Corresponding author: Z. Duan, Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117526. (firstname.lastname@example.org)
 Wirelessly coupled coils are crucial for efficient power transmission in various applications. Previous design methods are only eligible for improving the efficiency of circular or square coils. This paper presents a method of characterizing and optimizing rectangular coils used in inductively coupled systems. After setting up a lumped component model for inductive coils, the efficiency can be expressed in terms of geometrical parameters of the coils. Subsequently, the power efficiency can be plotted versus these parameters in Matlab, thus getting the desired coils for optimum power transfer. With this design procedure from mathematical optimization, we eventually designed two rectangular coils spaced 10 mm apart, which achieves a power transmission efficiency of 46.4% at a frequency of 3 MHz. The design methodology is verified by simulation and measurement.
 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.
 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.
 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.
 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].
 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 .
 For the secondary coil, we do a parallel-to-series conversion (fromFigures 2a to 2b), which is called narrowband approximation, and it is shown that the error caused by this approximation is negligible [Silay et al., 2008]. From Lee , we can get
where Q2= ωL2/Rs2 is usually much larger than 1 (During our application, the typical value for Q is around 20∼60).
 From equation (5), after some mathematical manipulations, we can arrive at
 Therefore the reflected resistance from secondary side to primary side is
 The power efficiency is defined as the power received by the load divided by the power provided by the source, and because reactive component will not dissipate power, the efficiency can be expressed as
where k is the coils coupling coefficient defined as
 When ω reaches , η reaches maximum
where Q1′ = ωL1/R1 and Q2′ = R2/ωLp2 ≈ R2/ωL2. Because maximum efficiency is desired for the power transfer, the whole circuit operates at its resonance mode.
2.2. Effect of C1 on the Inductive Link
 It can be seen from (12) that the efficiency at resonance is independent of the primary capacitor C1. However, the voltage transfer ratio ∣VL/Vs∣ is a function of C1. The expression of ∣VL/Vs∣ with respect to C1 is too lengthy to be listed here, but the optimum C1 where maximum ∣VL/Vs∣ is achieved can be found mathematically by a simple expression. At resonance, C1 can be determined by the following equation.
 During the design process, a fixed RL is assumed. However, after the coils according to a particular RL are designed, we can further maximize the efficiency by an adequate changing of RL.
 From (12), the inverse of ηmax can be expressed as a function of RL
where f(RL)min occurs at
 Therefore, if a matching circuit is designed to change the actual load to the value specified in (15), the efficiency will be enhanced. This method has been proved effective in Silay et al. .
2.4. Effect of Rsrc on the Inductive Link
 As the Rsrc increases, the efficiency will reduce. Therefore, this is no optimal value for Rsrc. However, Rsrc will have an influence on determining the geometrical parameters of the coils, for instance, the width of the coil trace. This would be explained in Step 3 of the design procedure elaborated in section 4.
 Previous papers dealing with inductive coils only covered circular coils made of Litz wires [Kendir et al., 2005; Baker and Sarpeshkar, 2007; Yang et al., 2007; Ghovanloo and Atluri, 2007] or square coils [Jow and Ghovanloo, 2007, 2009; Laskovski et al., 2009; Kilinc et al., 2010]. Circular coils made of Litz wire presents a smaller effective series resistance (ESR) and a larger quality factor, therefore boosting the final power transmission efficiency. However, due to the fact that this type of coils cannot be batch-fabricated without sophisticated fabrication technology, planar printed spiral coils are preferred. For this type of coils, considering the same horizontal and vertical dimensions, square coils have a larger coupling area than the circular ones and therefore found their application in many published papers. But still, square coils with same side lengths have their restrictions, limiting their use in the application when the space left for inductive coils design presents a different shape. In this circumstance, rectangular coils serves as a more general and favorable option. For instance, previous expression for the self inductance of square coils is expressed as [Jow and Ghovanloo, 2007; Kilinc et al., 2010]
where n is the number of turns, do and di are the outer and inner diameters of the coil, and davg = (do + di)/2.
 For this self inductance equation to be true, we should assume a uniform dofor two side lengths, which is not the case for rectangular inductive power coils. Consequently, this equation is not applicable to rectangular cases. In addition, previous equations for mutual inductance of square coils adopt an experiment-based coefficient adapted from circular coils, which may cause error and become unusable for the case of rectangular coils. Therefore, for the calculation of self and mutual inductance of rectangular coils, we should use filament method based on the Greenhouse method [Greenhouse, 1974].
3.1. Self Inductance
 Based on the Greenhouse method, the inductance of a rectangular coil can be obtained by summing up the self inductance of each segment, the positive and the negative mutual inductance between all pairs of segments. Also, this method is further used to calculate the inductance taking into account of the substrate eddy currents [Kang et al., 2007]. In this paper, we further extend this method to calculate the mutual inductance between the primary rectangular coil and the secondary one, which is a novel approach and can be programmed into Matlab to determine the power efficiency quickly instead of time-consuming HFSS simulation.
