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For an LCR circuit driven at frequency omega, the di equation reads L (di)/(dt)+Ri+ q/C=V_i=V_m sin omegat (a) Multiply the equation by i and simplify where possible. (b) Interpret each term physically. (c) Cast the equation in the form of a conservation of energy statement. (d) Integrate the equation over one cycle to find that the phase difference between v and i must be acute. |
Answer» Solution :Let circuit L-C-R as shown in figure.  From Kirchhoff.s law of closed circuit, `V_L+V_C+V_R=V_m sin omegat` `L(di)/(dt)+q/C+iR=V_m sin omegat`…(1) Multiplying I on both side, `Li(di)/(dt) + q/C i+i^2 R=V_m I sin omegat`...(2) where `Li(di)/(dt)=d/(dt)(1/2Li^2)` indicates the rate of change of ENERGY stored in an inductor. `i^2R=P` JOULE heat loss `q/C i=d/(dt)(q^2/(2C))` indicates the rate of change of energy stored in the CAPACITOR and Vi = rate at which driving force pours in energy. Hence, equation (2) is in the form of conservation of energy statement. Integrating both sides of equation (2) with respect to time over one full cycle (0 - T) we may write `int_0^T d/(dt)(1/2Li^2+q^2/(2C))dt+int_0^TRi^2dt=1/2int_0^T Vidt` where, `V_m sin omegat=V` `THEREFORE [0+1/2i^2Rt]_0^T =1/2 int_0^T Vi dt` `therefore [0+1/2i^2RT]=1/2int_0^T Vi dt` `therefore` 0+(Positive)=`int_0^T Vi dt [because i^2RT gt 0]` `therefore int_0^T Vi dt gt 0` if phase difference between V andi is a constant and acute angle. |
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