Abstract—This paper studies an inductive link optimization for implantable systems. By considering the relationship between wire diameter, number of turns, quality factor, coupling coefficient, and shape of coil, we can optimize the inductively coupled coils to maximize the power transfer efficiency in the face of geometric constraints. In this work, an intraocular prosthesis is used for a design example and power transfer efficiency of 45.8% is achieved where the secondary coil has a single layer of 11 turns, 0.064cm wire diameter, a self-inductance of 3.8μH, and Q of 65. Index Terms – Inductive power link, coil optimization, retina prosthesis, implantable device.

I. INTRODUCTION Over the past decade, short distance wireless

communication technologies have been applied to contactless smart card, wireless sensor networks and bio-medical [1-4]. Especially, wireless communication based on inductively coupled coils makes it possible to simultaneously transmit wireless power and data in using magnetic fields and is suitable for bio-medical applications where the devices are implanted inside body and any embedded power sources are not desired. Fig. 1 illustrates an architecture for such a wireless implantable system based on an inductive link [1-6].

Wireless implantable systems can require high power transfer efficiency (PTE) as well as high data rates for the purpose of controlling various functions [2]. Usually, it is more critical to accomplish high PTE rather than a high data rate in implantable systems because the power loss in the coils gives rise to heat that may impair normal tissues nearby. In order to increase PTE, therefore, optimal coil configurations have been explored in the literature [7, 8]. In [7], maximum PTE can be achieved by finding out an optimum equivalent AC series resistance of the secondary coil and by substituting it into PTE equation (η). However, that analysis didn’t consider that quality factor (Q) of the secondary coil is a function of the AC series resistance. Another way to increase the efficiency is to use Litz wire that can minimize AC series resistance due to proximity-effect losses [8]. While the use of Litz wire is very effective below 500 KHz it is seldom employed at higher frequencies due to its parasitic capacitance [9]. Thus, Litz wire is not well suited for implantable systems, e.g. retinal prostheses, which require over 1Mbps data rate and the reuse of the power carrier for its clock [8].

In this paper, a coil optimization technique is proposed to maximize PTE by considering the relationship between wire diameter, number of turns, quality factor, coupling coefficient, and sizes of the coils. This paper is organized as follows.

Section II describes PTE of the inductive link, Section III presents a design example and simulation results. In section IV we present our conclusions.

II. POWER TRANSFER EFFICIENCY Fig. 2 presents the simplified equivalent circuit of an

inductive link based on a resonant tank. Due to the series resonant tank in the primary side, at a resonant frequency (ωR), the load’s impedance that a power amplifier drives becomes low which makes it possible to deliver the maximum power from the source to the primary coil (Coil 1) [4]. On the other hand, at ωR, a parallel resonant tank in the secondary side functions as a current source. In the primary side, RS, RES1, and ZR represent the output resistance of power amplifier, the equivalent series resistance of Coil 1, and the reflected impedance from the secondary coil (Coil 2), respectively. Similarly, VIND, RES2, and RL,AC account for the induced voltage, the equivalent series resistance of Coil 2, and the load impedance. In [6], the PTE equation based on these factors is given as;

( ) ( )22

2,22

2,1

,212

2,3

212

1 ESACLESACLES

SACL

ESACL

S

O

RQRRQRRR

RQQk

RRQQKPP

⋅+×⎥⎥⎦

⎤

⎢⎢⎣

⎡⋅+⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛++⋅⋅

⋅⋅⋅==η

….. (1) where K is the coupling coefficient between Coil 1 and Coil 2.

For simplicity, in this work, we take the output resistance (RS) of power amplifier and the load impedance (RL,AC) as 50Ω and 1KΩ, respectively [12]. In order to achieve optimal PTE, the relationship between Q1, Q2, RES2, and K need to be considered in detail.

