ECE 1311 Ch4 - Farah Isa

ECE 1311

Chapter 4 Circuit Theorems

1

Outlines

Linearity Property

Superposition

Source transformation

Thevenins theorem

Nortons theorem

Maximum power transfer

2

Linearity Property

3

Learn theorems/methods for analysing electric circuits.

i.e. superposition, source transformation, thevenin and norton

are applicable to linear circuits.

Linearity property is a combination of both the

homogeneity (scaling) property and the additivity

property.

Homogeneity property:

If the input is multiplied by a constant, then the output is multiplied by

the same constant.

i.e. kvkiRiRv

Linearity Property

4

Additivity property:

Requires that the response to a sum of inputs is the sum of the

responses to each input applied separately.

i.e.

A circuit is linear if it satisfies both the homogeneity

property and additivity property.

A linear circuit consists of linear elements, linear

dependent sources and linear independent sources.

21212121

2211

vvRiRiRiivgives

iiiapplyingthen

RivandRiv

Example 1

5

For the circuit shown, find v0 when is=15 A and is=30 A.

The Mc-Graw-Hill Companies

Answers:

20V and 40V

Example 2

6

Assume that V0 =1 V and use the linearity to calculate the

actual value of V0 in the circuit shown.


Answer:

12 V

Superposition

7

Another method for analysing circuits.

Superposition principle states that:

The voltage across (or current through) an element in a linear

circuit is the algebraic sum of the voltages across (or currents

through) that element due to each independent source acting

alone.

Steps to Apply Superposition

8

1. Turn off all independent sources except one source.

Turn off VOLTAGE source by replacing the element with a SHORT circuit.

Turn off CURRENT source by replacing the element with a OPEN circuit.

2. Find the output (voltage or current) due to that active source.

3. Repeat Step 1 and Step 2 for each of the other independent sources.

4. Find the total contribution by adding algebraically all the contributions due to the independent sources.

Note: Dependent sources are left intact because they are controlled by circuit variables.

Example 3

9

Find v0 using superposition.


Answer:

6V

Example 4

10

Find vx using superposition.


Answer:

25V

Example 5

11

Find I using superposition.


Answer:

750mA

Source Transformation

12

Another method for simplifying electric circuits.

What is it? The process of replacing a voltage source vs in series with a resistor R

with a current source Is in parallel with a resistor R, or vice versa.

Note: The arrow of the current source is directed toward the positive terminal

of the voltage source.

Source transformation is not possible when R=0 or R=infinity.

Independent sources Dependent sources


Example 6

13

Find I0 using source transformation.


Answer:

1.78A

Example 7

14

Find ix using source transformation.


Answer:

7.059mA

Example 8

15

Find vx using source transformation.


Answer:

3.65V

Thevenins Theorem

16

The theorem states that:

A linear two-terminal circuit can be replaced by an equivalent

circuit consisting of a voltage source VTh in series with a resistor

RTh. VTh is the open circuit voltage at the terminals whereas RTh is the input or equivalent resistance at the terminals when the independent

sources are turned off.

To determine VTh, set terminals a-b to open circuit.

To determine RTh, turn off all independent sources.


Thevenins Theorem

17

Two cases to consider for RTh:

Case 1: No dependent sources in the network.

Turn of all independent sources.

RTh is the input resistance looking between terminal a-b.


Example 9

18

Determine Thevenins equivalent.


Answer:

VTh=9V

RTh=3 I=2.25A

Thevenins Theorem

19

Case 2: A network with dependent sources.

Turn of all INdependent sources.

Apply a voltage source v0 at terminals a-b and determine the resulting

current i0. Then RTh= v0 / i0.

Alternatively, insert a current i0 and determine v0.

May assume any values of v0 and i0 (i.e. v0 =1V or i0=1A).


Example 10

20



Answer:

VTh=5.333V

RTh=444.4m

Example 11

21



Answer:

VTh=0V

RTh=-7.5

Thevenins Theorem

22

A linear circuit with a variable load can be replaced by

the Thevenin equivalent.

Consider a linear circuit is terminated by a load RL, the

current through the load IL and the voltage across the

load VL can be determined as follows:


Th

LTh

LLLL

L

ThL

VRR

RIRV

RRTh

VI

Nortons Theorem

23

The theorem states that:

A linear two terminal circuit can be replaced by an

equivalent circuit consisting of a current source IN in

parallel with a resistor RN IN is the short circuit current through the terminals.

RN is the input or equivalent resistance at the terminals when the

independent sources are turned off.


Nortons Theorem

24

To determine IN, set the terminal a-b to short circuit. Thus:

IN=isc

Dependent and independent sources are treated the same way as in Thevenins Theorem.

.


Th

ThN

ThN

R

VI

RR

Example 12

25

Determine Nortons equivalent.


Answer:

IN=4.5A

RN=3

Example 13

26

Determine Nortons equivalent.


Answer:

IN=10A

RN=1

Maximum Power Transfer

27

Thevenins equivalent is useful in finding the maximum

power in a linear circuit.

If the entire circuit is replaced by its Thevenin

equivalent except for the load, the power delivered

to the load is:


L

LTh

ThL R

RR

VRiP

2

2

Maximum Power Transfer

28

For a given circuit, VTh and RTh are fixed.

The power delivered can be varied by varying RL.

Power is maximum when RL = RTh.

Th

Th

Th

ThTh

Th

ThL

R

Vp

RRR

Vp

RRwhen

4max

max

2

2

The power transfer profile with different RL

Example 14

29

The variable resistor R is adjusted until it absorbs the

maximum power from the circuit below.

Calculate the value of R for maximum power.

Determine the maximum power absorbed by R.


Answer:

R=25 P=7.84W

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ECE 1311 Ch4 - Farah Isa