228 IEEE TRANSACTIONS ON BIOMEDICAL CIRCUITS AND SYSTEMS, VOL. 6, NO. 3, JUNE 2012 Maximum Achievable Efficiency in Near-Field Coupled Power-Transfer Systems Meysam Zargham, Student Member, IEEE, and P. Glenn Gulak, Senior Member, IEEE Abstract—Wireless power transfer is commonly realized by means of near-field inductive coupling and is critical to many existing and emerging applications in biomedical engineering. This paper presents a closed form analytical solution for the optimum load that achieves the maximum possible power efficiency under arbitrary input impedance conditions based on the general two-port parameters of the network. The two-port approach allows one to predict the power transfer efficiency at any frequency, any type of coil geometry and through any type of media surrounding the coils. Moreover, the results are applicable to any form of passive power transfer such as provided by inductive or capacitive coupling. Our results generalize several well-known special cases. The formulation allows the design of an optimized wireless power transfer link through biological media using readily available EM simulation software. The proposed method effectively decouples the design of the inductive coupling two-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 harvesting, 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 IRELESS POWER TRANSFER (WPT) is critical to many emerging applications and is commonly realized by means of near-field inductive coupling. This type of power delivery system is advantageously used for biomedical implants – neural activity monitoring/stimulation –, emerging lab-on-chip (LoC) applications, RFID  and non-contact testing . In this system the circuits contained in the implant, the LoC or the silicon substrate are remotely powered by means of a power amplifier operating at a fixed carrier frequency. Additional functionality is achieved by modulating the carrier frequency in some manner to realize unidirectional 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, W Manuscript received May 19, 2011; revised July 30, 2011 and October 03, 2011; accepted October 29, 2011. Date of publication January 06, 2012; date of current version May 22, 2012. This work was supported by NSERC. This paper was recommended by Associate Editor E. M. Drakakis. The authors are with the Department of Electrical and Computer Engineering, 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 online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TBCAS.2011.2174794 (ii) the absorbed EM energy in tissue1 that causes the local temperature to increase possibly harming the biological tissue, and (iii) how often the battery has to be recharged when used in the context of portable medical devices. Hence in the case of implants, low-efficiency WPT implementations may cause discomfort and possible complications for the patients using it. Similar issues occur in the case of LoC applications where the local temperature of a small 10 to 100uL biological sample being measured needs to be held within strict tolerances (often within one Centigrade degree). Hence it is not possible to arbitrarily increase the strength of the EM fields to realize greater power transfer to the embedded system. In most applications, achieving high power-efficiency is extremely challenging due to the restriction on the geometry of the inductive media. Therefore, a great deal of attention in the literature has been devoted to optimization of near-field inductively coupled links. Previous authors have addressed the issue of link optimization using a simple inductor model in air for fixed load impedance at low frequencies –. Throughout the paper we refer to an inductively coupled link in air as a simple two-port model. In most practical applications, the inductive two-port is designed using numerical electromagnetic simulation software packages such as HFSS  or Momentum  that returns S parameters. Extracting the simple R, L model from these parameters, especially at high frequency, is quite challenging. In addition, and of central concern in this paper, many wireless power transfer applications require the EM waves to pass through biological material such as skin, muscle, fat, buffer solutions, etc., which we refer to as a general two-port model. These media are conductive and have higher relative permittivity constants than air , . Hence optimizing the link using a simple two-port model alone and ignoring the media during the optimization phase incurs large penalties in terms of achievable power efficiency. It is highly desirable for the output voltage to be insensitive to small changes in the distance between the two coils as well as lateral or angular misalignments. The main focus of this paper is the efficiency of these links using aligned coils. However the effects of lateral, vertical and angular misalignment specific to our discussions is briefly discussed in Appendix D and is more generally addressed in –. Recently ,  and others have realized the shortcomings of the simple two-port model at high frequencies and proposed the use of S-parameters under simultaneous conjugate matching to address these issues. However, it is well known that matching results in maximum power transfer but not necessarily, max1The rate at which energy is absorbed by biological tissue is known as SAR (Specific Absorption Rate). FCC regulates the acceptable maximum SAR for RF devices. 1932-4545/$31.00 © 2012 British Crown Copyright ZARGHAM AND GULAK: MAXIMUM ACHIEVABLE EFFICIENCY IN NEAR-FIELD COUPLED POWER-TRANSFER SYSTEMS imum efficiency . In fact, conjugate matching has a theoretical upper bound of 50% efficiency while a general two-port can be designed to have power efficiencies approaching 100%. The mathematical derivations presented in this paper prove that, unlike conjugate matching, the optimum load is independent of the source impedance and solely depends on two-port parameters. Another short-coming in the published classical link optimization techniques is the assumption of fixed load impedance. This assumption forces an extra unnecessary constraint on the design of the coupled inductors that could result in sub-optimal coil parameters. The reported power efficiency in such systems ,  is between 30 to 50%. By introducing the concept of optimum load and source impedance one effectively adds new design parameters to the system beneficially decoupling the problem of loading effect from the optimization process of the link. Our proposed approach achieves power efficiencies of greater than 80% at much greater coil separations to significant advantage in practical realizations. An interesting feedback approach was used by  to analyze the simple two-inductor model. However their proposed optimum load and efficiency is an approximation and is not especially accurate at low efficiencies though is increasingly accurate at high efficiency values. R. Harrison  suggested guidelines for maximizing the power efficiency of a simple two-port case. However no specific optimum load was presented. Simrad et al.  concluded that there exists an optimum load for which the efficiency is maximized but resorted to numerical methods to find the optimum load. Silay et al.  studied the effect of loading on maximizing the power efficiency of the link for a simple two-inductor model. However they did not decouple the input impedance from the load and hence their stated maximum achievable power efficiency 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 ,  a four-coil coupled system has been proposed in an attempt to add a degree of freedom to the effect of load and source impedance on the power efficiency of the system. However any method of impedance transformation introduces additional losses due to the finite quality factor of the components. In the case of four-coil systems, the transformation is carried out using coils with Q values up to 150. As we will see in Section III, the proposed method of matching networks uses discrete capacitors and inductors. The capacitors have Q values higher than 