Comprehensive Analysis and Measurement of Frequency-Tuned and Impedance-Tuned Wireless Non-Radiative Power Transfer Systems
Jason D. Heebl, Erin M. Thomas, Robert P. Penno, Anthony Grbic
11 Comprehensive Analysis and Measurement ofFrequency-Tuned and Impedance-Tuned WirelessNon-Radiative Power Transfer Systems
Jason D. Heebl,
Member, IEEE,
Erin M. Thomas,
Member, IEEE,
Robert P. Penno,
Senior Member, IEEE, andAnthony Grbic,
Member, IEEE
Abstract —This paper theoretically and experimentally in-vestigates frequency-tuned and impedance-tuned wireless non-radiative power transfer (WNPT) systems. Closed-form expres-sions for the efficiencies of both systems, as a function offrequency and system (circuit) parameters, are presented. Inthe frequency-tuned system, the operating frequency is adjustedto compensate for changes in mutual inductance that occur forvariations of transmitter and receiver loop positions. Frequency-tuning is employed for a range of distances over which the loopsare strongly coupled. In contrast, the impedance-tuned systememploys varactor-based matching networks to compensate forchanges in mutual inductance and achieve a simultaneous conju-gate impedance match over a range of distances. The frequency-tuned system is simpler to implement, while the impedance-tunedsystem is more complex but can achieve higher efficiencies. Bothof the experimental WNPT systems studied employ resonantshielded loops as transmitting and receiving devices.
I. I
NTRODUCTION A TTEMPTS to transfer power wirelessly have generallyrelied on transmitting and receiving radiated power [1],[2]. In addition to these far-field systems, wireless non-radiative power transfer (WNPT) via quasi-static magneticfields has garnered significant interest in recent years. Wirelesspower transfer using conventional magnetic induction as wellas resonant magnetic induction for increased range (mid-range distances) are actively being pursued in academia andindustry [3], [4], [5], [6], [7], [8]. WNPT by resonant magneticinduction can trace its roots to the work of Nikola Teslain the early twentieth century [9], [10], [11]. Much of itsrecent popularity, however, can be attributed to work at MITin 2007, where power was transferred between magnetically-coupled resonant coils [12]. The transmitting and receivingcoils formed a transformer, and resonance was used to improvepower transfer efficiency. The transformer in such a schemeis non-ideal given that the loop inductances are finite, thecoefficient of coupling is notably less than unity, and the loopsare lossy [13].
J. D. Heebl is with the Space and Missiles Systems Center, Satellite ControlNetwork, Los Angeles Air Force Base, El Segundo, CA 90245 USA (email:[email protected]).E. M. Thomas is with SRI International in the Communications, Radar andSensing Lab.R. P. Penno is with the Department of Electrical and Computer Engineering,University of Dayton, Dayton, OH 45469 USA.A. Grbic is with the Radiation Laboratory, Department of ElectricalEngineering and Computer Science, University of Michigan, Ann Arbor, MI48109-2122 USA (email: [email protected]).
In [14], [15], WNPT utilizing two coupled shielded-loopresonators, instead of resonant coils, was reported. Closed-form expressions for the loops’ circuit parameters, input andoutput impedances, and efficiency were developed from acircuit model of the system. Fixed matching networks wereused to optimize efficiency for specific distances, that isparticular values of mutual inductance. Experimental resultsshowed close agreement with theoretical predictions of optimalefficiency.In recent years, there have been attempts to develop im-proved WNPT systems with increased efficiency for a rangeof distances, or equivalently mutual coupling. Techniques suchas tuning the frequency of operation, employing impedance-matching networks [16], [17], [18], transponder configurations[19], [20], coil arrays [21], and the use of a superlens toenhance coupling [22] have been explored. In [23], [24],[25], constant efficiencies were maintained by tuning a WNPTsystem’s operating frequency while the transmit and receiveresonators were strongly coupled. In [26], the performanceof such a frequency-tuned system was compared in simula-tion with an impedance-matched system. However, analyticalmodels explaining frequency-tuning with fixed source and loadimpedances have not been developed to date. In addition,experiments that satisfactorily compare the full operation ofboth approaches have not been reported.In this paper, we present an analytical investigation of bothfrequency-tuned and impedance-matched WNPT systems. Thegoverning equations of a WNPT system are investigated understrong, critical, and weak coupling between transmit andreceive resonators. Conditions for optimal frequency-tunedoperation, with fixed source and load impedances, are pre-sented. The expressions analytically demonstrate why constantefficiencies can be attained in a strongly-coupled system.In addition, it is analytically shown that impedance-matchedsystems are capable of providing optimal power transfer forall mutual inductance values.Finally, the analytical findings are experimentally validatedfor a frequency-tuned and impedance-tuned WNPT system.Frequency-tuning is demonstrated for three fixed matchingnetworks. Each matching network provides a simultaneousconjugate match to source and load impedances at a fixeddistance. The system is frequency-tuned for spacings shorterthan this fixed distance. In addition, varactor-based matchingnetworks are used to demonstrate an impedance-tuned WNPTsystem. It is shown that such a system exhibits maximum a r X i v : . [ c s . OH ] J a n efficiency. The maximum efficiencies achieved using theimpedance-tuned WNPT system are compared to those of thefrequency-tuned system.II. C IRCUIT M ODEL
The electrically-small loops of WNPT systems are typicallymodeled using series resonator circuits [27]. Fig. 1 showsthe lumped-element circuit model of a WNPT system basedon resonant magnetic coupling between two electrically smallloops [14], [15]. The variable M = M denotes the mutualinductance between two loops. Fig. 1. Equivalent circuit for two inductively coupled resonant loopsemployed in a wireless non-radiative power transfer system.
