Nonharmonic Oscillations as Caused by Magnetic Saturation

It is shown that the distorted curve shapes of natural oscillations can be rigorously derived from the saturation characteristic of the oscillatory circuit by means of simple integrations. The natu-ral frequency is not constant as under linear conditions, but varies greatly with the amount of saturation. For forced oscillations, a concise dif-ferential equation is given covering all possible cases by only four parameters. The solution can be evaluated for any initial conditions by a straight-forward step-by-step construction. It shows graphically strange curve shapes of flux, current, and voltages, of no regularity with respect to periodic repetition or sym-metrical behavior during the transient state. In the steady state, a rigorous lineariza-tion of the differential equation allows considering the effect of saturation quan-titatively as a distortion of the impressed voltage. The final effect on magnitude and curve shape can be evaluated by repeated superposition of the residual dis-torting voltage. Intense higher har-monics are produced in this way. Consideration of the resistance voltage as actually present in the state of free oscillations shows that natural oscilla-tions can be sustained in true resonance by an impressed voltage of definite magni-tude and curve shape, requiring in series circuits a highly peaked voltage curve. Hence, the saturated circuit can respond to any constituent harmonic of this shape, leading to the forced development of sub-harmonics to the frequency of the supply voltage within certain ranges of its magni-tude. If the ohmic resistance is small a multitude of such subharmonics and all their higher harmonics may develop.