## 1. Introduction

[2] During the recent years, wireless sensor networks [*Viani et al.*, 2011] and near-field communication systems [*Chou et al.*, 2011a, 2011b] have been gradually drawing the attention of researchers. In all these systems, the communication link is either realized by far-field wideband antennas [*Qing et al.*, 2006; *Klemm et al.*, 2008] or near-field communication link. For the near-field communication link realized either by near-field antennas or inductive coils, the wireless power transfer plays an important role. For instance, the wireless power transfer can be applied in portable-telephone battery chargers [*Jang and Jovanovic*, 2003]. Also, they can be realized in a system for monitoring conduit obstruction [*Najafi et al.*, 2007], and sometimes they are in the shape of a wireless capsule for endoscopy [*Chen et al.*, 2009]. Furthermore, wireless power transfer can be utilized in an online electric vehicle [*Ahn and Kim*, 2011]. During the early years, the wireless link was achieved by only one pair of inductively couple coils, intended for both power and data transmission [*Zierhofer and Hochmair*, 1990; *Troyk and Schwan*, 1992]. However, to increase the efficiency during power transmission, coils with larger quality factor (Q) are required, therefore leading to narrower bandwidth. While for the data transmission, larger bandwidth would be an advantage, presenting a contradictory requirement. Due to this reason, there is a trend to implement the wireless link through separate links [*Liu et al.*, 2005; *Ghovanloo and Atluri*, 2007; *Simard et al.*, 2009], which can be optimized independently.

[3] In this paper, we focus on the link intended for power transmission. The wireless power transfer is achieved by two inductively coupled coils transferring energy from one coil to the other. Also, if a rechargeable battery is connected to the secondary coil, this power link acts as a vital part for wireless charging. It is obvious that how to enhance the power efficiency between these two coupled coils is the critical part during the power transmission. Some early design works of inductive links were constructed by circular spiral coils made of filament wires in the form of single or multiple individually insulated strands [*Kendir et al.*, 2005; *Baker and Sarpeshkar*, 2007; *Yang et al.*, 2007; *Ghovanloo and Atluri*, 2007]. This type of filament wire is called Litz wire, which presents a smaller effective series resistance (ESR) and a larger quality factor, therefore enhancing the power transmission efficiency. However, this type of coil cannot be batch-fabricated without sophisticated fabrication technology. Other wireless links were constructed by lithographically defined square spiral coils [*Jow and Ghovanloo*, 2007, 2009; *Laskovski et al.*, 2009]. Some of the coils are based on rigid substrates such as printed circuit board (PCB), and others are based on flexible substrates such as polyimide [*Shah et al.*, 1998] or parylene [*Li et al.*, 2005]. For the design procedure, some systematic design methods for optimizing the coils were proposed [*Jow and Ghovanloo*, 2007; *Silay et al.*, 2008]. However, none of them are suitable for improving the efficiency between rectangular coils. Because for one thing, rectangular or square coils has larger coupling area compared with circular or elliptic coils with the same horizontal and vertical dimensions. For another, during some practical applications, the space left for power coils design presents a certain shape other than spiral and square. In this case, rectangular coils serves as a more general and favorable alternative. Additionally, previous expression of mutual inductance for square coils is based on an experiment-based coefficient adapted from circular coils [*Jow and Ghovanloo*, 2007], which proves inefficient when applied in the case of rectangular coils. In this paper, we propose a new method for calculating the mutual inductance and present a method of how to characterize and optimize rectangular coils used in inductive link, and a transferring frequency of 3 MHz is assumed.

[4] Because the simulation of multiturn coil pairs in HFSS consumes a very large memory and a considerable amount of time even for the work stations, therefore we can first build up some lumped component models for the coils. Subsequently, these models are programmed into Matlab and we utilize these Matlab codes to determine the initial values of the coils' geometrical parameters, which is much quicker than Finite element-Method (FEM) based HFSS simulation. Eventually, we can use HFSS to do the final adjustments to further improve the efficiency. Therefore the advantage of our design method is due to the fact that it can speed up the design process and help us determine the geometrical parameters of the coils intended for power transfer efficiently.

[5] With the coil being modeled as an inductor in series with a resistor, the usually adopted schematic for an inductive link is a serial-parallel type circuit, as shown inFigure 1. The primary circuit is in serial resonance to provide a low-impedance load to the source connected before the primary coil, and the secondary circuit is in parallel resonance at the same frequency to better drive a nonlinear rectifier load [*Hu and Sawan*, 2005].

[6] In Section 2, we propose a simple equation for calculating the efficiency and evaluate the effect of various lumped component on the inductive link. Then in Section 3, we give the equations for the modeling of self inductance, mutual inductance, resistance due to skin effect and parasitic capacitance. Subsequently, in Section 4, a systematic design procedure executed in Matlab (MathWorks, Natik, MA) has been put forward for optimizing rectangular coils, with verification from simulation of HFSS (Ansoft, Pittsburgh, PA) and measurement results. The comparison of results from three approaches is presented in Section 5, followed by conclusion remarks in Section 6. The preliminary results have been presented in *Duan and Guo* [2011].