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5. RC and RL First-Order Circuits

5. RC and RL First-Order Circuits. CIRCUITS by Ulaby & Maharbiz. Overview. Transient Response. Non-Periodic Waveforms. Step Function. Ramp Function. Square Pulse. Exponential. Non-Periodic Waveforms: Step Function. Non-Periodic Waveforms: Ramp Function.

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5. RC and RL First-Order Circuits

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  1. 5. RC and RL First-Order Circuits CIRCUITS by Ulaby & Maharbiz

  2. Overview

  3. Transient Response

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

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

  12. Various types of capacitors

  13. Capacitors in Fingerprint Imager

  14. 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.

  15. Energy Stored in Capacitor

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

  17. RC Circuits at dc • At dc no currents flow through capacitors: open circuits

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

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

  20. Voltage Division

  21. 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

  22. Inductor Response to

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

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

  25. RL Circuits at dc • At dc no voltage across inductors: short circuit

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

  27. 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

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

  29. Natural Response of Charged Capacitor

  30. General Response of RC Circuit

  31. Solution of

  32. Example 5-9: Determine Capacitor Voltage

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

  34. Example 5-10: Charge/Discharge Action

  35. Example 5-10 (cont.)

  36. Example 5-11: Rectangular Pulse

  37. Natural Response of the RL Circuit

  38. General Response of the RL Circuit

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

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

  41. RC Op-Amp Circuits:Ideal Integrator

  42. Example 5-14: Square-Wave Signal

  43. RC Op-Amp Circuits: Ideal Differentiator

  44. Example 5-15: Pulse Response

  45. Multisim Example

  46. Summary

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