Segmented Coil Design Powering the Next Generation of High-efficiency Robust Micro-implants

Abstract

The next generation of Micro Active Implantable Medical Devices (M-AIMD) are small (< 1 cc), wireless, as well as battery-less. They are located in different parts of the body ranging from brain computer interface electrode arrays (e.g., Blackrock Neurotech Utah Array) to multi-chamber cardiac pacemakers (e.g., Abbott dual chamber Nanostim device). These devices require efficient charging and powering solutions that are very challenging to design. Such solutions require the careful balancing of multiple design parameters such as size, separation distance, orientation, and regulatory limits for emission and tissue safety. In this article, we introduce unique optimisation metrics for designing efficient transmit and receive coils for near-field magnetics-based charging solutions. We elaborate on how the metrics need to be altered depending on the regulatory limits. We discuss the impact of body tissue loading on transmit and receive coil performance using circuit analysis. We introduce a novel “segmented” transmit coil arrangement. We discuss the physics of segmentation, and we build a full wave simulation model, with practical design procedure, which is verified with measurements. Finally, we compare the near fields with and without tissue loading to show that segmented coils offer significant improvement to the performance and robustness of a wireless power transfer system.

1. Introduction

Progress in semiconductor technology has led to the development of substantially miniaturised Micro Active Implantable Medical Devices (also called M-AIMD) that are significantly smaller in size and are implanted in difficult to reach interstitial spaces within the human body, thereby permitting direct interaction with organ systems. This reduction in size facilitates the use of delivery systems (e.g., via catheter or hypodermic needle) that significantly reduce procedure time and burden of care for patients [1, 2, 3, 4]. M-AIMDs are either battery-less or have small batteries necessitating the need for efficient charging and powering solutions [5, 6, 7, 8]. The most common method is power transferred from an on-body transmitter to an in-body AIMD equipped with a receiver using near-field magnetic induction [9]. This is very challenging as it requires carefully balancing multiple design parameters such as size, depth of implant, orientation of implant (and associated misalignment), and regulatory limits for emission and tissue safety [10]. The need to efficiently deliver power to a small target volume (<1 cc) inside the body requires careful design of the transmit coil system and the receive coil system [11].

There is a lot of work in the literature identifying various parameters that need to be optimised to maximise power delivered to a load (therapy delivering M-AIMD) [12, 13, 14, 15]. For example, Fu et al. [16] studied the SIMO (single input multiple output) resonant inductive system and derived an expression for the optimal load and efficiency. Monti et al. [17] concentrated on deriving the solution for the SIMO system that is not necessarily a resonant inductive system. The authors approached the problem of maximising efficiency as a generalised eigenvalue problem. Zargham and Gulak [18] focused their attention on SISO (single input single output) systems. They focused on power transfer through CMOS substrates and lossy biological tissue. Minnaert and Stevens [19] described the three optimisation approaches (efficiency, delivered power and conjugate matching) for SISO systems. Their derivation was based on a generalised 2-port system and was not specific to an inductive resonant system. They suggested that the efficiency of power transmission is a monotonic function of an “extended kQ product, α” which was first introduced by the works of Ohira [20, 21]. Cho et al. [22] studied specific coil designs for wireless power transmission and compared the performance of the designed coils by using a figure of merit defined by Shinohara et al. [23]. In [24], Sharma derived the formulas for efficiency and the figure of merit of a two-coil resonant system. While there are many more relevant articles in the literature, to the best of our knowledge, none of the articles provides metrics to efficiently design the transmitter and receiver coils independently, taking into account the most important regulatory limits for designing these coils for delivering wireless power to medical implants. This is one of the two novel contributions of this article.

This article also focuses on the design of efficient transmit and receive coils where the coil segments are separated by lumped capacitors. These coils are called segmented coils. Segmentation of coils using lumped or distributed capacitors is not new and has been heavily used in Magnetic Resonance Imaging (MRI), for reducing Specific Absorption Rate (SAR) (for transmit coils) [25] and improving coil sensitivity (for receive coils) [26]. Mirbozorgi et al. [27] mentioned that segmentation helps achieve homogeneous power transfer efficiency. Tang et al. [28, 29] stated that the segmentation can significantly reduce the power loss (including the dielectric loss) and required voltage. Stoecklin et al. [30] concluded that capacitive coil segmentation can effectively suppress dielectric losses and non-uniform current distribution. Mark et al. [31] demonstrated that the segmentation results in decrease of the electric field above the transmit coil thereby reducing SAR in the nearby tissue and permitting higher power transfer efficiency. Pokharel et al. [32] use lumped capacitors to segment printed coils and subsequently develop a stacked metamaterial inspired wireless power transfer (WPT) system for efficient and robust power delivery to M-AIMDs. Most of the literature have discussed the positive outcomes of segmentation of coils, but to the best of our knowledge, no one has provided a detailed analytical and numerical (full wave) explanation, as to why segmented coils have lower dielectric losses and significant reduction in SAR when they are near (<1 cm) lossy body tissue. In this article, we address those gaps in knowledge and define our figures of merit (FoMs) to highlight the positive impact of segmenting the transmitter and receiver coils separately. To further validate our novel FoMs, we build and test the transmitter and receiver coils and compare our calculations with measurement results. This is the second novel contribution of this article.

2. Organisation of this work

This article is organised into the following sections:

First, we present a brief overview of the pertinent regulations (exposure and radiation) that limit the performance of WPT systems for medical implants. For a chosen design frequency, we identify the critical parameters that bound the maximum currents that can be carried by a transmit coil and a receive coil. These maximum currents dictate the maximum power that can be delivered by a WPT system.

Second, for a two-coil system, we derive, using circuit analysis, the optimal load resistance needed to maximise (a) delivered power and (b) efficiency. For both cases, we find the receive coil current, delivered power and efficiency.

Third, we derive an optimisation metric we term, system figure of merit, for a two-coil WPT system and show that the popular system link efficiency used in the literature, is a monotonic function of the system figure of merit. We split the system figure of merit into two parts: transmit figure of merit (characterising the transmit coil) and receive figure of merit (characterising the receive coil). We demonstrate that an increase of any of these two figures of merit results in an increase of the overall system link efficiency.

Fourth, we identify two mechanisms that cause proximity of lossy dielectric tissue to impact the impedance of a transmit coil. We identify the first mechanism to be associated with the interaction of the coil current with the tissue and the second mechanism to be associated with the interaction of the charges accumulated in the coil with the tissue.

Fifth, we study the effects of introducing lumped capacitors in series with coil wiring to break the coil turns into segments. We study the effect of segmentation on the resistance and reactance of coils. We investigate the impact that segmentation capacitors have on the transmit and receive figure of merit of the coil.

Finally, we validate our circuit models and associated transmit and receive figures of merit with measurements and full wave simulations in HFSS. We perform measurements and full wave simulations of the electrical properties of transmit coil design to demonstrate that the introduction of the segmentation capacitors improves the figure of merit of the transmit coil when it is both unloaded (in air) and loaded with lossy tissue.

3. Review of regulations

Figure 1 diagrammatically illustrates that the regulations associated with wireless power transmission to an implant from an external transmitter can be divided into two groups: radiation (EMC) and exposure. We note that in this article we have examined the regulatory limits for USA and Europe only.

Figure 1.

Diagrammatic representation of the pertinent FCC and EU regulations.

3.1 Exposure

Ensuring the safety of the human body during exposure to electromagnetic waves is an indisputable fact. SAR is a measuring factor for electromagnetic wave absorption. SAR is calculated as

$$\mathit{SAR}=\frac{{\sigma }_{\mathit{\text{tissue}}}}{{\rho }_{\mathit{\text{tissue}}}}{\left|E\right|}^2$$E1

where ${\sigma }_{\mathit{\text{tissue}}}$ is the conductivity of the tissue in S/m, ${\rho }_{\mathit{\text{tissue}}}$ is its mass density in kg/m³, and |E| is the RMS magnitude of the induced electric field in the tissue due to exposure to these EM waves. FCC [33] limits the peak average SAR to 1.6 W/kg, averaged over 1 gram of tissue. EN 1999/519/EC [34] limits SAR, as well as the volumetric current in the tissue. The actual limit values of SAR depend on the body part exposed to the RF energy. The limit on the induced current depends on the frequency.