 For an n-turn rectangular coil shown inFigure 3, the length of one segment can be given by
where < > denotes the integer part of the expression in brackets and i denotes the segment number (from outmost to innermost). The total self inductance can be determined by [Greenhouse, 1974]
where Li denotes one segment's self inductance. From Grove , we can get the mutual inductance between two parallel wire segments with lengths li and lj
 Consequently, the positive mutual inductance between segments can be calculated by
where i, j indicate the turn number (from outermost to innermost), k is the segment number in that turn. The distance between them can be given by
 The negative mutual inductance between segments can be calculated by
 The distance between them can be given by
 Therefore, the total inductance of rectangular coil can be obtained by
3.2. Mutual Inductance
 Subsequently, we can get the mutual inductance between primary and secondary coil by summing up all the mutual inductance between all pairs of segment. From Figure 4, we can see that those segments with current of same direction would have a positive mutual inductance, and segments with current of opposite direction would have a negative mutual inductance. For example, lp2, lp6, lp10 and ls2, ls6, ls10 would have positive mutual inductance, lp2, lp6, lp10 and ls4, ls8, ls12 would have negative mutual inductance.
 Now we can get the distance between one segment and center of the rectangular coil, for the primary coil
 For the secondary coil
where subscript 1 or p denotes the parameters of the primary coil, and 2 or s denotes those of the secondary coil.
 From Figure 4, we can see that positive mutual inductance (the mutual inductance between traces with current of same direction) can be calculated by
 Assuming that the coils are perfectly aligned along the center, the distance between segments can be given by
where D is the distance between primary coil and secondary coil along the center axis.
 Negative mutual inductance can be calculated by
 The distance between them can be given by
 During the calculation of mutual inductance between a pair of segments, the widths of these two segments have to be the same (28). However, in actual cases, the widths of segments for the secondary and primary coil are not always the same. Therefore, we made an approximation here by assuming
 Eventually, the total mutual inductance of two rectangular coils would be obtained by
3.3. Serial Resistance
 For the resistance of coils, two factors needs to be taken into consideration. One is the resistance caused by the skin effect, and the other one is the proximity effect or current crowding effect. Resistance caused by skin effect has been included in our model. The proximity effect has been investigated [Kuhn and Ibrahim, 2001; Jow and Ghovanloo, 2009], which is more pronounced when the operating frequency approaches 10 MHz for the coil size of our case. Because our operating frequency is only 3 MHz, we neglect this in our model. The AC resistance caused by skin effect can be expressed as [Jow and Ghovanloo, 2009; Eo and Eisenstadt, 1993]
where t is the thickness and w is the width of the copper, and teff is the effective thickness due to the skin effect. li is defined in (18), and δ is the skin depth of the conductor given by
where Cpc is the capacitance through air and Cps is the capacitance through substrate. εrc and εr are the dielectric constant of the copper and substrate respectively. α and β are empirically determined coefficient for the capacitance proportion. lg is the length of the gap, given by the following equation
3.5. Efficiency calculation
 After the model for the coil has been set up, we can do an equivalent transformation as shown in Figure 5 and substitute Leff and Reff into (12) to calculate the final efficiency, where
 Design constraints are generally imposed by fabrication technology (minimum width) and the space available for the secondary coils. Here we list all the parameters in Table 1.
Table 1. Design Constraints
Maximum dimension for secondary coil
a × b
25 mm × 10 mm
Minimum conductor width
Minimum conductor spacing
Distance between the coils
1 oz ≈ 35 μm
Conductor conductivity (copper)
5.8 × 107 S·m−1
Substrate thickness (FR4)
Substrate dielectric constant
 From HFSS simulation, we found that the substrate has insignificant effect on the value of mutual inductance and other parameters that would cause a difference in the final efficiency. Therefore we used the substrate of commonly available FR4 to support the coils. This may be due to the fact that near-field induction is largely a magnetic coupling, considering most materials are non-magnetic, any supporting substrate won't cause any significant change on the simulation and measurement result.
4.2. Step 2: Initial Values
 We first assume that the two coils are identical. And we initialize the parameters as s1= s2= 0.1 mm, lp1= ls1= 25 mm and lp2= ls2= 10 mm. For the spacing between turns, we just keep it at the minimum value for improvement of efficiency and coupling [Jow and Ghovanloo, 2007; Silay et al., 2008]. Also, we introduce a ratio r to ensure the length of innermost segment is larger than 0, which is defined as
 For a rectangular coil, there are five geometrical parameters: l1, l2, w, s and r. And the parameter n (number of turns) is interrelated with the defined ratio r, as shown in (44). Now we can plot the efficiency versus the left two variables w and r in Matlab to find out the optimum geometrical parameters.
