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1,2
1,2
2,1
1 National ICT Australia (NICTA), VRL, Melbourne 3010, Australia
2 The University of Melbourne, Melbourne 3010, Australia
In this paper, we investigated how frequency related effects influence the resistance of inductive coils used in wireless links for implantable electronics. In particular we studied proximity and skin effect by employing numerical simulation. In order to elicit these effects, simple 3-turn coils with varying fill-factors were studied. We found that proximity effect depends strongly on the fill-factor ratio. This is verified experimentally with a 7 and 17-turn coils of the same inner and outer diameter. We found that the coil with lower fill-factor has better performance in terms of its quality factor at higher frequencies. A maximum Q increase of 29% was observed for the 7-turn coil compared to the 17-turn coil.
Index Terms — Inductive power links, printed spiral coils (PSC), current crowding, skin-effect, proximity-effect, fill-factor
I.
I NTRODUCTION
M AGNETIC coupling is a means for wireless power transfer in implantable electronics. Printed spiral coils
(PSCs) have been widely used for wireless power transmission and data transmission due to their simplicity, small form factor and light weight. An important parameter for PSCs is quality factor. Quality factor contributes to the power transfer efficiency between coils. This means that the PSC must be optimized to obtain highest possible power-transfer efficiency and supply electronic circuitry with the required amount of power.
Many papers have dealt with the design and optimization of spiral coils. In particular PSC based technology for biomedical implants has been thoroughly studied and analyzed in [1-5].
The main goal in designing inductive wireless links is to maximize the coupling efficiency between the primary and secondary coils which are loosely coupled. This can be achieved by the increased quality factor of primary and secondary coils which depends on geometrical and physical parameters such as coil diameter, number of turns, spacing between primary and secondary coil and the properties of the environment surrounding the coils. The quality factor of a multi-turn spiral coil when used as an inductor can be significantly reduced from the value that is predicted by using a simple calculation of reactance divided by the direct current
(DC) resistance due to the proximity effect or current crowding. This phenomenon is studied in [6] and [7] since the increased demand for the low cost radio frequency integrated circuits (RFIC) generated interest in on-chip passive components such as inductors and transformers. In order to avoid numerical simulations a fast analytical method for the current crowding mechanism that guides the design of spiral inductors without the need for repetitive and time consuming simulations is proposed [6]. The same method also applies to inductive coils used as wireless links in implantable electronics [2]. However, analytical methods introduce errors due to the necessary approximations of the proximity effect for the spiral coil geometry.
The quality factor is defined in terms of power dissipation and in [7] the authors argue that typical quality factor plots in literature as the simple ratio between the imaginary and real part of the inductor impedance leads to incorrect inductor
(spiral) performance predictions in some applications. We provide an insight into this in our analysis of quality factor.
Since the PSC quality factor depends on the geometrical and physical parameters it is necessary to gain insight on how these influence the resistive losses due to the proximity and skin effect.
In this paper, the focus is on the numerical simulations and assessment of the proximity or current crowding of PSCs.
Since the measurements of PSC parameters are often not practical or reliable due to parasitic effects of the measurement set-up, the use of numerical methods for accurate studies of current crowding and skin effects can improve the design. In general, the design of passive components and inductors is based on simulations and many electromagnetic solvers include optimization and parametric search modules to aid the design process. Although numerical simulations can be costly in terms of computational time and memory, accurate determination of the PSC parameters including proximity effects can only be obtained from properly set-up numerical simulations. Focusing on the insight into skin and proximity effects in PSCs for implantable electronics, this paper presents a detailed investigation of the resistance and current density as a function of frequency and PSC geometry. Previously, we have reported on wireless technologies and designed PSCs for closed-loop retinal prostheses, [8]. There the design was based on the analytical formulae. In this paper we investigate the relationship between the geometric parameters and current crowding in the PSCs. The application of numerical methods makes it possible to compute and quantify frequency related effects without the limitations common to analytical methods.
Since the constraint on optimal design is the physical size and operating frequency an accurate and insightful numerical simulation in the frequency domain can be a reliable design tool.

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II.
