circuit A

circuit A - LAPLACE TRANSFORMS Chapter 14 Cha LAPLACE...

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APLACE TRANSFORMS LAPLACE TRANSFORMS Chapter 14
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LAPLACE TRANSFORMS Chapter 14 in Chapter 14 in D &S D &S ood reference for Laplace transforms Good reference for Laplace transforms Sample problems and clickable answers http://www.intmath.com/Laplace-transformation/Intro.php
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Why Do We Use Laplace Transforms? 2 1 in J & J) (12.1 in J & J) We use Laplace Transforms to transform a circuit problem into the s-domain domain , which makes the problem easier to solve. nce done we use the inverse s    j Complex frequency Once done, we use the inverse transform to obtain the solution to the riginal problem in the time domain original problem in the time domain.
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The Complex Plane he s omain) (The s-Domain) Imaginary axis (j) =x+ 1 tan y u r y u=x+jy Real axis 2 2 | | | | y x u r u x x y x complex conjugate u =x-jy
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Methodology solution problem Inverse Laplace Laplace ansform Solution in time domain in time domain transform transform in s-domain
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Methodology
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Linear Differential Equations linear differential quation time domain lution equation solution time domain Laplace transform Laplace or complex inverse Laplace transform Laplace transformed quation Laplace solution frequency domain algebra equation Key point! Recall that we have previously used (phasors). s j
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Example of a Laplace Transform f ( t ) e 2 t F ( s ) 1 L s 2 Laplace Transform t is a real variable s is complex variable (s) a complex -1 f(t) is a real function F(s) is a complex valued function Inverse Time Domain Frequency Domain
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Definition Convert time-domain functions and operations 0 ) ( ) ( )] ( [ dt e t f s F t f st L into frequency-domain C) s R, (t F(s) f(t) Linear differential equation (LDE) algebraic expression in complex plane Examine certain graphs in the s-domain for key LDE characteristics Important! Discrete systems use the analogous z-transform
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Examples of Laplace Transforms of ome Common Functions Some Common Functions Name f(t) F(s) mpulse 0 1 ) ( t t f y Impulse Step 1 1 1 ) ( t f 0 0 t gularit nctions Ramp s 2 1 s t t f ) ( Sin g Fu n Exponential a s 1 at e t f ) ( n( Sinusoid 2 s 2 ) sin( ) ( t t f (See p.500, 501 See p.500, 501) in J&J ) in J&J
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Linearity & Multiplication Properties f (t) ± f (t) F (s) ± F (s) Linearity f 1 (t) f 2 (t) f(t) F 1 (s) F 2 (s) F(s) ultiplication by constant af(t) a F(s) Multiplication by constant af 1 (t) ± bf 2 (t) aF 1 (s) ± bF 2 (s) Addition & Scaling Based on the properties of an integral.
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Example: Laplace Transform f Sine or Cos Function of Sine or Cos Function ecall j j e ) sin( ) cos( Recall j j e ) sin( ) cos(   j j e e 1 ) cos( Adding above j j 1 n( 2 Subtracting above   e e j 2 ) sin(
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Sine Function f ( t ) sin( t ) F ( s ) s 2   2 _______________________________________ F(s) f(t)e st dt 0 sin( t ) e dt 0 1 2 j e j t e j t  e dt 1 2 j e j t st dt e j t st dt
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circuit A - LAPLACE TRANSFORMS Chapter 14 Cha LAPLACE...

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