Zoran Blažević and Maja Škiljo IMAGE LICENSED BY INGRAM PUBLISHING Resonant Near-Field Power Transfer Revisiting the frequency-splitting phenomenon using the spherical mode theory antenna model. n this article, the resonant coupling phenomenon is revisited using the antenna theory. The methods for extracting the maximum power by the load (i.e., the frequency tracking, conjugate matching, and critical coupling adjustment) are addressed. To obtain the best performances in terms of the highest power and the greatest transmission range, the system’s tuning needs to be very sharp regardless of the method applied. Conjugate matching is shown to be the best method for reaching the highest possible antenna separations and the maximum efficiency, but the critical coupling adjustment turns out to be more practical in the sense of the bandwidth when the antenna separations are not too large. I BACKGROUND State-of-the-art technology is characterized by myriad smalland middle-size gadgets and electrocommunication devices, such as sensors, smartphones and cell phones, and laptops and tablets, all equipped with some rechargeable power source, such as a battery or capacitor. Additionally, emerging new technologies, such as electric cars and the Internet of Things (IoT), require charging methods that are much more convenient than the usage of many cords that lead to sockets. For this reason, wireless power transfer (WPT) becomes a promising solution to deal with the challenge. As the aforementioned devices require vastly different amounts of energy, ranging from very small (as IoT sensors do) to quite large (as electric vehicles do), different WPT principles can be adopted for this purpose. Digital Object Identifier 10.1109/MAP.2019.2920102 Date of publication: 27 June 2019 IEEE ANTENNAS & PROPAGATION MAGAZINE AUGUST 2019 For instance, in ultrahigh-frequency (UHF) radio-frequency (RF) identification systems, low-power passive tags are supplied by radio waves emanating from the reader antenna in the far field. Another example of the same method is a low-power IoT sensor charged by Wi-Fi router RF power. For the purposes of larger PTs, such as for charging smartphones, laptops, or electric cars, the best approach is the application of electrically small antennas (ESAs), inductively or resonantly coupled in the near field. The definition of ESA was first proposed by Wheeler [1] as an antenna whose maximum dimension is less than a radianlength (m/2r), m being the wavelength. Chu [2] gave an equivalent definition of ESA as the antenna enclosed inside a Chu sphere (i.e., the minimum sphere that encloses whole antenna, with a diameter less than a radianlength). The first exploration and application of ESAs for the purpose of WPT was done more than a century ago within the scope of radio research by Nikola Tesla in Colorado Springs, Colorado [3]. More details about the WPT development through history can be found in [4] and [5]. One of the successful more-recent experiments on Tesla’s WPT technology was done by Soljačić and his team from the Massachusetts Institute of Technology [6], who proposed an HF version of Tesla’s resonant four-coil system and analyzed the resonant magnetic coupling by coupled-mode theory (CMT) [7]. Another HF version of the magnetically coupled four-coil system is investigated in [8] using the equivalent-circuit scheme. The authors provide the definitions of coupling zones based on the degree of coupling between the resonators, analyzing the frequency-splitting phenomenon while proposing the 1045-9243/19©2019IEEE 39 method of critical-coupling operating-regime system setup and control to obtain the highest WPT performances. On the other hand, the problem of the resonant WPT at midranges is considered from the standpoint of the antenna theory and Z-matrix in [9] and [10]. There, the conjugate-matched near-field WPT system is analyzed, applying the approximation of the spherical mode theory antenna model (SMT-AM) that deduces the PT to the interaction of the lowest-order transverse electric (TE) and transverse magnetic (TM) spherical modes. In the literature, those two are often referred to as even mode and odd mode. However, because they switch the position on the frequency axis depending on the propagation geometry, this nomenclature is omitted in this article. The approach is applied to the