Iterative Solution to Resonant Near-Field Coupling Adjustments Between Different Electrically Small Antennas

In this communication, iterative procedures based on Banach fixed-point theorem of adjusting transmitter and receiver impedances for achieving the maximum near-field power transfer performances are explored. As the background of proposed methods for controlling frequency splitting, an important phenomenon that occurs when two antennas are coupled in the near-field, the analysis of the resonant near-field coupling using a spherical mode theory-antenna model (SMT-AM) is applied. Methods of recursive alternate system tuning, critical coupling, and conjugate matching are mutually compared, and selection among them is given on disposal to the user to achieve aimed performances.


I. INTRODUCTION
Once only a dream of radio engineers and enthusiasts sparked by Nikola Tesla's radio experiments in his New York laboratory and in Colorado Springs at the turnover of XIX/XX century [1], transmission of electrical energy without wires nowadays becomes a necessity.It is not only because of the myriad of small electronics in everyday use, gadgets and sensors that need to be freed from attaching cables or even their batteries, but also for the purposes of charging the larger consumers such as electric cars and trains.
A version of Tesla's four-coil resonant near-field wireless power transfer (WPT) system suitable for the existing devices, on the basis of the coupled-mode theory (CMT) [2], is proposed at MIT [3].The phenomenon of splitting the resonant frequency observed in the coupled antenna systems and the critical coupling adjustment for the best WPT efficiency by controlling the coupling between the driving loop and the antenna body are explored using the circuit theory in [4], whereas the conjugate matched WPT system is analyzed by spherical mode theory-antenna model (SMT-AM) in [5].The optimum WPT performance methods are compared on the basis of the frequency splitting phenomenon analysis by SMT-AM in [6].
However, to apply these results in a practical system, the transmitter (Tx) and receiver (Rx) circuitry needs to be embedded by some feasible procedure that drives it to the optimum point.The purpose of this communication is to analyze SMT-AM and to propose iterative adjustment methods based on Banach fixed-point theorem [7] that could be applied by a proper electrical circuit design.An automatic matching network as one depicted in [8] and [9] can be used for that purpose.
After the description of the theory in Section II, followed by some necessary generalization of the analysis from [5] and [6] related to WPT between different antennas in Section III, in Section IV, we depict the marching procedures toward the best efficiency.A WPT  example is discussed in Section V, after which the main conclusion is summarized in Section VI.

II. WPT MODELING
Consider the power transfer on some angular frequency ω = βc = 2π f and f = (c/λ) (λ being the wavelength) between two arbitrary positioned antennas Tx(1) and Rx (2).Let them be immersed in a homogeneous, isotropic medium with the propagation velocity c and separated by a center-to-center distance d, as shown in Fig. 1.The antennas' impedances measured in free space are Z 1,2 A = R 1,2 + j X 1,2 , whereas their radiation efficiencies are η R AD ).The antennas' radiation resistances consist of TE and TM mode resistances R T E and R T M , respectively: R R AD 2 , and the mode ratios are 2 ).The equivalent two-port network of coupling between different directly fed antennas based on the Z-matrix from [10] is drawn in Fig. 2. For simplicity, in the antenna reactances X 1 and X 2 , we include ideal (lossless) tuning reactance at the transmitter (Tx) and the receiver (Rx) side, respectively.
As the resonant WPT involves the transmission between electrically small antennas (ESAs) [11], the scheme can be applied in a simple fashion.Assuming that the representation of ESA as the minimum scattering antenna [10] is valid, the impedances Z 1,2 are equal to the antennas' impedances measured in free-space, Z 1,2 = Z 1,2 A , whereas the mutual impedance is [5] where T is the transmission coefficient of the lowest-order spherical modes given by (22) in Appendix B.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
The power-wave transmission coefficient S 21 [12] is given by where U G is the generator and U 2 output voltage.R L and R G are the resistances of the load terminating the Rx port and that of the generator, respectively.
In the scenario of coupling between inductively fed ESAs (fourcoil WPT systems), the Y-matrix can be applied, resulting in dual relations to that of the Z-matrix with impedances exchanged with corresponding admittances [10].