F REQUENCY -R ELATED EFFECTS IN PSC
To understand frequency-related effects in spiral inductors and PSC such as proximity effect and skin effect, analytical and numerical methods can be applied to guide and optimize the spiral coil design. In this study, we use finite element method (FEM) based numerical simulations to analyze the spiral coil designs and quantify the effect of current crowding on the resistance. ANSYS HFSS software simulations in the frequency domain are suitable for the investigation of the frequency-related effects. However, the accuracy of the simulations depends on the mesh size (number of finite elements in the computation) and the mesh refinement in the regions where significant variations in the electromagnetic fields can be expected. Since the multi-turn coils require the mesh refinement in the regions between the turns and inside the conductors the mesh refinement around the coil and inside the coil wire is required for accurate simulations, as shown for the three-turn coil example in Fig. 1. Overall, this reduces the total mesh size and computation time. Also, the use of curvilinear elements (elements with curved edges and curved faces) which are a more general type of FEM elements than their rectilinear counterparts reduce the total number of elements and generate more accurate solutions. conductor of Fig. 1 (conductor thickness, h=35 μ m) is 8.75
μ m which is less than half the skin depth at 10 MHz.
For effective analysis of multi-turn spiral coils we introduce a fill-factor ratio which is defined as the ratio between track width and spacing, ( w / s ). This definition is adequate to analyze the effect of varying width and spacing for a given outer and inner diameter of the coil.
A.
Skin Effect Analysis
2
In order to compare the resistance contribution due the skin effect with the contribution due to the proximity effect, a single turn coil (isolated wire) model is also analyzed. The radius of the single turn coil is relatively large and the assumption is that effect of current crowding due to the winding (spiral) is low. In this analysis, we first analyze the current density distribution in the cross section of the coil conductor and then compare the equivalent increase in resistance versus frequency for different conductor widths.
1) Current Density Distribution
As pointed out by Kuhn in [6], skin effect can be safely ignored for low frequencies. However, how low should this frequency be? In order to define a threshold frequency where skin effect starts to be significant, we look at the current density distribution in Fig. 2. It shows the current density distribution at 10 MHz for a single turn coil of the inner radius, r
0
=35.5mm and the outer radius, r
N
=35.83 mm. The height and width of the conductor is h =0.07mm and w =0.1 mm respectively. The DC current density of this conductor is
141 A/m 2 (based on the 1 μ A excitation current in the coil). At
10 MHz the current density at the inner and outer edges increases to 241.17 A/m 2 . It can also be observed that current crowding is more apparent at the longer edges of the rectangular cross section where current density is higher. The shorter edge is not as affected by skin effect at lower frequencies. This can be explained by considering the skin depth δ which is defined as ⁄ , where ρ is resistivity, μ permeability and f is frequency. At 10 MHz the skin depth of the coil conductors is δ=20.9 μ m. If we equate skin depth to half the dimension of the cross-section (due to symmetry), the resultant frequency which we call skin frequency is lower across the width compared to the height (or thickness) [9]. In other words, the skin frequency where skin effect starts to become significant i s lower fo r the er mension. Hence,
4 4
, (1)
Fig. 1 Refined mesh profile (cross sectional view) and the FEM model of the three-turn coil.
The mesh size (mesh seeding) is compatible with the skin depth and/or the highest frequency in which the spiral coil can operate (10 MHz). For example, the skin depth at 10 MHz is
20.9 μ m so that the mesh element (tetrahedron) in the whichever is the lower. For the conductor above ( w =0.1 and h =0.07 mm) f skin
is 1.69 MHz. Figure 3 shows the current density distribution along the cut-line through middle of the conductor at frequencies from 1 MHz to 10 MHz. The current density value at 1 MHz is approximately equal to the DC current density. As the frequency increases, the current density markedly decreases at the centre of the conductor.

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Fig. 2 Current density distribution at 10 MHz in a conductor (w-0.1 mm, h=0.07mm).
250
200
150
100
50
1
7
MHz
MHz
3
9
MHz
MHz
5
10
MHz
MHz
3
As we are interested in simulation results at frequencies of
10 MHz and below, we ignore substrate losses and use this simplified model to extract the resistance and determine the quality factor. If the self-resonance frequency of the coil is much higher than these frequencies the real component of the impedance Re( Z ) can be considered as equal to the resistance
R (see Appendix for analysis). This is valid for small coils of small number of turns (1-3 turns) and narrow conductors relative to the coil diameter.
