Systematic methods for labeling circuits and finding a solvable set of equations, Operational Amplifiers
Kevin D. Donohue, University of Kentucky 1

 Simple circuits with single loops or node-pairs can result in one equation with one unknown, when properly labeled.
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
Is 3A
+
Vx
-
Kevin D. Donohue, University of Kentucky 3

For more complex circuits a set of labels and equations only in terms of loop currents can be developed.
3I
2
R
5
R0
15
I
2
Kevin D. Donohue, University of Kentucky 4

 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
 If any branch contains a voltage source
 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

 Perform nodal analysis on circuits with current sources and resistors.
 Perform nodal analysis on circuits with voltage sources and and resistors.
Kevin D. Donohue, University of Kentucky 6

 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.
 If any branch contains a current source,
 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

 Perform loop analysis on circuits with voltage sources and resistors.
 Perform loop analysis on circuits with current sources and and resistors.
Kevin D. Donohue, University of Kentucky 8

 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
Vi
Ri
1K
+
Vo
-
Kevin D. Donohue, University of Kentucky 9

 The actual op amp is composed of many transistors, but can be approximated with a simpler circuit model:
V
+
V
-
Ri
Ro
A(V
+
- V
-
) +
Vo
-
V
+
V
-
+
Vo
-
Kevin D. Donohue, University of Kentucky 10



Typical values for Ri = 2 M

, A=10 6 , and Ro=50

.
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
-
V d

0
+
I
+

0
Kevin D. Donohue, University of Kentucky 11

 The simplifications for the op amp model suggest that nodal analysis will often be the best method of analyzing op amp circuits.
 Do examples of circuits with op amps, independent sources, and resistors.
Kevin D. Donohue, University of Kentucky 12

 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

 2 Microphones with sensitivities of 3 mV/dB and 6 mV/dB and are denoted as independent voltage sources V
1 and V
2
, 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.
R
2
V
1
R
1
R
3
R
5
R
6
+
Vo
-
R
4
V
2
Kevin D. Donohue, University of Kentucky 14

It can be shown that for the subcircuit:
V
1
R
1
R
3
R
2
+
Va
-
V a

V
2





1

1

R
1
R
2
R
3
R
4






V
1
R
2
R
1
R
4
V
2
Kevin D. Donohue, University of Kentucky 15

It can be shown that for subcircuit:
+
Va
-
R
5
R
6
+
Vo
-
V o
V a




1

R
6
R
5



Kevin D. Donohue, University of Kentucky 16

V
1
It can be shown that for the entire circuit:
R
2
R
1
R
3
V
2
R
4
R
6
+
Vo
-
R
5
V o


 1

R
6
R
5



 V
2





1

1

R
1
R
2
R
3
R
4






V
1
R
2
R
1


Kevin D. Donohue, University of Kentucky 17

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

 Scale the 3 mV/dB microphone circuit 2 times the amount as the other:



2 


1

1

R
1
R
2
R
3
R
4







R
2
R
1
 Scale the 3 mV/dB microphone through the first and second circuit to achieve a 30 mV/dB scale:
30 mV / dB


 1

R
6
R
5




R
2
R
1
3 mV / dB


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