the circuit theory behind coupled-mode magnetic

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 59, NO. 9, SEPTEMBER 2012 2065

The Circuit Theory Behind Coupled-Mode MagneticResonance-Based Wireless Power Transmission

Mehdi Kiani , Student Member, IEEE , and Maysam Ghovanloo , Senior Member, IEE E

Abstract— Inductive coupling is a viable scheme to wirel esslyenergize devices with a wide range of power requirements fromnanowatts in radio frequency identi cation tags to milliwatts inimplantable microelectronic devices, watts in mobil e electronics,and kilowatts in electric cars. Several analytical methods forestimating the power transfer ef ciency (PTE) across inductivepower transmission links have been devised b ased on circuit andelectromagnetic theories by electrical engineers and physicists,respectively. However, a direct side-by-side comparison betweenthese two approaches is lacking. Here, we h ave analyzed the PTEof a pair of capacitively loaded inductors via re ected load theory(RLT) and compared it with a method known as coupled-modetheory (CMT). We have also derived PTE equations for multiple

capacitively loaded inductors based on both RLT and CMT. Wehave proven that both methods basically result in the same setof equations in steady state and ei ther method can be appliedfor short- or midrange coupling conditions. We have veri edthe accuracy of both methods through measurements, and alsoanalyzed the transient response of a pair of capacitively loadedinductors. Our analysis shows that the CMT is only applicableto coils with high quality factor and large coupling distance.It simpli es the analysis by reducing the order of the differentialequations by half compared to the circuit theory.

Index Terms— Coupled-mode theory (CMT), near eld, qualityfactor, power transfer ef ciency (PTE), re ected load theory, res-onance circuits, wirele ss power transmission.

I. I NTRODUCTION

I NDUCTIVE power transmission links that utilize a pair of mutually coupled coils have been in use over the lastdecades to power up radio frequency identi cation (RFID)transponders and cochlear implants with power consumptionsin the range of sub-micro to milliwatts [1], [2]. The use of thistechnique to wirelessly transfer energy across a short distanceis, however, expected to see an explosive growth over thenext decade in a much broader range of applications fromadvanced implantable microelectronic devices (IMD), such asretinal implants and brain–computer interfaces (BCI) to cut the

power cord in charging mobile electronic devices, operatingsmall home appliances, and energizing electric cars, whichhave higher power consumptions in the order of hundreds of

Manuscript received July 14, 2011; revised October 20, 2011; accepted November 21, 2011. Date of publication January 24, 2012; date of currentversion August 24, 2012. This work was supported in part by the NationalInstitutes of Health, NIBIB, under Grant 1R21EB009437-01A1, and in part bythe National Science Foundation under Award ECCS-824199. This paper wasrecommended by Associate Editor E. Alarcon

The authors are with the GT-Bionics Lab, School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30308 USA (e-mail:[email protected]).

Color versions of one or more of the gures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identi er 10.1109/TCSI.2011.2180446

milliwatts to kilowatts [3]–[12]. As a result, improvin g thewireless power transmission ef ciency (PTE) in inductivelinks particularly at larger coupling distances (coi l separations)without increasing the volume and weight of the coupled coilshas gained considerable attention [13]–[19] .

Achieving high PTE is necessary in high-power IMDs to notonly reduce the size of the external energy sou rce (battery) thatshould be carried around by the patient but al so to limit thetissue exposure to the AC magnetic eld, which can result inexcessive heat dissipation if it surpasses safe limits, and to min-imize interference with nearby electronics [20]–[22]. In near eld RFID applications, the bottlen eck in increasing the readingrange without changing the coil size in many cases is the PTEwhen the received power is no longer suf cient to operate thetransponder [2]. Increasing the P TE in higher power applica-tions is also important for generating less heat, cost saving, re-ducing interference, and impro ving safety.

