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PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION C H A P T E R 5 Transient Analysis McGraw-Hill 1 GIORGIO RIZZONI © The McGraw-Hill Companies, Inc. 2000 PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 5.1 Examples of transient response 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1.0 1.2 t (s) (a) Transient DC voltage 1.4 1.6 1.8 2.0 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 t (s) (b) Transient sinusoidal voltage 1.6 1.8 2.0 1 0.5 0 –0.5 –1 McGraw-Hill 2 GIORGIO RIZZONI © The McGraw-Hill Companies, Inc. 2000 PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 5.2 Circuit with switched DC excitation t =0 R Switch 12 V C L Complex load McGraw-Hill 3 GIORGIO RIZZONI © The McGraw-Hill Companies, Inc. 2000 PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 5.3 A general model of the transient analysis problem RS Switch t=0 Vs McGraw-Hill Circuit containing R L /RC combinations 4 GIORGIO RIZZONI © The McGraw-Hill Companies, Inc. 2000 PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 5.5 Circuit containing energy-storage element A circuit containing energy-storage elements is described by a differential equation. The differential equation describing the series RC circuit shown is diC dt vS ( t ) McGraw-Hill +_ + 1 RC + vR R iR dv S dt iC = _ iC C 5 GIORGIO RIZZONI + v C (t ) _ © The McGraw-Hill Companies, Inc. 2000 PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 5.9 Differential equations of first-order circuits R + + _ v S (t) C v C (t) _ RC circuit: dv C dt – 1 RC vC – 1 RC vS = 0 R + _ v S (t) RL circuit: McGraw-Hill iL ( t ) di L dt – L R 1 iL – vS = 0 L L 6 GIORGIO RIZZONI © The McGraw-Hill Companies, Inc. 2000 PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 5.10 Decay through a resistor of energy stored in a capacitor t= 0 t= 0 Switch vC VB Switch C i(t) R Exponential decay of capacitor current 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time, s McGraw-Hill 7 GIORGIO RIZZONI © The McGraw-Hill Companies, Inc. 2000 PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 5.15 t=0 iL IS McGraw-Hill L 8 GIORGIO RIZZONI R © The McGraw-Hill Companies, Inc. 2000 PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 5.16 iL (0 + ) L iL(t) – v + R ( t) iL( t) 10 mA 0 McGraw-Hill t 9 GIORGIO RIZZONI © The McGraw-Hill Companies, Inc. 2000 PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 5.22 A more involved RC circuit t= 0 R1 R2 + V1 R3 vC C V2 _ McGraw-Hill 10 GIORGIO RIZZONI © The McGraw-Hill Companies, Inc. 2000 PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 5.23 The circuit of Figure 6.45 for t > 0 + R R2 1 vC R3 V1 McGraw-Hill V2 11 GIORGIO RIZZONI C _ © The McGraw-Hill Companies, Inc. 2000 PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 5.24 Reduction of the circuit of Figure 5.23 to Thevenin equivalent form R1 V1 R1 V R 1 1 R3 R2 V2 /R 2 + V R 2 R R T = R 1 R R V McGraw-Hill T 2 T 2 R 3 T + _ 12 GIORGIO RIZZONI © The McGraw-Hill Companies, Inc. 2000 PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 5.25 The circuit of Figure 5.22 in equivalent form for t > 0 RT + VT + _ VC C _ McGraw-Hill 13 GIORGIO RIZZONI © The McGraw-Hill Companies, Inc. 2000 PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 5.39 Second-order circuits R _+ v T (t) T C L Parallel case (a) R T L + _ v T (t) C Series case (b) McGraw-Hill 14 GIORGIO RIZZONI © The McGraw-Hill Companies, Inc. 2000 PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 5.43 Response of overdamped second-order circuit 2 1.5 xN ( t) 1 e– 0.5 e– 2t 1t 0 0 McGraw-Hill 0.1 0.2 0.3 0.4 0.5 0.6 t (s) 15 GIORGIO RIZZONI 0.7 0.8 0.9 1 © The McGraw-Hill Companies, Inc. 2000 PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 5.44 Response of critically damped second-order circuit 1 0.8 0.6 xN (t) 0.4 e– 0.2 te– t t 0 0 McGraw-Hill 0.1 0.2 0.3 0.4 0.5 0.6 t (s) 0.7 16 GIORGIO RIZZONI 0.8 0.9 1 © The McGraw-Hill Companies, Inc. 2000 PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 5.45 Response of underdamped second-order circuit McGraw-Hill 17 GIORGIO RIZZONI © The McGraw-Hill Companies, Inc. 2000 PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 5.46 t =0 + v C ( t) _ R C + v R ( t) _ + + _ V S R = 5000 McGraw-Hill vL ( t) _ i(t) L=1H C =1 V S = 25 V 18 GIORGIO RIZZONI L F © The McGraw-Hill Companies, Inc. 2000 PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 5.48 t =0 + v( t) C R iC ( t ) L iR ( t ) i L( t) IS _ McGraw-Hill L =2H C =2 R = 500 IS = 5 A 19 GIORGIO RIZZONI F © The McGraw-Hill Companies, Inc. 2000 PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 5.52 N2 = 100 N1 N1 N2 = 100 N1 N2 N1 LP,RP + VB – VB spark plug C – switch closed McGraw-Hill L P, R P + i 20 GIORGIO RIZZONI N2 i spark plug C switch closed © The McGraw-Hill Companies, Inc. 2000