III. MAXIMUM WPT PERFORMANCES
The maximum WPT performances for the scenario in Fig. 2 can be analyzed using an approach as in [6].The total differential of the power-wave transmission coefficient S 21 magnitude is As X 1 = X 1 (ω) and X 2 = X 2 (ω), the total derivative over angular frequency ω is given by Taking that (d X 1 /dω) ̸ = 0 and (d X 2 /dω) ̸ = 0, using (4) and putting (d|S 21 |/dω) = 0, the conditions for the extremum |S 21 | at given frequency are obtained by solving the system of two coupled equations: Hence, when the load and generator resistances are kept fixed, the calculation of the maximum |S 21 | by ( 2)-( 4) with respect to the antenna reactances X 1,2 leads to a system of two coupled nonlinear equations in X 1 and X 2 for the optimum reactances X 1 = X extr 1 and X 2 = X extr 2 given by It can be easily transformed into the system of two uncoupled depressed cubic equations in X 1 and X 2 given as (19) in Appendix A, by extracting X 1 from (5) or X 2 from (6) followed by inserting it into (6) or (5), respectively.They are solved for the optimum reactances by (20).Note that both the input and output of the system are tuned simultaneously (I m(Z in ) = I m(Z out ) = 0), that is, brought to resonance at the selected frequency, by adjusting the reactances to one of the real solutions pairs (X extr 1 , X extr 2 ) of ( 5) and (6).By multiplication of ( 5) and ( 6) with X 1 ̸ = 0 and X 2 ̸ = 0, respectively, followed by the subtraction of one from another, one obtains the following relationship between the optimum antenna reactances X 1,2 = X extr 1,2 that bring the WPT system in resonance at both ports and the given resistances as follows: It turns out that Tx and Rx impedances in a coupled resonant system have equal phase angles of either −(π/2) < < 0 (odd mode), or else 0 < < (π/2) (even mode).
The number of the reactance values that bring the system in resonance (i.e., the number of the system resonant frequencies) depends on the discriminants 1 and 2 given by (21) in Appendix A. As ( 1 / 2 ) = K 6 > 0, they have same sign.When 1 , 2 < 0, there is one pair (X extr 1 , X extr 2 ) of single real solutions to ( 5) and ( 6), and the resonant WPT system operates in under-coupled regime.The special case of 1 = 2 = 0 determines the critically coupled system, where one single solution representing the maximum, and one double real solution in the form of the inflection point exist.When R L and R G are such that 1 , 2 > 0, there are three real solution pairs (two maximums and one local minimum in between).Then, the frequency splitting occurs, which characterizes the over-coupled regime [4], [6].
As it can be seen from ( 5) and ( 6), strictly speaking, the WPT system cannot operate in resonance (I m(Z in ) = I m(Z out ) = 0) while either of the antennas is in self-resonance (X 1 = 0 or/and X 2 = 0), unless their separation is infinite (far-field).Moreover, as X extr it is clear that the system can be brought to resonance by the ESAs forced to self-resonate at different frequencies obeying (7).
Furthermore, by applying the maximization procedures of |S 21 | by R L and R G given in [6], and using (7), one easily finds the relationship among the antennas' reactances and the generator and the load resistance in the cases of conjugate matching and the critical coupling.In the former case where the optimum reactances are X extr 1,2 = X O P T 1,2 , and the optimum resistances whereas in the latter where X extr 1,2 = X crit 1,2 , due to the critical coupling condition 3 that stems from 1 = 2 = 0 and (20), we have that that is, K crit depends on the critical resistances R crit G and R crit L selected arbitrary as to obey (9).The basic characteristics of all the optimization procedures are summarized in Table I.

IV. ITERATIVE NEAR-FIELD TUNING SCHEME
The iterative solution to ( 5) and (6), that is, the recursive tuning to be depicted in the text that follows, is based on an application of the Banach fixed-point theorem.As, taking into consideration (7), systems ( 5) and ( 6) describe the contraction mapping f (x) = x, where x is the fixed point, the WPT system can be brought to resonance step-by-step by a process of the alternate tuning of the Tx and Rx.Commencing the adjustment at the Tx side x = x 1 , taking that in the first instance the receiver has certain free-space reactance x 2 (1) = X 2 (ω), for each tuning step n > 1, the following recursive formulas can be applied: The attractive fixed point of ( 11) and ( 12) among the possible solutions (20) (with lim n→∞ x 1,2 (n) = X extr 1,2 ) is determined as the one for which |(d[ f (x)])/d x| < 1 [7].As in the over-coupled system, the condition is not fulfilled only for the local minimum X extr , due to the shape of f (x) = x and depending on x 2 (1), the convergence point settles at odd (when x 2 (1) < X M I N 2 ) or at even mode (when x 2 (1) > X M I N