Design and optimization of inductive power transmissionlinks has been extensively st udied in the literature over thelast three decades [2], [5], [23]–[34]. The majority of theseapproaches model and analyze the inductive link from a circuit perspective, which differ s, at least on the surface, from thecoupled-mode theory (CMT) that was recently presented in a

new form by physicists a t MIT [35]. They utilized the CMTapproach to propose mu lti-coil inductive links, which canincrease the PTE considerably at large coupling distances [13],[14]. Their 4-coil in ductive link has so far been studied froma circuit perspective for power transmission to multiple smallreceivers, transc utaneous powering, and recharging mobiledevices [17]–[19]. We have presented the analysis, modeling,design, and opti mization of multi-coil inductive links usingthe circuit bas ed re ected load theory (RLT) in [36]. How-ever, an in-depth comparison between the coupled-mode andcircuit-base d theories, which would clarify the relationship be-tween these two methods, often used by physicist and electricalengineers, r espectively, is still lacking.

In this paper , rst we review the inductive link steady-stateanalysis using CMT and then prove that both CMT and the moreconventio nal RLT result in the same formulation for the induc-tive links’ key performance measures, particularly the PTE. For the rst time, we have derived the PTE equations for multi-coilinductive links via CMT. Our analysis shows that in the steadystate mo de, contrary to popular belief, both CMT and RLT areapplic able to small and large coupling distances, , as long asthe coils interact in the near eld regime. Because they are ba-sical ly the same [37]! We have also presented the measurementresults, which show the accuracy of both RLT and CMT modelsin estimating the PTE. On the other hand, our comparative tran-

sient analysis of a pair of capacitively loaded inductors reveals

1549-8328/$31.00 © 2012 IEEE


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2066 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 59, NO. 9, SEPTEMBER 2012

Fig. 1. A pair of capacitively loaded inductors used for ef ciency calculationusing the coupled-mode theory (CMT).

that CMT is accurate only if the mutual coupling, , is small,e.g., due to large , and coil quality factors, , are large, whichlimits the applicability of the CMT-based transient analysis tomidrange coupling distances of large coils. We have also shownthat utilizing CMT reduces the order of the differential equa-tions by half compared to the circuit theory as it only considersa rst order equation for each resonant object. Hence, the CMTseems to be helpful in analyzing the transient behavior of com- plicated multi-coil inductive links despite being less accuratethan its circuit-based counterpart. For all other conditions, theRLT and similar circuit based methods will do just ne.

In the following section, the CMT analysis for a pair of capac-itively loaded inductors is presented. Section III describes theRLT analysis for 2-coil inductive links. The correspondence be-tween CMT and RLT has been presented in Section IV. ThePTEanalysis of the multi-coil inductive links utilizing both CMT andRLT are presented in Section V. The calculation and measure-ment resultsare compared in Section VI followed by concludingremarks in Section VII.

II. COUPLED -MODE THEORYCMT is a framework to analyze energy exchange between

two resonating objects [35]. Based on the CMT, the time-do-main eld amplitudes of two objects, and , whichare de ned so that the energy contained in them are and

, respectively, at distance can be found from [37]

(1)

where and are the eigen frequencies, and are theresonance widths or rate of intrinsic decay due to the objects’absorption (Ohmic) and radiative losses, is the resonancewidth due to load resistance connected to the second object(proportional to ), is the excitation applied to the rst object, and is the coupling rate between the two objects. The CMT method has been recently applied to a pair of capacitively loaded conducting-wire loops, spaced by , asshown in Fig. 1, forming a conventional inductive power trans-mission link, in which is the power transmitter and isthe power receiver inductors, both tuned at the same frequency,

[14], [37].

A. Steady-State Analysis via Coupled-Mode Theory

In steady state analysis, which applies to the conventionalinductive power transmission links, in (1) is a sinu-soidal signal described as . In this condition, thealternating eld amplitude in the primary inductor is constant,

, resulting in a constant eld amplitude in the

secondary inductor, . It can be shown from (1)that [37]. Therefore, the average power at different nodes of the power transmission systemcan be calculated. The power absorbed by and the power delivered to are and , re-spectively. The power delivered to is and,therefore, based on the energy conservation theory, if we ne-glect the radiated power in the near eld regime, the total power delivered to the system from source is .Hence, the PTE of the 2-coil system can be found from

