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ECE 1311 ECE 1311 Electric CircuitElectric Circuit

Chapter 4Chapter 4

Circuit TheoremsCircuit Theorems

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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Circuit Theorems - Circuit Theorems - Chapter 4Chapter 4

4.1 Motivation4.2 Linearity Property4.3 Superposition4.4 Source Transformation4.5 Thevenin’s Theorem4.6 Norton’s Theorem4.7 Maximum Power Transfer

4.1 Motivation (1)4.1 Motivation (1)• Solve for v0 as a function of Vs.

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08:52 PM Dr. AHM Zahirul Alam

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4.2 Linearity Property (1)4.2 Linearity Property (1)It is the property of an element describing a linear relationship between cause and effect.

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

Homogeneity (scaling) property

v = i R → k v = k i R

Additive property

v1 = i1 R and v2 = i2 R

→ v = (i1 + i2) R = v1 + v2

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4.2 Linearity Property (2)4.2 Linearity Property (2)Example 1

Assume Io = 1 A, use linearity to find the actual value of Io in the circuit shown below.

*Refer to in-class illustration, text book, answer Io = 3A

Assume that Vo = 1V and use linearity property to calculate the actual value of Vo in the circuit.

08:52 PM Dr. AHM Zahirul Alam

08:52 PM Dr. AHM Zahirul Alam

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4.3 Superposition Theorem (1)4.3 Superposition Theorem (1)It states that the voltage across (or current through) an element in a linear circuit is the algebraic sum of the voltage across (or currents through) that element due to EACH independent source acting alone.

The principle of superposition helps us to analyze a linear circuit with more than one independent source by calculating the contribution of each independent source separately.

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We consider the effects of 8A and 20V one by one, then add the two effects together for final vo.

4.3 Superposition Theorem (2)4.3 Superposition Theorem (2)

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4.3 Superposition Theorem (3)4.3 Superposition Theorem (3)

Steps 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.

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4.3 Superposition Theorem (4)4.3 Superposition Theorem (4)

Two things have to be keep in mind:

1. When we say turn off all other independent sources: Independent voltage sources are replaced

by 0 V (short circuit) and Independent current sources are replaced

by 0 A (open circuit).

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

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4.3 Superposition Theorem (5)4.3 Superposition Theorem (5)Example 2: Use the superposition theorem to find v in the circuit shown below.

3A is discarded by open-circuit

6V is discarded by short-circuit

*Refer to in-class illustration, text book, answer v = 10V

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4.3 Superposition Theorem (5)4.3 Superposition Theorem (5)Example 2

Use the superposition theorem to find v in the circuit shown below.

3A is discarded by open-circuit

6V is discarded by short-circuit

*Refer to in-class illustration, text book, answer v = 10V

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4.3 Superposition Theorem (6)4.3 Superposition Theorem (6)Example 3Use superposition to find vx in the circuit below.

*Refer to in-class illustration, text book, answer Vx = 12.5V

2A is discarded by open-circuit

20 v1

4 10 V+

(a)

0.1v14

2 A

(b)

20

0.1v2

v2

10V is discarded by open-circuit

Dependant source keep unchanged

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4.3 Superposition Theorem (6)4.3 Superposition Theorem (6)Example 3

Use superposition to find vx in the circuit below.

*Refer to in-class illustration, text book, answer Vx = 12.5V

2A is discarded by open-circuit

20 v1

4 10 V+

(a)

0.1v14

2 A

(b)

20

0.1v2

v2

10V is discarded by open-circuit

Dependant source keep unchanged

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4.4 Source Transformation (1)4.4 Source Transformation (1)

•The arrow of the current source is

directed toward the positive

terminal of the voltage source.

•The source transformation is not possible when R = 0 for voltage source and R = ∞ for current source.

(a) Independent source transform

(b) Dependent source transform

++

++

--

--

Req

Req

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4.4 Source Transformation (2)4.4 Source Transformation (2)• An equivalent circuit is one whose v-i

characteristics are identical with the original circuit.

• It is the process of replacing a voltage source vS in series with a resistor R by a current source iS in parallel with a resistor R, or vice versa.

R

viorRiv ssss

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4.4 Source Transformation (3)4.4 Source Transformation (3)

(a) Independent source transform

(b) Dependent source transform

•The arrow of the current source is

directed toward the positive

terminal of the voltage source.

•The source transformation is not possible when R = 0 for voltage source and R = ∞ for current source.

++

++

--

--

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4.4 Source Transformation (4)4.4 Source Transformation (4)Example 4

Find io in the circuit shown below using source transformation.

*Refer to in-class illustration, textbook, answer io = 1.78A

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4.4 Source Transformation (5)4.4 Source Transformation (5)Example

Find ix in the circuit shown below using source transformation.

*Refer to in-class illustration, textbook, answer ix = 7.056 mA

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4.5 Thevenin’s Theorem (1)4.5 Thevenin’s Theorem (1)It states that a linear two-terminal circuit (Fig. a) can be replaced by an equivalent circuit (Fig. b) consisting of a voltage source VTh in series with a resistor RTh,

where

• VTh is the open-circuit voltage at the terminals.

