Wireless Power Transfer in Loosely Coupled Links: Coil ... - IEEE Xplore

416

IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 2, FEBRUARY 2011

Wireless Power Transfer in Loosely Coupled Links: Coil Misalignment Model Kyriaki Fotopoulou1 and Brian W. Flynn2 NXP Semiconductors, Central Research & Development, 3001 Leuven, Belgium Institute for Integrated Micro and Nano Systems, The University of Edinburgh, EH9 3JL Edinburgh, U.K. A novel analytical model of inductively coupled wireless power transfer is presented. For the first time, the effects of coil misalignment and geometry are addressed in a single mathematical expression. In the applications envisaged, such as radio frequency identification (RFID) and biomedical implants, the receiving coil is normally significantly smaller than the transmitting coil. Formulas are derived for the magnetic field at the receiving coil when it is laterally and angularly misaligned from the transmitting coil. Incorporating this magnetic field solution with an equivalent circuit for the inductive link allows us to introduce a power transfer formula that combines coil characteristics and misalignment factors. The coil geometries considered are spiral and short solenoid structures which are currently popular in the RFID and biomedical domains. The novel analytical power transfer efficiency expressions introduced in this study allow the optimization of coil geometry for maximum power transfer and misalignment tolerance. The experimental results show close correlation with the theoretical predictions. This analytic technique can be widely applied to inductive wireless power transfer links without the limitations imposed by numerical methods. Index Terms—Inductive link, misalignment analysis, RF coils, RFID, wireless power transfer.

I. INTRODUCTION

L

OW-POWER inductive links are used extensively for wireless powering of passive near-field radio frequency identification (RFID) systems, [1]. Other applications include embedded biomedical devices for drug delivery, neural prosthesis, and cochlear implants [2]–[4]. In inductive or magnetic coupled systems, energy is transfered from a primary transmitter (TX) coil to a secondary receiver (RX) coil with the aid of an alternating magnetic field as is illustrated in Fig. 1. Although improved designs have emerged in the industry, the impact of coil misalignment on the link efficiency and the optimization of the coil design has received less attention by researchers. The inductive links considered in this study are loosely coupled and are characterized by extremely low coupling coeffi[5]. This happens when the distance of sepacients ration between the TX and RX coils is much larger than the dimensions of the RX coil as is the case in most practical systems [6]. A comprehensive review of embedded electronic identification, monitoring and stimulation systems has been carried out by Troyk [7]. Referring to this study, the coils are typically separated by a layer of skin and tissues in the region of 2–6 cm. Also, a typical implanted microcoil has a diameter of less than 3 mm whereas an external transmitter coil has a diameter of at least 9 cm. Ideally, the TX and RX coils would be coaxially orientated so that maximum coupling results. However in practice, in RFID and biomedical applications, misalignment of the coils can easily occur due to anatomical conditions such as skin mobility and variations in the thickness of subcutaneous

Manuscript received January 14, 2010; revised October 29, 2010; accepted November 02, 2010. Date of publication November 18, 2010; date of current version January 26, 2011. Corresponding author: K. Fotopoulou (e-mail: k. [email protected]). 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.2010.2093534

Fig. 1. A low-power inductive link for implanted sensor systems, where d is the coil separation distance, 1 represents lateral misalignment, and # is the angular misalignment angle.

fatty tissue. In passive near-field RFID systems the reader (TX) and tag (RX) coils are separated by a distance , typically in the range of a few centimeters. In this case coil misalignment is the norm and can be easily demonstrated in spatially selective antennas for very close proximity 13.56–27 MHz ISM1 RFID applications. Classic examples of this type of RFID system include contact-less smart cards for access control, e-ticketing, and label item tracking. Here the mutual reader-tag alignment can vary drastically and for issues like anti-collision, safety, and reliability it is critical to be able to predict the misalignment tolerance of such systems and specify the geometric boundaries of operation. There have been several approaches to the analysis and design of inductively coupled transcutaneous links for optimal efficiency. Previous work mainly concentrated on steady-state circuit analysis and validation through experiment. Transcutaneous links have been analyzed by Donaldson et al. [8], Galbraith et al. [9], Heetderks et al. [10], and Ko et al. [11]. Although more limited, finite-element analysis and electromag1Industrial

Scientific Medical.

0018-9464/$26.00 © 2010 IEEE

FOTOPOULOU AND FLYNN: WIRELESS POWER TRANSFER IN LOOSELY COUPLED LINKS

netic simulators have also been used for transcutaneous link modeling [12]. While they can provide accurate results, they often involve cumbersome and computationally demanding calculations that mask conceptual understanding. To the best of the authors knowledge, the practical issues of coil misalignment and orientation and their effect on the transmission characteristics of RF links have largely been overlooked by researchers. Papers presented by Flack et al. [13] and Hochmair [14] consider only the effects of lateral displacement of coils on the mutual inductance. The study by Flack et al. is based on experimental data whereas Hochmair et al. resort to numerical integration for the mutual inductance. Neither of these works take into account the geometric characteristics of the coils and their shape, focusing only on parallel circular loops of zero thickness, essentially flat filaments, making these methods unsuitable for optimization of coil size and shape. The definitive work on the coupling of air-core coils was done by Grover and Terman, where it was demonstrated that the cou[15], pling coefficient is related to the mutual inductance [16]. The greater part of this work concentrated on the mutual induction calculation. Any theoretical investigation of the mutual inductance in arbitrary coil configurations is extremely complex due to the lack of symmetry and the tedious work required to solve the double integral in Neumann’s formula [17]. The most complete study of misalignment effects to date was carried out by Soma et al. [18]. In this paper, upper and lower bounds of the mutual inductance are provided between two idealized circular rings and shape correction factors introduced by Grover and Terman are used to account for solenoidal coils. This is an uneasy alliance between analytical results and empirical data and due to its semi-analytic nature the estimated bounds for the mutual inductance tend to be conservative. A straightforward application of Neumann’s formula for more complex coil geometries under misalignment is mathematically intractable which ultimately limits Soma’s approach. More recently, Babic et al. attempted to solve the mutual inductance of inclined circular coils but also resorted to cumbersome numerical integration, [19], [20]. Clearly, the majority of the available solutions are semi-analytical and mathematically complex. The aim of this work is to derive a closed form analytical solution for the calculation of the power transfer efficiency under different coil orientations and characteristics avoiding the involved mathematical treatment of the mutual inductance. In Fig. 2, the mutual inductance of two coaxial solenoid coils is plotted using the simplified Neumann’s expression treated in numerous text books [17]. It is evident from Fig. 2 that even at an ideal coaxial coil orientation when maximum coupling is naturally expected, for the coil dimensions considered in this study, the mutual inductance is still very low. This observation allows a purely electromagnetic approach to the problem. This coupled with a simple equivalent circuit for a loosely coupled resonant link yields a novel analytical approach, which offers understanding of the tradeoffs between coil position, shape, and size. An alternative approach for practical coils under misalignment, which avoids the complications introduced by the mutual inductance calculation is presented henceforth. The model introduced in this paper provides a tool for understanding the effects of coil misalignment and geometry on the

