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A Voltage V = V0 Sin ωt is Applied to a Series LCR Circuit. Derive the Expression for the Average Power Dissipated Over a Cycle. - CBSE (Science) Class 12 - Physics

ConceptAc Voltage Applied to a Series Lcr Circuit

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 (i) no power is dissipated even though the current flows through the circuit, (ii) maximum power is dissipated in the circuit?

Solution

Voltage V=V0sinωt is applied to an series LCR circuit.

Current is  π2" role="presentation" style="position: relative;">π2

I_0=V_0/Z

phi=tan^(-1)((X_C-X_L)/R)

Instantaneous power supplied by the source is

P=VI

=(V0sinωt)×(I0sin(ωt+ϕ)

=(V_0I_0)/2[cosphi-cos(2omegat+phi)]

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.

P=(V_0I_0)/2cosphi

=(V_0I_0)/(sqrt2sqr2)cosphi

=VIcosϕ

P=I2Zcosϕ

cosϕ  is called the power factor.

Case I

For pure inductive circuit or pure capacitive circuit, the phase difference between current and voltage is pi/2

:.phi=pi/2,cosphi=0

Therefore, no power is dissipated. This current is sometimes referred to as wattless current.

Case II

For power dissipated at resonance in an LCR circuit,

X_C-X_L=0, phi=0

∴ cos ϕ = 1

So, maximum power is dissipated.

Is there an error in this question or solution?
Solution A Voltage V = V0 Sin ωt is Applied to a Series LCR Circuit. Derive the Expression for the Average Power Dissipated Over a Cycle. Concept: Ac Voltage Applied to a Series Lcr Circuit.
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