6. Circuit Analysis by Laplace

6. Circuit Analysis by Laplace
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press CIRCUITS by Ulaby & Maharbiz

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

Initial/Final Conditions
Guidelines vC, iL do not change instantaneously Get derivatives dvC/dt and diL/dt from iC , vL Capacitor open, Inductor short at dc All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Example 6-2: Determine Initial/Final Conditions
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Example 6-2: Determine Initial/Final Conditions Circuit t = 0‒

Example 6-2: Initial/Final Conditions (cont.)
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Example 6-2: Initial/Final Conditions (cont.) t = 0+ Given:

Example 6-2: Initial/Final Conditions (cont.)
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Example 6-2: Initial/Final Conditions (cont.) t  

Series RLC Circuit: General Response

Series RLC Circuit : General Solution
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
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Series RLC Circuit: Natural Response Find Natural Response Of RLC Circuit 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 Solution of Diff. Equation Assume: It follows that: All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Solution of Diff. Equation (cont.)
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Solution of Diff. Equation (cont.) Invoke Initial Conditions to determine A1 and A2

Circuit Response: Damping Conditions
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Circuit Response: Damping Conditions s1 and s2 are real s1 = s2 Damping coefficient s1 and s2 are complex Resonant frequency

Overdamped Response Overdamped, a > w0
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Overdamped Response Overdamped, a > w0

Underdamped Response Damping: loss of stored energy
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Underdamped Response Damping: loss of stored energy Underdamped a < w0 Damped natural frequency

Critically Damped Response
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Critically Damped Response Critically damped a = w0

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

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

Parallel RLC Circuit Same form of diff. equation as series RLC
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Parallel RLC Circuit Overdamped (a > w0) Same form of diff. equation as series RLC Critically Damped (a = w0) Underdamped (a < w0)

Analysis Techniques Circuit Excitation Method of Solution Chapters
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Analysis Techniques Circuit Excitation Method of Solution Chapters 1. dc (w/ switches) Transient analysis 5 & 6 2. ac Phasor-domain analysis ( steady state only) 3. any waveform Laplace Transform This Chapter (single-sided) (transient + steady state) 4. Any waveform Fourier Transform (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 All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Singularity Functions (cont.)
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Singularity Functions (cont.) Unit Impulse Function For any function f(t): Sampling Property

Review of Complex Numbers
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Review of Complex Numbers We will find it is useful to represent sinusoids as complex numbers Rectangular coordinates Polar coordinates Relations based on Euler’s Identity

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

Laplace Transform Technique

Laplace Transform Definition

Laplace Transform of Singularity Functions
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Laplace Transform of Singularity Functions For A = 1 and T = 0:

Laplace Transform of Delta Function
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Laplace Transform of Delta Function For A = 1 and T = 0:

Properties of Laplace Transform
Time Scaling Example Time Shift All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

Properties of Laplace Transform (cont.)
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Properties of Laplace Transform (cont.) Frequency Shift Example Time Differentiation

Properties of Laplace Transform (cont.)
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Properties of Laplace Transform (cont.) Time Integration Frequency Differentiation Frequency Integration

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

1.Partial Fractions Distinct Real Poles

1. Partial Fractions Distinct Real Poles
Example The poles of F(s) are s = 0, s = −1, and s = −3. All three poles are real and distinct. All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press

2. Partial Fractions Repeated Real Poles

3. Distinct Complex Poles
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press 3. Distinct Complex Poles 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 Example Note that B2 is the complex conjugate of B1.

3. Distinct Complex Poles (Cont.)
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press 3. Distinct Complex Poles (Cont.) Next, we combine the last two terms:

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

Example : Interrupted Voltage Source
All rights reserved. Do not copy or distribute. © 2013 National Technology and Science Press Example : Interrupted Voltage Source Initial conditions: Voltage Source (s-domain) Cont.

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