From this circuit model, the input and output impedancescan be found: Z in = jωL − jωC + R + ( ωM ) R + Z L + jωL − jωC (1) Z out = jωL − jωC + R + ( ωM ) R + Z S + jωL − jωC . (2)For simplicity, it will be assumed that the loops depicted inFig. 1 are identical: R = R = R, L = L = L, C = C = C. (3)The efficiency, η , will be defined as the ratio of powerdelivered to the load ( P L ) to power available from the source( P AV S ) as follows: η = P L P AV S = (cid:16) − | Γ in | (cid:17) η (cid:48) (4) = (cid:16) − | Γ in | (cid:17) × ( ωM ) R L R (cid:104) ( R + R L ) + (cid:0) ωL − ωC (cid:1) (cid:105) + ( ωM ) ( R + R L ) , (5) where Γ in = Z in − Z S Z in + Z S is the input reflection coefficient. Thepower available from the source is the maximum power thatcan be delivered to the network: P AV S = | V S | (cid:60) e { Z S } . It doesnot depend on the input impedance, Z in , of the network [28].The reflected power at the input is proportional to | Γ in | givenby (6).When (6) is substituted into (5), the efficiency equationsimplifies to the expression shown in (7). The efficiency, givenby (5),(7), is the transducer gain of the two-port network whenthe source and load impedances are identical, real quantities[29]. The quantity η (cid:48) represents the power gain.At the resonant frequency of the isolated loops ( ω =1 / √ LC ), the efficiency (5) reduces to: η = (cid:16) − | Γ in | (cid:17) η (cid:48) (8) = (cid:16) − | Γ in | (cid:17) × ( ω M ) R L ( R + R L ) (cid:104) R ( R + R L ) + ( ω M ) (cid:105) . (9)In addition, at ω , the square of the reflection coefficientmagnitude reduces from (6) to: | Γ in | = (cid:34) R − R L + ( ω M ) ( R + R L ) + ( ω M ) (cid:35) . (10)Therefore, the complete expression for efficiency at resonance ω simplifies to: η = (cid:16) − | Γ in | (cid:17) η (cid:48) = (cid:34) R L ( ω M )( R + R L ) + ( ω M ) (cid:35) . (11)These expressions account for both dissipation and reflections.All the equations reported in this section were derivedfrom from the circuit shown in Fig. 1, under the conditionsspecified. They are in agreement recent papers on the topic [8],[30], [31], [32]. The following sections analytically discusstuning techniques to improve efficiency, and then validate theexpressions through experimentation.III. A NALYSIS OF T UNING T ECHNIQUES
Coupled-mode theory asserts that there are three regions ofoperation for two coupled resonators [12], [13]. These regionsare based on the strength of the mutual coupling between thetwo loop resonators relative to losses: • strong coupling: ( ωM ) > ( R + R L ) • critical coupling: ( ωM ) = ( R + R L ) • weak coupling: ( ωM ) < ( R + R L ) | Γ in | = (cid:104) ( R − R L ) ( R + R L ) − (cid:0) ωL − ωC (cid:1) + ( ωM ) (cid:105) + 4 R (cid:0) ωL − ωC (cid:1) (cid:104) ( R + R L ) − (cid:0) ωL − ωC (cid:1) + ( ωM ) (cid:105) + 4 ( R + R L ) (cid:0) ωL − ωC (cid:1) (6) η = (cid:16) − | Γ in | (cid:17) η (cid:48) = [2 R L ( ωM )] (cid:104) ( R + R L ) − (cid:0) ωL − ωC (cid:1) + ( ωM ) (cid:105) + 4 ( R + R L ) (cid:0) ωL − ωC (cid:1) (7) To obtain the resonances of the WNPT system, the imaginarypart of the input impedance (1) is set to zero: (cid:0) ω LC − (cid:1) × (cid:34) ( R + R L ) + (cid:18) ωL − ωC (cid:19) − ( ωM ) (cid:35) . (12)Equation (12) identifies three resonances of the coupled loops:conditions under which imag( Z in ) = 0 . The first term definesthe resonant frequency, ω = √ LC , of the loops in isolation.At this frequency, ω , the currents in the two loops are 90degrees out of phase. The expression in square parenthesesdefines the even and odd mode resonances of the coupled loopsystem: (cid:18) ωL − ωC (cid:19) = ( ωM ) − ( R + R L ) . (13)The currents in the two loops are in phase for the even mode,while 180 degrees out of phase for the odd mode. The even andodd mode resonances separate in frequency as the couplingbetween the two loops becomes stronger. An approximateexpression for the frequency separation between these twomodes can be found by setting ω = ω ± ∆ ω and solving(13) for ∆ ω : ∆ ω ≈ (cid:115)(cid:18) ω o M L (cid:19) − (cid:18) R + R L L (cid:19) . (14)When the system is strongly coupled, ∆ ω is positive and real,producing two resonant frequencies: an even and odd mode.At critical coupling, ∆ ω = 0 and the even and odd modefrequencies merge to ω . When the system is weakly coupled, ω = ω ± ∆ ω = ω ± j (cid:0) R + R L L (cid:1) . As a result, the resonantfrequency remains ω . A. Frequency-Tuned WNPT Systems
Now let us consider frequency-tuned WNPT systems.In a frequency-tuned WNPT system, the source and loadimpedances are fixed. Generally, the coupled loops are con-jugately matched to both source and load impedances at afixed distance, and the system is frequency-tuned for differentdistances within this range. The ratio of currents in twoidentical loops (see Fig. 1) can be written as: i = − jωM j (cid:0) ωL − ωC (cid:1) + ( R + R L ) i . (15)According to (13), the even and odd mode frequencies aregiven by: (cid:18) ωL − ωC (cid:19) = ± (cid:113) ( ωM ) + ( R + R L ) . (16)The + and − solutions identify the modes above and belowthe resonant frequency ( ω ) of the isolated loops, respectively.Substituting (16) into (15), yields the following ratio ofcurrents: i = i − j R + R L ωM ± (cid:104) − ( R + R L ) ( ωM ) (cid:105) . (17) It is clear from the expression above that the magnitude of thedenominator of (17) is equal to 1, and | i | = | i | . Furthermore,an analysis of the phase of (17) yields: φ + = R + R L ωM (18) φ − = − π − R + R L ωM . (19)Therefore, the higher frequency mode ( φ + ) is the evenmode, and the lower frequency mode ( φ − ) is the odd mode.It should be noted that these analytical findings were verifiedthrough circuit simulation in Agilent Advanced Design System(ADS).Frequency-tuned systems maintain the frequency of oper-ation at either the even or odd mode resonance in order tosustain a constant efficiency for distances within the stronglycoupled region [23], [24], [25], [26]. In these systems, thefrequency is tuned to maintain (13), since M changes withdistance. The odd mode is generally preferred, since thecurrents in the loops are out of phase and their radiationcancels in the far field.
