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Methods of Analysis ET 162 Circuit Analysis Electrical and Telecommunication Engineering Technology Professor Jang

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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.

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OUTLINES Introduction to Network Theorems Superposition Thevenin’s Theorem Norton’s Theorem Maximum Power Transfer Theorem ET162 Circuit Analysis – Network Theorems Boylestad 2 Key Words: Network Theorem, Superposition, Thevenin, Norton, Maximum Power

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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. Introduction to Network Theorems ET162 Circuit Analysis – Network Theorems Boylestad 3

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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 networksNumber of to be analyzedindependent sources = ET162 Circuit Analysis – Network Theorems Boylestad 4

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FIGURE 9.2 Removing the effects of ideal sources 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 ET162 Circuit Analysis – Network Theorems Boylestad 5

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Ex. 9-1 Determine I 1 for the network of Fig. 9.3. FIGURE 9.3 FIGURE 9.4 ET162 Circuit Analysis – Network Theorems Boylestad 6

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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 ET162 Circuit Analysis – Network Theorems Boylestad 7

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FIGURE 9.7 FIGURE 9.8 ET162 Circuit Analysis – Network Theorems Boylestad 8

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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 9.11 FIGURE 9.10 ET162 Circuit Analysis – Network Theorems Boylestad 9

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FIGURE 9.12 ET162 Circuit Analysis – Network Theorems10

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FIGURE 9.13 Ex. 9-4 Using the principle of superposition, find the current through the 12-kΩ resistor of Fig. 9.13. FIGURE 9.14 ET162 Circuit Analysis – Network Theorems Boylestad 11

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12 FIGURE 9.15

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FIGURE 9.16 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 9.18FIGURE 9.19 FIGURE 9.17 ET162 Circuit Analysis – Network Theorems Boylestad 13

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ET162 Circuit Analysis – Methods of Analysis Boylestad 14 FIGURE 9.20

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Thevenin’s Theorem FIGURE 9.21 Thevenin equivalent circuit 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 9.22 The effect of applying Thevenin’s theorem. ET162 Circuit Analysis – Network Theorems Boylestad 15

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ET162 Circuit Analysis – Methods of AnalysisBoylestad16 FIGURE 9.23 Substituting the Thevenin equivalent circuit for a complex network. 1.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 R L be temporary removed from the network. 2.Make the terminals of the remaining two-terminal network. 3.Calculate R TH 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. 4.Calculate E TH by first returning all sources to their original position and finding the open-circuit voltage between the marked terminals. 5.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 9.24 FIGURE 9.25 Identifying the terminals of particular importance when applying Thevenin’s theorem. FIGURE 9.26 Determining R TH for the network of Fig. 9.25. ET162 Circuit Analysis – Network Theorems Boylestad 17

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18 FIGURE 9.27 FIGURE 9.29 Substituting the Thevenin equivalent circuit for the network external to R L in Fig. 9.23. FIGURE 9.28

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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 9.30FIGURE 9.31 FIGURE 9.32 ET162 Circuit Analysis – Network Theorems Boylestad 19

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FIGURE 9.33 FIGURE 9.34 Substituting the Thevenin equivalent circuit in the network external to the resistor R 3 of Fig. 9.30. ET162 Circuit Analysis – Network Theorems Boylestad 20

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ET162 Circuit Analysis – Methods of Analysis Boylestad 21 FIGURE 9.35 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 9.37 FIGURE 9.36

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FIGURE 9.38 FIGURE 9.40 Substituting the Thevenin equivalent circuit in the network external to the resistor R 4 of Fig. 9.35. FIGURE 9.39 ET162 Circuit Analysis – Network Theorems Boylestad 22

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ET162 Circuit Analysis – Methods of Analysis Boylestad 23 Ex. 9-9 Find the Thevenin equivalent circuit for the network in the shaded area of the network of Fig. 9.41. FIGURE 9.43 FIGURE 9.42 FIGURE 9.41

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FIGURE 9.46 Substituting the Thevenin equivalent circuit in the network external to the resistor R L of Fig. 9.41. FIGURE 9.45 FIGURE 9.44 Boylestad 24

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ET162 Circuit Analysis – Methods of Analysis Boylestad 25 FIGURE 9.47 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 9.49

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ET162 Circuit Analysis – Methods of Analysis Boylestad 26 FIGURE 9.52 Substituting the Thevenin equivalent circuit in the network external to the resistor R L of Fig. 9.47. FIGURE 9.51 FIGURE 9.50

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Experimental Procedures FIGURE 9.53 FIGURE 9.54 FIGURE 9.55 ET162 Circuit Analysis – Network Theorems Boylestad 27

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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 ET162 Circuit Analysis – Network Theorems Boylestad 28

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1.Remove that portion of the network across which the Thevenin equivalent circuit is found. 2.Make the terminals of the remaining two-terminal network. 3.Calculate R N 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. 4.Calculate I N by first returning all sources to their original position and finding the short-circuit current between the marked terminals. 5.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. 29

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Ex. 9-11 Find the Norton equivalent circuit for the network in the shaded area of Fig. 9.58. FIGURE 9.59 Identifying the terminals of particular interest for the network of Fig. 9.58. FIGURE 9.58 FIGURE 9.60 Determining R N for the network of Fig. 9.59. Boylestad 30

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FIGURE 9.61 Determining R N for the network of Fig. 9.59. FIGURE 9.62 Substituting the Norton equivalent circuit for the network external to the resistor R L of Fig. 9.58. FIGURE 9.63 Converting the Norton equivalent circuit of Fig. 9.62 to a Thevenin’s equivalent circuit.

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Ex. 9-12 Find the Norton equivalent circuit for the network external to the 9-Ω resistor in Fig. 9.64. R N = R 1 + R 2 = 5 Ω + 4 Ω = 9 Ω FIGURE 9.64FIGURE 9.65 FIGURE 9.66 Boylestad 32

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FIGURE 9.67 Determining I N for the network of Fig. 9.65. FIGURE 9.68 Substituting the Norton equivalent circuit for the network external to the resistor R L of Fig. 9.64. ET162 Circuit Analysis – Network Theorems Boylestad 33

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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 in Fig. 9.69. FIGURE 9.69 FIGURE 9.70 FIGURE 9.71 Boylestad 34

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FIGURE 9.74 Substituting the Norton equivalent circuit for the network to the left of terminals a-b in Fig. 9.69. FIGURE 9.72 FIGURE 9.73 Boylestad 35

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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. 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. R L = R TH R L = R N ET162 Circuit Analysis – Network Theorems Boylestad 36

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