228 ieee transactions on biomedical circuits and …gulak/papers/zargham12d.pdf · 228 ieee...

228 IEEE TRANSACTIONS ON BIOMEDICAL CIRCUITS AND SYSTEMS, VOL. 6, NO. 3, JUNE 2012

Maximum Achievable Efficiency in Near-FieldCoupled Power-Transfer Systems

Meysam Zargham, Student Member, IEEE, and P. Glenn Gulak, Senior Member, IEEE

Abstract—Wireless power transfer is commonly realized bymeans of near-field inductive coupling and is critical to manyexisting and emerging applications in biomedical engineering.This paper presents a closed form analytical solution for theoptimum load that achieves the maximum possible power ef-ficiency under arbitrary input impedance conditions based onthe general two-port parameters of the network. The two-portapproach allows one to predict the power transfer efficiency atany frequency, any type of coil geometry and through any typeof media surrounding the coils. Moreover, the results are appli-cable to any form of passive power transfer such as provided byinductive or capacitive coupling. Our results generalize severalwell-known special cases. The formulation allows the design of anoptimized wireless power transfer link through biological mediausing readily available EM simulation software. The proposedmethod effectively decouples the design of the inductive couplingtwo-port from the problem of loading and power amplifier design.Several case studies are provided for typical applications.

Index Terms—CMOS coil, conjugate matching, energy har-vesting, inductive coupling, lab-on-chip, matching networks,medical implant, near-field, neural implant, on-chip receiver,optimum frequency, optimum load, power transfer efficiency,RFID, wireless power transfer.

I. INTRODUCTION

W IRELESS POWER TRANSFER (WPT) is critical tomany emerging applications and is commonly real-

ized by means of near-field inductive coupling. This type ofpower delivery system is advantageously used for biomed-ical implants [1]–[3] neural activity monitoring/stimulation[4]–[7], emerging lab-on-chip (LoC) applications, RFID [8]and non-contact testing [9]. In this system the circuits containedin the implant, the LoC or the silicon substrate are remotelypowered by means of a power amplifier operating at a fixedcarrier frequency. Additional functionality is achieved bymodulating the carrier frequency in some manner to realizeunidirectional or bidirectional command and data transfer.

The power efficiency of the near-field link is a measure of: (i)the power loss in circuits both at the transmitter and receiver,

Manuscript received May 19, 2011; revised July 30, 2011 and October 03,2011; accepted October 29, 2011. Date of publication January 06, 2012; date ofcurrent version May 22, 2012. This work was supported by NSERC. This paperwas recommended by Associate Editor E. M. Drakakis.

The authors are with the Department of Electrical and Computer En-gineering, University of Toronto, Toronto, ON M5S3G4, Canada (e-mail:[email protected]).

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

Digital Object Identifier 10.1109/TBCAS.2011.2174794

(ii) the absorbed EM energy in tissue1 that causes the localtemperature to increase possibly harming the biological tissue,and (iii) how often the battery has to be recharged when usedin the context of portable medical devices. Hence in the caseof implants, low-efficiency WPT implementations may causediscomfort and possible complications for the patients usingit. Similar issues occur in the case of LoC applications wherethe local temperature of a small 10 to 100uL biological samplebeing measured needs to be held within strict tolerances (oftenwithin one Centigrade degree). Hence it is not possible to arbi-trarily increase the strength of the EM fields to realize greaterpower transfer to the embedded system.

In most applications, achieving high power-efficiency is ex-tremely challenging due to the restriction on the geometry of theinductive media. Therefore, a great deal of attention in the liter-ature has been devoted to optimization of near-field inductivelycoupled links. Previous authors have addressed the issue of linkoptimization using a simple inductor model in air for fixed loadimpedance at low frequencies [10]–[18]. Throughout the paperwe refer to an inductively coupled link in air as a simple two-portmodel. In most practical applications, the inductive two-port isdesigned using numerical electromagnetic simulation softwarepackages such as HFSS [19] or Momentum [20] that returns Sparameters. Extracting the simple R, L model from these param-eters, especially at high frequency, is quite challenging. In addi-tion, and of central concern in this paper, many wireless powertransfer applications require the EM waves to pass through bio-logical material such as skin, muscle, fat, buffer solutions, etc.,which we refer to as a general two-port model. These media areconductive and have higher relative permittivity constants thanair [21], [22]. Hence optimizing the link using a simple two-portmodel alone and ignoring the media during the optimizationphase incurs large penalties in terms of achievable power ef-ficiency. It is highly desirable for the output voltage to be insen-sitive to small changes in the distance between the two coils aswell as lateral or angular misalignments. The main focus of thispaper is the efficiency of these links using aligned coils. How-ever the effects of lateral, vertical and angular misalignment spe-cific to our discussions is briefly discussed in Appendix D andis more generally addressed in [23]–[25].

Recently [26], [27] and others have realized the shortcomingsof the simple two-port model at high frequencies and proposedthe use of S-parameters under simultaneous conjugate matchingto address these issues. However, it is well known that matchingresults in maximum power transfer but not necessarily, max-

1The rate at which energy is absorbed by biological tissue is known as SAR(Specific Absorption Rate). FCC regulates the acceptable maximum SAR forRF devices.

1932-4545/$31.00 © 2012 British Crown Copyright

ZARGHAM AND GULAK: MAXIMUM ACHIEVABLE EFFICIENCY IN NEAR-FIELD COUPLED POWER-TRANSFER SYSTEMS 229

imum efficiency [28]. In fact, conjugate matching has a theoret-ical upper bound of 50% efficiency while a general two-port canbe designed to have power efficiencies approaching 100%. Themathematical derivations presented in this paper prove that, un-like conjugate matching, the optimum load is independent of thesource impedance and solely depends on two-port parameters.

Another short-coming in the published classical link opti-mization techniques is the assumption of fixed load impedance.This assumption forces an extra unnecessary constraint on thedesign of the coupled inductors that could result in sub-optimalcoil parameters. The reported power efficiency in such systems[10], [14] is between 30 to 50%. By introducing the conceptof optimum load and source impedance one effectively addsnew design parameters to the system beneficially decouplingthe problem of loading effect from the optimization process ofthe link. Our proposed approach achieves power efficiencies ofgreater than 80% at much greater coil separations to significantadvantage in practical realizations.

An interesting feedback approach was used by [17] to an-alyze the simple two-inductor model. However their proposedoptimum load and efficiency is an approximation and is not es-pecially accurate at low efficiencies though is increasingly ac-curate at high efficiency values.

R. Harrison [18] suggested guidelines for maximizing thepower efficiency of a simple two-port case. However no spe-cific optimum load was presented.

Simrad et al. [16] concluded that there exists an optimum loadfor which the efficiency is maximized but resorted to numericalmethods to find the optimum load.

Silay et al. [29] studied the effect of loading on maximizingthe power efficiency of the link for a simple two-inductor model.However they did not decouple the input impedance from theload and hence their stated maximum achievable power effi-ciency of 67%, is lower than the theoretically achievable bound.In addition they all used a simple two-inductor model in air,which suffers from the same shortcomings stated earlier.