1000. Therefore the matching networks using only capacitors 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 on the inductor geometry. This paper presents the first published result that optimizes the near-field link based on the general two-port parameters of the network. In this approach we introduce the concept of optimum load for any passive two-port network. We also derive a simple closed-form expression for the maximum achievable power efficiency of the given two-port and show that it is theoretically possible to approach 100% power transfer efficiency. These results also provide insight into the design of such links 229 Fig. 1. General two-port power transfer system model. by introducing a simple criterion on the two-port parameters to maximize power transfer efficiency. Moreover, the results are applicable to any form of passive power transfer such as inductive or capacitive coupling. These derivations provide a powerful tool for modifying the simple two-port inductor model to the more complicated but realistic general form (e.g. adding the conductance between the two coils to model the conductivity of media) and quickly observing the effects on the efficiency and optimum loading in the system. Therefore, it is easy to optimize a realistic wireless power transfer link through biological media using readily available EM simulation software. The optimum load is realized using matching networks. However, these matching networks are usually lossy and affect the maximum achievable power efficiency. In this paper, we address these issues and comment on the design of the matching stages to achieve optimum efficiency. The remainder of the paper is organized as follows. In Section II, we introduce simple optimization criteria to achieve maximum power transfer efficiency through a general passive two-port network. Using these criteria we introduce the maximum achievable power efficiency and optimum loading condition for a general passive two port. In Section III, we discuss how to mitigate loss of efficiency in matching networks and the resulting optimum number of stages for matching. Section IV, provides several case studies on inductive coupling through air, biological tissue encountered in implants and blood for lab-on-chip applications. Throughout we make quantitative comparisons with measured published results whenever possible. II. THE POWER EFFICIENCY OF THE TWO-PORT In this section we will derive the power transfer efficiency, or simply power efficiency, of a general passive two-port network from the source to the load. Fig. 1 shows the block diagram for a general inductive-coupled power transfer system. The power efficiency, or simply the efficiency, of the system is defined as (1) where is the power delivered to the load and is the power delivered by the source . The value of depends on var, the source impedance ious parameters such as the load , the impedance loading the source and the two-port parameters. Therefore to achieve the maximum possible efficiency 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 230 IEEE TRANSACTIONS ON BIOMEDICAL CIRCUITS AND SYSTEMS, VOL. 6, NO. 3, JUNE 2012 networks. In order to obtain the maximum possible efficiency of a two-port we first derive the efficiency for Fig. 1, then we and that would result in the introduce the conditions on maximum possible value for (1). A general linear two-port is represented in terms of its ABCD parameters , . The value of efficiency in (10) is given by and , and that allows for the maximum (12) (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, the voltage at the input of the two-port due to the source is (5) is the impedance at the input of the two-port and and denote the real and the imaginary parts of the expression. In (5) we have assumed that the matching networks are lossless. This assumption is revisited in detail in Section III. is then transformed by the two-port gain and The voltage hence shows up at the second port as where (13) As seen from (12) and (13), in general, the proposed optimum load is not matched to the two-port. Therefore, the optimum power efficiency does not happen when the load is matched to the two-port. In fact matching would never result in efficiencies higher than 50% while (10) can theoretically be as high as 100%. in (10) is a function of and represents The term the efficiency from the source to the input of the two-port for a depends on the input linear voltage source. The choice for driver . In practice, the two-port is driven by a class-E power should be replaced by the effiamplifier. Therefore, ciency of the employed power amplifier. Thus a more realistic form of (10) is given by (14) where We can then simplify the expression by substituting its ABCD parameters (6) with (7) Hence (6) simplifies to Using (8) the power efficiency from is the admittance of the lated, where to (8) can be calcu- (9) As expected, is a function of . Hence, there exists an opthat would maximize . Therefore by maxtimum load imizing (9) with respect to the real and imaginary parts of the and replacing the ABCD parameters with Z-paload rameters, we can show that the maximum achievable efficiency under optimum loading conditions in any passive two-port network is represents the power amplifier efficiency and (15) is the two-port efficiency. The efficiency of a power amplifier is a function of its load and this is what drives the choice behind . It is a well-known fact that there exists an optimum , which maximizes the power load, usually referred to as is delivery efficiency of a power amplifier. The value of completely different from the small-signal output impedance of the power amplifier and is generally found using load-pull techniques . Therefore, to maximize the power from the source to the load it is essential that the two-port would provide the aploading for the power amplifier. The propriate is theoretically efficiency of a class-E power amplifier 100% and in practice efficiencies higher than 75% are achievable. The second term in (10) is a function of two-port parameters. In order to maximize the power efficiency of the two-port, we need to maximize . Fig. 2 shows the maximum possible power efficiency from the two-port to the load as a function of the variable . Equations (12) and (13) represent the optimum series load. The equivalent parallel load is calculated in (16), (17). These quantities are best represented in terms of the network Y parameters (10) where (16) (11) (17) ZARGHAM AND GULAK: MAXIMUM ACHIEVABLE EFFICIENCY IN NEAR-FIELD COUPLED POWER-TRANSFER SYSTEMS 231 Fig. 3. Simple model for inductive power transfer. It is no surprise that (22) exactly matches with the maximum power efficiency derived in  for the same simple circuit. In order to gain some understanding about the optimum load, we will further simplify the model and assume that the capacitances are cancelled out by the matching network, the optimum load for the network in this case is Fig. 2. Maximum two-port efficiency as function of . Appendix F studies the variations in power transfer efficiency as the load deviates from the optimum load. It is interesting to note that the well-established simultaneous conjugate matching, that is widely used in microwave amplifiers  and guarantees 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 happens under transducer gain. The maximum value for the simultaneous conjugate matching condition and it is usually stated in terms of S-parameters . For a passive two-port (18) where (19) By simplifying (18) in terms of the two-port parameters it can be shown that (20) Therefore, our result agrees with the well-established , has 50% as its under the matched source condition. upper bound for the efficiency. In order to provide more insight into what each of these quantities represent we can relate them to a first-order, simple inductive coupling two-port model. Fig. 3 shows 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 inductors and k is the coupling factor between the two coils. This shows that for a simplified model, in order