A. Equivalent Series Resistance The equivalent series resistance (RES) of coil includes DC

resistance determined by diameter and length of wire and AC resistance terms by skin-effect and proximity-effect losses at radio frequencies. Since the RES influences the PTE as well as the quality factor of the coil, an appropriate RES is required for

Power-Efficient Inductive Link Optimization for Implantable Systems

Jungsuk Kim, Hyunchul Kim, and Kenneth D. Pedrotti

Electrical Engineering, University of California, Santa Cruz, CA 95064, U.S.A

Fig. 1. General architecture of wireless implantable systems based on an inductive link.

978-1-4244-7685-5/11/$26.00 © 2011 IEEE RWS 2011418

maximum PTE. Each resistance is derived in the following subsections.

DC Resistance

DC resistance is defined as; resistivity × wire’s length / wire’s area. In this work, a copper wire is employed and its resistivity (ρ) at 37.5° is 1.795μΩ/cm. Fig. 3 shows the cross section of both coils and its geometric factors used to define the coil. If a is large enough compared to c then the total length of wire is approximately 2π·a·n where n is the number of turns. Its area is π×(WD/2)2 where WD is the wire diameter. As a result, the DC resistance can be expressed as;

22

2

2

2⎟⎠⎞

⎜⎝⎛⋅

⋅⋅⋅≅

⎟⎠⎞

⎜⎝⎛⋅

⋅=⋅=DD

DCW

naW

LenthAreaLenthR

π

πρπ

ρρ ….. (2)

AC Resistance due to Ski-effect and Proximity-effect losses

As the carrier frequency increases, the flow of current on the wire is impeded by skin effect and proximity effect [10]. Due to the limited room inside body, wires should be compactly wound to save an area. However, this results in higher proximity-effect losses that tend to dominant over skin-effect ones in implantable bio-medical applications. Nonetheless, in this work, we include all losses for completeness.

The skin depth (σ) of copper wire at a radio frequency is 6.74/√frequency in centimeter (cm) and so the AC resistance [9] can be expressed as;

Ω⎥⎦

⎤⎢⎣

⎡ ××⋅= −9108.20 fWRR DDCskineff ρπ ….. (3)

In addition to skin-effect losses, proximity-effect losses based on Dowell’s assumption [10] are given as;

( ) ⎥⎦

⎤⋅+⋅⋅−⋅⋅−+

⎢⎣

⎡⋅−⋅⋅+⋅×⋅=

)2cos()2cosh()2sin()2sinh(1

32

)2cos()2cosh()2sin()2sinh(

''

''2

''

'''

ζζζζ

ζζζζζ

m

RR DCeffproximitty

….. (4) where `ζ is:

( ) bWnW DLD ⋅⋅= πσζ 2` Here, nL and m indicates the number of wires per layer and the number of layers, and these have the following relationship;

b=nL×WD, c=m×WD, and thus n= m×nL=(b×c)/WD2 ….. (5)

In Eq. (5), a square area, WD2, is used because we assume

that the coil is evenly wound as illustrated in Fig. 3. In terms of Eq. (2) through (4) the RES including DC

resistance and AC correction terms is now;

( )⎪⎪⎭

⎪⎪⎬

⎫

⎥⎦

⎤⎢⎣

⎡⋅+⋅⋅−⋅⋅−+

⋅−⋅⋅+⋅⋅+

⎪⎪⎩

⎪⎪⎨

⎧

⎥⎦

⎤⎢⎣

⎡ ××+×= −

effectproximitty

effectskin

DDCES

m

fWRR

)2cos()2cosh()2sin()2sinh(1

32

)2cos()2cosh()2sin()2sinh(

108.201

''

''2

''

'''

9

ζζζζ

ζζζζζ

ρπ

….. (6)

B. Self-inductance Depending on the coil’s shape, three different equations for

a single layer, a multilayer with multiple-rows, and a multilayer with single-row can be used to estimate each self-inductance. In the case where the coil is placed in the eyeball, for instance, a multilayer coil with multiple-rows is applied for generality. Eq. (7) presents the approximate self-inductance for such a coil [9].

