A series LCR resonant circuit is an a.c. circuit in which a series combination of an inductor, capacitor and resistor is connected to a source of an alternating e.m.f.
Consider an alternating e.m.f. Applied to a series of inductance L, pure capacitor of capacitance C and a resistor of resistance R.
Let, `V_(L), V_(C)` and `V_(R)` be the voltage of inductor capacitor and resistor respectively . Let `i` be the current flowing in the circuit.
`V_(R_) = iR, V_(L) = iX_(L) V_(C) = iX_(C)`
`X_(L)` = inductive reactance
`X_(C)` = capacitive reactance
From the phasor diagram :
`V_(C)^(2) = VR^(2) +(V_(L) - V_(C))^(2)`
But `V_(R) = iR, V_(L)= iX_(L) , V_(C) = iX_(C)`
`:. V^(2) = i^(2)R^(2) +(iX_(L) -X_(C))^(2)`
`V^(2) = i^(2) [R^(2) (X_(L) - X_(C))^(2)]`
`V= i sqrt(R^(2) +(X_(L) + X_(C))^(2))`
`V/i = sqrt(R^(2)+ (X_(L) - X_(C))^(2))`
Compairing with `R = V/I`
Thus the expression `sqrt(R^(2) + (X_(L) - X_(C))^(2))` gives the impedance of LCR circuit.
`:.Z= sqrt(R^(2) + (X_(L) - X_(C))^(2))`
Considition for resonance :
If the frequency of source is varied and a frequency is obtained for which inductive reactance `X_(L)` becomes equal to capacitive reactance `X_(C )` the impedance circuit becomes minimum `Z = R` and current flowing circuit is maximum.
Resonant frequency :
The frequency of a.c. source at which the impedance of circuit is minimum and r.m.s. current is maximum is called series resonant frequency.
At resonant frequency.
`X_(L) = X_(C)`
`:. omegaL = (1)/(omegaC)`
`:. omega^(2) = 1/(LC)`
`:. omega = (1)/(sqrt(LC))`
`:. 2pif_(r) = 1/(sqrt(LC))`
`:. f_(r) = 1/(2pisqrt(LC))`
Numerical :
Given : `C = 10muF = 10 xx 10^(-6) F, V = 100V`
As we know, `U = 1/2CV^(2)`
`U= 1/2 xx 10xx 10^(-6) xx (100)^(2)`
`U = 0.05 J`