to increase the efficiency we have to increase the mutual inductance and minimize the resistance. Using (15) the maximum achievable power efficiency from the two-port to the load is given by (22) (23) (24) where is the reflected from the source side. As is evident in (24), the imaginary part of the optimum load would completely ignore the impedances transferred from the input side and only resonate out the imaginary part of the coil, hence maximizing the voltage on the load. Once again (24) perfectly matches the common practice of resonating out the load presented in . Unfortunately, (24) only holds as long as the coupling is purely reactive. A good approximation of this case is when the two inductors are coupled through air, similar to this simple example. The picture changes, when the media in between the coils is conductive, , e.g tissue or biological media. In such scenarios the optimum load should be calculated using (12) and (13). The optimum load impedance balances the current between the conductive and inductive path such that the efficiency is maximized. Hence the resulting load is 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 the matching networks were neglected. In Section III, we consider the effect of these non-idealities on the total efficiency of the two-port. III. MATCHING NETWORKS AND EFFICIENCY The conversion of load impedance to the optimum load and the input impedance to the desired impedance has to be conducted through a filter commonly referred to as a matching network. Matching networks can transform any impedance with non-zero resistance to any desired resistance. The reactive part of the desired load is then easily adjusted by adding a reactive component in series or parallel. Therefore without loss of generality we will assume that the matching network is transforming a general complex load to a purely resistive desired load. There are different types of matching networks such as , T or L to choose from . In situations where the quality factor of the matching network is not enforced and efficiency is of primary concern, L-match is a good choice . Therefore in this section, the analysis are based on multi-section L-match networks. This being said, similar derivations can easily be developed for other types of matching 232 IEEE TRANSACTIONS ON BIOMEDICAL CIRCUITS AND SYSTEMS, VOL. 6, NO. 3, JUNE 2012 (33) Using (32), (33) the loss in the matching network is found to be (34) (35) Fig. 4. L-match sections for two different conversion cases (a) and (b). Assuming , networks. L-match networks are aptly named as they consist of two elements that form an L-shape circuit. A. L-Matching Network Analysis Depending on whether we need to increase or decrease the real part of the impedance, one of the two L-Section circuits in Fig. 4 is used. The efficiency loss through the matching network is due to the resistance of components used in the matching network. In the following analysis we assume the losses are small enough not to affect the impedance conversion operation of the network. . The Q of the We will first address case (a) where L-match network is defined as (36) On the load side, usually we need to step down the load (com. On the source side, on the other monly case b) and hand, the load resistance is the series resistance of the trans, and theremitter coil and is small. We will refer to this as fore we are commonly dealing with case (a). Equation (21) suggests that high efficiency occurs when the coils have high Q. . In such scenarios the loss through the Therefore matching networks can be simplified to (37) (38) (25) Using (25), the value of the reactance X and susceptance B are given by (26) (27) Using (26), (27), the portion of power loss due to is calculated to be and (28) Therefore, it is vital to use very high Q components. The transmitter inductors made using PCB traces have a Q between 50 and 250 in air, therefore, the series matching component on the source side needs to be a capacitor with a very high Q. On the load side however we can improve the efficiency by reducing the effective Q. B. Optimum number of stages According to (25), (31) the Q of the matching network using one stage may become large which as we saw in (30), (36) can hurt the power efficiency of the conversion. A remedy can be found by using multiple stages, each stage having . Using (38) the efficiency of each section i is (29) (39) where and are the Q of the series and parallel components used in the matching network is quality factor of the load. Assuming and , the total efficiency through the matching network 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 . There exist an optimum number of stages that maximizes the total efficiency (40) Using the proof presented in Appendix C, all stages should provide equal impedance conversion and the optimum number of stages for large Q is (41) (31) The value of reactance X and the susceptance B for case (b) are given by (32) IV. CASE STUDIES A natural question at this point is, what is a practical achievable value for and how much does the efficiency degrade with ZARGHAM AND GULAK: MAXIMUM ACHIEVABLE EFFICIENCY IN NEAR-FIELD COUPLED POWER-TRANSFER SYSTEMS a non-optimal load. How does biological tissue or relevant liquids such as blood affect the optimal coil design strategy? Is it possible to integrate the receiver coil on-chip using a standard CMOS process? Is there an optimum frequency of operation to maximize power transfer efficiency? In order to address these questions, we present three different case studies to demonstrate the power of the derived equations, verify the derivations in Section II, and provide intuition on possible achievable power efficiencies in different media as well as insight into coil design. It will become evident through the examples that the established conventional wisdom regarding coil design for biological tissue needs to be revisited. 233 TABLE I COIL GEOMETRY FOR  AND OPTIMIZED COILS A. Case Study 1: Two Coil Power Transfer through Air and Muscle Media Question: How does a conductive medium with higher relative permittivity between two coils affect WPT design? It is generally understood that losses through biological tissues and liquids are negligible at frequencies between several hundred kiloHertz to 20 MHz. As a result designers often ignore the effect of biological tissue during the coil design process , , , ,  and optimize the coils assuming air as the media between the two coils. In this example we challenge this conventional wisdom and re-evaluate coil design for conductive media such as muscle. In order to highlight the effect of biological tissue in between the two coils and to validate our simulation results we use the measurement data from , . Jow et al.  proposed a coil design strategy for powering up biological implants. The paper presents measurement results for air and claims that the design when used in the context of a biological implant would not significantly affect the efficiency under 20 MHz. In order to further investigate this claim and answer the question of whether or not the coil design has to be revisited, we first use Momentum  along with our method to reconstruct the measured data from  for the case of a simple two-port with air between the two coils. Then we use the same set of coils for the case of a general two-port to power up an implant buried under layers of skin, muscle and fat. The experiment  uses two 1-oz copper FR4 PCB pancake coils with 10 mm of separation. They report a measured efficiency of 75% at 5 MHz from the input of the two-port to the load. The summary of the coil parameters is shown in Table I. Fig. 5 shows how these coil parameters map to the specific geometry of the implemented coils. We used Momentum  to simulate the coils. We used (9) to consider the effect of load impedance on the efficiency. The real part of the load impedance was set to 500  and the imaginary part was set to cancel from the two-port. the Fig. 6 shows our simulated data versus the measurement results from . There is very good agreement