Evaluation of the Inductive Coupling between Coplanar Concentric Coils in the Presence of the Ground

. An analytical approach is presented that allows deriving an exact series-form representation for the fux linkage between two physically large concentric circular coils located on a lossy soil. Te expression comes from a three-step analytical procedure. First, the integral expression for the fux linkage is converted into a double integral consisting of a fnite and a semi-infnite integral. Next, the semi-infnite integral is recognized to be a well-known tabulated Sommerfeld integral, which may be analytically evaluated straightforwardly. Finally, applying Lommel’s expansion allows rewriting the remaining fnite integral as a sum of elementary integrals amenable to analytical evaluation. As a result, the fux linkage between the two coils is given as a sum of spherical Hankel functions of the wavenumber in the air and in the ground, multiplied by a coefcient depending on the geometrical dimensions of the coils. Te accuracy and robustness of the proposed formulation is tested by comparing its outcomes with those generated by numerical integration of the complete integral representation for the fux linkage and with the results provided by previous analytical approaches to the same problem. It is found that the use of the derived expression for the inductance makes it possible to obtain signifcant time savings as compared to numerical quadrature schemes.


Introduction
In recent years, current-carrying coils of wire have been extensively used in a number of felds of scientifc and technological interest, including electromagnetic sounding for earth exploration [1,2], magnetic resonance imaging [3][4][5], radio direction fnding [6][7][8][9], diathermy [10,11], and wireless power transfer (WPT) [12].In several engineering applications, coil antennas are arranged in pairs or organized within an array structure, with the scope of transmitting electric power, measuring the direction of arrival of a radio signal, or acquiring information about the subsurface structure of a terrestrial area and detecting shallow buried objects.For instance, with the growing demand for battery electric vehicles (BEVs), two-coil WPT systems capable of efcient energy transfer are increasingly requested.Basically, WPT systems for battery electric vehicles are used to transfer electrical energy from a source coil to a receiving coil connected to a charging system.Te peculiarity of this process lies in the fact that the energy transfer is performed via magnetic induction, without wires and cables [13][14][15][16][17].It turns out that the magnetic coupling between the coils plays a fundamental role in the efciency of the energy transfer, and the determination of the mutual inductance between the emitter and the receiver becomes necessary to get helpful knowledge about the coupling strength.
In literature, many works have addressed the problem of rigorously calculating the mutual inductance in a free-space environment [18][19][20][21][22][23][24][25][26][27].Unfortunately, such solutions cannot be used every time that the coils are located in close proximity to or on the top surface of a terrestrial area, which happens, for instance, in applications like WPT systems for BEV, where the coils operate close to the bottom part of the vehicle.Here, the presence of the soil has a considerable impact on the mutual coupling and, hence, on the performance of the energy transfer.
Despite the importance of evaluating the fux linkage between coaxial coils located in proximity to terrestrial areas, there is scarcity of analytical solutions that allow accomplishing this task.An attempt in this direction has been made in [28], where the problem of two coils placed at the air-ground interface is considered.Unfortunately, the derived solution for the mutual impedance is valid only for equally sized coils separated by a non-negligible radial distance and when taking the limit as the distance approaches zero (that is, when the coils share the same axis), the obtained formula diverges.To author's knowledge, the sole analytical approaches that efciently treat this problem are the well-established quasistatic explicit expression for the inductance [29] or the improved analytical solution derived in [30], valid in a wider frequency range.Yet, both these solutions have the disadvantage of being subject to the condition that the receiving coil is physically and electrically small (small-loop assumption) and, as a consequence, they cannot be used for arbitrarily sized coils.
Tus, the computation of the mutual inductance of coaxial coils located above an earth structure is usually carried out by using three-dimensional simulation techniques such as the boundary element method (BEM) and the fnite element method (FEM) or via numerical integration of its integral expression.However, especially if the coil system is supposed to operate very close to or on the top surface of the ground, these methods involve signifcant computational eforts, which, in turn, imply non-negligible time costs.
Te scope of the present work is to derive an exact explicit expression for the fux linkage between two physically large concentric circular coils positioned on the top surface of a homogeneous material medium, which responds to the following two requirements.First, it must be valid in the quasistatic as well as nonquasistatic frequency regions and regardless of the coils size.Second, it must exhibit advantages in terms of time cost with respect to numerical techniques.Starting from the infnite integral representation for the fux linkage, the expression is obtained by converting the product of the two Bessel functions contained in the integrand into the fnite integral of one Bessel function.Ten, the infnite and the fnite integral signs are interchanged, and the infnite integration is recognized to be a well-known tabulated Sommerfeld integral, whose analytical evaluation is straightforward.Finally, applying Lommel's expansion permits to turn the remaining fnite integral into a sum of simpler integrals amenable to analytical evaluation.As a result of the developed theory, the mutual inductance is expressed as a sum of spherical Hankel functions of the wavenumber in the air and in the conducting medium, multiplied by a coefcient depending on geometrical parameters.Te derived formula holds under the thin-wire assumption, which prescribes that the radii of the coils must be far greater than the wire radius.Te validity of the proposed approach is tested through comparison with the data provided by numerical integration of the integral representation for the mutual inductance, as well as with the outcomes from the previously published quasistatic solution to the same problem.What emerges is that the derived exact expression for the inductance is advantageous over numerical quadrature schemes as, accuracy being equal, its usage ensures considerable time savings with respect to numerical integration.On the other hand, the proposed expression allows relaxing the quasistatic assumption and the hypothesis of small receiving coil, both underlying the previous solution to the problem.

