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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. |
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Answer» <P> Solution :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)` |
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