between our simulation results (b) and the measured data (a) presented in . Next, we replace the air media in our simulations with 1 mm of skin, 2 mm of fat and 7 mm of muscle media as shown in Fig. 7. The frequency dependent permittivity and conductivity of the tissue can be modelled using the four parameter cole-cole model presented in . The summary of dielectric properties at 5 MHz is presented in Table I. In order to obtain the maximum possible efficiency, we used the optimum load (12), (13) for Fig. 5. Geometry of the coil based on d , w and s. Fig. 6. (a) Measurement results from  for Coils A1/A2 with 10 mm of Air separation and 500 load. (b) Momentum simulation using (9) for Coils A1/A2 with 10 mm of air separation and 500 load. (c) Simulation results for Coils A1/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 depicts the maximum achievable efficiency using the coils A1/A2 from  in the presence of tissue. The results indicate that the maximum achievable efficiency drops to 1% at 5 MHz. However, the huge loss can easily be overcome by redesigning the coils. Our simulations show, contrary to intuition, and unlike coupling through air, a greater number of turns would strongly degrade the efficiency. Hence coils designed for power transmission through conductive biological tissue have only a few turns. In order to demonstrate this point, a new set of coils B1/B2 234 IEEE TRANSACTIONS ON BIOMEDICAL CIRCUITS AND SYSTEMS, VOL. 6, NO. 3, JUNE 2012 TABLE II MEDIA ARRANGEMENT AND DIMENSIONS FOR  Fig. 7. 3D perspective of the biological media between the two coils. were designed for the new tissue environment. The coils were designed under the same constraint of 40 mm and 20 mm outer diameters. The geometry of the new proposed coil design can be found in Table I. Notice the large reduction in the number of turns. As shown in Fig. 6 the efficiency of the proposed coils is 77%. In order to find the effect of tissue on the maximum power transfer efficiency, we need to compare the results with an air optimized set of coils under the same constraints. Our simulations shows that the maximum power transfer efficiency for such coils is 83.8%, which is only 6.8% higher than the case when tissue exists between the two coils. The results indicate that the losses through biological sample media can indeed be made negligible at low frequencies. However the design of the coils needs to be revisited. The presence of muscle has a major impact on the self-resonance-frequency (SRF) of the coils. The measured SRF in air of coils A1 and A2 are 28.2 MHz and 24.5 MHz , respectively. We simulated the new value for the SRF when the coils are surrounded by muscle and the SRF was degraded to 5.5 MHz and 12.5 MHz, respectively. Therefore by increasing the SRF in coils B1, B2 the power efficiency is restored. Coils B1, B2 have a SRF of 63.3 MHz and 125 MHz, respectively. The authors in  revisited their approach for tissue in  and proposed a new set of coils with higher SRF for powering up an implant buried under 10 mm of muscle. The experiment used two coils fabricated on separate 1-oz FR4 substrates. The coils had 10 mm of separation. The gap between the two coils was filled with medical grade silicone, muscle and plastic bags and achieved an efficiency of 31%. The authors propose using medical grade silicon on top of the coils to mitigate the lowering of SRF due to the muscle. In the remainder of this case study, we first simulate the measurement results from  and then comment on how we can further improve the efficiency. Table II shows the media used for simulating the measurement results from  and Table III shows the geometry for the coils as well as the properties of the substrate. The data in Table III were extracted from . In order to reconstruct the measurement results in , we simulated the coils C1/C2 using Momentum. The measured power efficiencies represent the power efficiency from the two-port to the 500 load. Table IV presents a summary of the results. Our simulation shows that under optimum loading condition, using a fewer number of turns and larger spacing between the traces, efficiencies up to 73% are achievable even without the TABLE III COIL GEOMETRY FOR  AND OPTIMIZED COILS TABLE IV PERFORMANCE 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. However, the parallel optimum loading for the new coil geometry is close to 11 , which is much smaller than the nominal load, 500 . The 11 resister value was calculated by substituting the parameters from Table III into (23) and (24) and converting the the calculated series impedance to parallel. Therefore matching networks are essential for harvesting the higher efficiency. It is evident that adding matching networks introduces an extra degree of freedom in the design, which can be exploited to our advantage. In fact using the coils D1/D2 under the traditional resonant tuning condition and 500 load would reduce the efficiency down to 12%. Fig. 8 shows the achievable efficiency as a function of frequency for both of these scenarios. The load impedance is set to 500 and we compare resonant tuning versus optimal load using matching networks. In conclusion, we observe that the losses through tissue can indeed be ZARGHAM AND GULAK: MAXIMUM ACHIEVABLE EFFICIENCY IN NEAR-FIELD COUPLED POWER-TRANSFER SYSTEMS 235 TABLE V COIL GEOMETRY FOR CASE STUDY 2 PART B.1 Fig. 8. Power efficiency from the two-port to the load through muscle using the set 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 to be considered during the design procedure. Unlike the coils optimized for air media, coils optimized for the general two-port model tend to have a very few number of turns (usually under 3) and larger spacing (s) in between the traces. As a result the design space of optimum coil design is quite small compared to the case for air alone. Therefore, because of the reduced parametric design space for the coils the design can quickly be optimized in a few iterations using an EM simulator and (15). The optimization process only needs to consider (15) and can ignore the load, as the optimum load can always be realized using matching networks with only a few percent penalty in efficiency. Fig. 9. The measurement setup. (a) Micrograph of the on-chip coil and CMOS core 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. B. Case Study 2: Receiver Coil on CMOS Silicon Substrate. Question: Can a WPT receiver coil, integrated on a lossy CMOS silicon substrate, be designed with high power transfer efficiency? If so, what circuit design insights need to be followed? Integrating the receiver coil using a standard CMOS fabrication process would significantly reduce the total system cost, especially in the case of embedded implant and lab-on-chip applications. In this example we explore the possibility of having the receiver coil integrated on a CMOS silicon substrate while the transmitter coil is realized of copper on FR4 substrate. By fully integrating the receiver coil with an on-chip matching network the conventional chip package and requisite encapsulation can be eliminated. Inductive coupling works on the basis of Faraday’s law of induction, and as such is therefore, to first order, proportional to the area of the receiver coil. Hence it is immediately evident that by integrating the receiver coil the expected efficiency will suffer. In addition to this, the silicon substrate has higher loss associated with it compared to an FR4 substrate and the CMOS metal layers are thinner and hence more resistive compared to 1-oz copper traces. Therefore, is it possible to practically realize such systems at high transfer power efficiency using on-chip coils? What is the optimum frequency of operation? What is the optimum geometry for the coils? Can we integrate the matching networks on-chip? The following case study has been designed to answer these questions in the two sub-sections that follow. 