A BOUND ON WIRELESS POWER TRANSFER TO SUBWAVELENGTH DEVICES IN THE BODY

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A BOUND ON WIRELESS POWER TRANSFER TO

SUBWAVELENGTH DEVICES IN THE BODY

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF ELECTRICAL

ENGINEERING

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Sanghoek Kim August 2013

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http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/hw833jt5108

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

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I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Ada Poon, Primary Adviser

Teresa Meng

David Miller

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost for Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in

University Archives.

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Abstract

Wireless power transfer to medical devices in the body is desirable for removal of bulky energy storage components. Most existing power transfer systems are concep-tually based on coils linked by slowly varying magnetic fields (less than 10 MHz). These systems have many important capabilities, but they are poorly suited for tiny, millimeter-scale devices where extreme asymmetry between the source and the re-ceiver results in weak coupling.

In general, system parameters such as operating frequency and the structure of source that optimizes efficiency should be determined based on the receiver structure and position, as well as the tissue property of the body. Although many researchers attempt to optimize the system through numerical electromagnetic simulations and experiments of specific source structures and operating frequencies, it remains un-known which design parameters yield the highest global efficiency.

This dissertation instead analytically solves the problem. With a proper modeling of tissue, we prove that for a given receiver structure and position, there exists an upper bound on power transfer efficiency and a solution for the current distribution that achieves this bound. The bound on efficiency and the optimal current distri-bution are evaluated for various medical applications – a cardiovascular device to sense and stimulate heart signals, a wireless endoscope in the small intestine, and a neurostimulator in the brain to record neural activity.

Reminiscent of Shannon’s channel capacity in information theory, this theory pre-dicts that there is significant room for improvement in existing power transfer systems and facilitates the design of structures to approach this optimal bound. The design procedure of the wireless power delivery system for a given receive structure in the

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body is demonstrated. The theory is verified with both numerical simulation and experimental results.

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Acknowledgement

I thank God for placing in my life teachers, colleagues, friends, and family members who have guided and supported me throughout the completion of my PhD studies.

I would like to thank my advisor, Prof. Ada Poon, a great mentor of mine. From her, I first learned the attitude of scholars–the physical insights she earns from studying a subject deeply constantly astonish me. She also taught me how to present and write new ideas as a researcher. All the lessons I learned from her will be great resources of mine throughout my life.

I would like to thank Prof. David Miller, for his keen insights that have inspired me in many ways, for serving on my oral committee, and for reading this thesis. I am grateful to Prof. Teresa Meng for her continual support and great encouragement during my course of study. I would also like to thank Prof. Mark Horowitz and Prof. John Pauly, for serving on my committee.

I cherish the fond memories with my colleagues at Stanford. I have enjoyed and benefited from the collaboration with John Ho. He made the figures in my paper self-explainatory and aesthetically pleasing, and he has also been my best English teacher, who always proposes better sentence to express my ideas. I am grateful to have worked with him. I also appreciate other colleagues in Poon’s group, Jihoon Jang, Andrew Ma, Alex Yeh, Anatoly Yakovlev, Yuji Tanabe, Kamal Agggawal, Stephenie Hsu, Ming Tsang, Vivien Tsao, and Steve Wong. Thanks to them, I enjoyed fun-filled lab life.

I was really blessed to have a great church community at Stanford. All of the community members have supported and cared each other sincerely. I thank Pastor Chris, Pastor Jaeho Chung, Hochan Lee, Hannah Cho, Byoung Choi, Seulgi Moon,

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Kyuho Han, Wonuk Jo, Joohong Choi, Crystal Han, and Rick Kim, to name just a few. I cherish the good memories with friends I usually have had lunch with, Youngsik Kim, Jaewon Yang, Myounghwan Kim, Kietae Kwon, and Sunghee Park.

This research would not have been possible without the generous support from Kwanjeong Scholarship, Samsung Advanced Institute of Technology, Center for Cir-cuit and System Solution (C2S2), Autonomic Technologies. I also thank June Wang and Ann Guerra in CIS for their administrative support.

I give my heartfelt thanks to my parents, who taught me math until my middle school years, for their endless support and prayer. I also appreciate parents in law who have not spared anything to support my research. My wife, Eunyoung Ji, is the best gift I have received from God. Before marriage, she used to be a very promising employee in the biggest fashion company in Korea. She pressed a hard ‘Pause’ button in her career to marry and live with me in the United States. I can’t thank her enough for her brave decision and sacrifice. Without any hesitation, I dedicate this thesis to her, and our baby-to-be, Choan Kim.

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List of Tables

1.1 Frequency, Range, and Coil Size of Select Implants . . . 6 1.2 Tuning Configurations of Select Studies . . . 8 1.3 Approximate Optimal Frequency at 5 cm Depth for Select Types of

Human Tissue . . . 10 2.1 Optimal power transfer efficiency for typical values of R22 and

mini-mum load resistanceRL= 10 Ω . . . 31

4.1 Improvement of the efficiency by the optimization over the coil-based sources . . . 43 4.2 Tissue composition for several medical applications. Unit for the

thick-ness t is millimeter. . . 43

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List of Figures

1.1 In inductive coupling, currentI1 is passed through the external

trans-mit coil creating a magnetic field H1in the space around the coil. Some

of the field passes through the receive coil, which induces emfV2 in the

coil. . . 3 1.2 Lumped circuit model for inductive coupling. The coils can be shunt

tuned by inserting capacitors Cp1 and Cp2 in parallel or series tuned

with Cs1 and Cs2. . . 6

1.3 Coupling parameter versus frequency between two small coils in ho-mogenous muscle positioned 5 cm apart at different orientations. . . . 10 2.1 A planar current sourceM1z(x, y) on top of a multilayer inhomogeneous

tissue model delivers power to an implanted antenna atz =_−zf. . . . 14

2.2 Overall wireless power transfer system. This work focuses on the analy-sis and the optimization of the shaded region. (J₁, M1) are the electric

and the magnetic current distributions on the external antenna struc-ture while (J2, M2) are those on the implant antenna structure. . . . 15

2.3 Coupling parameter versus frequency for a vertical magnetic dipole at zf = 5 cm for different source distributionsd1 = 1 cm above the interface. 22

2.4 The coupling parameter γ versus frequency for a vertical magnetic dipole receiver at depths zf − d1 = 2 cm, 4 cm, and 8 cm for (a) the

optimized current distribution and (b) a uniform source. . . 24

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2.5 The distribution of Voc and SAR at y = 0 in the tissue for the point,

uniform, and optimal sources at 2 GHz. The receiver is a magnetic dipole with normal oriented along thez direction, and (d1, zf) = (1 cm,

5 cm). . . 25 2.6 Magnitude and phase of M1z,opt(x, y) at 2 MHz and 2 GHz for the

receive mangetic dipole where (d1, zf) = (1 cm, 5 cm). . . 26

2.7 Magnitude and phase of M1z,opt(x, y) at 2 MHz and 2 GHz for the

receive electric dipole where (d1, zf) = (1 cm, 5 cm). . . 27

2.8 ρ0.9 versus frequency for a magnetic dipole and electric dipole receiver

at depth zf − d1 = 4 cm. . . 28

2.9 ρ0.9 versus frequency for a magnetic dipole receiver at depthszf− d1 =

2 cm, 4 cm, and 8 cm. . . 29 3.1 The general configuration for wireless power transfer consists of a

source surface current density J1 and a receive coil positioned in tissue

with magnetic dipole moment ˆn at an angle θ. The tissue geometry is modeled by multilayers. By convention, the dipole moment aligns with the z-axis when θ = 0◦ _{and the} _{x-axis when θ = 90}◦_{. . . .} ₃₄