(2)

where is the distance-dependent gure-of-merit for energy transmission systems [37]. To maximize thePTE based on (2), fom should be maximized and an optimalvalue should be chosen for . Parameters that affect fomare obviously the coupling rate between the two objects, ,

which should be increased and the resonance width of each object (intrinsic loss of each inductor), , which should be de-creased. These are quite similar to the coupling coef cient, ,and inverse of the quality factor, , used in conventionalmethods for optimizing inductive links [33]. The other key pa-rameter, , which shows the effect of in opti-mizing the PTE, can be found by calculating the derivative of

in (2) with respect to , resulting in [13]

(3)

B. Transient Analysis via Coupled-Mode Theory

The CMT can also be utilized in analyzing the transient be-havior of the resonant-coupled inductors in Fig. 1 by setting

in (1) and considering an initial energy stored in. This analysis provides designers with better understanding

of the dynamics of energy exchange in such inductively coupled systems. As the rst step, we eliminate in (1) to ndan expression for the time varying eld in the primary coil

(4)

For the sake of simplicity, if similar to [14] we assume thatthe two inductors are identical, i .e., , and(no load condition, ), then can be found from (4):

(5)

where and are constants,whichvalues dependon the initialconditions. Similarly, can be calculated using (1) and (5)as

(6)

where and are also constants with values dependent on theinitial conditions. The total energy in and can be foundfrom and , respectively [24].


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KIANI AND GHOVANLOO: CIRCUIT THEORY BEHIND COUPLED-MODE MAGNETIC RESONANCE-BASED WIRELESS POWER TRANSMISSION 2067

Fig. 2. (a) Simpli ed model of a 2-coilpower transmission link with a resistiveload. (b) Equivalent circuit seen across the driver. (c) Re ected load onto the primary loop at resonance frequency.

If one starts with 100% of the total energy normalized and ini-tially stored in , i.e., and ,

the energy stored in each object over time can be found from

(7)

It should be noted that and are not the instantaneous butthe peak values (or envelopes) of the energy contents stored in

and , which are inresonance with and , respectively,at the rate of .

III. R EFLECTED LOAD THEORY

Fig. 2(a) shows a pair of inductively coupled coils, which will

be referred to as the primary and secondary . It can beshown that the highest PTE across such links can be achievedwhen both LC-tanks are tuned at the same resonance frequency,i.e., [31].

A. Steady State Analysis via Re ected Load Theory

The inductive link PTE is mainly dependent on the mutualcoupling between the coils, , and their quality factors,

and [33]. At resonance frequency,the secondary loop which is connected to can be re ectedonto the primary loop and represented by [2]. To nd ,the secondary loop is modeled with a parallel load as shown inFig. 2(a). , the parasitic resistance of , can be transformedto a parallel resistance equal to in parallel with

[30]. Hence, if we de ne , then we can re- ect the secondary impedance onto the primary side [Fig. 2(b)]:

(8)

where is the loaded quality factor of the sec-ondary coil [9], [30]. At resonance, resonates out with

, leaving only the re ected resistance, , on the pri-mary side, as shown in Fig. 2(c). In the simpli ed inductive link model of Fig. 2(c), and also resonate out, and the input power provided by simply divides between and . The power absorbed by is dissipated as heat in the primary coiland the power delivered to , i.e., the transferred power to

the secondary loop, divides between and , which are theonly power consuming components on the secondary side. Thiswill lead to

(9)

where is often referred to as the load qualityfactor, and [31]. It can be seen from(9) that large , and are needed to maximize the PTE.However, for a given set of and values, there is anoptimal load, , which can maximize thePTE. can be found by calculating the derivative of (9)with respect to from

(10)

B. Transient Analysis via Circuit TheoryIn this analysis, the primary and secondary loops are consid-

ered identical, i.e.,and , similar to the CMT criteria in [14]. We have

also set V (short circuit) to focus on the transient re-sponse while considering an initial condition (current) in .Primary and secondary loop currents can be found from

(11)

where is the mutual inductance between and. Taking the derivative of (11) after substituting and

with and , respectively, result in

(12)

Utilizing Laplace transform, (12) can be written as

(13)

Characteristic equation in the S-domain can be found by settingthe coef cient matrix determinant in (13) to zero:

(14)

The roots of (14) are

(15)


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where

(16)

Therefore, and can be calculated from

(17)

(18)

In order to nd the unknown coef cients in (17) and (18), oneshould have at least four initial conditions. Two of them are

, and A, which indicate that 100% of the total energy is initially stored in . The other two initialconditions can be found from (11) when :

(19)

IV. COUPLED MODE VERSUS R EFLECTED LOAD THEORIES

In this section, we compare the formulation derived fromCMT and RLT for the 2-coil inductive link PTE and transientresponse derived in the previous sections.

