A Derivation of the Bjerknes Logarithmic Decrement Formula

The Bjerknes method of determining the logarithmic decrement of a circuit consists in coupling loosely to it an oscillatory circuit containing a thermo indicator; the latter circuit having known constants. From the resonance curve obtained by detuning the latter circuit, the logarithmic decrement of the unknown circuit is calculable by formula (30) above. The current in the resonant circuit is the sum of two component currents; one forced (and of the same decrement and frequency as the exciting current) and one a transient or free oscillatory current. Using the convenient mathematical method of imaginary symbolic operators, the differential equation of the excited current is set up and solved for the resonance condition and for non-resonant conditions. The integrated effect of the square of the current is obtained for each case. Proceeding on the assumption that the detuning of the circuits is not large, the usual formula for obtaining the logarithmic decrement is derived.