Theory
Te geometric confguration of the problem is shown in Figure 1.Two concentric large circular loops, with radii a and (a > b), are placed on the surface of a homogeneous medium.Te wire that constitutes each coil is assumed to be thin, which implies that the wire radius r 0 is much smaller than the external radius of the coil (r 0 ≪ a, b).Te magnetic permeability is taken to be everywhere that of free space μ 0 , while the dielectric constant and electric conductivity of the material medium are denoted by ϵ 1 and σ 1 , respectively.
Our scope is to determine an exact explicit expression for the mutual inductance M of the two loops, which is not subject to any restrictive assumption.Tis will be done by introducing an efcient method that allows to evaluate the integral representation for M. If, for the purposes of the derivation, the primary coil is taken to be the larger loop of Figure 1, the mutual inductance is defned as the fux encompassed by the receiving coil per unit current fowing in the source coil.Tus, after setting I � 1 A, with the time factor e jωt assumed and suppressed for notational simplicity, it reads where r and ϑ are, as depicted in Figure 1(b), the polar coordinates of the generic feld point, while H z is the vertical magnetic feld generated by the source loop, given by [[29], 4.88] with J ] (•) being the ] th-order Bessel function, and with where σ 0 � 0 and μ 1 � μ 0 .Te use of ( 2) into (1) provides the following expression for M: and after introducing the well-known identity [[31], 5.52.1], and expression ( 4) is turned into 2 International Journal of Antennas and Propagation which is the exact integral representation for the fux linkage between the coils.Since it holds [ [32], 11.41.17] formula ( 6) may be expressed as Te semi-infnite integral in ( 9) may be evaluated by multiplying the numerator and denominator of the integrand by u 0 − u 1 and then making use of the identity [33] with It yields where the integral on the right-hand side is well-known and given by [ [34], p. 677, no.13] with h (2)  0 (ξ) being the zeroth-order spherical Hankel function of the second kind.Te use of (12) together with (13) into (9) provides and since it is not difcult to prove that one obtains where it has been set Te fnite integral may be analytically evaluated once g 1 is replaced with its Maclaurin expansion with respect to cos ϑ.If we let τ � −2ab cos ϑ, the expansion is given by International Journal of Antennas and Propagation and since from the diferentiation properties of the spherical Bessel functions [ [33], 10.1.24],it is easily deduced that one obtains the relation which may be applied recursively to give Te use of ( 21) into (18) provides and for l � 1, the following expansion is found: which, substituted for g 1 in ( 16), leads to Te integral on the right-hand side is tabulated and given by [ [31], 2.512.2−2.512.3] where m!! denotes the double factorial.Tus, after setting m � 2l − 1, expression (24) becomes where account has been taken that (2l − 1)! � (2l − 1)!! (2l − 2)!!, and with and [35] h (2) It should be noted that the obtained representation (26) for the fux linkage may be also used to obtain the self-loop inductance of each coil.In fact, for a loop with external radius a and wire radius r w , the self-inductance under the thin-wire assumption (that is, r w ≪ a) is given by L a, r w  � M a, a − r w . ( Finally, the expression for the mutual inductance of a pair of circular loops may be applied to the computation of the fux linkage between multiturn coils.If, as an example, the emitter is a single-turn coil with radius a, while the receiver is a pancake coil composed of N b turns, with radii denoted by b i (i � 1, 2, . . ., N b ), the mutual inductance between them reads [26] while the overall self-inductance of the receiver is given by It should be observed that ( 26) may be used provided that the source loop supports a uniform current.In fact, expression (2), underlying the proposed approach, is based on the uniform current assumption which, in principle, holds if the length of the wire that constitutes the source loop is less than or equal to 0.3 times the free-space wavelength λ [36].For electrically larger antennas, the variations of the current along the loop become large and (2) cannot be used any longer, unless a uniform current is forced by the feeding system.As explained in [37,38], this may be done by dividing the source coil into electrically short segments and then by driving them in parallel by means of radial transmission lines.

