EE203-SUNYBuffalo-15-Chapter12-01

EE203-SUNYBuffalo-15-Chapter12-01 - SMALL for Big Things...

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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 EE 203 Circuit Analysis 2 Lecture 15 Chapter 12.1 Definition of Laplace Transform Laplace Transform “The Laplace transform of a function f(t) ” is given The Laplace transform of f(t) is also denoted F(s) 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 15 | Chapter 12 | 1/5 | 1/16 EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 15 | Chapter 12 | 1/5 | 2/16 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 Mathematical Transform: Why Laplace Transform? Mathematical Transforms Examples: Logarithms, Phasor, Laplace Transform They are designed to create a new domain to make the mathematical manipulation easier. manipulation easier. After finding the unknown in the new domain, we inverse-ransform it back to the original domain. Logarithms A = BC log A = log B + log C Logarithms are used to change a multiplication or division problem into a simpler addition or subtraction problem. Phasor It converts a sinusoidal signal into a complex number for easier, algebraic computation of circuit values. After determining the phasor value of a signal, we transform it back to its timedomain expression. EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 15 | Chapter 12 | 1/5 | 3/16 Mathematical Transform: Why Laplace Transform? Laplace Transform To transform a set of integrodifferential equations from the time domain to a set of algebraic equations in the frequency domain. Th Therefore, simplify the solution for an unknown quantit to the manipulation of a lif tity th set of algebraic equations. Frequency Domain Time Domain The one-sided Laplace transform ignores f(t) for t < 0- what happens prior to 0- is accounted for by the initial conditions. Lower limit 0Upper limit infinite The integration from 0- to 0+ zero. EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 15 | Chapter 12 | 1/5 | 4/16 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 Functional Tranforms Laplace Transform Step Function: Ku(t) Impulse Function: Kδ(t) Functional Transform the Laplace transform of a specific function, such as sin ωt, t, e-at, and so on. Operational Transform a general mathematical property of the Laplace transform, such as such finding the transform of the derivative of f(t). EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 15 | Chapter 12 | 1/5 | 5/16 EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 15 | Chapter 12 | 1/5 | 6/16 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 EE 203 Circuit Analysis 2 Lecture 15 Chapter 12.2 Step Function 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 15 | Chapter 12 | 1/5 | 7/16 EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 15 | Chapter 12 | 1/5 | 8/16 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 Step Function (1) Step Function (2) Ku(t): the symbol for the step function Discontinuity at t = a A discontinuity or jump at the origin (t – a < 0) (t – a > 0) Ku(t) is not defined at t = 0. In situations where we need to define the transition between 0- and 0+, we assume that it is linear and that K[u(t - 1) - u(t - 3)] Turning ON Turning OFF 0-: symmetric points arbitrarily close to the left of the origin 0+: symmetric points arbitrarily close to the the right of the origin EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo (a – t > 0) (a – t < 0) Lecture 15 | Chapter 12 | 1/5 | 9/16 EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 15 | Chapter 12 | 1/5 | 10/16 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 Example 12.1 EE 203 Circuit Analysis 2 Lecture 15 Chapter 12.3 Impulse Function 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 15 | Chapter 12 | 1/5 | 11/16 EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 15 | Chapter 12 | 1/5 | 12/16 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 Impulse Function Impulse Function The The concept of impulse function enables us to define the derivative at a discontinuity. Impulse Mathematical Mathematical Definition a signal of infinite amplitude and zero duration duration Does not exist in nature, but some circuit signals come very close to approximating this definition. this The area under the impulse function is constant. This area represents the strength This (K) of the impulse. The impulse is zero everywhere except at t = 0. An impulse that occurs at t = a is defined Kδ(t – a). δ(t): Unit Impulse Function Graphic symbol for the impulse function is an arrow. Sifting Kδ(t): Impulse Function the impulse function sifts out everything everything except the value of f(t) at t = a. K: strength of the impulse function EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 15 | Chapter 12 | 1/5 | 13/16 EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 15 | Chapter 12 | 1/5 | 14/16 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 Sifting Laplace Transform of the Impulse Function The The impulse function sifts out everything except the value of f(t) at t = a. Laplace Laplace Transform of the Impulse Function ∫ ∞ 0 − ∞ 0− −∞ −∞ δ (t )e st dt = ∫ δ (t )e − st dt − ∫ δ (t )e − st dt = e − s 0 − 0 = 1 The nth Derivative of the Impulse Function an Impulse Function = a Derivative of a Step Function EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 15 | Chapter 12 | 1/5 | 15/16 EE 203 Circuit Analysis 2 | Spring 2008 | Prof. Kwang W. Oh | EE@SUNY-Buffalo Lecture 15 | Chapter 12 | 1/5 | 16/16 ...
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This note was uploaded on 10/19/2011 for the course EE 203 taught by Professor Staff during the Spring '08 term at SUNY Buffalo.

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