3.2 (a) Theoretical bound on efficiency η for the θ = 90◦ _{(black) and the}

θ = 0◦_{(red) receive coil orientations. Coil-based source structures with}

diameter from 0.6 to 6 cm (error bars show the max, min, and mean) achieve efficiencies well below the theoretical bound. (b–c) Current distribution of the coil source (top) and the optimal source (bottom) at 2.6 GHz. (d–e) The magnetic field component aligned with the receiver dipole moment ˆx and the Poynting vector (white) generated by the coil source (top) and the optimal source (bottom) at 2.6 GHz. 39

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4.1 (a) Coupling parameterγ, (b) receiver self-resistance R22, and (c)

effi-ciencyη as a function of frequency for a coil receiver with radius 400 µm and θ = 0◦ in cardiovascular devices. (d) Efficiency for a coil receiver with radius 400 µm and θ = 0◦ _{in cardiovascular devices. Coil-based}

source structure with diameter from 0.6 to 6 cm (error bars show the max, min, and mean) achieve the efficiency well below the theoreti-cal bound. The efficiency of coil-based sources are computed from a commercial EM simulator. The optimal source at denoted points are shown in Fig. 4.2 . . . 44 4.2 Optimal electric current density. For a cardiovascular devices with (a)

θ = 0◦ _{at 10 MHz (b)} _{θ = 0}◦ _{at 1.4 GHz (c) θ = 90}◦ _{at 2.6 GHz. For}

a device in small intestine with (d) θ = 0◦ _{at 910 MHz (e)} _{θ = 90}◦ _at

1.7 GHz. For a device in brain with (d) θ = 0◦ _{at 2.1 GHz (e) θ = 90}◦

at 3.3 GHz. For all the above plots, movies showing the current flow according to time are uploaded as supplementals online. . . 45 4.3 Efficiencyη as a function of frequency for a receive coil of radius 400 µm

in (a) heart, (b) small intestine, and (c) brain. Coil-based source struc-ture with diameter from 0.6 to 6 cm (error bars show the max, min, and mean) achieve the efficiency well below the theoretical bound. The efficiency of coil-based sources are computed from a commercial EM simulator. . . 47 5.1 (a) The neurostimulator implant inside the head to relieve the headache

pain (b) Modeling of the receiver occupying same volume and in same position. . . 51 5.2 A bond on efficiency for a given receiver is shown with a solid black

curve. Due to finite conductivity of copper, the efficiency of loop (solid blue) is much lower than the bound. To achieve the efficiency as close to the bound as possible, one should operate at shaded frequency range. 52

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5.3 (a)The optimal current distribution for the frequency range of inter-ests. (b) Simulated loop structure resembling the optimal source (c) Fabricated loop structure with matching network for the measurement 53 5.4 (a) Multi-turn loop fits inside the same cover used in the ATI product

(b) Optical cable attached to detect LED lighting (c) Turkey breast for the experiment . . . 54 5.5 . . . 55 5.6 (a) A reduced view of the optimal current density composed of three

current paths. (b) Change in current density with time along an axial cut (dotted line) over a half-period. (c) Spatial phase variation shows that the center is stationary (constant region, shaded) while currents away from the center propagate inwards (linear region, non-shaded). . 56 5.7 (a) The slot-array efficiency (red) approaches the theoretical bound

(black) and exceeds the performance of coil sources (blue). Note that the efficiency includes structural resistive losses. The slot array can be configured to be insensitive to large lateral displacements of the device (error bars show max, min, and mean from 0 to 2 cm). The inset shows the change in efficiency at 2.9 GHz for the phase-adjusted (solid red) and non-phase-adjusted (dashed red) slot array. (b) Slot array design and (c) time snapshot of the vector current density overlaying an intensity colormap. . . 57 6.1 Body model for wireless power transfer to a cardiac implant. (a) Source

plane modeled by current density J1. (b) Source position 5 cm above a

small receive coil (white dot) on the heart. (c) Receive coil orientation at angle θ. . . 61 6.2 Theoretical coupling parameterγ, receiver self-resistance R2, and

cou-pling efficiencyκ/(Γ1Γ2) as a function of frequency for a receive coil of

radius 0.5, 1, and 2 mm. Results are obtained on the multilayer model. 62

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6.3 Efficiency η as a function of frequency. The receiver is a 1-mm radius coil oriented θ = 90◦ on the heart. Theory results are obtained on the multilayer model and the FDTD results on the body model. . . 63 6.4 Efficiency as a function of coil angular displacement at 1.7 GHz for

sources optimized for theθ = 0◦_{, 45}◦_{, and 90}◦ _{receive coil orientations.} ₆₄

6.5 Available open-circuit voltage _|Voc|2 and SAR distribution along the

y = 0 slice of the model at 200 MHz and 1.7 GHz. The receiver is a 1-mm coil tiltedθ = 90◦ is placed at a depth of _{−5 cm. . . .} 65 6.6 Received open circuit Voc and received power Pr as a function of the

receive coil radius at 1 GHz and 1.7 GHz. The coil is orientedθ = 90◦_. ₆₆

A.1 Geometry of the field equivalence representation of the source over multilayer tissue. For an arbitrary source located in the half-space above the planeS (Region I), there exist corresponding surface currents Js and Ms that generate identical fields in the half-space below the

plane (Region II). . . 71 C.1 Optimal source at 100 MHz for the cardiac implant configuration. . . 78 D.1 Intensity of the angular spectrum of the optimal source at 2.6 GHz.

The concentric circles have radii corresponding to the wavenumber in (a) air, (b) fat, (c) bone or skin, and (d) muscle or heart. Note that tissue types with similar dielectric properties at 2.6 GHz have been grouped together. . . 80

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Chapter 1 Introduction

Implantable medical devices such as cochlear implants, pacemakers, and cardiac de-fibrillators play an increasingly important role in managing a broad range of medical disorders. These devices perform sensing, drug delivery, and local stimulation inside the patient’s body and are necessary for continuous treatment and health monitoring. In order to provide unobtrusive care and minimize invasiveness, implantable de-vices need to be miniaturized to allow placement close to the area of treatment. While the electronic and mechanical components of implants continue to reduce dramatically in size, electrochemical energy storage has been much slower to miniaturize. With-out alternative means of power, the battery can dominate the size of the implant. This greatly constrains the placement of the implant and suffers from disadvantages such as risk of corrosion or wire-snapping and complications arising from battery replacement.

Wireless powering of implantable devices enables the removal of the battery alto-gether. In place of a battery, a receiver is placed on the implant to harvest energy transmitted by an external source. The receiver rather than battery then constrains the size of the implant. Various means of power delivery have been considered, in-cluding ultrasound and electromagnetic waves at near-infrared or radio frequencies. In this dissertation, we will consider wireless powering using electromagnetic waves.

In the past fifty years, most work on electromagnetic wireless powering considered frequencies below 10 MHz. At low frequencies, the propagation of electromagnetic

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CHAPTER 1. INTRODUCTION 2

fields in tissue can be approximated by the diffusion equation. On solving the diffusion equation, the source and receiver are found to be inductively coupled, and power is transferred through the means of coils. The transmit and receive coils can then be designed to maximize power transfer efficiency or robustness to variations in the link conditions.

Since tissue absorption increases with frequency, higher frequencies were not con-sidered in most analyses. Indeed, the diffusion equation reveals that the loss in tissue increases exponentially with frequency and suggests that better efficiency can be obtained by a low frequency carrier. However, at higher frequencies, the diffusion equation is not valid in tissue. Instead, the wave equation must be solved. Re-cently, Poon et al. [61, 63] showed that the optimal frequency is in the low GHz range for millimeter-scale receivers. Operating at higher frequencies could allow dramatic miniaturization and enable fully integrated implantable devices.

