4.5 - 4.5 Laplace Transforms Elements of Linear Circuit...

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57 4.5 Laplace Transforms Elements of Linear Circuit Theory Passive circuits no external energy supply Active circuits needs to be "powered up" to work, e.g. op-amps Laplace Transforms : if f ( t ), then ( ) 0 () st f tf t e d t ℑ= ⎡⎤ ⎣⎦ and f tF s , where s is complex, si σ ω = + Inverse Transform : -1 Fs ft Impedance Z (applied voltage)/(forcing current) Some basic components Resistor Capacitor Inductor R i C i L i et R it =⋅ 1 C itd t = di dt L = E sR I s 1 Cs Es Is = E sL s I s = ⋅⋅ Z R = 1 Zs Cs = Z s = in series: Z 1 Z 2 E out E in Total Impedance = Z 1 (s) + Z 2 (s) in parallel: Z 1 Z 2 E out E in Total Impedance = -1 12 11 Z(s)= + Z( ) Z(s) s
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58 Example: R i(t) C e (t) out e (t) in 0 1 () 1 ( ) [first order O.D.E] t RC RC in out out out in out t out in et R i i d t C de et i d t i t C Cd t de C e dt ee e ed τ ττ =⋅+ =⇒ = =+ = Simpler to use Laplace Transforms 1 1 1 1 in out out in Es I sR Cs s Cs Es R C Es s R C =⋅ = +
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59 Impedance Matching Thevenin's Theorem.: Any linear circuit can be reduced to a voltage source (the open circuit voltage at
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4.5 - 4.5 Laplace Transforms Elements of Linear Circuit...

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