Transactions on Biomedical Engineering - 1996 - Geometric Approach for Coil Coupling Enhancement

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708 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 43, NO. 7, JULY 1996

Geometric Approach for Coupling Enhance tof Magnetically Coupled Coilsc. M. Zierhofer,” Member, IEEE, and E. S . Hochmair, Member, IEEE

Abstract This paper presents a geometric approach for theenhancement of the coupling coefficient between two magneticallycoupled coils. It is demonstrated that the coupling coefficientcan be considerably enhanced if the turns of the coils are notconcentrated at the Circumferences but distributed across thediameters. For analysis each of the two coils is assumed to becomposed of concentric circular loops. The experimental resultsare in very good agreement with the theoretical results.

I. INTRODUCTION

NE possibility for transcutaneously providing an im-planted stimulator with power and information is to

transmit radio frequency (RF)-power via an inductively cou-pled coil system. Such a coil system consists of the primarycoil which is outside the body and the secondary coil implantedwith the stimulator. When facing each other, they form atransformer which allows energy transfer from the transmitterto the implant. The distance between the coils essentially de-termines their minimum geometric size, because with respectto the efficiency of the power transfer, the coupling coefficientbetween the coils has to be sufficiently high. If the coupling istoo low, a higher current in the primary coil has to be used toprovide the same output from the secondary coil. Thus morepower is wasted in the primary coil due to 12R-losses.

A considerable body of knowledge is available for thedesign of inductive links including magnetically coupled coils[11-[7]. In medical applications an inductive link shouldusually fulfil two requirements. First, the RF-voltage amplitudeat the secondary coil should be insensitive to variations ofthe relative position of the coupling coils, since this relativeposition is not very well defined. This property, for example, isnecessary in applications where the RF-voltage at the receivercoil is used to derive an implant supply voltage which hasto be kept within particular limits. Second, an optimizationof the efficiency of the power transfer is often desirable.However, these two requirements cannot be met independentlyof each other. For example, the diameter of the secondary coilcan be so chosen as to be smaller than that of the primarycoil. In general, such selection reduces the coupling at aparticular distance between the coils (compared to the casewhen the diameter of the secondary coil is equal that of the

Manuscript received September 26, 1994; revised February21, 1996. Thiswork was supported by the Austrian Science Foundation under GrantsP9616-TEC and P10169-OTE. Asterisk indicates corresponding author.

*C. M. Zierhofer is with the Institute of Experimental Physics, Universityof Innsbruck, Austria (e-mail:[email protected]).

E. S. Hochmair is with the Institute of Experimental Physics, UniversityofInnsbruck, Austria.

Publisher Itcm IdentifierS 001 8-9294(96)04832-X.

primary coil), and thereby reduces the power efficiency, but itmakes the coupling coefficient more insensitive to lateral coildisplacement, as long as the secondary coil remains withinthe diameter of the primary coil [3], [8]. This “geometric”approach aims at directly keeping the coupling coefficientconstant. However, the coupling coefficient is still sensitiveto variations of the coil separation.

Instead of trying to keep the coupling coefficient itselfconstant, the receiver RF-amplitude can be made insensitiveto varying coil coupling by employing resonant circuits inthe transmitter and the receiver, where the coils represent

the inductances. The characteristics of these resonant circuitscan be exploited to obtain a sufficiently constant receiverRF-amplitude within a defined range of coil coupling. Thisapproach is the one which is most commonly used (e.g.,

Most inductive link designs found in literature employtwo circular coils whose turns are concentrated at the coils’circumferences. Investigations of the coils’ geometry have ex-clusively concentrated on calculating the coupling coefficientor the mutual inductance as a function of the ratios of the coildiameters and distance parameters [ 3 ] , [5], and [8]. Attemptsto enhance the coil coupling in order to reduce 12R-lo sse s arelimited to the recommendation that ferrite backings be used

The fundamental question examined in this paper is, whetherit is possible to exploit the area within the outer circumferencesof primary and secondary coils to enhance the couplingcoefficient. It tums out that the coupling coefficient canbe significantly improved, if the turns of the coils are notconcentrated at the outer circumferences, but distributed acrossthe radii. This distribution can be achieved by using single con-centric circular turns to construct the coils, allowing a closed-form mathematical calculation of the coupling Coefficient. Asshown in Fig. 1, primary and secondary coils are composed N,and Nb circular coils, respectively. The connection betweenthese circular coils is assumed to accomplished in such a waythat:

1) the direction of the current is assumed to be equal inall the turns of a coil (and thus each turn enhances themagnitude of the mutual inductance between primaryand secondary coil), and

2 ) the overall length of the connection paths is sufficientlyshorter than the geometric dimensions of the coils, sothat the influence of the current through the paths on theself and mutual inductances is negligible.

