Feb 1

# Feb 1 - EE 102 WINTER 10 Laplace Transform Techniques...

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Unformatted text preview: EE 102 WINTER 10 Laplace Transform Techniques MONDAY FEB 1 Wed. Feb 3 is Mid Term Review Feb 8 is Midterm Logarithmic Decrement Log of Ratio of amplitude in one period Log e bt 2 e b t T 2 bT 2 b 2 Ν Ν damped natural frequency in Hertz Printed by Mathematica for Students Recall : Ω a a 1 2 4 Natural Frequency a 2 Π Damped Natural Frequency Ω 2 Π Damping Ratio : b 2 Ν These two determine the system usually small damping so we can use a 2 Π for Ν QUESTION What Input of given Finite power would you use to identify the syste m ? 2 Feb 1.nb Printed by Mathematica for Students FEBRUARY 1 SINUSOIDAL RESPONSE : SECOND ORDER SYSTEMS u t Sin 2 Π ft Φ U s 2 Π f s 2 4 Π 2 f 2 V s b 1 s b s 2 a 1 s a 2 Π f s 2 4 Π 2 f 2 We could find the inverse transform by partial fractions again but why? We already know it is given by the convolution : v t b 1 t W t Σ u Σ Σ b t W t Σ u Σ Σ b 1 t W t Σ Sin2 Π f Σ Σ b t W t Σ Sin2 Π f Σ Σ What can we say about the response ? The Steady State response is again sinusoidal with the same frequency as the input. This because we note that the system function is a sum of exponentials and hence so is the derivative so we can invoke our result for the first order system if the system is stable Indeed we I mean you can carry out the integration and get the answer explicitly. It is simplified if we only want the steady state value. For then, assuming again system stability : b 1 t W t Σ Sin2 Π f Σ Σ b t W t Σ Sin2 Π f Σ ΣΣ becomes the steady state version : b 1 W Σ Sin2 Π f t Σ Σ b W Σ Sin2 Π f t Σ Σ Note that here Sin2 Π ft has to be defined for...
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Feb 1 - EE 102 WINTER 10 Laplace Transform Techniques...

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