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Methods of Analysis

ET 162 Circuit Analysis

Electrical and Telecommunication

Engineering Technology

Professor Jang

Acknowledgement

I want to express my gratitude to Prentice Hall giving me the permission to use instructor’s material for developing this module. I would like to thank the Department of Electrical and Telecommunications Engineering Technology of NYCCT for giving me support to commence and complete this module. I hope this module is helpful to enhance our students’ academic performance.

OUTLINES

- Introduction to Network Theorems

- Superposition

- Thevenin’s Theorem

- Norton’s Theorem

- Maximum Power Transfer Theorem

Key Words: Network Theorem, Superposition, Thevenin, Norton, Maximum Power

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Introduction to Network Theorems

This chapter will introduce the important fundamental theorems of network analysis. Included are the superposition, Thevenin’s,Norton’s, and maximum power transfer theorems. We will consider a number of areas of application for each. A through understanding of each theorems will be applied repeatedly in the material to follow.

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Superposition Theorem

The superposition theorem can be used to find the solution to networks with two or more sources that are not in series or parallel. The most advantage of this method is that it does not require the use of a mathematical technique such as determinants to find the required voltages or currents.

The current through, or voltage across, an element in a linear bilateral network is equal to the algebraic sum of the current or voltages produced independently by each source.

Number of networks Number of

to be analyzed independent sources

=

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Figure 9.1 reviews the various substitutions required when removing an ideal source, and Figure 9.2 reviews the substitutions with practical sources that have an internal resistance.

FIGURE 9.1 Removing the effects of practical sources

FIGURE 9.2 Removing the effects of ideal sources

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Ex. 9-1 Determine I1 for the network of Fig. 9.3.

FIGURE 9.3

FIGURE 9.4

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Ex. 9-2 Using superposition, determine the current through the 4-Ω resistor of Fig. 9.5. Note that this is a two-source network of the type considered in chapter 8.

FIGURE 9.5

FIGURE 9.6

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Ex. 9-3 a.Using superposition, find the current through the 6-Ω resistor of Fig. 9.9. b. Determine that superposition is not applicable to power levels.

FIGURE 9.9

FIGURE 9.10

FIGURE 9.11

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Ex. 9-4 Using the principle of superposition, find the current through the 12-kΩ resistor of Fig. 9.13.

FIGURE 9.13

FIGURE 9.14

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Ex. 9-5 Find the current through the 2-Ω resistor of the network of Fig. 9.16. The presence of three sources will result in three different networks to be analyzed.

FIGURE 9.16

FIGURE 9.17

FIGURE 9.18

FIGURE 9.19

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Thevenin’s Theorem

Any two-terminal, linear bilateral dc network can be replaced by an equivalent circuit consisting of a voltage source and a series resistor, as shown in Fig. 9.21.

FIGURE 9.21 Thevenin equivalent circuit

FIGURE 9.22 The effect of applying Thevenin’s theorem.

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FIGURE 9.23 Substituting the Thevenin equivalent circuit for a complex network.

- Remove that portion of the network across which the Thevenin equivalent circuit is to be found. In Fig. 9.23(a), this requires that the road resistor RL be temporary removed from the network.
- Make the terminals of the remaining two-terminal network.
- Calculate RTH by first setting all sources to zero (voltage sources are replaced by short circuits, and current sources by open circuit) and then finding the resultant resistance between the two marked terminals.
- Calculate ETH by first returning all sources to their original position and finding the open-circuit voltage between the marked terminals.
- Draw the Thevenin equivalent circuit with the portion of the circuit previously removed replaced between the terminals of the equivalent circuit.

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Ex. 9-6 Find the Thevenin equivalent circuit for the network in the shaded area of the network of Fig. 9.24. Then find the current through RL for values of 2Ω, 10Ω, and 100Ω.

FIGURE 9.25 Identifying the terminals of particular importance when applying Thevenin’s theorem.

FIGURE 9.24

FIGURE 9.26 Determining RTH for the network of Fig. 9.25.

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FIGURE 9.27

FIGURE 9.29 Substituting the Thevenin equivalent circuit for the network external to RL in Fig. 9.23.

