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Transient Analysis

C H A P T E R. 5. Transient Analysis. 1. 0.8. 0.6. 0.4. 0.2. 0. 0. 0.2. 0.4. 0.6. 0.8. 1.0. 1.2. 1.4. 1.6. 1.8. 2.0. t. (s). (a) Transient DC voltage. 1. 0.5. 0. –. 0.5. –. 1. 0. 0.2. 0.4. 0.6. 0.8. 1.0. 1.2. 1.4. 1.6. 1.8. 2.0. t. (s).

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Transient Analysis

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  1. C H A P T E R 5 Transient Analysis

  2. 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 t (s) (a) Transient DC voltage 1 0.5 0 – 0.5 – 1 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 t (s) (b) Transient sinusoidal voltage Figure 5.1 Examples of transient response

  3. t = 0 R Switch C 12 V L Complex load Figure 5.2 Circuit with switched DC excitation

  4. Switch R S Circuit t = 0 containing V s R L / RC combinations Figure 5.3 A general model of the transient analysis problem

  5. A circuit containing energy-storage elements is described by a differential equation. The differential equation describing the series RC circuit shown is di dv 1 S C + i = C dt RC dt _ v + R i R C + i R + _ v ( t ) v ( t ) C S C _ Figure 5.5 Circuit containing energy-storage element

  6. R + + _ v ( t ) C v ( t ) S C _ dv 1 1 C – v – v = 0 RC circuit: C S dt RC RC R + _ v ( t ) i ( t ) L S L di R 1 L i – – v = 0 RL circuit: L S dt L L Figure 5.9 Differential equations of first-order circuits

  7. t = 0 Switch Switch v V C R C B i ( t ) Exponential decay of capacitor current 1 0.9 0.8 0.7 Capacitor voltage, V 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 Figure 5.10 Decay through a resistor of energy stored in a capacitor t = 0

  8. t = 0 i L I L R S Figure 5.15

  9. – + i (0 ) L v ( t ) R L ( ) i t L + i ( t ) L 10 mA 0 t Figure 5.16

  10. t = 0 R R 1 2 + v R V V C C 1 3 2 _ Figure 5.22 A more involved RC circuit

  11. Figure 5.23 The circuit of Figure 6.45 for t > 0 + R R 1 2 v R C C 3 V V 1 2 _

  12. R R R 1 3 2 V V / R 1 2 2 R 1 V V 1 2 + R T R R 1 2 R = R R R T 1 2 3 R T + V _ T Figure 5.24 Reduction of the circuit of Figure 5.23 to Thevenin equivalent form

  13. R T + + V V C _ T C _ Figure 5.25 The circuit of Figure 5.22 in equivalent form for t > 0

  14. R T + v (t) _ C L T Parallel case (a) R T L + _ v ( t ) T C Series case (b) Figure 5.39 Second-order circuits

  15. 2 1.5 x ( ) t N 1 – t e 2 0.5 e – t 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t (s) Figure 5.43 Response of overdamped second-order circuit

  16. 1 0.8 0.6 x ( t ) N 0.4 – t e 0.2 – t te 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t (s) Figure 5.44 Response of critically damped second-order circuit

  17. Figure 5.45 Response of underdamped second-order circuit

  18. t = 0 _ v ( t ) + C R _ + v ( t ) C R + + V v ( t ) L _ S i ( t ) L _ R = 5000 L = 1 H C = 1 F V = 25 V S Figure 5.46

  19. t = 0 + C R L v ( t ) i ( t ) i ( t ) i ( t ) C R L I _ S L = 2 H C = 2 F I R = 500 = 5 A S Figure 5.48

  20. N N 2 2 = 100 = 100 N N 1 1 N N N N 1 1 2 2 L R L , R , + + P P P P V V B B i i – – spark spark plug plug C C switch switch closed closed Figure 5.52

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