The L-C parallel resonant circuit

Explanation

Parallel Resonsant Cicruit : It is an a.c. circuit. In which a parallel combination of an inductor and capacitor is connected to a source of an alternating e.m.f

`i=I _(0)sin omega t`
`i= i_(C)-i_(L)`
`:. i_(L)=(e)/(X_(L)),i_(C)=(e)/(X_(C))`
`:. i=-(e)/(X_(C))-(e)/(X_(L))`
`=e((1)/(X_(C))-(1)/(X_(L)))`
`i=e((1)/(1//omegac)-(1)/(omegaL)) ( :. X_(C) =(1)/(omegaC))`
`i=e(omegac-(1)/(omegaL))`
`:. Z=(e)/(i)`
`Z=((1)/(omegaC-(1)/(omegaL)))`
For parallel resonance,
`X_(L)=X_(C)`
`i.e., (1)/(Z)=0`
`:. Z=oo`
The frquency at which impedence (Z) become infinite and current become zero is called resonant frequncy.

`omega C=(1)/(omegaL)=0`
`omega C=(1)/(omegaL)`
`:. omega^(2)=(1)/(LC)`
`:. omega =(1)/(sqrt(LC))`
`:. omega =2 pi f_(r)`
`:. 2pif""_(r) (1)/(sqrt(LC))`
`f_(r)=(1)/(2 pisqrt(LC))`