3.2 Radiation

The FCC rules and regulations are presented in Title 47 of the Code of Federal Regulations (CFR). Part 15 [35] covers the radio frequency devices. Part 18 [36] covers the Industrial, Scientific and Medical Equipment (ISM). Part 15 and Part 18 limit the radiated electric field at 3 m or 30 m depending on frequency.

EN 300330 [37] covers Short Range Devices (SRD) in the frequency range 9 kHz to 25 MHz and inductive loop systems in the frequency range 9 kHz to 30 MHz. It is a harmonised standard covering the essential requirements of article 3.2 of Directive 2014/53/EU. The standard limits the magnetic field at 10 m from the device. The most generous H-field limits are in three frequency bands containing 6.78 MHz, 13.56 MHz, 27.12 MHz.

EN 303417 [38] covers the wireless power transmission systems, using technologies other than radio frequency beam in the 19–21 kHz, 59–61 kHz, 79–90 kHz, 100–300 kHz, 6765–6795 kHz ranges. It is harmonised standard covering the essential requirements of article 3.2 of Directive 2014/53/EU.

EN 300220–2 [39] covers SRDs operating in the frequency range 25–1000 MHz. for non-specific radio equipment. The most generous H-field limits are in two frequency bands containing 27.12 MHz, 40.68 MHz.

EN 2013/572/EU [40] covers SRDs too. The emphasised frequency bands having higher limits are centred at 6.78 MHz, 13.56 MHz, 27.12 MHz, 40.68 MHz.

EN 55014–1 (“CISPR 14”) [41] covers household appliances, electric tools and similar apparatus. This regulation is very restrictive (3 dBμA/m at 3 m in 4–30 MHz range) when applied to the inductive loops and WPT devices.

EN 55011 (“CISPR 11”) [42] covers ISM equipment. The devices are sorted into two groups (Non-ISM and ISM equipment) and two classes (non-residential environment and residential environment).

4. Derivation for delivered power and efficiency

Most commonly, WPT circuits use electromagnetic coupling between coils. These WPT circuits use capacitors to reduce reactive power. Figure 2 is a commonly chosen series–series capacitor representation which has been widely used because the capacitances can be chosen independent of the load and coupling conditions.

Figure 2.

Schematic of the WPT system.

At resonance, ${\omega }^2=\frac{1}{L_t{C}_t}=\frac{1}{L_r{C}_r}$, the equation that links the currents in the transmit and in the receive coil is:

$$\mathit{j\omega M}{I}_t+\left(R_r+R_L\right)I_r=0$$E2

where $I_t$ is the current in the transmit coil, $I_r$ is the current in the receive coil, $R_r$ is the resistance of the receive coil, $R_L$ is the resistance of the load, and $M$ is the mutual inductance. In our derivations, we assume that the voltages and currents are strictly sinusoidal. Therefore, we replace the time derivatives $\partial /\partial t$ by $\mathit{j\omega }$ (multipliers).

From Eq. (2) we can see that there is a 90-degree phase shift between the transmit and receive currents. The $I_{t,\mathit{\text{limit}}}$ and $I_{r,\mathit{\text{limit}}}$ are the maximum allowed currents in the transmit coil and in the receive coil, correspondingly.

If we ignore the phase shift and redefine $j{I}_r$ as new $I_r$, then we get the following expression for the load resistance:

$$R_L=\frac{\mathit{\omega M}{I}_t}{I_r}-R_r$$E3

Additionally, the delivered power is:

$$P_L=\frac{1}{2}{R}_L{I}_r^2=\frac{1}{2}\left(\mathit{\omega M}{I}_t{I}_r-R_r{I}_r^2\right)$$E4

The power loss in the receive coil is:

$$P_r=\frac{1}{2}{R}_r{I}_r^2$$E5

The power loss in the transmit coil is:

$$P_t=\frac{1}{2}{R}_t{I}_t^2$$E6

The efficiency is:

$$\eta =\frac{P_L}{P_L+P_r+P_t}=\frac{1-\frac{R_r}{\mathit{\omega M}}\frac{I_r}{I_t}}{1+\frac{R_t}{\mathit{\omega M}}\frac{I_t}{I_r}}$$E7

We assume that the current in the transmit coil is fixed at its maximum value of $I_t=I_{t,\mathit{\text{limit}}}$.We can proceed with two ways: (a) to maximise the delivered power and (b) to maximise the efficiency.

(a) Maximising the delivered power.

We differentiate the delivered power with respect to I_r, equate it to zero and obtain the optimal current in the receive coil:

$$I_{r,\mathit{opt}}=\mathit{min}\left(\frac{\mathit{\omega M}}{2R_r}{I}_t,I_{r,\mathit{\text{limit}}}\right)$$E8

where the receive current is limited by $I_{r,\mathit{\text{limit}}}$.

(b) Maximising the efficiency.

We differentiate the efficiency, equate it to zero and obtain the optimal current in the receive coil:

$$I_{r,\mathit{opt}}=\mathit{min}\left(I_t·\frac{R_t}{\mathit{\omega M}}\left(\sqrt{1+Q_M^2}-1\right),I_{r,\mathit{\text{limit}}}\right)$$E9

where

$$Q_M=\frac{\mathit{\omega M}}{\sqrt{R_t{R}_r}},$$E10

which we call a mutual quality factor.

We use the optimal receive current to obtain the expressions for the optimal delivered power $P_L$ by using Eq. (4) and efficiency $\eta$ by using Eq. (7).

If the current limits are high (infinite), then both cases can be elaborated further:

1. Maximising the delivered power, with high current limits.

   Optimal current is:

   $$I_{r,\mathit{opt}}=\frac{\mathit{\omega M}}{2R_r}{I}_t$$E11

   Optimal load resistance is:

   $$R_{L,\mathit{opt}}=R_r$$E12

   Delivered power is:

   $$P_L=\frac{1}{8}\frac{{\left(\mathit{\omega M}\right)}^2}{R_r}{I}_t^2$$E13

   Efficiency is:

   $$\eta =\frac{1}{2+\frac{4R_t{R}_r}{{\left(\mathit{\omega M}\right)}^2}}=\frac{1}{2+\frac{4}{Q_M^2}}$$E14

2. Maximising the efficiency, with high current limits.

   Optimal current is:

   $$I_{r,\mathit{opt}}=\frac{R_t}{\mathit{\omega M}}\left(-1+\sqrt{1+Q_M^2}\right)I_t$$E15

   Optimal load resistance is:

   $$R_{L,\mathit{opt}}=R_r\sqrt{1+Q_M^2}$$E16

   Delivered power is:

   $$P_L=\frac{1}{2}\frac{R_t{R}_r}{\mathit{\omega M}}{I}_t^2\left(1+Q_M^2-\sqrt{1+Q_M^2}\right)$$E17

   Efficiency is:

   $$\eta =\frac{\sqrt{1+Q_M^2}-1}{\sqrt{1+Q_M^2}+1}$$E18

From the above formulas we see that $Q_M$ serves as the system figure of merit. Increase of the $Q_M$ leads to the increase of the efficiency of the transfer in both cases.

5. Figures of merit for transmit and receive coils

5.1 From efficiency perspective

For some WPT systems the magnetic field of the transmit coil is not changing significantly in the region of space that contains the receive coil. This happens if the receive coil is much smaller than the transmit coil and/or it is located far enough from the transmit coil. The induced voltage of the receive coil is $\mathit{j\omega M}{I}_t$ (where $M$ is the mutual inductance). However, according to Faraday’s law of induction, the induced voltage is $\mathit{j\omega }{B}_t{A}_r$, where $B_t$ is the magnetic field of the transmit coil, $A_r$ is the area of the receive coil.

The expression for the mutual quality factor can be expressed as follows:

$$Q_M=\frac{\mathit{\omega M}}{\sqrt{R_t{R}_r}}=\omega \frac{B_t/I_t·A_r}{\sqrt{R_t{R}_r}}=\omega \frac{B_t/I_t}{\sqrt{R_t}}·\frac{A_r}{\sqrt{R_r}}$$E19

We observe that the values of the transmit and receive coil can be separated. We can define the figures of merit for the transmit coil $F_t$ and the receive coil $F_r$

$$F_t\equiv \frac{B_t/I_t}{\sqrt{R_t}}=\frac{B_t}{\sqrt{2P_{\mathit{\text{in}}}}}$$E20

$$F_r\equiv \frac{A_r}{\sqrt{R_r}}$$E21

where $P_{\mathit{\text{in}}}=\frac{1}{2}{R}_t{I}_t^2$ is an input (transmitted) power into the transmit coil.

The expression for the receive figure of merit $F_r$ can be represented differently in the following way. From the expression for the delivered power (13):

$$P_L=\frac{1}{8}\frac{{\left(\omega B_t/I_t·A_r\right)}^2}{R_r}{I}_t^2=\frac{1}{8}{\omega }^2{B}_t^2·\frac{A_r^2}{R_r}$$E22

we get:

$$F_r\equiv \frac{A_r}{\sqrt{R_r}}=\frac{\sqrt{8P_L}}{\omega B_t}$$E23

From Eq. (19) we get the expression for the mutual inductance as:

$$Q_M=\omega ·F_t{F}_r$$E24

Increase in any of these two figures of merit ($F_t$ and $F_r$) leads to an increase in the efficiency. The mutual quality factor $Q_M$ can be seen as a figure of merit for the transmit-receive coil system. The system with higher $Q_M$ is more efficient.

The expressions for the transmit figures of merit defined as $F_t=\frac{B_t}{\sqrt{2P_{\mathit{\text{in}}}}}$ and $F_r=\frac{\sqrt{8P_L}}{\omega B_t}$ are more general than those defined using the transmit resistance $R_t$, receive resistance $R_r$ and receive coil area $A_r$. These definitions apply not only to coils, but also to any “structure” that can perform the following tasks: (a) generate magnetic field (if transmit structure), (b) harvest RF energy (if receive structure). The transmit and receive figures of merit are defined as follows:

• Transmit figure of merit $F_t=\frac{B_t}{\sqrt{2P_{\mathit{\text{in}}}}}$ is a measure of ability of the transmit structure (or coil) to generate the RF magnetic field given the consumed input power.

• Receive figure of merit $F_r=\frac{\sqrt{8P_L}}{\omega B_t}$ is a measure of ability of the receive structure (or coil) to harvest the RF power from the incident magnetic field.

It is worth mentioning that the $F_t$ and $F_r$ figures of merit are not the properties solely of the transmit coil and receive coil correspondingly. The coil resistances (and consequently the figures of merit) are affected by the nearby tissue. The coil-tissue separation distance clearly affects these figures of merit. These figures of merit also depend on frequency.

5.2 From delivered power perspective

The expression for the delivered power Eq. (22) can be modified as follows:

$$P_L=\frac{1}{8}{\omega }^2·B_t^2·F_r^2$$E25

The delivered power is proportional to the square of the receive figure of merit $F_r$. It is also proportional to the square of the magnetic field $B_t$ of the transmit coil. This seems to be an intuitive result: the higher the magnetic field is, the more power we can harvest from it.

5.2.1 Considering SAR limit

The magnetic field that we are able to generate at the location of the receive coil cannot be arbitrarily high: the current in the coil is limited by exposure and radiation limits. SAR limit is one of these limits. One can define a SAR figure of merit as a ratio of the magnetic field of the transmit coil to the square root of SAR:

$$F_{t,\mathit{SAR}}\equiv \frac{B_t}{\sqrt{\mathit{SAR}}}$$E26

By defining the SAR figure of merit using Eq. (26) the maximum achievable magnetic field would be calculated as $F_{t,\mathit{SAR}}·\sqrt{\mathit{SAR}}$. The FCC limit of SAR is 1.6 W/kg.

It is worth saying that the $F_{t,\mathit{SAR}}$ figure of merit is not a property solely of the transmit coil. It is a property of the combination of the transmit coil and the nearby tissue. The coil-tissue separation clearly affects the $F_{t,\mathit{SAR}}$. This figure of merit is also a function of frequency.

This figure of merit can also be used to compare the competing designs of the transmit coils. The transmit coil with higher $F_{t,\mathit{SAR}}$ can deliver more power to the receive coil.

5.2.2 Considering other limits

Apart from SAR, there are other regulations that limit the transmit coil current and the transmit coil magnetic field. For each one of them one can establish the corresponding figure of merit in the following way:

1. Volumetric current $J$, according to EN 1999/519/EC, if below 10 MHz. The corresponding figure of merit would be:

   $$F_{t,J}\equiv \frac{B_t}{J}$$E27

2. Electric field at the certain distance from the coil (d = 3 m, 30 m, 300 m). The corresponding figure of merit would be:

   $$F_{t,E}\equiv \frac{B_t}{E_d}$$E28

3. Magnetic field at the certain distance from the coil (d = 10 m). The corresponding figure of merit would be:

   $$F_{t,H}\equiv \frac{B_t}{H_d}$$E29

Figure 3 provides a visual representation of the development of figures of merit from the WPT formulas and the regulations.

Figure 3.

Figures of merit diagram.

6. Impact of tissue loading on the transmit and receive coils

The electric field of the transmit coil can be separated into two parts: the “current” electric field and the “charge” electric field:

$$\mathbf{E}={\mathbf{E}}_{\mathit{\text{current}}}+{\mathbf{E}}_{\mathit{\text{charge}}}=-\mathit{j\omega }\mathbf{A}-\nabla \mathrm{\Phi }$$E30

where $\mathbf{A}$ is the magnetic vector potential and $\mathrm{\Phi }$ is the electric scalar potential. Figure 4 shows the two components of the electric field when a WPT coil is close to lossy dielectric tissue (e.g. muscle).

Figure 4.

Electric field of the coil.

The ${\mathbf{E}}_{\mathit{\text{charge}}}$ mostly exists between the terminals of the coil. The ${\mathbf{E}}_{\mathit{\text{current}}}$ electric field exists as concentric circles above the coil.

6.1 ${\mathrm{E}}_{\mathit{\text{current}}}$ electric field

${\mathbf{E}}_{\mathit{\text{current}}}$ electric field infiltrates the tissue and excites current in it. The current in the tissue flows in self-terminating lines as shown in Figure 5. This leads to ohmic losses in the tissue and adds to the resistance of the transmit coil. Additionally, there is some amount of inductance associated with this current flow.

Figure 5.

Induced current in the tissue.

The effect of the current flow in the tissue may be crudely approximated by a shorted inductance. The Kirchhoff’s laws are:

$$\left(R+\mathit{j\omega L}\right)I+\mathit{j\omega }{M}_{\mathit{\text{tissue}}}{I}_{\mathit{\text{tissue}}}=V$$

$$\mathit{j\omega }{M}_{\mathit{\text{tissue}}}I+\left(R_{\mathit{\text{tissue}}}+\mathit{j\omega }{L}_{\mathit{\text{tissue}}}\right)I_{\mathit{\text{tissue}}}=0$$E31

where $M_{\mathit{\text{tissue}}}$ is the mutual inductance between the transmit coil and the shorted inductance.

