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www.GOALias.blogspot.comPhysics7.7 POWER IN AC CIRCUIT: THE POWER FACTORWe have seen that a voltage v = v msinωt applied to a series RLC circuitdrives a current in the circuit given by i = i msin(ωt + φ) whereivZmm= andφtan⎛ X⎝− XR⎞⎠−1C L= ⎜ ⎟Therefore, the instantaneous power p supplied by the source is( msin ω ) [ msin( ω φ)]p = vi = v t × i t +vmim= [ cos φ − cos(2 ωt+ φ)](7.37)2The average power over a cycle is given by the average of the two terms inR.H.S. of Eq. (7.37). It is only the second term which is time-dependent.Its average is zero (the positive half of the cosine cancels the negativehalf). Therefore,vmivmmimP = cosφ= cos φ22 2= VIcosφ[7.38(a)]This can also be written as,2P I Z cosφ= [7.38(b)]So, the average power dissipated depends not only on the voltage andcurrent but also on the cosine of the phase angle φ between them. Thequantity cosφ is called the power factor. Let us discuss the followingcases:Case (i) Resistive circuit: If the circuit contains only pure R, it is calledresistive. In that case φ = 0, cos φ = 1. There is maximum power dissipation.Case (ii) Purely inductive or capacitive circuit: If the circuit containsonly an inductor or capacitor, we know that the phase difference betweenvoltage and current is π/2. Therefore, cos φ = 0, and no power is dissipatedeven though a current is flowing in the circuit. This current is sometimesreferred to as wattless current.Case (iii) LCR series circuit: In an LCR series circuit, power dissipated isgiven by Eq. (7.38) where φ = tan –1 (X c– X L)/ R. So, φ may be non-zero ina RL or RC or RCL circuit. Even in such cases, power is dissipated only inthe resistor.Case (iv) Power dissipated at resonance in LCR circuit: At resonanceX c– X L= 0, and φ = 0. Therefore, cosφ = 1 and P = I 2 Z = I 2 R. That is,maximum power is dissipated in a circuit (through R) at resonance.252EXAMPLE 7.7Example7.7 (a) For circuits used for transporting electric power, alow power factor implies large power loss in transmission. Explain.(b) Power factor can often be improved by the use of a capacitor ofappropriate capacitance in the circuit. Explain.