5. rc and rl first-order circuits

5. RC AND RL FIRST-ORDER CIRCUITS

CIRCUITS by Ulaby & Maharbiz

Overview

Transient Response

Non-Periodic WaveformsStep

Function

Square Pulse

Ramp Function

Exponential

Non-Periodic Waveforms: Step Function

Non-Periodic Waveforms: Ramp Function

Waveform synthesis as sum of two ramp functions

Non-Periodic Waveforms: Pulses

Waveform Synthesis1. Pulse 2. Trapezoid

Non-Periodic Waveforms: Exponentials

CapacitorsPassive element that stores energy in

electric fieldParallel plate capacitor

dAC

0o

1 tdtiC

t

t

For DC, capacitor looks like open circuit

Voltage on capacitor must be continuous (no abrupt change)

Various types of capacitors

Capacitors in Fingerprint Imager

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.

Energy Stored in Capacitor

Capacitor Response: Given v(t), determine i(t), p(t), and w(t)

C =

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

Capacitors in SeriesUse KVL, current

same through each capacitor

Capacitors in Parallel

NCCCCC 321eq

Use KCL, voltage same across each

capacitor

Voltage Division

InductorsPassive element that stores energy in

magnetic field

0o

1 tidttvL

it

t

At dc, inductor looks like a short circuit

Current through inductor must be continuous (no abrupt change)

lANL 2

Solenoid Wound Inductor

Inductor Response to

Inductors in SeriesUse KVL, current is same through all

inductors

Inductors in Parallel

Voltage is same across all inductors

Inductors add together in the same

way resistors do

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

Response Terminology

Natural response – response in absence of sourcesForced 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

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

τ is called the time constant of the circuit.

Natural Response of Charged Capacitor

General Response of RC Circuit

Solution of

Example 5-9: Determine Capacitor Voltage

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

Example 5-10: Charge/Discharge Action

Example 5-10 (cont.)

Example 5-11: Rectangular Pulse

Natural Response of the RL Circuit

General Response of the RL Circuit

Example 5-12: Two RL Branches

At t=0-

Cont.

Example 5-12: Two RL Branches (cont.)

After t=0:

RC Op-Amp Circuits: Ideal Integrator

Example 5-14: Square-Wave Signal

RC Op-Amp Circuits: Ideal Differentiator

Example 5-15: Pulse Response

Multisim Example

Summary