417

Fig. 2. Mutual inductance versus coil radius and number of turns for two coupled short solenoid coils.

link efficiency. Hence, designers can now optimize practical systems for maximum efficiency without compromising on misalignment tolerance. II. INDUCTIVE POWER TRANSFER MODEL The operation of the inductive link resembles a loosely coupled transformer, represented by the equivalent circuit of Fig. 3, [21]. The circuital representation of Fig. 3 is appropriate as the interaction between the transmitter and receiver coils is entirely magnetoquasistatic. This is supported by the fact that for the applications considered in this study, the receiving coil is situated well within the reactive near-field of the transmitter coil since and wavelength of the resonant the coil separation distance obey the inequality . In this process, frequency energy is stored in the near-field of the TX coil and is not radiated in space. It follows that any radiation losses denoted are negligible and the total losses in the coil can be attributed to the ohmic resistance of the conductor represented as in Fig. 3. A coil can be considered to be electrically small if the total conductor length is less than a tenth of the wavelength of the operating frequency [22], [23]. Therefore, most practical coils used in the applications considered are electrically small. Hence, a uniform current distribution in the conductor can be assumed which in return allows for the transmitter and receiver coils to be treated as lumped elements [24]. Thus, the inducand , tance of the trasnitter and RX coils is denoted as respectively. It is also important to define the physical meaning of the power transfer efficiency studied in this work. The power transfer efficiency of an inductive link denoted, , is the ratio of the real power dissipated in the load of the RFID tag or , to the power provided by the source embedded sensor, driving the TX coil,

(1) , , , and define the voltage and current amplitudes at the resonant TX and RX tuned circuits, respectively. The phase differences between the voltage and current signals in the primary and secondary tuned circuits are given by

418


Following from (5) and (6), the received power is expressed in terms of the induced voltage across the RX coil: (7) The resulting magnetic field vector at the receiver coil can be obtained by integrating Biot-Savart’s law around the transmitter loop [17], [25]: (8)

Fig. 3. Equivalent circuital model of an inductive link assuming poor coupling. The mutual coupling is very small and as a result the presence of the RX coil has a negligible effect on the TX circuit.

and . It should be noted that the phase difference does not affect the power transfer function and therefore only the magnitude of (1) is of interest. The maximum power received by the RX coil is the power delivered to the resistive component of the load when conjugately matched. The resistive losses of the RX coil are then equal to the resistive load and the reactive components are canceled by resonance. Unintended magnetic coupling with nearby objects is minimized by the high selectivity of the resonant tuned TX and RX circuits. As demonstrated by Galbraith in [9], there are four combinations of series and parallel tuned TX and RX circuits, which can be equally well represented by the equivalent circuit of Fig. 3. is It follows that on the transmitter side the impedance assumed to resonate with the transmitter coil inductance as shown in Fig. 3: (2)

It is evident that the solution of (8) depends on the shape of the TX coil and the location as well as size and shape of the RX coil. The induced voltage at the terminals of a close conducting circuit, such as the receiver loop, is expressed by Faraday’s law as the rate of change of flux through the effective surface area : (9) An alternative equivalent representation of (9) is adopted here [26]: (10) where is the effective area of the RX coil, is the peris the magnetic field intensity meability of free space, and generated by the TX source and computed by (8). A uniform magnetic flux cutting through the effective area of the RX coil is necessary in order for Faraday’s law (10) to be applicable. This condition is satisfied by the poor coupling criterion. Combining (8), (10) and substituting in (3) and (7), for a pair of magnetically coupled short solenoid coils, yields (11)

where the real The TX coil is exited by a sinusoidal current input power under these conditions is given by (3) On the receiver side, the load impedance denoted in Fig. 3 should be conjugately matched to the impedance of the RX coil to achieve maximum power transfer (4) Under resonance, the reactive parts in the load and RX coil impedances cancel out. In this case, the available real power that is transferred across the inductive link is defined as

(5) Consequently, referring to Fig. 3 the RX circuit is transformed to a potential divider: (6)

The effect of (8) is now concealed in the power transfer expression (11). Nevertheless, this will become apparent as the form of the power transfer expression in (11) will adapt to account for other coil geometries as discussed in Section III that follows. The methodology for the power transfer efficiency presented in this section was first introduced by Yates et al., [21]. Equation (11) is only valid for coaxial solenoid coils. Clearly, this approach is of very limited use for the applications considered in this work. Even though more complex geometries such as spiral coils are widely employed in RFID and biomedical applications there has been little research on the influence of geometry on the power transfer efficiency. A study by Shah et al. derived some general guidelines based on experimental work but did not consider misalignment effects [27]. An extended analytical model that accounts both for lateral and angular coil misalignment as well as spiral coils is presented in Section III. III. MISALIGNMENT AND COIL GEOMETRY ANALYSIS Clearly, coil orientation is a key parameter in the design of inductively coupled systems. The rate of change of power coupled