design of very HF and UHF WPT systems with multiplearm folded ESAs in [11] and [12]. However, none of these papers dealing with the AMs address the frequency-splitting problem. However, the SMT-AM model is considered for the prediction of split resonant frequencies in [13]. The former (CMT) approach assumes that the coupling between two resonators is a consequence of interlapping and interleaving of their reactive fields, without considering a small radiation field contribution. As a result, WPT systems are, on many occasions, operated at the antenna (first) selfresonant frequency [8], [14], [15], applying techniques that avoid frequency splitting. However, it is shown in the latter (SMT-AM) approach that, due to its interaction with the reactive coupling component, the near-field radiative component can play a significant role in conjugate-matched WPT systems at midranges. In this article, the frequency-splitting phenomenon is investigated by adopting the SMT-AM model from [9]. We propose a definition of the coupling zones (an alternative to the one given z θ0 Tx y G ~ θ1 z′ d φ0 x Rx x′ φ1 L FIGURE 1. The radio propagation geometry. ZA – ZM RG = RL l1 U1 ZA – ZM l2 ZM = Re[ZA]T U2 y′ in [8]) in the sense of the antenna theory approach, and we reexamine both conjugate-matched and critically coupled WPT systems. For that purpose, the validity of the applied SMT-AM approximation is tested by comparison with the results of a fullwave model. WPT MODELING Let us consider the PT on some angular frequency ~ = 2rf, f = c/m (c being the velocity of the electromagnetic waves in the considered propagation medium) between two equal arbitrarily positioned ESAs separated by a center-to-center distance d, as shown in Figure 1. The ESAs’ free-space input impedance is Z A = R A + jX A, the frequency of the first self-resonance is ~ 0, and the radiation efficiency is h RAD = R RAD /R A, where R RAD is the radiation resistance. The case of different antennas can be approached in a similar manner to that shown here. When the two Chu spheres satisfying the condition b a 1 0.5 (where b = 2r/m and a is the radius of the minimum sphere that encloses whole antenna) did not intersect (d 2 2a), the problem was analyzed using SMT-AM in [9]. It is shown that, by supposing a uniform current phase distribution across the antenna, the coupling between two reciprocal ESAs can be approached by applying the addition and translation theorem on the TE10 and TM10 modes (the lowest-order spherical modes). Those two are represented by their radiation resistances R TE and R TM, respectively, R RAD = R TE + R TM, and the mode ratio is a = R TM /R TE . The assumption that there are no higher-mode interactions is shown not to be valid when two ESAs are very close to each other, but then the WPT efficiency is high even without proper receiver matching. The equivalent two-port network of two coupled ESAs is drawn accordingly, as in Figure 2. For simplicity, in the antenna reactance X A, we include an ideal (lossless) tuning reactance. ESAs come close to the concept of the minimum-scattering antenna (MSA) [16] (sometimes referred to as the equal-scattering antenna [17]). Such antennas have the property that, for the specific reactive termination at the antenna port, they do not scatter at all and thus become invisible for the transmitter. Whether or not the termination is open circuit as in the case of directly fed short dipoles, such an ESA is referred to as the canonical MSA. Hence, the two-antenna WPT system can be represented by a Z-matrix (i.e., by the scheme in Figure 2), and the ESAs’ free-space impedances can be applied [18]. In the case of four-coil systems, the inductively fed ESA can be approximated by an MSA that vanishes if terminated by a short circuit. In that case, a Y-matrix can be applied, resulting in dual relations to that of the Z-matrix with impedances exchanged with corresponding admittances. The power transmission coefficient S 21 is given by S 21 = 2 U 2 UG RL UG FIGURE 2. An equivalent circuit for WPT between two equal ESAs. 40 RG , RL (1) where U G and U 2 are the generator and the output voltage, respectively. R L is load terminating the receiver port, and RG = RL. AUGUST 2019 IEEE ANTENNAS & PROPAGATION MAGAZINE The reflection coefficient at the transmitter input port is given as S 11 = Z in - R L , Z in + R L (2) where Z in is the input impedance. Applying (1) to the scheme in Figure 2, one obtains S 21 = 2Z M R L 2 . (Z A + R L)2 - Z M 3 T = - 4 6(R A + R L)2 + Re (Z 2M)@ - 27 Im 2 (Z 2M) (R A + R L)2 . (6) (3) Z M is the mutual impedance between the antennas that, following the SMT-AM and applying the MSA concept, can be calculated as [9], [18] Z M = R A T, to the frequency tracking. Strictly speaking, there is no possibility that the passive WPT system can operate in resonance (Im (Z in) = 0) while the antennas are in self-resonance (X A = 0) unless the antenna separation is infinite (far field). The number of antenna reactance values that bring the system in resonance (i.e., the number of the system resonant frequencies) is determined by the discriminant T: (4) where T is the transmission coefficient of the spherical modes and it is a function of the electrical distance d/m between the antennas, their radiation efficiency, and the space orientation of the receiver relative to the transmitter. The complete relations for the transmission coefficient of the lowest-order spherical modes are given in “The Lowest-Order Spherical Modes Transmission Coefficient T.” MAXIMUM WPT PERFORMANCES To obtain the maximum WPT performances for a given scenario (i.e., to ensure the absorption of the maximum power by the load), ;S 21 ; must be maximized. With the load resistance fixed, the calculation of the maximum |S 21 | by (3) with respect to the antenna reactance X A leads to a determinantal equation in the form of a depressed cubic equation for the optimum antenna reactance given by 2 2 X 3A + 6(R A + R L)2 + Re (Z M )@ X A - Im (Z M ) (R A + R L) = 0. (5) It can be shown that the whole system is tuned (i.e., brought to resonance at the selected frequency) by adjusting the antenna reactance to one of the real solutions of (5). This is analogous When T 1 0, there is one single real solution to (5), and the resonant WPT system operates in an undercoupled regime. On the other hand, when R L is such that T 2 0, there are three real solutions. Then, the frequency splitting occurs, which characterizes the overcoupled zone of the receiver. In the overcoupled zone, there is one local minimum near X A = 0 (i.e., not far from ~ = ~ 0). Also, there are two maximums—the global one and the local one—each representing one PT mode. They are positioned on the left and right of the frequency axis relative to ~ = ~ 0 . The boundary between the two regions of the receiver (T = 0) is called the critical coupling, and it is characterized by ;S 21(~); with the global maximum and with one double extremum in the form of inflection point. At this point, the frequency splitting is still suppressed. When the system is tuned to the frecrit quency of the global maximum, then ;S 21 ; = ;S 21 ;, and the fixed load absorbs maximum power. The solutions to (5) are derived in “Tuning Reactances as Solutions to the Depressed Cubic Equation.” However, as R L can also be adjustable, there is an infinite number of (R L, X L) combinations that bring the system in resonance and provide peak WPT performances. Therefore, in the text that follows, two approaches for selecting the best possible (R L, X L) pairs that provide the maximum power are addressed. CONJUGATE MATCHING OPT The best combination (R OPT L , X L ) of the antenna reactance and the load resistance among all of the possible (R L, X L) pairs is the one that provides the maximum PT efficiency (PTE). PTE is the ratio of the power absorbed by the load at the THE LOWEST-ORDER SPHERICAL MODES TRANSMISSION COEFFICIENT T The transmission coefficient T of the lowest-order spherical modes between two equal minimum-scattering antennas (as in Figure 1) is derived in [9] as T = h RAD c Al10,10 + a 1 + a2 Bl10, 10 m . (S1) Al10, 10 and Bl10, 10 are translated and rotated spherical-mode coefficients between the lowest-order spherical transverse electric or transverse magnetic modes given by A10, 10 A10, 10e A10, 10o Al10, 10 ; E = cos i 1; E + sin i 1' cos z 1; E + sin z 1; E1, B 10, 10 B 10, 10e B 10, 10o Bl10, 10 (S2) IEEE ANTENNAS & PROPAGATION MAGAZINE AUGUST 2019 where A 10, 10 = P0 (cos i 0) h (02) ( bd ) + P 20 (cos i 0) h 2(2) ( bd ), (S3) B 10, 10 = 0, (S4) cos (2) A 11, 10eo = - 1 P 21 (cos i 0) z 0 h 2 ( bd ), 2 sin (S5) sin (2) B 11, 10eo = ! 3 P 11(cos i 0) z 0 h 1 ( bd ). 