In practical applications spiral-shaped coils are more useful.

PI-t41, [61, and U] .

(cf. t41).

0018-9294/96 05.00 996 IEEE

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ZIERHOFER AND HOCHMAIR: GEOMETRIC APPROACH FOR COUPLING ENHANCEMENT OF MAGNETICALLY COUPLEDCOILS 709

Fig. 1.composed of circular concentric loops.

Geometric arrangementand notation for primary and secondary coils

However, it is much more complicated to analyze the couplingbetween spiral-shaped coils. It is assumed that the resultsderived in the following approximate the case of spiral-shapedcoils sufficiently well.

The enhancement of coupling, as demonstrated in this paper,

may be exploited in two ways.1) For given (maximum) sizes of primary and secondary

coils and given self-inductances, an enhancement ofcoupling will reduce the 1’R-losses and thus improvethe power transmission efficiency of the inductive link.

2) For a given coupling coefficient, the size of primary andsecondary coils can be reduced. This property enablesa reduction of the overall size of an implanted device,if the size of the device is determined by the secondarycoil.

11. ANALYSIS

In general, the coupling coefficient k between two magnet-ically coupled coils is defined as

d, distance between the axes p can be expressed by the singleintegral

where Jo and Jl re the Bessel functions of zeroth- and first-order, respectively, [9]. This expression does not contain theradius R of the coil’s wire. It is assumed that the ratios andf are sufficiently small (cf. [ 3 ] ) .

For the case of perfect alignment, i.e., p = 0, (2) leads to

2M ( a ; , p = 0, d = [ : ;)K(r ; ) E ( ) ] ( 3 )

where

and K ( K ) nd E ( K ) re the complete elliptic integrals of thefirst and second kind, respectively.

Equation (3) can be used to derive a formula for the self-inductance of a single circular loop. As shown in [101, for thecondition << 1, the self-inductance of such a loop (radius U

and wire-radius R ) can be approximated by

As mentioned above, primary and secondary coils employedin an inductive link usually consist of a particular numberof single circular loops of approximately equal diameter. Theself-inductance of such coils is approximately equal to theself-inductance of a single loop [as derived in ( 5 ) ] ,multipliedby the square of the number of turns. For a coil composed ofNu concentric circular loops (cf. Fig. 1 with different radii,a i = I . . ,N a ) , and with wire-radius R , the overallself-inductance L , becomes

i=l

N , N ,

where 6 ~ , ~1 for i = j , and bz J = 0 otherwise.

coils, M b can be calculated usingThe mutual inductance between primary and secondary

111. COMPUTED RESULTS

where h f u b is the mutual inductance, and L,; L b are theself-inductances of the coils.

In the following, the relative permeability of the coil mate-rial and its surrounding medium is assumed to be pT = 1.

Following 191, the mutual inductance of two circular air-cored loops whose axes are parallel (radii a and b, coil distance

Equations (6) and (7) allow the computation of the couplingcoefficient as defined in (1). All calculations have been per-formed on an IBM-PC using MATLAB (The Mathwork s, Inc.,South Natick, MA). For convenience, the MATLAB notationis used here for the description of coil configurations. Coil “a”

is described by a = [amax: A: ami, , ] , where the first and the



710 IEEE TRANSACTIONSON BIOMEDICAL ENGINEERING,VOL. 43, NO. 7, JULY 1996

I

‘0 0 1 0 2 0 3 0 4 0 5 0.6 0 7 8 0 9 1

dCoil Distance-

Fig. 2. Coupling coefficient I as a function of spacing for various coilconfigurations. Primary and secondary coils are assumed to be identical. Thenormalized wire radius is 12 .5* l o p 3 .

a m a x

third number within the brackets are the radii of the maximumand minimum loops, respectively, and the second number, A ,is the increment between the radii. With this, a1 = amaxand C L N ~ = a m i n . For example, coil a = [ l : ~ O . l : O . l ]

denotes coil “a” which is composed of 10 loops with radii1, 0. 9, 0. 8, . . . , 0. 2, nd 0.1.