In [14], [30] a four-coil coupled system has been proposedin an attempt to add a degree of freedom to the effect of loadand source impedance on the power efficiency of the system.However any method of impedance transformation introducesadditional losses due to the finite quality factor of the compo-nents. In the case of four-coil systems, the transformation is car-ried out using coils with Q values up to 150. As we will seein Section III, the proposed method of matching networks usesdiscrete capacitors and inductors. The capacitors have Q valueshigher than 1000. Therefore the matching networks using onlycapacitors tend to have lower penalties in terms of efficiency.In addition to this, having four coupled coils increases the cost,size, complexity of design and enforces several constraints onthe inductor geometry.

This paper presents the first published result that optimizesthe near-field link based on the general two-port parameters ofthe network. In this approach we introduce the concept of op-timum load for any passive two-port network. We also derivea simple closed-form expression for the maximum achievablepower efficiency of the given two-port and show that it is the-oretically possible to approach 100% power transfer efficiency.These results also provide insight into the design of such links

Fig. 1. General two-port power transfer system model.

by introducing a simple criterion on the two-port parameters tomaximize power transfer efficiency. Moreover, the results areapplicable to any form of passive power transfer such as induc-tive or capacitive coupling. These derivations provide a pow-erful tool for modifying the simple two-port inductor model tothe more complicated but realistic general form (e.g. adding theconductance between the two coils to model the conductivity ofmedia) and quickly observing the effects on the efficiency andoptimum loading in the system. Therefore, it is easy to optimizea realistic wireless power transfer link through biological mediausing readily available EM simulation software.

The optimum load is realized using matching networks. How-ever, these matching networks are usually lossy and affect themaximum achievable power efficiency. In this paper, we addressthese issues and comment on the design of the matching stagesto achieve optimum efficiency. The remainder of the paper isorganized as follows. In Section II, we introduce simple opti-mization criteria to achieve maximum power transfer efficiencythrough a general passive two-port network. Using these cri-teria we introduce the maximum achievable power efficiencyand optimum loading condition for a general passive two port.In Section III, we discuss how to mitigate loss of efficiency inmatching networks and the resulting optimum number of stagesfor matching. Section IV, provides several case studies on in-ductive coupling through air, biological tissue encountered inimplants and blood for lab-on-chip applications. Throughout wemake quantitative comparisons with measured published resultswhenever possible.

II. THE POWER EFFICIENCY OF THE TWO-PORT

In this section we will derive the power transfer efficiency, orsimply power efficiency, of a general passive two-port networkfrom the source to the load. Fig. 1 shows the block diagram fora general inductive-coupled power transfer system. The powerefficiency, or simply the efficiency, of the system is defined as

(1)

where is the power delivered to the load and is the powerdelivered by the source . The value of depends on var-ious parameters such as the load , the source impedance

, the impedance loading the source and the two-portparameters. Therefore to achieve the maximum possible effi-ciency in the system we need to be able to freely choose the load

and the desired input loading . As shown in Fig. 1,these impedance conversions are realized using the matching


networks. In order to obtain the maximum possible efficiencyof a two-port we first derive the efficiency for Fig. 1, then weintroduce the conditions on and that would result in themaximum possible value for (1). A general linear two-port isrepresented in terms of its ABCD parameters

(2)

(3)

Without loss of generality we choose the desired impedance,, to be n times smaller than

(4)

where n is an arbitrary positive real number. Therefore, thevoltage at the input of the two-port due to the source is

(5)

where is the impedance at the input of the two-port andand denote the real and the imaginary parts of the

expression. In (5) we have assumed that the matching networksare lossless. This assumption is revisited in detail in Section III.The voltage is then transformed by the two-port gain andshows up at the second port as hence

(6)We can then simplify the expression by substituting with

its ABCD parameters

(7)

Hence (6) simplifies to

(8)Using (8) the power efficiency from to can be calcu-

lated, where is the admittance of the

(9)

As expected, is a function of . Hence, there exists an op-timum load that would maximize . Therefore by max-imizing (9) with respect to the real and imaginary parts of theload and replacing the ABCD parameters with Z-pa-rameters, we can show that the maximum achievable efficiencyunder optimum loading conditions in any passive two-port net-work is

(10)

where

(11)

and , , and. The value of that allows for the maximum

efficiency in (10) is given by

(12)

(13)

As seen from (12) and (13), in general, the proposed optimumload is not matched to the two-port. Therefore, the optimumpower efficiency does not happen when the load is matched tothe two-port. In fact matching would never result in efficiencieshigher than 50% while (10) can theoretically be as high as 100%.The term in (10) is a function of and representsthe efficiency from the source to the input of the two-port for alinear voltage source. The choice for depends on the inputdriver . In practice, the two-port is driven by a class-E poweramplifier. Therefore, should be replaced by the effi-ciency of the employed power amplifier. Thus a more realisticform of (10) is given by

(14)

where represents the power amplifier efficiency and

(15)

is the two-port efficiency. The efficiency of a power amplifieris a function of its load and this is what drives the choice be-hind . It is a well-known fact that there exists an optimumload, usually referred to as , which maximizes the powerdelivery efficiency of a power amplifier. The value of iscompletely different from the small-signal output impedance ofthe power amplifier and is generally found using load-pull tech-niques [28]. Therefore, to maximize the power from the sourceto the load it is essential that the two-port would provide the ap-propriate loading for the power amplifier. Theefficiency of a class-E power amplifier is theoretically100% and in practice efficiencies higher than 75% are achiev-able. The second term in (10) is a function of two-port parame-ters. In order to maximize the power efficiency of the two-port,we need to maximize . Fig. 2 shows the maximum possiblepower efficiency from the two-port to the load as a function ofthe variable .

Equations (12) and (13) represent the optimum series load.The equivalent parallel load is calculated in (16), (17). Thesequantities are best represented in terms of the network Yparameters

(16)

(17)


Fig. 2. Maximum two-port efficiency as function of �.

Appendix F studies the variations in power transfer efficiencyas the load deviates from the optimum load. It is interesting tonote that the well-established simultaneous conjugate matching,that is widely used in microwave amplifiers [31] and guaran-tees maximum power transfer, occurs in the special case where

. In this situation the, where is the maximum available power from

the source, which is realized under source matching condition. In the literature is commonly referred to as

transducer gain. The maximum value for happens underthe simultaneous conjugate matching condition and it is usuallystated in terms of S-parameters [31]. For a passive two-port

(18)

where

(19)

By simplifying (18) in terms of the two-port parameters it canbe shown that

(20)

Therefore, our result agrees with the well-established ,under the matched source condition. has 50% as itsupper bound for the efficiency. In order to provide more insightinto what each of these quantities represent we can relate themto a first-order, simple inductive coupling two-port model. Fig. 3shows the circuit block diagram for such a two-port system.Using Fig. 3 we find that

(21)

where and are the quality factor for each of the inductorsand k is the coupling factor between the two coils. This showsthat for a simplified model, in order to increase the efficiencywe have to increase the mutual inductance and minimize the re-sistance. Using (15) the maximum achievable power efficiencyfrom the two-port to the load is given by

(22)

Fig. 3. Simple model for inductive power transfer.

