5. RC and RL First-Order Circuits
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Transcript of 5. RC and RL First-Order Circuits
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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 WaveformsStep
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 Synthesis1. Pulse 2. Trapezoid
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Non-Periodic Waveforms: Exponentials
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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)
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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 SeriesUse KVL, current
same through each capacitor
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Capacitors in Parallel
NCCCCC 321eq
Use KCL, voltage same across each
capacitor
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Voltage Division
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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
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Inductor Response to
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Inductors in SeriesUse 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 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
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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
τ 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
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Example 5-10: Charge/Discharge Action
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Example 5-10 (cont.)
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Example 5-11: Rectangular Pulse
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Natural Response of the RL Circuit
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General Response of the RL Circuit
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Example 5-12: Two RL Branches
At t=0-
Cont.
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Example 5-12: Two RL Branches (cont.)
After t=0:
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RC Op-Amp Circuits: Ideal Integrator
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Example 5-14: Square-Wave Signal
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RC Op-Amp Circuits: Ideal Differentiator
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Example 5-15: Pulse Response
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Multisim Example
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Summary