Fig. 2 depicts the coupling coefficient I between two iden-tical coils as a function of the normalized spacing for p =

0. Various coil configurations with different minimum radiia m i n = bmin are considered. The minimum physically possibleincrement between the windings A = 2R = 0 . 0 2 5 ~ ~ ~ ~withwire radius = 12.5 * lop3) is chosen here (exceptfor the trivial case of the single turn coil configuration.Obviously, the lowest coupling coefficient is obtained for thesingle turn coil configuration, and the coupling coefficient isincreasing for decreasing minimum coil radii. For minimumradii smaller than 0.4am,,, the coupling coefficient remainsalmost unchanged.

The coupling coefficient is not very sensitive to variationsof increment A, when radii amax nd a m i n emain unchanged.For example, when equal maximum and minimum radii asin Fig. 2 are considered, and the increment chosen is A =

4R, the relative deviation from the results shown in Fig. 2is within +0.3 and -2.1%. For A = 8R he relativedeviation lies between +3.7 and -0.7 . This insensitivityis a strong indication for the assumption that the couplingcoefficient will also not change very much, if spiral coils (withcorresponding minimum and maximum radii) are used insteadof coils composed of circular turns.

Fig. 3(a), (b), and (c) depicts the coupling coefficient be-tween two identical coils as a function of the normalized lateraldisplacement for three different coil distances = 0.2,0.5,and 0.8, respectively. In each plot, four coil configurations areconsidered, the single-coil configuration a = b = amax), ndcoils with minimum radii = = 0.8,0.6, and 0.4. Asin Fig. 2, the minimum possible increment A = 2R is chosen.

m x a m a x

0 ‘ I0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1

Lateral Displacementa-

(a)

0.7Y

8 0 8

6 0 5

s

M

50 4

0 3

0 2

0 1

vO 0 1 0 2 0 3 0.4 0 5 0.6 0.7 0 8 0 9 1

Lateral Displacementam

(b)

Fig. 3. Coupling coefficientas a function of the normalized lateral displace-ment or identical primary and secondary coils. In each plot, four coil

configurations+ = [ l ] [l: 0. 025: 0. 8] 1: - 0 . 0 2 5 : 0 . 6 ] , a n d[ l : .025:0.4,a i th incrementA = 2R are considered. The normalizedwire radius is The coil distances are: (a) 0.2,(b) d 0.5

a m a x

= 1 2 . 5*m a y all lax

a m a x

Fig. 3 shows that the coupling is more sensitive to lateral

displacement for coils with smallera m i n ,

since the couplingcoefficient is considerably higher at p = 0. The couplingcoefficient of all configurations is about equal at = 1.In many practical applications, the lateral displacement can bekept small using positioning magnets in the centre of the coils.

An intuitive explanation of the coupling enhancement of“distributed” coils is given with the help of two equal coilsa , one primary and one secondary coil. Each of these coilsshall be composed of two windings ( N , = 2 ) which areconcentrated most closely to the circumferences, i.e., A = 2Rand a = [amax amax - ZR]. Self-inductance 150 of coils a



ZlERHOFER AND HO CHMAIR: GEOMETRIC APPROACH FOR COUPLING ENHANCEMENT OF MAGNETICALLY COUPLED COlLS 711

0 8 -

0 7

0 6 --

6 0 58 0.5U

0 7

5

6 0 5 -

0 6 -

0 4 L

0 3 -

0 2

0 1 -

[I : -0.025 : 0.610.3

0.2

0.1

[I : 0.025 : 0 81

i I0 0.1 0.2 0 3 0.4 0.5 0.6 0 7 0.8 0.9 ‘ 1

Lateral Displacement-?La,,

(C)