1) Strongly Coupled WNPT System:
Operating at an evenor odd mode frequency means that the system operates offof the resonant frequency, ω , of the isolated loops. Atan even or odd mode frequency, there exists an impedancemismatch and some power is reflected back to the source. Itcan be shown that the reflected power, proportional to (6), isminimized ( δ | Γ in | δω = 0 ) for a fixed source/load impedancewhen operating at the even or odd mode resonance definedby (13). Therefore, operating at either the even or odd modefrequency guarantees the highest efficiency for a frequency-tuned WNPT system.The reflection coefficient at the even or odd mode frequencycan be found by substituting (13) into (6), | Γ in | = (cid:18) RR + R L (cid:19) . (20)under the assumption that ( ωM ) ≥ ( R + R L ) . Substitut-ing (13) and (20) into the expression for efficiency (5) yields: η = (cid:16) − | Γ in | (cid:17) η (cid:48) (21) = (cid:34) − (cid:18) RR + R L (cid:19) (cid:35) (cid:18) R L R + R L (cid:19) (22) = (cid:18) R L R + R L (cid:19) . (23)Therefore, a constant efficiency is achieved by maintainingoperation at either the even or odd mode resonant frequencyof a strongly coupled WNPT system. The efficiency is not afunction of distance (or equivalently M ), provided that (13)is upheld by properly tuning the frequency ω and ensuringthat the loops are strongly coupled: ( ωM ) ≥ ( R + R L ) .Further, since Γ in is constant, the input impedance is alsoconstant for a frequency-tuned WNPT system.
2) Critically and Weakly Coupled WNPT System:
Sincecritically and weakly coupled WNPT systems do not exhibiteven or odd mode resonances, optimal efficiency is maintainedby operating at ω . The equations for the reflection coefficientand total efficiency at critical and weak coupling are givenby (10) and (11), respectively. Note that at critical coupling, ( ωM ) = ( R + R L ) , and (11) reduces to (23). Thus, asmooth transition in efficiency occurs at critical coupling: theboundary between weak and strong coupling.Finally, it should be noted that for fixed source and loadimpedances and operation at frequency ω , the efficiencypeaks at the distance (or equivalently the M value) corre-sponding to critical coupling. In other words, δηδM = 0 when ( ωM ) = ( R + R L ) for a fixed source and load impedance. B. Impedance-Tuned WNPT Systems
Next, let’s explore a WNPT system that is simultaneously,conjugately matched to both source and load impedances fora specific distance of operation. Under these conditions, theoptimal source and load impedances, Z in = Z ∗ S and Z out = Z ∗ L , become [14]: Z S = j (cid:18) ωC − ωL (cid:19) + (cid:114) R + R R ( ωM ) (24a) Z L = j (cid:18) ωC − ωL (cid:19) + (cid:114) R + R R ( ωM ) . (24b)At the resonant frequency ω , these expressions reduce to: R S = (cid:114) R + R R ( ω M ) (25a) R L = (cid:114) R + R R ( ω M ) . (25b)For identical loops, given by (3), the optimal source and loadresistances at ω become: R LS = R L = R S = (cid:113) R + ( ω M ) . (26)Equation (26) defines an optimal value of R L = R S at a givenmutual inductance value (distance). If the system operates at ω and (26) is satisfied, the system exhibits the maximumpossible efficiency: η max = η (cid:48) = (cid:20) ω M R + R LS (cid:21) (27a) = (cid:34) ω M R + (cid:112) R + ( ω M ) (cid:35) . (27b)This expression for maximum efficiency is obtained by sub-stituting (26) into either (9) or (11). Equation (27b) is theavailable gain ( G A ) at the resonant frequency of the loops( ω ) [29].The expression for the optimal load resistance (26) can berewritten as R L − R = ( ωM ) and directly compared tothe critical coupling condition: ( R + R L ) = ( ωM ) . Giventhat R L − R < ( R + R L ) , a conjugately matched WNPTsystem is always weakly coupled. Since it is weakly coupled,the denominator of (27b) is always less the numerator. In otherwords, the efficiency is less than unity. C. Discussion
Subsection A showed that frequency tuning allows a con-stant efficiency to be maintained for distances within thestrongly coupled range of operation. A higher source/loadresistance results in a higher efficiency but also a shorter rangeof distances over which the system is strongly coupled. Inother words, a higher efficiency can be maintained for close-in distances at the expense of a shorter range. Subsection Bshowed that maintaining a simultaneous conjugate match tosource and load impedance through impedance tuning resultsin a weakly coupled system. Nonetheless, such an impedancematching scheme results in maximum possible efficiency. Tosummarize, a frequency-tuned WNPT system is simple toimplement. An impedance-tuned system is more complex butcan achieve higher efficiencies.
Strong Critical WeakCoupling Coupling Coupling | Γ | (cid:16) RR + R L (cid:17) Expressions (cid:104) R − R L +( ωM ) ( R + R L ) +( ωM ) (cid:105) Frequency are equalTuned η (cid:16) R L R + R L (cid:17) Expressions (cid:104) R L ( ωM )( R + R L ) +( ωM ) (cid:105) are equalImpedance-Tuned η = (cid:20) ω M R + √ R +( ω M ) (cid:21) at ω TABLE IS
UMMARY OF THE EXPRESSIONS FOR | Γ in | , AND EFFICIENCY η , EXPERIENCED UNDER STRONG , CRITICAL , AND WEAK COUPLING . T
HEEXPRESSIONS FOR FREQUENCY - TUNING ARE DERIVED IN SECTION
III-A,
AND THOSE FOR IMPEDANCE - TUNING ARE DERIVED IN SECTION
III-B.