At higher frequencies, the wavelength is comparable to the distance of separation and the choice of source structure becomes important. To analytically determine the optimal source structure, the body is modeled as a planar multilayered medium comprising different tissue layers. In Chapter 2, a continuous rotational current source on an infinite plane is optimized. Most coil-based source structures are included in this scope of source. The scope of source under the consideration for optimization is extended from Chapter 2 to Chapter 3 to include both rotational and irrotational current source, which can be demonstrated to be the most general representation for any arbitrary physical source. In Chapter 4, the bounds on efficiency are solved for various medical applications, and the theory predicts that there is significant room for improvement in current power transfer systems and as such facilitates the design of source structures to approach this optimal bound. The wireless power delivery system design procedure to a given receive structure in the body is demonstrated in Chapter 5. Finally, our planar medium tissue modeling is validated by studying the wireless powering of a cardiac implant in Chapter 6 using a phantom of the human body and Finite Discrete Time Domain (FDTD) method. By comparing the theoretical results using planar medium modeling and the FDTD method results using the phantom of the human body, we confirm our multilayer tissue modeling.

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CHAPTER 1. INTRODUCTION 3 air body H1 I1 V2

Figure 1.1: In inductive coupling, currentI1 is passed through the external transmit

coil creating a magnetic field H1in the space around the coil. Some of the field passes

through the receive coil, which induces emf V2 in the coil.

In this chapter, we review the principles of wireless powering through inductive coupling in Section 1.1. In Section 1.2, we discuss powering at higher frequencies and review work showing that the optimal frequency lies in the low GHz region for tiny devices.

1.1 Inductive Coupling

1.1.1 Electromagnetic Basis

The use of RF electromagnetic waves to transfer energy for biomedical devices be-gan in the early 1960s, where Schuder et al. [69] demonstrated energy transfer over the chest wall of animals through inductive coupling. There has since been several comprehensive analyses on the inductive link over tissue [19, 24, 31, 44]. In inductive coupling, an external coil carries time-varying current that results in time-varying magnetic and electric fields in the space around the coil. A second coil is placed inside the patient’s body, as shown in Figure 1.1. The basis of energy transfer is

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Faraday’s law

V2 =iωµ0

Z

H1· ds (1.1)

where ω is the frequency of the fields. Wireless powering occurs when the transmit coil produces a magnetic field H1 that induces emf V2 at the receive coil. As the

magnetic field at the receiver varies with time, voltage proportional to the rate of change is induced in the implant.

For a transmit current distribution, the magnetic field H1 can be found by solving

Maxwell’s equations. Inductive coupling is based on the assumption that the fields are slowly changing. For low-frequency fields, the displacement current term can be ignored; this is known as the quasi-static approximation. Ignoring tissue, the field H1 under this approximation is governed by the diffusion equation whose solution is

given by the Biot-Savart law. On substituting this into (1.1), the emf at the receive coil V2 will have the form

V2 =iωM I1 (1.2)

where the mutual inductance M is a real quantity dependent only on the geometry of the coils. This can be even more compactly written asV2 =Z21I1, whereZ21 =iωM

is the mutual impedance between the coils.

Since same laws must apply from the receive to the transmit coil as well, there will be an opposing emfV1 induced in the transmit coil for a currentI2 in the receive

coil. In addition, there exists a self-inductance L that arises from a coil’s opposition to its own change in current. The total emf at the receiver is then

V2 =iω(M I1− L2I2). (1.3)

It can be shown that M = k√L1L2 where the constant of proportionality k < 1

is known as the coupling coefficient. Although the inductive coupling was described in air, tissue losses are negligible at low frequencies and the results can be directly applied.

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1.1.2 Two-port Network

A convenient way to summarize the various forms of coupling is the two-port network. In this model, the induced emf in both coils are given by

" V1 V2 # = " Z11 Z12 Z21 Z22 # " I1 I2 # (1.4)

where Z12 = Z21, Z11 = −iωL1 and Z22 = −iωL2. Note that the impedances are

all purely imaginary. This is a consequence of the quasi-static approximation and indicates that the energy is stored in the fields around the coil rather than radiated. In practice, the impedances contain real partsR to account for conductivity losses in the coil and tissue. The current I2 at the implant is also due to the induced emf at

the receive coil V2 as given by

I2 =−V2/ZL (1.5)

where ZL is the load impedance that depends on the circuitry that the system is

trying to power.

For implantable devices, the current in the receive coil is usually very small com-pared to the transmit coil. As such, the coils are loosely coupled and the induced emf at the transmit can be approximated as due only to self-inductance. The power transfer efficiency is then given by

η = Pr Pt = |Z21| 2 4R11R22 4R22RL |Z22+ZL|2 . (1.6)

In this expression, the load-dependent term on the right is the matching efficiency that depends on the relative impedance of the load and the antenna at the receiver. The term on the left is the coupling efficiency and is the ratio of power available sat the receiver to the input power. From the coupling efficiency, we extract the coupling parameter γ

γ = |Z21|

2

R11

, (1.7)

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Table 1.1: Frequency, Range, and Coil Size of Select Implants

Implant Carrier Range source Receiver

Type (MHz) Coil Coil

Generic [5] 1-2 0.5 cm 1 cm 0.47 cm Generic [80] 4 0.8 cm Generic [34] 10 1.5 cm 3.5 cm 2.7 cm Cortical visual [60] 5,10.33 1 cm Retinal [51] 1-10 Retinal [72] 2.5 0.575 cm 1.34 cm 1.34 cm Muscular [8, 52] 2 9-20 cm 0.12 cm Muscular [70] 6.78 Muscular [48] 2 Bio-sensor [35] 27 9 cm 5 cm Sensor [74] 0.133 < 1 m Sensor [9] 13.56 10 cm 15 cm 6 cm Sensor [13] 3.18 4 cm 4.8 cm 0.6 cm

The transmit power is usually limited to avoid interference with other devices, reduce the size of the source, or the extend the device lifetime if an external battery is used [37]. As such, the goal of most studies on wireless powering is to maximize the power transfer efficiency or minimize its sensitivity to variations in the link.

Cp1 Cs1 L2 Cs2 RL M L1 Cp2 RC

Figure 2: Illustrates the tuning circuit configuration.

Table 1: Z1and Z2in diﬀerent tuning circuit configurations.

Primary Secondary R1 X1 R2 X2 Untuned Untuned RC ωL1 RL ωL2 Untuned Series RC ωL1 RL ωL2−ωC12 Untuned Shunt RC ωL1 _1+ωR2_CL2 2R2L ωL2− ωC2R 2 L 1+ω2_C2 2R2L Series Series RC ωL1−_ωC1₁ RL ωL2−_ωC1₂ Series Shunt RC ωL1−ωC11 RL 1+ω2_C2 2R2L ωL2− ωC2RL2 1+ω2_C2 2R2L Shunt Series RC 1+( 1 ωL1−ωC1)2R2C ( 1 ωL1−ωC1)R 2 C 1+( 1 ωL1−ωC1)2R2C RL ωL2−_ωC1₂ Shunt Shunt RC 1+( 1 ωL1−ωC1)2R 2 C (_ωL11 −ωC1)R2C 1+( 1 ωL1−ωC1)2R 2 C RL 1+ω2_C2 2R2L ωL2− ωC2R 2 L 1+ω2_C2 2R2L 2

Figure 1.2: Lumped circuit model for inductive coupling. The coils can be shunt tuned by inserting capacitors Cp1 and Cp2 in parallel or series tuned with Cs1 and

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1.1.3 Link Optimization

Table 1.1 lists the frequency, range, and coil size of wirelessly powered implantable devices described in literature. The list is not exhaustive, but is representative of the typical implant parameters. When the secondary coil is small or not in close proximity to the primary coil, design techniques were proposed to maximize the efficiency such as the use of ferrite cores [68], impedance transformation with inductance tapping and voltage doubler circuits [44], and efficient rectifying circuits [38].

In practice, the load impedance ZL can be changed by introducing a matching

network between the receive coil and the load. A simple way to improve the matching efficiency is to insert a capacitor to adjust the imaginary part ofZLsuch that it cancels

that ofZ22at the frequency of interest as shown in Figure 1.2; this technique is known

as tuning.