419

Fig. 4. Lateral misalignment configuration of the TX, RX coils: (a) circular geometry representing the solenoid and the circular spiral coils, (b) square geometry representing the square spiral coils.

across the inductive link when the RX coil is displaced from the ideal coaxial orientation is evaluated by the study of the two following forms of misalignment: 1) Lateral Misalignment—In lateral misalignment a pair of TX and RX coupled coils are situated in parallel planes, separated by a distance and their centers are displaced by a distance , as shown in Fig. 4. 2) Angular Misalignment—In angular misalignment the plane of the RX coil is tilted to form an angle and the axis of one coil passes through the center of the other coil, as shown in Fig. 5 A general misalignment case which incorporates both lateral displacement and angular tilt of the coils is not considered in this paper. This approach is supported by the fact that there is no strong interaction between the two misalignment effects as demonstrated by Soma in [18]. At small lateral misalignments the angular effect dominates and at large lateral misalignments the lateral effect prevails. Hence, the two displacement configurations can be studied independently, which is advantageous from an optimization point of view. In this manner, separate limits can be set for the maximum permissible angular and lateral displacements for different applications. A. Notation, Coil Geometries and Coil Modeling Table I lists the coil and configuration parameters used in the model presented in this paper. The choice of a specific coil geometry for RFID and embedded devices depends on the intended application. The two most critical factors are frequency of operation and size for implanted devices. Solenoid coils are common for inductively coupled low-frequency (LF) passive devices operating in the range 20–135 kHz. In commercial RFID transponder systems glass-encapsulated ferrite cored solenoids are common as the ferromagnetic material increases the effective area of the RX coil. At high frequencies (HF) such as 6.7, 13.56, 27.125, and 40.688 MHz, planar spirals are the

norm, [28]. Typically, square or circular coils with 5 to 10 turns over a credit card size form factor are used. These coils are relatively inexpensive and can be used to provide a range in the order of tens of centimeters. HF planar coils are fabricated from copper or aluminum. The coils used in short-range low-power inductively coupled systems fall into two main categories: 1) Short Solenoids: cylindrical coils that have a diameter appreciably larger than length. Each turn has the same radius. Short solenoids can have air or ferromagnetic cores. 2) Spiral Inductors: the longitudinal thickness of these coils is small compared to the radial thickness and coil radius. They are essentially flat spirals. Spiral inductors can take many forms such as square, rectangular, circular or polygonal. Circular and square spirals made from enameled copper or fabricated on PCB and plastic substrates are the focus of this paper. In many applications spiral coils are often favored over solenoid coils, as they are compact, robust and lightweight. Printed spiral coils also offer more flexibility for optimizing their geometry and aspect ratio, a fact that makes them particularly attractive for implantation. The electromagnetic field generated by a current system of any complexity and shape can be evaluated by means of superimposing the field contributions generated by elementary structures. Such structures are infinitesimal current conductors or rings in which a constant current distribution is assumed [29], [30]. In view of this, each turn of a closely wound N-turn solenoid coil constructed of circular wire, can be approximated by a collection of vertically stacked equal concentric circular loops which span the length of the solenoid. Such an approximation is valid since, in practice, the diameter of the conductor is small in comparison with the solenoid radius and the coils are electrically small. For most spiral structures, the spacing between adjacent turns and the trace width are small compared to the inductor size. In

420


Fig. 5. Angular misalignment configuration of the TX, RX Coils: (a) circular geometry representing the solenoid and the circular spiral coils, (b) square geometry representing the square spiral coils.

the special case of a planar spiral coil of circular conductor, the diameter of the wire is considerably smaller than the dimensions of the spiral and the wavelength at the operating frequency. Assuming that the outer diameter of the spiral is not significantly larger than the inner one the approximation of having closed turns has a negligible effect on the results. It follows that for closely spaced turns it is possible to replace the -turn spiral with concentric circles in order to evaluate analytically the integral expressing the magnetic field of the spiral, [31]. At the receiving end of the inductive link, the total induced emf across the RX spiral can be expressed using Faraday’s law, as the sum of EMF induced in each concentric loop. Hence, the RX coil is being approximated as a number of concentric loops with , in a similar manner to the different radii approach adopted for modeling the TX coil. A coil of a square geometry is represented in Fig. 4(b) and Fig. 5(b). Again, as for the circular loop considered previously, the amplitude and phase of the current along the loop conductor is assumed to be constant as the coils are electrically small. It is known that the far-field components in the plane of an electrically small square loop coincide with the field components of a circular loop antenna of the same area, [23], [32]. However, the behavior of a square loop in the near-field varies significantly from that of the circular loop. For electrically small square loops, which are finite in comparison to the distance to the observation point, the inductive near-field will be influenced by the loop shape. The near-field of a circular loop is studied in a limited number of papers. Research focusing on the near-field of a square loop is even more limited, with the exception of a paper written by Levin [33]. Based on the closed turns approximation the square spiral can be represented by a collection of concentric square loops where each loop is partitioned into four