IV. A
NALYSIS OF C OUPLED S HIELDED L OOP R ESONATORS
Resonant shielded loops will be used to experimentallyinvestigate frequency-tuned and impedance-tuned WNPT. Ashielded loop is a coaxial, electrically-small loop antennawith a primarily magnetic response [33], [34]. The centralconductor is left open circuited at the termination point of theloop to provide a resonance (see Fig. 2a). A shielded loop canbe constructed from a semirigid coaxial cable by removing asmall portion of the outer conductor at a point πr from thefeed point, where r is the radius of the loop. This split inthe outer conductor provides a current path which allows theopen-circuited stub to be in series with the loop inductance.Currents supported by the resonant shielded loop are shownwith arrows in Fig. 2a.Due to the skin depth, the currents on the outer surfaceof the loops are isolated from those within the coaxial cablecomprising the loops [35]. As a result, the resonant shieldedloops can be broken down into a coaxial “feed” element, a loopinductance, and an open circuit transmission line. These threecomponents are depicted in Fig. 2b. Integrating these elementsinto the basic circuit model of Fig. 1 yields the model shownin Fig. 2c for magnetically coupled resonant shielded loops.A complete network representation for the circuit shown inFig. 2c can be derived by cascading the ABCD (transmission) Fig. 2. (a) Cross section of a shielded loop resonator. (b) Circuit model ofan isolated, resonant shielded loop. (c) Circuit model for two magneticallycoupled resonant shielded loops. parameters of the individual elements as follows: (cid:20) cos βl jZ o sin βl jZ o sin βl cos βl (cid:21) (cid:34) Z Z Z + Z + Z Z Z Z Z Z (cid:35) ∗ (cid:20) cos βl jZ o sin βl jZ o sin βl cos βl (cid:21) . (28)The first and last matrices in (28) represent the feedlines ofthe source and load loops, respectively. The central matrixrepresents the coupled loops shown in Fig. 2c [29] withvariables Z , Z , and Z defined: Z = Z − Z = R + 1 jωC + jω ( L − M ) (29a) Z = Z − Z = R + 1 jωC + jω ( L − M ) (29b) Z = Z = jωM . (29c)The matrices in (28) are multiplied to obtain the completetransmission matrix for the system. This complete ABCD-matrix is then converted to an impedance matrix [29] and usedto derive a T-equivalent circuit for the system, shown in Fig.3. The optimal source and load impedances are then found bysetting Z S = Z in ∗ and Z L = Z out ∗ , respectively.In [14], two shielded-loop resonators were constructed andare used in this experimental study. The experimentally ex-tracted parameters of the loops are given in Table II. Theoptimal Z S and Z L values were computed as a function ofdistance, or equivalently M (see Fig. 11). Fig. 3. General T-network representing the magnetically coupled, resonantshielded loops. The elements are expressed in terms of the impedancematrix elements ( Z , Z , Z , Z ) of the coupled loops. Source and loadmatching networks are shown connecting the loops to the 50 Ω signal sourceand power meter. Loop 1 Loop 2L ( µH ) ( pF ) Ω ) 0.23 0.2Feedline Length (cm) 38.5 39.5Radius (cm) 10.7 10.7TABLE IIC IRCUIT MODEL PARAMETERS FOR TWO SHIELDED - LOOP RESONATORS [14].
Capacitive L-section matching networks were designed tomatch the
50 Ω load and source to the coupled, shielded-loop system, as shown in Fig. 3. These fixed networks weredesigned for optimal power transfer at specific distances.The mutual inductance M was computed as a function ofdistance for the two axially aligned electrically small loopsusing expressions for filamentary current loops given in [36].The optimal capacitor values for the matching networks areplotted with respect to distance in Fig. 4a. Optimal impedancevalues for Z S and Z L at distances of 20, 35, and 50 cm areshown in Table III. Matching networks optimized for thesethree distances are referenced throughout this work. Fig. 4. (a) Plot showing the capacitance values for the L-section matchingnetworks needed to conjugately match the loops to 50 Ω . (b) Schematic ofthe capacitive L-section matching networks.Distance (cm) Z S (Ω) Z L (Ω) . − . j . − . j . − . j . − . j . − . j . − . j TABLE IIIC
ALCULATED OPTIMAL IMPEDANCE VALUES AT THREE DISCRETEDISTANCES FOR THE SHIELDED LOOP
WNPT
SYSTEM IN [14].
V. A N E XPERIMENTAL F REQUENCY -T UNED
WNPTS
YSTEM E MPLOYING R ESONANT S HIELDED L OOPS
In this section, the analysis from section III-A is validatedexperimentally. The shielded loop resonators and L-sectionmatching networks from [14], that conjugately match the cou-pled loops to
50 Ω source and load impedances for distancesof 20, 35, and 50 cm (values given in Fig. 4a), were tested overa range of distances and frequencies. The loops were placedon an automated, 3-axis translation stage and the two-portscattering parameters were measured for the coupled, shieldedloops (with impedance-matching networks in place) using anetwork analyzer (Hewlett Packard 3753D). The insertion lossassociated with the fixed matching networks over the rangeof frequencies needed for frequency tuning ( ω ± ∆ ω ) isnegligible. Therefore, the efficiency is expressed directly asmeasured | S | .For each pair of matching networks, Fig. 5 displays theexperimental efficiency ( | S | ) as a function of frequency anddistance. The volume encompassed by the dotted lines denotesthe strongly coupled region corresponding to each matchingnetwork. Note that similar characteristics were reported in[23] for a single load. By testing several different matchingnetworks, one can truly see the effect that the static-load hason the overall efficiency of the system. Fig. 5. Experimental efficiency ( | S | ) vs. distance and frequency fortwo magnetically coupled, resonant shielded loops. The coupled loops wereimpedance-matched to the 50 Ω source and load using matching networksdesigned for (a) 20 cm, (b) 35 cm, and (c) 50 cm loop separations. Theseparation into even and mode frequencies is evident as the two loops becomestrongly coupled. Dotted lines denote the regions of strong coupling. The experimental data shown in Fig. 5 was used to plot theefficiency curves of Fig. 6. Fig. 6a shows the experimentalefficiency at the resonant frequency ω versus distance, andcompares it to theory. The theoretical curves were generatedusing (11), the loop parameters given in Table II, and thesource and load impedances for the three distances givenby (25). A loss of efficiency due to reflections occurs fordistances less than the critical coupling point. Fig. 6b showsthe experimental efficiency when frequency tuning is em-ployed, and compares it to theory. The theoretical curves were generated using (23) for distances of strong coupling, and (11)for distances of critical and weak coupling. For critical andweak coupling distances, the system operates at the resonantfrequency ω . Once again, the actual loop parameters givenin Table II, as well as the source and load impedances givenby (25) for the three distances, were used in the theoreticalcalculations. Table IV compares the experimental efficienciesfor strong coupling to the theoretical values given by (23).For comparison purposes, Figs. 6a and 6b also show themaximum possible efficiency vs. distance given by (27b),which assumes a simultaneous conjugate match at all dis-tances. The expected critical coupling distances are also la-beled in both figures. These points were determined by findingthe distance where ( ωM ) = ( R + R L ) for each of the threematching networks. Fig. 6. Plots of experimental and theoretical efficiencies vs. distance usingfixed L-section matching networks. (a) The experimental and theoretical effi-ciencies at the self-resonant frequency of the loops ( ω ). (b) The experimentaland theoretical efficiencies under frequency-tuning. While in strong coupling,the system was frequency-tuned to operate at an even or odd mode frequencyin order to maintain a constant efficiency, as in (23). The (cid:3) , (cid:13) , and (cid:52) plotthe experimental efficiencies ( | S | ) vs. distance for three fixed matchingnetworks designed for loop separations of 20, 35 and 50 cm, respectively.The dotted, dash-dotted, and solid lines plot the corresponding theoreticalefficiencies. In (a), theoretical efficiencies were calculated using (11). In (b),efficiencies were calculated for weak coupling using (11), and for strongcoupling using (23). The dashed line plots the maximum achievable efficiencygiven by (27b). Distances corresponding to critical coupling are marked withan asterisk. Therefore, by tuning to the even or odd mode frequencyof the strongly coupled shielded loop system, constant experi-mental efficiencies are observed. These experimental efficien-cies are in excellent agreement with theory. Nevertheless, theefficiencies attained through frequency-tuning are lower thanthose that can be attained through a complex conjugate match(labeled Theoretical Absolute max efficiency in Fig. 6).
