  Question

# A voltage V=V0 sin ωt is applied to a series LCR circuit. Derive the expression for the average power dissipated over a cycle. Under what condition is (i) no power dissipated even though the current flows through the circuit, (ii) maximum power dissipated in the circuit?

Solution

## Voltage V=V0 sin ωt is applied to a series LCR circuit. Current is I=I0 sin (ωt+Φ) I0=V0Z Φ=tan−1(XC−XLR) Instantaneous power supplied by the source is  P=VI=(V0 sin ωt)×(I0 sin (ωt+Φ)) = V0I02[cos Φ−cos (2ωt+Φ)] The average power over a cycle is average of the two terms on the R.H.S. of the above equation. The second term is time dependent, so, its average is zero. So,        P=V0I02cos Φ = V0I0√2 √2cos Φ = VI cos Φ P=I2 ZcosΦ cos Φ is called the power factor. Case 1. For pure inductive circuit or pure capacitive circuit, the phase difference between current and voltage i.e., Φ is π2. ∴                    Φ=π2 So             Cos Φ=0  Therefore, no power is dissipated. Case 2. For power dissipated at resonance in an LCR circuit, XC−XL=0,            Φ=0 ∴                 Cos Φ=1  So, maximum power is dissipated.  Suggest corrections   