Tuning can be used to desensitize the power transfer efficiency to the link con-ditions. Table 1.2 lists the different tuning configurations that have been studied. In [21], it was shown that displacement tolerance could be improved with only a small degradation in efficiency from the untuned counterpart. By operating at k where the voltage transfer ratio is optimal, the dependency of the efficiency onk was reduced [12]. Using a similar idea, an inductive link was designed in [32] wherek was chosen to maximize transimpedance to desensitize its variation with k. In [24, 25], design procedures to desensitize the voltage transfer ratio to variation ink were pro-posed using stagger-tuned inductive links. Using coils at free-running oscillation it was shown that the efficiency is independent of k when k is greater than the inverse of the loaded quality factor of the secondary coil [14–16, 36]. In [82], a circuit ap-proach was described to improve the tolerance to coupling variation using coils at self oscillation.

In most of these studies, the idea of using resonant LC tanks on both coils to enhance the inductive link is deeply embedded. Recent techniques on using coupled resonance to achieve midrange power transfer for potential applications in consumer electronics are reminiscent of this idea [46].

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Table 1.2: Tuning Configurations of Select Studies Reference Transmit Tuning Receive Tuning Resonance

[21] – shunt unloaded

[12] series shunt loaded

[32] shunt shunt unloaded

[24, 25] stagger stagger –

[14–16, 36] series series –

[82] series-shunt shunt –

1.2 Higher Frequencies

1.2.1 Higher Frequency Field Behavior

Wireless powering at higher frequencies is motivated by the fact that the induced emf at the receive coil increases with frequency, as can be seen from (1.1). This allows smaller receive structures to be used while keeping the delivered power constant. At higher frequencies, however, tissue effects can no longer be ignored since loss also increases with frequency. In order to establish the behavior of the fields at higher frequencies, Maxwell’s equations must be solved in tissue.

At higher frequencies, the quasi-static approximation is valid for good conductors. However, tissue is better modeled as a low-loss dielectric in which the displacement current term is significant. The magnetic and electric fields are then governed by the wave equation rather than the diffusion equation. Since most of the energy is radiated rather than stored in the fields, the link cannot be described in terms of inductive coupling.

In [61, 63], the power transfer efficiency for a point source and receiver in ho-mogeneous tissue was derived at higher frequencies. While the diffusion equation shows that efficiency decreases exponentially with frequency, full-wave analysis re-veals that it actually increases with frequency. This increase, however, is eventually offset by relaxation loss that results from the time lag between the electric field and the polarization of the tissue. As such, there exists an optimal frequency for wireless powering.

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1.2.2 Point Source and Tissue Model

A current density Jscan be decomposed into an irrotational component J and a

rota-tional component M. The rotarota-tional component is indistinguishable from a fictitious magnetic current density and can be thought of as being composed of small loops of current. A point source can thus be modeled as either an M or J current density, which approximate a small dipole or a small coil respectively. Due to the vector nature of electromagnetic fields, a point source is characterized by its orientation as well as magnitude and phase. A small coil located at r0 can thus be written as

M(r) =iωµ0ArIδ(r− r0)α (1.8)

whereAris the area of the coil,I the current, and α the unit vector in the direction of

the coil. If the transmit and receive coil are assumed to be small and carry currentsI1

andI2 respectively, they can both be represented as a point magnetic current density.

In order to establish the optimal frequency, a model for dielectric relaxation in tissue is required. A popular model for biological media is the Debye relaxation model

r(ω) = ∞+ r0− ∞ 1_{− iωτ} +i σ ω0 (1.9) wherer is the frequency-dependent dielectric permittivity of tissue. The parameters

r0, ∞, σ, and τ are given in [22] for different types of tissue. The imaginary part

of r accounts for both conductivity and relaxation losses. For simplicity, both the

transmit and receive coil are assumed to be embedded in homogeneous tissue.

1.2.3 Optimal Frequency

In order to define the optimal frequency of operation, an expression for the power transfer efficiency is required. The efficiency based on the two-port model (1.6) can be used if the impedances are defined appropriately for higher frequencies. In general, R11 accounts for losses due to radiation, tissue, and conductivity in the transmit

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CHAPTER 1. INTRODUCTION 10 106 107 108 109 10−6 10−5 10−4 10−3 Frequency [Hz] Edge−to−edge ( θ=90°) Tilt ( θ=45°) Face−to−face ( θ=0°) γ

Figure 1.3: Coupling parameter versus frequency between two small coils in homoge-nous muscle positioned 5 cm apart at different orientations.

Table 1.3: Approximate Optimal Frequency at 5 cm Depth for Select Types of Human Tissue

Tissue Type Approximate fopt (GHz)

Blood 1.58

Bone (cancellous) 1.70

Bone (cortical) 2.01

Fat (infiltrated) 2.68

Fat (not infiltrated) 3.86

Heart 1.68 Liver 1.70 Lung (deinflated) 1.72 Muscle 1.76 Skin (dry) 1.98 Skin (wet) 1.79

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structure. At high frequencies, tissue is the dominant form of loss. As such, R11=

ω |I1|2

Z

Im(r)|E1(r)|2 dr. (1.10)

The mutual impedance Z21 can be defined in terms of the receive power. Suppose

that a magnetic current density M2 exists at the receiver. Then, normalizing by the

magnitude of the currents at the source and receiver, Z21=

1 I1I2

Z

M2(r)· E1(r)dr (1.11)

where the integral vanishes if M2 is a point source.

The fields due to the source can be expressed in terms of the dyadic Green’s functions ¯Gm and ¯Ge for the magnetic and electric field. The Green’s functions

act like the impulse response for point source and have a simple analytical form for homogeneous tissue. If matching considerations are neglected, the optimal frequency maximizes the coupling parameter

γ = _|Z21|2/R11. (1.12)

In [61], two orientations for the source and receiver were considered: edge-to-edge and face-to-face. An upper and lower bound for the optimal frequency was then obtained using asymptotic expressions for the dielectric permittivity. Alternatively, the Green’s functions can be decomposed into the sum of vector multipoles [10]. Considering only the lowest order multipoles, the optimal transmit orientation was found and the optimal frequency was shown to be approximately

fopt≈ 1 2π s c√r0 τ d(r0− ∞) (1.13)

where d is the distance between the source and receiver [63].

We now consider identical transmit and receive coils with areaAr = 4 mm2 facing

each other in muscle tissue. Figure 1.3 shows the coupling efficiency in muscle over a range of frequencies for the face-to-face, 45◦ tilt, and edge-to-edge orientations.

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In all cases, the optimal frequency is near the low GHz range. Table 1.3 lists the approximate optimal frequency for different types of tissue when the optimal orien-tation is used. Matching considerations doesn’t change optimal frequencies much for millimeter-scale receivers [43]. Although the optimal frequencies are in the low GHz range for tiny receivers, most existing implants operate exclusively in the low MHz range as Table 1.1 shows.

In addition to increased coupling efficiency, there are other practical reasons for operating at higher frequencies. Higher frequencies result in increased received open voltage which eases constraints on the power harvesting circuitry [56]. If data is modulated onto the power carrier, increasing the operating frequency also enables higher data rates [78]. Finally, smaller wavelengths allow the fields to be shaped by the structure of the source as will be described from the next chapter.

1.3 Summary

We review the inductive coupling analysis of the wireless link and design techniques to optimize the link. The analysis is extended to higher frequencies using a simple homogeneous model of tissue and the optimal frequency for wireless powering was found to be in the low GHz range for millimeter-scale receivers. Prototypes of im-plantable devices operating at higher frequencies have recently been demonstrated in literature [56] [78]. At higher frequencies where the wavelength is comparable to the distance of separation, the source can be optimized to improve power transfer efficiency as will be described from the next chapter.

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Chapter 2 Optimization of Continuous,

Infinite, Rotational Sources

In this chapter, we consider the problem of finding the optimal source for a small receiver in tissue. Given the greater degree of freedom allowed in the design of the external source, we remove restrictions on the dimension and structure of transmit antenna by modeling the source as an infinite sheet of magnetic current density. The distribution maximizing the power transfer efficiency is analytically solved for a mul-tilayer tissue model and a small receiver modeled as a combination of an electric and magnetic dipoles of arbitrary orientation. By finding the optimal current distribution, an upper limit on power transfer efficiency for a magnetic current distribution can be established for a given implant in tissue.

