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Discuss the phenomenon of resonance in a LCR series a.c. circuit. |
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Answer» Solution :We know that when an alternating voltage V=`V_(m) SIN omega t`is applied to a LCR series circuit, the current in the circuit is given by: `I = V_(m)/sqrt(R^(2) + X_(L) - X_(C))^(2).sin(omegat-phi) = V_(m)/Z sin(omega t-phi)` where impedance, `Z = sqrt(R^(2) + (X_(L)-X_(C))^(2))` and phase angle `phi = tan^(-1) (X_(L)-X_(C))/R` If it so happens that `X_(L) = X_(C)`then Z=R= minimum and `phi = tan^(-1)(0)= 0^(@)`i.e., the current will have a maximum VALUE given by I = and will be in same phase as that of applied voltage. Such a situation is called the phenomenon of electrical resonance. For electrical resonance, the necessary CONDITION is: `X_(L) = X_(C)` or `L omega_(0) = 1/(C omega_(0))` Obviously as L and C are fixed for a circuit, resonance occurs for a PARTICULAR frequency known as "resonant frequency". `v_(0)`(or resonant ANGULAR frequency `omega_(0)` ) given by: `Lomega_(0) = 1/(C omega_(0))` which leads us to `omega_(0) = 1/sqrt(LC)` `rArr v_(0)=1/(C omega_(0)) =1/(2pi sqrt(LC))` |
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