Solving this for impedance $Z=V/I$:

$$Z=R+\frac{{\omega }^2{M}_{\mathit{\text{tissue}}}^2{R}_{\mathit{\text{tissue}}}}{R_{\mathit{\text{tissue}}}^2+{\omega }^2{L}_{\mathit{\text{tissue}}}^2}+\mathit{j\omega }\left(L-\frac{{\omega }^2{M}_{\mathit{\text{tissue}}}^2{L}_{\mathit{\text{tissue}}}}{R_{\mathit{\text{tissue}}}^2+{\omega }^2{L}_{\mathit{\text{tissue}}}^2}\right)$$E32

The presence of the ${\mathbf{E}}_{\mathit{\text{current}}}$ electric field results in an increase of the resistance and a decrease of the inductance in the presence of the tissue. Generally, Eq. (32) can be written as:

$$Z=R+R_{\mathit{\text{eddy}}}+\mathit{j\omega }\left(L-L_{\mathit{\text{eddy}}}\right)$$E33

where the definitions of $R_{\mathit{\text{eddy}}}$ and $L_{\mathit{\text{eddy}}}$ can be inferred from the Eq. (32).

It can be observed that the tissue loading the coil leads to induced (eddy) currents in the tissue which causes power loss. This power loss in the tissue exhibits itself as an increased resistance and a decreased reactance of the transmit coil.

6.2 ${\mathrm{E}}_{\mathit{\text{charge}}}$ electric field

When we excite the transmit coil with voltage, there are electric charges that accumulate on the wiring near the coil terminals. When the coil is in close proximity to lossy tissue it can be modelled as a lossy dielectric between the plates of a parallel plate capacitor, as shown in Figure 6.

Figure 6.

Approaching tissue to the coil.

The Ampere’s law is:

$$\nabla \times \mathbf{H}=\mathbf{J}+\mathit{j\omega }\mathbf{D}$$E34

where $\mathbf{H}$ is the magnetic field, $\mathbf{J}$ is the current density, $\mathbf{D}$ is the electric displacement. Taking divergence on both sides of (34) we get:

$$div\left(\left({\sigma }_{\mathit{\text{tissue}}}+\mathit{j\omega }{\epsilon }_0{\epsilon }_{r,\mathit{\text{tissue}}}\right)\mathbf{E}\right)=0$$E35

where ${\sigma }_{\mathit{\text{tissue}}}$ is conductivity of tissue in S/m, ${\epsilon }_{r,\mathit{\text{tissue}}}$ is relative electric permittivity of tissue in F/m, ${\epsilon }_0$ is vacuum permittivity.

The normal component of the vector $\left({\sigma }_{\mathit{\text{tissue}}}+\mathit{j\omega }{\epsilon }_0{\epsilon }_{r,\mathit{\text{tissue}}}\right)\mathbf{E}$ is preserved in the lossy tissue as shown in Figure 7.

Figure 7.

Electric field inside the capacitor.

The electric fields inside the capacitor and outside of the tissue are related as follows:

$$E_{\mathit{\text{tissue}}}=\frac{E_0}{{\epsilon }_{r,\mathit{\text{tissue}}}+\frac{{\sigma }_{\mathit{\text{tissue}}}}{\mathit{j\omega }{\epsilon }_0}}$$E36

where $E_0$ is the electric field in air.

We denote the thickness of the tissue as $l$ and the remaining free space between the plates of the capacitor as $h$. Voltage across the capacitor plates is:

$$V=h{E}_0+l{E}_{\mathit{\text{tissue}}}=E_0\left(h+\frac{l}{{\epsilon }_{r,\mathit{\text{tissue}}}+\frac{{\sigma }_{\mathit{\text{tissue}}}}{\mathit{j\omega }{\epsilon }_0}}\right)$$E37

The electrical field in empty space between capacitor plates is:

$$E_0=\frac{Q}{{\epsilon }_{0}A}$$E38

where $Q$ is the charge on the capacitor plates and $A$ is the area of the capacitor plates. Capacitance is:

$$C=\frac{Q}{V}=\frac{{\epsilon }_{0}A}{h+\frac{l}{{\epsilon }_{r,\mathit{\text{tissue}}}+\frac{{\sigma }_{\mathit{\text{tissue}}}}{\mathit{j\omega }{\epsilon }_0}}}$$E39

We note that the capacitance has an imaginary component. The impedance associated with this capacitance is calculated as:

$$Z_c=\frac{1}{\mathit{j\omega C}}=\frac{1}{\mathit{j\omega }{\epsilon }_{0}A}\left(h+\frac{l}{{\epsilon }_{r,\mathit{\text{tissue}}}+\frac{{\sigma }_{\mathit{\text{tissue}}}}{\mathit{j\omega }{\epsilon }_0}}\right)$$E40

This can be elaborated as:

$$Z_c=\frac{1}{\mathit{j\omega }{\epsilon }_{0}A}\left(h+\frac{l}{{\epsilon }_{r,\mathit{\text{tissue}}}}\frac{1}{1+\frac{{{\sigma }_{\mathit{\text{tissue}}}}^2}{{\omega }^2{\epsilon }_0^2{\epsilon }_{r,\mathit{\text{tissue}}}^2}}\right)+\frac{l}{A{\sigma }_{\mathit{\text{tissue}}}}\frac{1}{1+\frac{{\omega }^2{\epsilon }_0^2{\epsilon }_{r,\mathit{\text{tissue}}}^2}{{{\sigma }_{\mathit{\text{tissue}}}}^2}}=\frac{1}{\mathit{j\omega }{C}_p}+R_p$$E41

where $C_p$ and $R_p$ are the effective capacitance and resistance and take the form:

$$C_p=\frac{{\epsilon }_{0}A}{h+\frac{l}{{\epsilon }_{r,\mathit{\text{tissue}}}}·\frac{{\omega }^2{\epsilon }_0^2{\epsilon }_{r,\mathit{\text{tissue}}}^2}{{{\sigma }_{\mathit{\text{tissue}}}}^2+{\omega }^2{\epsilon }_0^2{\epsilon }_{r,\mathit{\text{tissue}}}^2}}$$E42

$$R_p=\frac{l}{A{\sigma }_{\mathit{\text{tissue}}}}·\frac{{{\sigma }_{\mathit{\text{tissue}}}}^2}{{{\sigma }_{\mathit{\text{tissue}}}}^2+{\omega }^2{\epsilon }_0^2{\epsilon }_{r,\mathit{\text{tissue}}}^2}$$E43

As we see from these formulas, the presence of the tissue between the capacitor plates leads to an increase of the effective capacitance $C_p$ and the appearance of the effective resistance $R_p$. In the absence of tissue $R_p=0$. When the coil is closer to a lossy dielectric medium like body tissue (e.g. muscle), we observe that the resonance frequency of the coil drops (detuning) and the ohmic losses increase.

To determine the resistance and reactance of a coil in close proximity to lossy tissue we develop an equivalent circuit shown in Figure 8. The impedance of the circuit in Figure 8 is:

Figure 8.

Coil model with shunt capacitor and resistor.

$$Z=\frac{1}{\mathit{j\omega }{C}_p+\frac{1}{R+\mathit{j\omega L}}}$$E44

We assume that the capacitive reactance $\frac{1}{\omega C_p}$ far exceeds the resistance $R_p$, and we neglect the resistance $R_p$. To further simplify this expression, we assume that the quality factor of the coil is much higher than unity ($R\ll \mathit{j\omega L}$). The self-resonance frequency of the coil is defined as:

$${\omega }_s=\frac{1}{\sqrt{L{C}_p}}$$E45

We assume that the capacitive reactance $1/\omega C_p$ of the coil is much higher than the inductive reactance $\mathit{\omega L}$. This implies that the Self Resonance Frequency (SRF) of the coil is much higher than the operating frequency ($\omega \ll {\omega }_s$), which is considered favourable for most practical coil designs.