( ) Hcba

naLself μ⋅+⋅+⋅×

⋅⋅=109654.2

8.0 22 …. . (7)

Here, it is desired that n be large in order to achieve a high self-inductance. However, if the dimensions a, b, and c are assumed constrained by the available space inside body then the only way to increase the self-inductance is by decreasing the wire diameter (WD) in Eq. (5).

On the other hand, the equation for the self-inductance of a flat multilayer coil with single-row would be used if the coil is for an implantable neural recording system [3] where it would be located between the skull and dura mater.

C. Quality Factor The unloaded quality factor is defined as (2πf ×Lself)/RES. As

carrier frequencies increase, Q also increases. However, due to the skin absorption proportional to (carrier freq.)2, frequencies below 20MHz are preferred for implantable bio-medical applications [12]. To attain high Q, therefore, RES

Fig. 3. Cross section of primary and secondary coils.

Fig. 2. Simplified model of inductive link.

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should be minimized while Lself is maximized. Through Eq. (6) and (7), it has been shown that RES and Lself

are functions of the dimensions of coil and can be expressed as follows;

RES=r (a, b, m, WD, f), Lself= l (a, b, c, WD), So, Q =q (a, b, c, m, WD, f)

Because the maximum coil dimensions are constrained by practical applications, the coil, used in the intraocular prosthesis example below, should fit within the eyeball and be compact to allow surgical implantation. For this reason, we assume that the maximum exterior coil diameter and width will be limited by the constraints, thus fixing a and b at their geometric maximum. In addition, the carrier frequency is almost fixed below 20MHz because of the skin absorption. Hence, Q can be simplified as Q≈ q(m, WD) where m and WD are the only adjustable variables.

D. Coupling Coefficient The coupling coefficient is a function of coil dimensions

and is [9]; ( ) ( )

21

222111

64.01096109654.2

aacbacbaNK

⋅⋅++×++⋅⋅= ….. (8)

where N depends on ( ) ( ) 22

2122

2112 DaaDaarr +++−= In this equation, D indicates a distance between Coil 1 and Coil 2 and N is calculated by the lookup table presented in [9]. As a result, the coupling coefficient is expressed as K=k(a, b, c, D) which can approximate k(m, WD) if c is substituted with m×WD in terms of Eq. (5).

Through Eqs. (6), (7), and (8), it has been shown that Lself, RES, Q and K are functions of a, b, c, D, m, f, and WD. However, in implantable bio-medical applications, a, b, c, D, and f are almost fixed due to the restricted space inside body and the skin absorption. Hence, PTE in Eq. (1) can approximate;

η≈ p(m, WD) where m and WD should be optimized to maximize it.

III. DESIGN EXAMPLE AND SIMULATION RESULTS Using the following design example and simulation results

we next show how to decide optimal m and WD to achieve maximum PTE in an intraocular prosthesis. Because of this device’s high-density stimulator, typically more than 256 [2], the implant requires higher PTE than other biomedical implants, e.g. a pace maker and a neural recording system [3].

Fig. 4 illustrates the cross section of an eyeball and the coil potions. The diameter of Coil 2 is restricted by the adult eyeball size that is roughly 2.4cm in diameter. Thus, we assume that the radius of a2 is 1.1cm and the remainder (0.2cm) is filled with a coil holder and a biocompatible glue to fix it to the inside wall of the eyeball. The length of b2 is 2×√ (1.22-1.12) ≈0.96. However, we choose 0.76cm for b2 to allow room (0.2cm) for the holder and the glue and to ensure the coil is small enough to fit though a surgical incision. As described in Eq. (5), the height of c2 is dependent on m×WD which is

adjustable to maximize PTE. Coil 1, which is placed 2cm apart from Coil 2, transmits a radio frequency signal at 2MHz that would be reused as a system clock in the implant [13]. The geometry of Coil 1, a1 and b1, are determined here so as to fit roughly within the frame of a pair of eyeglasses. However, c1 was chosen to maximize PTE after 100 iterative simulations in which each m1 and m2 varied from 1 to 10. This resulted in a1, b1, and c1 values of 1.5cm, 0.3cm, and 0.039cm, respectively. In this case, Coil 1 has a triple layer coil with 70 turns and Q of 16.2 at 2MHz.

