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.