This chapter is organized as follows. Section 2.1 presents the source and the tissue models as well as the expression for the coupling parameter. Section 2.2 expresses the coupling parameter in terms of the transmit current distribution. Section 2.3 derives the optimal current distribution that maximizes the coupling parameter. Section 2.4 compares the performance of optimal source to that of a point source and a uniform source, and shows the resulting field distributions in tissue as well as the properties of the optimal source. Section 2.5 discusses receiver considerations and relates the optimization gain to power transfer efficiency. Finally, we summarize this chapter in Section 2.6.

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CHAPTER 2. CONTINUOUS, INFINITE, ROTATIONAL SOURCES 14 z 0 Medium 1 (Air) Medium 2 Medium . . . . . . −

Figure 2.1: A planar current source M1z(x, y) on top of a multilayer inhomogeneous

tissue model delivers power to an implanted antenna atz =_−zf.

2.1 Model and Problem Formulation

2.1.1 Source and Tissue Models

We model the inhomogeneity of the link as an air-tissue planarly multilayered medium, as illustrated in Fig. 2.1. The transmit antenna is modeled as an infinite sheet of mag-netic current density at z = 0 with distribution

M1(r) =M1z(x, y) δ(z) ˆz. (2.1)

Since the receive antenna is small, it can be modeled as a combination of magnetic and electric dipoles with arbitrary orientation located at r = rf:

M2(r) =iωµArI2δ(x, y, z + zf) α (2.2a)

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CHAPTER 2. CONTINUOUS, INFINITE, ROTATIONAL SOURCES 15

External Device

Matching Network

PA Matching_network Rectifier &_Regulator

Two-port Network

VDD

Implantable Device IC Transmission link

Figure 2.2: Overall wireless power transfer system. This work focuses on the analysis and the optimization of the shaded region. (J₁, M1) are the electric and the magnetic

current distributions on the external antenna structure while (J2, M2) are those on

the implant antenna structure.

where Ar is the area of the magnetic dipole and ArI2 is its magnetic moment, and

lr is the length of the electric dipole and lrI2 is its electric moment. The vectors ˆα

and ˆβ denote the orientation of the magnetic and electric dipoles respectively, and the relative contributions from the two dipoles are normalized such that α2₊_β2 _{= 1.}

For a given M2 and J2, we want to find M1z(x, y) that optimizes the power transfer

efficiency, as will be next defined.

2.1.2 Coupling Parameter

Fig. 2.2 shows a typical wireless power transfer system. In this work, we focus on the power transfer efficiency over the transmission link shown as the shaded region of Fig. 2.2. The coupling between the transmit and the receive structures can be abstracted as a two-port network:

V1 =Z11I1+Z12I2

V2 =Z21I1+Z22I2.

Denoting the equivalent input impedance of the power receiver as ZL, we have V2 =

−ZLI2 and hence,

I2 =−

Z21

Z22+ZL

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Now, the received power at the output of the two-port network can be written as Pr =RL I₂ 2 (2.3) where RL is the real part of ZL. Since receive structure is small, the source and

receiver are loosely coupled. The transmit power at the input of the two-port network can then be approximated by

Pt = Re{V1I1∗} ≈ R11 I₁ 2 . (2.4)

where R11 is the real part of Z11. The power transfer efficiency is thus given by

η := Pr Pt ≈ Z₂₁ 2 4R11R22 4R22RL Z₂₂+Z_L 2. (2.5)

In this expression, the efficiency is the product of two factors: the coupling efficiency ηc on the left and the matching efficiency ηm on the right. The coupling efficiency

is the ratio of the power available at the receiver to the input power. The matching efficiency is the ratio of the power delivered to the load to the available power. In this dissertation, we focus on optimizing the source for a given receiver. Since the matching efficiency is independent of the source structure, it suffices to maximize the coupling efficiency. From ηc, we extract the coupling parameter γ

γ = |Z21|

2

R11

, (2.6)

which is completely determined by the source. The optimal source is thus given by the current distribution M1z(x, y) that maximizes γ.

2.2 Self and Mutual Impedances

To maximize the coupling parameter for a given receiver, we will need to express the coupling parameter in terms of M1z(x, y) first. This is achieved by first defining

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the sources M₁, M₂, and J₂. We then derive these fields in terms of the source distributions.

2.2.1 Definitions

For a loosely coupled two-port network, the real part of Z11 accounts for the tissue

loss, the conduction loss in the transmit structure, and the radiation loss. Tissue loss usually dominates the radiation loss, since the radiation efficiency is poor due to the presence of lossy tissue in near-field region. We assume that the antenna efficiency is close to unity. Denoting the electric and the magnetic fields from the transmit source by E1(r) and H1(r) respectively, R11 = ω I₁ 2 Z z<−d1 Im(r)E1(r) 2 dr (2.7)

where (r) is the permittivity at r.

We define the mutual impedance via the concept of induced emf [28, Chapter 3]. It is given by Z21= 1 I1I2 Z M₂_{· H}₁dr + Z J₂_{· E}₁dr (2.8a) = Voc I1 , (2.8b)

where Voc is the received open circuit voltage of

Voc=iωµArα· H1(rf) +lrβ· E1(rf) (2.9)

for a small receive dipole. Putting these together,

γ = iωµArα· H1(rf) +lrβ· E1(rf) 2 ωR z<−d1Im(r) E1(r) 2 dr . (2.10)

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2.2.2 Expressions for the Fields

The electromagnetic fields can be expressed in terms of source through the Green’s functions: H1(r) =iω Z ¯ Ghm(r− r0)M1(r0)dr0 (2.11a) E1(r) =− Z ¯ Gem(r− r0)M1(r0)dr. (2.11b)

Taking the 2D Fourier transform with respect to (x, y) for a given depth z yields H1(kx, ky, z) = iω ¯Ghm(kx, ky, z)ˆzM1z(kx, ky) (2.12a)

E₁(kx, ky, z) =− ¯Gem(kx, ky, z)ˆzM1z(kx, ky). (2.12b)

In free-space, via the use of Weyl identity, the Green’s functions are given by ¯ Ghm,f s(kx, ky, z) = ie−ikzz 2kz ¯ I₋kk t k2 ¯ Gem,f s(kx, ky, z) =− e−ikzz 2kz k_{× ¯I} where kz = pk2− kx2− ky2, k = h kx ky −kz it

, and k is the wavenumber of free-space. In the multi-layered medium, we need to include the reflection and the trans-mission coefficients. From [11, Chapter 2], when z is in between−dn+1 and −dn, the

Green’s functions can be written as ¯ Ghm,n(kx, ky, z) = i 2k1z ¯_I₋ k1kt1 k2 1 · Ane−iknzz+ ˜RT En,n+1eiknz(z+2dn) (2.13a) ¯ G_em,n(kx, ky, z) = − 1 2k1z k₁_{× ¯I · A}_ne−iknzz_{+ ˜}_RT E n,n+1eiknz(z+2dn) (2.13b)

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where knz = pkn2 − kx2− k2y, kn =

h

kx ky −knz

it

, and kn is the wavenumber of

the nth layer. The term ˜RT E

n,n+1 is the generalized reflection coefficient while An can

be interpreted as the generalized transmission coefficient. Their expressions can be found in [11, Chapter 2]. Once _M1z(kx, ky) is known, performing the inverse Fourier

transform yields the electromagnetic fields at any point in space.

