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ET 162 Circuit Analysis Methods of 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 ET162 Circuit Analysis – Network Theorems Boylestad 2 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 Theorems Boylestad 3 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 to be analyzed ET162 Circuit Analysis – Network Theorems = Number of independent sources Boylestad 4 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 Theorems Boylestad 5 Ex. 9-1 Determine I1 for the network of Fig. 9.3. I 1' 0 A FIGURE 9.3 E 30V I 5A R1 6 '' 1 I 1 I 1' I 1'' 0A5A 5A ET162 Circuit Analysis – Network Theorems FIGURE 9.4 Boylestad 6 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. RT R1 R2 / / R3 24 12 / / 4 24 3 27 E1 54V I 2A RT 27 12 A2 A R2 I I 15 . A R2 R3 12 4 ' 3 FIGURE 9.5 FIGURE 9.6 ET162 Circuit Analysis – Network Theorems Boylestad 7 RT R3 R1 / / R2 4 24 / / 12 4 8 12 FIGURE 9.7 E 2 48V I 4A RT 12 '' 3 I 3 I 3'' I 3' 4 A 15 . A 2.5 A (direction of I 3'' ) FIGURE 9.8 ET162 Circuit Analysis – Network Theorems Boylestad 8 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 a. considering that the effect of the 36V source ( Fig. 9.10): I 2' 36V E E 2A RT R1 R2 12 6 considering that the effect of the 9 A source ( Fig. 9.11): I 2'' ET162 Circuit Analysis – Network Theorems FIGURE 9.11 R1 I (12 )(9 A) 6A R1 R2 12 6 Boylestad 9 The total current through the 6 resistor ( Fig 9.12) is I 2 I 2' I 2'' 2 A 6 A 8 A b. The power to the 6 resistor is Power I 2 R (8 A) 2 (6 ) 384 W FIGURE 9.12 The calculated power to the 6 resistor due to each source, misu sin g the principle of sup erposition, is P1 ( I 2' ) 2 R (2 A) 2 (6 ) 24 W P2 ( I 2'' ) 2 R (6 A) 2 (6 ) 216 W P1 P2 240 W 384 W because (2 A) (6 A) 2 (8 A) 2 2 ET162 Circuit Analysis – Network Theorems 10 Ex. 9-4 Using the principle of superposition, find the current through the 12-kΩ resistor of Fig. 9.13. R1 I (6 k)(6 mA) I 2 mA R1 R2 6 k 12 k ' 2 FIGURE 9.13 FIGURE 9.14 ET162 Circuit Analysis – Network Theorems Boylestad 11 FIGURE 9.15 considering that the effect of the 9V voltage source ( Fig. 9.15): 9V E I 0.5 mA R1 R2 6 k 12 k '' 2 Since I 2' and I 2'' have the same direction through R2 , the desired current is the sum of the two: I 2 I 2' I 2'' 2 mA 0.5 mA 2.5 mA 12 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 ET162 Circuit Analysis – Network Theorems FIGURE 9.17 9.18 Boylestad FIGURE 9.19 13 considering the effect of the 12 V source ( Fig. 9.17): I 1' E1 12 V 2A R1 R2 2 4 considering that the effect of the 6V source ( Fig. 9.18): I 1'' E2 6V 1A R1 R2 2 4 considering the effect of the 3 A source ( Fig. 9.19): I 1''' R2 I (4 )(3 A) 2A R1 R2 2 4 The total current through the 2 resistor FIGURE 9.20 appears in Fig.9.20, and '' ''' ' IET162 I I I AAnalysis 2 A 2 A 1A Circuit Analysis – Methods 1 1 1 1 1of Boylestad 14 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 Theorems Boylestad 15 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 RL be temporary removed from the network. 2. Make the terminals of the remaining two-terminal network. 3. 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. 