A. Two-Coil Inductive Link Power Transfer Ef ciency

Resonance widths, , and coupling rate, , in a pair of capacitively loaded inductors in Fig. 1 are equivalent in termsof circuit model parameters in Fig. 2 to and ,respectively [37]. Similarly, the load resonance width, , isequal to . By substituting these in fom and

(20)

In the next step, we substitute the CMT parameters from (20) in(2) and recalculate the PTE

(21)

After simpli cation and considering that, the PTE formula in (21) can be further simpli ed to

(22)

which is the same as (9) that was derived via RLT.Similarly, it is straightforward to show that the optimum

in (3) can be linked to in (10) by substitutingthe equivalent circuit parameters in (20), leading to

(23)

Thus, we have shown mathematically that the CMT and RLTequations forthe PTE and optimal loading of 2-coil power trans-mission links in steady state are basically the same.

B. Two-Coil Inductive Link Transient Response

In order to arrive at the CMT transient response in (7), twoassumptions were made. First, the coil quality factors were con-sidered large , resulting in and in(16). Second, the coupling distance, , was considered large,resulting in small , which simpli es the rest of (16) to

(24)

and (19) to

(25)

Thus, and in (17) and (18) can be approximated as

(26)

(27)

Unknown coef cients in (26) and (27) can be found by applyingthe initial conditions, which result in

, and .When substituting these in (26) and (27)

(28)


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(29)

The large assumption combined with the expansion of thesinusoidal functions in (28) and (29)

(30)

can further simplify the primary and secondary currents as

(31)

Considering that the energy stored in an inductor, , that carriesa current, , is , the normalized envelope of the energy

inside and can be expressed as [38]

(32)

By substituting , and in (32), wecan arrive at (7), which was derived using CMT. Therefore, incalculating the transient response of the inductive links, CMT isonly valid for midrange coupling distances (small ) betweenlarge coils (high- ), as also indicated in [37].

In order to validate our theoretical calculations and demon-strate the level of accuracy (or lack thereof) in the transientCMT analysis for various coupling coef cients and factors,the 2-coilinductivelink in Fig. 2 was simulated in the LT-SPICEcircuit simulator (Linear Technology, Milpitas, CA), while set-ting and initial current, , at 0 V and 1 A, respec-tively, for the two conditions summarized in Table I. Fig. 3(a)shows the percentage of the energy stored in primary, , andsecondary, , coils over time for a pair of large identical coilsthat are placed relatively far from each other (midrange, low

, high- ). It can be seen that both CMT (7) and circuit-based(12) equations, solved in MATLAB (MathWorks, Natick, MA),

match the LT-SPICE simulation results very well. In Fig. 3(b),however, which represents a small pair of coils that are veryclose to one another (short range, high , low- ) the CMT- based formulas have become quite inaccurate in predicting theenergy exchange, while the circuit analysis still matches theLT-SPICE simulation output. In this condition, the inductor cur-rents tend to have two harmonics, and in (17) and (18),one of which has not been predicted by the CMT. Nonetheless,the simplicity of the CMT in analyzing the transient behavior of inductive links can be useful particularly in midrange high-conditions. This stems from the fact that CMT models each res-onant object (e.g., RLC tank) with a rst-order differential equa-tion, while in the circuit analysis the order of the equations in-creases by each independent energy storage element regardlessof being an inductor or a capacitor.

Fig. 3. Energy stored in the primary and secondary coils ( and ) versustime starting from an initial condition, A, based on CMT (7), circuittheory (12), and SPICE simulations for 2-coil inductive links in: (a) midrange,high- , and (b) short range, low- conditions, as speci ed in Table I.