2.3 Optimal Source Distribution

We will now express the coupling parameter in (2.10) in terms of _M1z(kx, ky). For a

Fourier transform pairg(t) and G(ω), g(0) = 1

2πR G(ω) dω. Therefore, H1(0, 0,−zf) = 1 4π2 Z Z H1(kx, ky,−zf)dkxdky (2.14) and hence, H1(0, 0,−zf) =iω1 4π2 Z Z ¯ G_hm,j(kx, ky,−zf)ˆzM1z(kx, ky)dkxdky. (2.15) Similarly, E₁(0, 0,_−zf) =₋ 1 4π2 Z Z ¯ Gem,j(kx, ky,−zf)ˆzM1z(kx, ky)dkxdky. (2.16)

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CHAPTER 2. CONTINUOUS, INFINITE, ROTATIONAL SOURCES 20 By Parseval’s theorem, _{R |g(t)|}2 dt = 1 2πR |G(ω)| 2 dω. Therefore, Z z<−d1 Im(z)E1(r) 2 dr = 1 4π2 Z Z Z z<−d1 Im(z) E1(kx, ky, z) 2 dkxdkydz (2.17a) = 1 4π2 Z Z _hZ z<−d1 Im(z) ¯Gem(kx, ky, z)ˆz 2 dzi (2.17b) · M_1z(k_x, k_y) 2 dkxdky. Defining h(kx, ky) = 1 4π2 h k₁2ArαtG¯hm,j(kx, ky,−zf)ˆz +lrβtG¯em,j(kx, ky,−zf)ˆz i f (kx, ky) = s ω 4π2 Z z<−d1 Im(r) ¯G_em(k_x, k_y, z)ˆz 2 dz,

the coupling parameter in (2.10) can be written as

γ = RR h(k_x, k_y)M_1z(k_x, k_y)dk_xdk_y 2 RR f (k_x, k_y)M_1z(k_x, k_y) 2 dkxdky . (2.18)

The optimization problem is to find_M1z(kx, ky) such that the expression in (2.18)

is maximized. By the Cauchy-Schwarz inequality, (2.18) is maximized when

M1z,opt(kx, ky) = h∗(kx, ky) f (k_x, k_y) 2 (2.19)

and the optimal value for the coupling parameter in (2.18) is

γopt = Z Z h(kx, ky) f (kx, ky) 2 dkxdky. (2.20)

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along the z direction (α = ˆz and β = 0), the optimal source distribution is given by

M1z,opt(kx, ky) =

Ar(k1z +k2z) Imk2zeik ∗

2z(−zf+d1)

2π2_ωeik1zd1 (2.21)

and the corresponding coupling parameter in (2.18) is

γopt = A2 r πω Im 2 Z ∞ 0 Imk2ze2 Im k2z(−zf+d1)kρ3dkρ (2.22)

where kρ = pkx2+ky2. As another example, when the receiver is an electric dipole

oriented along the x direction (α = 0 and β = ˆx), the optimal source distribution is given by M1z,opt(kx, ky) = lrky(k1z+k2z) Imk2zeik ∗ 2z(−zf+d1) 4π2_ωk2 ρeik1zd1 (2.23) and the corresponding coupling parameter is

γopt = l2 r 2πω Im 2 Z ∞ 0 Imk2ze2 Im k2z(−zf+d1)kρdkρ. (2.24)

2.4 Results

2.4.1 Comparison with Point and Uniform Sources

We compare the optimal transmit current distribution to a point source and an uni-form source in terms of the resulting coupling parameter and field distributions in tissue. The receiver and tissue model must be fixed in order to perform a comparison. For simplicity, we consider a magnetic dipole with an area of Ar = π mm2 oriented

in the z direction. For a vertical magnetic dipole receiver, the received open-circuit voltage is given by

Voc=iωµAr|H1z(rf)| (2.25)

which is dependent only on the z component of the magnetic field. We also consider a simple tissue model composed of an air-muscle half-space where the source is placed at d1 = 1 cm above the air-muscle interface. The tissue properties are modeled by

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CHAPTER 2. CONTINUOUS, INFINITE, ROTATIONAL SOURCES 22 10−5 10−4 10−3 10−2 [Ω] 106 ₁₀7 ₁₀8 ₁₀9 Frequency [Hz] Optimal Uniform Point

Figure 2.3: Coupling parameter versus frequency for a vertical magnetic dipole at zf = 5 cm for different source distributions d1 = 1 cm above the interface.

assigning a dielectric permittivity to each layer. The dependence of with frequency is modeled by the 4-term Cole-Cole relaxation model [23] in the same manner as [62].

Coupling Parameter

The coupling parameter of a source can be obtained by writing an expression for its current distribution M1z(x, y) and substituting its Fourier transform M1z(kx, ky)

into (2.18). The point source has the form

M1z(x, y) = δ(x, y). (2.26)

The Fourier transform of the point source is then simply_M1z(kx, ky) = 4π2. Similarly,

the uniform source is modeled as a circle function with a fixed radius R,

M1z(x, y) =    1 if px2₊_y2 _{< R} 0 otherwise . (2.27)

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The Fourier transform of the circle function is given by

M1z(kx, ky) =

2πRJ1(Rpk2x+ky2)

pk2 x+k2y

(2.28)

where J1 is a Bessel function of the first kind and of the first order. For the uniform

source, we choose a radius R = 1 cm. Further increasing the size of the uniform source actually reduces the coupling parameter due to increased tissue loss.

Fig. 2.3 shows the coupling parameter versus frequency of each source for an implant at zf = 5 cm. At low frequencies (<100 MHz), the improvement of the

optimal source coupling over the point and uniform sources is negligible. At higher frequencies, however, the gain obtained by optimization is significant. For example, at 2 GHz, the optimal source outperforms the point source and the uniform source by about 11 dB. Although the uniform source covers a larger area than point source, its coupling parameter is only slightly higher since the current distribution over the area has not been optimized.

The coupling parameter versus frequency at three different depths in tissue is shown in Fig. 2.4 for the optimal and the uniform source. The coupling parameter drops with depth much more quickly for the uniform source than the optimized source. This suggests that the gain obtained by source optimization increases with the depth of the implant.

Field Distributions

For a given source, the E and H fields can be computed everywhere in tissue. The magnitude of the E field is responsible for tissue heating while, for the vertical mag-netic dipole, power is delivered by the z-component of the H field. The absorbed power in tissue is measured by the specific absorption rate (SAR), which is defined as the absorbed power spatially averaged over a volume of 1 cm3. The IEEE safety guidelines require that the SAR not exceed 1.6 mW/cm3 [4].

Fig. 2.5 shows the distribution of the received open-circuit voltage and SAR dis-tribution at 2 GHz for the uniform and optimal sources. The open-circuit voltage distribution represents the emf induced in a receiver located at a given position. As

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CHAPTER 2. CONTINUOUS, INFINITE, ROTATIONAL SOURCES 24 106 ₁₀7 ₁₀8 ₁₀9 10−8 10 10 10 100 Frequency [Hz] −6 −4 −2 106 ₁₀7 ₁₀8 ₁₀9 Frequency [Hz] (a) (b) 2 cm 4 cm 8 cm [Ω] 2 cm 4 cm 8 cm

Figure 2.4: The coupling parameterγ versus frequency for a vertical magnetic dipole receiver at depths zf − d1 = 2 cm, 4 cm, and 8 cm for (a) the optimized current

distribution and (b) a uniform source.

a basis of comparison between sources, the transmit power is normalized such that the peak SAR is equal to the safety guideline for each source. The field distributions of the point source and uniform source are highly similar. However, the fields due to the optimal source exhibit focusing, the effect where the fields are redistributed such that they interfere constructively at the focal point and destructively otherwise. This enables the optimized source to achieve a 11 dB improvement as compared to the uniform source.

2.4.2 Optimal Source Distribution

Using the same tissue model, we consider the optimal source distribution for both a magnetic and electric dipole. The magnetic dipole is again oriented in thez direction with area Ar =π mm2. The electric dipole is lying parallel to the x direction with

length lr = 2 mm.

