1. E E 2315 Lecture 10 Natural and Step Responses of RL and RC Circuits

2. Conservation of Charge (1/4) Energy transferred if v 10  v 20 Total system charge is conserved

3. Conservation of Charge (2/4) Initial stored energy: At equilibrium:

4. Conservation of Charge (3/4) Initial Charge: Final Charge: Since

5. Conservation of Charge (4/4) Final stored energy: Energy consumed in R:

6. Conservation of Flux Linkage (1/3) Energy transferred if i 10  i 20 Total system flux linkage is conserved. Initial stored energy: At equilibrium:

7. Conservation of Flux Linkage (2/3) Initial flux linkage: Final flux linkage: Since

8. Final stored energy: Energy consumed in R: Conservation of Flux Linkage (3/3)

9. Natural RL Response (1/2) Inductor has initial current, i o. Switch opens at t = 0 Inductor current can’t change instantaneously

10. Natural RL Response (2/2) KVL: Separate the variables: Integrate: Exponential of both sides:

11. Natural RC Response (1/2) Capacitor has initial voltage, v o. Switch closes at t = 0. Capacitor voltage can’t change instantaneously KCL: Separate the variables:

12. Natural RC Response (2/2) Integrate: Exponential of both sides:

13. RL Step Response (1/4) Make-before-break switch changes from position a to b at t = 0. For t < 0, I o circulates unchanged through inductor.

14. RL Step Response (2/4) For t > 0, circuit is as below. Initial value of inductor current, i, is I o. The KVL equation provides the differential equation.

15. RL Step Response (3/4) Solution has two parts: Steady State Response Transient Response Determine k by initial conditions:

16. RL Step Response (4/4) Inductor behaves as a short circuit to DC in steady state mode

17. RC Step Response (1/3) Switch closes at t = 0. Capacitor has initial voltage, V o. v-i relationship: By KVL & Ohm’s Law:

18. RC Step Response (2/3) Response has two parts –steady state –transient Use initial voltage to determine transient Steady State ResponseTransient Response

19. RC Step Response (3/3) Capacitor becomes an open circuit to DC after the transient response has decayed.

20. Unbounded Response (1/5) Need Thévenin equivalent circuit from terminal pair connected to inductor Let initial current = 0A in this example.

21. Unbounded Response (2/5) Voltage divider to get v x : Then Thévenin voltage

22. Unbounded Response (3/5) Therefore:

23. Unbounded Response (4/5) Steady state: Transient:

24. Unbounded Response (5/5) Use initial conditions to determine k. Complete response is unbounded: