EE2010 Fundamentals of Electric Circuits

Lecture 09Network Theorems

Introduction

A largecomplex circuits

A largecomplex circuits

Simplifycircuit analysis

Simplifycircuit analysis

Circuit TheoremsCircuit Theorems

‧Thevenin’s theorem Norton theorem‧ ‧ Superposition max. power transfer‧‧Thevenin’s theorem Norton theorem‧ ‧ Superposition max. power transfer‧

linear circuit

A linear circuit is one whose output is linearly related (or directly proportional) to its input

Superposition principle

The 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.

Turn off, killed, inactive source: independent voltage source: 0 V (short circuit) independent current source: 0 A (open circuit)

Dependent sources are left intact.

Superposition principleSteps to apply superposition principle:

1. Turn off all independent sources except one source. Find the output (voltage or current) due to that active source using nodal or mesh analysis.

2. Repeat step 1 for each of the other independent sources.3. Find the total contribution by adding algebraically all the

contributions due to the independent sources. Turn off voltages sources = short voltage sources; make it

equal to zero voltage Turn off current sources = open current sources; make it

equal to zero current Superposition involves more work but simpler circuits. Superposition is not applicable to the effect on power.

Removing the effect of ideal sources

Voltage source is replaced by a S/C

Current source is replaced by a O/C

Removing the effect of practical sources

Superposition principle

(b) Dependent Current Source

A current source whose parameters are controlled by voltage/current else where in the system

v = αVx VDCS(Voltage Dependent Current source)

v = βix CDCS(Current Dependent Current source)

(a) Dependent Voltage Source

v = µVx VDVS(Voltage Dependent Voltage source)

v = ρix CDVS(Current Dependent Voltage source)

A voltage source whose parameters are controlled by voltage/current else where in the system

Dependent Source

For Superposition, All dependent sources must be left intact!!You can’t apply O/C and S/C on dependent sources

Superposition principle

Example -1

Using the superposition theorem, determine current I1 for the network in Fig.

Solution: Since two sources are present, there are two networks to be analyzed. First let us determine the effects of the voltage source by setting the current source to zero amperes as shown in Fig.

Example -1

Since I1’ and I1’’ have the same defined direction, the total current is defined by

The voltage source is in parallel with the current source and load resistor R1, so the voltage across each must be 30 V. The result is that I1 must be determined solely by

Using the superposition theorem, determine the current through the 12Ω resistor in Fig.

Example -2

Considering the effects of the 54 V source requires replacing the 48 V source by a short-circuit equivalent as shown in Fig. The result is that the 12Ω and 4Ω resistors are in parallel.

The total resistance seen by the source is

Example -2

and the source current is

Using the current divider rule results in the contribution to I2 due to the 54 V source:

Example -2

The total resistance seen by the 48 V source is

Applying the current divider rule results in

13

Use the superposition theorem to find v in the circuit in Fig.

Example -3

14

Since there are two sources,

let

Voltage division to get

Current division, to get

Hence

And we find

21 vvv

V2)6(84

41

v

A2)3(848

3

i

V84 32 iv

V108221 vvv

Example -3

a. Using the superposition theorem, determine the current through resistor R2 for the network in Fig.

b. Demonstrate that the superposition theorem is not applicable to power levels.

Example -4

Solutions:(a) Simple series circuit with a current equal to

Example -4

Parallel combination of resistors R1 and R2. Applying the current divider rule

The total solution for current I2 is thesum of the currents established by the two sources.

(b) The power delivered to the 6 Ω resistor is

Example -4

Using the total resultant current

Using the principle of superposition, find the current l2 through the 12 k resistor in Fig.

Example -5

Solution: Considering the effect of the 6 mA current source

Example -5

Current divider rule:

Considering the effect of the 9 V voltage source

Example -5