For the magnetic dipole receiver, the optimal source distribution M1z,opt is

circu-larly symmetric. Fig. 2.6 shows the magnitude and phase of a radial slice ofM1z,opt at

2 MHz and 2 GHz for an implant at zf = 5 cm. At both frequencies, the magnitude

decays quickly and is negligible at large radial distances. At low frequencies, the magnitude is negligible outside ρ = 2 cm while the phase is almost constant within

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CHAPTER 2. CONTINUOUS, INFINITE, ROTATIONAL SOURCES 25 Optimal z [cm] −4 −2 0 2 4 −6 −5 −4 −3 −2 x [cm] Uniform z [cm] −4 −2 0 2 4 −6 −5 −4 −3 −2 −4 −2 0 2 4 −6 −5 −4 −3 −2 −4 −2 0 2 4 −6 −5 −4 −3 −2 SAR [mW/cm3_] 0 0.4 0.8 1.2 1.6 |Voc|2 [mV2] 0 0.5 1 1.5 2 V (r ) = 19.7 mV_{oc f} V (r ) = 3.5 mV_{oc f} P = 63 mW_t P = 158 mW_t Point z [cm] −6 −5 −4 −3 −2 −4 −2 0 2 4 −4 −2 0 2 4 −6 −5 −4 −3 −2 V (r ) = 2.3 mV_{oc f} P = 45 mW_t x [cm]

Figure 2.5: The distribution of Voc and SAR at y = 0 in the tissue for the point,

uniform, and optimal sources at 2 GHz. The receiver is a magnetic dipole with normal oriented along thez direction, and (d1, zf) = (1 cm, 5 cm).

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CHAPTER 2. CONTINUOUS, INFINITE, ROTATIONAL SOURCES 26 2 MHz 2 GHz 8 4 0 −4 −8 [rad] [V/m2_] 8 4 0 −4 −8 [cm] [cm] −8 −4 0 4 8 −8 −4 0 4 8 −8 −4 0 4 8 x [cm] −8 −4 x [cm]0 4 8 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 8 4 0 −4 −8 8 4 0 −4 −8 1 0.5 0 1 0.5 0

Figure 2.6: Magnitude and phase ofM1z,opt(x, y) at 2 MHz and 2 GHz for the receive

mangetic dipole where (d1, zf) = (1 cm, 5 cm).

the circular region. Due to the relatively long wavelengths in tissue, there are no interference effects and optimal source is obtained by placing all of the energy at the point closest to the receiver. At high frequencies, however, the magnitude of the optimal source distribution extends a much larger radius and the phase varies quickly with a period of wavelength in tissue. The distribution resembles a ring and leads to constructive interference at the implant and destructive interference elsewhere.

For a receive electric dipole, the optimal source distribution is not circularly sym-metrical. Since the source is composed of magnetic dipoles, circularly symmetric sources, such as the point and uniform source, result in zero E field along the x-axis so no power is delivered to an electric dipole receiver. Instead, Fig. 2.7 shows that the distribution is conjugate symmetric across the x-axis such that the E fields add constructively along the direction of the dipole in x. The source distribution also exhibits decaying behavior similar to the magnetic dipole at both frequencies.

The optimal transmit current distribution was found along an infinite sheet. In practice, the source can be realized only within a limited area. An important property of the optimal current distribution is that the magnitude decays rapidly as the radial distance ρ increases, as shown in both Fig. 2.6 and 2.7. Since the contribution of

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CHAPTER 2. CONTINUOUS, INFINITE, ROTATIONAL SOURCES 27 2 GHz −8 −4 0 4 8 x [cm] −8 −4 x [cm]0 4 8 8 4 0 −4 −8 [rad] [V/m2_] 8 4 0 −4 −8 8 4 0 −4 −8 −3 −2 −10 1 2 3 −3 −2 −10 1 2 3 2 MHz 8 4 0 −4 −8 [cm] −8 −4 0 4 8 −8 −4 0 4 8 [cm] 1 0.5 0 1 0.5 0

Figure 2.7: Magnitude and phase ofM1z,opt(x, y) at 2 MHz and 2 GHz for the receive

electric dipole where (d1, zf) = (1 cm, 5 cm).

the current becomes negligible at large radial distances, an optimal source dimension can be defined for which beyond there is a diminishing return in performance. The ν-radius of the source is defined as the radius ρν where

Z √ x2_+y2_≤ρ_ν Mz,opt(x, y) 2 dxdy = ν Z Mz,opt(x, y) 2 dxdy (2.29)

for 0 < ν ≤ 1. For example, ρ0.9 gives the radius of the transmit current

distribu-tion that it contains 90% energy of in the optimal source distribudistribu-tion. The current outside the ν-radius can be safely ignored with minimal impact on the coupling pa-rameter [40].

Fig. 2.8 shows ρ0.9 for a receiving magnetic and electric dipole at zf = 5 cm.

Somewhat counterintuitively, theν-radius is small for low frequencies, which suggests that in this range the optimal current distribution resembles a small uniform source. This is consistent with the results in Fig. 2.3 where the uniform and optimal sources were found to have comparable performance at low frequencies. The ν-radius then increases with frequency up to the low GHz range, beyond which it decreases again due the excessive tissue loss past the low-GHz range. Fig. 2.9 showsρ0.9 with varying

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CHAPTER 2. CONTINUOUS, INFINITE, ROTATIONAL SOURCES 28 Magnetic dipole Electric dipole 0 2 4 6 8 10 [cm] ρ_0.9 107 ₁₀8 ₁₀9 Frequency [Hz] 106

Figure 2.8: ρ0.9 versus frequency for a magnetic dipole and electric dipole receiver at

depth zf − d1 = 4 cm.

implant.

2.5 Receiver Considerations

Optimizing the transmit current distribution allows the maximum coupling parameter γopt to be obtained, and hence the maximum coupling efficiency ηc,opt=γopt/4R22 as

well for a given receiver. The total power transfer efficiency, however, is given by the product of the coupling efficiency ηc and the matching efficiency ηm. In this section,

we show how the matching efficiency can be maximized subject to practical limitations that arise in an integrated circuit (IC) implementation of the receiver [57, 79]. This establishes an upper-bound on the power transfer efficiency that can be obtained for a given receiver in tissue.

From (2.5), the matching efficiency is given by

ηm = 4R22RL Z₂₂+Z_L 2 (2.30)

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CHAPTER 2. CONTINUOUS, INFINITE, ROTATIONAL SOURCES 29 107 ₁₀8 ₁₀9 Frequency [Hz] 106 0 2 4 6 8 10 [cm] 2 cm 4 cm 8 cm ρ_0.9

Figure 2.9: ρ0.9 versus frequency for a magnetic dipole receiver at depths zf − d1 =

2 cm, 4 cm, and 8 cm.

whereZ22is the self-impedance of the receiving antenna andZLis the load impedance.

For a fixed frequency, Z22 is determined by the antenna dimensions and material as

well as the surrounding tissue. The load impedanceZL, however, can be controlled by

introducing a matching network between antenna and the load as shown in Fig. 2.2. Note that the conjugate matching condition

ZL=Z22∗ (2.31)

yields the maximum matching efficiency ηm = 1. Conjugate matching requires both

resonance, which occurs when the imaginary part of ZL cancels that of Z22, and

matched resistance, the condition where RL=R22.

Practical limitations to conjugate matching arise from the limited transformation range of the matching network. Supposing that the resonance condition is met, the matching network must be able to perform impedance transformation between the antenna and the load in order to achieve matched resistance. For typical implants, the load impedance is determined by the rectifier, which has values on the order of 1 kΩ [57, 79]. On an IC, however, the Q-factor is typically limited to <10, which

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yields a maximum transformation ratio of 1:100. As such, we have the minimum load resistance condition

RL> 10 Ω. (2.32)

This limits our ability to perform conjugate matching when the antenna self-resistance is small.

