Calculation of the Magnetic Forces Between Planar Spiral Coils using Concentric Rings Abbas Shiri and Abbas Shoulaie Department of Electrical Engineering Iran University of Science and Technology, Tehran, Iran abbas_shiri@iust.ac.ir, shoulaie@iust.ac.ir Abstract─ In this paper, the forces between current carrying planar spiral coils are calculated. In order to facilitate the calculation process, the coils have been replaced by concentric rings and using first and second order complete elliptic integrals, the forces between them have been calculated. The comparison of the calculations resulting from the replaced rings method and the direct method shows that the former is more effective in both simplicity and calculation time. To evaluate the precision of the calculations, planar spiral coils have been constructed and tested. The experimental results validate the results of the calculations. Index Terms─ Planar spiral coils, magnetic force, vector magnetic potential, concentric rings. I. INTRODUCTION Planar spiral coils are used extensively in different applications such as communications, power electronics, and casting industries [1-3]. In these systems, to have a high inductance and flat configuration, spiral windings are employed. In DC/DC converters, because of flatness and special configuration, planar spiral coils are a better replacement for the ordinary inductances in order to reduce the volume of the converter. To calculate the magnetic force between these coils, some methods have been reported in literature. In [2] these forces are obtained just by test. In [3] the finite difference method is employed to calculate the force between them; furthermore, in this reference to calculate the magnetic force, spiral coils are replaced by concentric rings, but there is no study and discussion on the precision of the method. In [4] the force between circular coaxial coils has been investigated. Recently, the above authors employed mesh-matrix method in order to calculate the force between spiral coils [5]. In this paper, using concentric rings instead of spiral coils, an effective and simple procedure is developed to calculate the magnetic force between these coils. Using the results obtained from the numerical solution of the direct calculation method, the precision of the proposed method is investigated and finally compared with experimental results. II. DIRECT CALCULATION METHOD Consider a system of two spiral coils as shown in Fig. 1. To calculate the magnetic force between them, we should first calculate the vector magnetic potential resulting from one of the coils in any given point like P (see Fig. 2). Vector magnetic potential of spiral coil 1 in any given point P is obtained by the following equation [6]:      1 1 0 4 R l d I A (1) where 1 I is the current of the coil, l d  is the longitudinal differential component, and 1 R is the distance between this differential component and point P. The coordinates marked by prime are related to the source. With suitable substitutions for l d  , the following equation for vector magnetic potential is obtained: . ] sin cos [ ] cos sin [ 4 1 1 0              R r d a a d r a a I A y x y x        (2) 1054-4887 © 2010 ACES 468 ACES JOURNAL, VOL. 25, NO. 5, MAY 2010