TABLE II NDUCTIVE LINK SPECIFICATIONS FOR TRANSIENT A NALYSES

V. MULTI -COIL I NDUCTIVE POWER TRANSFER

CMT-based analysis took circuit designers by surprisewhen the group physicists at MIT demonstrated a method of achieving high PTE by utilizing multiple coils (3- and 4-coils)for wireless power transmission [13], [14]. Nonetheless, theclosed-form CMT formulation presented in the literature was


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limited to 2-coils. In this section, we derive the closed-formPTE equations for multi-coil inductive links based on the CMTand compare them with parallel equations derived from RLT, particularly for 3- and 4-coil links [36]. We prove that bothCMT and RLT result in the same set of equations.

Equations for a pair of capacitively loaded inductors in Fig. 1can be extended to inductors, in which the rst and th in-ductors are connected to the energy source and load, respec-tively, while all inductors are tuned at the same resonance fre-quency, [14]. The time-domain eld amplitudes of each in-ductor, , can be expressed as

(33)

where and are the coupling rate between the th andth inductor and resonance width of the th inductor, re-

spectively. For the sake of simplicity, the coupling betweennon-neighboring inductors has been considered negligible. For the rst and th inductors, the eld amplitudes are

(34)

In the steady state mode, the eld amplitudes in each inductor is considered constant, i.e., . Therefore, thedifferential equations in (33) and (34) result in a set of

equations

(35)

One can solve (35) to nd constants based on the load eldamplitude, . From these values, the average power at dif-ferent nodes of the inductive power transmission link can becalculated. The absorbed power by the th inductor and the de-livered power to can be expressed as and

, respectively, from which the total delivered

power to the system from source can be found from, using the law of conservation of energy. Finally,

the PTE of the -coil system can be found from

(36)

In an -coil link, the re ected load from the th coilonto the th coil can be found from

(37)

where is the coupling coef cient between the th andth coils. is the loaded quality factor of the th

coil, which can be found from

(38)

where and are the unloaded quality factor and parasitic series resistance of the th coil , respectively.It should be noted that for the last coil, which is connected tothe load in series, . Assuming thatthe coupling between non-neighboring coils is negligible, the

partial PTE from the th coil to th coil can be written as

(39)

Using (37)–(39), the overall PTE in such a multi-coil inductivelink can be found from

(40)

A. Three-Coil Power Transfer Inductive Links

The 3-coil inductive power transfer link, shown in Fig. 4(a),was initially proposed in [14] and analyzed based on the CMT.Ifweignore due to large separation between and , the eld amplitudes at each inductor can be calculated by solving aset of two equations in (35), which leads to

(41)

PTE of the 3-coil link can then be found by substituting (41) in(36), which leads to (42) after some minor simpli cations, asshown at the bottom of the page.

The lumped circuit model for 3-coil inductive link has beenshown in Fig. 4(b). In this type of inductive links, can beadjusted by changing the distance or geometry of to matchthe actual with the optimal in (10) for the – inductive link. It should be noted that , which is the seriesresistance of , can also include the source output resistance[36]. The PTE of this circuit can be calculated by re ecting theresistive components of each loop from the load back towards

(42)


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Fig. 4. (a) Three capacitively loaded, mutually coupled, inductors for wireless power transmission in which and serve as impedance matching ele-ments to improve the PTE. (b) Lumped circuit model of the 3-coil inductivelink.

the primary coil loop, one stage at a time, using (37) and cal-culating the percentage of the power that is delivered from onestage to the next, using (39), until it reaches . According to(40), this procedure leads to

(43)

where has been ignored due to large separation betweenand , and

(44)

The resonance widths, , and coupling rates, , inCMT based on circuit parameters are de ned as and

, respectively [37]. By substituting these parametersin (42) and multiplying both numerator and denominator with

, the 3-coil PTE can be found from

(45)

where . It can be seen that (45), whichis derived from the CMT is the same as (43), which is based onthe RLT. Therefore, these two formulations are not different inthe steady state analysis.

