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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=tan1(XCXLR)

Instantaneous power supplied by the source is 

P=VI=(V0 sin ωt)×(I0 sin (ωt+Φ))

= V0I02[cos Φcos (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

= V0I02 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,

XCXL=0,            Φ=0

                 Cos Φ=1 

So, maximum power is dissipated.

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