Laplace

Laplace - WEDNESDAY OCTOBER 7 2009 Class Notes 102 EE Topic...

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Unformatted text preview: WEDNESDAY OCTOBER 7 2009 Class Notes 102 EE Topic : LAPLACE TRANSFORMS Motivation RECALL u H t L = R Q L H t L + G Q H t L , t > v H t L = G Q H t L = ' Output' voltage Input- Output Relation dv dt = G dQ dt = G u H t L R- v H t L R dv dt + k v H t L = k u H t L d k dt k = 1 for k = d n v H t L dt n + 1 n- 1 a k d k v H t L dt k = m d k u H t L dt k n th order differential equation. The coefficient of the hhighest degree is nonzero. Comes from EQUILIBRIUM LAWS OF SCIENCE EXAMPLE : d 2 v H t L dt 2 + a 1 d v H t L dt + a v H t L = d u H t L dt + b u H t L Printed by Mathematica for Students CIRCUIT EXAMPLE : L d 2 Q H t L dt 2 + R dQ H t L dt + G Q H t L = u H t L v H t L = G Q H t L Output volts u H t L Input Volts d 2 dt 2 Q H t L + R L d dt Q H t L + G L Q H t L = u H t L L d 2 dt 2 v H t L + R L d dt v H t L + G L v H t L = G L u H t L This is our IP OP RELATION ! HOW DO SOLVE FOR v H . L given u H . L ?? LAPLACE TRANSFORMS ** ** ** ** ** ** ** ** ** ** ** ** ** ** Recall v H t L = t-¥ W H t- s L u H s L ds where W H t L is ' generalises' e- kt , t > W H . L defined on the " positive Half- line" Make the change of variable Σ = t- s d Σ = - ds t-¥ W H t- s L u H s L ds = ¥ W H Σ L u H t- Σ L d Σ CHANGE THE SUBJECT ... .. COMPLEX PLANE z = x + i y- ¥ < x, y < ¥ RIGHT HALF PLANE Consists of all points z such that Re z > Σ ‡ PICTURE HERE ON THE WHITE BOARD 2 Laplace.nb Printed by Mathematica for Students We prefer to use the letter s instead of z : Right HalfPlane = 8 s Re s ‡ Σ ‡ < Specifying sigma specifies the half plane....
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Laplace - WEDNESDAY OCTOBER 7 2009 Class Notes 102 EE Topic...

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