4. Calculate ETH 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 the equivalent circuit. ET162 Circuit Analysis – Methodsbetween of Analysis the terminals ofBoylestad 16 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. RTH R1 / / R2 (3 )(6 ) 3 6 2 FIGURE 9.26 Determining RTH for the network of Fig. 9.25. ET162 Circuit Analysis – Network Theorems Boylestad 17 FIGURE 9.28 FIGURE 9.27 RTH IL FIGURE 9.29 Substituting the Thevenin equivalent circuit for the network external to RL in Fig. 9.23. R2 E1 (6 )(9V ) 6V R2 R1 6 3 E TH RTH R L 6V 15 . A 2 2 R L 2 : IL R L 10 : 6V IL 0.5 A 2 10 R L 100 : I L 6V 0.059 A 2 100 18 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 RTH R1 R2 4 2 FIGURE 9.32 ET162 Circuit Analysis – Network Theorems Boylestad 6 19 V2 I 2 R2 (0) R2 0V E TH V1 I 1 R1 I R1 (12 A)(4 ) 48 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 Theorems Boylestad 20 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 RTH R1 / / R2 (6 )(4 ) 6 4 2.4 ET162 Circuit Analysis – Methods of Analysis FIGURE 9.37 Boylestad 21 FIGURE 9.38 FIGURE 9.39 E TH R1 E1 R1 R2 (6 )(8V ) 6 4 4.8V FIGURE 9.40 Substituting the Thevenin equivalent circuit in the network external to the resistor R4 of Fig. 9.35. ET162 Circuit Analysis – Network Theorems Boylestad 22 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 RTH R1 / / R3 R2 / / R4 6 / / 3 4 / / 12 2 3 5 9.43 ET162 Circuit Analysis – FIGURE Methods of Analysis Boylestad 23 FIGURE 9.45 FIGURE 9.44 V1 R1 E (6 )(72V ) 48V R1 R3 6 3 R2 E (12 )(72V ) V2 54V R2 R4 12 4 V E TH V1 V2 0 E TH V2 V1 54V 48V 6V FIGURE 9.46 Substituting the Thevenin equivalent circuit in the network external to the resistor RL of Fig. 9.41. Boylestad 24 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 RTH R4 R1 / / R2 / / R3 14 . k 0.8 k / / 4 k / / 6 k 14 . k 0.8 k / / 2.4 k 14 . k 0.6 k 2 k FIGURE ET162 Circuit Analysis – Methods 9.49 of Analysis Boylestad 25 V4 I 4 R4 (0) R4 0V ' E TH V3 RT' R2 / / R3 4 k / / 6 k 2.4 k FIGURE 9.50 RT' E1 (2.4 k)(6V ) V3 ' 4.5V RT R1 2.4 k 0.8 k ' E TH V3 4.5V V4 I 4 R4 (0) R4 0V FIGURE 9.51 '' E TH V3 RT' R1 / / R3 0.8 k / / 6 k 0.706 k RT' E 2 (0.706 k)(10V ) V3 ' 15 .V 0.706 k 4 k RT R2 ' '' '' E TH V3 15 . V E TH E TH E TH 4.5V 15 .V FIGURE 9.52 Substituting the Thevenin equivalent circuit Circuit Analysis to – Methods of Analysis in theET162 network external the resistor RL of Fig. 9.47. Boylestad ' 3V ( polarity of E TH 26 ) Experimental Procedures FIGURE 9.53 FIGURE 9.54 FIGURE 9.55 ET162 Circuit Analysis – Network Theorems I SC E TH RTH RTH E TH I SC RTH VOC I SC Boylestad where E TH VOC 27 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 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 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. 4. Calculate IN 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 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. R N R1 / / R2 3 / / 6 (3 )(6 ) 3 6 2 FIGURE 9.60 Determining RN for the network of Fig. 9.59. Boylestad 30 V2 I 2 R2 (0) 6 0 V IN E 9V 3 A R1 3 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 in Fig. 9.64. FIGURE 9.64 FIGURE 9.65 RN = R 1 + R 2 =5Ω+4Ω =9Ω FIGURE 9.66 Boylestad 32 IN R1 I R1 R2 (5 )(10 ) 5 4 5556 . A FIGURE 9.67 Determining IN for the network of Fig. 9.65. ET162 Circuit Analysis – Network Theorems FIGURE 9.68 Substituting the Norton equivalent circuit for the network external to the resistor RL of Fig. 9.64. Boylestad 33 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 9.70 FIGURE 9.69 R N R1 / / R2 4 / / 6 (4 )(6 ) 4 6 2.4 FIGURE 9.71 Boylestad 34 FIGURE 9.72 FIGURE 9.73 I ' N E1 7 V 175 . A R1 4 I N'' I 8 A I N I N'' I N' 8 A 175 . A 6.25 A FIGURE 9.74 Substituting the Norton equivalent circuit for the network to the left of terminals a-b in Fig. 9.69. Boylestad 35 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 FIGURE 9.75 Defining the conditions for maximum power to a load using the Thevenin equivalent circuit. power to a load using the Norton equivalent circuit. ET162 Circuit Analysis – Network Theorems Boylestad 36