Fig. 3. Cont inued) . Coupling coefficient as a function of thenormalized lateral displacement for identical primary

and secondary coils. In each plot, four coil configurationsA 1 = [ l ] , 11: 0.025:0.8] , [ l : 0.025:0.6] , and[l: .025:0.43: with increment A = 2R are considered. The normalizedwire radius is - = 12.5 *

a m a x

U m . l x a,,

The coil distance is: (c) 0. 8.a m a x a m a

0.8O’gl, = b = [60 50 451mm

\a = b = [60 50 451mma = b = 6 0 mm

0 10 20 30 40 50 60

Coil Distanced (mm)

0 ‘

Fig. 4. Coupling coefficient as a function of the coil distance for coil con-figurations a = b = GO 451 mm (upper curve) and a = b = 60mm (lower curve). Solid curves are the computed results, circles indicateexperimental data. The wire radius is R = 0.2 mm.

55 50

is calculated with Lo = Lo1 + 502 + 2M12, where Lol, L o 2

are the self-inductances of the single loops, and Mlz is themutual inductance between them. If the wire radius R is small

compared to amax, hen Lo1 M Lo2 M M n M L O O ,nd thus

With the mutual inductance, MO, between primary and sec-ondary coils at a particular separation, the coupling coefficientis simply IC0 = 5 cf. ( l ) ] . Now the radius of the inner turnin both coils is reduced, i.e., a’ = [amax amax 2Ra], witha > 1. This in general will reduce both, MO and Lo. However,

Lo M 4&0(= N ~ L o o ) .

1

;a = b = 60 m m

0 9 t

V0 3

0 2

0 1

10 20 30 40 50

Lateral Displacementp mm)

(a)

1

= b [60 55 50 451 inm

O 9

O a t

0; 10 20 30 40 50 60

I.atcral 1 isplacernetit p min)

(b)

Fig. 5. Coupling coefficient as a function of the lateral displacement forcoil separations d = 10,30, and 60 mm. Solid curves are the theoreticalresults, circles indicate experimental data. The wire radius is R = 0.2mm: (a) Single-tum coils a = b = 60 mm and (b) “Distributed’ coilsa = b = [60 55 50 451 mm.

the effect on Lo will be stronger than on M O , ince especiallythe mutual inductance between the single turns is rapidlydecreasing, and thus the coupling factor ko, as defined above,is increased.

Iv. EXPERIMENTAL ESULTS

The theoretical considerations presented above were verifiedby experiment. Measurements were performed with coppercoils attached to a construction made of plexiglass. The wireradius was R = 0.2 mm. For measurement, a sinusoidalvoltage with an amplitude of uo = lV, and a frequencyof fo = 2 MHz was applied to the primary coil, and thevoltage at the unloaded secondary coil was dletected. Equalcoil configurations for primary and secondarry coils wereinvestigated. In this case, the coupling coefficient is simply



712 IEEE TRANSACTIONSON BIOMEDICALENGINEERING, VOL. 43, NO. 7, JULY 1996

TABLE IDISTRIBUTEDOILCONFIGURATIONSITH a m a x = 12 mm TO I M P L E M E X T

COILSW I T H SELF- INDUCTANCEA = 0.88 pH ( R = 0. 125 mm)

Wire Length Relative UnloadedQuality Reduction

l Q L i % )

Number of Turns Radiusn=, 1 (mm)“14 11.25 294 05

9.11 3372.8

6 6.79 365 9.57 4.79 384 -23.48 3.09 397 -25.99 1.66 407 -27.8

equal to the amplitude ratio of secondary and primary voltage(this can be easily verified by using the T-equivalent circuitof two coupled coils).

Fig. 4 depicts the coupling coefficient as a function of thecoil distance for p = 0. Two coil configurations a = b =

[BO 55 50 451 mm and a = b = 60 mm are investigated.As expected, a considerably enhanced coupling coefficient isobtained for the “distributed” coil s. For example, at d = 0mm, the coupling for the “nondistributed” and “distributed”coils are knondlst = 0.109 and kdlst = 0.184, respectively(theoretical values), which means an enhancement by 68.8 .The experimental data are in very good agreement with thetheoretical results.

Fig. 5 shows the coupling coefficient as a function of thelateral displacement for various coil distances d = 10.30, and60 mm. Again, the values measured are very close to thecalculated results.

V. DISCUSSION

In the previous sections of this paper it has been shown thatdistributing the turns of coils across the radii considerablyenhances the coupling coefficient. The question arises, howthis improvement can be exploited in practical applications,either to enhance the power transfer efficiency, or to reducethe geometric size of the coils.