With the aforementioned assumptions, the impedance of the coil simplifies to:

$$Z\approx \frac{R}{{\left(1-\frac{{\omega }^2}{{\omega }_s^2}\right)}^2}+\mathit{j\omega }\frac{L}{1-\frac{{\omega }^2}{{\omega }_s^2}}$$E46

From the above equation it can be observed that the proximity of lossy dielectric tissue results in an increase of the parasitic capacitance $C_p$ and lowers the SRF of the coil ${\omega }_s$ due to an appearance of the parasitic resistance in series with the parasitic capacitance. This always results in an increase in the resistance of the coil. Depending on the coil geometry, dielectric properties of tissue near the coil and frequency of operation, the reactance may either decrease or increase when the coil is near lossy tissue.

7. Segmentation

Segmentation is a process of inserting additional capacitors in between the coil windings (see Figure 9). The capacitor placement is roughly equidistant throughout the windings of the coil. The purpose of the segmentation capacitors is to decrease the voltages between the terminals of the coil and between the turns of the coil.

Figure 9.

Schematic of non-segmented and segmented loaded coils.

The values of the segmentation capacitors are chosen to significantly decrease the visible inductance of the coil. There is no exact formula for the values of the segmentation capacitors, but our recommendation is as follows:

$$C_{\mathit{seg}}=\frac{N}{{\omega }^{2}L}$$E47

where N is the number of segments and L is the coil inductance. If we have N segments, then we have N-1 segmentation capacitors. The cumulative effect of N-1 segmentation capacitors placed in series is represented as the cumulative segmentation capacitance:

$$C_{\mathit{seg},c}=\frac{N}{\left(N-1\right){\omega }^{2}L}$$E48

7.1 Effect of segmentation on the coil resistance and inductance

Figure 10 shows the equivalent circuit of a non-segmented and segmented coil when the coil is loaded by body tissue. The segmentation affects the coil impedance by reducing the electric charges on the wiring of the coil. Mathematically, the effect of segmentation capacitors can be introduced by modifying Eq. (46) as follows:

Figure 10.

Non-segmented and segmented loaded coils.

$$Z\approx \frac{R+R_{\mathit{\text{eddy}}}}{{\left(1-\frac{{\omega }^2}{{\omega }_s^2}\left(1-\frac{1}{{\omega }^{2}L{C}_{\mathit{seg},c}}\right)\right)}^2}+\mathit{j\omega }\left(\frac{L\left(1-\frac{1}{{\omega }^{2}L{C}_{\mathit{seg},c}}\right)}{1-\frac{{\omega }^2}{{\omega }_s^2}\left(1-\frac{1}{{\omega }^{2}L{C}_{\mathit{seg},c}}\right)}-L_{\mathit{\text{eddy}}}\right)$$E49

The SRF ${\omega }_s$ depends on whether the coil is loaded or not: loaded value ${\omega }_{s,\mathit{\text{loaded}}}$ is smaller than the unloaded value ${\omega }_{s,\mathit{\text{unloaded}}}$. We consider the ratio ${\omega }^2/{\omega }_{s,\mathit{\text{loaded}}}^2$ much less than unity, otherwise the coil would not be functioning correctly.

We will now study the effect of tissue loading on both the non-segmented and the segmented coils. For the non-segmented coil, the expression$\left(1-\frac{1}{{\omega }^{2}L{C}_{\mathit{seg},c}}\right)$ is unity. The coil is tuned under unloaded condition by placing a tuning capacitor $C_{\mathit{\text{tune}}}$ in series with it. So, the reactance of the unloaded coil is zero.

For the tuned non-segmented coil, the unloaded and loaded impedances are:

$$Z_{\begin{array}{c}\mathit{\text{unloaded}}\\ \mathit{\text{non}}-\mathit{seg}\end{array}}\approx \frac{R}{{\left(1-\frac{{\omega }^2}{{\omega }_{s,\mathit{\text{unloaded}}}^2}\right)}^2}+\mathit{j\omega }\left(\frac{L}{1-\frac{{\omega }^2}{{\omega }_{s,\mathit{\text{unloaded}}}^2}}-\frac{1}{{\omega }^2{C}_{\mathit{\text{tune}}}}\right)$$E50

$$Z_{\begin{array}{c}\mathit{\text{loaded}}\\ \mathit{\text{non}}-\mathit{seg}\end{array}}\approx \frac{R+R_{\mathit{\text{eddy}}}}{{\left(1-\frac{{\omega }^2}{{\omega }_{s,\mathit{\text{loaded}}}^2}\right)}^2}+\mathit{j\omega }\left(\frac{L}{1-\frac{{\omega }^2}{{\omega }_{s,\mathit{\text{loaded}}}^2}}-L_{\mathit{\text{eddy}}}-\frac{1}{{\omega }^2{C}_{\mathit{\text{tune}}}}\right)$$E51

The difference between these values is:

$$\begin{array}{c}{\mathit{Z}}_{\begin{array}{c}\mathit{loaded}\\ \mathit{non}-seg\end{array}}-{\mathit{Z}}_{\begin{array}{c}\mathit{unloaded}\\ \mathit{non}-seg\end{array}}\approx \mathit{R}\left(\frac{2{\mathit{\omega }}^2}{{\mathit{\omega }}_{\mathit{s},\mathit{loaded}}^2}-\frac{2{\mathit{\omega }}^2}{{\mathit{\omega }}_{\mathit{s},\mathit{unloaded}}^2}\right)+{\mathit{R}}_{\mathit{eddy}}\left(1+\frac{2{\mathit{\omega }}^2}{{\mathit{\omega }}_{{s},\mathit{loaded}}^2}\right)\\ +\mathit{j\omega L}\left(\frac{{\omega }^2}{{\omega }_{s,\mathit{\text{loaded}}}^2}-\frac{{\omega }^2}{{\omega }_{s,\mathit{\text{unloaded}}}^2}\right)\end{array}$$E52

For the segmented coil, the expression $\left(1-\frac{1}{{\omega }^{2}L{C}_{\mathit{seg},c}}\right)$ simplifies to $1/N$ for a coil with $N$ segments. Again, we tune the coil when it is not loaded, so the reactance of the unloaded coil is zero.

For the tuned segmented coil, the unloaded and loaded impedances are:

$$Z_{\begin{array}{c}\mathit{\text{unloaded}}\\ \mathit{seg}\end{array}}\approx \frac{R}{{\left(1-\frac{{\omega }^2}{{\omega }_{s,\mathit{\text{unloaded}}}^2}·\frac{1}{N}\right)}^2}+\mathit{j\omega }\left(\frac{\frac{1}{N}·L}{1-\frac{{\omega }^2}{{\omega }_{s,\mathit{\text{unloaded}}}^2}·\frac{1}{N}}-\frac{1}{{\omega }^2{C}_{\mathit{\text{tune}}}}\right)$$E53

$$Z_{\begin{array}{c}\mathit{\text{loaded}}\\ \mathit{seg}\end{array}}\approx \frac{R+R_{\mathit{\text{eddy}}}}{{\left(1-\frac{{\omega }^2}{{\omega }_{s,\mathit{\text{loaded}}}^2}·\frac{1}{N}\right)}^2}+\mathit{j\omega }\left(\frac{\frac{1}{N}·L}{1-\frac{{\omega }^2}{{\omega }_{s,\mathit{\text{loaded}}}^2}·\frac{1}{N}}-L_{\mathit{\text{eddy}}}-\frac{1}{{\omega }^2{C}_{\mathit{\text{tune}}}}\right)$$E54

The difference between these values is:

$$\begin{array}{c}{Z}_{\begin{array}{c}\mathit{\text{loaded}}\\ \mathit{seg}\end{array}}-Z_{\begin{array}{c}\mathit{\text{unloaded}}\\ \mathit{seg}\end{array}}\approx \frac{R}{N}\left(\frac{2{\omega }^2}{{\omega }_{s,\mathit{\text{loaded}}}^2}-\frac{2{\omega }^2}{{\omega }_{s,\mathit{\text{unloaded}}}^2}\right)+R_{\mathit{\text{eddy}}}\left(1+\frac{2{\omega }^2}{N{\omega }_{s,\mathit{\text{loaded}}}^2}\right)\\ +\mathit{j\omega }\frac{L}{N^2}\left(\frac{{\omega }^2}{{\omega }_{s,\mathit{\text{loaded}}}^2}-\frac{{\omega }^2}{{\omega }_{s,\mathit{\text{unloaded}}}^2}\right)\end{array}$$E55

Comparing Eqs. (52) and (55) we observe that for a segmented coil: (a) the resistance increase due to proximity of lossy tissue is lower than that for unsegmented coil, (b) the reactance increase due to the proximity of lossy tissue is lower than that for unsegmented coil. This is clearly due to the $1/N$ and$1/N^2$ factors responsible for this effect. Therefore, segmenting the coil significantly improves the robustness of the coil to the deleterious effects of the lossy body tissue.

7.2 Effect of segmentation on the transmit coil figure of merit

Figure 11 shows the equivalent circuit for non-segmented and segmented loaded transmit coils.

Figure 11.

Non-segmented and segmented loaded transmit coils.

The input current splits into two branches: current $I$ that flows through the ideal inductor $L$ and parasitic current $I_p$ that flows through the capacitor $C_p$ and resistor $R_p$

These two currents are related as follows:

$$I_p\approx -\frac{{\omega }^2}{{\omega }_s^2}\left(1-\frac{1}{{\omega }^{2}L{C}_{\mathit{seg}},c}\right)I$$E56

where we neglected the resistances $R$ and $R_p$.

In the transmit figure of merit ($\frac{B_t}{\sqrt{2P_{\mathit{\text{in}}}}}$), the magnetic flux density $B_t$ depends on the current $I$ in the coil. If we keep the current $I$ fixed, then the magnetic flux density will also remain fixed.

The power needed to generate the current $I$ (and magnetic flux density $B_t$) is:

$${\mathit{P}}_{\mathit{in}}=\frac{1}{2}\left(\mathit{R}+{\mathit{R}}_{\mathit{eddy}}\right){\mathit{I}}^2+\frac{1}{2}{\mathit{R}}_{\mathit{p}}{\mathit{I}}_{\mathit{p}}^2$$

$$=\frac{1}{2}\left(R+R_{\mathit{\text{eddy}}}\right)I^2+\frac{1}{2}{R}_p\frac{{\mathit{\omega }}^4}{{\mathit{\omega }}_s^4}{\left(1-\frac{1}{{\omega }^{2}L{C}_{\mathit{seg},c}}\right)}^2{I}^2$$E57

The figure of merit is then:

$$F_t=\frac{B_t}{\sqrt{2P_{\mathit{\text{in}}}}}=\frac{B_t/I}{\sqrt{\left(R+R_{\mathit{\text{eddy}}}\right)+R_p\frac{{\omega }^4}{{\omega }_s^4}{\left(1-\frac{1}{{\omega }^{2}L{C}_{\mathit{seg},c}}\right)}^2}}$$E58

We observe that segmentation leads to an increase in the transmit figure of merit of a coil. This is because for the segmented coil, the voltage V across the terminals of the coil is reduced by $\frac{I}{\mathit{j\omega }{C}_{\mathit{seg},c}}$. This means that the current through the parasitic resistance $R_p$ will be less and, therefore, the corresponding ohmic loss will be less.

7.3 Effect of segmentation on the receive figure of merit

Figure 12 shows the equivalent circuit for non-segmented and segmented loaded receive coils. In the figure, $A_r$ is the effective aperture area of the receive coil, $B_t$ is the incident magnetic field from the transmit coil and $\omega B_t{A}_r$ is the voltage appearing across the receive coil terminals.

Figure 12.

Non-segmented and segmented loaded receive coils.

For the non-segmented coil, the optimal loaded resistance is:

$$R_L\approx \frac{R_r+R_{\mathit{\text{eddy}}}}{{\left(1-\frac{{\omega }^2}{{\omega }_s^2}\left(1-\frac{1}{{\omega }^{2}L{C}_{\mathit{seg},c}}\right)\right)}^2}$$E59

Currents through the voltage source $\omega B_t{A}_r$ and through the load $R_L$

$$I_r\approx \frac{1}{2}·\frac{\omega B_t{A}_r}{R_r+R_{\mathit{\text{eddy}}}}$$E60

$$I_L\approx \frac{1}{2}·\frac{\omega B_t{A}_r}{R_r+R_{\mathit{\text{eddy}}}}·\left(1-\frac{{\omega }^2}{{\omega }_s^2}\left(1-\frac{1}{{\omega }^{2}L{C}_{\mathit{seg},c}}\right)\right)$$E61

The delivered power is:

$$P_L\approx \frac{{\left(\omega B_t{A}_r\right)}^2}{8\left(R_r+R_{\mathit{\text{eddy}}}\right)}·\left(1-\frac{R_p}{R_r+R_{\mathit{\text{eddy}}}}\frac{{\omega }^4}{{\omega }_s^4}{\left(1-\frac{1}{{\omega }^{2}L{C}_{\mathit{seg},c}}\right)}^2\right)$$E62

Again, we observe that the receive figure of merit $\sqrt{8P_L}/\omega B_t$ without the segmentation capacitor ($C_{\mathit{seg},c}\to \mathit{inf}$) is lower than the receive figure of merit with segmentation capacitor. Segmentation, therefore, leads to the increase of the receive figure of merit.

8. Full wave simulations and measurements

To verify the theory presented in the previous sections, a PCB spiral coil is modelled in Ansys HFSS, as shown in Figure 13. A trace on the bottom layer is used to connect the inner terminal of the coil through a via, to form a closed loop. The locations of the segmentation capacitors are indicated for different segmentation numbers. The dimensions of the coil are listed in Table 1. The substrate is a 1.5 mm FR-4 with 1 oz. copper.

Figure 13.

Top (left) and bottom (right) view of the spiral coil. ($a$ is the tuning capacitor, $b$ is the capacitor for 2 segments and $c$ is the capacitors for 4 segments).

The coil is firstly simulated without any capacitors. The inductance of the coil can be obtained as:

$$L=\frac{\mathrm{im}\left(Z_{11}\right)}{\omega }$$E63

where $Z_{11}$ is the input impedance of the coil, and $\omega$ is the radian frequency. The tuning capacitor can be calculated as:

$$C_{\mathit{\text{tune}}}=\frac{1}{\omega \mathrm{im}\left(Z_{11}\right)}$$E64

The values of the segmentation capacitors are calculated using Eq. (47). In practice, the values of the segmentation capacitors would be a little higher due to the parasitic capacitance of the coil itself.

Once the coil is tuned to resonate at the desired frequency, either with or without segmentation, the resistance can be obtained as:

$$R=\mathrm{re}\left(Z_{11}\right)$$E65

To evaluate the effect of the segmentation on the resistance, three cases are compared by simulation and verified with measurement: (a) coil without segmentation with one series capacitor to resonate the coil; (b) coil with one segmentation capacitor splitting the coil wiring into two equal segments; (c) coil with three segmentation capacitors splitting the coil wiring into four equal segments.

8.1 Coil resistance

The fabricated coils with and without segmentation are shown in Figure 14. All the coils are tuned to resonate at 27.12 MHz. A comparison of the simulated and the measured resistance with and without segmentation is shown in Table 2. Excellent agreement is found between the simulations and the measurements. The values of the capacitors needed to resonate the coil at 27.12 MHz are higher than the values calculated using Eq. (64). The measured resistances of the Printed Circuit Board (PCB) coils are higher than the simulated ones because of the extra capacitance and loss from the testing cable and connector which is not included in the simulations. What is clear from both simulation and measurement is that the addition of segmentation capacitors significantly reduces the coil resistance and the associated power loss in the coil.

