# For an LCR circuit driven at frequency ω,

Question:

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.

Solution:

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.