Experimental Theoretical PredictedMatching Results Results CriticalNetwork from Fig. 6 from (23) Coupling Point20 cm (1.87 r ) .
85% 79 . r ) .
5% 49 . r ) .
54% 31 . COMPARISON OF EXPERIMENTAL , FREQUENCY - TUNED EFFICIENCIESAND THOSE THEORETICALLY FOUND FOR STRONG COUPLING USING (23).
It should be noted that the further the distance at which thesystem is matched, the lower the optimal R L will be. Selectionof a lower R L results in lower efficiencies at close distancesbut extends the distances over which the system is stronglycoupled (see Table IV). This expands the range over whichfrequency-tuning applies, and results in higher efficiencies atfurther distances.VI. A N E XPERIMENTAL I MPEDANCE -T UNED
WNPTS
YSTEM E MPLOYING V ARACTOR -B ASED M ATCHING N ETWORKS
A complex conjugate match, given by (26), can be satisfiedfor a given M using fixed matching networks. However, as M changes (relative position of the loops varies), this condi-tion is no longer maintained. In this section, the analysis of theimpedance-tuned system presented in section III-B is validatedexperimentally through the use of tunable impedance-matchingnetworks.The efficacy of varactor-based impedance-matching net-works has been demonstrated in earlier works [37], [38]. Here,varactors are used to implement tunable L-section matchingnetworks for a WNPT system. The capacitance of a varactorvaries directly with an applied reverse voltage bias. When a RFsignal is applied atop this bias, non-linearities due to harmonicdistortion can occur. Earlier works have analyzed this [39],[40], [41], and proposed linearized topologies to mitigate theseeffects. In these proposed solutions, an anti-parallel varactorpair forms the feed point for DC-bias and ground, and ananti-series varactor pair forms the variable capacitor. Thesenetworks are desired for third harmonic suppression in order toreduce distortion. For the suppression to occur, a diode gradingcoefficient of M ≈ . is desired [39]. A low-distortionvaractor network in the form of this anti-parallel anti-seriescombination was used in this work (Fig. 7). Fig. 7. Linearized topology suggested in [40], [41]. The anti-series varactorpair functions as a single variable capacitor, and an anti-parallel varactor pairprovides a high zero-bias impedance for the DC biasing supply as in [42].
A. Circuit Development
This section describes how varactor-based tunable capaci-tors were integrated into L-section matching networks to yielda tunable WNPT system. The total capacitance of a diode isgiven by: C V = C jo (cid:16) V R V J (cid:17) M + C P , (30)where C jo is the junction capacitance of the diode, V R is theapplied reverse voltage (the biasing voltage), V J is the built-indiode junction voltage, C P is the package capacitance, and M is the grading coefficient.By placing several anti-series varactor pairs in parallel, theequivalent capacitances needed to meet the matching networkcriteria were achieved (see Fig. 4a). An abrupt-junction var-actor diode (Skyworks SMV1494) was selected. This varactorhas a grading coefficient M = 0 . , thus meeting the criteriafor linear operation. In addition, the SMV1494 has C jo = 58 pF and comes in the SC-79 package for reduced printedcircuit board footprint, low parasitic inductance, and a nearlynegligible package capacitance ( C P ≈ pF). The number ofanti-series varactor pairs placed in parallel was chosen to be 3and 8 for C S and C P , respectively (see Fig. 8). This situatesthe reverse bias voltage ( V R ) required to properly bias the net-works within the operational range of the SMV1949 varactors( ∼ V). The 3 anti-series varactor pairs comprising C S were able to achieve a tunable capacitance ranging from 86.7pF to to 22 pF, for V R = 0 V to V R = 10 V, respectively.The 8 anti-series varactor pairs comprising C P had a tunablerange of 232 pF to 58.8 pF, for V R = 0 V to at V R = 10 V, respectively. The final circuit schematic of the matchingnetwork is shown in Fig. 8.
Fig. 8. A circuit schematic of the varactor-based L-section matching network.Coilcraft 1812LS 22.3 µ H inductors were used as RF chokes to isolate theRF signal from the bias voltage and ground. Murata GCM1882C1H 470nF capacitors serve as DC blocks, to constrain the biasing voltage from theresonant loops. The anti-series varactor pairs acting as matching capacitors C S and C P (see Fig. 4b) are noted on the image. All varactors are SkyworksSMV1494 abrupt junction varactors. B. Experimental Validation
The circuit in Fig. 8 was simulated using the AgilentAdvanced Design System (ADS) harmonic balance solver.One of the two fabricated varactor-based L-section matchingnetworks is depicted in Fig. 9. Two such tunable matching
Fig. 9. Photograph of a varactor-based L-section matching network. Thisis the fabricated realization of the circuit schematic shown in Fig. 8. Furthercomponent data is given in [43]. networks were tested on the shielded-loop system. An analogsignal generator (Agilent N5183A) was used to supply 10 mWof power at 38 MHz to the source loop. The output power wasmeasured with a power meter (Agilent E4416A) and powersensor (Agilent N8485A), as depicted in Fig. 3. The two DCbias points on the networks were tuned independently with aDC voltage supply (Agilent E3648A/E3631A).The theoretical DC bias values were derived by interpolatingthe required capacitances vs. distance from Fig. 4a with the V R vs. C V characteristics of the varactor given by (30).The experimental efficiency of the impedance-tuned, coupledshielded loops is plotted with respect to distance in Fig. 10.It is compared to the efficiency curves for the fixed-capacitormatching networks (see Fig. 6). The experimental efficiencyis approximately lower than the theoretical maximumefficiency assuming a perfect conjugately matched systemgiven by (26). This slight loss in efficiency is due to parasitics,tolerances, and losses of the manually constructed varactor-based matching networks.Fig. 11 plots the optimal load and source impedance valuesfor a conjugately matched shielded-loop system (as they werederived in section IV). The shaded grey area is the region thatcan be matched by the varactor-based L-section matching net-works used in this work. The required source/load impedancesstray from this region for distances < cm, giving rise toan impedance mismatch. This explains the significant dropin efficiency at these distances (see Fig. 10). This could bemitigated by using varactors with a larger tuning range, orusing a modified network that is tailored to a different range Fig. 10. A plot of efficiency vs. distance for the fixed-capacitor and varactor-based tunable L-section matching networks. The solid line indicates the resultsfrom the varactor network. The (cid:3) , (cid:13) , and (cid:52) plots show the results from Fig.6 for comparison. The dash-dot line plots the theoretical maximum achievableefficiency given by (27b). of distances. Alternatively, for distances < cm, the tunablenetwork can be biased to establish a strongly coupled system(satisfying ωM > R + R L ) and frequency-tuning employedto maintain a constant efficiency at close-in distances. Fig. 11. Plotted on the Smith chart are the optimal Z S and Z L valuesfor varying distances, given by (24a) and (24b) respectively. For the shieldedloops presented in [14], the values are nearly the same, indicating identicalloops. The shaded grey region represents the impedances that can be matchedby the varactor network used in this work. In this section, it was shown that by maintaining a conju-gate impedance-match with tunable varactor-based matchingnetworks, one can achieve higher efficiencies than with afrequency-tuned WNPT system. This however comes at thecost of added complexity.