1) WPT to a CMOS Receiver Coil through Air: In order to demonstrate the validity of our simulation results in the presence coil in top-layer of a CMOS substrate we fabricated a 1 TSMC CMOS process. The area in metal in a standard 0.18 the middle of the coil was occupied by 0.95 of active and passive CMOS circuitry. The CMOS integrated coil was then powered up by a PCB board held 10 mm above the CMOS integrated coil with air as the media between the two coils. Table V shows the details of each coil. The power transfer efficiency was measured using a Verigy (Agilent) 93000 SOC tester. This test environment dictated that the nominal load during the measurement was 50 . We used matching networks to convert the 50 load to the optimum load (126.1–195.58i) for the on-chip coil. On the source (PCB) side to be the same as the source we set the desired impedance impedance of the tester 50 . The value of the optimum load and the input impedance were calculated using (12), (13). The simulations were performed through Momentum. We simulated the TSMC CMOS parameters full CMOS substrate using 0.18 as well as the top three metal layers and the results showed an excellent match between simulations and measurement. Fig. 9 shows the geometrical setup of the simulation as well as the die photo and Fig. 10 shows the measurement results versus the simulations obtained from Momentum. The measured efficiency shows the ratio of the power delivered to the 50 load in the 236 IEEE TRANSACTIONS ON BIOMEDICAL CIRCUITS AND SYSTEMS, VOL. 6, NO. 3, JUNE 2012 (a) (b) Fig. 11. (a) Two parallel coils. (b) Two series coils manufactured using the top two metal layers. Fig. 10. Simulation versus measurement results for Case Study 2 part B.1. TABLE VI COIL GEOMETRY FOR CASE STUDY 2 PART B.2 USING FR4 COILS presence of the matching networks versus the power delivered to the two-port. The coils were not optimized for the distance separation or frequency chosen, but nevertheless, it illustrates the predictive nature and the ease of use for the method proposed in this paper even in the presence of a lossy CMOS substrate. 2) WPT to a CMOS Receiver Coil through Tissue: In this part we first recreate the scenario presented in , , . The authors state that the optimum frequency for powering a typical set of miniature coils, mm-sized, through tissue is in the GHz range, and that this frequency drops to a couple of hundred MHz for the case where one of the coils is larger (cm-sized). They off-chip receiver coil present measurement results for a 4 transmitter coil. The power is transferred through and a 4 air and 15 mm of muscle. Their measurement results achieve a total power efficiency of 28.4 dB at 915 MHz. Though the exact geometry of the coils and their substrate are not presented in the paper, we infer a set of parameters where our simulation results are very close to the measurement results from . In order to recreate the measurement results we assumed an FR4 substrate for both the receiver and transmitter. The simulation setup uses a receiver coil adjacent to 15 mm of muscle  as well as 10 mm of spacing of air between the transmitter and the muscle. Table VI summarizes the geometry, the substrate, the simulation and the measurement results. The achieved power efficiency of 28 dB is acceptable for some biomedical applications. However, in this example both coils were fabricated on an FR4 substrate. It is natural to ask how the result would change if the receiver coil is manufactured on-chip in a standard CMOS process. Hence, in order to answer this question, we simulated an on-chip receiver coil fabricated IBM CMOS process. The simulation modeled the in a 0.13 IBM CMOS substrate with 13 different layers of full 0.13 dielectric. In order to make a fair comparison between the two and the cases, we limited the size of the receiver coil to 4 maximum metal width was enforced by the design rules of the . The coil inductance was increased CMOS process to be 140 using two different metal layers in series. Fig. 11 demonstrates the series on-chip coil. Using this new setup the maximum possible efficiency we could achieve at 915 MHz was 46.055 dB, which is too small for practical applications. The huge loss in power efficiency is due to the losses through the silicon substrate. The substrate also reduces the SRF of the receiver coil. Therefore it is obvious that lower frequencies are more suitable for on-chip power receivers and 915 MHz is not the optimum frequency of operation for this case. In a search for the optimum frequency for on-chip coils, we simulated a wide range of frequencies from 40 MHz to 950 MHz and the maximum power efficiency of 33.1 dB occurred around 115 MHz. As is evident the loss increased due to the silicon substrate at 915 MHz and thus reduced the power efficiency by 18 dB. However, by lowering the frequency to 115 MHz we can recoup most of these losses. Even in the scenario where the design is restricted to an ISM band our simulations shows that a frequency of 40.68 MHz results in 34.75 dB which is still 11.3 dB higher than the achieved power transfer efficiency at 915 MHz. Further improvement in the power transfer is realized by reducing the size of the transmitter as well as the 10 mm gap between the transmitter and the muscle. The resistance of the coils was reduced using two different metal layers in parallel as shown in Fig. 11. The optimum frequency for new setup is now 120 MHz yielding a power transfer efficiency of 26.17 dB. Table VII specifies the geometries used for this simulation. Therefore, to conclude it is possible to integrate the receiver in a standard CMOS process. However, in the presence of biological tissue the optimum frequency is approximately 100 MHz and not in the GHz range. Finally in order to make a comparison between optimum loading condition presented in this paper and the traditional conjugate matching, we have plotted the power transfer efficiency from a source with 50 impedance load in Fig. 12. The maximum possible theoretical to a 1.4 efficiency curve (c) represents (10) with zero source impedance for frequencies between 115 MHz to 125 MHz. The optimum ZARGHAM AND GULAK: MAXIMUM ACHIEVABLE EFFICIENCY IN NEAR-FIELD COUPLED POWER-TRANSFER SYSTEMS TABLE VII COIL GEOMETRY FOR CASE STUDY 2 PART B.2 USING ON-CHIP RECEIVER COIL 237 TABLE VIII COIL GEOMETRY FOR CASE STUDY 3 The CMOS coil consists of two turns of top metal in parallel with two turns of second top metal layer. The CMOS coil consists of two turns of top metal in series with two turns of second top metal layer. Fig. 12. Power transfer efficiency through muscle from source to the load using coil set F1/F2 (a) under optimum loading conditions R = 500 and matching networks tuned to 120 MHz, (b) under simultaneous conjugate matching tuned to 120 MHz, (c) maximum possible theoretical efficiency R = 0. loading condition (a) shows the same two-port including the matching networks tuned to 120 MHz and with the desired re, set to 500 . Finally the simultaneous conjugate sistance, matching (b) represents the power efficiency in the presence of matching networks tuned to 120 MHz. As is evident the optimum loading conditions results in higher efficiency compared to simultaneous conjugate matching. C. Case Study 3: WPT to Fully-Integrated CMOS Receiver Coil and On-Chip Matching Network Immersed in Blood Question: What circuit design compromises are needed to fully integrate in CMOS the on-chip matching network with the receiver coil? In Case Study 2 part B.2 we saw that it is possible to integrate the receiver coil and reduce the cost of fabricating a miniature 4 coil with a metal width and spacing on the order of 10’s of micrometers. However the matching networks at 120 MHz involve component values that are too large to be implemented on-chip. In this case study, we consider the possibility of a fully integrated lab-on-chip receiver immersed in blood capable of delivering 1 mW of power to a 1.4 load at 1.2 V supply using a 150 mW transmitter. Table VIII specifies the properties of the transmitter and receiver along with the media. In order to maximize the efficiency we started with a 1-turn transmitter coil G1 on Rogers RT/duriod 5880 and two-turn