2 cos (S6) In (S3)–(S6) h (n2) are n-order spherical Hankel functions of the second kind, and P nm are the spherical Legendre polynomials. 41 TUNING REACTANCES AS SOLUTIONS TO THE DEPRESSED CUBIC EQUATION The determinantal (5) in the form X 3A + pX A + q = 0, where p = (R A + R L)2 + Re (Z M2 ) and q = -Im (Z 2M)(R A + R L) can be solved by deducing it to the quadratic equation, applying the substitution X A = x - (p/3x) . By solving the obtained quadratic equation in x followed by calculating X A(x) = X extr A , the first solution is obtained as 3 X extr A = q + 2 q - T +3 - 108 2 - T , 108 (19) receiver port and the power that enters the system through the transmitter antenna port. It can be written using (1) and (2) as PTE = |S 21 |2 . 1 - |S 11 |2 (7) The optimum antenna reactance (tuned antenna) and the optimum resistance for the maximum PT between two identical antennas separated by a distance can be derived by 2PTE/2X A = 0 as X OPT A Im (Z 2M) = . 2R A (8) The optimum resistance can be found by inserting (8) into (7), and then 2PTE/ 2R L |X A = XOPT = 0. It is given by A 2 2 R OPT = Re 6(Z OPT L A ) @ - Re (Z M) , (9) where Z OPT = R A + jX OPT A A . When the system is conjugate matched at the selected frequency, S11 = 0. And, according to (7), the maximum PTE is equal to the maximum |S 21 |2. Also, the system radiation loss is then at the minimum, as is the influence of the WPT system on living beings in its surroundings [19]. Inserting (9) into the Vieta’s formulas for (5), the other two solutions for the extremum |S 21 | in the case of R L = R OPT are L derived as X EXT = A 2 ) -Im (Z M )1 ! 4R A 1 - 32R 3A R A + R OPT L 3, 2 Im 2 (Z M ) (10) where the negative sign in front of the square root in (10) corresponds to the local minimum, whereas the positive sign corresponds to the local maximum of |S 21 |. For the load resistance determined by (9), the frequency of transmission can be set to each of these points (the maximum PTE, the local maximum |S 21 |, or the local minimum near the antenna resonance) by setting the antenna reactance at values given by (8) and (10), respectively. CRITICAL COUPLING The critical coupling method of adjustment must provide the crit critical mutual impedance Z M = Z M for the maximum |S 21 |, at which frequency splitting ceases to exist. Considering (5) and (6) 42 where T is given by (6). The other two solutions are obtained by using Vieta’s formulas, or by multiplying each term on the right side of (19) with a different complex solution to 3 -1 , one with - (1/2)(1 - j 3 ) and Y 0, depending the other with - (1/2)(1 + j 3 ). As in our case q = on the sign of T, the depressed cubic (5) can have either three single real solutions (T 2 0), one single and one double real solution (T = 0), or else one real and two conjugate complex solutions (T 1 0). at a given antenna separation, the adjustment is accomplished by satisfying the condition T = 0 for one single real solution of (5) representing the maximum |S 21 | settled at the operating frequency: first by the proper choice of the load resistance R L = R crit L that fulfills it and then by selecting the tuning reactance X A = X crit A that emerges as one single solution of (5). After some mathematical manipulations, one obtains 2 3 3 R crit Re 2 (Z M) @ - R A, L = 6 Im (Z M) - (11) X crit A (12) 3 = 3 4 Im (Z 2M) (R A + R crit L ). Additionally, there also exists one double real solution of (5) representing the inflection point of the |S 21 | characteristics, and crit the adjustment of the antenna reactance to X inf A = -(1/2) X A settles the operating frequency at the inflection point. The conditions for critical coupling can be established efficiently in the four-coil systems by controlling the couplings between the driving loop and the antenna body on both the transmitter and the receiver side, as shown in [8], followed by a fine tuning with the aid of (12). As a matter of fact, the analysis of the four-coil system would lead to a dual equation for critical susceptance Bcrit [i.e., with Z M exchanged with the mutual admittance YM in (12), and with R A and R crit L exchanged with conductivities G A = Re (1/ Z A) and G Lcrit, respectively.] Obviously, a theoretical possibility for the critical coupling exists within the near-field zone determined by R crit L $ 0. Following (4), this range can be expressed by 3 Im 2 ^T h - 3 Re 2 (T) $ 1. Note that the maximum electrical distance limit of the critical coupling method depends on the antennas’ radiation efficiency, and it varies with the receiver position and orientation relative to the transmitter. Inside this area, any terminating resistance that falls below the value of the critical resistance causes the overcoupling of the resonant WPT system. One may also note that, because Y ~ 0, the coupling zone definition proposed of tuning at ~ = herein diverges from those given in [14] and [15]. Furthermore, by inserting (11), (12), and (4) into (3), the critically coupled |S 21 | can be expressed in the form |S 21 |crit = 1/h RAD {[Im 2/3 (T) - Re 2/3 ^T h]3/2 - 1} G (bd), where G is not a function of the radiation efficiency. The critical transmission efficiency at some near-field distance from the transmitter is inversely proportional to -h RAD . The radiation efficiency of the knee point 2|S 21 |crit / 2h RAD = 1 rises with the antenna AUGUST 2019 IEEE ANTENNAS & PROPAGATION MAGAZINE separation and frequency as h knee RAD = G (bd). Thus, to reach high critical coupling performances (above the knee value) at a given electrical distance, the WPT antenna design should aim for the radiation efficiency h RAD 2 h knee RAD . For example, in the case of the antenna separation of 0.1 m, this value for the coaxially arranged nonrotated antennas is h knee RAD = 50%. However, for a given radius of the Chu sphere a 1 m /4r, as R TM + f 2, R TE + f 4 and the antenna loss resistance R LOSS + f , the increment of the radiation efficiency is directly opposed to lowering the frequency of transmission. The necessary tradeoff is investigated in [20], whereas some other aspects of WPT antenna characterization regarding matching network are analyzed in [21] and [22]. very low at 0.05 m). Moreover, it is found that this discrepancy is due to the error in the imaginary part of the mutual impedance exclusively. The calculation of the real part gives accurate results regardless of the size of the gap (d 2 2a) between the loops. From the results in Figure 4, although the real component Re(Z M) has a small influence at low antenna separations, it has a crucial role at midranges. When a precise tuning is executed, the critical |S crit 21 | decays much slower with distance until the knee point than in the case of ~ = ~ 0 (X A = 0). Even at the IEEE ANTENNAS & PROPAGATION MAGAZINE AUGUST 2019 S21 NUMERICAL EXAMPLE AND DISCUSSION 0.2 Critical Coupling Versus Conjugate Matching 1 Coaxial PEC 0.9 Loops D = 0.1 λ 0.8 at 13.56 MHz 0.7 S21 Take, for example, the scenario where a conjugate-matched mobile receiver is entering the overcritical zone, determined by the condition that the term under the square root in (10) is lower than zero, which is analogous to T 1 0 (R L = R OPT 1 R crit L L ). Let the antennas be equal perfectly conducting (PEC) loops of 20-cm diameter, each connected serially to the capacitor C = 206.6 pF that settles the antennas’ self-resonance near an industry, science, medicine frequency of 13.56 MHz. The operating frequency is selected to be exactly 13.56 MHz. The magnitude of S 21 [calculated by SMT-AM using (3), (4) and (13)] for the case of the nonrotated loops in a coaxial position is given in Figure 3. These results are compared to the ones obtained by a full-wave model using FEKO, where the antennas are concluded with the Linville impedance Z Linville in the case of the conjugate matching. crit When the receiver arrives on the boundary Z M = Z M OPT crit (T = 0), then R L = R L . Besides the existing maximum PTE point on the operating frequency f = 13.56 MHz tuned EXT = X crit by X A = X OPT = A A , the inflection point X A = X A OPT inf -(1/2) X A = X A appears at the opposite side on the frequency axis relative to ~ = ~ 0 . This case is illustrated in Figure 3(a). When the receiver finds itself inside the overcoupled zone, the WPT system can be maintained at optimum performance by continuing to adjust X A and R L to the conjugate-matched condition with (8) and (9). Alternatively, it can be turned back to the critical conditions by adjusting R L and X A with (11) and (12). This is illustrated in Figure 3(b). In the former case, the frequency splitting exists while the peaks of the |S 21 | are sharp. The latter choice ensures a larger fractional bandwidth and suppressed