It is no surprise that (22) exactly matches with the maximumpower efficiency derived in [10] for the same simple circuit. Inorder to gain some understanding about the optimum load, wewill further simplify the model and assume that the capacitancesare cancelled out by the matching network, the optimum load forthe network in this case is

(23)

(24)

where is the reflected from the sourceside. As is evident in (24), the imaginary part of the optimumload would completely ignore the impedances transferred fromthe input side and only resonate out the imaginary part of thecoil, hence maximizing the voltage on the load. Once again (24)perfectly matches the common practice of resonating out theload presented in [10]. Unfortunately, (24) only holds as longas the coupling is purely reactive. A good approximation of thiscase is when the two inductors are coupled through air, similarto this simple example. The picture changes, when the mediain between the coils is conductive, , e.g tissue orbiological media. In such scenarios the optimum load should becalculated using (12) and (13). The optimum load impedancebalances the current between the conductive and inductive pathsuch that the efficiency is maximized. Hence the resulting loadis different from what is commonly practiced (resonant tuning)in the design of implantable wireless power delivery systems.In the derivations presented up to this point, the losses of thematching networks were neglected. In Section III, we considerthe effect of these non-idealities on the total efficiency of thetwo-port.

III. MATCHING NETWORKS AND EFFICIENCY

The conversion of load impedance to the optimum loadand the input impedance to the desired impedance

has to be conducted through a filter commonly referred toas a matching network. Matching networks can transform anyimpedance with non-zero resistance to any desired resistance.The reactive part of the desired load is then easily adjusted byadding a reactive component in series or parallel. Thereforewithout loss of generality we will assume that the matchingnetwork is transforming a general complex load to a purelyresistive desired load. There are different types of matchingnetworks such as , T or L to choose from [31]. In situationswhere the quality factor of the matching network is not en-forced and efficiency is of primary concern, L-match is a goodchoice [28]. Therefore in this section, the analysis are basedon multi-section L-match networks. This being said, similarderivations can easily be developed for other types of matching


Fig. 4. L-match sections for two different conversion cases (a) and (b).

networks. L-match networks are aptly named as they consist oftwo elements that form an L-shape circuit.

A. L-Matching Network Analysis

Depending on whether we need to increase or decrease thereal part of the impedance, one of the two L-Section circuits inFig. 4 is used.

The efficiency loss through the matching network is due to theresistance of components used in the matching network. In thefollowing analysis we assume the losses are small enough notto affect the impedance conversion operation of the network.We will first address case (a) where . The Q of theL-match network is defined as

(25)

Using (25), the value of the reactance X and susceptance Bare given by

(26)

(27)

Using (26), (27), the portion of power loss due to andis calculated to be

(28)

(29)

where and are the Q of theseries and parallel components used in the matching networkand is quality factor of the load. Assuming

, the total efficiency through the matchingnetwork for case (a) is found to be

(30)

We can follow the same procedure for case (b) where .The quality factor in this case is given by

(31)

The value of reactance X and the susceptance B for case (b) aregiven by

(32)

(33)

Using (32), (33) the loss in the matching network is found to be

(34)

(35)

Assuming ,

(36)

On the load side, usually we need to step down the load (com-monly case b) and . On the source side, on the otherhand, the load resistance is the series resistance of the trans-mitter coil and is small. We will refer to this as , and there-fore we are commonly dealing with case (a). Equation (21) sug-gests that high efficiency occurs when the coils have high Q.Therefore . In such scenarios the loss through thematching networks can be simplified to

(37)

(38)

Therefore, it is vital to use very high Q components. Thetransmitter inductors made using PCB traces have a Q between50 and 250 in air, therefore, the series matching component onthe source side needs to be a capacitor with a very high Q. Onthe load side however we can improve the efficiency by reducingthe effective Q.

B. Optimum number of stages

According to (25), (31) the Q of the matching network usingone stage may become large which as we saw in (30), (36) canhurt the power efficiency of the conversion. A remedy can befound by using multiple stages, each stage having . Using (38)the efficiency of each section i is

(39)

There exist an optimum number of stages that maximizes thetotal efficiency

(40)

Using the proof presented in Appendix C, all stages should pro-vide equal impedance conversion and the optimum number ofstages for large Q is

(41)

IV. CASE STUDIES

A natural question at this point is, what is a practical achiev-able value for and how much does the efficiency degrade with


a non-optimal load. How does biological tissue or relevant liq-uids such as blood affect the optimal coil design strategy? Is itpossible to integrate the receiver coil on-chip using a standardCMOS process? Is there an optimum frequency of operation tomaximize power transfer efficiency? In order to address thesequestions, we present three different case studies to demonstratethe power of the derived equations, verify the derivations inSection II, and provide intuition on possible achievable powerefficiencies in different media as well as insight into coil design.It will become evident through the examples that the establishedconventional wisdom regarding coil design for biological tissueneeds to be revisited.

A. Case Study 1: Two Coil Power Transfer through Air andMuscle Media

Question: How does a conductive medium with higher rel-ative permittivity between two coils affect WPT design? It isgenerally understood that losses through biological tissues andliquids are negligible at frequencies between several hundredkiloHertz to 20 MHz. As a result designers often ignore the ef-fect of biological tissue during the coil design process [11], [12],[14], [32], [33] and optimize the coils assuming air as the mediabetween the two coils. In this example we challenge this conven-tional wisdom and re-evaluate coil design for conductive mediasuch as muscle. In order to highlight the effect of biologicaltissue in between the two coils and to validate our simulationresults we use the measurement data from [11], [34].

Jow et al. [11] proposed a coil design strategy for poweringup biological implants. The paper presents measurement resultsfor air and claims that the design when used in the context of abiological implant would not significantly affect the efficiencyunder 20 MHz. In order to further investigate this claim andanswer the question of whether or not the coil design has to berevisited, we first use Momentum [20] along with our methodto reconstruct the measured data from [11] for the case of asimple two-port with air between the two coils. Then we usethe same set of coils for the case of a general two-port to powerup an implant buried under layers of skin, muscle and fat. Theexperiment [11] uses two 1-oz copper FR4 PCB pancake coilswith 10 mm of separation. They report a measured efficiency of75% at 5 MHz from the input of the two-port to the load. Thesummary of the coil parameters is shown in Table I.

Fig. 5 shows how these coil parameters map to the specificgeometry of the implemented coils. We used Momentum [20]to simulate the coils. We used (9) to consider the effect of loadimpedance on the efficiency. The real part of the load impedancewas set to 500 [11] and the imaginary part was set to cancelthe from the two-port.