top-layer metal coil G2 on a CMOS substrate. Using (15) we searched for the optimum frequency of operation between 80 MHz to 150 MHz. The maximum efficiency occurred at 140 MHz. However the optimum load (12), (13) at this frequency required a capacitance of 126 pF, which is too large to be implemented on-chip. Also the DC resistance of the optimum load was determined to be only 280 . Converting the 1.4 load resistance to 280 requires an on-chip matching network with inductors that occupy large area and have very low Q and hence are not suitable for our design. One possible solution to this problem is to increase the frequency of operation. This would increase the DC resistance of the optimum load and reduce its capacitance but at the cost of lower efficiency. At 400 MHz the load capacitance is reduced to 16 pF and the optimum load is 800 at the cost of 5 dB loss in power efficiency. Using (12), (13) we can see that another way to address the issue of the optimum load problem is by increasing the inductance and resistance of the receiver coil using more metal layers in series for the receiver coil as shown in Fig. 11. Using this new receiver coil configuration we were able to increase the DC resistance and reduce the capacitance without increasing the frequency much higher than the optimum. Table VIII shows the final choice for frequency as well as the geometry of the coils that provides an attractive compromise in design parameters. At 180 MHz the efficiency is 0.5 dB lower than the optimum frequency but now the DC part of the optimum load matches our desired load impedance and the required capacitance for the optimum load is 15.1 pF, which can easily be implemented on-chip. Generally speaking, optimum planar coils with on-chip receivers can easily be designed for power transfer through biological tissues by following a few simple guidelines: 1) The optimum frequency of operation is around 100 MHz with 40.68 MHz as the closest ISM band. 2) The outer radius of 238 IEEE TRANSACTIONS ON BIOMEDICAL CIRCUITS AND SYSTEMS, VOL. 6, NO. 3, JUNE 2012 the transmitter, in the case of circular loops, should satisfy where is the distance between the coils . This constraint is modified to (42) for the case of square coils (see Appendix E). 3) The optimum . 4) The trace transmitter has only a few turns, typically to achieve width for the transmitter is generally high Q. 5) The outer dimension of the receiver coil (on a silicon die) should be the largest value permitted by the die area. 6) In CMOS processes with DRC rules that constraint maximum wire width, use two or three top metal layers in series for the receiver coil with maximum allowed wire width. 7) The last step is to optimize (15) by sweeping the trace width, spacing and the number of turns in an EM simulator. Usually, this process quickly converges due to the constrained design space. 8) Once the optimum geometry has been found, the optimum load and desired loading for the power amplifier can be independently realized using matching networks. The case studies presented in this section use planar structures. Such structures are becoming more popular due to the low fabrication cost and more flexible geometry. However some biomedical circuits use non-planar coils. These helical coils can easily be simulated using 3D EM simulators such as HFSS . HFSS also produces S parameters which can easily be used to calculate the optimum load and predict the maximum achievable efficiency. Helical solenoids tend to have lower self-resonance frequency compared to planar coils and hence are usually operated in the kHz to low MHz range , . V. CONCLUSION In this paper we have studied inductive power transfer through a media in its most general form using two-port parameters. The two-port approach makes no simplifying assumption about the type of media or the characteristics of the coils. Therefore it is capable of correctly predicting the power transfer efficiency at any frequency, through CMOS substrate or biological media. We presented a closed form analytical solution for the optimum load that would maximize the efficiency of power transfer. Using this optimum load we have found the closed form solution for the maximum possible power efficiency under arbitrary input impedance conditions. The concept of optimum load decouples the design of the coils from the load. Therefore the coils can be optimized independent of the load while fully considering the media surrounding the coils. However realizing the optimum load requires matching networks which tend to be lossy. We introduced simple equations that can predict the efficiency loss due to the matching networks as well as the optimum number of matching stages for achieving minimum efficiency loss. Finally, we introduce measurement and simulation results for several case studies such as power transfer through air, muscle and blood using coils integrated on FR4 and CMOS substrates. The case studies demonstrate the insight provided by the optimum load condition as well as the ease of design using the equations derived in Section II. The results show that optimum coils for biological and lab on chip applications tend to have only two or fewer number of turns and hence can be quickly optimized due to the constrained design space. We also showed that it is possible to fully integrate the receiver coil and the appropriate matching network on a standard CMOS process without a significant loss in power transfer efficiency relative to the predicted optimal value. APPENDIX A DERIVATION OF (5) Assuming that we have lossless matching networks, the input power to the matching network should be equal to the output power hence (43) Using (4) we can simplify (43) to (44) Finally, a simple algebraic manipulation produces (45) APPENDIX B DERIVATION OF EQUATIONS (12) & (13) In order to calculate we need to take the partial , and derivative of (9) with respect to the real, imaginary, , parts of . We will ignore the term in (9) during the optimization process. The first step is to simplify (9) in terms of ABCD parameters (46) where rA, rB, rC, rD represent the real part and represent the imaginary part of the ABCD parameters. Next we term take the partial derivatives of (46) ignoring the (47) (48) where (49) (50) (51) ZARGHAM AND GULAK: MAXIMUM ACHIEVABLE EFFICIENCY IN NEAR-FIELD COUPLED POWER-TRANSFER SYSTEMS Next we need to simultaneously set the two partial derivatives to zero 239 TABLE IX DESIGN EXAMPLE FOR FIG. 4 (52) (53) The solution to (52), (53) is shown as follows: (54) (55) Converting (54) and (55) from ABCD to Z-parameters results in (56) (57) APPENDIX C OPTIMUM NUMBER OF MATCHING STAGES The optimum number of stages, N, is given by The overall efficiency that we are trying to maximize, using N stages is given by (63) (58) where k is a function of 0.05 for and and is approximately 2 , therefore for large Q values The matching network has to realize a total impedance converhence the following constraint exists on the sion ratio of impedance conversion of the subsections: (59) Using Lagrange multipliers method, we can maximize the following equation: (60) Using simple algebraic manipulations we find that the maximum of the function occurs when (61) Now assuming (62) (64) A. Example for Optimum Number of Stages in Fig. 4 In this example a 5 load is being up-converted to 442 using structure (a) in Fig. 4 and a 2 load is being down-converted to 10 using the second configuration. Table IX summarizes problem for different numbers of matching stages. As you can see the equation for the optimum number of stages (64) provides the highest power efficiency for the matching networks. The derivations in Section III assumed that the parasitic resistance of the components has no impact on the impedance conversion. However these unwanted resistances can be comor the desired resistance . Hence the parable to the load would deviate from the desired actual input impedance, impedance, . Table IX shows that the mismatch improves with larger number of stages. 