Using MATLAB, we calculate RES, Lself, Q, K, and PTE of Coil 2 with respect to m and WD, where m varies from 1 to 10 and WD are commercially available from 0.013cm to 0.163cm [14], and the results are shown in Figs. 5. From Fig. 5 (e), we can obtain a single layer coil (m=1) to optimize the PTE with the design parameters shown in Table. I.

This result can be understood physically as follows. For the single layer coil as the WD decreases from the optimum, n can increase in terms of Eq. (5), leading to increasing both the RES and Lself of Coil 2 as shown in Figs. 5 (a) and (b). Because the RES2 dominates in this case, the Q2 gets small and ultimately the PTE is decreased. On the other hand, as the WD is increased form the optimum, n is decreased, resulting in decreasing RES2 and Lself2. In this case, the Lself2 dominates over RES2 which cause Q2 to increase as shown in Fig. 5 (c). However, the reduced Lself2 limits the flux linkage and accordingly decreases the efficiency.

If the number of layers is increased (m>1), n can be increased allowing greater coefficient coupling as depicted in Fig.5 (d). However, the losses by the proximity effect are reinforced thus decreasing the PTE. Interestingly, even though Q2 increases for thicker WD, the PTE decreases. It shows that Q optimization alone cannot be used as a proxy for PTE optimization.

IV. CONCLUSION This paper has been presented equations for the PTE

between two loosely coupled coils in terms of the coil and wire geometry. Assuming that spatial constraints exist in an intraocular implant used as a design example, some of the

Fig. 4. Cross section of an eyeball and the coil positions.

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Table I. Design Parameters of Coil 2 Coil 2

Resonant Freq. (ωR) 2MHz

a 1.1 cm

b 0.76 cm

c 0.064 cm

Series resistance (RES2) 0.74 Ω

Number of turns (n) 11

Number of layers (m) 1

Wire diameter (WD) 0.064 cm

Q 65

Self-inductance (Lself) 3.8μH

Coupling coefficient (k) 0.089

Transfer Efficiency (η) 45.8%

parameters can be fixed and optimized by inspection. Therefore, we could accomplish a PTE of 45.8% within the limited size of eyeball. For the optimum PTE, Coil 2 is composed of a single layer with 65 turns and 0.064cm diameter while Coil 1 has a triple layer with 70 turns and 0.013cm diameter. In this work, we have shown that maximum PTE was fundamentally associated with a wire diameter and number of layers rather than a quality factor alone.

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[3] R.R. Harrison, et al, “A low-power integrated circuit for a wireless 100-electrode neural recording system,” IEEE J. Solid-State Circuits, vol. 42, No. 1, pp. 123-133, Jan. 2007.

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[9] F. E. Terman, Radio Engineers Handbook. New York: McGraw-Hill, 1943.

[10] X. Nan and C. Sullivan, “An improved calculation of proximity-effect loss in high-frequency windings of round conductors,” in Proc. of IEEE Power Elect. Special. Conf., vol. 2, pp. 853-860, Jun. 2003

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[14] http://www.bulkwire.com/wire-cable/magnet-wire.html.

Fig. 5. MATLAB simulation results; (a) Equivalent series resistance (RES2), (b) Self-inductance (Lself2), (c) Quality factor (Q2), (d) Coupling coefficient (K), (e) Power transfer efficiency (PTE).

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