Fig. 6 shows our simulated data versus the measurement re-sults from [11]. There is very good agreement between our sim-ulation results (b) and the measured data (a) presented in [11].Next, we replace the air media in our simulations with 1 mmof skin, 2 mm of fat and 7 mm of muscle media as shown inFig. 7. The frequency dependent permittivity and conductivityof the tissue can be modelled using the four parameter cole-colemodel presented in [22]. The summary of dielectric propertiesat 5 MHz is presented in Table I. In order to obtain the maximumpossible efficiency, we used the optimum load (12), (13) for

TABLE ICOIL GEOMETRY FOR [11] AND OPTIMIZED COILS

Fig. 5. Geometry of the coil based on � , w and s.

Fig. 6. (a) Measurement results from [11] for Coils A1/A2 with 10 mm of Airseparation and 500� load. (b) Momentum simulation using (9) for Coils A1/A2with 10 mm of air separation and 500 � load. (c) Simulation results for CoilsA1/A2 with 10 mm of (skin+fat+muscle) separation and optimum load (12),(13). (d) Simulation results for Coils B1/B2 with 10 mm of (skin+fat+muscle)separation and optimum load.

the simulations with biological tissue as the medium. Fig. 6 de-picts the maximum achievable efficiency using the coils A1/A2from [11] in the presence of tissue. The results indicate that themaximum achievable efficiency drops to 1% at 5 MHz. How-ever, the huge loss can easily be overcome by redesigning thecoils. Our simulations show, contrary to intuition, and unlikecoupling through air, a greater number of turns would stronglydegrade the efficiency. Hence coils designed for power trans-mission through conductive biological tissue have only a fewturns. In order to demonstrate this point, a new set of coils B1/B2


Fig. 7. 3D perspective of the biological media between the two coils.

were designed for the new tissue environment. The coils weredesigned under the same constraint of 40 mm and 20 mm outerdiameters. The geometry of the new proposed coil design canbe found in Table I. Notice the large reduction in the number ofturns. As shown in Fig. 6 the efficiency of the proposed coilsis 77%. In order to find the effect of tissue on the maximumpower transfer efficiency, we need to compare the results withan air optimized set of coils under the same constraints. Oursimulations shows that the maximum power transfer efficiencyfor such coils is 83.8%, which is only 6.8% higher than the casewhen tissue exists between the two coils.

The results indicate that the losses through biological samplemedia can indeed be made negligible at low frequencies. How-ever the design of the coils needs to be revisited. The presenceof muscle has a major impact on the self-resonance-frequency(SRF) of the coils. The measured SRF in air of coils A1 and A2are 28.2 MHz and 24.5 MHz [11], respectively. We simulatedthe new value for the SRF when the coils are surrounded bymuscle and the SRF was degraded to 5.5 MHz and 12.5 MHz,respectively. Therefore by increasing the SRF in coils B1, B2the power efficiency is restored. Coils B1, B2 have a SRF of63.3 MHz and 125 MHz, respectively.

The authors in [11] revisited their approach for tissue in [34]and proposed a new set of coils with higher SRF for poweringup an implant buried under 10 mm of muscle. The experimentused two coils fabricated on separate 1-oz FR4 substrates. Thecoils had 10 mm of separation. The gap between the two coilswas filled with medical grade silicone, muscle and plastic bagsand achieved an efficiency of 31%. The authors propose usingmedical grade silicon on top of the coils to mitigate the low-ering of SRF due to the muscle. In the remainder of this casestudy, we first simulate the measurement results from [34] andthen comment on how we can further improve the efficiency.Table II shows the media used for simulating the measurementresults from [34] and Table III shows the geometry for the coilsas well as the properties of the substrate. The data in Table IIIwere extracted from [34]. In order to reconstruct the measure-ment results in [34], we simulated the coils C1/C2 using Mo-mentum. The measured power efficiencies represent the powerefficiency from the two-port to the 500 load. Table IV presentsa summary of the results.

Our simulation shows that under optimum loading condition,using a fewer number of turns and larger spacing between thetraces, efficiencies up to 73% are achievable even without the

TABLE IIMEDIA ARRANGEMENT AND DIMENSIONS FOR [34]

TABLE IIICOIL GEOMETRY FOR [34] AND OPTIMIZED COILS

TABLE IVPERFORMANCE SUMMARY FOR COILS C1/C2 AND D1/D2

The stated efficiency includes the losses due to matching network.Without medical graded silicon coating on top of the metal traces.

presence of the expensive medical grade silicon coating. How-ever, the parallel optimum loading for the new coil geometryis close to 11 , which is much smaller than the nominal load,500 . The 11 resister value was calculated by substituting theparameters from Table III into (23) and (24) and converting thethe calculated series impedance to parallel. Therefore matchingnetworks are essential for harvesting the higher efficiency. It isevident that adding matching networks introduces an extra de-gree of freedom in the design, which can be exploited to ouradvantage. In fact using the coils D1/D2 under the traditionalresonant tuning condition and 500 load would reduce the ef-ficiency down to 12%. Fig. 8 shows the achievable efficiency asa function of frequency for both of these scenarios.

The load impedance is set to 500 and we compare resonanttuning versus optimal load using matching networks. In conclu-sion, we observe that the losses through tissue can indeed be


Fig. 8. Power efficiency from the two-port to the load through muscle using theset of coils D1/D2. (a) Using matching networks for optimum load at 13.6 MHz.(b) Using resonant tuning for the 500 � load.

made negligible at low frequencies, however the media has tobe considered during the design procedure. Unlike the coils op-timized for air media, coils optimized for the general two-portmodel tend to have a very few number of turns (usually under 3)and larger spacing (s) in between the traces. As a result the de-sign space of optimum coil design is quite small compared to thecase for air alone. Therefore, because of the reduced parametricdesign space for the coils the design can quickly be optimized ina few iterations using an EM simulator and (15). The optimiza-tion process only needs to consider (15) and can ignore the load,as the optimum load can always be realized using matching net-works with only a few percent penalty in efficiency.

B. Case Study 2: Receiver Coil on CMOS Silicon Substrate.

Question: Can a WPT receiver coil, integrated on a lossyCMOS silicon substrate, be designed with high power transferefficiency? If so, what circuit design insights need to be fol-lowed? Integrating the receiver coil using a standard CMOS fab-rication process would significantly reduce the total system cost,especially in the case of embedded implant and lab-on-chip ap-plications. In this example we explore the possibility of havingthe receiver coil integrated on a CMOS silicon substrate whilethe transmitter coil is realized of copper on FR4 substrate. Byfully integrating the receiver coil with an on-chip matching net-work the conventional chip package and requisite encapsulationcan be eliminated.

Inductive coupling works on the basis of Faraday’s law ofinduction, and as such is therefore, to first order, proportionalto the area of the receiver coil. Hence it is immediately evidentthat by integrating the receiver coil the expected efficiency willsuffer. In addition to this, the silicon substrate has higher lossassociated with it compared to an FR4 substrate and the CMOSmetal layers are thinner and hence more resistive compared to1-oz copper traces. Therefore, is it possible to practically realizesuch systems at high transfer power efficiency using on-chipcoils? What is the optimum frequency of operation? What is theoptimum geometry for the coils? Can we integrate the matchingnetworks on-chip? The following case study has been designedto answer these questions in the two sub-sections that follow.