240 IEEE TRANSACTIONS ON BIOMEDICAL CIRCUITS AND SYSTEMS, VOL. 6, NO. 3, JUNE 2012 Fig. 14. B component of the magnetic field of a wire carrying current I at point (x; y; z ). Fig. 13. B component of the magnetic field of a bent wire carrying current I at point (x; y; z ). I. The component of the magnetic field at an arbitrary point in the space is given by APPENDIX D COIL ALIGNMENT SENSITIVITY Using (9) we can approximate the change in the efficiency due to misalignments, change in the distance between the coils or tilting. The following derivations assume that the structure is using the optimum load for the no misalignment case. These derivations are based on the simple two inductor model shown in Fig. 3. Any misalignment would result in change (usually reduction) in the mutual inductance between the coils. However at low frequencies the other two-port parameters tend to stay constant. Hence by taking advantage of this fact and in order to capture the deviation from the optimum power efficiency we derive the Taylor series for (9) in terms of Z parameters with re. We can use the following transformations spect to from ABCD parameters to circuit parameters for the simple case shown in Fig. 3: (67) where and . Now using (67) we can derive the equation for the current loop in Fig. 14. (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 where both coils have square spiral shape. Similar derivations can be performed on circular structures. Calculating the mutual inductance requires knowledge of the magnetic field generated by one of the coils at each point in the space. Without loss of generality, we will assume that the coils are in the plane. Hence we need the component of the field. A square loop consists of four wires. Fig. 14 shows such a square loop carrying current I. The component of magnetic field, , generated by such a loop at an arbitrary point can be found using Biot-Savart’s law. In order to find the field we can break the loop into two segments and derive the magnetic field for each portion. Fig. 13 shows two wire segments of length 2W and 2K, carrying current ZARGHAM AND GULAK: MAXIMUM ACHIEVABLE EFFICIENCY IN NEAR-FIELD COUPLED POWER-TRANSFER SYSTEMS 241 TABLE X COIL GEOMETRY FOR MISALIGNMENT SIMULATION Fig. 15. Efficiency as a function of lateral misalignment using Momentum (ADS) and MATLAB (66). A misalignment of 15 mm corresponds to the right outer edge of coil H2 being aligned with the right outer edge of coil H1. (68) where is the length of the edge of the square loop as shown in Fig. 14. In practical cases where the wire traces have finite . For the genwidth, eral case where the coil has turns, each turn can be treated as an individual loop carrying current I. Hence the generated magnetic field at each point in space is the superposition of the fields due to each individual turn (69) with The mutual inductance between two spiral coils and with turns, respectively, can now be easily calculated using (68), (69) (70) represents integration over the area of each where and every loop at the receiver. This integral can easily be evaluated numerically in MATLAB. In order to show the utility of (66) we have simulated the efficiency of two square spiral coils as a function of lateral misalignment between the two coils. The simulations were performed in ADS (Momentum). The properties of the coils are presented in Table X. The assumed load during the efficiency simulations is the optimum load for no misalignment case. Therefore, we do not update the load to the optimum load as we introduce misalignment to the structure. As is obvious from Fig. 5 these coils are not fully symmetric with respect to and axis. Therefore the efficiency simulations presented in Fig. 15 for each misalignment value is the average of the power efficiency when the center of the smaller coil is moved up, down, right and left with respect to the center of larger coil. The calculated values of mutual inductance using (70) had up to 25% error with respect to the simulation results from ADS however the maximum error in was only 8%. Fig. 15 illustrates the results. Fig. 15 shows that the efficiency drops by 45% when the right edge of the smaller coil is aligned with the right outer edge of transmitter coils (15 mm misalignment). A similar simulation was done for Case Study 3 and the efficiency was reduced by 40% under the same condition. In addition to this, the loss in efficiency due to deviation from the optimum load even at the extreme case of 15 mm lateral misalignment or 10 mm of vertical misalignment was less than 1%. The exact same steps could be applied to the vertical misalignment scenario. However in situations where we are only dealing with vertical misalignment, we can simplify (70) by assuming that the magnetic field produced at the receiver coil is uniform and equals the field at the center of the coil. The mutual inductance between a square spiral transturns and a receiver coil with turns can mitter coil with be approximated using the following equation: (71) where represents the effective area of the receiver coil. This area can be calculated using the following formula: (72) where is the pitch between two consecutive turns and is half of the outer diameter of the receiver coil. Equation (71) over-estimates the mutual inductance. Nevertheless, the simulation results are in a good agreement with calculations from (71). Fig. 16 shows the simulation versus calculation results for vertical misalignment. The last case addressed in this section is angular misalignment. We assume that the normal vector of the receiver coil is tilted degrees with respect to the Z axis in Fig. 14. As a result the effective area in (72) should be updated to (73) where for small values of this can be approximated by (74) 242 IEEE TRANSACTIONS ON BIOMEDICAL CIRCUITS AND SYSTEMS, VOL. 6, NO. 3, JUNE 2012 Fig. 17. Loss in efficiency due to deviation of B from B Fig. 16. Efficiency as a function of vertical misalignment using Momentum (ADS) and MATLAB (66) where coils H1 and H2 are normally separated by a distance of 10 mm. Therefore (opt). Since we interested in the percentage of change from the maximum achievable efficiency given an percent deviation from the imaginary part of the optimum load. We can set from (16) and set can be approximated by (80) (75) Rewriting (77) we have APPENDIX E PROOF OF (42) (81) We want to find the value of W in Fig. 14 that would maximize the field at a distance from the center of the coil. We can maximize this value by maximizing (68) at (0, 0, ), hence (76) Hence the optimum outer edge follows: Hence the loss in efficiency as a function of is given by (82) where . F can also be represented in terms of Z parameters APPENDIX F COIL SENSITIVITY TO CHANGES IN LOAD In real world applications there are always some uncertainty and time variance associated with the real and imaginary parts of the load. In addition to this, it is extremely difficult to accurately predict the inductance of the fabricated coils or the thickness of different biological layers. As a result our load would deviate from the optimum load (16), (17). In this section we study the effect of deviation from the optimum load. First we will consider the change in the imaginary part of the load. The Efficiency of the two-port for an arbitrary load in terms of Y parameters is given by (77) It is interesting to note that the sensitivity decreases for larger values of . In order to get an idea on how sensitive we are , Fig. 17 uses the coils from the with respect to changes in first and the third case study, presented earlier to show the loss deviates up to 50% from its in achievable efficiency when optimal value. Fig. 17 confirms our expectation that the sensitivity to variations in is much higher when dealing with low efficiency coupling. Starting from (77) we can derive a similar set of equations for the percentage of loss in two-port efficiency due to the variations in the real part of the parallel load where (78) (83) and (79) Fig. 18 shows the reduction in efficiency due to changes in the real part of the load. , the Similar to