable students!)
http://www.st.com/stonline/books/pdf/docs/5304.p
df (data sheet)
Kevin D. Donohue, University of Kentucky
13
Ø
2 Microphones with sensitivities of 3 mV/dB and 6 mV/dB and are
denoted as independent voltage sources V1 and V2, respectively.
Determine resistor values so that Vo is the difference between the
microphone sound pressure levels (SPL) such that a 1 dB change in
SPL corresponds to a 30 mV change in Vo.
R2
R1
R6
R3
V1
V2
R4
+
Vo
-
R5
Kevin D. Donohue, University of Kentucky
14
It can be shown that for the subcircuit:
R2
R1
R3
V1
V2
R4
+
Va
-
æ
ç 1+
Va = V2 ç
ç
ç 1+
è
Kevin D. Donohue, University of Kentucky
R1
R2
R3
R4
ö
÷
÷ −V
÷ 1
÷
ø
R2
R1
15
It can be shown that for subcircuit:
+
Va
-
R6
+
Vo
-
Vo æ
= çç1 +
Va è
R6 ö
÷
R5 ÷ø
R5
Kevin D. Donohue, University of Kentucky
16
It can be shown that for the entire circuit:
R2
R1
R6
R3
V1
V2
R4
+
Vo
-
R5
æ æ
ç ç1+
æ R6 öç ç
÷ V2
Vo = çç1 +
÷ç ç
è R5 øç ç 1 +
ç
è è
Kevin D. Donohue, University of Kentucky
R1
R2
R3
R4
ö
÷
÷ −V
÷ 1
÷
ø
ö
÷
R2 ÷
R1 ÷
÷÷
ø
17
Ø
Ø
The structure of the formula suggests breaking the
problem into 2 parts:
Part 1: Let the part of the formula associated with
the first subcircuit take care of the scaling the
microphone sensitivities (to make both
sensitivities equal) and then subtracting them.
Part 2: Let the part of formula associated with
second subcircuit take care of the scaling the
difference signal to the 30 mV/dB specification.
Kevin D. Donohue, University of Kentucky
18
Ø
Scale the 3 mV/dB microphone circuit 2 times the
amount as the other: æç 1 + R1 ö÷
2ç
ç
ç 1+
è
Ø
R2
R3
R4
÷=
÷
÷
ø
R2
R1
Scale the 3 mV/dB microphone through the first
and second circuit to achieve a 30 mV/dB scale:
æ
30mV / dB = çç1 +
è
ö
R6 öæ R2
֍
3mV / dB ÷÷
ç
÷
R5 øè R1
ø
Kevin D. Donohue, University of Kentucky
19
Note in the last example there are many
solutions and several ways to set up the
design equations and solve. Can you
determine if one way is better than the
other? What additional criteria may be
added to this problem?
Kevin D. Donohue, University of Kentucky
20