and linear dipole elements, with lengths for the TX and RX coils respectively. B. Power Transfer Expressions The mathematical expressions governing the power transfer from the transmitter to the receiver vary depending upon the coil shape and orientation. Each individual coil geometry will be discussed in the following sections. The model discussed in Section II is extended here to account for coupled solenoids, square and circular spirals for the receiver coil considered both in the coaxial and misaligned configurations. The power transfer functions introduced in this section provide insight to the design of inductively coupled embedded sensor systems with respect to a number of parameters summarized in Table I. 1) Power Transfer for Coils in Perfect Alignment: Starting with the ideal coaxial orientation, the magnetic coupling between the TX and RX coils discussed in the following sections comprises of two parts. Part one is focused on the evaluation of the magnetic field generated by the the TX coil for solenoid, circular, and spiral coil geometries using Biot-Savart Law as given in (8). Part two computes the induced voltage across the RX coil due to the presence of the alternating magnetic flux generated by the resonant TX coil. The induced voltage across the RX coil is computed using Faraday’s law as given by (10), for solenoid, circular and spiral coil geometries. magnetic field In the coaxial configuration the dominant component at the center of the RX coil is computed using (8) and evaluated in Appendix A and Appendix B, for circular and square geometries, respectively. The resultant magnetic field at the center of the RX spiral coil is computed as discussed in for Section III-A. Therefore, the magnetic field component


421

TABLE I COIL AND CONFIGURATION PARAMETERS

a circular spiral transmitter coil of lowing expression:

turns is given by the fol-

(12) At the receiving end of the inductive link, the total induced EMF across the receiver spiral can be expressed as the sum of the individual contributions of each concentric loop given by (10):

coils given by expression: (15) in (15) is the effective area of the square spiral RX coil derived as the summation of the area of each consecutive turn of the spiral and estimated as [34]: (16)

(13) Consequently, the power transfer function describing the coupling between circular spiral coils is given by the following expression:

(14) The square spiral coil structure is now considered, where the is evaluated by (72). Hence, the magnetic field component power transfer function for a set of loosely coupled square spiral

where is the spacing between each successive turn, is the internal diameter of the spiral and is the number of turns. 2) Power Transfer Under Lateral Misalignment: This section presents the power transfer functions, for each coil geometry, in the lateral misalignment configuration depicted in Fig. 4(a). In the lateral misalignment case, we can ignore the and components of the magnetic field vector since they are parallel to the RX plane and do not contribute to the flux lines cutting through the RX coil. To derive the power transfer under lateral misalignment, the same methodology is used as was employed for coaxially aligned coils. Three expressions are derived for each coil geometry using the method discussed in Section II. The magnetic field component strength at

422


the center of the RX coil varies for each geometry and it also depends on the position of the RX with respect to the TX coil. is computed by (31) Starting with the short-solenoid coils in Appendix A. Using the configuration parameters of Fig. 4 the efficiency of two laterally displaced coupled solenoids is computed by equation

(17) For circular spiral coils, laterally misaligned, the same principle applies. The component of the magnetic field strength denoted is computed by (32) in Appendix B and the power transfer function for this geometry is now given by

(18)

in (18) represents the modulus of the elliptic In this case, integral and is given by (33). Finally, referring to Fig. 4(b) for the square spiral coil geometry, the magnetic field at the RX coil is computed by (72). In this expression, the and factors represent the lateral displacement of the RX from the center of the TX coil across the and -axis, respectively. Therefore, in the same manner the power transfer across an inductive link comprising of two laterally misaligned square spiral coils is derived by (19):

(19) 3) Power Transfer for Tilted Coils: This section presents the power transfer functions for coupled solenoid and spiral coils when the RX is tilted as illustrated in Fig. 5. In this configuracomponent of the magnetic field vector dominates tion the at the center of the RX coil. It is evident from Fig. 5, that the magnetic field component computed using expression (8) should be modified to account for the angular tilt of the RX coil. As the RX coil is tilted by an angle , the new component of the magnetic field strength vertical to the plane of the tilted coil can be determined by the dot product of the unit vector vertical to the plane of the rotated coil and the magnetic field vector at the center of the RX prior to its rotation: (20) As the RX coil is rotated with respect to the y-axis in Fig. 5, the unit vector is defined as . It should be noted that in the angular misalignment configuration of Fig. 5 only a tilt angle is considered. Hence, the and factors in (72) are equated to zero, and the remaining expression is modified according to (20) above. Thus, the power transfer function for the solenoid coils in the angular misalignment case is (21) In a similar manner, for the circular spiral structures the efficiency of the link is given by the expression (22) Finally, for the square structure the efficiency of the inductive link comprised of two coupled square spiral coils is evaluated by equation

(23) It follows from the power transfer expressions presented in this section that the efficiency of the link is directly proportional to the square of the resonant frequency. Nevertheless, an optimization with respect to frequency is not considered critical in this


423

TABLE II PROTOTYPE SHORT SOLENOID COIL CHARACTERISTICS

for each set of prototype coils was calculated using the following formula: (25) The rectified DC voltage induced at the RX coil denoted as was measured by a Fluke 77 digital multimeter. For each RX coil the load resistance is calculated by using (25), hence, the received power at the RX side can be easily derived as

Fig. 6. Schematic diagram of the experimental setup.

(26) work since RFID systems can only operate at predefined specified ISM frequency bands as discussed in Section III-A. IV. EXPERIMENTAL VERIFICATION In this section, we discuss the experimental verification of the analytical expressions derived in Section III-B. A. Experimental Setup The power transfer model developed in this work was tested with the aid of the experimental setup depicted Fig. 6. On the transmitter side parallel resonance was employed to maximize the current and hence the magnetic field strength generated by the TX coil source. The main drawback of this method is that the impedance of the parallel tuned TX circuit goes to infinity at resonance complicating the matching between the power amplifier used and the coil. To overcome this, the TX coil was tapped at a low impedance point and power was fed in at this point. A further refinement was the use of a variable matching network to match the 50 output impedance of the signal generator to the coil minimizing reflected power. On the receiving side of the loosely coupled transformer shown in Fig. 6, the coil was matched to the equivalent resistance presented by the rectifier and the load impedance. Despite the nonlinear nature of the Schottky rectifier used in the experimental setup, it can be across the RX tank treated as an effective resistive load circuit as discussed in [35]–[37]:

