IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 1, JANUARY 2014 7000209 Accurate Computation of Mutual Inductance of Two Air Core Square Coils with Lateral and Angular Misalignments in a Flat Planar Surface Ezhil Reena Joy, Ankit Dalal, and Praveen Kumar Department of Electronics and Electrical Engineering, Indian Institute of Technology, Guwahati 781039, Assam, India This paper presents an analytical approach for computing the mutual inductance between two air core square coils placed in a flat planar surface. The mutual inductance of the coils is calculated for all possible variations including lateral and angular misalignments in space. In contrast to conventional approximated formulae, the straightforward approach based on Biot-Savart principle is used and their integrals are computed numerically. The results of computed mutual inductance by analytical method are validated by finite element analysis and an experimental setup. Finally, the analysis compares the three mutual inductance calculations: an analytical method, the finite-element model and an experimental results. The values computed by three methods in all cases are in good agreement. Index Terms—Analytical models, coils, coupled mode analysis, finite element methods, inductance measurement. I. INTRODUCTION T HE computation of mutual inductance (MI) between two coils is a classical problem in electrical engineering that remains important to this day for a wide variety of physical disciplines [1]. Some important applications such as transcuta- neous transformers, coil guns, linear motors, contactless based electric vehicle charging systems etc., are modeled with induc- tively coupled circuits such as transformers and contactless en- ergy transfer (CET) systems [2]–[6]. Recent developments in CET systems have prompted more the requirement to investi- gate the MI of the coil [7]–[9]. The design of such CET coils are very complex as the coils are usually misaligned due to varia- tions in the system and worsen the coupling between the coils [9], [10]. These misalignments could cause fluctuations in the output voltage and affect the stability of the system [4]. There- fore, the computation of MI with all its lateral and angular mis- alignments must be fully addressed, which is the first step for studying the characteristics of such systems. A survey of past literature shows that, Grover’s tabular data remains the most standard for calculating the MI for a wide variety of coils and wire forms [11]–[15]. However, its use has mostly been restricted to zeroth and first-order calculations and it is proved to be inaccurate for loosely coupled and short coils [12]–[15]. Several contributions for MI computation are found in the literature [11], [13]–[22]. In some work, MI is calculated by means of approximated formulas [12], [16], [17], Heuman’s lambda function [1], [18]–[20], Bessel and Struve functions [21], [22] and in other works using Biot-Savart law [23], [24]. However, much of the earlier works are devoted for circular and coaxial coils [1], [12]–[22]. These coil geometries are well suited for fixed coil systems and they are not tolerant of misalignments in the coils. Square and rectangular coil geometries are found to be well suited for mid-distance CET Manuscript received February 23, 2013; revised May 04, 2013; accepted Au- gust 08, 2013. Date of publication August 22, 2013; date of current version De- cember 23, 2013. Corresponding author: A. Dalal (e-mail: d.ankit@iitg.ernet. in). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2013.2279130 Fig. 1. Block diagram of contactless system. systems, as they show better tolerance for misalignments [3], [25]. Although, there are numerous works carried out in this area, only very few works have studied these coil geometries with misalignment effects [3], [5], [7]–[10], [25]–[28]. In [9] and [10], an analysis is presented based on rectangular coil geometry. However, these works mainly focused on the design of charger based on resonant magnetic coupling to transfer power wirelessly over a long distance. It has been observed from the literature, the effects of misalignments of the coils have not been investigated in detail from both geometrical and circuit design standpoints. This paper describes an analytical approach to compute the MI between two coils. A detailed investigation of all possible lateral and angular misalignments with horizontal and vertical variations is presented. Square coil geometry has been chosen here to analyze the misalignments of the coils. The proposed an- alytical approach is capable of calculating MI for all positions of the coil, thus reducing complex mathematical equations. The results of the analytical model are compared with 3-D finite ele- ment analysis (FEA) and an experimental setup. Fig. 1 shows the main building blocks used for MI computation. The coil which is excited is referred as excitation coil (EC) and the coil where the output variations are observed is referred as obser- vation coil (OC). The finite element results and experimental evaluation justifies the accuracy of the analytical model in all cases. II. POSSIBLE VARIATIONS OF SQUARE COILS The analysis presented in this paper computes MI between two air core square coils, placed in a flat planar surface coin- ciding in space. As the MI of the coil varies with the change in position of the coils, different variations of the coils, i.e., mis- alignments are analyzed throughout this paper. Different cases of variations of OC with respect to EC are taken into account, which are shown in Fig. 2 and its corresponding schematics are 0018-9464 © 2013 IEEE