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Ex. 9-7 Find the Thevenin equivalent circuit for the network in the shaded area of the network of Fig. 9.30.

FIGURE 9.30

FIGURE 9.31

FIGURE 9.32

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FIGURE 9.34 Substituting the Thevenin equivalent circuit in the network external to the resistor R3 of Fig. 9.30.

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Ex. 9-8 Find the Thevenin equivalent circuit for the network in the shaded area of the network of Fig. 9.35. Note in this example that there is no need for the section of the network to be at preserved to be at the “end” of the configuration.

FIGURE 9.36

FIGURE 9.35

FIGURE 9.37

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FIGURE 9.39

FIGURE 9.40 Substituting the Thevenin equivalent circuit in the network external to the resistor R4 of Fig. 9.35.

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Ex. 9-9 Find the Thevenin equivalent circuit for the network in the shaded area of the network of Fig. 9.41.

FIGURE 9.42

FIGURE 9.41

FIGURE 9.43

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FIGURE 9.44

FIGURE 9.46 Substituting the Thevenin equivalent circuit in the network external to the resistor RL of Fig. 9.41.

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Ex. 9-10 (Two sources) Find the Thevenin equivalent circuit for the network within the shaded area of Fig. 9.47.

FIGURE 9.48

FIGURE 9.47

FIGURE 9.49

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FIGURE 9.51

FIGURE 9.52 Substituting the Thevenin equivalent circuit in the network external to the resistor RL of Fig. 9.47.

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Experimental Procedures

FIGURE 9.53

FIGURE 9.54

FIGURE 9.55

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Norton’s Theorem

Any two-terminal, linear bilateral dc network can be replaced by an equivalent circuit consisting of a current source and a parallel resistor, as shown in Fig. 9.56.

FIGURE 9.56 Norton equivalent circuit

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Remove that portion of the network across which the Thevenin equivalent circuit is found.

- Make the terminals of the remaining two-terminal network.
- Calculate RN by first setting all sources to zero (voltage sources are replaced by short circuits, and current sources by open circuit) and then finding the resultant resistance between the two marked terminals.
- Calculate IN by first returning all sources to their original position and finding the short-circuit current between the marked terminals.
- Draw the Norton equivalent circuit with the portion of the circuit previously removed replaced between the terminals of the equivalent circuit.

FIGURE 9.57 Substituting the Norton equivalent circuit for a complex network.

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Ex. 9-11 Find the Norton equivalent circuit for the network in the shaded area of Fig. 9.58.

FIGURE 9.58

FIGURE 9.59 Identifying the terminals of particular interest for the network of Fig. 9.58.

FIGURE 9.60 Determining RN for the network of Fig. 9.59.

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FIGURE 9.61 Determining RN for the network of Fig. 9.59.

FIGURE 9.62 Substituting the Norton equivalent circuit for the network external to the resistor RL of Fig. 9.58.

FIGURE 9.63 Converting the Norton equivalent circuit of Fig. 9.62 to a Thevenin’s equivalent circuit.

Ex. 9-12 Find the Norton equivalent circuit for the network external to the 9-Ω resistor inFig. 9.64.

FIGURE 9.64

FIGURE 9.65

RN = R1 + R2

= 5 Ω + 4 Ω

= 9 Ω

FIGURE 9.66

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FIGURE 9.67 Determining IN for the network of Fig. 9.65.

FIGURE 9.68 Substituting the Norton equivalent circuit for the network external to the resistor RL of Fig. 9.64.

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Ex. 9-13 (Two sources)Find the Norton equivalent circuit for the portion of the network external to the left of a-b inFig. 9.69.

FIGURE 9.70

FIGURE 9.69

FIGURE 9.71

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FIGURE 9.73

FIGURE 9.74 Substituting the Norton equivalent circuit for the network to the left of terminals a-b in Fig. 9.69.

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Maximum Power Transfer Theorem

A load will receive maximum power from a linear bilateral dc network when its total resistive value is exactly equal to the Thevenin resistance of the network as “seen” by the load.

RL = RTH

RL = RN

FIGURE 9.74 Defining the conditions for maximum power to a load using the Thevenin equivalent circuit.

FIGURE 9.75 Defining the conditions for maximum power to a load using the Norton equivalent circuit.

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