The experimental results are presented in Fig. 8–Fig. 11. Overall, the analytical model solution shows very good correlation with the measured results. B. Prototype Coils For the experimental verification of the analytical model discussed in the previous section, power transfer measurements were made using prototype solenoid, flat and printed circular and square spiral coils. Tables II and III list the geometric and electrical characteristics of these coils. The simplest coil structure of a short solenoid coil with cylidrical conductor is investigated first. The design consists of 8 turns of 19 SWG copper enameled wire. The decision to use 19 SWG wire is well justified since a conductor with larger cross-sectional area is required to minimize the internal resistance of the coil and at the same time provide a flexible enough conductor for hand winding of a solenoid coil. In addition, the number of turns was chosen to provide enough inductance when the length of the coil remains , to conform to significantly smaller than its radius , a short-coil structure. In the closely wound approach, adjacent turns are touching to ensure that the ratio of the spacing between . The diadjacent conductors and the wire radius is ameters of the prototype coils where selected to comply with the loosely coupling approximation and provide the required inductance and internal impedance for matching and resonant frequency selection. For the coils employed in this work operation was confined to frequencies below the self-resonant frequency.

(24) The equivalent resistance seen by the RX coil, denoted as is the shunt combination of the rectifier and the load impedance. The series resistance of the RX coil is transformed to a parwhere where the quality allel one denoted must be greater than 10 [38]. Once the factor is known the rectifier and the load impedance can value of be matched to the impedance of the resonant RX coil. Hence,

In addition, printed spirals of strip conductor were fabricated using PCB techniques on single sided copper clad epoxy glass . For alu(FR4) boards with a conductor thickness of 35 minium and copper coils, a track thickness of m even is required to achieve a sufficient quality factor for a small track width , [39]. In fact the quality factor of the coil deteriorates as the track width reduces as is experimentally demonstrated by Shah et al. in [27]. Furthermore, every effort

424


TABLE III PROTOTYPE FLAT AND PRINTED SPIRAL COIL CHARACTERISTICS

was made to concentrate the turns of the coils on the outline of the spiral essentially creating coils with small fill ratios in order to maximize the effective area of the RX coil and generating maximum magnetic flux though the TX coil. Clearly, there is a tradeoff between maximizing the quality factor of the coils and augmenting the amount of magnetic flux that passes through the enclosed collective area of the turns in the spiral. However, according to Mohan et al. in [40], a smaller spacing improves the inter-winding magnetic coupling which in return increases the inductance of the spiral and reduces the area consumed by the spiral. In essence, the only reason supporting a large spacing originates from the need to reduce the inter-winding capacitance. However, Yue et al. in [41] demonstrates that this is not a major concern as the inter-winding capacitance is shadowed by the underpass capacitance. The coils inter-turn capacitance is dependent upon the manufacturing technology of the coils and for printed coils it is estimated in the region of 2–4 pF. Subsequently, a track width of 2 mm and 5 mm with an equal track spacing are selected for the TX and RX coils, respectively. Therefore, the simple structure for a spiral inductor coil appears to be deceptive since the design of such a coil for efficient power transfer is a rather complex task. Coil geometry is an important factor in the design of the spiral configuration as it has a direct effect in the electrical characteristics. In addition to the printed spiral coils, conventional wire-wound flat spirals were constructed using 19 SWG enameled copper cylidrical conductor. The turns of the prototype coils are close wound and bonded by embedding the coils in a layer of epoxy. The coil dimensions were chosen to satisfy the weakly coupled assumption of the theoretical model presented in Section III-B. A contour plot of the coupling coefficient between two coaxial circuilar loops is shown in (27), as is compluted in MATLAB based on the following empirical expression given in [42]:

Fig. 7. Countour plot of the coupling coefficient between two coaxial short solenoid coils. The radius of the TX coil is 0.06 m and the radius of the RX coil varies from 0.005 m to 0.04 m.

is possible to set an empirical ratio between the dimensions of the TX and RX coils for which a loosely coupled approximation is valid. Therefore, it can be shown from Fig. 7 that for a TX to , the minimum coil separaRX coil radius ratio of 3 tion distance necessary for a loosely coupled link with should be equal to the TX coil diameter. Following this, the dimensions of the coils have been selected to comply with the loosely coupled criteria but still close to the loosely coupled approximation boundary to evaluate the accuracy of the model. For all the prototype coil pairs constructed, the TX coil was twice the size of the RX. The size of the coils was decided first. Once the geometrical parameters were set, then the impedance of the coils was calculated so that they could be reasonably matched to the transmitting and receiving electronics of the experimental setup in Fig. 6. C. Electrical Properties of Coils

(27) While this expression does not represent the coupling factor for the misaligned cases and is not as accurate as finite-element modeling, this still provides a decent idea of the magnitude of the coupling coefficient. Since the coupling factor will decrease even more for the misaligned scenarios, (27) is a good first approximation for estimating the maximum RX coil radius that will still obey the loosely coupled principle. Based on Fig. 7, it

Several closed form expressions exist for the self inductance of solenoid, printed, and spiral coils. However, compact modeling expressions that account for the resistive losses of the coil geometries studied in this paper, including the effects of skin and proximity losses are more limited. In order to compute the self inductance of the prototype printed spirals coils discussed in the previous section, the analytical solutions given by Mohan et al. were employed [40]. The series resistance of the printed spiral coils, both square and circular were calculated using the


425

Fig. 8. Power transfer efficiency for coupled solenoid coils: (a) Ideal case, (b) lateral misalignment, and (c) angular misalignment. Dotted line represents the experimental data and the solid line is the analytical model solution.

Fig. 9. Power transfer efficiency for coupled printed circular spiral coils: (a) Ideal case, (b) lateral misalignment, and (c) angular misalignment. Dotted line represents the experimental data and the solid line is the analytical model solution.

