EE203-SUNYBuffalo-17-Chapter12-03

EE203-SUNYBuffalo-17-Chapter12-03 - SMALL for Big Things...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: SMALL for Big Things University at Buffalo SMALL for Big Things University at Buffalo nanobioSensors & MicroActuators Learning Lab The State University of New York nanobioSensors & MicroActuators Learning Lab The State University of New York Why the Laplace Transform: Details are in Chapter 13 EE 203 Circuit Analysis 2 Lecture 17 Chapter 12.6 Applying Laplace Kwang W. Oh, Ph.D., Assistant Professor SMALL (nanobioSensors and MicroActuators Learning Lab) Department of Electrical Engineering University at Buffalo, The State University of New York 215E Bonner Hall, SUNY-Buffalo, Buffalo, NY 14260-1920 Tel: (716) 645-3115 Ext. 1149, Fax: (716) 645-3656 E-mail: kwangoh@buffalo.edu, http://www.SMALL.Buffalo.edu EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 17 | Chapter 12 | 3/5 | 1/11 Opening Opening of the switch in the step jump of the source current from zero to Idc Laplace Transform Inverse Transform : Notation EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 17 | Chapter 12 | 3/5 | 2/11 SMALL for Big Things University at Buffalo SMALL for Big Things University at Buffalo nanobioSensors & MicroActuators Learning Lab The State University of New York nanobioSensors & MicroActuators Learning Lab The State University of New York Inverse Transforms EE 203 Circuit Analysis 2 Lecture 17 Chapter 12.7 Inverse Transforms Kwang W. Oh, Ph.D., Assistant Professor SMALL (nanobioSensors and MicroActuators Learning Lab) Department of Electrical Engineering University at Buffalo, The State University of New York 215E Bonner Hall, SUNY-Buffalo, Buffalo, NY 14260-1920 Tel: (716) 645-3115 Ext. 1149, Fax: (716) 645-3656 E-mail: kwangoh@buffalo.edu, http://www.SMALL.Buffalo.edu EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 17 | Chapter 12 | 3/5 | 3/11 General Form for the Inverse Transforms: a Rational Function of s A rational function is formed when a polynomial is divided by another polynomial. A proper rational function: if m > n can be expanded as a sum of partial fractions An improper rational function: if m <= n In fact, for linear, lumped-parameter circuits whose component values are constant, the s-domain expressions for the unknown voltages and currents are always rational functions of s. If we can inverse-transform rational functions of s, we can solve for the time-domain expressions for the voltages and currents. Examples EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 17 | Chapter 12 | 3/5 | 4/11 SMALL for Big Things University at Buffalo SMALL for Big Things University at Buffalo nanobioSensors & MicroActuators Learning Lab The State University of New York nanobioSensors & MicroActuators Learning Lab The State University of New York The Roots of D(s) Partial Fraction Expansion: Proper Rational Functions The The roots of D(s) are either are 1. Real and distinct For example, D(s) D(s) must be in factored form before we can make a partial fraction expansion. The denominator has 4 roots 2. Complex and distinct Two distincts: s = 0 , s = -3 for each distinct root of D(s), a single term appears in the sum of partial fractions. A multiple root of multiplicity 2, s = -1 for each multiple root of D(s) of multiplicity r, the expansion contains r terms. 3. Real and repeated 4. Complex and repeated EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 17 | Chapter 12 | 3/5 | 5/11 EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 17 | Chapter 12 | 3/5 | 6/11 SMALL for Big Things University at Buffalo SMALL for Big Things University at Buffalo nanobioSensors & MicroActuators Learning Lab The State University of New York nanobioSensors & MicroActuators Learning Lab The State University of New York 1. Partial Fraction Expansion: Distinct Real Roots of D(s) (1) 1. Partial Fraction Expansion: Distinct Real Roots of D(s) (2) Testing the results F(s) x (s + 0) 0) x (s + 0) F(s) x (s + 8) x (s + 8) F(s) x (s + 6) 6) x (s + 6) 6) A partial fraction expansion creates an identity; thus both sides must be the same for all s values hence we choose values that are easy to verify EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 17 | Chapter 12 | 3/5 | 7/11 The inverse-transform EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 17 | Chapter 12 | 3/5 | 8/11 SMALL for Big Things University at Buffalo SMALL for Big Things University at Buffalo nanobioSensors & MicroActuators Learning Lab The State University of New York nanobioSensors & MicroActuators Learning Lab The State University of New York 2. Partial Fraction Expansion: Distinct Complex Roots of D(s) (1) s 2 + 6s + 25 = s 2 + 2 ⋅ 3s + 32 + 16 = ( s + 3) 2 + 42 = ( s + 3 − j 4)( s + 3 + j 4) 2. Partial Fraction Expansion: Distinct Complex Roots of D(s) (2) K3 is the conjugate of K2 for complex conjugate roots, you actually need to you calculate only half the coefficients Testing the results: s = -3 with the denominator in factored form Inverse-Transform In general, having the function in the time domain contain imaginary components is undesirable. Fortunately, because the terms involving imaginary components always come in conjugate pairs, we can eliminate the imaginary components simply by adding the pairs. EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 17 | Chapter 12 | 3/5 | 9/11 SMALL for Big Things University at Buffalo nanobioSensors & MicroActuators Learning Lab The State University of New York 2. Partial Fraction Expansion: Distinct Complex Roots of D(s) (3) Distinct Distinct Complex Roots a pair of terms of the form appears in the partial fraction expansion, where the partial fraction coefficient is a complex number Inverse-Transform for the complex conjugate pair EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 17 | Chapter 12 | 3/5 | 11/11 EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 17 | Chapter 12 | 3/5 | 10/11 ...
View Full Document

Ask a homework question - tutors are online