Question

A circuit consists of a capacitor with capacitance `C` and a coil with active resistance `R` and inductance `L` connected in parallel. Find the impedane of the circuit at frequency `omega` of alternating voltage.

Answer 1

`i_(1) = (V)/(X_(C)` current leads the voltage by `pi//2`
`i_(2) = V//sqrt(R^(2) + X_(L)^(2))` current lags the voltage by phi
where `tan phi = X_(L) //R`
`i = sqrt(i_(1)^(2) + _(2)^(2) + 2i_(1) i_(2) cos(90 + phi)) = sqrt(i_(1)^(2) + i_(2)^(2) -2i_(1) i_(2) sin phi)`
`(V)/(Z) = sqrt(V^(2)/(X_(C)^(2)) + (V^(2))/(R^(2) + X_(L)^(2)) -2(V)/(X_(C)) .(V)/(sqrt(R^(2) + X_(C)^(2))). (X_(L))/(sqrt(R^(2) + X_(L)^ (2))`
`(1)/(Z) = [(1)/X_(C)^(2) + (1)/(R^(2) + X_(L)^(2)) - (2(X_(L)//X_(C)))/((R^(2) + X_(L)^(2)))]^(1/2)`
`= [(R^(2) + X_(L)^(2) + X_(C)^(2) -2X_(L) X_(C))/(X_(C)^(2) (R^(2) + X_(L)^(2) )]]^(1/2)`
`=[((R^(2))/(X_(C)^(2))+((X_(L)-X_(C))/(X_(C)))^(2))/((R^(2)+X_(L)^(2)))]^((1)/(2))`
`Z = sqrt( (R^(2) + omega^(2) L^(2))/((omegaCR)^(2) + (omega^(2) LC -1)^(2)))`

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