Figure 5 shows the ac resistance and influence of the skin effect on a single-turn coil (isolated wire) for different conductor widths. In order to show the influence of skin-effect the results are given as ratio between the R and R dc
versus frequency. As the width of the conductor increases the change in R / R dc
over frequency is greater. For example, for mm, the increase of R / R dc w =0.1 at 5 MHz is 20% and for w =0.2mm the increase is 60%. However, due to the dominant DC resistance the ac total resistance, R for w =0.1 mm is greater than the total ac resistance for w =0.2 mm conductor.
1 5
0.8
w=0.1
mm w=0.15
mm w=0.2
mm w=0.25
mm
4
0.6
3
0.4
Edge Centre Edge
0
0.5 0.52 0.54 0.56 0.58 0.6 0.62 0.64 0.66 0.68 0.7
0.2
2
Position,   mm
Fig. 3 Current density distribution in a conductor versus distance
(cut-line along the width, w ) at frequencies from 1-10 MHz.
2) Resistance
Resistance R is estimated from the S -parameter matrix by converting it into Y -matrix. Then the resistance is extracted by fitting the model shown in Fig. 4.
The procedure to extract the frequency dependent resistance values from the RLC model where R represents the coils resistance, L is the inductance and
C is the parasitic capacitance of the coil is explained in [7] and in the appendix. This model of the PSC is adequate since the dominant power loss is in the metal conductors. The assumption is that values of C and L are relatively constant for the entire frequency range and the resistance increases significantly with frequency [1-6]. The parasitic capacitance C is estimated from the resonant frequency ω frequency inductance of the coil L where C =1 / ω converted to Z ( ω ).
0
and low
0
2 L . Then the admittance of C is subtracted from Y ( ω ) and the resulting data
L
C
R ( f )
Fig. 4 Simple three-element model of PSC, [1].
0 1
0 1 2 3 4 5 6 7 8 9 10
Frequency,   MHz
Fig. 5 Resistance, R and ratio of R / R dc
versus frequency.
B.
Proximity Effect Analysis
When the conductors in a spiral coil are comparatively close together, there is an increase in their alternating current resistance caused by the current crowding on the inside edge of the coil due to the proximity of conductors. This effect has been studied in [6-7] for the on-chip spiral inductor applications and it can be explained by two parallel wires carrying the same alternating current. The magnetic field of one wire will induce longitudinal eddy currents in the adjacent wire, that flow in loops along the wire, in the same direction as the main current on the side of the wire facing away from the other wire, and back in the opposite direction on the side near the other wire. Thus the eddy current will support the main current on the side facing away from the first wire, and oppose the main current on the side facing the first wire. The net effect is to redistribute the current in the cross section of the wire into a thin strip on the side facing away from the other wire. Since the current is concentrated into a smaller area of the wire, the resistance is increased. In this paper we study how the current crowding affects the ac resistance of onsubstrate printed spiral coils for implantable electronics.
In the design of the implantable electronic devices small dimension of the power supply and data coils is a required

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, AB-02 (DOUBLE-CLICK HERE) < feature. However, this limits the available widths of the conductor wire and spacing between the turns. Therefore, an optimal coil design includes trade-off between the coil size and p ower loss (resistive loss). If the diameter (radius) and size of the coil are limited, then the only possible variations are the conductor width and thickness and spacing between the turns. To keep the coil radius, width, w
0 r
0
fixed when the conductor
increases the spacing between the turns, s
0
will decrease (Fig. 6). Smaller spacing between the turns causes current crowding at the inner side of the coil and increases ac resistance at higher frequencies. Figure 7 shows a three-turn coil prototype printed on the flexible polyimide substrate (150
μ m thickness) to allow the structure to conform to the shape of the eye. The prototype is not made of biocompatible material since it was used to verify the design concept. More appropriate materials for biomedical applications are parylene or polyurethane.
1) Current density distribution
4
In order to analyze proximity-effect and compare it with the skin-effect, a straight wire (single-turn coil) is twisted three times to form a three-turn coil (Fig. 7). This makes the DC resistance of the three-turn coil and single-turn coil equal. r
0 r
0 w
0 s
0 w
0 s
0 w
0 h r
0 w
0
Δ w s
0
+ Δ s w
0
Δ w s
0
+ Δ s w
0
Δ w h Fig. 7 Three-turn coil printed on flexible polyimide.
Fig. 6 Spiral coil topology with different w and s .
Fig. 8 Current density distribution in the coil conductors. Coil 1: w =0.1 mm, s =0.127mm, h =0.07mm, Coil 2: w =0.15 mm, s =0.06 mm, h =0.07mm, Coil 3: w =0.2 mm, s =0.035 mm, h =0.07mm.
As mentioned in §II.A, the assumption is that a single-turn coil with the large diameter generates the same current density distribution and resistance as the straight wire of the same length and cross sectional area. Then the single-turn coil can be used to determine the resistance due to the proximity and distinguish it from the total resistance. Figure 8 shows the current density distribution in the three-turn coil conductors for three different geometries. All coils have the same inner

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, AB-02 (DOUBLE-CLICK HERE) < 5 and outer radius of r
0
=11.5mm and r and h =0.07 mm is 142.85 A/m 2
N
=12.28 mm. The value of the direct current density for the conductor with w =0.1 mm
(estimate is based on 1 μ A current in the coil). At 10 MHz this value increases to about
333.6 A/m 2 at the inner edge of the coil. At the outer edge of the coil it increases to around 303.3 A/m 2 . As the conductor width increases and the spacing decreases the current density at the outer edge of the coil decreases more.
2) Resistance
Figure 9 shows the computed resistance ratio ( R / R dc
) when the conductor width varies from 0.1 mm to 0.25 mm for the single-turn coil, N =1 and three-turn coil, N =3. The outer and inner diameter of the three-turn coil is kept constant. Thus, the spacing between the turns varies from 0.3 down to 0.1 mm. As mentioned above the total length, volume and the DC resistance of the single-turn coil and multi-turn coil are the same. It can be observed that the increase in the resistance due to the wire winding increases as the width of the conductor increases and the space between the turns decreases. For the relatively large space between the turns, the ac resistance is mainly due to the skin effect and the contribution due to the proximity effect is negligible. The coils with w =0.1 mm and w =0.15 ( s =0.3 mm and s =0.233 mm) and N=3 have R / R dc
ratio almost equal to the ratio obtained from the single-turn coil
( N =1). This is due to the larger space between the turns.
In order to quantify the changes in resistance when the multi-turn coil is used the proximity factor is introduced by normalizing the resistance, R of the multi-turn coil with
3-turn respect to resistance of the single-turn coil, R
1-turn
.
Figure 10 shows the proximity factor, R
3-turn
/ R
1-turn
versus frequency for different conductor widths, w and turn-to-turn spaces, s ( w =0.1 s =0.3; w =0.15 s =0.233; w =0.2 s =0.166 and w =0.25 mm s =0.1mm). When the inner turn separation is comparable with or smaller than the conductor width the current crowding on the inner sides of the coil wires increases the total ac resistance at higher frequencies (above 1 MHz).
For example, the increase in resistance of 3-turn coil is below
5% for the conductor, w =0.15 mm and s =0.233 at all frequencies up to 10 MHz. The resistance for the coil with smaller spacing between the turns, s =0.1 and conductor width, w =0.25 increases for up to 15 %. This means that a coil with smaller w to s ratio (fill-factor) is a better design solution to keep the resistance contribution due to the proximity low at higher frequencies. The simulation results in Fig. 9 show that at frequencies below 1 MHz, proximity-effect has small impact on the total resistance. The increase in total resistance is less than 10 % of the DC resistance. As the frequency increases the proximity resistance contribution becomes greater than skin effect resistance contribution. As shown in
[2], the proximity related resistance is proportional to the frequency squared while the skin effect related resistance is proportional to the square root of frequency. However, the proximity resistance follows a square of frequency relationship to the limiting frequency [6]. After this point this relationship no longer holds.
As it can be seen from Fig. 10 the proximity factors of the coils with wider conductor reach the peak values at the lower frequencies. For example, the proximity factor of the coil with w =0.25 mm has the peak value at 2.5 MHz, while the coil with w =0.15 mm reaches its peak value at 3.5 MHz.