Fig. 10. Power transfer efficiency for coupled planar circular spiral coils: (a) Ideal case, (b) lateral misalignment, and (c) angular misalignment. Dotted line represents the experimental data and the solid line is the analytical model solution.

expression developed by Yue et al. in [41], [43]. The calculated electrical parameters where used in the design phase to estimate the inductance and resistance of coils for the frequencies of interest and coil dimensions selected. Once constructed, all the prototype coils where characterized using the HP 8753C Network Analyzer and HP 85046A S-Parameter Test Set. The measured values for the inductance and resistance were used for calculating the capacitance required to resonate the coils at the specified frequency. In addition, measured values were also used for matching the RX to the rectifier and dummy load in power transfer measurements. Tables II and III list both the calculated and measured electrical characteristics as well as dimensions for all the prototype coils used in the experimental verification of the analytical model. V. CONCLUSION Coil misalignment is an inherent problem of inductive coupled links that significantly impairs the wireless power transfer

efficiency. This paper introduced a novel analytical derivation for the near-field power transfer efficiency of loosely coupled inductive links under lateral and angular coil misalignment for three different coil geometries. The analytical power transfer formulations proposed show good agreement with the experimental data. The advantage of this new approach lies in the fact that the trend of the link efficiency as a result of variation of a specific parameter in the model can be quickly identified. This model can predict the response of the system under misalignment conditions, different coil geometries, and operating frequency without the need of repeated and costly simulation runs. APPENDIX A MAGNETIC FIELD OF CIRCULAR GEOMETRY The magnetic field intensity at the center of a receiver coil of circular shape, laterally misaligned by a distance , can be evaluated using Biot-Savart’s law as expressed in (8). Referring to Fig. 4(a), the vectors and can be derived. Consequently,

426


Fig. 11. Power transfer efficiency for coupled printed square spiral coils: (a) Ideal case, (b) lateral misalignment, and (c) angular misalignment. Dotted line represents the experimental data and the solid line is the analytical model solution.

the field components can be defined using (8). For the geometry component can be calculated from depicted in Fig. 4(a), the (28) A solution for an integral of this form, which can be recognized as a standard elliptic integral, is described by Good [44] and also in the extensive collection of elliptic integrals and special functions compiled by Byrd and Friedman in [45]. By equating to zero yield the field in the coaxial coil configuration. Hence, the z-component of the magnetic field intensity at the center of the RX coil can be expressed in terms of the lateral misalignment distance as

where is the lateral misalignment on the y-axis and modulus of the elliptic integrals expressed as

APPENDIX B MAGNETIC FIELD OF SQUARE GEOMETRY The square loop can be treated as four short linear dipoles. In this way, we can compute the magnetic field generated by each dipole and add the resulting expressions. Applying LBS yields for the dipole (34) Starting with the vector definition in the integrand of the LBS, as given in (34) (35)

(29)

(36)

is the

(37)

(30) Therefore, the z-component of the magnetic field generated by short solenoids and circular spiral TX coils can be evaluated by superimposing the contributions of the individual loops that constitute these more complex TX geometries. Consequently, the z-component of the radial magnetic field produced by a short solenoid TX coil of N-turns is given by (31) where the modulus of the elliptic integrals can be expressed by (30). Finally, the component for the magnetic field generated by , is expressed as a circular spiral TX coil,

(32) In this case, the modulus of the elliptic integrals and varies for each concentric loop. Hence, it is more appropriate to define as follows:

The cross product of vectors nant:

and is given by their determi(38)

Substituting in (34) it follows that: (39)

This integral can be simplified by an appropriate substitution. Now, let in the integrand of (39). Immediately, and the new limits of integration are this yields . Consequently, by substituting for and in (39) the new integral becomes

In the same manner, applying LBS for to the dipole

(40) :

(33) (41) where

is the radius of each concentric loop.


Starting with the vector definition in the integrand of the LBS as given in (41)

Applying LBS for the dipole

427

: (55)

(42) (43) (44) The cross product of vectors minant:

Starting by the vector definition in the integrand of the LBS as given in (55) (56)

and is given by their deter-

(57) (58)

(45) The cross product of vectors determinant:

Substituting in (41) it follows that:

and

is given by their (59)

(46) Substituting in (55) it follows that: By applying the substitution becomes

, the integral in (46)

(60)

In a similar manner to the previous square sections, substiin (60) becomes tuting for (47) Applying LBS for the dipole

:

(61) (48)

(49)

The integrals in (40), (47), (54), and (61) can be evaluated based on the integration by substitution technique which can simplify the integrand until it resembles an expression that is integrable. In this manner, the expressions for the magnetic field due to each side of the square loop can be represented by the following general form:

(50)

(62)

Starting with the vector definition in the integrand of the LBS as given in (48)

(51) The cross product of vectors determinant:

and

is given by their

(52)

Careful inspection of the above integral, (62) yields that by a double substitution the integral can be evaluated analytically. Initially a substitution of the form , where is a constant and can take place in the denominator of the integrand in (62) resulting in

Substituting in (48) it follows: (63) (53)

By applying the substitution becomes

, the integral in (53)

It is critical to notice that the limits of integration change with each successive integration. A further substitution of the form , where is a constant and , returns

(54)

(64)

428


Applying a trigonometric substitution of the form and , in (64), gives

(65)

(69) for each integral term in (70) leads to the analytic expression for the magnetic field of the square spiral. For example, the first integral term in (62), representing the magnetic field strength due to the contribution from one side of the square loop, is

The final integral in (65) is a standard cosine integral which can now be solved, giving the following expression: where

is

, as

and

represents

(66) By employing trigonometric identities, the Pythagorean identity can be written as follows: (67) Thus, substitution of (67) into the expression (66), yields the solution of the integral (62) in the form