 From the curve shown in Figure 7a and the result of Matlab code, we can see that at w = 150 μm, r = 0.21, and a maximum efficiency of 24.2% is achieved. In order to validate the results, we assume a fixed value 150 μm for w, and vary r from 0.1 to 0.7. We get the comparison curves from three approaches shown in Figure 7b. The calculated, simulated and measured efficiency agree with each other reasonably. And from the results, we can see that as r is further reduced below a certain value (in our case, it's around 0.2–0.3), which means more turns in the center of coils, the efficiency remain almost the same or reduce by a small amount. This agrees with the conclusion in Ghovanloo and Atluri  and Jow and Ghovanloo  that turn very close to the center of the coils does not help in increasing the coupling and the final efficiency.
 For the impact of Rsrc on the power efficiency, we changed the Rsrc from 0 to 10 Ω, and we found that the best w where maximum efficiency is achieved varies from 0.22 mm to 0.12 mm. This means that when the outer dimension is fixed, if there is a larger source resistance, we should decrease the w to increase the overall quality factor. This is because that when a large source resistance is present, a smaller w means both a larger resistance and a larger inductance, but the inductance increasing is faster compared to the resistance increasing, therefore leading to a larger overall quality factor and a large efficiency accordingly.
4.4. Step 4: Optimizing Primary Coil
 As the size limitation for primary coil is less stringent, after the secondary coil is fixed, we can increase the dimension of primary coil to further increase the efficiency. In this step, we assume r1= r2= 0.21 and the ratio of l2/l1 for the primary coil remains at 0.4 as the secondary coil.
 From the curve shown in Figure 8a, we can see that as we increase the outer dimension lp1, the efficiency increases gradually. However, beyond a certain point, in our case lp1= 60 mm, the efficiency remain almost the same, which is around 48%, as shown in Figure 8b.
4.5. Step 5: Optimized Design
 For our design iteration, the further increase in the size of external coil doesn't help a lot in increasing the efficiency, and the optimized parameters for the coupling coils are summarized in Table 2.
Table 2. Geometrical Parameters of Optimized Coils
5. Measured Performance
 The fabricated coupling coils are shown in Figure 9. From the figure, we can see some holes at the corners of the substrate, through which we can use some plastic fixtures to support and separate the coupling coils. The measurement setup is shown in Figure 10. For the measurement process, first, the network analyzer (HP 8753D) was used to measure the S-parameters of the coupling coils. Subsequently, the S-parameters were converted to Z-parameters which contain the information of self-inductance, mutual inductance and serial resistance. Finally, an equivalent circuit ofFigure 2a was set up in ADS to find the power efficiency. The comparison results from three approaches are shown in Table 3.
Table 3. Comparison Results From Three Approaches of Optimized Coils
 From Table 3, we can see that except a slight overestimation of L1 and therefore an overestimated Q1, all the other parameters from three different approaches agree with each other excellently. The quality factor of the secondary coil is only around 25, due to the size limitation, and the quality factor of the primary coil is between 45 and 55. Both of them satisfy the condition for the simplification process in equations (1) and (2) of Section 2.
 The final optimized efficiency given by the measurement result is 46.4%, presenting a slight difference from the Matlab calculation and HFSS simulation. The main reason is that for the self inductance and mutual inductance, the results from three methods agree well with each other, however, for the resistance, simulation and calculation method lead to underestimated result when compared to measurement, leading to a smaller measured efficiency. In all, the difference between different approaches may be caused by the following factors: 1) the inherent inaccuracy of the equation proposed to model the rectangular coil. Some secondary effect, for instance, the current crowding effect has not been included in our model 2) The SMA connector will introduce some resistance which was neglected both in simulation and calculation method 3) the inevitable misalignment between the coils will cause some reduction in efficiency.
 Unlike previous methods which are only suitable for optimizing square or circular coil, we have proposed a method of how to model and optimize the geometrical parameters of rectangular coils for power transmission, which can be used in a wider scope of application. Our major contribution lies on two aspects: a. we provide a new and simple method for calculating the power efficiency. b. we propose a method of solving the practical problem for the optimization of rectangular coils by using the filament method of calculating the self and mutual inductance.
 The design procedure was executed in Matlab, and validated by simulation from HFSS and measurement from network analyzer. The advantage of this design method lies on the fact that with the help of Matlab codes, we can first determine the initial values for geometrical parameters of the coupled coils in a more rapid way, sparing the effort of the time-consuming HFSS simulation. After the initial parameters of the coils have been decided, then we can use HFSS to do some final adjustments to the coils' geometrical parameters to further enhance the efficiency.
 This work is funded by A*STAR (Agency for Science, Technology and Research) SERC (Science and Engineer Research Council), Singapore under grant 1021710161. The authors would like to thank Ruifeng Xue and Minkyu Je from Institute of Microelectronics, A*STAR for their valuable technical discussions.