2
).The number of steps n = N required to reach the convergence depends on the intensity of frequency splitting, that is, on the electrical distance of the transmission and the resistances involved.Also, x 2 (1) = X M I N 2 is the repelling point, in the sense that the probability for the system to find an equilibrium in this point equals zero.In other words, the nearer a WPT system is to the repelling point, the larger number of steps are required for convergence toward the attractive fixed point.
Furthermore, by allowing varying resistances R G and R L , the recursive tuning method by (11) and ( 12) is easily expanded to provide the conjugate matching in a manner that the input and output are conjugate matched alternately until both sides become adjusted simultaneously, that is, until {|S 21 | 2 } ≤ 1.Alternatively, the critical coupling adjustment can be approached by alternately setting Tx and Rx impedance step-by-step for given K crit , obeying (9).The input impedance in the nth step (n > 1) is (see Fig. 2 and Table I) and the generator resistance is adjusted to be while the input is tuned as with (11) according to When the stage of input tuning is finished, the output can adjusted following the same logic, using formulas: The process repeats itself until a satisfactory precision is reached, that is, until the difference in the adjusted impedance between two succeeding steps becomes small enough.The derivative of the contraction function in the inflection point X extr 1 = X in f 1 equals one, but it shows not to be an attractive fixed point.For given K = K crit , regardless of the initial conditions, the system finds equilibrium in the critical coupling point (maximum) 1 .Note that at each iteration, unlike the tuning and conjugate matching methods that require only the information about the system impedance measured at the input of the antenna to be adjusted, in the critical coupling procedure, the adjusting side needs data about the current impedance state of the opposite one also.

V. EXAMPLE
To test the presented procedures, we selected a two-coil WPT system consisting of a wide 10-turn helix transmitter with a diameter of 15 cm and height of 4 cm, and a smaller 10-turn conical helical receiver of 2-cm height with an inner radius of 2 cm and outer one of 5 cm.Both are made of copper conductors with a diameter of 2 mm, and each is placed in the loop connection with a variable capacitor C 1,2 to be set for system resonance at an ISM frequency of f = 13.56MHz.
The free-space parameters are calculated by numerical software 4nec2.The results show that ESAs are predominantly TE mode, high Q, and of low efficiency η R AD The system frequency characteristics by (2) without and with tuning, with critical coupling at K crit = K O P T = 5.42, and with conjugate matching are depicted in Fig. 3.Even with a simple even mode tuning, WPT efficiency at 13.56 MHz is significantly improved.Also, this procedure has a clear advantage over the frequency tracking method, as the maximum |S 21 | is higher.It is easily shown that this stands regardless of the selected transfer mode (odd or even).
However, much higher efficiency can be achieved by satisfying critical coupling conditions [4], [6] and the highest by the conjugate matching [5], [6].In this scenario, the maximum |S 21 | obtained by the two latter methods are virtually equal, and the differences in their frequency characteristics are within the thickness of the curve, which enables their fair comparison for iterative adjustments.
Figs. 4-6 compare the results calculated by ( 2) for ( 11)-(18) and for (20).For all the iterative procedures, Rx is initially set to self-resonate at 13.56 MHz (x 2 (1) = 0).For the conjugate matching, the far-field matched receiver is initially assumed, that is, r L (1) = R 2 = 0.40 .When the tuning method is observed, as x 2 (1) > (the worst-case scenario), the system needs some time to figure this out and to start to converge toward the even mode maximum faster.Then, the maximum |S 21 | is reached by the fewest number of iterations.
The critical coupling method in this example ensures only negligibly less performance than the conjugate matching one, but the system Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.needs a greater number of iterations to converge to this point.Note that the capacity adjustment, that is, the adjusted resonance offset in frequency for the conjugate matching is negligible, especially when compared to the one for critical coupling.Hence, it seems that the adjustment method by controlling the coupling between the driving

