Q. 305.0( 1 Vote )

# For an LCR circui

Given:
Equation given :

(i)
Multiplying the equation by I,

Multiplying by (1/2) and substituting i=dq/dt

Hence, the simplified equation.
(ii)
Here, first term:
means the rate of change of energy stored in the inductor.
Second term :
represents the Joule’s Heating Loss.
Third Term:
Represents rate of change of energy in Capacitor.
Fourth term:
represents the energy due to driving force. It increases stored energy in inductor and capacitor.
(iii)
Conservation of Energy states that:
The sum of the individual energies across the individual components is equal to the total power of the circuit.
The equation in the form of a conservation of energy statement is

(iv)
Integrating the equation from 0 to T with resected to dt.

The integral of first term and third term would be zero as d/dt and dt gets cancelled. and V=Vmsinωt

LHS is positive as i, R and T are always positive, hence RHS is also positive.
Hence, phase difference between v and I must be acute for the cycle to be positive.

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