3 w=0.15,   N=3 w=0.15,   N=1
2.5
2 w=0.2,   N=3 w=0.2,   N=1 w=0.25,   N=3
1.5
1
0 1 2 3 4 5 6 7 8 9 10
Frequency,   MHz
Fig. 9 Resistance divided by dc value versus frequency for different cases of width w and s . In all cases, the outer and inner radius is set to
11.5 mm and 12.8 mm respectively.
The proximity factor decreases after the limiting frequency since the proximity resistance becomes almost proportional to the frequency (Fig. 9) and the skin resistance dependence on frequency remains the same through the whole frequency range. The proximity factor of the coils with smaller width and larger inner turn spacing is relatively low. For example, the proximity effect contribution to the ac resistance of the coil with w =0.1 mm and s =0.3 mm is less than 2 % for the frequencies up to 10 MHz (Fig. 10). The resistance plots of coils with smaller conductors (widths) and larger spacing show this proportionality up to the higher frequencies than the coils with larger conductors. Figure 10 also shows that the high fill-factor coils (N=3) have higher quality factor for the frequencies up to 10 MHz. This is due to the similar values of the inductances for the 3-turn coils so that the quality factor is only influenced by the resistance, R .
Figure 11 shows how the fill-factor influences the R / R dc dc ratio. As the frequency and the fill-factor decrease R / R becomes close to unity. As the frequency increases even the coils with small fill-factor have the increased ac resistance due to the increased skin-effect at higher frequencies.
3) Measurements
To verify the accuracy of numerical simulations, two PSCs for the closed-loop retinal prostheses [2, 8] with different number of turns, conductor widths and spacing were fabricated and tested (see Fig. 12, Table I). The inner and outer diameters of the coils are about 23 mm and 35 mm, respectively. The conductive trace consists of 35 μ m thick Cu plated on one side with thin Ni and Au layers. However, due to difficulty in maintaining tight manufacturing tolerances, the actual thickness obtained is usually less than the specified value.
We employed flexible printed circuit technology which enables multiple isolated layers of conductive traces to be sandwiched in between polyimide layers and connected through vias through the polyimide. This allows a compact and thin PSC consisting of the spiral coil and return trace to be fabricated [8].

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1.5
1.4
1.3
1.2
1.1
1 w=0.25
w=0.2
w=0.15
w=0.1
mm mm mm mm
Fig. 10 Proximity factor ( R
3-turn
/ R
1-turn
) versus frequency.
3
0 1 2 3 4 5 6 7 8 9 10
Frequency,   MHz
2   MHz 4   MHz
60
50
40
30
20
10
0
6 measured re f , was fitted sistance whi h is a wi
str n of frequency
1 ⁄ (2) where α , β and γ are fitting coefficients. R dc
is the resistance extrapolated to 0 Hz (DC). The first term of this equation is self explanatory. The second term is contribution due to skin effect while the last two terms are contribution due to proximity effects. It is noted that (2) can be simplified by
0 that leaves us with the equating the coefficients same approximation for effective resistance as suggested by
Kuhn [6]. Using (2), we found reasonable good fitting with measurement data. The measured and fitted resistance data are compared to simulated values as shown in Fig. 13 and 14.
2.5
6   MHz
10   MHz
8   MHz
2
1.5
1
0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5
w/s
Fig. 11 Resistance vs w / s ratio (fill-factor).
Turns
TABLE I
PSC PARAMETERS
Diameter [mm]
Width
Inner Outer [mm]
Spacing
[mm]
Fill- factor
10 mm
Impedance measurement was performed using a vector network analyzer (HP E5071C) as described in [10]. Before measurements were made, calibration of the setup is necessary by measuring a short circuit, open circuit and known impedance standard. This was done for a frequency range between 9 kHz and 100 MHz. Calibration removes the effect of cable and connectors from the measured impedance giving true coil only measurement. To determine the coil impedance and resistance S -parameters were measured using the network analyzer and then converted to Z -parameters (see Appendix).
The RLC lumped parameter circuit shown in Fig. 4 is used to extract the resistance from the measured impedance data by following the procedure described in [7] and in §II of this paper. From the simulations, we verified that the polyimide substrate does not vary the impedance value of the PSC for the frequencies below 10 MHz. As we are interested in frequencies of 10 MHz and below, we can safely ignore substrate losses and use this simplified model.