Let us consider a typical inductive link for transcutaneouspower and da ta transfer as shown in Fig. 6(a). This linkconsists of a parallel tuned transmitter and a parallel tunedreceiver circuit, which are magnetically coupled. Resistor RIrepresents the output resistance of the RF-amplifier drivingthe link, and resistor Rz is the load. Resistors Rsl and R s zare the series resistances of transmitter and receiver coils,respectively. For convenience, circuit Fig. 6(a) is rearrangedas shown in Fig. 6(b), where the coil losses are representedby the parallel resistors Rpl and R p 2 . Series-and parallel coilresistances are related via

with angular frequency w , and index i = 1 , 2 .

receiver resonant circuits is defined asThe center frequency W O 01 o z of transmitter and

The overall loaded qualities, Q i are defined via the relation

where the qualities Q R ~nd Q L ~epresent the qualities of theresonant circuits either exclusively due to resistances Ri, or

exclusively due to resistances R,,, respectively. Qualities Q L ~are often called the unloaded qualities of the coils.Following [3], efficiency ~ 1 1 2 as the ratio between the

power delivered to the receiver circuit and the overall powerconsumption is given by

1

712 = 1 (1 1)

where k is the coupling coefficient. For this equation it isassumed that the system is operated at the center frequency,i.e., lu‘ = W O .

Within the receiver circuit, the power is split up between theload, R 2 and the parallel coil resistor, R p 2 . This is describedby efficiency 7722 defined as

1 1

with Q R ~nd Q L ~rom (10). Thus, the overall efficiency,7 s the ratio between the power in load R2 and the overallpower consumption is given by

= 71277221 1

(p++(k+ )(k+ )) I + % ) .(13)

Obviously, efficiency 7 keeps increasing with increasingcoupling coefficient k , and therefore any coupling enhance-ment-at a particular coil separation-results in an improvedefficiency. It can also be seen that for a high efficiency theunloaded qualities Q L ~hould he as high as possible. As

suggested in this paper, a coupling enhancement is achievedby using “distributed” coils. However, the distribution ofthe coil windings across the radii results in a particularreduction of the unloaded qualities. This is because the wirelengths necessary to achieve a particular inductance for the“distributed” coils-and with it the series resistances-aresomewhat greater than for coils, where the windings areconcentrated at the circumferences. Looking at (8) and (10)it can be seen, that the series resistances are inverselyproportional to the unloaded qualities. The wire length I

of coil a composed of N , turns can be calculated byN

I = 27r a2 + 2(am,, m i n ) . . (14)i = l

where the term 2( amax a m i n ) epresents the overall wirelength necessary to connect the circular turns of the coil witheach other. A typical example is demonstrated with the helpof Table I. Coil a = [ l a : .25: 11.251 mm with R = 0.125m m is composed of N , = 4 turns which are concentratedmost closely to the circumference (outer radius: amax = 1 2



ZIERHOFER AND HOCHMAIR: GEOMETRIC APPROACH FOR COUPLING ENHANCEMENTOF MAGNETICALLY COUPLED COILS 713

(b)

Fig. 6 . (a) Circuit diagram of an inductive RF-link and (b) approximated circuit

mm, inner radius: a m i n = 11.25 mm). The resulting self-

inductance is L,=

0.88 pH. The same self-inductance canbe achieved with N , = 5,6 ,7 ,8 , or 9 turns (first column),where the distances between the turns are increasing and theminimum radii a m i n are decreasing (second column). Columns3 and 4 depict the resulting wire lengths [according to (14)]and the relative reduction of the unloaded quality AQL inpercent referred to the coil composed of four turns.

To summarize, “distributed’ coils yield, on the one hand,an increased coupling between transmitter and receiver coils,but reduce the unloaded qualities on the other. Nevertheless, anet improvement of the overall efficiency can be obtained,as demonstrated with the help of Fig. 7. Here, the overallefficiency 7 is computed for three different implementationsof transmitter and receiver coils (for each case, transmitter

and receiver coils are identical a = b) with inductancesLa = Lb = 0.88 pH (wire radius R = 0.125 mm). Thequalities are chosen Q n l = 20 and Q R Z = 5.

