6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

6. CIRCUIT ANALYSIS BY LAPLACE

CIRCUITS by Ulaby & MaharbizAll rights reserved. Do not copy or

distribute. © 2013 National Technology and

Science Press

All rights reserved. Do not

copy or distribute. © 2013 National

Technology and Science Press

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

All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

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

All rights reserved. Do not copy or distribute.

© 2013 National Technology and Science

Press

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 Response

All rights reserved. Do not copy or distribute. ©

2013 National Technology and Science Press

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


All rights reserved. Do not copy or distribute. © 2013 National


Example 6-3: Overdamped RLC Circuit

Cont.


Example 6-3: Overdamped RLC Circuit All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

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



Example 6-5 (cont.)



Analysis Techniques

Circuit Excitation Method of SolutionChapters

1. dc (w/ switches) Transient analysis 5 & 62. ac Phasor-domain analysis 7 -9 ( steady state only)

3. any waveform Laplace Transform This Chapter

(single-sided) (transient + steady state)

4. Any waveform Fourier Transform 12 (double-sided) (transient + steady state)

Single-sided: defined over [0,∞] Double-sided: defined over [−∞,∞]


Singularity Functions

A singularity function is a function that either itself is not finite everywhere or one (or more) of its derivatives is (are) not finite everywhere.

Unit Step Function



Singularity Functions (cont.)

Unit Impulse Function

For any function f(t): Sampling Property


Review of Complex Numbers

We will find it is useful to represent sinusoids as complex numbers

jyxz jezzz

1j

Rectangular coordinatesPolar coordinates

sincos je j

Relations based on Euler’s Identity

yz

xz

)Im(

Re


Relations for Complex Numbers

Learn how to perform these with your calculator/computer


© 2013 National Technology and Science Press

Laplace Transform Technique


Laplace Transform Definition


copy or distribute.


Laplace Transform of Singularity Functions

For A = 1 and T = 0:


Laplace Transform of Delta Function

For A = 1 and T = 0:




Properties of Laplace Transform

Time Scaling

Time Shift Example


Properties of Laplace Transform (cont.)

Frequency Shift

Time Differentiation

Example


Properties of Laplace Transform (cont.)

Time Integration

Frequency Differentiation

Frequency Integration



© 2013 National Technology and Science

Press

All rights reserved. Do not copy or distribute. © 2013

National Technology and Science Press

Circuit Analysis


ExampleAll rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Partial Fraction Expansion Partial fraction expansion facilitates

inversion of the final s-domain expression for the variable of interest back to the time domain. The goal is to cast the expression as the sum of terms, each of which has an analog in Table 10-2.

Example



1.Partial Fractions Distinct Real Poles



1. Partial Fractions Distinct Real Poles

The poles of F(s) are s = 0, s = −1, and s = −3. All three poles are real and distinct.

Example



2. Partial Fractions Repeated Real Poles

All rights reserved.

Do not copy or distribute. © 2013 National



Example

Cont.




Example cont.


3. Distinct Complex Poles

Example

Note that B2 is the complex conjugate of B1.

Procedure similar to “Distinct Real Poles,” but with complex values for s

Complex poles always appear in conjugate pairs

Expansion coefficients of conjugate poles are conjugate pairs themselves


3. Distinct Complex Poles (Cont.)

Next, we combine the last two terms:


4. Repeated Complex Poles: Same procedure as for repeated real poles


Property #3a in Table 10-2:

Hence:

All rights reserved. Do not copy or distribute. © 2013 National Technology and Science

Press

s-Domain Circuit Models

Under zero initial conditions:





copy or distribute. © 2013 National








Example : Interrupted Voltage Source

Initial conditions:

Voltage Source

(s-domain)Cont.


Example : Interrupted Voltage Source (cont.)

Cont.



Cont.










Multisim Example of RLC Circuit


RFID CircuitAll rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Tech Brief 10:

Micromechanical Sensors and Actuators



Tech Brief 11: Touchscreens and Active Digitizers


© 2013 National

Technology and Science

Press

Summary


Documents

6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press