The measurement data is listed in Table II. They are found by following the procedure outline in the Appendix. The
10 mm
Fig. 12 Photographs of fabricated 7-turn (above) and 17-turn coil
(below).
As it can be observed in Fig. 13 the discrepancies between the measured and simulated data for 17-turn coil are small.
However, the measured resistance values for the 7-turn coil
(Fig. 14) are higher than the simulated resistance values at frequencies above 3 MHz. This may be due to the calibration errors which affect the small resistance of 7-turn coil more than 17-turn coil. It is observed that the resistance of PSC with
7 turns exhibits a more linear relationship with frequency, compared to the 17 turn PSC.

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TABLE II
PSC MEASUREMENT
100
Measurement 7-turn 17-turn
80
L [ μ H] 2.502
C [pF] 3.949 f
0
R dc
[ Ω ] 1.118
R
0
[ Ω ] 27.39
[MHz] 50.6
Q @ 5MHz
Q @ 10 MHz
Q max f @ Q max [MHz]
54.3
83.0
89.1
14.3
15.14
4.198
4.889
66.61
20.0
66.5
58.8
68.8
6.27
60
40
20
17 ‐ turn
7 ‐ turn
20
15
10
5
0
17 ‐ turn   (measured)
17 ‐ turn   (simulation)
17 ‐ turn   (Fit)
0 1 2 3 4 5 6 7 8 9 10
Frequency,   MHz
Fig 13 Measured and simulated resistance for 17-turn (above).
3
2.5
2
1.5
1
0.5
7 ‐ turn   (measured)
7 ‐ turn   (simulation)
7 ‐ turn   (Fit)
7
0
0 5 10 15
Frequency,   MHz
20 25 30
Fig 15 Q for 7-turn and 17-turn PSC.
Since the quality factor is defined for resonant frequencies
[11], the plot of Fig. 15 is only valid if the PSCs are operating in resonance at these frequencies. It can be observed that at resonant frequencies below 7 MHz quality factor, Q of 17-turn PSC is higher than Q of 7-turn PSC.
For example, at 5 MHz Q of 17-turn coil is 66.5 and Q of 7turn coil is 54.3. This is due to the higher inductance of the
17-turn coil. However, as the frequency increases the proximity effect becomes more dominant and the resistance of 17-turn coil which has higher fill-factor increases faster than the resistance of 7-turn coil (Fig. 13 and 14). This trend of resistance (proximity factor) increase with fillfactor is observed from the simulations shown in Fig. 10 and 11. Beyond 7 MHz, the quality factor of the 17-turn coil falls below that of the 7-turn coil and at 10 MHz the quality factor values for 17-turn and 7-turn PSC are 58.8 and 83, respectively. The quality factor for the 17-turn coil reaches a peak of 68.8 at 6.27 MHz while Q of the 7-turn coil continue to rise to a maximum value of 89.1 at 14.3
MHz. This maximum quality factor increase of 29% for 7turn coil suggests that a lower fill-factor may be suitable for optimal design of PSCs at higher operating (resonance) frequencies.
0
0 1 2 3 4 5 6 7 8 9 10
Frequency,   MHz
Fig 14 Measured and simulated resistance for 7-turn (below).
This can be attributed to the lower fill-factor of the 7-turn PSC which in turn reduces proximity effect. The maximum quality factor for the 7-turn PSC is found to be higher compared to the
17-turn PSC.
Figure 15 plots the quality factor of 7-turn and 17- turn unloaded PSCs which is based on a simple expression given in
[7]
, (3) where L is the induc tance of t he coil at low frequencies and
R ( f ) is frequency d epend measured impedance. ent resistance extracted from the
III.
C ONCLUSION
This paper presents a study of the frequency related effects in inductive coils for implantable electronics. The main focus is the effect of current crowding (proximity effects) and the related increase of coil resistance. We demonstrate how the proximity effects and the related resistance can be analyzed with electromagnetic simulations based on the FE method.