Results and Discussion
Te validity and robustness of the proposed approach is tested by applying the derived expression (26) to the computation of the fux linkage between two coaxial coils International Journal of Antennas and Propagation placed on a clay soil.Te electrical conductivity and dielectric permittivity of the soil are taken to be equal to σ 1 � σ � 10 mS/m and ϵ 1 � 10 ϵ 0 , respectively [39].At frst, the real and imaginary parts of M are computed against the operating frequency, with b � 0.5 m and a � 5 m.Te obtained results, illustrated in Figure 2, are compared with the data provided by both numerical integration of (6) and the fnite diference time domain (FDTD) method.In particular, numerical integration is carried out by using an adaptive Gauss-Kronrod G7-K15 scheme, arising from the combination of a 7-point Gauss rule with a 15-point Kronrod rule.On the other hand, the FDTD lattice is confgured with 300 × 300 × 20 cubic cells with side lengths of 5 cm, spanning a computational domain of 15 × 15 × 1 m enclosing the coils.Te mesh is terminated with PML absorbing boundary conditions in the three Cartesian directions.
In addition, for the sake of comparison, the results are also compared with those obtained by using the quasistatic approximation for the produced magnetic feld, derived under the further assumption that the receiver is both electrically and physically small.In this situation, the mutual inductance is simply given by the vertical magnetic feld H 0 z generated by the emitter at its center, multiplied by the magnetic permeability in free space and the area of the receiving coil.It reads M � μ 0 2 H 0 z , where the quasistatic approximation for H 0 z is given by [29] As is shown by Figure 2, the curves arising from (26) approach the data originating from the numerical integration of the complete representation (6) as L is increased, with L being the index of truncation of the sum in (26).In particular, excellent agreement is observed for L � 3 all over the considered frequency range, which implies that these trends may be referred to as exact curves.Conversely, the profles of the real and imaginary parts of M arising from the quasistatic solution (32) are in agreement with the respective exact curves only in the low-frequency range, up to about 10 MHz (real part of M ) and 3 MHz (imaginary part of M ).Tese frequency limits are consistent with the theoretical upper limit of validity of the quasistatic assumption.To convince on this point, it sufces to consider that the quasistatic assumption is valid as long as the largest dimension D of the considered system is electrically small, that is, for λ ≫ D, with λ being the free-space wavelength.Tis ensures that the overall size of the two-loop system is sufciently small for electromagnetic retardation to have negligible impact on the feld distribution.Since, in the present confguration, D coincides with the diagonal of the square bounding box enclosing the loops, which is 2a long, the quasistatic feld assumption implies that and, as a consequence, it holds as long as the operating frequency f obeys the inequality with c being the speed of light in vacuum.Tus, f max is the frequency below which (32) must provide exact results, and this is confrmed by the curves plotted in Figure 1.Tereinafter, the profles of the real and the imaginary parts of (32) will deviate from the corresponding exact curves, and this may happen at about f max , as in the case of the trend of the imaginary part, or at a higher frequency.While (26) and numerical integration schemes have been shown to be comparable to each other as far as the exhibited accuracy is concerned, the same cannot be said when it comes to computation time.In fact, the proposed formulation allows achieving signifcant time savings with respect to purely numerical procedures.Tis aspect is clarifed by Table 1, which shows the average CPU time taken by the proposed approach and Gaussian integration to calculate the profles depicted in Figure 2. Te entries of Table 1 tell us that the speedup ofered by ( 26) is equal to about 376 when L � 3, that is, when the curves arising from the two methods coincide.Tis confrms that the proposed method is signifcantly faster than Gaussian integration.