Case 1: Coils with amax = 12 mm, a m i n = 11.25 mmcomposed of N , = 4 turns are assumed (cf. firstrow in Table I), and the unloaded qualities areassumed to be Q L = Q L ~ Q L Z= 80.

Case 2: Coils with the same inductances as in Case 1,but with N , = 6 turns (third row in Table I) areselected, resulting in increased wire-lengths of thecoils and thus in reduced unloaded qualities Q L =

Q L ~ Q L ~ 80 * 0.805 = 64.4. Nevertheless,the overall efficiency is considerably higher than

in Case 1. For example, at d=

10 mm, 7=

0.52in Case 1, and rl = 0.61 in Case 2, which meansa relative improvement of 17.3%.

Case 3: Here the calculation of 7 is based on a coupling co-

alizable) “concentration” of turns and shows-at

least for the present example-the greatest possibleimprovement of “distributed” coils compared to“concentrated” coils. In this case, the wire lengthis N * 12 * 2 * T = 267 mm, which means alength reduction of Al, = -8.8% and thus aslight enhancement of the unloaded (qualities Q L =

Q L ~ Q L ~ = 87.9. The overall efficiencyis clearly below the efficiencies of ICases 1 and 2.At d = 10 mm, 7 = 0.45 in Case 3, which meansan efficiency reduction of 13.5% referred to thecorresponding efficiency in Case 1.

So far, only the overall efficiency of the power transferof the inductive link Fig. 6 has been considered. The ideaspresented in this paper may also be exploited to reduce the

overall size of transmitter and receiver coils in applications,where the voltage at the receiver coil has to be mensitiveto coupling variations. Following [ 3 ] , a maximum of thereceiver voltage as a function of coupling coefficient 5 occursat “critical coupling,” i.e., when k d m quals one. Atcritical coupling, efficiency 712 = 0.5 (note, ithat 7 1 2 is theefficiency of the power transfer fr om the transmitter to thereceiver circuit). Assuming the same circuit pammeters as inCase 1 of above (i.e., La = LI, = 0.88 ,uH, R = 0.125 mmQ R ~ 20, Qnz = 5, and coils with amax = b = 12mm, amln = b,,, = 11.25 mm, N, = 4 , Q L = SO),critical coupling occurs at a distance dc,,t = 10.8 mm. Thesame distance for critical coupling and thus for efficiencyvla = 0.5 is achieved for coils aInax = b = 10.3 mm,

umln = b = 6.8 mm, N , = 6. This means that theouter radii can be reduced from 12 mm to 10.3 mm, whichcorresponds to a size reduction of the coils by -14%.

efficient k derived from single turn coils with radiia = b = 1 2 mm. It is assumed that for both coilsN = 3.55 turns are concentrated at the same radiusto achieve equal inductances as in Cases 1 and 2.This assumption might be of theoretical interest,since it represents the most extreme (but nonre-

VI. CONCLUSION

This paper presents a geometric approach for enhancingthe coupling between two magnetically coupled coils. Thisenhancement is achieved by distributing the turns of the coilsacross the radii instead of concentrating them at the outer



714

IF-----0 9

0 7

r

9 0 6 -

c0 5 -

0 4 -

0 3 -

0 2 -

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING,VOL. 43, NO. 7, JULY 1996

0 ‘5 10 15 20 25

Coil Distanced (mm)

Fig. 7. Overall efficiency 7 as a function of coil separationd . For all cases,transmitter and receiver coils are identical, andL , = Lb = 0 .88 p H (wireradius R = 0.125 mm ), and qualities Q R ~ 20 and C ) R ~= 5. Case 1:a m a x = b,n,, = 12 mm, amin = b,n;n = 11.25 mm, .Ya = 4. Q L = 80.0:

Case 2: arrlax=

h=

12 mm,a m i n =

brrlin G.79 mm,Ya =

6.Q L = 64.4; Case 3: a m a X = h = = b,,, = 12 mm, = 3..>.5.I, = 87.9.

circumferences. As shown in Section 111 for identical primaryand secondary coils, for given maximum and minimum coilradii, the coupling of distributed coils is insensitive to thespacing between the windings of the coils. Thus, in practicalapplications, the self-inductances can be chosen according tothe requirements of the link, relatively independent from thecoupling coefficient.