B. Four-Coil Power Transfer Inductive Links

Fig. 5(a) shows an inductive power transfer link consistingof four capacitively loaded inductors, in which and arethe main coils responsible for power transmission, similar to the2-coil link, while and are added for impedance matching[18], [36], [37]. The eld amplitudes at each inductor can befound by solving a set of three equations in (35). and can be found based on from (41) and can be found from

(46)

To simplify the analysis, and have been neglectedin comparison to coupling rates between neighboring coils. Onecan nd the 4-coil PTE utilizing CMT by substituting (41) and(46) in (36), which after simpli cation leads to

(47)

where

(48)

In the 4-coil lumped circuit schematic, shown in Fig. 5(b),the PTE can be calculated from (40) if we ignore , and

in comparison to and [18], shown in (49) at the bottom of the page.

We can prove that CMT and RLT equations in (47) and(49) are basically the same by substituting the resonancewidths, , and coupling rates, , and in(47) and (48) with their equivalent circuit parameters,

, and , respectively [37].Once both numerator and denominator of (47) are multiplied by , the CMT-based 4-coil PTE leads to (50),shown at the bottom of the next page, where and are

(51)

The third and fourth terms of the denominator in (50) can bewritten in terms of as

(52)

(49)


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Fig. 5. (a) Four capacitively loaded, mutually coupled, inductors for wireless power transmission, in which and serve as impedancematching elements to improve the PTE. (b) Lumped circuit model of the 4-coil

inductive link.

By further manipulation of the numerator and denominator

(53)

Once and are substituted from (51) in (53), it will be iden-tical to (49), which was derived from the RLT.

VI. MODELING VERSUS MEASUREMENT R ESULTS

In order to verify the accuracy of the PTE equations derivedfrom the RLT and CMT in measurements, we designed and fab-ricated three sets of inductive links with 2, 3, and 4 coils. and

in all three links were identical printed spiral coils (PSCs),fabricated on 1.5-mm-thick FR4 printed circuit boards (PCB)with 1-oz copper (35.6 m thick). and in the 3- and 4-coillinks were made of single lament solid copper wires, placed inthe middle of and , respectively [see Fig. 6(b)]. Given

and geometries, the identical diameter of and wereoptimized based on the procedure that we presented in [36] for the nominal coupling distance of cm, ,and MHz. Table II shows speci cations these threeinductive links.

Fig. 6(a) shows the measurement setup that we used for thePTE, which is quite suitable for multi-coil inductive links [36].In this method, resonance capacitors and are added to thedriver and load coils, and the entire circuit is considered a 2-port

Fig. 6. (a) PTE measurement setup using a network analyzer with all the coilstuned at the carrier frequency and connected to the load coil. (b) Four-coilinductive link used to measure the PTE (speci cations in Table II).

system including the multi-coil inductive link in between. A net-work analyzer is used to measure the S-parameters, from whichthe Z-parameters are derived [39]. The PTE can then be foundfrom 2-port equations

(54)

where and are derived when .The requirement in calculating the Z-parameters ensuresthat the network analyzer loading (often 50 ) on the inductivelink does not affect the measurement results. Fig. 6(b) shows the4-coil inductive link, which coils were held in parallel and per-

fectly aligned using sheets of Plexiglas. and were placedin the middle of and in a coplanar fashion, respectively,to achieve .

Fig. 7 compares the measured versus calculated values of thePTE versus coupling distance in the 2-, 3-, and 4-coil induc-tive links. Calculated PTE values are from the RLT and CMTmodels described in previous sections, while the circuit param-eters, such as and , are extracted from the models presentedin [33] and [36] for PSCs and wire-wound coils, respectively. It

(50)


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TABLE III NDUCTIVE LINK SPECIFICATIONS FOR POWER TRANSFER

EFFICIENCY MEASUREMENTS

Fig. 7. Comparison between measured and calculated (using RLT or CMTequations) values of the PTE versus the coupling distance, , for 2-, 3-, and4-coil inductive links speci ed in the Table II.

can be seen that both RLT and CMT models, which are basicallythe same in steady state, estimate the PTE with a high level of accuracy.