frequency splitting but at the expense of a somewhat smaller PTE than in the former case. The maximum WPT performances obtained by the two methods are compared in Figure 4. While deriving the results, we separately tested the discrepancies of mutual impedance produced by neglecting highermodes interactions in the SMT-AM approximation by the full-wave model. Note that the agreement between the SMTAM approximative model and the results of the full-wave model in Figures 3 and 4 are quite satisfactory. The conclusion from [9] about the rise of error when the antenna separation becomes less than 0.1 m is confirmed here as well. However, unless the separations are very small, this error is acceptable (e.g., it is still Conjugate Matching at Critical Distance 1 0.9 Coaxial PEC Loops D = 0.14 λ at 13.56 MHz 0.8 0.7 SMT-AM 0.6 ω = ω0 f = 13.56 MHz 0.5 FEKO 0.4 0.3 0.2 0.1 0 –0.6 –0.5 –0.4 –0.3 –0.2 –0.1 0 0.1 Frequency Offset (kHz) (a) 0.6 0.5 0.4 0.3 0.2 0.1 0 –0.6 –0.5 –0.4 –0.3 –0.2 –0.1 0 Frequency Offset (kHz) 0.1 0.2 SMT-AM, Conjugate Matching at 13.56 MHz ω = ω0, Conjugate Matching f = 13.56 MHz, Conjugate Matching FEKO, Conjugate Matching at 13.56 MHz SMT-AM, Critical Coupling at 13.56 MHz ω = ω0, Critical Coupling f = 13.56 MHz, Critical Coupling FEKO, Critical Coupling (b) FIGURE 3. The transmission coefficient ; S 21 ; versus frequency f for WPT between PEC loops separated by d and conjugate matched for f = 13.56 MHz. (a) The receiver is very near the critical coupling border, and the differences between the two characteristics are within the thickness of the curve, and (b) the conjugate-matched WPT system is overcoupled. 43 separations larger than the knee one, the tuned system reaches higher values of |S crit 21 | than the detuned one as the knee of the former is settled at a significantly larger antenna separation than the one of the latter. Although, according to (5) and (6), the real component is not the cause of the frequency splitting [it is caused solely by the existence of Im (Z M)], it has a profound influence on the S 21(~) characteristic, displacing the position of the minimum from the antenna’s self-resonant frequency ~ 0 in the overcritical zone and producing the inflection point in terms of the critical coupling. Moreover, the real component is responsible for the frequency splitting in the conjugate-matched condition. Without its influence, the conjugate-matched WPT system would be locked in an undercoupled regime regardless of the antenna separation, as previously noted in [15]. This can be deduced from the comparison of (9) and (11) by neglecting Re (Z M). Maximum WPT Performances 1 AN EXPERIMENTAL OBSERVATION OF FREQUENCYSPLITTING PHENOMENON 0.9 Max (S21) 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.02 0.05 0.1 0.15 0.2 Antenna Separation (λ) 0.25 SMT-AM; RL = RLOPT, XA = X AOPT SMT-AM; RL = RLOPT, XA = X AEXT FEKO; ZL = Z Linville SMT-AM; RL = RLCRIT, XA = XACRIT FEKO, Critical Impedance SMT-AM; RL = RLCRIT, XA = 0 FIGURE 4. The maximum WPT performances between two coaxial PEC loops tuned at 13.56 MHz versus distance. d = 2.5 cm VNA ω = ω0 0.9 d = 5 cm VNA ω = ω0 0.9 d = 7.5 cm VNA ω = ω0 0.9 0.8 0.8 0.8 0.7 0.7 0.7 0.6 0.6 0.6 0.5 0.4 1 S21 1 S21 S21 1 0.5 0.4 0.5 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0 10 15 20 Frequency (MHz) 0 10 15 20 Frequency (MHz) 0 10 15 20 Frequency (MHz) (a) (b) (c) FIGURE 5. The measured magnitude of S 21 between two coaxial electrically small two-turn HF cylindrical coils terminated with 50 X: (a) Taken at 2.5-cm separation, (b) measured very close to the critical distance, and (c) typical for the system operating in terms of undercritical coupling. 44 To validate the observed results further, we selected the measurement setup similar to the one of the Tesla’s earliest experiments with resonant coupling [14], where WPT measurements between two coaxially positioned electrically small coils were conducted by a vector network analyzer (VNA). For that purpose, two turns of approximately equal lengths of 2-mm isolated copper wire with a 10-cm radius and a 1-cm height were wound for each coil. The coils were made to antiresonate on approximately equal frequencies near 34 MHz. The self-resonant frequency of each coil was adjusted to 13.56 MHz by a variable capacitor C (C . 