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LC-other

# LC-other - Elements of Linear Circuit Theory Passive...

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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 t f t e = dt and ( ) ( ) f t F = s , where s is complex, s i σ ω = + Inverse Transform : = ( ) -1 ( ) F s f t Impedance Z (applied voltage)/(forcing current) Some basic components Resistor Capacitor Inductor R i C i L i ( ) ( ) e t R i t = 1 ( ) ( ) C e t i t dt = ( ) di dt e t L = ( ) ( ) E s R I s = 1 ( ) ( ) Cs E s I s = ( ) ( ) E s L s I s = Z R = 1 ( ) Z s Cs = ( ) Z s L 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 1 2 1 1 Z(s)= + Z ( ) Z (s) s 1

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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 e t R i idt C de e t idt i t C C d de e t RC e dt e e e e d τ τ τ = ⋅ + = = = + = t Simpler to use Laplace Transforms 1 ( ) ( ) 1 ( ) ( ) ( ) 1 ( ) 1 in out out in E s I s R Cs
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