TABLE VCOIL GEOMETRY FOR CASE STUDY 2 PART B.1

Fig. 9. The measurement setup. (a) Micrograph of the on-chip coil and CMOScore circuitry. (b) Details of CMOS structures used in the simulation model.(c) Overall geometry of FR4 Tx and CMOS Rx coils as specified in Table V.

1) WPT to a CMOS Receiver Coil through Air: In order todemonstrate the validity of our simulation results in the presenceof a CMOS substrate we fabricated a 1 coil in top-layermetal in a standard 0.18 TSMC CMOS process. The area inthe middle of the coil was occupied by 0.95 of active andpassive CMOS circuitry. The CMOS integrated coil was thenpowered up by a PCB board held 10 mm above the CMOS inte-grated coil with air as the media between the two coils. Table Vshows the details of each coil.

The power transfer efficiency was measured using a Verigy(Agilent) 93000 SOC tester. This test environment dictated thatthe nominal load during the measurement was 50 . We usedmatching networks to convert the 50 load to the optimum load(126.1–195.58i) for the on-chip coil. On the source (PCB) sidewe set the desired impedance to be the same as the sourceimpedance of the tester 50 . The value of the optimum load andthe input impedance were calculated using (12), (13). The sim-ulations were performed through Momentum. We simulated thefull CMOS substrate using 0.18 TSMC CMOS parametersas well as the top three metal layers and the results showed anexcellent match between simulations and measurement. Fig. 9shows the geometrical setup of the simulation as well as the diephoto and Fig. 10 shows the measurement results versus the sim-ulations obtained from Momentum. The measured efficiencyshows the ratio of the power delivered to the 50 load in the


Fig. 10. Simulation versus measurement results for Case Study 2 part B.1.

TABLE VICOIL GEOMETRY FOR CASE STUDY 2 PART B.2 USING FR4 COILS

presence of the matching networks versus the power delivered tothe two-port. The coils were not optimized for the distance sep-aration or frequency chosen, but nevertheless, it illustrates thepredictive nature and the ease of use for the method proposed inthis paper even in the presence of a lossy CMOS substrate.

2) WPT to a CMOS Receiver Coil through Tissue: In this partwe first recreate the scenario presented in [26], [35], [36]. Theauthors state that the optimum frequency for powering a typicalset of miniature coils, mm-sized, through tissue is in the GHzrange, and that this frequency drops to a couple of hundred MHzfor the case where one of the coils is larger (cm-sized). Theypresent measurement results for a 4 off-chip receiver coiland a 4 transmitter coil. The power is transferred throughair and 15 mm of muscle. Their measurement results achievea total power efficiency of 28.4 dB at 915 MHz. Though theexact geometry of the coils and their substrate are not presentedin the paper, we infer a set of parameters where our simulationresults are very close to the measurement results from [36]. Inorder to recreate the measurement results we assumed an FR4substrate for both the receiver and transmitter. The simulationsetup uses a receiver coil adjacent to 15 mm of muscle [22] aswell as 10 mm of spacing of air between the transmitter and themuscle. Table VI summarizes the geometry, the substrate, thesimulation and the measurement results.

The achieved power efficiency of 28 dB is acceptable forsome biomedical applications. However, in this example bothcoils were fabricated on an FR4 substrate. It is natural to ask

(a) (b)

Fig. 11. (a) Two parallel coils. (b) Two series coils manufactured using the toptwo metal layers.

how the result would change if the receiver coil is manufacturedon-chip in a standard CMOS process. Hence, in order to answerthis question, we simulated an on-chip receiver coil fabricatedin a 0.13 IBM CMOS process. The simulation modeled thefull 0.13 IBM CMOS substrate with 13 different layers ofdielectric. In order to make a fair comparison between the twocases, we limited the size of the receiver coil to 4 and themaximum metal width was enforced by the design rules of theCMOS process to be 140 . The coil inductance was increasedusing two different metal layers in series. Fig. 11 demonstratesthe series on-chip coil. Using this new setup the maximum pos-sible efficiency we could achieve at 915 MHz was 46.055 dB,which is too small for practical applications. The huge loss inpower efficiency is due to the losses through the silicon sub-strate. The substrate also reduces the SRF of the receiver coil.Therefore it is obvious that lower frequencies are more suitablefor on-chip power receivers and 915 MHz is not the optimumfrequency of operation for this case. In a search for the optimumfrequency for on-chip coils, we simulated a wide range of fre-quencies from 40 MHz to 950 MHz and the maximum powerefficiency of 33.1 dB occurred around 115 MHz. As is evi-dent the loss increased due to the silicon substrate at 915 MHzand thus reduced the power efficiency by 18 dB. However, bylowering the frequency to 115 MHz we can recoup most of theselosses. Even in the scenario where the design is restricted to anISM band our simulations shows that a frequency of 40.68 MHzresults in 34.75 dB which is still 11.3 dB higher than theachieved power transfer efficiency at 915 MHz. Further im-provement in the power transfer is realized by reducing the sizeof the transmitter as well as the 10 mm gap between the trans-mitter and the muscle. The resistance of the coils was reducedusing two different metal layers in parallel as shown in Fig. 11.The optimum frequency for new setup is now 120 MHz yieldinga power transfer efficiency of 26.17 dB. Table VII specifies thegeometries used for this simulation.

Therefore, to conclude it is possible to integrate the receiver ina standard CMOS process. However, in the presence of biolog-ical tissue the optimum frequency is approximately 100 MHzand not in the GHz range. Finally in order to make a com-parison between optimum loading condition presented in thispaper and the traditional conjugate matching, we have plottedthe power transfer efficiency from a source with 50 impedanceto a 1.4 load in Fig. 12. The maximum possible theoreticalefficiency curve (c) represents (10) with zero source impedancefor frequencies between 115 MHz to 125 MHz. The optimum


TABLE VIICOIL GEOMETRY FOR CASE STUDY 2 PART B.2

USING ON-CHIP RECEIVER COIL

The CMOS coil consists of two turns of top metal in parallelwith two turns of second top metal layer.

Fig. 12. Power transfer efficiency through muscle from source to the load usingcoil set F1/F2 (a) under optimum loading conditions� � �� and matchingnetworks tuned to 120 MHz, (b) under simultaneous conjugate matching tunedto 120 MHz, (c) maximum possible theoretical efficiency � � �.

loading condition (a) shows the same two-port including thematching networks tuned to 120 MHz and with the desired re-sistance, , set to 500 . Finally the simultaneous conjugatematching (b) represents the power efficiency in the presence ofmatching networks tuned to 120 MHz.

As is evident the optimum loading conditions results in higherefficiency compared to simultaneous conjugate matching.

