Transcript EECS 215: Introduction to Circuits

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
Waveform synthesis as sum of two ramp functions
Non-Periodic Waveforms: Pulses
Waveform Synthesis
1. Pulse
2. Trapezoid
Non-Periodic Waveforms: Exponentials
Capacitors
Passive element that stores energy in electric field
Parallel plate capacitor
C
1 t
   i dt   t0 
C to

For DC, capacitor looks
like open circuit

Voltage on capacitor
must be continuous (no
abrupt change)
A
d
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 Series
Use KVL, current same
through each capacitor
Capacitors in Parallel
Use KCL, voltage same
across each capacitor
Ceq  C1  C2  C3    C N
Voltage Division
Inductors
Passive element that stores energy in magnetic field
Solenoid Wound Inductor
1 t
i   vt  dt  i t0 
L to

At dc, inductor looks like
a short circuit

Current through inductor
must be continuous (no
abrupt change)
N 2 A
L
l
Inductor Response to
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.
Natural Response of Charged Capacitor
General Response of RC Circuit
Solution of
Example 5-9: Determine Capacitor Voltage
Example 5-9 Solution
(a) Switch was moved at t = 0
At t = 0
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