Electrcal Crcut II (ECE33b) Applcaton of Laplace Tranform to Crcut Analy Anet Dounav The Unverty of Wetern Ontaro Faculty of Engneerng Scence
Crcut Element Retance Tme Doman (t) v(t) R v(t) = R(t) Frequency Doman Lv(t) V L(t) I V I R V = RI Smlar to phaor equvalent crcut
Crcut Element Inductor Tme Doman (t) v(t) Frequency Doman L Dfferental Form: Integral Form: v(t) V() L(I() ()) V I L L L() L (t) (t) d(t) v(t) L () dt V L t I() I v(x)dx V() L L () () ()/
Crcut Element Capactor Tme Doman (t) v(t) Frequency Doman L Dfferental Form: Integral Form: (t) I() C(V() v()) V I C /(C) Cv() v(t) L dv(t) (t) C v() v dt v(t) C t I (x)dx v() V() V I() C /(C) v()/ v()
Crcut Element Magnetcally Coupled Inductor Tme Doman (t) M v (t) L L (t) v (t) d(t) d (t) v(t) L M dt dt d (t) d(t) v (t) L M dt dt Frequency Doman V () LI () L () MI() M() V () LI () L () MI() M() I () V () L () M () L () M() M L L I () V ()
Crcut Element The advantage of the Laplace tranform on crcut element :. Intal condton are pat of a equvalent crcut (they appear a ource). any form of exctaton can be repreented 3. In the doman (or Laplacedoman) all method of analy and concept can be ued a before (.e. KCL, KVL, uperpoton, etc)
Example For the network hown, draw the doman equvalent crcut and fnd out the voltage n both the and tme doman. Alo verfy the ntal and fnal value theorem. Cae A) v () =, Cae B) v () = V (t)=3e t u(t) ma K 5F v (t)
Example For the network hown, determne the output voltage v (t) and current (t) (no energy tored at t=). v 3V.5F.8H.96
Example 3 For the network hown, determne the output voltage v (t). u(t).5f (t) (t) H Vo(t)
Example 4 Determne the output voltage for the followng crcut. t= t= H.5F V v (t)
Tranfer Functon The tranfer functon or network functon of crcut can be expreed a m m am am... a H() n n b b... b n where () YO () H()X () and X () the nput gnal Y O () the output repone If x (t)=(t) then X ()=, referred to a the mpule repone By knowng the mpule repone of a network we can fnd the repone due to ome forcng functon ung equaton () The repone of network are governed by real and complex conjugate pole r K K H() p a jb a jb n
Tranfer Functon jb a K jb a K p r H() k q p r charactertc equaton where the dampng coeffcent and o the undamped natural frequency Thee two factor, and o control the repone of the network, a follow
Tranfer Functon charactertc equaton Cae, >: Overdamped Network The root of the charactertc equaton are: p, p Contrbuton of econdorder tranfer functon X () K p K p x( t) K e ( ) t K e ( ) t Locaton of pole j Tmedoman repone x(t) x x t
Tranfer Functon charactertc equaton Cae, <: Underdamped Network The root of the charactertc equaton are: X p p j, Contrbuton of econdorder tranfer functon K () p K p x( t) K e t co( t ) Locaton of pole x j Tmedoman repone x(t) x t
Tranfer Functon charactertc equaton Cae, <: Underdamped Network The root of the charactertc equaton are: p p j, Locaton of pole x j j co x
Tranfer Functon charactertc equaton Cae 3, =: Underdamped Network The root of the charactertc equaton are: p, p Contrbuton of econdorder tranfer functon X () K p Locaton of pole j K p t t x( t) Kte Ke Tmedoman repone x(t) x pole t
Example 5 If the mpule repone of a network h(t)=e t, determne the repone v o (t) to an nput v (t)=e t u(t) V. Example 6 Determne v o (t) for C=8F, 6F and 3F. H u(t)v C Vo(t)
Example 7 The tranfer functon of the network H() 4 8 Determne the polezero plot of H(), the type of dampng exhbted by the network and the unt tep repone of the network.
Steady State Repone We have een that Laplace tranform technque can be ued to determne the complete repone of crcut Complete repone of crcut compoed of:. Tranent or natural repone, whch dappear a. Steady tate or forced repone, whch preent at all tme Recall the network repone can be expreed a YO () H()X () where X () the nput gnal () the output repone Y O () (pole of The tranent porton reult from the pole of H() H() pole of Y O H() tranfer functon X ()) The teadytate porton reult from the pole of X () t Note: that H()X () form a product, therefore each effected by each other
Example 8 For the crcut hown, determne the teady tate voltage v o (t) for t> f the ntal condton are zero..5f co(t)u(t)v H Vo(t)