1. 5. RC AND RL FIRST-ORDER CIRCUITS CIRCUITS by Ulaby & Maharbiz

2. Overview

3. Transient Response

4. Non-Periodic Waveforms Step Function Square Pulse Ramp Function Exponential

5. Non-Periodic Waveforms: Step Function

6. Non-Periodic Waveforms: Ramp Function

7. Waveform synthesis as sum of two ramp functions

8. Non-Periodic Waveforms: Pulses

9. Waveform Synthesis 1. Pulse 2. Trapezoid

10. Non-Periodic Waveforms: Exponentials

12. Capacitors Passive element that stores energy in electric field Parallel plate capacitor  For DC, capacitor looks like open circuit  Voltage on capacitor must be continuous (no abrupt change)

13. Various types of capacitors

14. Capacitors in Fingerprint Imager

15. Tech Brief 11: Supercapacitors A new generation of capacitor technologies, termed supercapacitors or ultracapacitors, is narrowing the gap between capacitors and batteries. These capacitors can have sufficiently high energy densities to approach within 10 percent of battery storage densities, and additional improvements may increase this even more. Importantly, supercapacitors can absorb or release energy much faster than a chemical battery of identical volume. This helps immensely during recharging. Moreover, most batteries can be recharged only a few hundred times before they are degraded completely; supercapacitors can be charged and discharged millions of times before they wear out. Supercapacitors also have a much smaller environmental footprint than conventional chemical batteries, making them particularly attractive for green energy solutions.

16. Energy Stored in Capacitor

17. Capacitor Response: Given v(t), determine i(t), p(t), and w(t) C =

18. RC Circuits at dc  At dc no currents flow through capacitors: open circuits

19. Capacitors in Series Use KVL, current same through each capacitor

20. Capacitors in Parallel Use KCL, voltage same across each capacitor

21. Voltage Division

22. Inductors Passive element that stores energy in magnetic field  At dc, inductor looks like a short circuit  Current through inductor must be continuous (no abrupt change) Solenoid Wound Inductor

23. Inductor Response to

24. Inductors in Series Use KVL, current is same through all inductors

25. Inductors in Parallel Voltage is same across all inductors Inductors add together in the same way resistors do

26. RL Circuits at dc  At dc no voltage across inductors: short circuit

28. Response Terminology Natural response – response in absence of sources Forced response – response due to external source Complete response = Natural + Forced Transient response – time-varying response (temporary) Steady state response – time-independent or periodic (permanent) Complete response = Transient + Steady State Source dependence Time dependence

29. Natural Response of Charged Capacitor (a) t = 0 − is the instant just before the switch is moved from terminal 1 to terminal 2 (b) t = 0 is the instant just after it was moved; t = 0 is synonymous with t = 0 + since the voltage across the capacitor cannot change instantaneously, it follows that

30. Solution of First-Order Diff. Equations  τ is called the time constant of the circuit.

31. Natural Response of Charged Capacitor

32. General Response of RC Circuit

33. Solution of

34. Example 5-9: Determine Capacitor Voltage

35. Example 5-9 Solution At t = 0 At t > 0 (a) Switch was moved at t = 0 (b) Switch was moved at t = 3 s

36. Example 5-10: Charge/Discharge Action

37. Example 5-10 (cont.)

38. Example 5-11: Rectangular Pulse

39. Natural Response of the RL Circuit

40. General Response of the RL Circuit

41. Example 5-12: Two RL Branches At t=0 - Cont.

42. Example 5-12: Two RL Branches (cont.) After t=0:

43. RC Op-Amp Circuits: Ideal Integrator

44. Example 5-14: Square-Wave Signal

45. RC Op-Amp Circuits: Ideal Differentiator

46. Example 5-15: Pulse Response

47. Multisim Example

48. Summary