Design and optimization of printed spiral coils for wireless passive sensors

National Science Foundation, Grant/Award Number: 1637250 Abstract Monitoring physiological signals during regular life might provide many benefits including early detection of abnormalities and tracking the severities of diseases. A wireless connection between the passive sensor and the scanner eliminates the obtrusive wires, resolves battery‐related issues, and makes it easy‐to‐use. We have previously proposed a wireless resistive analogue passive sensor technique that operates with the help of inductive coupling. The variation of resistive physiological transducer (secondary side) leads to amplitude modulation on the scanner coil (primary side). The design of printed spiral coil (PSC) on printed circuit board, significantly affects the performance of the overall system in terms of sensitivity, the output voltage change as a reflection of the transducer change. To optimize the PSC's profile and maximize the sensitivity, we employ three methods: iterative, analytical, and genetic algorithm (GA). The GA optimized PSCs, as the best result, have been fabricated and the measurement showed a sensitivity of 0.72 mƱ which has 5% (8.8%) deviation from the simulation (theoretical) results. This method can be utilized to design a PSC pair in near‐field applications to transfer amplitude modulation with various sizes and fabrication constraints.


| INTRODUCTION
Wearables can provide continuous long-term monitoring of physiological parameters. It allows the detection of abnormality and the variation of symptoms severity at the natural life setting instead of confined clinical setup. This is feasible due to advancements that produce lightweight and low-power wearable medical devices and sensors. Fully passive (battery-less) sensors are a promising technology for this case. The optimization of a fully passive wireless sensor is essential, which is addressed here as a progress of our previous research [1].
Bluetooth and WiFi are two common wireless techniques that have been dominantly used in the medical wearable sensors researches and applications [2,3]. The battery requirement, power consumption, hardware complexity, and cost are the major barriers of these active wireless sensors that led to exclude these methods from the scope of this study. With regard to fully passive sensors, radio frequency identifier (RFID) as another wireless technique is also utilized in the wearable system researches and applications [4]. The active chip on the RFID tag requires a power source that is supplied by a wireless signal, an embedded battery, or power harvesting. Such existing passive technologies can share the findings from this study.
We have previously reported [1] a novel wireless resistive analogue passive (WRAP) sensor that a pair of planar spiral coils (PSCs) provide an inductive coupling (magnetic connection) at 13.56 MHz between the scanner (reader) and the sensor. In this approach, the variation of a resistive transducer causes the variation of inducting coupling (Q) to the scanner (primary side). The overall system performance is measured by sensitivity of the link, i.e., the primary voltage changes in response to the transducer variations. A wireless capacitive analogue passive (WCAP) transducer that responds to the body signal employs the same idea of magnetic inductive connection between a fully passive sensor and a scanner circuit [5]. The transducer variation in response to the bio-signal modulates the resonance frequency of the LC resonator on the sensor side. In WCAP method, the transducer capacitance range affects the coil self-inductance optimization, and as a result, the coil design, overall sensitivity, and resonance frequency cannot be independently designed. On the other hand, in the WRAP [1], the coil characteristics can be optimized for desired resonance frequency and maximum sensitivity independently from the transducer. In addition, a WCAP is more vulnerable to the capacitance disturbances due to body vicinity that affects the probed signal and declines the signal-to-noise ratio (SNR) [6]. Moreover, the WCAP frequency modulation (FM) technique requires a larger bandwidth and complex FM demodulator in the scanner compared to WRAP. Furthermore, the availability of capacitive transducers for different physical signals limits their application further. Towe [7] and Schwerdt [8] reported probing bio-potentials by a varactor transducer through the radio frequency backscattered signal. The major drawback associated with this method in addition to the discussed concerns, is their limited application to the biopotentials.
In contrast, WRAP sensor method uses resistive variations for both physical signals and bio-potentials, with the help of a resistive transducer or a transadmittance (e.g. MOSFET) device, respectively [9]. Here we attempt to maximize the sensitivity, namely the primary's modulated amplitude in response to a unit change in the transducer resistance. To maximize the sensitivity, a pair of PSCs are optimized by considering the size and fabrication constraints. To the best of our knowledge, existing literature on the inductive wireless connection mainly focuses on the magnetically coupled resonant wireless power transfer in both industrial and medical fields [10].
Near-field antenna design has been also studied in the RFID and near-field communication (NFC) researches. However, their concern about impedance matching, power transfer, data transmission rate, and metal proximity is addressed by component adjustment, antenna shape, and shielding with no discussion on the antenna detailed design [11]. Previously, the measurement showed that a maximum sensitivity can be achieved even by taking the power transfer efficiency as the objective function [1]. The theoretical result of genetic algorithm (GA) showed a better performance in comparison with the four other deterministic optimization methods sensitivity objective function [12].
In this paper, The optimization results of GA, iterative, and analytical methods are compared, and it is shown that the results can be improved by optimizing the individual components prior to the coil optimization. The GA result showed the highest sensitivity and due to its fast and less complex algorithm, its result has been considered as the optimum PSC pair and has been fabricated in three samples. The experimental sensitivity results show less than 9% and 5% disagreement with analytical and simulation results, respectively.
The remainder of this article is organized as follows. In Sections 2 and 3, the hardware concept, schematic model, and the analytical equations are explained. Section 3 describes the different methods for coil optimization and the results are presented in Sections 4 and 5. Section 6 concludes the article. The susceptibility to the capacitors and coil profile tolerances are discussed in supplementary documents. Figure 1 shows the concept of a WRAP system. It consists of primary and secondary circuits that are also known as scanner and sensor, respectively. The resistive transducer variation in the passive secondary side modulates the primary coil voltage through the magnetic field between a pair of PSCs. A sinusoidal AC supply drives the primary circuit at 13.56 MHz within the industry, scientific, and medical (ISM) frequency band. Here, 'sensitivity' is defined as the normalized primary voltage (V OUT /V OSC in Figure 1) change to a unit change of transducer resistance as follows:

