Methods of Analysis

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

2. 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.

3. 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 ET162 Circuit Analysis – Network TheoremsBoylestad2

4. 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. ET162 Circuit Analysis – Network TheoremsBoylestad3

5. 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 = ET162 Circuit Analysis – Network TheoremsBoylestad4

6. 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 ET162 Circuit Analysis – Network TheoremsBoylestad5

7. Ex. 9-1 Determine I1 for the network of Fig. 9.3. FIGURE 9.3 FIGURE 9.4 ET162 Circuit Analysis – Network TheoremsBoylestad6

8. 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 TheoremsBoylestad7

9. FIGURE 9.7 FIGURE 9.8 ET162 Circuit Analysis – Network TheoremsBoylestad8

10. 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 ET162 Circuit Analysis – Network TheoremsBoylestad9

11. FIGURE 9.12 ET162 Circuit Analysis – Network Theorems10

12. 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 ET162 Circuit Analysis – Network TheoremsBoylestad11

13. FIGURE 9.15 12

14. 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 ET162 Circuit Analysis – Network TheoremsBoylestad13

15. FIGURE 9.20 ET162 Circuit Analysis – Methods of AnalysisBoylestad14

16. 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. ET162 Circuit Analysis – Network TheoremsBoylestad15

17. 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. ET162 Circuit Analysis – Methods of AnalysisBoylestad16

18. 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. ET162 Circuit Analysis – Network TheoremsBoylestad17

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

20. 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 ET162 Circuit Analysis – Network TheoremsBoylestad19

21. FIGURE 9.33 FIGURE 9.34 Substituting the Thevenin equivalent circuit in the network external to the resistor R3 of Fig. 9.30. ET162 Circuit Analysis – Network TheoremsBoylestad20

22. 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 ET162 Circuit Analysis – Methods of AnalysisBoylestad21

23. FIGURE 9.38 FIGURE 9.39 FIGURE 9.40 Substituting the Thevenin equivalent circuit in the network external to the resistor R4 of Fig. 9.35. ET162 Circuit Analysis – Network TheoremsBoylestad22

24. 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 ET162 Circuit Analysis – Methods of AnalysisBoylestad23

25. FIGURE 9.45 FIGURE 9.44 FIGURE 9.46 Substituting the Thevenin equivalent circuit in the network external to the resistor RL of Fig. 9.41. Boylestad24

26. 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 ET162 Circuit Analysis – Methods of AnalysisBoylestad25

27. FIGURE 9.50 FIGURE 9.51 FIGURE 9.52 Substituting the Thevenin equivalent circuit in the network external to the resistor RL of Fig. 9.47. ET162 Circuit Analysis – Methods of AnalysisBoylestad26

28. Experimental Procedures FIGURE 9.53 FIGURE 9.54 FIGURE 9.55 ET162 Circuit Analysis – Network TheoremsBoylestad27

29. 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 TheoremsBoylestad28

30. 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. 29

31. 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. Boylestad30

32. 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.

33. 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 Boylestad32

34. 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. ET162 Circuit Analysis – Network TheoremsBoylestad33

35. 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 Boylestad34

36. FIGURE 9.72 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. Boylestad35

37. 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. ET162 Circuit Analysis – Network TheoremsBoylestad36