The price paid for such a coupling enhancement are mod-erate decreases of the unloaded qualities of the coils due toincreased wire lengths (assuming equal inductances and equalouter radii of concentrated and distributed coils). Typically,the unloaded qualities are reduced by about 20%. However, inmost applications, the coils are parts of transmitter and receiverresonant circuits. If the unloaded qualities a re sufficiently highcompared to loaded qualities, the effect of coupling enhance-ment on the overall efficiency q predominates over the effectof unloaded quality reduction and thus a net improvement ofoverall efficiency is obtained.

The results presented here are calculated for identical pri-mary and secondary coil. However, the formulas derived canbe applied for arbitrary coil configurations.

Some experiments have been performed to verify the theo-retical results. The experimental data are in very good agree-ment with the Dredicted values.

REFERENCES

J . C. Shuder, H. E. Stephenson, and J. F. Townsend, “High-leveltlectromagnetic energy transfer through a closed chest wall,” inIRElnr. Conv. Rec., vol. 9, pt. 9, pp. 119-126, 1961.W. H. KO,S . P. Liang, and C . D. F. F ung, “Design of radio-frequencypowered coils for implant instruments,”Med. Biol. Eng. Comput., vol.15, pp. 634 64 0, 1977.E. S . Hochmair, ‘System optimization for improved accuracy in tran-scutaneous signal and power transmission,”IEEE Trans. Biomed. Eng.,vol. 31, pp. 177-186, Feb. 1984.D . C. Galbraith, M. Soma, and R. L. White, “A wide-band efficientinductive transdermal power and data link with coupling insensitivegain,” IEEE Trans. Biomed . Eng.,vol. BME-34, pp. 265-275, Apr. 1987.M . Soma, D. C. Galbraith, and R. L. White, “Radio-frequency coilsin implantable devices: Misalignment analysis and design procedure,”IEEE Trans. Biomed. Eng.,vol. BME-34, pp. 276-282, Apr. 1987.C. M. Zierhofer and E. S. Hochmair, “High-efficiency coupling-insensitive transcutaneous power and data transmission via an inductivelink,’‘ IEEE Trans. Biomed. Eng., vol. 37, pp. 716-722, July 1990.N. de N. Donaldson and T. A. Perkins, “Analysis of resonant coupledcoils in the design of radio-frequency transcutaneous links,”Med. B i d .Eng. Coinput., vol. 21, pp. 612-627, 1983.F. C . Flack, E. D. James, and D. M . Schlapp, “Mutual inductance ofair-cored coils: Effect on design of RF-coupled implants,” Med. Biol.Eng.. vol. 9. pp. 79-85, 197 1.L. Hannakam, “Berechnung der Gegeninduktivitat achsenparallelerZylinderspulen,” Archivfiir Elektrotechnik,vol. 51, no. 3, pp. 141-154,1967.W . Greiner, Theoretische Physik, vol. 3. Frankfurt am Main: H.Deutsch, 1991.

C. M. Zierhofer (M’91) was born in Innsbruck,Austria, in 1962. He received the DipLIng. andDr.Techn. degrees in electrical engineering from theTechnical U niversity of Vienna, Austria, in 1985 and1989, respectively.

In 1986 he joined the Institute of Experimen-tal Physics, University o f Innsbruck, Austria. Hisresearch interests include cochlear implant design,analog and digital signal processing, and RF -poweramplifiers.

rwin S. Hochmair (M’83) was born in Vienna,Austria, in 1940. He received the DipLIng. andDr.Tech. degrees in electrical engineering from theTechnical U niversity of Vienna, Austria, in 1964 and1967, respectively.

In 1965, he joined the Institutefor Physical Elec-tronics, Technical University of Vienna, Austria.From 1970 to 1972, he was with the NASA MarshallSpace Flight Center, AL, as a Research Associate.During 1979, he was a Visiting Associate Professorat Stanford University, Stanford, CA. Since 1986 ,

he has been a Full Professor at the Institute of Experimental Physics at theUniversitv of Innsbruck. Austria. His current research interests are circuitdesign, signal processing, and cochlear implant design

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Transactions on Biomedical Engineering - 1996 - Geometric Approach for Coil Coupling Enhancement