(68) Finally, by algebraic simplification of the previous expression, the solution to the integral of (62) is (69)

Therefore, the total magnetic field due to the square current loop shown in Fig. 4(b) and Fig. 5(b) can be expressed by superimposing the contributions of each linear dipole that compose the four sides of the square. The total field denoted is thus

(71) (70) Consequently, the integral function (70) is composed of the individual contributions of the four sides of the square loop, each being an integral of the general form shown by (62). The solution to this integral was evaluated earlier in this section and the result is shown in (69). The substitution of the and term in

Referring back to the equivalent representation of a square spiral, as discussed in Section III-A, the coil is modeled as a number of concentric square loops. The integral equation method described above for the square loop can be employed here to solve the three-dimensional magnetic field problem for the square spiral. The algorithm in (71) computes the dominant


component of the magnetic field generated by the square loop of Fig. 4(b) and Fig. 5(b). Applying this algorithm to each consecutive loop in Fig. 4(b) and Fig. 5(b) and summing the results yields the component of the induced field due to the TX square spiral coil as

(72)

ACKNOWLEDGMENT The authors acknowledge the support from the Scottish Funding Council for the Joint Research Institute with the Heriot-Watt University, which is a part of the Edinburgh Research Partnership in Engineering and Mathematics (ERPem). The authors also wish to thank Dr. Konstantinos Drakakis for his expert advice on mathematical problems. REFERENCES [1] K. Finkenzeller, RFID Handbook—Radio-Frequency Identification Fundamentals and Applications. New York: Wiley, 1999.

429

[2] S. Smith, T. Tang, J. Terry, J. Stevenson, B. Flynn, H. Reekie, A. Murray, A. Gundlach, D. Renshaw, B. Dhillon, A. Ohtori, Y. Inone, and A. J. Walton, “Development of a miniaturised drug delivery system with wireless power transfer and communication,” IET Nanobiotechnology, vol. 1, no. 5, pp. 80–86, 2007. [3] J. Coulombe, M. Sawan, and J. F. Gervais, “A highly flexible system for microstimulation of the visual cortex: Design and implementation,” IEEE Trans. Biomed. Circuits Syst., vol. 1, no. 4, pp. 258–269, 2007. [4] F.-G. Zeng, S. Rebscher, W. Harrison, X. Sun, and H. Feng, “Cochlear implants: System design, integration, and evaluation,” IEEE Rev. Biomed. Eng., vol. 1, pp. 115–142, 2008. [5] G. Vandevoorde and R. Puers, “Wireless energy transfer for stand-alone systems: A comparison between low and high power applicability,” Sens. Actuators A: Phys., vol. 92, no. 1, pp. 305–311, 2001. [6] S. F. Pichorim and P. J. Abatti, “Design of coils for millimeter- and submillimeter-sized biotelemetry,” IEEE Trans. Biomed. Eng., vol. 51, no. 8, pp. 1487–1489, 2004. [7] P. R. Troyk, “Injectable electronic identification, monitoring, and stimulation systems,” Annual. Rev. Biomed. Eng., vol. 1, pp. 177–209, 1999. [8] N. N. Donaldson and T. A. Perkins, “Analysis of resonant coupled coils in the design of radio frequency transcutaneous links,” Med. Biol. Eng. Comput., vol. 21, no. 5, pp. 612–627, 1983. [9] D. C. Galbraith, M. Soma, and R. L. White, “A wide-band efficient inductive transdermal power and data link with coupling insensitive gain,” IEEE Trans. Biomed. Eng., vol. 34, no. 4, pp. 265–275, 1987. [10] W. J. Heetderks, “Rf powering of millimeter and submillimeter sized neural prosthetic implants,” IEEE Trans. Biomed. Eng., vol. 35, no. 5, pp. 323–327, 1988. [11] S. L. W. Ko and C. Fung, “Design of radio-frequency powered coils for implant instruments,” Med. Biol. Eng. Comput., vol. 15, no. 6, pp. 634–640, 1977. [12] S. Schmidt and G. Lazzi, “Use of FDTD thin-strut formalism for biomedical telemetry coil designs,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 8, pp. 1952–1956, 2004. [13] F. C. Flack, E. D. James, and D. M. Schlapp, “Mutual inductance of air-cored coils: Effect on design of radio-frequency coupled implants,” Med. Biol. Eng. Comput., vol. 9, pp. 79–85, 1971. [14] E. S. Hochmair, “System optimization for improved accuracy in transcutaneous signal and power transmission,” IEEE Trans. Biomed. Eng., vol. BME-31, no. 2, pp. 177–186, 1984. [15] F. W. Grover, Inductance Calculations, Working Formulas and Tables, 2nd ed. New York: Van Nostrand, 1946. [16] F. Terman, Radio Engineers Handbook. New York: McGraw-Hill, 1943. [17] S. Ramo, J. R. Whinnery, and T. VanDuzer, Fields and Waves in Communication Electronics, 3rd ed. New York: Wiley, 1994. [18] M. Soma, C. D. Galbraith, and R. White, “Radio-frequency coils in implantable devices: Misalignment analysis and design procedure,” IEEE Trans. Biomed. Eng., vol. BME-34, no. 4, pp. 276–282, 1987. [19] S. Babic and C. Akyel, “Improvement in calculation of the self- and mutual inductance of thin-wall solenoids and disk coils,” IEEE Trans. Magn., vol. 36, no. 4, pp. 1970–1975, Jul. 2000. [20] S. Babic and C. Akyel, “Calculating mutual inductance between circular coils with inclined axes in air,” IEEE Trans. Magn., vol. 44, no. 7, pp. 1743–1750, Jul. 2008. [21] D. C. Yates, A. S. Holmes, and A. J. Burdett, “Optimal transmission frequency for ultralow-power short range radio links,” IEEE Trans. Circuits Syst. I, vol. 51, no. 7, pp. 1405–1413, 2004. [22] C. Balanis, Antenna Theory: Analysis and Design, 2nd ed. New York: Wiley, 1997. [23] J. D. Kraus, Antennas, 2nd ed. New York: McGraw-Hill, 1988. [24] G. S. Smith, Loop Antennas, in Antennas Engineering Handbook. New York: McGraw-Hill, 1984. [25] W. R. Smythe, Static and Dynamic Electricity, 2nd ed. New York: McGraw-Hill, 1950. [26] E. C. Snelling, Soft Ferrites: Properties and Applications, 2nd ed. London, U.K.: Butterworths, 1988. [27] M. R. Shah, R. P. Phillips, and R. A. Norman, “A study of printed spiral coils for neuroprosthetic transcranial telemetry applications,” IEEE Trans. Biomed. Eng., vol. 45, no. 7, pp. 867–876, 1998. [28] S. Y. Leung and D. C. Lam, “Performance of printed polymer-based RFID antenna on curvilinear surface,” IEEE Trans. Electron. Packag. Manuf., vol. 30, no. 3, pp. 200–205, 2007. [29] L. K. Urankar, “Vector potential and magnetic field of current-carrying finite arc segment in analytical form, part I: Filament approximation,” IEEE Trans. Magn., vol. MAG-16, no. 5, pp. 1283–1288, Sep. 1980.

