5. rc and rl first-order circuits
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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