. P OWER IN AC C IRCUIT : T HE P OWER F ACTOR We have seen that a voltage v = v m sin ω t applied to a series RLC circuit drives a current in the circuit given by i = i m sin( ω t + φ ) where v i Z and tan L X X φ ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ Therefore, the instantaneous power p supplied by the source is ( [ ] sin sin( p v i v i φ [ ] cos cos( m m v i φ φ ( . ) The average power over a cycle is given by the average of the two terms in R.H.S. of Eq.
( . ). It is only the second term which is time-dependent. Its average is zero (the positive half of the cosine cancels the negative half).
Therefore, cos m m v i P φ cos v i φ cos V I φ [ . (a)] This can also be written as, cos P I Z φ [ . (b)] So, the average power dissipated depends not only on the voltage and current but also on the cosine of the phase angle φ between them. The quantity cos φ is called the power factor .
Let us discuss the following cases: Case (i) Resistive circuit : If the circuit contains only pure R , it is called resistive . In that case φ = , cos φ = . There is maximum power dissipation. Case (ii) Purely inductive or capacitive circuit : If the circuit contains only an inductor or capacitor, we know that the phase difference between voltage and current is π / .
Therefore, cos φ = , and no power is dissipated even though a current is flowing in the circuit. This current is sometimes referred to as wattless current . Case (iii) LCR series circuit : In an LCR series circuit, power dissipated is given by Eq. ( .
) where φ = tan – ( X c – X L )/ R . So, φ may be non-zero in a RL or RC or RCL circuit. Even in such cases, power is dissipated only in the resistor. Case (iv) Power dissipated at resonance in LCR circuit : At resonance X c – X L = , and φ = .
Therefore, cos φ = and P = I Z = I R . That is, maximum power is dissipated in a circuit (through R ) at resonance. Example7. (a) For circuits used for transporting electric power, a low power factor implies large power loss in transmission.
Explain. (b) Power factor can often be improved by the use of a capacitor of appropriate capacitance in the circuit. Explain.