InterviewSolution
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InterviewSolution
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A series circuit consisting of a capacitor with capacitance with capacitance C, a rasistance R, and a coil with inductance L and negligible activeresistance is connected to an oscillator whose frequency can be varied without changing the voltag amplitude. Find the frequency at which the voltage amplitude is maximum (a) across the capacitor , (b) across the coil. |
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Answer» Solution :`(a) v_(c)=(1)/( omegaC) (V_(m))/( sqrt(R^(2)+(omegaL-(1)/( omegaC))^(2)))` `=( V_(m))/( sqrt( (omegaRC)^(2)+ ( omega^(2) LC-1)^(2)))=(V_(m))/( sqrt(((omega^(2))/( omega_(0)^(2))-1)^(2)+ 4 beta^(2) omega^(2) // omega_(0)^(4)))` `=(V_(m))/( sqrt(((omega^(2))/( omega_(0)^(2))-1-( 2 beta)/( omega_90)^(2))^(2)+(4 beta^(2))/( omega_(0)^(2))-( 4 beta^(4))/( omega_(0)^( 4))))` Thisis maximum when `omega^(2)=omega_(0)^(2)- 2 beta^(2) = (1)/( LC)-(R^(2))/( 2L^(2))` `(b)` `V_(L)=I_(m) omegaL=Vm(omegaL)/( sqrt(R^(2)(omegaL-(1)/( omegaL))^(2)))` `=( V_(m)L)/( sqrt((R^(2))/(omega^(2))+(L-(1)/( omega^(2)C))^(2)))=(V_(m)L)/(sqrt(L^(2)-(1)/( omega^(2))((2L)/(C)-R^(2))+(1)/( omega^(4)C^(2))))` `=(V_(m)L)/( sqrt(((1)/( omega^(2)C)-(L- ( CR^(2))/( 2)) )^(2)+L^(2)- ( L- (1)/( 2) CR^(2))^(2)))` This is maximum when `(1)/( omega^(2)C)=L-(1)/( 2) CR^(2)` or ` omega^(2) =( 1)/( LC-(1)/(2) C^(2) R^(2))=(1)/( (1)/( omega_(0)^(2))-( 2 beta^(2))/( omega_(0)^(4)))` `=( omega_(0)^(4))/( omega_(0)^(2)-2 beta^(2)) `or ` omega=( omega_(0)^(2))/( sqrt( omega_(0)^(2)-2 beta^(2)))` |
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