430


[30] L. K. Urankar, “Compact extended algorithms for elliptic integrals in electromagnetic field and potential computations part II: Elliptic integral of third kind with extended integration range,” IEEE Trans. Magn., vol. 30, no. 3, pp. 1236–1241, May 1994. [31] R. Rodriguez, J. Dishman, F. Dickens, and E. Whelan, “Modeling of two-dimensional spiral inductors,” IEEE Trans. Compon., Hybrids, Manuf. Technol., vol. 3, no. 4, pp. 535–541, 1980. [32] R. King, “The rectangular loop antenna as a dipole,” IEEE Trans. Antennas Propag., vol. 7, no. 1, pp. 53–61, 1959. [33] B. Levin, “Field of a rectangular loop,” IEEE Trans. Antennas Propag., vol. 52, no. 4, pp. 948–952, 2004. [34] H. Lee, “A high-bandwidth induction sensor coil,” J. Phys. E: Sci. Instrum., vol. 15, no. 10, pp. 1017–1019, 1982. [35] R. W. Erickson, Fundamentals of Power Electronics. Norwell, MA: Kluwer, 1999. [36] G. B. Joung and B. H. Cho, “An energy transmission system for an artificial heart using leakage inductance compensation of transcutaneous transformer,” IEEE Trans. Power Electron., vol. 13, no. 6, pp. 1013–1022, 1998. [37] A. Ghahary and B. H. Cho, “Design of a transcutaneous energy transmission system using a series resonant converter,” IEEE Trans. Power Electron., vol. 7, no. 2, pp. 261–269, 1992. [38] C. Bowick, RF Circuit Design, 2nd ed. Boston, MA: Newnes, 2008. [39] “ICODE coil design guide,” Philips Semiconductors, Application Note, Product Specification Rev. 3.0 Sep. 2002. [40] S. Mohan, M. Hershenson, S. Boyd, and T. Lee, “Simple accurate expressions for planar spiral inductances,” IEEE J. Solid-State Circuits, vol. 34, no. 10, pp. 1419–1424, 1999. [41] C. P. Yue and S. S. Wong, “On-chip spiral inductors with patterned ground shields for Si-based RF IC’s,” IEEE J. Solid-State Circuits, vol. 33, no. 5, pp. 743–752, 1998. [42] C. Sauer, M. Stanac´evic´, G. Cauwenberghs, and N. Thakor, “Power harvesting and telemetry in CMOS for implanted devices,” IEEE Trans. Circuits Syst. I, vol. 52, no. 12, pp. 2605–2613, 2005. [43] C. P. Yue and S. S. Wong, “Physical modeling of spiral inductors on silicon,” IEEE Trans. Electron Devices, vol. 47, no. 3, pp. 560–568, 2000.

[44] R. H. Good, “Elliptic integrals, the forgotten functions,” Eur. J. Phys., vol. 22, pp. 119–126, 2001. [45] P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists. New York: Springer-Verlag, 1954.

Kyriaki Fotopoulou was born in Athens, Greece, in December 1981. She received the M.Eng and Ph.D. degrees in electrical and electronic engineering from the University of Edinburgh, Edinburgh, U.K., in 2004 and 2008, respectively. She is presently a Research Scientist for NXP Semiconductors based at IMEC, Leuven, Belgium. Her research interests include near-field electromagnetics, RF coil design for inductive power transfer, and III-V semiconductors for power electronics. She is currently a Development Engineer for GaN HEMT, Schottky diodes, and Silicon on Insulator LDMOS devices for RF and power electronic applications.

Brian W. Flynn was born in Edinburgh, U.K., in 1949. He received the B.Sc. and Ph.D. degrees in electrical engineering from the University of Edinburgh, Edinburgh, U.K. He is presently a Senior Lecturer in the Department of Electrical Engineering, University of Edinburgh. After a period working on a Research Fellowship at the University of Edinburgh, he moved to Marconi Communications Systems, Ltd., as a Development Engineer working on microwave radio link systems. In 1980, he was appointed to a lectureship in Electrical Engineering at the University of Edinburgh and has taught courses in electromagnetics and RF/microwave design. He is the coauthor of a textbook on the design of switched-mode power supplies and has worked on the design of switched-mode power supplies. His current research interests include machine wear debris monitoring using inductive sensors and wireless power transfer. Dr. Flynn is a member of the Institution of Engineering Technology and a Chartered Engineer in the U.K.