C. Case Study 3: WPT to Fully-Integrated CMOS ReceiverCoil and On-Chip Matching Network Immersed in Blood

Question: What circuit design compromises are needed tofully integrate in CMOS the on-chip matching network with thereceiver coil? In Case Study 2 part B.2 we saw that it is pos-sible to integrate the receiver coil and reduce the cost of fabri-cating a miniature 4 coil with a metal width and spacingon the order of 10’s of micrometers. However the matching net-works at 120 MHz involve component values that are too largeto be implemented on-chip. In this case study, we consider thepossibility of a fully integrated lab-on-chip receiver immersedin blood capable of delivering 1 mW of power to a 1.4load at 1.2 V supply using a 150 mW transmitter. Table VIII

TABLE VIIICOIL GEOMETRY FOR CASE STUDY 3

The CMOS coil consists of two turns of top metal in series with twoturns of second top metal layer.

specifies the properties of the transmitter and receiver alongwith the media. In order to maximize the efficiency we startedwith a 1-turn transmitter coil G1 on Rogers RT/duriod 5880 andtwo-turn top-layer metal coil G2 on a CMOS substrate. Using(15) we searched for the optimum frequency of operation be-tween 80 MHz to 150 MHz. The maximum efficiency occurredat 140 MHz. However the optimum load (12), (13) at this fre-quency required a capacitance of 126 pF, which is too large tobe implemented on-chip. Also the DC resistance of the optimumload was determined to be only 280 . Converting the 1.4load resistance to 280 requires an on-chip matching networkwith inductors that occupy large area and have very low Q andhence are not suitable for our design.

One possible solution to this problem is to increase the fre-quency of operation. This would increase the DC resistance ofthe optimum load and reduce its capacitance but at the cost oflower efficiency. At 400 MHz the load capacitance is reducedto 16 pF and the optimum load is 800 at the cost of 5 dBloss in power efficiency. Using (12), (13) we can see that an-other way to address the issue of the optimum load problemis by increasing the inductance and resistance of the receivercoil using more metal layers in series for the receiver coil asshown in Fig. 11. Using this new receiver coil configuration wewere able to increase the DC resistance and reduce the capaci-tance without increasing the frequency much higher than the op-timum. Table VIII shows the final choice for frequency as wellas the geometry of the coils that provides an attractive compro-mise in design parameters. At 180 MHz the efficiency is 0.5 dBlower than the optimum frequency but now the DC part of theoptimum load matches our desired load impedance and the re-quired capacitance for the optimum load is 15.1 pF, which caneasily be implemented on-chip.

Generally speaking, optimum planar coils with on-chipreceivers can easily be designed for power transfer throughbiological tissues by following a few simple guidelines: 1) Theoptimum frequency of operation is around 100 MHz with40.68 MHz as the closest ISM band. 2) The outer radius of


the transmitter, in the case of circular loops, should satisfywhere is the distance between the coils [37].

This constraint is modified to

(42)

for the case of square coils (see Appendix E). 3) The optimumtransmitter has only a few turns, typically . 4) The tracewidth for the transmitter is generally to achievehigh Q. 5) The outer dimension of the receiver coil (on a sil-icon die) should be the largest value permitted by the die area.6) In CMOS processes with DRC rules that constraint maximumwire width, use two or three top metal layers in series for the re-ceiver coil with maximum allowed wire width. 7) The last stepis to optimize (15) by sweeping the trace width, spacing andthe number of turns in an EM simulator. Usually, this processquickly converges due to the constrained design space. 8) Oncethe optimum geometry has been found, the optimum load anddesired loading for the power amplifier can be independently re-alized using matching networks.

The case studies presented in this section use planar struc-tures. Such structures are becoming more popular due to thelow fabrication cost and more flexible geometry. However somebiomedical circuits use non-planar coils. These helical coils caneasily be simulated using 3D EM simulators such as HFSS [19].HFSS also produces S parameters which can easily be used tocalculate the optimum load and predict the maximum achievableefficiency. Helical solenoids tend to have lower self-resonancefrequency compared to planar coils and hence are usually oper-ated in the kHz to low MHz range [12], [38].

V. CONCLUSION

In this paper we have studied inductive power transferthrough a media in its most general form using two-portparameters. The two-port approach makes no simplifyingassumption about the type of media or the characteristics ofthe coils. Therefore it is capable of correctly predicting thepower transfer efficiency at any frequency, through CMOSsubstrate or biological media. We presented a closed formanalytical solution for the optimum load that would maximizethe efficiency of power transfer. Using this optimum load wehave found the closed form solution for the maximum possiblepower efficiency under arbitrary input impedance conditions.The concept of optimum load decouples the design of the coilsfrom the load. Therefore the coils can be optimized independentof the load while fully considering the media surrounding thecoils. However realizing the optimum load requires matchingnetworks which tend to be lossy. We introduced simple equa-tions that can predict the efficiency loss due to the matchingnetworks as well as the optimum number of matching stagesfor achieving minimum efficiency loss.

Finally, we introduce measurement and simulation results forseveral case studies such as power transfer through air, muscleand blood using coils integrated on FR4 and CMOS substrates.The case studies demonstrate the insight provided by the op-timum load condition as well as the ease of design using theequations derived in Section II. The results show that optimumcoils for biological and lab on chip applications tend to have

only two or fewer number of turns and hence can be quickly op-timized due to the constrained design space. We also showedthat it is possible to fully integrate the receiver coil and theappropriate matching network on a standard CMOS processwithout a significant loss in power transfer efficiency relativeto the predicted optimal value.

APPENDIX ADERIVATION OF (5)

Assuming that we have lossless matching networks, the inputpower to the matching network should be equal to the outputpower hence

(43)

Using (4) we can simplify (43) to

(44)

Finally, a simple algebraic manipulation produces

(45)

APPENDIX BDERIVATION OF EQUATIONS (12) & (13)

In order to calculate we need to take the partialderivative of (9) with respect to the real, , andimaginary, , parts of . We will ignore the

term in (9) during the optimization process. The firststep is to simplify (9) in terms of ABCD parameters

(46)

where rA, rB, rC, rD represent the real part andrepresent the imaginary part of the ABCD parameters. Next wetake the partial derivatives of (46) ignoring the term

(47)

(48)

where

(49)

(50)

(51)


Next we need to simultaneously set the two partial derivativesto zero

(52)

(53)

The solution to (52), (53) is shown as follows:

(54)

(55)

Converting (54) and (55) from ABCD to Z-parameters resultsin

(56)

(57)

APPENDIX COPTIMUM NUMBER OF MATCHING STAGES

The overall efficiency that we are trying to maximize, usingN stages is given by

(58)

The matching network has to realize a total impedance conver-sion ratio of hence the following constraint exists on theimpedance conversion of the subsections:

(59)

Using Lagrange multipliers method, we can maximize the fol-lowing equation:

(60)

Using simple algebraic manipulations we find that the max-imum of the function occurs when

(61)

Now assuming

(62)

TABLE IXDESIGN EXAMPLE FOR FIG. 4

The optimum number of stages, N, is given by

(63)

where k is a function of and and is approximately 20.05 for , therefore for large Q values

(64)

A. Example for Optimum Number of Stages in Fig. 4

In this example a 5 load is being up-converted to 442using structure (a) in Fig. 4 and a 2 load is being down-con-verted to 10 using the second configuration. Table IX summa-rizes problem for different numbers of matching stages. As youcan see the equation for the optimum number of stages (64) pro-vides the highest power efficiency for the matching networks.

