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1Engr228 Zybooks Chapter 6.2 – Series RLC
Circuits
Chapter 6.2Series RLC Circuits
Engr228 - Circuit Analysis
Spring 2020
Dr Curtis Nelson
Section 6.2 Objective
• Be able to determine the natural and step responses of series RLC circuits.
2Engr228 Zybooks Chapter 6.2 – Series RLC
Circuits
Equations for Analysing the Natural Response of Parallel RLC Circuits
Summary of Transient Responses
3Engr228 Zybooks Chapter 6.2 – Series RLC
Circuits
Source - Free Series RLC Circuit
0)(1)()(
0)(1)()(
0
2
2
=++
=++
=++
ò
tiCdt
tdiRdttidL
dttiCdt
tdiLRti
vvv CLR
LR +
vL(t)-
+ vR(t) -
C
i(t)
-vC(t)
+
Comparing Series and Parallel RLC Circuits
0)(1)()(2
2
=++ tiCdt
tdiRdttidL
Series RLCParallel RLC
0)(1)(1)(2
2
=++ tvLdt
tdvRdt
tvdC
tsts eAeAtv 2121)( +=
LCRCRCs 1
21
21 2
2,1 -÷øö
çèæ±-=
20
22,1 waa -±-=s
RC21
=aLC1
0 =w
tsts eAeAti 2121)( +=
LCLR
LRs 1
22
2
2,1 -÷øö
çèæ±-=
20
22,1 waa -±-=s
LR2
=aLC1
0 =w
4Engr228 Zybooks Chapter 6.2 – Series RLC
Circuits
Series RLC Circuit Solution
If:α > ω0 (overdamped):
α = ω0 (critically damped):
α < ω0 (underdamped):
w 0 = 1LC
a =R2L
tsts eAeAti 2121)( +=
( )21)( AtAeti t += -a
( ) ( )( )tBtBeti ddt wwa sincos)( 21 += -
s1 = -a + a 2 -w02
s2 = -a - a 2 -w02
Equations for Analysing the Natural Response of Series RLC Circuits
5Engr228 Zybooks Chapter 6.2 – Series RLC
Circuits
Equations for Analysing the Step Response of Series RLC Circuits
Zybooks Response Summary
6Engr228 Zybooks Chapter 6.2 – Series RLC
Circuits
Textbook Problem 8.50 (Nilsson 11th)
vo(t) = 16 - 16e−400tcos300t − 21.33e−400tsin300t V
The circuit contains no initial energy. Find vo(t) for t ≥ 0.
Zybooks Participation Exercise 6.5.3
7Engr228 Zybooks Chapter 6.2 – Series RLC
Circuits
Summary: Solving RLC Circuits
1. Identify the series or parallel RLC circuit;2. Find α and ω0;3. Determine whether the circuit is overdamped, critically
damped, or underdamped;4. Assume a solution (natural response + forced response):
fstst VeAteA ++ 21
ftsts VeAeA ++ 21
21
fddt VtBtBe ++- )sincos( 21 wwa
Overdamped
Critically damped
Underdamped
5. Find A, B, and Vf using initial and final conditions.
Equations for Analysing the Natural Response of Parallel RLC Circuits
8Engr228 Zybooks Chapter 6.2 – Series RLC
Circuits
Equations for Analysing the Step Response of Parallel RLC Circuits
Section 6.2 Summary
• Showed how to determine the natural and step responses of series RLC circuits.