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