Figure 14.

Fabricated coils with and without segmentation capacitors.

Radius $R$, mm	Trace width $w$, mm	Inter-trace distance $g$, mm	No. of turns
35	3	2	4

Table 1.

Coil dimensions.

Coil	Simulated/Measured	Capacitor values (pF)	Resistance (Ω)
Coil non-segmented	Simulated	C1 = 30	0.69
	Measured	C1 = 35	0.86
Coil with 2 segments	Simulated	C1 = C2 = 64	0.50
	Measured	C1 = C2 = 68	0.53
Coil with 4 segments	Simulated	C1 = C2 = C3 = C4 = 134	0.39
	Measured	C1 = C2 = C3 = C4 = 139	0.42

Table 2.

Simulated and measured resistance for segmented and non-segmented coils.

8.2 Figure of merit $F_t$

The electromagnetic (EM) fields generated from the coils can be simulated in HFSS. The FoM $F_t$ is used to compare the coils with and without segmentation. For ease of comparison, the fields along the X, Y, Z directions are plotted, where the origin of the coordinate system is the center of the coil, and the coil is placed at the XY-plane, as indicated in Figure 13. The electric fields normalised by the input power as $\frac{E}{\sqrt{P_{\mathit{\text{in}}}}}$ are also plotted. From Figure 15 we can observe the following trends:

1. Segmentation can increase the magnetic field strength without changing the field distribution generated by the coil. So, the designer can start with the non-segmented coil to optimise the field coverage first.

2. Segmentation can suppress the electric field in the direction perpendicular to the coil surface (moving away from the coil).

3. Segmentation can significantly alter the electric field distribution near the coil surface with localised maxima close to the capacitors.

Figure 15.

FoM Ft and normalised electric field plots along Z, X, and Y directions. The plots for X and Y directions are at z = 4 mm height.

Figure 16 plots the heat map of the magnitude of the electric field in the PCB substrate indicating that, as the electric fields are concentrated around the segmentation capacitors, the dielectric loss in the substrate is reduced.

Figure 16.

Electric field in the substrate. (left: no segment; middle: 2 segments; right: 4 segments).

8.3 Transfer efficiency

In this section, the effect of the segmentation on the power transfer efficiency is evaluated in both simulations and measurements. The receive coil shares the same HFSS model as the transmit coil, only with different dimensions and number of turns. The parameters of the receive coil that is simulated is shown in Table 3. The substrate is 0.8 mm FR-4 with 1 oz. copper.

Radius $R$, mm	Trace width $w$, mm	Inter-trace distance $g$, mm	No. of turns
4.88	0.24	0.26	5

Table 3.

Geometry of the receive coil.

The receive coil is placed 10 mm above the transmit coil with its center aligned with the center of the transmit coil, as shown in Figure 17. To investigate the loading effect of the human body, a hand is placed close to the coil. To measure the transfer efficiency, we perform the following steps:

1. Calibrate two ports of network analyser at the frequency of interest.

2. Tune the transmit coil with series capacitor, connect it to the network analyser and measure its resistance. Repeat for the receive coil.

3. Place the two coils in proximity of each other and connect them to the network analyser.

4. Measure and save the S-matrix of the system (2 × 2 matrix).

5. Convert S-matrix to Z-matrix using:

   $$Z=Z_0\left(U+S\right){\left(U-S\right)}^{-1}$$E66

   where $U$ is the unity matrix, $Z_0$=50 Ω

6. Calculate mutual inductance using:

   $$M=\frac{1}{\omega }\mathrm{im}\left(Z_{21}\right)$$E67

7. Calculate the mutual quality factor using Eq. (10).

8. Calculate the efficiency using Eqs. (14) or (18), depending on if we choose to maximise delivered power or transfer efficiency.

Figure 17.

Measurement setup for the transfer efficiency (left: top view; middle: side view; right: a hand is close to the coils).

While designing a WPT system for medical implants, care must be taken to understand the various use cases and user interactions and its implications on power delivery. An important decision that needs be made is whether a design is maximised for delivered power to an implant or efficiency of the WPT link. On one hand, if it is challenging for the receive coil inside of the implant to harvest the needed amount of power, then maximising the delivered power is preferential. On the other hand, most body worn charging systems are battery-powered and have a limited amount of available power to deliver to the implant. So, maximising the efficiency directly results in longer duration before the battery runs out on the charger and needs to be recharged by the patient or the caregiver.

As an example for this article, we chose to calculate the transfer efficiency using Eq. (18) for the coils with and without segmentation. The transfer efficiency is also simulated in HFSS for comparison. Furthermore, a 200 mm × 200 mm 3-layered tissue stack model is placed 2 mm above the transmit coil and the receive coil is embedded in the fat layer with the same 10 mm distance to the transmit coil in HFSS, as shown in Figure 18. The thickness of the skin, fat and muscle is 2 mm, 23 mm and 20 mm, respectively.

Figure 18.

Transfer efficiency simulation with tissue stack.

The simulated and measured transfer efficiencies are summarised in Table 4. We observe that the transfer efficiency both in air and in tissue can be improved with segmentation. Although we have done the calibration to minimise the effect of the cables and connectors, the measured efficiency is still a little lower than the simulated one, which is not surprising. However, with segmentation, we can see that the measured efficiency is much closer to the simulated one. It implies that the segmentation can reduce the loading effect of the environment (e.g. cables). The measured efficiency of a coil without segmentation in the presence of body tissue (hand) shows a significant drop from 61.9% to 46.2%, while the measured efficiency of a coil with two and four segmentations shows only a drop from 65.0% to 62.3% and 67.2% to 66.5%, respectively. This clearly indicates that the segmented coils are more robust to the presence of lossy tissue. In case of the simulated coils, the tissue of Figure 18 has a much larger effect on the coil, because there is large drop in efficiency when the tissue is nearby for non-segmented and segmented coils.

Segmentation	no segments	2 segments	4 segments
Simulated in air	67.4	68.6	70.7
Simulated with tissue stack	36.3	41.6	42.8
Measured in air	61.9	65.0	67.2
Measured with a hand nearby	46.2	62.3	66.5

Table 4.

Transfer Efficiency (%).

8.4 Figure of merit $F_{t,\mathit{SAR}}$

The SAR value in tissue is simulated in HFSS using the same tissue model as in the previous section. For a fair comparison, the SAR is also normalised by the input power as:

$$\overline{\mathit{SAR}}\equiv \frac{\mathit{SAR}}{P_{\mathit{\text{in}}}}$$E68

Figure 19 compares the distribution of the peak average SAR where the SAR value has been normalised to the peak SAR value for each of the three coil designs presented. The IEC/IEEE 62704-4 method is used to calculate the peak average 1 g SAR. Without segmentation, the regions of high SAR value occur at the overlapping area between the trace on the top and bottom layers of the PCB. This is because there is high stored electric field between the layers resulting in high parasitic capacitance. For the coil with two segments, the regions of high SAR value are between the segmentation capacitor and the areas of overlap between the top and the bottom layers of the PCB. Both these regions have high parasitic capacitance. For the coil with four segments, the 3 segmentation capacitors are lined close to each other resulting in a region of high stored electric field. This results in the coil with four segments having higher peak average SAR compared to the coil with two segments, but still lower than the coil with no segmentation capacitors. The results also clearly indicate that the locations of the segmentation capacitors play a critical role in reducing the peak average SAR.

Figure 19.

The SAR maps in the tissue. (top: coil with no segments; middle: coil with two segments; bottom: coil with four segments).