On the other hand, investigation of the time cost of (32) would not be useful since this expression cannot provide the same accurate results as the other two methods.In particular, the loss of accuracy of (32) as frequency is increased may be further analyzed by taking a glance at Figure 3, which depicts the relative errors resulting from calculating the real and imaginary parts of M by using (32) instead of numerical integration of (6).From the analysis of the curves plotted in Figure 3(a), it is clearly seen that the relative error of the real part of (32) is less than 1% at frequencies smaller than 1 MHz and is equal to 2% at 3 MHz.Tereinafter, the error rapidly increases as frequency grows up, and beyond 10 MHz, it ranges from 10% to 100% and over.Similar conclusions may be deduced from the analysis of the results depicted in Figure 3(b).Here, the error generated by the imaginary part of ( 32) is slightly smaller than 0.2% up to 1 MHz.Beyond that frequency, it rapidly grows up, and at 10 MHz, it exceeds 10%.
Additional information on the limitations implied by (32) may be acquired from the investigation of the behavior of the fux linkage between the coils as the radius b of the receiver is changed.Tis point is illustrated in Figure 4, which depicts the trends of the real and imaginary parts of the mutual inductance against b, calculated by using the same methods as in the previous example.Te radius of the emitter and the ground conductivity are taken to be a � 5 m and σ 1 � 10 mS/m, respectively, while the operating frequency is assumed to be f � 10 MHz.Te received radius is assumed to span the 0.05 m ≤ b ≤ 2 m interval.On one hand, the trends shown in Figure 4 confrm that the data provided by (26) converge to the exact results as L is increased.On the other hand, the plotted data clearly point out that the smallloop approximation (32) cannot entirely capture the exact International Journal of Antennas and Propagation trend of the inductance, and that, in particular, it fails as soon as b becomes large enough to be comparable to the radius of the emitter.In fact, at the frequency of 10 MHz, the receiver ceases to be physically small well before its size becomes comparable to the free-space wavelength, which event occurs when the length of the wire constituting the receiver, that is, 2πb, is at least equal to λ/3 [36].Tis latter condition implies that the receiver is no longer electrically small when while it ceases to be physically small for smaller values of b, that is, when b becomes larger than a/10 � 50 cm.As shown by the curves plotted in Figure 4, the 50-cm limit is approximately where the trends for the real and imaginary parts of M arising from (32) start to deviate from the exact data.Te curves of Figure 4 also allow to conclude that at intermediate or high frequencies, the real and the imaginary parts of the fux linkage have diferent behaviors with increasing b.In particular, while the imaginary part is always negative in sign and monotonically decreases (i.e., increases in absolute value), the real part of M does the same only if the radius of the receiver is sufciently small with respect to that of the emitter.When this is no longer true, the slope of the curve starts to diminish in absolute value as b is increased, with the result that the real part of M soon reaches its minimum value.Tereinafter, the real part starts to increase and, after zero-crossing, becomes positive in sign.
Finally, one would ask to what extent changing the soil properties afects the magnetic coupling between concentric coils.Tis point is clarifed by Figure 5, which depicts b− profles of the amplitude of the mutual inductance of two single-turn coils, with the ground conductivity σ taken as a parameter.Te calculation has been performed by using (26), and the index at which the infnite sum is truncated is taken to be L � 9. Te source loop, with radius a � 20 m, is assumed to operate at the frequency of 10 MHz, and three diferent values for σ are considered, while the relative dielectric permittivity of the ground is still assumed to be equal to 10. Te obtained trends are compared to that arising from neglecting the presence of the ground, which comes from applying the well-known analytical expression [26] where and with K(κ) and E(κ) being the complete elliptic integrals of the frst and second kind, respectively.Te curves plotted in Figure 5 point out how increasing the conductivity of the medium leads to reduce the mutual inductance of the loops.

