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Find the damped oscillation frequency of the circuit shown in figure. The capacitance C, inductance L, and active resistance R are supposed to be known. Find how must C,L, and R be interrelated to make oscillations possible. |
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Answer» Solution :GIVEN `q=q_(1)+q_(2)` `I_(1)=-dot(q_(1)), I_(2)=-dot(q_(2))` `LI_(1)=RI_(2)=(q)/(C)`. Thus `CL ddot(q_(1))+(q_(1)+q_(2))=0` `RC dot(q_(2))+q_(1)+q_(2)=0` Putting `q_(1)=A e^(iomegat)` `q_(2)=Be^(+iomegat)` `(1-omegaLC)A+B=0` `A+(1+iomegaRC)B=0` Asolutiion EXISTS only if `(1- omega^(2)LC)(1+iomegaRC)=1` or `iomegaRC-omega^(2)LC-IOMEGA^(3)LRC^(2)=0` or `LRC^(2) omega^(2)-i omegaLC-RC=0` `omega^(2)-iomega(1)/(RC)-(1)/(LC)=0` `omega=(i)/(2RC)+-sqrt((1)/(LC)-(1)/(4R^(2)C^(2)))~=ibeta+-omega_(0)` Thus `q_(1)=(A_(1)cos omega_(0)t+A_(2) sin omega_(0) t ) e^(-BETAT) `etc `omega_(0)` is the oscillation FREQUENCY. Oscillations are possible only if `omega_(0)^(2)gt0` `i.e.` `(1)/(4R^(2))lt(C)/(L)` 
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