5. RC and RL First-Order Circuits CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

All rights reserved. Do not copy or distribute All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Transient Response All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Non-Periodic Waveforms All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Non-Periodic Waveforms Step Function Ramp Function Square Pulse Exponential

Non-Periodic Waveforms: Step Function All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Non-Periodic Waveforms: Step Function

All rights reserved. Do not copy or distribute All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Non-Periodic Waveforms: Ramp Function All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Non-Periodic Waveforms: Ramp Function

Waveform synthesis as sum of two ramp functions All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Non-Periodic Waveforms: Pulses All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Non-Periodic Waveforms: Pulses

Waveform Synthesis 1. Pulse 2. Trapezoid All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Non-Periodic Waveforms: Exponentials All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Non-Periodic Waveforms: Exponentials

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Passive element that stores energy in electric field All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press 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)

Various types of capacitors All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Various types of capacitors

Capacitors in Fingerprint Imager All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

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. All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Energy Stored in Capacitor All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Energy Storage Comparison All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Capacitor Response: Given v(t), determine i(t), p(t), and w(t) All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Capacitor Response: Given v(t), determine i(t), p(t), and w(t) C =

All rights reserved. Do not copy or distribute All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press RC Circuits at dc At dc no currents flow through capacitors: open circuits

Use KVL, current same through each capacitor Capacitors in Series Use KVL, current same through each capacitor All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Capacitors in Parallel Use KCL, voltage same across each capacitor All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Voltage Division All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Passive element that stores energy in magnetic field All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Inductors Passive element that stores energy in magnetic field Solenoid Wound Inductor At dc, inductor looks like a short circuit Current through inductor must be continuous (no abrupt change) Note: voltage divider with capacitors

Inductor Response to All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Use KVL, current is same through all inductors Inductors in Series Use KVL, current is same through all inductors All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Inductors in Parallel Voltage is same across all inductors Inductors add together in the same way resistors do All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

RL Circuits at dc At dc no voltage across inductors: short circuit All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press RL Circuits at dc At dc no voltage across inductors: short circuit

All rights reserved. Do not copy or distribute All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Response Terminology Source dependence Time dependence All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press 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

Natural Response of Charged Capacitor All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press 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

Solution of First-Order Diff. Equations All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Solution of First-Order Diff. Equations τ is called the time constant of the circuit.

Natural Response of Charged Capacitor All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Natural Response of Charged Capacitor

General Response of RC Circuit All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press General Response of RC Circuit

All rights reserved. Do not copy or distribute All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Solution of

Example 5-10: Determine Capacitor Voltage All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Example 5-10: Determine Capacitor Voltage

Example 5-10 Solution (a) Switch was moved at t = 0 At t = 0 All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Example 5-10 Solution (a) Switch was moved at t = 0 At t = 0 (b) Switch was moved at t = 3 s At t > 0

Example 5-11: Charge/Discharge Action All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Example 5-11: Charge/Discharge Action

All rights reserved. Do not copy or distribute All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Example 5-11 (cont.)

Example 5-12: Rectangular Pulse All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Example 5-12: Rectangular Pulse

Natural Response of the RL Circuit All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Natural Response of the RL Circuit

General Response of the RL Circuit All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press General Response of the RL Circuit

Example 5-13: Two RL Branches All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press - At t=0 Cont.

Example 5-13: Two RL Branches (cont.) After t=0: All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

RC Op-Amp Circuits: Ideal Integrator All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Example 5-15: Square-Wave Signal All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

RC Op-Amp Circuits: Ideal Differentiator All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Example 5-16: Pulse Response All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Example 5-16: Pulse Response

Multisim Example All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

All rights reserved. Do not copy or distribute All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Summary