Jan 25

# Jan 25 - EE 102 JAN 13 Printed by Mathematica for Students...

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Unformatted text preview: EE 102 JAN 13 JAN 25 27 Printed by Mathematica for Students FINALLY THE APPLICATION OF THE TRANSFORM Let us solve the the input output relation : dv t dt k v t u t t Technique : Take Laplace Transforms on both sides. Let U . , V . denote the Laplace Transforms of u . and v . respectively just here . Then by our rules s V s v k V s U s Or, V s 1 k s U s v OR V s v k s U s k s After this elementary arithmetic All we need to do now is find the function of time whose Laplace Transform is the right side Which we call the Inverse Laplace Transform. Symbolically : Find 1 V s 1 v k s U s k s 1 v k s 1 U s k s But this is actually the MORE DIFFICULT PROBLEM Given the Laplace Transform wannabee how do you find the the time domain function of which it is the Transform ?? BUILD UP ' TABLE' REPERTOIRE OF TRANSFORMS that you can dip into p46 of Book 2 Jan 25.nb Printed by Mathematica for Students Examples : F s a what is the Laplace Transform of the Delta Function ? e st Δ t t 1 Hence the Inverse Laplace Transform of F s a is a Δ t t 1 k s F s , k Inverse Transform : f t e kt t By direct verification u t Step Function a t W' s s W s W W Lim s a s k s Re.s Now use Derivative Formula W' s s W s a k s W' t a e kt Hence integrate to get : W t t a e ks ds a 1 e kt k Hence total response : v t e k t v a 1 e kt k which checks with our previous answer. Jan 25.nb 3 Printed by Mathematica for Students Back to 1 v k s 1 U s k s Generally then we need to find : 1 U s k s HERE WE HAVE ' BIG THEOREM ' THE CONVOLUTION THEOREM Let v t t W t Σ u Σ Σ t The Right Side is called the ' Convolution of the functions W t and u t t Denoted W u which is then u W since by a change of variable Σ t Τ we have v t t W t Σ u Σ Σ t W Τ u t Τ Τ Theorem The Laplace Transform of a Convolution of two Laplace Transformable Functions is the Product of the Laplace Transforms Since both transforms have to be defined, the abscissa of convergence of the Transform of the convolution will be the larger of the two abscissae of the convolvey functions. Outline of Proof : e st t W t Σ u Σ Σ t 4 Jan 25.nb Printed by Mathematica for Students e st W t Σ u Σ Σ t defining W t t W t t Now change order of integration i the double integral : e st W t Σ u Σ Σ t W t Σ e st dt u Σ Σ change variable : Τ t Σ Σ W Τ e s Σ Τ d Τ u Σ Σ W Τ e s Σ Τ d Τ u Σ Σ W Τ e s Σ Τ d Τ u Σ...
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Jan 25 - EE 102 JAN 13 Printed by Mathematica for Students...

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