Fall 2001 ENGR201 Circuits I - Chapter 4 1

Chapter 4 - Additional Analysis Techniques(for Linear Circuits): Linearity and Superposition

Definition: A linear circuit is a circuit constructed using elements that have linear voltage-current relationships. That is, V = f(I) and I = f(V) are linear relationships.

Read Chapter 4, pages 113 - 120

Fall 2001 ENGR201 Circuits I - Chapter 4 2

Examples of linear functions:dx

y mx y y xdxdt

Examples of nonlinear functions: 2 lny mx y x y x

Note that Ohm’s law is a linear relationship: V = IR

Q: What is a linear function?A: A linear function is a function which satisfies both of the following properties:• additivity

if y = f(x) and y1 = f(x1) and y2 = f(x2) theny = f(x1 + x2) = y1 + y2

• homogeneity:if y = f(x) then y1 = f(Kx) =Ky

•additivity and homgeneity combined:if y = f(x) and y1 = a1f(x1) and y2 = a21f(x2) then

y = f(a1x1 +a2x2) = a1y1 + a2y2

Fall 2001 ENGR201 Circuits I - Chapter 4 3

Using Linearity to Analyze Circuits

+Vo

-

What is the value of Vo in the circuit shown? by the VDR, Vo = (1/3) 12V = 4v

What is the value of Vo if the 12-volt source is doubled to 24v? Vo = (1/3) 24V = 8v Vo is doubled

What is the value of Vo if the 12-volt source cut in half (6v)? Vo = (1/3) 6V = 2v Vo is halved

Because the circuit is linear, Vo is linear function of the input (Vo = Vin/3)multiplying the input by a constant results in the output being multiplied by the same constant.

Fall 2001 ENGR201 Circuits I - Chapter 4 4

+Vo

-

Example 4.1 Use linearity to find Vo

Solution: Assume that Vo is known and that Vs is unknown and workthe problem “backwards”.

Assume a convenient value for Vo: assume Vo = 1v Io = 1v/2k = 0.5ma

Io

By KVL VA = 4k Io + 1v =4k 0.5ma + 1v = 3v

+VA

-

I1 = 3v/3k = 1ma

I1

By KCL, Is = I1 + Io = 1.5ma

Is

Vs = 2k Is + VA = 2k 1.5ma + 3v = 6v

Since the actual value of Vs is 12v (2 6v), then the actual value of Vo is 2 1v = 2v

If we know the value of Vo for one value of Vs , we know Vo for any value of Vs.

Fall 2001 ENGR201 Circuits I - Chapter 4 5

Superposition

Superposition is based upon the additivity property of linear circuits and applies if the circuit has more than one independent source.

Superposition: • For a source with multiple independent sources, determine the desired

voltage or current when each source is applied with all other sources zeroed out.

– a voltage source is zeroed by replacing it with a short circuit– a current source is zeroed by replacing it with an open circuit

• The response when all independent sources are applied simultaneously is the sum of the responses to the individual sources.

Fall 2001 ENGR201 Circuits I - Chapter 4 6

Apply 2mA source:

2

2 2

3 22

3 6 3

26 4

36

o

o o o

kI mA mA

k k

V mA k v

V V V V

Example 4.3 Find Vo using superposition

+Vo

-

Apply 3V source:

+Vo1

-

1

63 2

6 2 1o

kV V V

k

I

+Vo2

-

Fall 2001 ENGR201 Circuits I - Chapter 4 7

Voltage Sources

Observations and definitions:• RL, IL, and VL are the load resistance, current, and voltage

• Ri is the internal resistance of the voltage source

• no load is defined as the condition IL = 0 A (open circuit PL = 0 W)

• by KVL, VL = VS – ILRi

• the difference between the available voltage, VS, and the load voltage, VL, is the drop across the internal resistance

• for an ideal voltage source, Ri = 0

Fall 2001 ENGR201 Circuits I - Chapter 4 8

Ideal Versus Practical Voltage Sources

VL = VS – ILRi

VL

IL

VS

no load

Ideal Voltage Source

VL

IL

VS

no load

Practical Voltage Source

full load

(slope = -Ri)

Fall 2001 ENGR201 Circuits I - Chapter 4 9

Current Sources

Observations and definitions:• RL, IL, and VL are the load resistance, current, and voltage

• Ri is the internal resistance of the current source

• no load is defined as the condition VL = 0 V (short circuit PL = 0 W)

• by KCL, IL = IS – IL

• the difference between the available current, IS, and the load current, IL, is the amount of current flowing through the internal resistance

• for an ideal current source, Ri =

Fall 2001 ENGR201 Circuits I - Chapter 4 10

Ideal Versus Practical Current Sources

IL = IS – Ii

IL

VL

IS

no load

Ideal Current Source

IL

VL

IS

no load

Practical Current Source

full load

(slope = -1/Ri)i

L Si L

RI I

R R