5. RC AND RL FIRST-ORDER CIRCUITS CIRCUITS by Ulaby &
Maharbiz
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Overview
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Transient Response
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Non-Periodic Waveforms Step Function Square Pulse Ramp Function
Exponential
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Non-Periodic Waveforms: Step Function
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Non-Periodic Waveforms: Ramp Function
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Waveform synthesis as sum of two ramp functions
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Non-Periodic Waveforms: Pulses
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Waveform Synthesis 1. Pulse 2. Trapezoid
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Non-Periodic Waveforms: Exponentials
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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)
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Various types of capacitors
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Capacitors in Fingerprint Imager
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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.
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Energy Stored in Capacitor
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Capacitor Response: Given v(t), determine i(t), p(t), and w(t)
C =
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RC Circuits at dc At dc no currents flow through capacitors:
open circuits
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Capacitors in Series Use KVL, current same through each
capacitor
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Capacitors in Parallel Use KCL, voltage same across each
capacitor
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Voltage Division
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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
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Inductor Response to
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Inductors in Series Use KVL, current is same through all
inductors
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Inductors in Parallel Voltage is same across all inductors
Inductors add together in the same way resistors do
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RL Circuits at dc At dc no voltage across inductors: short
circuit
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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
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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
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Solution of First-Order Diff. Equations is called the time
constant of the circuit.
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Natural Response of Charged Capacitor
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General Response of RC Circuit
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Solution of
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Example 5-9: Determine Capacitor Voltage
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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