The derivations in Section III assumed that the parasitic re-sistance of the components has no impact on the impedanceconversion. However these unwanted resistances can be com-parable to the load or the desired resistance . Hence theactual input impedance, would deviate from the desiredimpedance, . Table IX shows that the mismatch improveswith larger number of stages.


Fig. 13. � component of the magnetic field of a bent wire carrying current Iat point �� .

APPENDIX DCOIL ALIGNMENT SENSITIVITY

Using (9) we can approximate the change in the efficiencydue to misalignments, change in the distance between the coilsor tilting. The following derivations assume that the structureis using the optimum load for the no misalignment case. Thesederivations are based on the simple two inductor model shownin Fig. 3. Any misalignment would result in change (usually re-duction) in the mutual inductance between the coils. Howeverat low frequencies the other two-port parameters tend to stayconstant. Hence by taking advantage of this fact and in orderto capture the deviation from the optimum power efficiency wederive the Taylor series for (9) in terms of Z parameters with re-spect to . We can use the following transformationsfrom ABCD parameters to circuit parameters for the simple caseshown in Fig. 3:

(65)

Hence the Taylor series is given by

(66)The next step is to find as a function of geometry

and misalignment. In this appendix we present the case whereboth coils have square spiral shape. Similar derivations can beperformed on circular structures. Calculating the mutual induc-tance requires knowledge of the magnetic field generated by oneof the coils at each point in the space. Without loss of generality,we will assume that the coils are in the plane. Hence we needthe component of the field. A square loop consists of fourwires. Fig. 14 shows such a square loop carrying current I. The

component of magnetic field, , generated by such a loopat an arbitrary point can be found using Biot-Savart’slaw. In order to find the field we can break the loop into twosegments and derive the magnetic field for each portion. Fig. 13shows two wire segments of length 2W and 2K, carrying current

Fig. 14. � component of the magnetic field of a wire carrying current I atpoint �� .

I. The component of the magnetic field at an arbitrary pointin the space is given by

(67)

where and .Now using (67) we can derive the equation for the current loopin Fig. 14.


TABLE XCOIL GEOMETRY FOR MISALIGNMENT SIMULATION

(68)

where is the length of the edge of the square loop as shownin Fig. 14. In practical cases where the wire traces have finitewidth, . For the gen-eral case where the coil has turns, each turn can be treated asan individual loop carrying current I. Hence the generated mag-netic field at each point in space is the superposition of the fieldsdue to each individual turn

(69)

The mutual inductance between two spiral coils withand with turns, respectively, can now be easily calculatedusing (68), (69)

(70)

where represents integration over the area of eachand every loop at the receiver. This integral can easily be eval-uated numerically in MATLAB. In order to show the utility of(66) we have simulated the efficiency of two square spiral coilsas a function of lateral misalignment between the two coils. Thesimulations were performed in ADS (Momentum). The prop-erties of the coils are presented in Table X. The assumed loadduring the efficiency simulations is the optimum load for no mis-alignment case. Therefore, we do not update the load to the op-timum load as we introduce misalignment to the structure.

As is obvious from Fig. 5 these coils are not fully symmetricwith respect to and axis. Therefore the efficiency simulationspresented in Fig. 15 for each misalignment value is the averageof the power efficiency when the center of the smaller coil ismoved up, down, right and left with respect to the center oflarger coil. The calculated values of mutual inductance using(70) had up to 25% error with respect to the simulation resultsfrom ADS however the maximum error in was only8%. Fig. 15 illustrates the results.

Fig. 15. Efficiency as a function of lateral misalignment using Momentum(ADS) and MATLAB (66). A misalignment of 15 mm corresponds to the rightouter edge of coil H2 being aligned with the right outer edge of coil H1.

Fig. 15 shows that the efficiency drops by 45% when the rightedge of the smaller coil is aligned with the right outer edge oftransmitter coils (15 mm misalignment). A similar simulationwas done for Case Study 3 and the efficiency was reduced by40% under the same condition. In addition to this, the loss in ef-ficiency due to deviation from the optimum load even at the ex-treme case of 15 mm lateral misalignment or 10 mm of verticalmisalignment was less than 1%. The exact same steps could beapplied to the vertical misalignment scenario. However in situa-tions where we are only dealing with vertical misalignment, wecan simplify (70) by assuming that the magnetic field producedat the receiver coil is uniform and equals the field at the center ofthe coil. The mutual inductance between a square spiral trans-mitter coil with turns and a receiver coil with turns canbe approximated using the following equation:

(71)

where represents the effective area of the receiver coil. Thisarea can be calculated using the following formula:

(72)where is the pitch between two consecu-tive turns and is half of the outer diameter of the receivercoil. Equation (71) over-estimates the mutual inductance. Nev-ertheless, the simulation results are in a good agreement withcalculations from (71). Fig. 16 shows the simulation versus cal-culation results for vertical misalignment.

The last case addressed in this section is angular misalign-ment. We assume that the normal vector of the receiver coil istilted degrees with respect to the Z axis in Fig. 14. As a resultthe effective area in (72) should be updated to

(73)

where for small values of this can be approximated by

(74)


Fig. 16. Efficiency as a function of vertical misalignment using Momentum(ADS) and MATLAB (66) where coils H1 and H2 are normally separated by adistance of 10 mm.

Therefore can be approximated by

(75)

APPENDIX EPROOF OF (42)

We want to find the value of W in Fig. 14 that would maxi-mize the field at a distance from the center of the coil. We canmaximize this value by maximizing (68) at (0, 0, ), hence

(76)

Hence the optimum outer edge follows: .

APPENDIX FCOIL SENSITIVITY TO CHANGES IN LOAD

In real world applications there are always some uncertaintyand time variance associated with the real and imaginary parts ofthe load. In addition to this, it is extremely difficult to accuratelypredict the inductance of the fabricated coils or the thickness ofdifferent biological layers. As a result our load would deviatefrom the optimum load (16), (17). In this section we study theeffect of deviation from the optimum load. First we will considerthe change in the imaginary part of the load. The Efficiency ofthe two-port for an arbitrary load in terms of Y parameters isgiven by

(77)

where

(78)

and

(79)

Fig. 17. Loss in efficiency due to deviation of � from � ��.

Since we interested in the percentage of change from the max-imum achievable efficiency given an percent deviation fromthe imaginary part of the optimum load. We can set from(16) and set

(80)

Rewriting (77) we have

(81)

Hence the loss in efficiency as a function of is given by

(82)

where

F can also be represented in terms of Z parameters

It is interesting to note that the sensitivity decreases for largervalues of . In order to get an idea on how sensitive we arewith respect to changes in , Fig. 17 uses the coils from thefirst and the third case study, presented earlier to show the lossin achievable efficiency when deviates up to 50% from itsoptimal value.