| Circuit schematic
The matching/resonator block in Figure 1 consists of capacitors to tune both primary and secondary resonance frequencies on 13.56 MHz as well as matching the impedance between the primary coil and the AC supply. Figure 2 shows the complete circuit schematic where R P (R S ), C P (C S ), and L P (L S ) are the primary (secondary) PSC equivalent components, C tp (C ts ) is the tuning capacitor for primary (secondary) resonator, R in is the AC supply internal resistor, and C in is the matching impedance. This configuration in which the primary and secondary tuning capacitors, C tp and C ts (Figure 2), are in parallel with the primary and secondary coils, is called parallelparallel (PP). We have shown that PP arrangement as compared with three other possible arrangements (SP, SS, and PS) shows the best performance (highest sensitivity and lowest susceptibility to tolerance) [13]. The sensitivity in Equation (1) depends on the component values and the mutual inductance between the primary and the secondary coils, M, that is defined by the primary and secondary self-inductances (L P and L S ) and the coupling coefficient in Equation (2).
Coupling coefficient (k) depends on the coil physical relative position, size, physical shape, and environment objects. The coupling factor is assumed as a constant parameter in this study and 0.08 is the best experimental value for the setting of this work. Therefore, the sensitivity becomes just a function of the circuit components, as shown in Figure 2

| Equations
There are two groups of equations: the circuit analytical equations and the PSC's physical equations. The analytical equations determine the sensitivity as a function of the electrical components shown in Figure 2. The PSC's physical equations describe the relation between the PSC equivalent electrical components and its physical profile.
According to Figure 3, the analytical equations are defined in Equations (3)-(8) to calculate the normalized primary voltage (output voltage) derivative versus transducer resistance. Figure 4 shows the PSC physical characteristics and equivalent circuit. Coil equivalent series resistor is defined in Equations (9)-(12) [14].
where t is conductor's thickness and according to the printed circuit board (PCB) fabricator it is 0.96 mm, ρ is conductor's resistivity (1.68�10 -8 Ω·m for copper), w is conductor's width (as shown in Figure 4), n is number of turns in the coil, µ is the magnetic permeability of conductor (1.26 � 10 −6 H/m for copper), δ and L C are the skin depth and conductor length, respectively, and are defined in Equations (10) and (11).
Two adjacent tracks form a capacitor that is defined in Equation (12) [14]. Its dielectric splits between the board coating (air) and the PCB substrate (FR4) with the relative dielectric constant ε rc =1 and ε rs ¼ 4:4, respectively.
Since the PSC self-inductance has a critical influence on the sensitivity, the comparison of the calculation results between the different methods: Current Sheet [15], Modified Wheeler [15], Modified Grover [16], and Monomial Fit [16] showed that the Current Sheet has the minimum deviation from the experimental results. Therefore, the Current Sheet with Equation (14) has been adopted for the PSC self-inductance calculation [15].
The concept of sensitivity and (R) ΔV Out in response to ΔR T F I G U R E 2 Circuit schematic of wireless resistive analogue passive system. C tp and C ts are primary and secondary trimmer capacitors, respectively, and k is the coupling coefficient F I G U R E 3 Equivalent circuit of the schematic shown in Figure 2. Here: C 1 = C tp +C p and C 2 = C ts +C s , V P = V Out , and R T = R Transducer F I G U R E 4 (L) Printed spiral coil physical characteristics and (R) the equivalent circuit NOROOZI AND MORSHED where µ 0 is the air magnetic permeability and φ is called fillratio: If s < 3w, the maximum error for Equation (14) is less than 8% [15].

| METHODS
The sensitivity in Equation (8) is a multivariable function including PSC physical specifications ( Figure 4): number of turns (n), conductor width (w), space between two tracks (s), PSC's outer and inner sizes or diameters (d O and d i ) for both primary and secondary as well as circuit components: C tp , C ts , and C in . A sensor with 1 KΩ resistance is considered as the target transducer. The objective function in Equation (8) along with Equations (9), (12), and (14) form a complex multivariable function.
By choosing the secondary coil size as small as 20 mm for ease-of-use of the sensors and users' comfort, the variables reduce to 10: d O1 , n 1 , s 1 , w 1 , n 2 , s 2 , w 2 , C tp , C ts , C in . The coil's internal size is defined from the conductor's width, the gap between two conductors, number of turns, and the PSC's external size as follows: To make the optimization faster and more accurate, we break the optimization process into two steps. In the first step, the best components including primary/secondary self-inductances and the capacitors are specified. In the second step, the coil profiles which generate the best match to the step 1 self-inductance results are determined.

