Laplace.2

Laplace.2 - u H t L = R Q L H t L + G Q H t L , t...

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Unformatted text preview: 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 b k 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 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 ! Printed by Mathematica for Students 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 = t- s d = - ds t- W H t- s L u H s L ds = W H L u H t- L d CHANGE THE SUBJECT ... .. COMPLEX PLANE z = x + i y- < x, y < RIGHT HALF PLANE Consists of all points z such that Re z > PICTURE HERE ON THE WHITE BOARD 2 Laplace.2.nb Printed by Mathematica for Students We prefer to use the letter s instead of z : Right HalfPlane = 8 s Re s < Specifying sigma specifies the half plane. Laplace Transform : of a function W H t L of time t defined on the HalfLine : t F H s L = Limit T T e- st W H t L dt = e- st W H t L dt Re s > is called the "Abcissa of Convergence" H Forget it ! L In Engineering we just calculate- if the answer does not make sense-- We just throw it away ! HERE AND BELOW F H s L s in righthalf plane will denote a Laplace Transform : F H . L = L H W H . LL A function in the Time Domain becomes a function in the Laplace Domain viz a Right Half Plane. Example W H t L = e 4 t t = 4 For Res > 4, F H s L = Lim T e- st e 4 t dt = Lim T e- H s- 4 L t- H s- 4 L = Lim 1- e- H s- 4 L T s- 4 = 1 s- 4 For s NOT EQUAL TO 4 Blows up if s = 4 ! s = 4 Re.s = 4 is the barrier ! W H t L = t k k t k = F H s L = Lim T e- s t 1 ds = Lim I 1- e- sT M 1 s 1 s with = Laplace.2.nb 3 Printed by Mathematica for Students Arbitrary k Positive Integer e- st t k dt Recall d e- st ds = - t e- s t d n e- st d n s = H- t L n e- st Hence e- st t k dt = H- 1 L k d k ds k e- st dt = H- 1 L k d k ds k 1 s k = 1 F H s L = 1 s 2 k = 2 F H s L = 2 s 3 k = 3 F H s L = 3.2 s 4 k F H s L = k ! s k + 1 How about NEGATIVE INTEGERS : W H t L = 1 t ?? What about W H t L = t 1 4 4 Laplace.2.nb Printed by Mathematica for Students Learn about the ' Gamma Function " G H L = e- t t - 1 dt e- st t k dt make change of variable: s t = This yields e- st t k dt = 1 s k + 1 e- t t k dt = G H k + 1 L s k + 1 where we can now use noninteger values for k....
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Laplace.2 - u H t L = R Q L H t L + G Q H t L , t...

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