C. Angular and Axial Misalignment
Up to this point, mutual inductance ( M ) was varied onlyas a function of axial distance. In reality, M is affected byvariations in axial distance ( d ), axial misalignments ( c ), andthe angle ( θ ) between the two planes of the loops, as shownin Fig. 12. These variations all affect the efficiency of powertransfer.Using [44], [45], analytical values for M under thesevariations were found, and the corresponding efficiencies were Fig. 12. Relative position of the resonant shielded loops with respect to eachother. The plots identifies variations in axial distance, axial misalignment andangle between the two planes of the loops. computed. Theoretical and experimental results were obtainedby aligning the loops co-axially at one of three discretedistances ( d =
20, 35, or 50 cm). They were then furthersubjected to either: • Axial misalignments ( d fixed, θ = 0 , and c varied) • Angular variations ( d fixed, c = 0 , and θ varied) Fig. 13. Plots show M at 38 MHz as a function of (a) axial misalignment( ∆ c ) of parallel loops (b) angular misalignment ( ∆ θ ) between the two loopsfor 3 different axial distances ( d ). The plots in Figs. 13a and 13b depict the changes in M for movements off-axis ( ∆ c ) and changes in angle ( ∆ θ ),respectively. Note that at far distances ( d (cid:29) r ), ∆ M ismuch smaller for changes in c and θ than at close distances.The experimental results in Fig. 14 depict the changes inefficiency ( η ) under the conditions described above. Results forboth fixed-capacitor matching networks and tunable varactor-based networks are in good agreement with theory. At d = M are negligible. Therefore, results for the tunable networkare only displayed at 20 cm, where they can adapt to changesin mutual inductance, M , and show appreciable gains inefficiency. Fig. 14. Power transfer efficiency as a function of (a) distance off-axis ( ∆ c )for parallel loops, and (b) rotation ( ∆ θ ) between two loops aligned along acentral axis. The solid lines depict theoretical maximum efficiencies with theassumption that optimal source (24a) and load (24b) are used. The dotted linesdepict the theoretical efficiencies for resonant shielded-loop WNPT systememploying fixed-capacitor matching networks optimized for 20, 35, or 50 cmdistances. The (cid:3) plots represents measured efficiencies as the system withfixed-capacitor matching networks undergoes axial or angular misalignments.The (cid:13) plots represents measured efficiencies as the system with varactor-based matching networks undergoes axial or angular misalignments and thenetworks are tuned for maximum efficiency. Distances ( d ) are fixed for eachset of measurements, and are denoted on the plots. VII. A H
IGHER P OWER M ATCHING N ETWORK
In the experiments of section VI, 10 mW of power was sup-plied to the tunable matching networks and resonant shieldedloops. In practice, it is necessary to scale power levels to thoseused by modern electronic devices. In this section, tunablevaractor-based matching networks capable of handling higherpower levels are considered. Varactor networks with highpower handling capability have been demonstrated in [38],[40].Commercially available Micrometrics MTV4045-10 andMTV4060-16 abrupt junction tuning varactors were selectedfor capacitances C S and C P (see Fig. 4a), respectively. Thesevaractors have high breakdown voltages ( B V ) of 45 V and 60V, respectively, and a grading coefficient of M = 0 . .It was determined through simulation and interpolationthat 44 and 40 pairs of anti-series varactor pairs met therequirements from for C S and C P , respectively (see Fig.4a). The results of the interpolation can be seen in Fig. 15.Simulations showed that the network could handle an appliedpower of approximately W. Beyond this power level, thevaractor specifications are exceeded and harmonic distortiondegrades the WNPT system’s performance.These higher power L-section matching networks weretested on the shielded loops. The 38 MHz signal from thesource (Hewlett Packard 8654) was amplified by a Minicircuits52 dB RF Class A amplifier (ZHL-ED12128A/1) to producethe desired input power level P in . The output power wasmeasured with a power meter (Agilent E4416A) and high-power sensor (Agilent N8481). The varactor-based L-sectionmatching networks were independently tuned with a DCvoltage source (Agilent E3648A/E3631A). The experimentalefficiencies of the high power network over distance are shown Fig. 15. The capacitances required as a function of distance for thevaractor-based L-section matching networks. The C V vs. The curves show thetheoretical reverse bias voltages needed to achieve the required capacitancesfor the high power matching networks. in Fig. 16, for multiple input power levels. Fig. 16. A plot of efficiency vs. distance for the resonant shielded-loopWNPT system employing high-power varactor-based L-section matching net-works for various source power levels. The dash-dot line plots the theoreticalmaximum achievable efficiency given by 27b.