It can be seen from Table II that the calculated PTEs of the3- and 4-coil links at cm are 31.4% and 31.3%, whichare very close to the measured results, 29.7% and 27.9%, respec-tively. On the other hand, the measured 2-coil PTE at the samedistance (1.37%) shows a greater difference on a percentage basis from the calculated value of 1.1%. This is most likelydue to the measurement errors and equipment non-idealities atvery low PTE levels. Because, as shown in Fig. 7, the calculatedand measured PTEs for the same 2-coil link match very well at

higher PTE values. The measured and calculated PTEs in 3- and4-coil links show slightly more differences at smaller cm. This is because for the sake of simplicity, we have neglectednon-neighboring coils’ coupling in our models in Section V,which are more signi cant when is small. We would alsolike to point out that the 2-coil PTE at cm istimes smaller than that of 3- and 4-coil links because cannot provide values close to the in (10) for in the 2-coil link. On the other hand, the 3- and 4-coil linkscan achieve the optimal owing to their extra degree of freedom in impedance transformation, provided by , whichdetermines the optimal sizing of and [36].

VII. CONCLUSION

Electrical engineers analyze inductively coupled coils, trans-formers, and RLC tank circuits using lumped models, wellknown Kirchhoff’s current/voltage laws, and a method oftenreferred to as the RLT. Physicists, on the other hand, prefer toconsider the energy stored in the system and exchanged be-tween two or more resonating objects (in this case capacitivelyloaded conductive-wire loops) and use a method known as theCMT. We have presented detailed formulation for calculatingthe PTE based on both theories for a conventional 2-coilinductive power transmission link, and extended the solutionsin the steady state to 3-, 4-, and -coil systems. In each case,we have proven that despite using different parameters andterminologies, both CMT and RLT lead to the exact same setof equations, and therefore, both are applicable to short- andmidrange coil arrangements as long as near- eld non-radiativeconditions are applicable to the resonating circuits. Moreover,

we have derived equations describing the transient behavior of the 2-coil inductive power transmission links using both theCMT and circuit theory. Theoretical analysis and simulationresults show that in this case, CMT is only accurate whencoils have small coupling and large quality factors. However, itsimpli es the analysis by reducing the order of the differentialequations by half, compared to the circuit-based approach. Themeasurement results show that the RLT and CMT equationsare both suf ciently accurate for the PTE calculation.

ACKNOWLEDGMENT

The authors would like to thank members of the GT-Bionicslab for their comments and constructive discussions.

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Mehdi Kiani (S’09) received the B.S. degree fromShiraz University, Shiraz, Iran, in 2005 and theM.S. degree from Sharif University of Technology,Tehran, Iran in 2008. He is currently pursuing thePh.D. degree in the GT-Bionics Lab at the GeorgiaInstitute of Technology, Atlanta.

Maysam Ghovanloo (S’00-M’04-SM’10) was bornin Tehran, Iran, in 1973. He received the B.S. degreein electrical engineering from the University of Tehran, Tehran, in 1994, the M.S. degree in biomed-ical engineering from the Amirkabir Universityof Technology, Tehran, in 1997, and the M.S. andPh.D. degrees in electrical engineering from theUniversity of Michigan, Ann Arbor, in 2003 and2004, respectively.

From 2004 to 2007, he was an Assistant Professor in the Department of Electrical and Computer Engi-

neering, North Carolina State University, Raleigh. In June 2007, he joined thefaculty of Georgia Institute of Technology, Atlanta, where he is currently an As-sociate Professor and the Founding Director of the GT-Bionics Laboratory inthe School of Electrical and Computer Engineering. He has authored or coau-thored more than 90 peer-reviewed conference and journal publications.

Dr. Ghovanloo is an Associate Editor of the IEEE T RANSACTIONS ONCIRCUITS AND SYSTEMS II, IEEE TRANSACTIONS ON B IOMEDICAL CIRCUITSAND SYSTEMS , and a member of the Imagers, MEMS, Medical, and Displays(IMMD) subcommittee at the International Solid-State Circuits Conference(ISSCC). He is the 2010 recipient of a CAREER award from the NationalScience Foundation. He has also received awards in the 40th and 41st DesignAutomation Conference (DAC)/ISSCC Student Design Contest in 2003 and2004, respectively. He has organized several special sessions and was a member of Technical Review Committees for major conferences in the areas of circuits,systems, sensors, and biomedical engineering. He is a member of the Tau BetaPi, AAAS, Sigma Xi, and the IEEE Solid-State Circuits Society, IEEE Circuitsand Systems Society, and IEEE Engineering in Medicine and Biology Society.

the circuit theory behind coupled-mode magnetic

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