Fig. 17 confirms our expectation that the sensitivity to vari-ations in is much higher when dealing with low efficiencycoupling. Starting from (77) we can derive a similar set of equa-tions for the percentage of loss in two-port efficiency due to thevariations in the real part of the parallel load

(83)

Fig. 18 shows the reduction in efficiency due to changes inthe real part of the load.

Similar to the case where we had deviation in , thestructure is more sensitive at low coupling scenarios. Howeverthe change in efficiency with the increase in the real part of the


Fig. 18. Loss in efficiency due to deviation of � from � ��.

Fig. 19. Output voltage and the delivered power for coils G1/G2 from the ear-lier example assuming 100% rectifier efficiency.

load is not of any concern as larger real loads require less cur-rent and hence lower power to begin with. Therefore the systemshould consider the possible lower bound on the real part of theload. The next graph in Fig. 19 demonstrates the same conceptin terms of current consumption. The figure depicts the changein voltage and delivered power as the circuits draw differentamounts of current from the supply. The data are calculated fortwo different power levels of 0.5 W and 1 W at the input of thetwo-port. Here we are considering the DC current provided tothe load by the rectifier. We also assumed that the rectifier has100% efficiency and hence the ac resistive load seen from theinput of the rectifier is half the DC resistive load: .

As you can see the optimum current is 2.64 mA and 1.86 mAfor 1 W and 0.5 W of input power respectively. The efficiencydegrades as the current consumption deviates from this value.However the system would still be functional at lower currentvalues since the voltage and power levels meet the minimumrequirement of the circuit. Higher current values will result inlower voltage and hence might not be operational. Therefore thepower transfer two-port should consider the worst case currentconsumption. It also requires a circuit to clip the voltage at lowcurrent consumption levels to protect the circuits. The next log-ical question at this point is how accurately can we predict theoptimum load. The uncertainty in the imaginary part of the op-timum load can be associated with variations in the thicknessand electrical properties of the layers surrounding the two coils

as well as the inductance of the coils. In order to study the ef-fect of the variations in the media we first used the coils G1/G2from case study 3. The blood depth was reduced to 6 mm fromthe original 9 mm while the total distance between the two coilswas kept at 10 mm. Simulation shows that the change in theimaginary part of the load for the new setup is less than 0.5%.Next we used case study 3 from [39]. In this case study the re-ceiver is buried under 10 mm of various biological tissue suchas fat, skull and Dura. We increased the thickness of every bio-logical tissue by 20% hence increasing the distance between thetwo coils from 10 mm to 12 mm. We also increased the thick-ness of the brain layers underneath the CMOS coil by the same20%. As the result, the imaginary part of the load deviated fromthe predicted value by less than 1% which resulted in less that1% deviation from maximum possible achievable efficiency inthe two-port. However the achievable efficiency dropped to 11%from the original 18% due to the extra 2 mm of separation. Inconclusion, the main contributing factor in uncertaintyis the variation in the inductor value. On-chip inductors havearound 5 percent variations [40], [41] which at weak couplingsituations can lead to 15% loss in power efficiency.

The above analysis studied the variation of efficiency withchanges in load. However they did not consider the effect ofmatching networks. Insertion of matching networks between theload and the two-port affects the load variations. These varia-tions depend on the quality factor of the matching networks. Inall cases however if the real part of the load of the matching net-work is changed by 50% the change in the real part of the desiredload is less than 50%. But the new mismatched load would haveextra reactance. Whether or not the added imaginary part is crit-ical depends on the initial susceptance of the desired load (17).Nevertheless by changing the quality factor of the matching net-work the designer can control the undesired added susceptance.However the change in real part is always attenuated which isa desirable property. In an L-match network the quality factordepends on the load and the desired impedance and thereforethe designer has no control over the value. But using Pi or Tmatching networks the designer would be able to control thequality factor of the matching network. Here we will show howthe desired load would change with variations in the load for thecase of an L-match network shown in Fig. 4. For simplicity wewill assume that the load is purely resistive. This network wouldconvert the load to the desired load value of . Now if theload changes by a factor of , the new valueof the desired load for case (a) is given by

(84)

which is always smaller than m and the added susceptance isgiven by

(85)

For case (b) in Fig. 4 we have

(86)

(87)


The added susceptance is usually small enough to be ignored.Nonetheless in cases where the optimal susceptance (17) is com-parable to (84) or (86), one has to choose a Pi matching networkand adjust the quality factor.

ACKNOWLEDGMENT

The authors would like to thank CMC for fabrication support.

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Meysam Zargham (S’06) received the B.Sc. degreefrom Sharif University of Technology, Tehran, Iran,in 2005 and the M.Sc. degree in electrical engi-neering from the University of Alberta, Edmonton,AB, Canada in 2008.

He is currently working toward the Ph.D. degreeat the University of Toronto, Toronto, ON, Canada,where his research is in the area of CMOS integratedcircuits for biomedical applications. He was amember of the icore High Capacity Digital Com-munications Laboratory. While working toward the

M.Sc. degree, he was involved in many different projects in a variety of groups,including the design of analog LDPC decoders, micro-fluidic lab-on-a-chipdesign, and the modeling of carbon nanotube transistors.

P. Glenn Gulak (S’82–M’83–SM’96) received thePh.D. degree from the University of Manitoba, Win-nipeg, MB, Canada.

While at the University of Manitoba, he helda Natural Sciences and Engineering ResearchCouncil of Canada Postgraduate Scholarship. He isa Professor with the Department of Electrical andComputer Engineering, University of Toronto, ON,Canada, as well as a registered Professional Engineerin the Province of Ontario. His present research in-terests are currently focused on algorithms, circuits,

and system-on-chip architectures for digital communication systems; and forbiological lab-on-chip microsystems. He has authored or coauthored more than100 publications in refereed journal and refereed conference proceedings. Inaddition, he has received numerous teaching awards for undergraduate coursestaught in both the Department of Computer Science and the Department ofElectrical and Computer Engineering at the University of Toronto. He held theL. Lau Chair in Electrical and Computer Engineering for the five-year periodfrom 19992004. He currently holds the Canada Research Chair in SignalProcessing Microsystems and the Edward S. Rogers Sr. Chair in Engineering.From January 1985 to January 1988, he was a Research Associate in theInformation Systems Laboratory and the Computer Systems Laboratory atStanford University, Stanford, CA. From March 2001 to March 2003, he wasthe Chief Technical Officer and Senior Vice President of LSI Engineering, afabless semiconductor startup headquartered in Irvine, CA with $70M USD offinancing that focused on wireline and wireless communication ICs

Dr. Gulak served on the ISSCC Signal Processing Technical Subcommitteefrom 1990 to 1999, was ISSCC Technical Vice-Chair in 2000, and served asthe Technical Program Chair for ISSCC 2001. He was the recipient of the IEEEMillennium Medal in 2001. He currently serves on the Technology DirectionsSubcommittee for ISSCC and as Editor-at-Large for ISSCC 2012.

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