Smooth frequency change effects in electrical circuits: explained by the Fresnel integral

Abstract

AbstractThe paper presents the problem of the relation between ac voltage frequency supplying a circuit and the resulting current frequency in that circuit in the case when the voltage frequency changes smoothly following a $${f}_{{v}}\left(t\right)$$ f v t function. Contrary to the expectations, the resulting current frequency fc(t) changes in a different way that this of the voltage. One can say that the current frequency variations are considerably amplified compared with these of the supply voltage. This problem is important because we have to deal with voltage frequency changes that are introduced deliberately, like in case of starting ac motors, and voltage frequency variations that result from external influences, like in case of electrical grids with significant component from wind generating units. Paper solves this problem in analytical way by presenting a relation between supply voltage frequency variations $${f}_{v}\left(t\right)$$ f v t and resulting current frequency fc(t). It also contains solution of the inverse problem which can find the frequency function of the supply voltage that will result in a required frequency of the current. According to the best author’s knowledge, this problem was not reported and examined before and potentially has a number of practical implications, one of which – the induction motor frequency starting – is presented as for example purposes in the paper. The significance of the problem discussed in this paper has still to be examined and established, since frequency variations apply to many engineering branches and until now were counteracted by frequency control without a comprehensive identification of the problem.Article Highlights The supply voltage frequency variations have an important effect on the respective current frequency of the loads. One can forecast that a frequency of the current changes so that it follows exactly the frequency change of the supply voltage. However, in reality the situation looks much different. The frequency of the current changes occur much more intensively than the frequency in the supply voltage, that is the frequency changes of the current amplify the variations of the supply voltage. This phenomenon has neither been examined nor described in technological literature yet, according to best author’s knowledge. The paper presents, explains and solves the problem in an analytical way. It presents a formula to calculate the frequency of a current when a function of the frequency of the supply voltage is given. The paper also contains a solution of the inverse of the problem described above. There is presented general form of an ordinary differential equation that serves to find a supply voltage frequency function which results in required frequency of the current in a circuit. Presented solutions are general and hold for any lumped parameters circuits: single phase, multi-phase as well as for ac electrical motors supply. In the paper this is demonstrated for single phase circuit supplied by the three exemplary voltage frequency functions as well as for the case of induction motor frequency starting.