The simulations results then can be used to aid the design of practical printed spiral coils. Smaller coils with low dc resistance can be achieved with larger conductor width and smaller spacing between the turns of the coil. As observed in the simulation results, small spacing between the coil turns increases the total resistance at higher frequencies due to the current crowding effects. Therefore lower w to s ratio can provide a better printed coil design at higher operating
(resonance) frequencies.
The proper and efficient use of the FEM simulations makes it possible to compute and quantify frequency related effects and therefore it can aid the design process of finding a better

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IV.
A CKNOWLEDGMENT
The authors would like to thank Shun Bai for his assistance with the impedance measurement of the coils. We would also like to thank Thomas Benke of LEAP Australia Pty Ltd for his help with the simulations.
NICTA is funded by the Australian Government as represented by the Department of Broadband, Communications and the
Digital Economy and the Australian Research Council through the ICT Centre of Excellence program.
8
Im
1
1⁄
⁄ where ω =2 π f . It is observed that Re( Z ) approximates to R dc when frequency f tends to 0. Likewise, Im( Z ) approximates to
and 1⁄ as f tends to 0 and ∞ , respectively. These observations provide a good basis for estimating the values for
R dc
, L and C of the PSCs.
Due to the presence of L and C , the circuit will resonate at resonance frequency f
0
which can be found by equating the imaginary component of the impedance in (A2) to zero. This is found to be
1
(A3)
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[8] D. Ng, S. Bai, J. Yang, N. Tran and E. Skafidas,”Wireless technologies for closed-loop retinal prostheses,” Journal of Neural Engineering 6, pp. 1-10, 2009.
[9] H. A. Wheeler, “Formulas for the skin effect,” Proc. IRE , vol. 30, no. 9, pp. 412-424, Sep. 1942.
[10] R. Rodriguez, J.M. Dishman, F.T. Dickens and E.W. Whelan,
“Modeling of Two-Dimensional Spiral Inductors,” IEEE Transactions on Components, Hybrids and Manufacturing Technology , Vol. CHMT-
3, No. 4, December 1980, pp. 535-541
[11] D. M. Pozar, “Microwave Engineering,” Addison Wesley 1993. where R
·
1
. This approximation is used to verify the value of C when the inductance is known. In practice, it is difficult to make measurements at dc and extremely high frequencies. We found that measurement made at frequencies much smaller or larger than the resonant frequenc y g ives a good approximation for R , L and C . This can be sh own by rearranging Re( Z ) in (A2) as follows
Re
0
is the resistance at resonance. Usually, the second term is much smaller than the first term. Hence, (A3) can be approximated as 1 √
1
1
(A4) where is the resonance frequency in rad/s. When
⁄ and , Re . Similarly,
Im
1
·
1 .
(A5 )
A PPENDIX
The measured S -parameters can be transformed into impedance Z using the follow ing equ ations. Note that both S and Z are complex num bers. where Z
0
1
1
(A1)
is an arbitrary characteristic resistance value used during the measurement. A value of 50 Ω is often used for Z
0
.
Based on the induc tor model shown in Fig. 4, the impedanc e of the circuit an e derived to be
Re
1
1⁄
(A2)
Hence, when or ∞ , Im
or 0 , Im
1⁄ .
and when
In order to obtain R ( f ) from measured Re( Z ), we remove the effect of C from the measured data. This is done by considering a new circuit model without the parallel connected capacitor C . We let the new circuit impedance to be Z
1
. The real component of the impedance of the new circuit can then be defined from the impedance of the original circuit. This real comp t R ( f ). Hence,
Re
Re Re Im
(A5)
Re Im Re Im
The quality factor giv en by (3) is often approximated by
Im
Re
(A6)
Although (A6) is a convenient method to calculate Q when impedance values are immediately known from measurement,

> FOR CONFERENCE-RELATED PAPERS, REPLACE THIS LINE WITH YOUR SESSION NUMBER, E.G.
, AB-02 (DOUBLE-CLICK HERE) < it should be noted that it gives a good estimate of Q only when
. This is numerically verified when Q is plot using (A6) compared to (3). Q obtained from (A6) gives an unrealistic value of 0 at f
0
, whereas Q from (3) never falls below 0 for all frequencies. Hence, the use of (A6) to estimate Q at low frequencies is justified whilst a more rigorous method using
(3) is required when the frequency of operation approaches higher values.
9