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, SUNYBuffalo, Buffalo, NY 142601920
Tel: (716) 6453115 Ext. 1149, Fax: (716) 6453656
Email: [email protected], http://www.SMALL.Buffalo.edu EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  [email protected] 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  [email protected] 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, SUNYBuffalo, Buffalo, NY 142601920
Tel: (716) 6453115 Ext. 1149, Fax: (716) 6453656
Email: [email protected], http://www.SMALL.Buffalo.edu EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  [email protected] 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, lumpedparameter circuits whose component values are
constant, the sdomain expressions for the unknown voltages and currents are
always rational functions of s. If we can inversetransform rational functions of s,
we can solve for the timedomain expressions for the voltages and currents.
Examples EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  [email protected] 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  [email protected] Lecture 17  Chapter 12  3/5  5/11 EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  [email protected] 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  [email protected] Lecture 17  Chapter 12  3/5  7/11 The inversetransform EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  [email protected] 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 InverseTransform 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  [email protected] 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 InverseTransform for the complex conjugate pair EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  [email protected] Lecture 17  Chapter 12  3/5  11/11 EE 203 Circuit Analysis 2  Spring 2008  Prof. Kwang W. Oh  [email protected] Lecture 17  Chapter 12  3/5  10/11 ...
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 Spring '08
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 Fraction, Complex number, Rational function, Partial fractions in integration

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