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• Analytic approximation by multiple Taylor series expansions. • Unlimited spatial area with high accuracy is achieved. • A fast evaluation of the magnetic field for every point in space can be done. • Accuracy depends on the distance of the evaluation point to the current loop. In the wide field of contactless energy transfer over a large distance, e.g. for charging electric vehicles or for supplying energy to implanted medical devices in the human body, different types of coils are used. For being able to determine the properties of the path of energy transfer, the spatial distribution of the generated magnetic field is very important. An analytic expression has several advantages for determining the spatial distribution of the magnetic field in an efficient way. In this paper, an analytic approximation for the vector potential and the magnetic flux density of a filamentary circular current loop is proposed. The derivation of the fundamental equations and the concept of the approximation is presented and the range of parameters, in which an analytic solution with sufficient precision is obtained, is determined and evaluated dependent on the parameters of the design and the approximation. The results of the analytic approximation for the magnetic flux density are presented based on an exemplary filamentary circular current loop. Finally, the results are verified by a FEM simulation. The analytic approximation shows a sufficient accuracy at large distances as well as in the close vicinity of the current loop and turns out to be a fast and efficient option to evaluate the magnetic field of one or more circular coils. [ABSTRACT FROM AUTHOR]