Keeping the source frequency equal to the resonating frequency of the series LCR circuit, if the three elements, L, C and R are arranged in parallel, show that the total current in the parallel LCR circuit is minimum at this frequency. Obtain the current rms value in each branch of the circuit for the elements and source specified in Exercise for this frequency.

Explanation

Here, `Z_(L ) = j omega L = j X_(Ll)` = inductive reactance
`Z_(C ) = - ( j )/( omega C ) = - j X_(C )` = capacitive reactance

Here, R, `Z_(L)` and `Z_(C 0` are connected in parallel . If effective impedance of this parallel connection is Z then according to law of parallel combination of resistance , we can write,
`(1)/(Z) = ( 1)/( R ) + ( 1)/(Z_(L ) ) + (1)/( Z _(C ))`
`:. (1)/(Z ) = ( 1)/( R ) +| ( 1)/( j ) (( 1)/( X_(L )) - ( 1)/( X_(C )))`
`:. (1)/(2) = (( 1)/( R )) + j { - ((1)/( X_(L)) - ( 1)/(X_(C)))}` `( :. j^(2) = - 1 " " 1 =-j^(2) )`
`:. ( 1)/( | Z|^(2)) = ((1)/(R))^(2) + ((1)/( X_(L ) ) - (1)/( X_(C)))^(2)`
`:. (1)/( |Z|)= sqrt(((1)/( R ))^(2) + ( (1)/( X_(L))-(1)/( X_(C ))^(2)))` ....(1)
At resonance,
`X_(L) = X_(C ) rArr (1)/(X_(L)) = (1)/(X_(C)) rArr (1)/( X_(L)) - ( 1)/( X_(C )) = 0`
`:. (1)/( |Z|) =` minimum `rArr |Z| =` minimum = R
`rArr I_(m) = ` minimum
now, in exercise sum no. , we had
`omega = omega_()) = 50 ( rad)/(s)`. At this angular frequency, here parallel resonance takes place with `|Z| = ` max =R
At this resonance,
(i) `I_(R ) = ( V_(rms))/( R ) = ( 230)/( 40) = 5.75 A`
(ii) `I_(L) = ( V_(rms))/( X_(L)) = ( V_(rms))/( omega_(0) L ) = ( 230)/( 50 xx 5 ) = 0.92A`
(iii) `I_(C ) =(V_(rms))/(X_(C)) = ( (V_(rms))/( ( 1)/( omega_(0)C)) ) = V_(rms) omega_(0) C `
`:. ` `I_(C ) = = ( 230) ( 50 ) ( 80 xx 10^(-8)) = 0.92A`
Here rms value of current through the source is given by,
`I_(rms) = sqrt(I_(R )^(2) + ( I_(L ) - I_(c ))^(2))`
`= I_(R ) ( :. I_(L ) = I_(C ) ` at resonance )

Thus, at resonance we can ignore the presence of L and C combination because the circuit behave as if only resistance R is connected across the source.