International Journal of Antennas and Propagation
Tis means that the amplitude of the mutual inductance reaches its maximum value when the two loops are embedded in free space.
Te efect of changing the ground conductivity on the real and imaginary parts of the mutual inductance is illustrated by Figure 6.In this example, it is assumed that a � 5 m and b � 2 m, and the operating frequency is still taken to be equal to 10 MHz.Te ground is characterized by relative dielectric permittivity equal to 10, while the electrical conductivity σ is comprised between 1 mS/m and 1 S/m.Again, the σ− profles of M arising from (26) and (32) and numerical integration of (6) are computed and plotted.From the analysis of the curves depicted in Figure 6, it is evident that at the frequency of 10 MHz, the impact of changing σ on the magnetic coupling is not negligible up to about 100 mS/ m.Tis is true for both the components of M. In particular, for σ ≤ 100 mS/m, the real part may assume positive as well as negative values, and its absolute value decreases almost everywhere as σ is increased.Conversely, the imaginary part of M is negative in sign all over the considered range of σ, and its absolute value monotonically decreases as σ grows up.

Conclusions
Te purpose of the present work has been to derive an exact explicit analytical solution describing the fux linkage between two concentric coils positioned on a conducting terrestrial area.Te solution has been derived through a purely analytical approach based on converting the complete single-integral expression for the fux linkage into a double integral consisting of a fnite and a semi-infnite integral.Ten, the semi-infnite integral is recognized to be a well-known tabulated Sommerfeld integral, which may be evaluated straightforwardly.Finally, Lommel's expansion is applied to convert the remaining fnite integral into a combination of simpler integrals amenable to analytical evaluation.As a result, the fux linkage between the two coils is given as a sum of spherical Hankel functions of the wavenumber in the air and in the ground, multiplied by a coefcient depending on the geometrical dimensions of the coils.Numerical tests have been performed to test the validity of the proposed expression.It has been shown that with accuracy being equal, the time cost of the proposed approach is signifcantly smaller than that implied by numerical integration schemes.

Figure 1 :
Figure 1: (a) Isometric and (b) top views of the geometry under study.

Figure 3 :Figure 2 :
Figure3: Relative errors of the real and imaginary parts of (32) as compared to the numerical integration of (6) plotted against the operating frequency.(a) Relative error of the real part of (32) against frequency.(b) Relative error of the imaginary part of (32) against frequency.

Figure 5 :
Figure 5: Magnitude of the mutual inductance of two coils placed on a conducting soil plotted against the receiver radius with the soil conductivity taken as a parameter.

Figure 4 :
Figure 4: Flux linkage between two coils on a conducting soil plotted against the receiver radius b.(a) Real part of the fux linkage vs. b.(b) Imaginary part of the fux linkage vs. b.

Table 1 :
CPU time comparisons for the calculation of M against frequency.