Obtain an expression for the average power supplied to a series RLC circuit. Discuss the average power when the series RLC circuit behaves as a pure resistive, inductive or capacitive circuit.

Explanation

For a series RLC circuit, instantaneous power supplied is given by
`P=vi`
i.e., `P=(v_(m)sinomegat)(i_(m)sin(omegat+phi))`
`=(v_(m)i_(m))/(2)[cosphi-cos(2omegat+phi)]`
The average power supplied,
`barp=p_(av)=(v_(m)i_(m))/(2)cosphi`
because `ltcos(2omegat)gt =0,v_(m)=sqrt(2)v_(rms),i_(m)=sqrt(2)i_(rms)`
i.e., `barp=v_(rms)i_(rms)cosphi`
Power factor of an ac circuit,
`cosphi=(R)/(Z)`
so `barp=v_(rms)i_(rms)=(R)/(Z)`
Case (i):
For a series RLC to behave as a resistive circuit:-
`X_(L)=X_(C)` so that `Z=Z_(min)=R,X_(L)-X_(C)=0`
`therefore` Power factor `cosphi=(R)/(Z_(min))=1`
Average power dissipation at resonance = `barp=v_(rms)i_(rms)`
Case (ii):
For a series RLC to behave as an inductive circuit:-
`X_(L)gtX_(C)`
i.e., `2pif_(H)Lgt(1)/(2pif_(H)C)`
or `f_(H)gt(1)/(2pisqrt(LC))orf_(H)gtf_(0)` where, `f_(0)=(1)/(2pisqrt(LC))`
Power factor `cosphi=(R)/(Z)=(R)/(sqrt(R^(2)+X_(L)^(2)))~~(R)/(X_(L))" for "X_(L)gtR`.
Case (iii):
For a series RLC to behave as a capacitive circuit :-
`P.f=cosphi=(R)/(sqrt(R^(2)+X_(C)^(2)))" for "X_(C)gtX_(L)andf_(L)ltf_(0)`