6. circuit analysis by laplace circuits by ulaby & maharbiz all rights reserved. do not copy or...
TRANSCRIPT
![Page 1: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/1.jpg)
6. CIRCUIT ANALYSIS BY LAPLACE
CIRCUITS by Ulaby & MaharbizAll rights reserved. Do not copy or
distribute. © 2013 National Technology and
Science Press
![Page 2: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/2.jpg)
All rights reserved. Do not
copy or distribute. © 2013 National
Technology and Science Press
![Page 3: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/3.jpg)
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
![Page 4: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/4.jpg)
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
![Page 5: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/5.jpg)
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
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 6: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/6.jpg)
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:
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 7: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/7.jpg)
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
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 8: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/8.jpg)
Series RLC Circuit: General Response
All rights reserved. Do not copy or distribute. ©
2013 National Technology and Science Press
![Page 9: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/9.jpg)
Series RLC Circuit : General Solution
Solution Outline
Transient solution
Steady State solution
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 10: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/10.jpg)
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.
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 11: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/11.jpg)
Series RLC Circuit: Natural Response
Find Natural Response Of RLC Circuit
0
Solution of Diff. Equation
Assume:
It follows that:
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 12: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/12.jpg)
Solution of Diff. Equation (cont.)
0
Invoke Initial Conditions to determine A1 and A2
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 13: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/13.jpg)
Circuit Response: Damping Conditions
Damping coefficient
Resonant frequency
s1 and s2 are real
s1 = s2
s1 and s2 are complex
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 14: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/14.jpg)
Overdamped Response
Overdamped, a > w0 tsts eAeAtv 21
21
factor damping
frequencyresonant 0
20
22,1 s
L
R
2
LC
10
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 15: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/15.jpg)
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
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 16: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/16.jpg)
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 Technology and Science Press
![Page 17: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/17.jpg)
All rights reserved. Do not copy or distribute. © 2013 National
Technology and Science Press
![Page 18: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/18.jpg)
Example 6-3: Overdamped RLC Circuit
Cont.
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 19: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/19.jpg)
Example 6-3: Overdamped RLC Circuit All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 20: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/20.jpg)
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
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 21: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/21.jpg)
Oscillators
If R=0 in a series or parallel RLC circuit, the circuit becomes an oscillator
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 22: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/22.jpg)
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 23: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/23.jpg)
Example 6-5 (cont.)
All rights reserved. Do not copy or distribute. © 2013 National
Technology and Science Press
![Page 24: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/24.jpg)
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 [−∞,∞]
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 25: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/25.jpg)
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
All rights reserved. Do not copy or distribute. ©
2013 National Technology and Science Press
![Page 26: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/26.jpg)
Singularity Functions (cont.)
Unit Impulse Function
For any function f(t): Sampling Property
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 27: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/27.jpg)
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
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 28: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/28.jpg)
Relations for Complex Numbers
Learn how to perform these with your calculator/computer
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
![Page 29: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/29.jpg)
Laplace Transform Technique
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 30: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/30.jpg)
Laplace Transform Definition
All rights reserved. Do not
copy or distribute.
© 2013 National Technology and Science Press
![Page 31: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/31.jpg)
Laplace Transform of Singularity Functions
For A = 1 and T = 0:
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 32: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/32.jpg)
Laplace Transform of Delta Function
For A = 1 and T = 0:
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 33: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/33.jpg)
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
![Page 34: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/34.jpg)
Properties of Laplace Transform
Time Scaling
Time Shift Example
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 35: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/35.jpg)
Properties of Laplace Transform (cont.)
Frequency Shift
Time Differentiation
Example
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 36: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/36.jpg)
Properties of Laplace Transform (cont.)
Time Integration
Frequency Differentiation
Frequency Integration
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 37: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/37.jpg)
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science
Press
![Page 38: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/38.jpg)
All rights reserved. Do not copy or distribute. © 2013
National Technology and Science Press
![Page 39: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/39.jpg)
Circuit Analysis
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 40: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/40.jpg)
ExampleAll rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 41: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/41.jpg)
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
All rights reserved. Do not copy or distribute. ©
2013 National Technology and Science Press
![Page 42: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/42.jpg)
1.Partial Fractions Distinct Real Poles
All rights reserved. Do not copy or distribute. © 2013
National Technology and Science Press
![Page 43: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/43.jpg)
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
All rights reserved. Do not copy or distribute. © 2013
National Technology and Science Press
![Page 44: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/44.jpg)
2. Partial Fractions Repeated Real Poles
All rights reserved.
Do not copy or distribute. © 2013 National
Technology and Science Press
![Page 45: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/45.jpg)
2. Partial Fractions Repeated Real Poles
Example
Cont.
All rights reserved. Do not copy or distribute. ©
2013 National Technology and Science Press
![Page 46: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/46.jpg)
2. Partial Fractions Repeated Real Poles
Example cont.
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 47: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/47.jpg)
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
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 48: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/48.jpg)
3. Distinct Complex Poles (Cont.)
Next, we combine the last two terms:
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 49: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/49.jpg)
4. Repeated Complex Poles: Same procedure as for repeated real poles
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 50: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/50.jpg)
Property #3a in Table 10-2:
Hence:
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science
Press
![Page 51: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/51.jpg)
s-Domain Circuit Models
Under zero initial conditions:
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 52: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/52.jpg)
All rights reserved. Do not copy or distribute. ©
2013 National Technology and Science Press
![Page 53: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/53.jpg)
All rights reserved. Do not
copy or distribute. © 2013 National
Technology and Science Press
![Page 54: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/54.jpg)
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
![Page 55: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/55.jpg)
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
![Page 56: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/56.jpg)
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
![Page 57: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/57.jpg)
Example : Interrupted Voltage Source
Initial conditions:
Voltage Source
(s-domain)Cont.
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 58: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/58.jpg)
Example : Interrupted Voltage Source (cont.)
Cont.
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 59: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/59.jpg)
Example : Interrupted Voltage Source (cont.)
Cont.
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 60: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/60.jpg)
Example : Interrupted Voltage Source (cont.)
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 61: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/61.jpg)
All rights reserved. Do not copy or distribute. © 2013
National Technology and Science Press
![Page 62: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/62.jpg)
All rights reserved. Do not copy or distribute. © 2013
National Technology and Science Press
![Page 63: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/63.jpg)
All rights reserved. Do not copy or distribute. © 2013
National Technology and Science Press
![Page 64: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/64.jpg)
Multisim Example of RLC Circuit
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 65: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/65.jpg)
RFID CircuitAll rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press
![Page 66: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/66.jpg)
Tech Brief 10:
Micromechanical Sensors and Actuators
All rights reserved. Do not copy or distribute. © 2013 National
Technology and Science Press
![Page 67: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/67.jpg)
Tech Brief 11: Touchscreens and Active Digitizers
All rights reserved. Do not copy or distribute.
© 2013 National
Technology and Science
Press
![Page 68: 6. CIRCUIT ANALYSIS BY LAPLACE CIRCUITS by Ulaby & Maharbiz All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press](https://reader034.vdocuments.us/reader034/viewer/2022052207/56649d755503460f94a56359/html5/thumbnails/68.jpg)
Summary
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press