TABLE II COMPARISON BETWEEN PROPOSED ITERATIVE METHOD AND NONLINEAR
OPTIMIZATION OF THE TWO-COIL SYSTEM AT f = 13.56MHZ loop and the antenna body in four-coil systems depicted in [4] leads the system closer to the conjugate matching point than to the critical coupling one.
For the purpose of validation, the optimum solutions of (2) obtained by the proposed iterative adjustment procedures are compared to the results obtained by interior-point nonlinear optimization algorithm embedded in MATLAB.First, the global optimization by two reactance parameters X 1 and X 2 is examined by the interior-point method imposing no constraints on (2).The procedure provides the results for the two optimizing reactance parameters equal to (20), indicating that the optimum point is the system resonance.By enforcing resonance introducing the constraint ( 7) on (2), the same optimizing reactances are obtained, but with a fewer number of iterations.
In the next step, it is allowed that all four parameters X 1 , X 2 , R G , and R L may vary in the optimization algorithm.The conjugate-matched optimization is examined first, applying the constraint (8) K = K O P T = 5.42.Then, the critical coupling optimization is accomplished by selecting equal K = K crit = K O P T by (10) together with introducing an additional constraint (9) on (2).They both give the optimization solutions exceedingly close to the exact ones as well.
In the end, the global solution that maximize |S 21 | is sought by imposing no constraints on (2).In accordance with the powermatching theorem, it is found that the obtained solution corresponds to the solution for the conjugate-matching optimization, but it is achieved with a greater number of iterations.
The comparison of the number of iterations n = N in the proposed method versus the number of iterations M reported by MATLAB needed to achieve each of the optimum solutions within the optimization tolerance of 10 −6 is depicted in Table II.The listed values of the maximum power-transmission coefficient |S 21 | 2 calculated by (2) can also be read out from Figs. 3 and 6.
It may be noted that the proposed methods provide a faster convergence in comparison to the used black-box algorithm, except the iterative critical coupling one that requires much more iterations.An additional fact that the black-box optimization with critical coupling constraint (9) needs a significantly larger number of iterations than with other applied constraints, reflects the complexity of the critical coupling adjustment as well.However, before judgment about the method, two things should be considered.The first is that it reaches high performance level |S 21 | 2 > 95 % immediately after the first step, n = 2, as it is visible in Fig. 6.The second is that the selection of critical K = K crit , that is, the critical resistances according to (10) is arbitrary.For instance, the selection of K crit = 1 leads to a quite different conclusion, as then N = 14, whereas M = 23.
Hence, the performance of the iterative algorithm by the fixed-point method is shown to be comparable to that of the interior-point optimization.Whereas the simple tuning shows a fast convergence but the lowest maximum performances among the considered, the conjugate matching exhibits vice versa.It also turns out that the critical coupling adjustment can be set to provide simultaneously the fastest convergence and almost the best performances, unlike the other two approaches.

VI. CONCLUSION
The automatic near-field adjustment schemes presented herein provide a possibility for obtaining the best WPT performances in an iterative fashion, implying minimum or no information exchange within the system.The simple tuning and the conjugate matching require only a time synchronization protocol between the Tx and the Rx.Besides the synchronization, in the critical coupling method, they both have to calibrate themselves according to the current impedance state of the opposite side for selected K crit .It is shown that the system can be brought to the selected point of resonance by each of these SMT-AM-based procedures, in a manner feasible to be implemented in electrical circuits by application of adaptive matching networks.

APPENDIX A SOLUTIONS TO THE DEPRESSED CUBIC EQUATION
The determinantal equation in the form of a depressed cubic where M )], and 2 )/(R L + R 2 ) can be solved applying the substitution X 1,2 = y − ( p/3y).By solving the obtained quadratic equation in y, followed by calculating X 1,2 (y) = X extr 1,2 , the first solution is obtained as follows:
and there is only one attractive fixed point Z d[ f (z)])/dz| z=Z O P T G | = max 2

1 = 0 . 6 %, η R AD 2 =
0.06 % at 13.56 MHz, with only a slight mode mismatch between them.The input impedances of the transmitter and the receiver are Z 1 = (2.16+ j2629.05)and Z 2 = (0.40 + j615.08), respectively.Initially, they are set to self-resonate at 13.56 MHz by serial tuning capacitors (C 1 = 4.46 pF, C 2 = 19.08 pF), and arranged coaxially at d = 15 cm.The mutual impedance calculated by (1) for this WPT system configuration is Re(Z M ) = 1.67 m and I m(Z M ) = 67.74 .For the frequency tracking and in the tuning simulation, R G = 50 and R L = 10 are selected, which ensure the system is overcoupled ( 1,2 > 0) at the selected antenna separation.The resistances R G = R O P T G = 150.75 and R L = R O P T L = 27.81 would set the conjugate matched system to the undercoupled regime ( 1,2 < 0).