The results demonstrate the ability of the varactor-based L-section matching networks to provide an impedance-match forvarying values of M , in order to maximize efficiency. How-ever, once the input power is increased beyond 5 W, a decreasein the experimental efficiencies occurs. This is due to a fewreasons. First, the varactor specifications are exceeded; mostnotably the maximum current ratings. In addition, harmonicdistortions begin to affect efficiency at these higher powerlevels. Furthermore, an impedance mismatch is incurred atfarther distances. Fig. 15, shows that the required reverse biasvoltage V R approaches B V = 45 V (breakdown voltage for theMTV4045-10) as the distance is increased. Therefore, V R waslowered as V RF increased to ensure that V R + V RF < B V [38].This in turn resulted in a higher than desired capacitance C S .Using a varactor with a higher B V would alleviate this [46].Varactors with high linearity and tuning range [47], as wellas varactors with higher power handling capabilities [48] havebeen proposed in the past, but are not commercially available.VIII. C ONCLUSION
In this paper, frequency tuning and impedance tuning tech-niques were explored for increasing the efficiency of wirelessnon-radiative power transfer (WNPT) systems. Both tech-niques were investigated analytically using a lumped-elementcircuit model. Expressions for the input reflection coefficientsand efficiencies were derived. It was shown that a frequency-tuned system can maintain constant efficiency in the strongly coupled region of operation. This constant efficiency, however,is limited by reflections which prevent optimal operation. Inaddition, an impedance-tuned WNPT system was investigated.It was shown that impedance-matched WNPT system canachieve optimal/maximum possible efficiency.Both frequency tuning and impedance tuning were experi-mentally demonstrated using a resonant shielded-loop WNPTsystem. The shielded-loop system exhibited even and oddmodes in the strong coupling region. By tuning the operatingfrequency to one of these modes, a constant efficiency wasachieved for distances within the strongly coupled region.The experimental impedance-tuned WNPT system employedvaractor-based matching networks to achieve maximum ef-ficiency. The matching networks maintained a simultaneousconjugate match to source and load over a range of distances.Further, it was shown that such matching networks can com-pensate for changes in the mutual inductance, M , whichoccur with axial misalignments and angular variations of thetransmitter and receiver loops.Finally, higher power matching networks were developedand experimentally demonstrated for the impedance-tunedWNPT system. The matching networks were realized usingmultiple varactors to withstand power levels exceeding 5 W. Areduction in efficiency due to device limitations and harmonicdistortion was observed as power levels increased. A varactorwith a higher reverse bias voltage could alleviate these issues.Control circuity can be developed to create a smart WNPTsystem capable of dynamically improving efficiency over awide range of practical distances. This smart system couldemploy dynamic frequency tuning [49], impedance tuning ora combination of both. For example, a switch matrix couldswitch in or out static matching networks designed for variousfixed distances. Frequency tuning could then be employed fordistances between these fixed values.R EFERENCES[1] W. C. Brown, “The History of Power Transmission by Radio Waves,”
IEEE Transactions on Microwave Theory and Techniques , vol. 32, no.9, pp. 1230–1242, September 1984.[2] Z. Popovic, E. A. Falkenstein, D. Costinett, and R. Zane, “Low-PowerFar-Field Wireless Powering for Wireless Sensors,”
Proceedings of theIEEE , vol. 101, no. 6, pp. 1397–1409, 2013.[3] Fulton Innovation, eCoupled Wireless Power Technology
Setting the Interna-tional Standard for Interoperable Wireless Charging
Proceedingsof the IEEE , vol. 101, no. 6, pp. 1276–1289, 2013.[8] J. Kim, J. Kim, S. Kong, H. Kim, I.-S. Suh, N. P. Suh, D.-H. Cho,J. Kim, and S. Ahn, “Coil Design and Shielding Methods for a MagneticResonant Wireless Power Transfer System,”
Proceedings of the IEEE ,vol. 101, no. 6, pp. 1332–1342, 2013.[9] Nikola Tesla,
Apparatus for Transmission of Electrical Energy , 1900,US Patent No. 649,576.[10] Nikola Tesla,
System of Transmission of Electrical Energy , 1900, USPatent No. 649,621.[11] H. W. Secor, “Tesla Apparatus and Experiments - How to Build BothLarge and Small Tesla and Oudin Coils and How to Carry on SpectacularExperiments with Them,”
Practical Electrics , Nov 1921.[12] A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, andM. Soljaˇci´c, “Wireless Power Transfer via Strongly Coupled MagneticResonances,”
SCIENCE , vol. 317, no. 5834, pp. 83–86, July 2007. [13] W. L. Everitt and G. E. Anner, Communication Engineering , McGraw-Hill, 1956.[14] E. M. Thomas, J. D. Heebl, C. Pfeiffer, and A. Grbic, “A PowerLink Study of Wireless Non-Radiative Power Transfer Systems UsingResonant Shielded Loops,”
IEEE Transactions on Circuits and SystemsI , vol. PP, no. 99, 2012.[15] E. M. Thomas, J. D. Heebl, and A. Grbic, “Shielded Loops for WirelessNon-Radiative Power Transfer,” in
IEEE International Symposium onAntennas and Propagation , July 2010.[16] C. Chen, T. Chu, C. Lin, and Z. Jou, “A Study of Loosely CoupledCoils for Wireless Power Transfer,” in
IEEE Transactions on Circuitsand Systems II , July 2010, pp. 536–540.[17] M. Zargham and P. Gulak, “Maximum Achievable Efficiency inNear-Field Coupled Power-Transfer Systems,”
IEEE Transactions onBiomedical Circuits and Systems , vol. 6, no. 3, pp. 228–245, June 2012.[18] K. E. Koh, T. C. Beh, T. Imara, and Y. Hori, “Multi-Receiver andRepeater Wireless Power Transfer via Magnetic Resonance Coupling -Impedance Matching and Power Division Utilizing Impedance Inverter,”in , 2012, pp. 1–6.[19] M. Ettorre and A. Grbic, “A Transponder-Based, Nonradiative WirelessPower Transfer,”
IEEE Antennas and Wireless Propagation Letters , vol.11, pp. 1150–1153, Sept 2012.[20] S. Hui, W. Zhong, and C. Lee, “A Critical Review of Recent Progressin Mid-Range Wireless Power Transfer,”