the case where we had deviation in structure is more sensitive at low coupling scenarios. However the change in efficiency with the increase in the real part of the ZARGHAM AND GULAK: MAXIMUM ACHIEVABLE EFFICIENCY IN NEAR-FIELD COUPLED POWER-TRANSFER SYSTEMS Fig. 18. Loss in efficiency due to deviation of R from R (opt). Fig. 19. Output voltage and the delivered power for coils G1/G2 from the earlier example assuming 100% rectifier efficiency. load is not of any concern as larger real loads require less current and hence lower power to begin with. Therefore the system should consider the possible lower bound on the real part of the load. The next graph in Fig. 19 demonstrates the same concept in terms of current consumption. The figure depicts the change in voltage and delivered power as the circuits draw different amounts of current from the supply. The data are calculated for two different power levels of 0.5 W and 1 W at the input of the two-port. Here we are considering the DC current provided to the load by the rectifier. We also assumed that the rectifier has 100% efficiency and hence the ac resistive load seen from the . input of the rectifier is half the DC resistive load: As you can see the optimum current is 2.64 mA and 1.86 mA for 1 W and 0.5 W of input power respectively. The efficiency degrades as the current consumption deviates from this value. However the system would still be functional at lower current values since the voltage and power levels meet the minimum requirement of the circuit. Higher current values will result in lower voltage and hence might not be operational. Therefore the power transfer two-port should consider the worst case current consumption. It also requires a circuit to clip the voltage at low current consumption levels to protect the circuits. The next logical question at this point is how accurately can we predict the optimum load. The uncertainty in the imaginary part of the optimum load can be associated with variations in the thickness and electrical properties of the layers surrounding the two coils 243 as well as the inductance of the coils. In order to study the effect of the variations in the media we first used the coils G1/G2 from case study 3. The blood depth was reduced to 6 mm from the original 9 mm while the total distance between the two coils was kept at 10 mm. Simulation shows that the change in the imaginary part of the load for the new setup is less than 0.5%. Next we used case study 3 from . In this case study the receiver is buried under 10 mm of various biological tissue such as fat, skull and Dura. We increased the thickness of every biological tissue by 20% hence increasing the distance between the two coils from 10 mm to 12 mm. We also increased the thickness of the brain layers underneath the CMOS coil by the same 20%. As the result, the imaginary part of the load deviated from the predicted value by less than 1% which resulted in less that 1% deviation from maximum possible achievable efficiency in the two-port. However the achievable efficiency dropped to 11% from the original 18% due to the extra 2 mm of separation. In uncertainty conclusion, the main contributing factor in is the variation in the inductor value. On-chip inductors have around 5 percent variations ,  which at weak coupling situations can lead to 15% loss in power efficiency. The above analysis studied the variation of efficiency with changes in load. However they did not consider the effect of matching networks. Insertion of matching networks between the load and the two-port affects the load variations. These variations depend on the quality factor of the matching networks. In all cases however if the real part of the load of the matching network is changed by 50% the change in the real part of the desired load is less than 50%. But the new mismatched load would have extra reactance. Whether or not the added imaginary part is critical depends on the initial susceptance of the desired load (17). Nevertheless by changing the quality factor of the matching network the designer can control the undesired added susceptance. However the change in real part is always attenuated which is a desirable property. In an L-match network the quality factor depends on the load and the desired impedance and therefore the designer has no control over the value. But using Pi or T matching networks the designer would be able to control the quality factor of the matching network. Here we will show how the desired load would change with variations in the load for the case of an L-match network shown in Fig. 4. For simplicity we will assume that the load is purely resistive. This network would to the desired load value of . Now if the convert the load , the new value load changes by a factor of of the desired load for case (a) is given by (84) which is always smaller than m and the added susceptance is given by (85) For case (b) in Fig. 4 we have (86) (87) 244 IEEE TRANSACTIONS ON BIOMEDICAL CIRCUITS AND SYSTEMS, VOL. 6, NO. 3, JUNE 2012 The added susceptance is usually small enough to be ignored. 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Solid-State Circuits, vol. 35, no. 3, pp. 356–361, Mar. 2000.  J. Yoon, Y. Choi, B. Kim, Y. Eo, and E. Yoon, “CMOS-compatible surface-micromachined suspended-spiral inductors for multi-GHz silicon RF ICs,” IEEE J. Solid-State Circuits, vol. 23, no. 10, pp. 591–593, Oct. 2002. ZARGHAM AND GULAK: MAXIMUM ACHIEVABLE EFFICIENCY IN NEAR-FIELD COUPLED POWER-TRANSFER SYSTEMS Meysam Zargham (S’06) received the B.Sc. degree from Sharif University of Technology, Tehran, Iran, in 2005 and the M.Sc. degree in electrical engineering from the University of Alberta, Edmonton, AB, Canada in 2008. He is currently working toward the Ph.D. degree at the University of Toronto, Toronto, ON, Canada, where his research is in the area of CMOS integrated circuits for biomedical applications. He was a member of the icore High Capacity Digital Communications 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-chip design, and the modeling of carbon nanotube transistors. 245 P. Glenn Gulak (S’82–M’83–SM’96) received the Ph.D. degree from the University of Manitoba, Winnipeg, MB, Canada. While at the University of Manitoba, he held a Natural Sciences and Engineering Research Council of Canada Postgraduate Scholarship. He is a Professor with the Department of Electrical and Computer Engineering, University of Toronto, ON, Canada, as well as a registered Professional Engineer in the Province of Ontario. His present research interests are currently focused on algorithms, circuits, and system-on-chip architectures for digital communication systems; and for biological lab-on-chip microsystems. He has authored or coauthored more than 100 publications in refereed journal and refereed conference proceedings. In addition, he has received numerous teaching awards for undergraduate courses taught in both the Department of Computer Science and the Department of Electrical and Computer Engineering at the University of Toronto. He held the L. Lau Chair in Electrical and Computer Engineering for the five-year period from 19992004. He currently holds the Canada Research Chair in Signal Processing Microsystems and the Edward S. Rogers Sr. Chair in Engineering. From January 1985 to January 1988, he was a Research Associate in the Information Systems Laboratory and the Computer Systems Laboratory at Stanford University, Stanford, CA. From March 2001 to March 2003, he was the Chief Technical Officer and Senior Vice President of LSI Engineering, a fabless semiconductor startup headquartered in Irvine, CA with $70M USD of financing that focused on wireline and wireless communication ICs Dr. Gulak served on the ISSCC Signal Processing Technical Subcommittee from 1990 to 1999, was ISSCC Technical Vice-Chair in 2000, and served as the Technical Program Chair for ISSCC 2001. He was the recipient of the IEEE Millennium Medal in 2001. He currently serves on the Technology Directions Subcommittee for ISSCC and as Editor-at-Large for ISSCC 2012.