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6-1 Electricity Principles & Applications Eighth Edition Richard J. Fowler Chapter 6 Complex-Circuit Analysis (student version) McGraw-Hill © 2013 The McGraw-Hill Companies Inc. All rights reserved. 6-2 INTRODUCTION • Superpositon Theorem (Page 156) • Simultaneous Equations (Page 136) • Voltage Sources (Page 160) • Thevenin’s Theorem (Page 161) • Norton’s Theorem (Page 169) McGraw-Hill © 2013 The McGraw-Hill Companies Inc. All rights reserved. 6-3 Dear Student: This presentation is arranged in segments. Each segment is preceded by a Concept Preview slide and is followed by a Concept Review slide. When you reach a Concept Review slide, you can return to the beginning of that segment by clicking on the Repeat Segment button. This will allow you to view that segment again, if you want to. McGraw-Hill © 2013 The McGraw-Hill Companies Inc. All rights reserved. 6-4 Concept Preview • Two or more sources are required to use the superposition theorem. (Page 156) • Superposition theorem requires only series-parallel rules and procedures to determine values in a complex circuits. (Page 157) • Superposition theorem will solve for all currents and voltages in the circuit. (Page 157) McGraw-Hill © 2013 The McGraw-Hill Companies Inc. All rights reserved. 6-5 Superposition Technique 20 W B1 2A 110 V 5A 10 W 30 W 3A B2 Rty 10 V V u 1110 Replace B2 with a short and calculate the currents. 20 W Rty 110 V u 110 30 W 3A 10 W 4A 110 V 1A Replace B1 with a short and calculate the currents. 20 W 110 V 1A 8A 10 W 30 W 7A 110 V Algebraically add the two currents for each resistor. McGraw-Hill © 2013 The McGraw-Hill Companies Inc. All rights reserved. 6-6 Verification of Superposition Results Using Circuit Measurements. The dark bars on the ammeters are the negative terminals. Notice that both the directions and magnitudes of the currents agree with those obtained by the superposition method. McGraw-Hill © 2013 The McGraw-Hill Companies Inc. All rights reserved. 6-7 Simultaneous equations A procedure for determining the values of I1 and I2 using two independent equations is shown below. The independent equations are: #1 #2 8.5V = 5W(I1) - 1W(I2) 2.5V = -1W(I1) + 3W(I2) Multiply equation # 2 by 5 and add the results to equation #1. #2 [2.5V = -1W(I1) + 3W(I2)] x 5 yields 12.5V = -5W(I1) + 15W(I2) #1 8.5V = 5W(I1) - 1W(I2) 21V = 14W(I2) Now solve for I2 which yields I2 = 21V / 14 W = 1.5A Finally, using this value of I2 in either equation #1 or #2, solve for I1 which yields I1 = (8.5V + 1.5V) / 5W = 2A In summary the values are: I1 = 2A McGraw-Hill and I2 = 1.5A © 2013 The McGraw-Hill Companies Inc. All rights reserved. 6-8 Concept Review • Two or more sources are required to use the superposition theorem. • Superposition theorem requires only series-parallel rules and procedures to determine values in a complex circuits. • Superposition theorem will solve for all currents and voltages in the circuit. • Simultaneous equations techniques can be used to solve loop equations. Repeat Segment McGraw-Hill © 2013 The McGraw-Hill Companies Inc. All rights reserved. 6-9 Concept Preview • Thevenin’s theorem reduces a complex circuit to an equivalent-circuit voltage source with a load. (Page 161) • Norton’s theorem reduces a complex circuit to an equivalent-circuit current source with a load. (Page 169 • Norton’s theorem and Thevenin’s theorem may not solve for all values of current and voltage. (Page 171) McGraw-Hill © 2013 The McGraw-Hill Companies Inc. All rights reserved. 6 - 10 Voltage Sources (Page 160) Ideal Source 15 V V Voltage does not change when loaded. V 15 V No load Loaded Equivalent-Circuit Source V 15 V No load McGraw-Hill Voltage changes when loaded. V 13 V Loaded © 2013 The McGraw-Hill Companies Inc. All rights reserved. 6 - 11 Thevenizing A Complex Circuit 20 W B1 110 V R R2TH 2 = 36.7 V V 30 RTHW= 6.7 W 10 W (Page 161) B2 110 V Replace Remove B1 and R2 and Be2 calculate with shorts VTHand . calculate RTH. RTH 6.7 W - VTH 36.7 V + R2 30 W 30 V 1A Draw the Thevenin equivalent circuit. Load the circuit with R2. Calculate the load V and I. 20 W 1A 110 V 8A 10 W R2 7A 30 W 30 V 110 V Return to the original circuit and calculate the other currents. © 2013 The McGraw-Hill Companies Inc. All rights reserved. McGraw-Hill 6 - 12 Applying Norton’s Theorem to a Complex Circuit (Page 169) •Select a resistor to be the load. •Replace the load with a short. •Calculate the current through the short. This is IN. •Remove the short from the load terminals. •Replace all sources with shorts. •Calculate the resistance between the load terminals. This is RN. •Use IN and RN for the Norton circuit. •Connect the load to the Norton circuit. •Calculate Vload and Iload. McGraw-Hill © 2013 The McGraw-Hill Companies Inc. All rights reserved. 6 - 13 Equivalency Of Norton and Thevenin Circuits Page (172) + + + RTH 8W V 16 V VTH 16 V The open-circuit voltages are equal. RTH 8W A 2A VTH 16 V The short-circuit currents are equal. RTH 8W VTH 16 V McGraw-Hill IN 2A RN 8W IN 2A RN 8W RL When equally loaded, the voltages 24 W IN and currents 12 V 2A 0.5 A are equal. RN 8W V 16 V A 2A RL 24 W 12 V 0.5 A © 2013 The McGraw-Hill Companies Inc. All rights reserved. 6 - 14 Complex-Circuits Quiz Which theorem(s) require(s) algebraically adding the currents for each resistor? ____ Superposition Which theorem(s) require(s) shorting all voltage sources at the same time? ____ Thevenin Norton Which theorem(s) use(s) an equivalentcircuit current source? ____ Norton Which theorem(s) use(s) an equivalentcircuit voltage source? ____ Thevenin Which theorem(s) determine(s) the current for all resistors in the circuit? ____ The voltage of a(n) ____ voltage source is independent of the load current. Solutions by simultaneous equations require ____ equations. McGraw-Hill Superposition ideal independent © 2013 The McGraw-Hill Companies Inc. All rights reserved. 6 - 15 Concept Review • Thevenin’s theorem reduces a complex circuit to an equivalent-circuit voltage source with a load. • Norton’s theorem reduces a complex circuit to an equivalent-circuit current source with a load. • Norton’s theorem and Thevenin’s theorem may not solve for all values of current and voltage. Repeat Segment McGraw-Hill © 2013 The McGraw-Hill Companies Inc. All rights reserved. 6 - 16 REVIEW • Superpositon Theorem • Simultaneous Equations • Voltage Sources • Thevenin’s Theorem • Norton’s Theorem McGraw-Hill © 2013 The McGraw-Hill Companies Inc. All rights reserved.