• RTh is the input or equivalent resistance at the terminals when the independent sources are turned off.

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4.5 Thevenin’s Theorem (2)4.5 Thevenin’s Theorem (2)Example 5Using Thevenin’s theorem, find the equivalent circuit to the left of the terminals in the circuit shown below. Hence find i.

*Refer to in-class illustration, textbook, answer VTH = 6V, RTH = 3, i = 1.5A

6

4

(a)

RTh

6

2A

6

4

(b)

6 2A

+VTh

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4.5 Thevenin’s Theorem (3)4.5 Thevenin’s Theorem (3)

Example 5Using Thevenin’s theorem, find the equivalent circuit to the left of the terminals in the circuit shown below. Hence find i.

*Refer to in-class illustration, textbook, answer VTH = 6V, RTH = 3, i = 1.5A

6

4

(a)

RTh

6

2A

6

4

(b)

6 2A

+VTh

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4.5 Thevenin’s Theorem (4)4.5 Thevenin’s Theorem (4)

Example 6Find the Thevenin equivalent circuit of the circuit shown below to the left of the terminals.

*Refer to in-class illustration, textbook, answer VTH = 5.33V, RTH = 3

6 V

5 Ix

4 +

(a)

1.5Ix

i1

i2

i1 i2

3

o

+VTh

b

a

1.5Ix 1 V+

3 0.5Ix

5

(b)

a

b

4

Ix i

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4.5 Thevenin’s Theorem (4)4.5 Thevenin’s Theorem (4)

Example 6

Find the Thevenin equivalent circuit of the circuit shown below to the left of the terminals.

*Refer to in-class illustration, textbook, answer VTH = 5.33V, RTH = 3

6 V

5 Ix

4 +

(a)

1.5Ix

i1

i2

i1 i2

3

o

+VTh

b

a

1.5Ix 1 V+

3 0.5Ix

5

(b)

a

b

4

Ix i

ExampleExample• Find the Thevenin equivalent at terminals

a-b of the circuit in Fig. 4.107.

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Assignment#4Assignment#4Due: 24-25/01/2011Due: 24-25/01/2011

• 4.4, 4.19, 4.26, 4.34, 4.39

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ReminderReminder• Mid-Term exam

• Friday Jan 28 from 8:00-10:00pm• Venue: Architecture Gallery • Time: 8:00 pm – 10:00 pm

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4.6 Norton’s Theorem (1)4.6 Norton’s Theorem (1)It states that a linear two-terminal circuit can be replaced by an equivalent circuit of a current source IN in parallel with a resistor RN,

Where • 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.

The Thevenin’s and Norton equivalent circuits are related by a source transformation.

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4.6 Norton’s Theorem (2)4.6 Norton’s Theorem (2)

Example 7Find the Norton equivalent circuit of the circuit shown below.

*Refer to in-class illustration, textbook, RN = 1, IN = 10A.

2

(a)

6

2vx

+

+vx

+vx

1V+ix

i

2

(b)

6 10 A

2vx

+

+vx

Isc

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4.6 Norton’s Theorem (3)4.6 Norton’s Theorem (3)

Example 7

Find the Norton equivalent circuit of the circuit shown below.

*Refer to in-class illustration, textbook, RN = 1, IN = 10A.

2

(a)

6

2vx

+

+vx

+vx

1V+ix

i

2

(b)

6 10 A

2vx

+

+vx

Isc

ExampleExample• Find the Thèvenin and Norton

equivalents at terminals a-b of the circuit shown in Fig. 4.108.

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4.7 Maximum Power Transfer (1)4.7 Maximum Power Transfer (1)

L

ThTHL R

VPRR

4

2

max

If the entire circuit is replaced by its Thevenin equivalent except for the load, the power delivered to the load is:

The power transfer profile with different RL

For maximum power dissipated in RL, Pmax, for a given RTH, and VTH,

LLTh

ThL R

RR

VRiP

2

2

4.7 Maximum Power Transfer (2)4.7 Maximum Power Transfer (2)

• Find the value of RL for maximum power transfer in the circuit and determine the maximum power

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Example 8Determine the value of RL that will draw the maximum power from the rest of the circuit shown below. Calculate the maximum power.

2

4

1 V+

(a)

1

3vx

+

i

v0+ vx

9 V+

io

1 +VTh

+

3vx

2

+ vx 4

(b)

Fig. a

=> To determine RTH

Fig. b

=> To determine VTH

*Refer to in-class illustration, textbook, RL = 4.22, Pm = 2.901W

4.7 Maximum Power Transfer (3)4.7 Maximum Power Transfer (3)

Assignment#5Assignment#5Due Thursday 26-27/01/2011Due Thursday 26-27/01/2011

• 4.47, 4.48, 4.60, 4.66, 4.71

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Quiz

• Use Thevenin’s theorem to find V0

Quiz

• Use Norton’s theorem to find V0