IEEE Transactions on PowerElectronics , vol. PP, no. 99, 2013.[21] D. Liang, H. T. Hui, and T. S. Yeo, “A Phased Coil Array for EfficientWireless Power Transmission,” in
IEEE International Symposium onAntennas and Propagation , 2012, pp. 1–2.[22] D. Huang, Y. Urzhumov, D. Smith, K. H. Teo, and J. Zhang, “Magneticsuperlens-enhanced inductive coupling for wireless power transfer,”
Journal of Applied Physics , vol. 111, no. 6, pp. 064902–1–8, 2012.[23] A. P. Sample, D. T. Meyer, and J. R. Smith, “Analysis, ExperimentalResults, and Range Adaptation of Magnetically Coupled Resonators forWireless Power Transfer,”
IEEE Transactions on Industrial Electronics ,vol. 58, no. 2, pp. 544–554, February 2011.[24] J. Park, Y. Tak, Y. Kim, and S. Nam, “Investigation of AdaptiveMatching by the Frequency Tracking Method for Near Field WirelessPower Transfer,” in
IEEE Transactions on Antennas and Propagation .IEEE, July 2011.[25] T. Imura, H. Okabe, and Y. Hori, “Basic Experimental Study on HelicalAntennas of Wireless Power Transfer for Electric Vehicles by UsingMagnetic Resonant Couplings,” in
IEEE Vehicle Power and PropulsionConference , September 2009, pp. 936–940.[26] J. Park, Y. Tak, Y. Kim, and S. Nam, “Investigation of AdaptiveMatching Methods for Near-Field Wireless Power Transfer,”
IEEETransactions on Antennas and Propagation , vol. 59, no. 5, pp. 1769–1773, May 2011.[27] J. Lee and S. Nam, “Fundamental Aspects of Near-Field Coupling SmallAntennas for Wireless Power Transfer,”
IEEE Transactions on Antennasand Propagation , vol. 58, no. 11, pp. 3442–3449, Nov 2010.[28] G. Gonzalez,
Microwave Transistor Amplifiers: Analysis and Design ,Prentice Hall, second edition, 1997.[29] D. M. Pozar,
Microwave Engineering , John Wiley and Sons, Inc.,second edition, 1998.[30] J. Garnica, R. A. Chinga, and Jenshan Lin, “Wireless Power Transmis-sion: From Far Field to Near Field,”
Proceedings of the IEEE , vol. 101,no. 6, pp. 1321–1331, 2013.[31] J. S. Ho, S. Kim, and A. S. Y. Poon, “Midfield Wireless Powering forImplantable Systems,”
Proceedings of the IEEE , vol. 101, no. 6, pp.1369–1378, 2013.[32] J. Garnica, J. Casanova, and J. Lin, “High Efficiency Midrange WirelessPower Transfer System,” in
IEEE Microwave Theory and TechniquesSociety International Microwave Workshop Series on Innovative WirelessPower Transmission: Technologies, Systems, and Applications (IMWS) ,2011, pp. 73–76.[33] A. Stensgaard, “Planar Quadrature Coil Design Using Shielded-LoopResonators,”
Journal of Magnetic Resonance , vol. 125, no. 1, pp. 84–91,March 1997.[34] M. D. Harpen, “The Theory of Shielded-Loop Resonators,”
MagneticResonance in Medicine , vol. 32, no. 6, pp. 785–788, December 1994.[35] A. Stensgaard, “Optimized Design of the Shielded-Loop Resonator,”
Journal of Magnetic Resonance , vol. 122A, no. 2, pp. 120–125, October1996.[36] D. L. Sengupta and V. V. Liepa,
Applied Electromagetics and Electro-magnetic Compatibility , John Wiley and Sons, Inc., 2006. [37] C. Hoarau, N. Corrao, J. D. Arnould, P. Ferrari, and P. Xavier, “CompleteDesign and Measurement Methodology for a Tunable RF Impedance-Matching Network,”
IEEE Transactions on Microwave Theory andTechniques , vol. 56, no. 11, pp. 751–754, November 2008.[38] H. M. Nemati, C. Fager, U. Gustavsson, R. Jos, and H. Zirath, “Designof Varactor-Based Tunable Matching Networks for Dynamic Load Mod-ulation of High Power Amplifiers,”
IEEE Transactions on MicrowaveTheory and Techniques , vol. 57, no. 5, pp. 1110–1118, 2009.[39] R. G. Meyer and M. L. Stephens, “Distortion in Variable-CapacitanceDiodes,”
IEEE Journal of Solid-State Circuits , vol. 10, no. 1, pp. 47–54,February 1975.[40] K. Buisman, C. Huang, A. Akhnoukh, M. Marchetti, L. C. N. de Vreede,L. E. Larson, and L. K. Nanver, “Varactor Topologies for RF Adaptivitywith Improved Power Handling and Linearity,”
IEEE Microwave Theoryand Techniques Society International Microwave Symposium , pp. 319–322, June 2007.[41] K. Buisman, L. C. N. de Vreede, L. E. Larson, M. Spirito, A. Akhnoukh,T. L. M. Scholtes, and L. K. Nanver, “‘Distortion Free’ VaractorDiode Topologies for RF Adaptivity,” in
IEEE Microwave Theory andTechniques Society International Microwave Symposium Digest , June2005.[42] K. Buisman, L. C. N. de Vreede, L. E. Larson, M. Spirito, A. Akhnoukh,Y. Lin, X. Liu, and L. K. Nanver, “A Monolithic Low-Distortion Low-Loss Silicon-on-Glass Varactor-Tuned Filter With Optimized Biasing,”
IEEE Microwave and Wireless Components Letters , vol. 17, no. 1, pp.58 – 60, Jan 2007.[43] “Note on Components,” Components used for board construction wereSkyline SMV1494 Varactors, Coilcraft 1812LS-223XJLB Inductors,Murata GCM1882C1H Capacitors, J657-ND SMA end launch, andRogers RO4003C 35 µ m copper foil, 0.813 mm dielectric substrate.[44] F. Grover, “The Calculations of the Mutual Inductance of CircularFilaments in Any Desired Positions,” Proceedings of the Institute ofRadio Engineers , vol. 32, no. 10, pp. 620–629, October 1944.[45] S. I. Babic, F. Sirois, and C. Akyel, “Validity Check of MutualInductance Formulas for Circular Filaments with Lateral and AngularMisalignments,”
Progress In Electromagnetics Research M , vol. 8, pp.15–26, 2009.[46] “Note on Components,” MTV4045-10 and MTV4060-16 varactors havea reverse breakdown voltage B V =
45 V and 60 V, respectively. TheMTV4090 series has a breakdown voltage of V B =
90 V, and is desired.However, the varactors in this study were chosen for their product-lineavailability.[47] K. Buisman, L. K. Nanver, T. L. M. Scholtes, H. Schellevis, and L. C. N.de Vreede, “High Performance Varactor Diodes Integrated in a Silicon-on-Glass Technology,”
Proceedings of the 35th European Solid-StateDevices Research Conference , pp. 117–120, September 2005.[48] Sudow, M., Nemati, H. M., Thorsell, M., Gustavsson, U., Andersson,K., Fager, C., Nilsson, P.-A., ul Hassan, J., Henry, A., Janzen, E., Jos,R., and Rorsman, N., “SiC Varactors for Dynamic Load Modulation ofHigh Power Amplifiers,”
IEEE Electron Device Letters , vol. 29, no. 7,pp. 728–730, July 2008.[49] J. de Mingo, A. Valdovinos, A. Crespo, D. Navarro, and P. Garcia,“An RF Electronically Controlled Impedance Tuning Network Designand its Application to an Antenna Input Impedance Automatic MatchingSystem,”