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5. RC and RL First-Order Circuits. CIRCUITS by Ulaby & Maharbiz. Overview. Transient Response. Non-Periodic Waveforms. Step Function. Ramp Function. Square Pulse. Exponential. Non-Periodic Waveforms: Step Function. Non-Periodic Waveforms: Ramp Function.
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5. RC and RL First-Order Circuits CIRCUITS by Ulaby & Maharbiz
Non-Periodic Waveforms Step Function Ramp Function Square Pulse Exponential
Waveform Synthesis 1. Pulse 2. Trapezoid
Capacitors Passive element that stores energy in electric field • For DC, capacitor looks like open circuit • Voltage on capacitor must be continuous (no abrupt change) Parallel plate capacitor
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.
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 Series Use KVL, current same through each capacitor
Capacitors in Parallel Use KCL, voltage same across each capacitor
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
Inductors in Series Use 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 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 (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.
Example 5-9 Solution (a) Switch was moved at t = 0 At t = 0 (b) Switch was moved at t = 3 s At t > 0
Example 5-12: Two RL Branches - At t=0 Cont.
Example 5-12: Two RL Branches (cont.) After t=0:
