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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 (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.
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