Table 2.1 lists the optimal power transfer efficiency for different frequencies. The values of R22 were obtained for a magnetic dipole of radius 1 mm and an electric

dipole of length 2 mm in muscle, and are typical of the antenna self-impedance. Interestingly, R22 of electric dipole is high at low frequencies, which is opposite of

the result in free space. This is due to the high dielectric loss around the dipole in a lossy medium. For the magnetic dipole, conjugate matching cannot be achieved at low frequencies due to small values ofR22.

When receiver considerations are taken into account, Table 4.1 shows that the optimal frequency remains in the low-GHz range. Although the optimal coupling pa-rameterγopt of the electric dipole is much higher than the magnetic dipole, the power

transfer efficiency is somewhat worse due to high dielectric loss in the surrounding tissue. Since matching efficiency is independent of source, the effect of the receiver considerations will be identical for other sources. The gain obtained by optimiza-tion of the coupling parameter directly translates to an increase in power transfer efficiency.

2.6 Summary

We studied the optimal source for wireless power transfer to small receiver embedded in multiple planar layers of tissue. We considered a general source composed of an arbitrary magnetic current sheet and a receiver modeled by a combination of magnetic and electric dipoles with arbitrary orientations. By abstracting the coupling between the source and the receiver as a two-port network, an expression for the power transfer efficiency was derived and found to be the product of the coupling and matching efficiency. We expressed the coupling efficiency in terms of the fields in

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Table 2.1: Optimal power transfer efficiency for typical values of R22 and minimum

load resistance RL= 10 Ω

Receiver Frequency R22 γopt ηc,opt ηm ηopt

[MHz] [Ω] [dB(Ω)] [dB] [dB] [dB] 2 0.02 _−43.8 _{−32.8 −21.0 −53.8} Magnetic 20 0.02 _−44.4 _{−33.4 −21.0 −54.4} dipole 200 0.08 _−43.2 _{−38.3 −15.0 −53.3} 2000 10.6 _−26.1 _−42.4 0 _−42.4 2 1307 _−23.4 _−60.6 0 _−60.6 Electric 20 1105 _−24.1 _−60.6 0 _−60.6 dipole 200 546 _−23.3 _−56.7 0 _−56.7 2000 29 _−17.5 _−38.1 0 _−38.2

tissue via a plane wave decomposition and derive the magnetic current distribution that maximizes the coupling efficiency for a given receiver. The optimal source was then compared to the point and uniform source. Finally, receiver considerations were taken into account to find the improvement in power transfer efficiency obtained by source optimization.

The optimal source distribution achieves the highest power transfer efficiency at the low-GHz range. At the low-GHz range, we find that the optimal source dis-tribution does not resemble the uniform disdis-tribution, but is more complicated in shape. Consequently, the optimal source invokes focusing of electromagnetic fields to concentrate fields at the receive implant while reducing of the heating in the surround-ing tissue, which results in substantial improvement in the power transfer efficiency. Lastly, the optimal source distribution informs us a dimension of the transmit antenna beyond which there is a diminishing return in performance.

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Chapter 3 Bound on Wireless Power Transfer

Efficiency

In Chapter 2, we observed that the power transfer efficiency can be significantly improved by optimizing the infinitesimal coils (rotational current) confined to an in-finite plane. Similarly to the optimization done in Chapter 2, this chapter obtains an explicit solution for the source that maximizes power transfer efficiency for the general configuration shown in Fig. 3.1. However, the source is now represented by a general surface current density confined to an infinite plane. It can be shown from the field equivalence principle [28] that an electric current density comprising both rotational and irrotational components is sufficient to represent an arbitrary three-dimensional source (see Appendix A). The current density that is optimal in the resulting parameter space thus bounds the performance attainable by any physical realization of the source, including apertures enhanced with near-field plates [27] or metamaterials [55, 73]. In contrast to direct optimization, a solution can be found in the equivalence form without recourse to non-analytical techniques (e.g. evolutionary optimization) [18, 65] or limitations in the investigated space. As a numerical demon-stration, we solve the optimal source for parameters clinically relevant to a medical cardiac device. The theory reveals the existence of sources that outperform the best coil-based design by a factor of 4. Based on analyses of the solution in this chapter, we design a physical source structure that closely approaches this optimal efficiency

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CHAPTER 3. BOUND ON EFFICIENCY 33

in Chapter 5.

3.1 Model and Problem Formulation

3.1.1 Tissue, Source, and Receiver Model

As in previous chapters, we model the inhomogeneity of the tissue as a planarly multilayered medium, as illustrated in Fig. 3.1. The tissue properties are modeled by assigning a dielectric permittivity to each layer. The dependence of with frequency is obtained from the Debye relaxation model [62].

Over the multilayer structure, we look for a source that maximizes the power transfer efficiency. It is difficult, however, to optimize it, since the shape of the source can be arbitrary in three dimensional space. The problem can be simplified by invoking the equivalence principle [28]. According to equivalence principle, any arbitrary source can be represented by an equivalent surface (tangential) current density, J1, along a plane Ssrc between the source and medium as shown in Fig. 3.1

(See Appendix A). For the convenience, Ssrc is assumed to be placed atz = 0.

As a result, without loss of generality, we model the source with surface electric current J1 on Ssrc in the rest of the paper:

J1(r) =J1x(rs)δ(z) ˆx +J1y(rs)δ(z) ˆy (3.1)

where rs=xˆx+yˆy. Lastly, the receiver of miniature devices is modeled as a magnetic

dipole with arbitrary orientation located at rf = (0, 0,−zf) (Fig. 3.1):

M2(r) =iωµArI2δ(x, y, z + zf) ˆα (3.2)

where ArI2 is the magnetic moment of the dipole, ˆα denotes the orientation of the

magnetic dipole, which is tilted by θ from the z-axis. For a given rf and ˆα, we want

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CHAPTER 3. BOUND ON EFFICIENCY 34 x z y 2 j rf θ . . . n_ˆ . . . J₁ 1

Figure 3.1: The general configuration for wireless power transfer consists of a source surface current density J1 and a receive coil positioned in tissue with magnetic dipole

moment ˆn at an angle θ. The tissue geometry is modeled by multilayers. By con-vention, the dipole moment aligns with the z-axis when θ = 0◦ and the x-axis when θ = 90◦.

3.1.2 Coupling Parameter

As we did in Chapter 2, we abstract the coupling between the source structure and receive coil as a two-port network :

V1 =Z11I1+Z12I2

V2 =Z21I1+Z22I2.

Since the receive coil is small, the source structure and receive coil are loosely coupled. The power transfer efficiency is then given by [43]

η = Pr Pt ≈ Z₂₁ 2 4R11R22 4R22RL Z22+ZL 2. (3.3)

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CHAPTER 3. BOUND ON EFFICIENCY 35

where Pr is the received power at the output and Pt is the transmit power at the

input of the two-port network. The efficiency (3.3) is the product of two factors: the coupling efficiency ηc on the left and the matching efficiency ηm on the right.

The coupling efficiency is the ratio of the power available at the receiver to the input power. The matching efficiency is the ratio of the power delivered to the load to the available power, and is independent of transmit antenna structure. It is maximized when the resistance of the load and the receiver are equal RL = R2. The mismatch

betweenRLand R2 can in principle be corrected by an impedance matching network

under the same physical constraints explained in Section 2.5.

In this dissertation, as we focus on optimizing the source structure for a given receive coil, it suffices to maximize the coupling efficiency. From ηc, we extract the

coupling parameter γ

γ = |Z21|

2

R11

(3.4) which is completely determined by the source. The optimal source is thus given by the current distribution J1(rs) that maximizes γ.

3.2 The Optimization Problem and its Solution

To find the optimal source for a given receive coil, we need to express the coupling parameter in terms of J₁ first. This is achieved by first defining the impedances of the two-port network in terms of the electromagnetic fields from the sources J1. Then,

we will derive these fields in terms of the source distributions, and find the source that maximizes the coupling parameter for the receive coil.

A BOUND ON WIRELESS POWER TRANSFER TO SUBWAVELENGTH DEVICES IN THE BODY