55 pF). The WPT on 4,001 VNA frequency points between 5 and 35 MHz was measured at various antenna separations between 2.5 and 15 cm in steps of 2.5 cm. The VNA ports’ impedance was set to R L = 50 X, and a 50-X 1:1 balun was applied on each side of the WPT link. The total insertion loss was estimated to be approximately 1 dB. Typical examples of |S 21 | frequency characteristics are depicted in Figure 5. Each is the average of 16 consecutive VNA measurements at the distance. Figure 5(a) was taken at 2.5-cm separation. Two resonant peaks are observed, and the system operates in an overcoupled regime. Note that the position of the local minimum is displaced from the antennas’ self-resonant frequency. Figure 5(b) was measured very close to the critical distance, where the critical resistance is approximately 50 Ω. The resonant frequency (i.e., that of the maximum) is 14.42 MHz, at which the received power is 0.5% higher relative to the one at ~ = ~ 0 . Figure 5(c) is typical for the system operating in terms of undercritical coupling. One clear peak very near the antennas’ self-resonant frequency can be observed. Results of the measurements are summarized in Figure 6. Note that as the antenna separation becomes larger, all three monitored frequencies converge to the antenna-self-resonant frequency. However, immediately after the 5-cm separation, only one resonant frequency exists, and the frequency-splitting phenomenon ceases. This is followed by a rapid decrease of the received power. Thus, in these simple measurements of WPT between two real coils, one can notice all of the effects that confirm the conclusions from the “Maximum WPT Performances” section. CONCLUSIONS Despite the fact that the SMT-AM approximative model is not capable of making a completely accurate prediction of the performances of the near-field antenna system at very AUGUST 2019 IEEE ANTENNAS & PROPAGATION MAGAZINE VNA Measurements 19 18 0.8 17 S21 0.7 16 15 0.6 14 0.5 13 0.4 0.3 1.5 2.5 12 5 7.5 10 12.5 Coil Separation (cm) Frequency of Extremum (MHz) 0.9 11 15 Global Maximum S21 S21 (f = 13.56 MHz) Local Maximum S21 Global Maximum Frequency f = 13.56 MHz Local Maximum Frequency Local Minimum Frequency FIGURE 6. The measured magnitudes of S 21 (blue) and the extrema frequencies (red) versus coil separation. low electrical distances, it nevertheless reveals the true potential of resonant WPT technology. Furthermore, by applying the antenna theory, the frequency-splitting phenomenon can be predicted and controlled. The two methods for obtaining maximum WPT performances are presented and compared. The method of critical coupling is more convenient to be used at low ranges because of a larger fractional bandwidth. Nonetheless, as a result of the SMT-AM-based analysis, it turns out that the critically coupled WPT system is the most effective when it is tuned properly, which means that the antennas (including the tuning reactance) cannot be in self-resonance at the operating frequency. To reach the largest distances of transmission possible (relative to the given frequency), the conjugate-matching approach must be applied, while the used antennas’ loss resistance should be as low as possible. ACKNOWLEDGMENT This work has been supported in part by the Croatian Science Foundation under the project “Internet of Things: Research and Applications,” UIP-2017-05-4206. AUTHOR INFORMATION Zoran Blažević (zblaz@fesb.hr) is with the Department of Electrical and Computer Engineering, University of Split, Croatia. His current research interests include resonant wireless power transfer, radio-frequency identification systems, antennas, and radio-channel modeling. He is a Member of the IEEE. Maja Škiljo (msekelja@fesb.hr) is with the Department of Electrical and Computer Engineering, University of Split, Croatia. His current research interests include resonant wire- IEEE ANTENNAS & PROPAGATION MAGAZINE AUGUST 2019 less power transfer, ground-penetrating radars, radio-frequency identification systems, and antennas. He is a Member of the IEEE. REFERENCES [1] H. A. Wheeler, “Fundamental limitations of small antennas,” Proc. IRE, vol. 35, no. 12, pp. 1479–1484, 1947. doi: 10.1109/JRPROC.1947.226199. [2] L. J. Chu, “Physical limitations of omni-directional antennas,” J. Appl. Phys., vol. 19, no. 12, pp. 1163–1175, 1948. doi: 10.1063/1.1715038. [3] N. 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