Another important advantage of introducing segmentation capacitors in the wiring of the coils (or along the coil traces) is that the distribution of the averaged SAR and its maximum value can be significantly altered by optimising the locations of the segmentation capacitors along the coil. For example, the 4-segment coil (in Figure 14) has the three segmentation capacitors in close proximity, all in the same sector of the circular coil. For the same coil Figure 20 shows a significantly different SAR distribution and reduced maximum SAR value when the three segmentation capacitors are spread along the coil with 90-degree separation. The heat map shows that the regions of high SAR value are shaped like a circle and the peak value of the SAR is reduced by 40%. It should be noted that the tuning capacitor is not shown in this plot because it is placed at the far end of the coil input.

Figure 20.

SAR map of the coil with spread segmentation capacitors.

With the normalised SAR and the coil resistance, the maximum allowed current within FCC limit can be calculated as

$$I_{\mathit{max}}=\sqrt{2P_{\mathit{max}}/R_{\mathit{\text{loaded}}}}$$E69

where $P_{\mathit{max}}=1.6/\overline{\mathit{SAR}}$, and $R_{\mathit{\text{loaded}}}$ is the coil resistance when the coil is in close proximity to lossy tissue.

Table 5 summarises the coil impedance, SAR, $F_{t,\mathit{SAR}}$ and maximum current compliant to FCC limit with different segmentations. The coil without segmentation is also listed for comparison. It is noted that when the segmentation capacitors are spread along the coil, the peak SAR is significantly reduced, and the maximum current within FCC limit is increased. From the Table 5 we observe that coil resistance decreases as we increase coil segmentation: from 0.69 Ω to 0.39 Ω if in air (−44%) and from 1.88 Ω to 0.94 Ω if near tissue (−50%).

Coil	Non-segmented	Segmented with 2 segments	Segmented with 4 segments	Segmented with 4 segments, spread
$Z_{11}$ ($\mathrm{\Omega }$) in air	0.69 + j1.27	0.50 + j0.94	0.39 + j4.69	0.42 + j4.28
$Z_{11}$ ($\mathrm{\Omega }$) with tissue	1.88 + j9.65	1.13 + j2.17	0.97 + j5.51	0.94 + j4.40
$\overline{\mathit{SAR}}$ (1/kg)	13.93	7.26	10.13	6.00
$F_t\left(\mathrm{mT}/\left(\mathrm{A}\sqrt{\mathrm{\Omega }}\right)\right)$ with tissue (z = 10 mm)	0.0938	0.1015	0.1089	0.1070
$F_{t,\mathit{SAR}}$ ($\mathrm{mT}/\left(\sqrt{\mathrm{W}/\mathrm{kg}}\right)$) (z = 10 mm)	0.0251	0.0377	0.0342	0.0437
Max allowed current within FCC 1.6 W/kg limit (mA)	349.6	624.6	570.7	753.2

Table 5.

Comparison of coil impedance, SAR,$F_t$,$F_{t,\mathit{SAR}}$ and the max current compliant to FCC limit with segmentations.

When we compare the “$Z_{11}$ in air” and “$Z_{11}$ with tissue”, we observe that both resistance (real part of impedance) and reactance (imaginary part of impedance) increase when the tissue is in the proximity of the transmit coil. For example, the resistance grows from 0.69 Ω to 1.88 Ω (+172%) for the non-segmented coil when the tissue is approached.

The figure of merit $F_t$ grows by 16% and $F_{t,\mathit{SAR}}$ grows by 74% as we increase coil segmentation.

Let us now compare the change in resistance $∆R$ and the change in reactance $∆X$ for the four coils as shown in Table 6. We observe that change in resistance $∆R$ is decreasing with the progressing segmentation, up to 56%. In the following table we show the measured change in reactance $∆X$.

Coil	Non-segmented	Segmented with 2 segments	Segmented with 4 segments	Segmented with 4 segments, spread
$∆R$, (Ω)	1.19	0.63 (−47%)	0.58 (−51%)	0.52 (−56%)

Table 6.

Change in resistance.

From the Table 7 we observe that there is a decrease in the change in reactance $∆X$ as the number of segments in the transmit coil, N increases. Assuming that Leddy contribution is negligible, the Eq. (55) predicts that the ratio of non-segmented $∆X$ to segmented $∆X$ would grow as $N^2$ . For the “4-segment spread” the ratio of changes in reactance far exceeds the prediction of $N^2$. Spreading the segmentation capacitors away from one another, significantly helps to stabilise the transmit coil. While the numbers do not exactly match, the trend showing the increase in the ratio is as predicted in Eq. (55).

Coil	Non-segmented	Segmented with 2 segments	Segmented with 4 segments	Segmented with 4 segments, spread
$∆X$, (Ω)	8.38	1.23 (−85%)	0.82 (−90%)	0.12 (−98.6%)
$\frac{∆X_{\mathit{\text{non}}-\mathit{\text{segmented}}}}{∆X_{\mathit{\text{segmented}}}}$	1	6.8	10.2	69.8
$N^2$ factor	1	4	16	16

Table 7.

Comparing the change in reactance with the predicted one.

9. Conclusions

In this work, we introduced and derived unique optimisation metrics for designing efficient transmit and receive coils for magnetics based WPT solutions for medical implants. We reviewed the regulations imposed on WPT systems for medical implants in the US and EU regions and determined the most limiting parameters that place a bound on the maximum current that can be driven into a coil. We derived the expressions for delivered power and efficiency considering the identified regulatory limits for the transmit and the receive coil currents. We demonstrated that, under certain conditions, the system figure of merit can be “split” into transmit figure of merit and receive figure of merit permitting independent evaluation of transmit and receive coils.

We studied the effect of lossy tissue on the performance of transmit coils from a circuit theory perspective. We showed that the resistance of the transmit coil increases in the presence of tissue because of two types of electromagnetic phenomena: (i) increase in parasitic capacitance between the opposite charges accumulating in the surfaces of the coil (charge contribution); (ii) the eddy currents in the tissue (current distribution). We showed that the change in reactance of the coil due to the presence of lossy tissue is dependent on which contribution (charge or current) is more significant.

With this improved understanding of the effect of lossy tissue on coils we introduced the concept of segmented on-body transmit coils. We hypothesised that the resistance and reactance of a transmit coil with segmentation capacitors is less sensitive to the presence of lossy tissue. We derived the impact of segmentation on the transmit figure of merit and the receive figure of merit of a coil using circuit theory. We showed analytically that segmented coils have the potential to significantly improve both (transmit and receive) figures of merit, thereby positively affecting the efficiency of a WPT system.

To validate our hypothesis and assertions we built PCB coil prototypes at 27.12 MHz with and without segmentation. We performed full wave simulations using HFSS models of the same coils. We showed through simulations and measurements that the resistance of the transmit coil reduces substantially (as much as 50%) when we went from no segmentation to up to four segments (with three segmentation capacitors). We also confirmed that the proximity of lossy tissue has a significantly smaller effect on segmented transmit coil. We noted that, on the specific coils we built, we measured that the change in reactance of a coil between air and close proximity of tissue reduced from 4.2% (for non-segmented coil) to 0.06% (for segmented coil with capacitors uniformly spread). We also confirmed that the transmit figures of merit ($F_t$ and $F_{t,\mathit{SAR}}$) of the segmented coil are higher than those of the non-segmented coil. $F_t$ grew by 16% and $F_{t,\mathit{SAR}}$ grew by 74% as we increased the level of segmentation. We have found that the way the segmentation capacitors are spaced on the coil has a significant effect on coil performance and the distribution of electric field close to the wiring of the coil. This is an important result as the number of segmentation capacitors and their distribution to break up the coil wiring controls the distribution of electric field and will be very useful in controlling not just SAR but also to reduce coupling with the internal electronics of a charger.

Acknowledgments

We are grateful to Mark Norris, Richard Davies, Matthew Armean-Jones, and Olympia Karadima for their useful feedback that helped in writing this chapter.