Write an equation of average power for L-C-R series AC circuit and discuss its various cases.

Explanation

Average power for a given circuit.
`P = VI cos phi`
where `V = (V_(m))/(sqrt(2)), I = ( I_(m))/( sqrt(2))`
Special cases `:`
(1) Resistive circuit ( Circuit containing only pure resistor ) `:`
If the circuit contains only pure R, it is called resistive.
In such circuit current and voltage are in same phase. So phase difference `phi = 0^(@)`
`:.` Average power `P = VI cos 0^(@)`
`:. P =VI`
There is maximum power dissipation.
(2) Purely inductive or capacitive circuit OR circuit contains only an inductor OR Capacitor C `:`
If the circuit contains only an inductor or only capacitor, such a circuit is called inductive and capacitve respectively.
In purely inductive circuit, current is `( pi )/(2)` behind of voltage and in purely capacitive circuit, current is `( pi )/(2)` ahead of voltage. means in both this type of circuit phase difference between current and voltage is `( pi )/(2)`.
Average power P `= VI cos phi`
`= VI cos ""(pi )/(2)`
`=0 [ :. cos "" ( pi )/(2) = 0 ]`
Even when current flows in circuit calculation of power is zero. This current called wattless current.
(3) L-C-R series circuit `:`
In an L-C-R series circuit power dissicpated is `P =VI cos phi` where ` phi = tan^(-1) ((X_(C ) - X_(L))/(R ))`
So, sometime `phi` may be non-zero in a RL or RC, or L-C-R circuits. Even in such case, power is dissipated only in the resistor.
(4) Power dissipated at resonance in L-C-R circuit `:`
At resonance `X_(C ) - X_(L) = 0`, and `phi = 0^(@)`.
Therefore,` cosphi =1` and `P = I^(2)Z = I^(2) R` `( :. P = VI cos phi , Z = R )`
Hence, maximum power is dissipated in a circuit at resonance.