For an LCR circuit driven at frequency ω, the equation reads L di/dt + Ri + q/C = vi = vm sin ꞷ t
(i) Multiply the equation by i and simplify where possible.
(ii) Interpret each term physically.
(iii) Cast the equation in the form of a conservation of energy statement.
(iv) Integrate the equation over one cycle to find that the phase difference between v and i must be acute.
L di/dt + Ri + q/C = vi = vm sin ꞷ t
(i) Multiplying the above equation with I, we get
d(1/2 Li2)/dt + 1/2C dq2/dt + i2R/2 = ½ Vm i sin ꞷt
(ii) d(1/2 Li2)/dt represents the rate of change of potential energy in inductance L
d/dt q2/2C represents the energy stored in dt time in the capacitor
i2R represents the joules heating loss
½ Vm i sin ꞷt is the rate of driving force
(iii) The first equation is in the form of conservation of energy
(iv) Integrating the equation from o to T we get dt as positive which is possible when the phase difference is constant and the angle made is acute.
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