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6. RLC CIRCUITS
CIRCUITS by Ulaby & Maharbiz
Overview
Second Order Circuits
A second order circuit is characterized by a second order differential equation
Resistors and two energy storage elements
Determine voltage/current as a function of time
Initial/final values of voltage/current, and their derivatives are needed
Initial/Final Conditions
vC, iL do not change instantaneously
Get derivatives dvC/dt and diL/dt from iC , vL
Capacitor open, Inductor short at dc
Guidelines
Example 6-2: Determine Initial/Final Conditions
Circuit
t = 0‒
mF8
H2.0
6
4
2
A4
V24
3
2
1
0
0
C
L
R
R
R
I
V
Example 6-2: Initial/Final Conditions (cont.)
mF8
H2.0
6
4
2
A4
V24
3
2
1
0
0
C
L
R
R
R
I
Vt = 0+
Given:
Example 6-2: Initial/Final Conditions (cont.)
t
mF8
H2.0
6
4
2
A4
V24
3
2
1
0
0
C
L
R
R
R
I
V
Series RLC Circuit : General Solution
Solution Outline
Transient solution
Steady State solution
Series RLC Circuit: Natural Response
Find Natural Response Of RLC Circuit
0
Natural response occurs when no active sources are present, which is the case at t > 0.
Series RLC Circuit: Natural Response
Find Natural Response Of RLC Circuit
0
Solution of Diff. Equation
Assume:
It follows that:
Solution of Diff. Equation (cont.)
0
Invoke Initial Conditions to determine A1 and A2
Circuit Response: Damping Conditions
Damping coefficient
Resonant frequency
s1 and s2 are real
s1 = s2
s1 and s2 are complex
Overdamped Response
Overdamped, a > w0 tsts eAeAtv 21
21
factor damping
frequencyresonant 0
20
22,1 s
L
R
2
LC
10
Underdamped Response
Underdamped a < w0
Damping: loss of stored energy
tDtDetv ddt sincos 21
factor dampingfrequencyresonant 0
20
22,1 s
L
R
2
LC
10
220 d Damped natural frequency
Critically Damped Response
Critically damped a = w0
21tetBBtv
factor damping
frequencyresonant 0
20
22,1 s
L
R
2
LC
10
Total Response of Series RLC CircuitNeed to add Forced/Steady State Solution tvvtv tss
Natural solution represents transient response, decays to 0 as t .v() represents forced/steady state solution.Overdamped ( a > w0)
Critically Damped ( a = w0)
Underdamped ( a < w0)
tsts eAeAvtv 2121
21
tetBBvtv
tDtDevtv ddt sincos 21
Now find unknown constants from initial conditions v(0+) and
dv/dt at t = 0+
Example 6-7: Overdamped RLC Circuit
Cont.
Example 6-7: Overdamped RLC Circuit
Example 6-8: Pulse Excitation
Example 6-9: Determine Capacitor Response
Circuit
t = 0‒
At t = 0 ‒ :
Example 6-9: Capacitor Response (cont.)
t = 0+
Initial values of the capacitor voltage and its derivative will be needed to evaluate constants D1 and D2
Example 6-9: Capacitor Response (cont.)
t > 0 This is just a
series RLC circuit!
Example 6-9: Capacitor Response (cont.)
Parallel RLC Circuit
LC
I
LC
i
dt
di
RCdt
id s2
2 1
sIdt
dvCi
R
v
dt
diLv
20
22,1 s
RC2
1
LC
10
Overdamped (a > w0)
Critically Damped (a = w0)
Underdamped ( a < w0)
tsts eAeAiti 2121
21
tetBBiti
tDtDeiti td2d1
sincos
Same form of diff. equation as series RLC
Oscillators
If R=0 in a series or parallel RLC circuit, the circuit becomes an oscillator
General Second Order Circuits
Setup differential equation
Determine a, w Natural solution Forced solution
(steady state) Unknowns from
initial conditions
Example 6-13: Op-Amp Circuit
0321 iiii n
00 out
2
out
1
S
dt
dvC
R
v
R
V
dt
diLiRv L
L3out
LCR
Vi
LCR
R
dt
di
L
R
CRdt
id
1
sL
2
3L3
22L
2 1
Substitute vout into KCL expression, rearrange for diff. equation in terms of iL
L
R
CR 22
1 3
2
LCR
R
2
30
Example 6-13: Op-Amp Circuit (cont.)
LCR
Vi
LCR
R
dt
di
L
R
CRdt
id
1
sL
2
3L3
22L
2 1
L
R
CR 22
1 3
2
0 R3
R2LC
Cont.
Example 6-13: Op-Amp Circuit (cont.)
Cont.
Example 6-13: Op-Amp Circuit (cont.)
Multisim Example of RLC Circuit
RFID Circuit
Tech Brief 12:
Micromechanical Sensors and Actuators
Tech Brief 13: Touchscreens and Active Digitizers
Summary