5. rc and rl first-order circuitszyang/teaching/20182019springe...rc and rl first-order circuits...
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5. RC AND RL FIRST-ORDER CIRCUITSCircuit Analysis & Design by Ulaby, Maharbiz & Furse
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Capacitors
Passive element that stores energy in electric field
Parallel 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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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
NCCCCC 321eq
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
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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Various Styles of Inductors
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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
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 5-10: Determine Capacitor Voltage
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Example 5-10 Solution
At t = 0
At t > 0
(a) Switch was moved at t = 0
(b) If Switch was moved at t = 3 s instead of t=0
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Example 5-11: Charge/Discharge Action
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Example 5-11 (cont.)
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Example 5-12: 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 5-13: Two RL Branches
At t=0-
Cont.
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Example 5-13: Two RL Branches (cont.)
After t=0:
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RC Op-Amp Circuits: Ideal Integrator
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Example 5-15: Square-Wave Signal
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RC Op-Amp Circuits: Ideal Differentiator
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Example 5-16: Pulse Response
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Summary