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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, SUNYBuffalo, Buffalo, NY 142601920
Tel: (716) 6453115 Ext. 1149, Fax: (716) 6453656
Email: kwangoh@buffalo.edu, http://www.SMALL.Buffalo.edu EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  EE@SUNYBuffalo Lecture 15  Chapter 12  1/5  1/16 EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  EE@SUNYBuffalo 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 inverseransform 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@SUNYBuffalo 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 onesided 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@SUNYBuffalo 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, eat, 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@SUNYBuffalo Lecture 15  Chapter 12  1/5  5/16 EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  EE@SUNYBuffalo 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, SUNYBuffalo, Buffalo, NY 142601920
Tel: (716) 6453115 Ext. 1149, Fax: (716) 6453656
Email: kwangoh@buffalo.edu, http://www.SMALL.Buffalo.edu EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  EE@SUNYBuffalo Lecture 15  Chapter 12  1/5  7/16 EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  EE@SUNYBuffalo 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@SUNYBuffalo (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@SUNYBuffalo 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, SUNYBuffalo, Buffalo, NY 142601920
Tel: (716) 6453115 Ext. 1149, Fax: (716) 6453656
Email: kwangoh@buffalo.edu, http://www.SMALL.Buffalo.edu EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  EE@SUNYBuffalo Lecture 15  Chapter 12  1/5  11/16 EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  EE@SUNYBuffalo 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@SUNYBuffalo Lecture 15  Chapter 12  1/5  13/16 EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  EE@SUNYBuffalo 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@SUNYBuffalo Lecture 15  Chapter 12  1/5  15/16 EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  EE@SUNYBuffalo 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.
 Spring '08
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