| Electrical components optimization
According to Figures 2 and 3, the tuning and the coil intrinsic capacitors form two parallel capacitors in the primary and secondary are denoted as C 1 and C 2 . For any L P and L S pair, a unique combination of C 1 , C 2 , and C in can be found to maximize the sensitivity. Table A1 shows the maximum sensitivity and the optimum component for two self-inductance ranges. For each pair of (L P , L S ) within the range, the circuit components are optimized for the maximum sensitivity by employing the GA technique (for more details, see the supplementary documents).

| Optimization of PSC structure
In the second step, we maximize the sensitivity in Equation (8) for the PSC primary and secondary profiles, (d O1 , n 1 , s 1 , w 1 , n 2 , s 2 , w 2 ) using the results of the first step in Table A1 . Through the first step, the components for all possible (L P , L S ) pairs within the range, are optimized and the results are utilized in the second step, coil profile optimization. We use three approaches to find the optimum coil's profile: GA, iterative, and analytic. GA as a fast and appropriate method for a multivariable and complex function does not provide any circuitry and (fabrication) tolerances insight. Analytic approach for the circuitry analysis and the iterative approach as the method that we have used before are also utilized for comparison [1]. In all the following methods, the condition in Equation (17) is satisfied to keep both coil's selfinductance and quality factor at the maximum level, simultaneously [17].
where j=1 for primary and j=2 for secondary coil.

| Iterative optimization method
We had adopted an iterative method for coil's profile optimization previously [1], and we use it here with some modifications as shown in Figure 5. In this method, each step maximizes the sensitivity objective function (8) for two variables out of seven within the defined range. The iteration repeats if the objective function growth is good enough or greater than a specified value (known as error). Figure 6 shows the flowchart of analytical method. This method gives an analytic insight to the coil design. In the other two optimization methods, the best primary and secondary coil profiles are found without tolerance analysis. However, the analytical method starts with optimum self-inductances in Table A1and finds the coil profiles that generate the optimum self-inductances with minimum susceptibility to the tolerances. The optimum L P and L S in Table A1are not precisely feasible due to the discrete nature of the coil physical elements: n, s, and w. In addition, a PSC with a specific self-inductance does not have a unique profile (d O , n, s, w). Moreover, the optimum design is the one that has the minimum change due to the fabrication tolerances: Δs ¼ ±1 mil and Δw ¼ ±0:5 mil, according to the PCB fabrication service (Oshpark LLC). Hence, the best profiles are those that make the closest values to the optimized L P and L S and show the minimum susceptibility to the fabrication tolerance. According to Figure 6, Steps 2 and 3 indicate excluding the coil profiles whose selfinductances do not meet the minimum error to the target self-inductance. Steps 4 and 5 in Figure 6 rank the effect of fabrication tolerances on the sensitivity and adopt the profiles with the minimum vulnerability to the tolerances.

| GA optimization method
A typical classic optimization algorithm generates a single point and through an iterative deterministic computation tries to approach to the optimum point. In contrast, GA generates a broad random population and by using different methods of mutation, crossover, scaling, and selection, finds the next generation to converge to the optimum point. In addition, GA is fast and appropriate method for a discrete function with no need for straightforward derivative. GA is described with the flowchart in Figure 7. In the coil optimization problem, all the variables are specified as discrete with the lower and upper bounds specified in Table A2 . The nonlinear constraints in Equation (17) define a minimum limit for the primary and secondary internal coil sizes. The important GA setting in the coil optimization problem is shown in Table A3. The parameters are set based on GA average performance over multiple runs with the selected parameters.

| THEORETICAL RESULTS
This section presents the theoretical results of the optimization methods described in Section 3.2. The optimum capacitors for range (L P : 3-5 µH and L S : 3-6 µH) in Table A1 , are applied for the three optimization methods. For all calculations the coupling factor (k), transducer resistance, and the secondary outer size are considered as 0.08, 1 kΩ, and 20 mm, respectively. For the analytical method, due to the discrete nature of the coil parameters, the minimum errors of feasible L P and L S comparing to the optimum values in Table A1 are 0.03% and 1.08%, respectively (Step 3 in Figure 6). The possible primary and secondary PSC profiles are shown in Table (A4) . Table shows the optimum sensitivity for all combination of nine primary profiles with one secondary profile in Table A4. According to  Table A5 , design 6 shows the highest sensitivity and then, it is considered as the analytical optimum design.
The optimum results and the maximum influence of tolerance are shown in Table A6 for all three methods. The PSC physical specifications affect the coupling factor. However, if we ignore the effect of coil parameters on the coupling factor, the GA results with the highest sensitivity are selected for the rest of this article (Table A6).

| Measuring the coupling factor (k)
The PCB shown in Figure 8 has been designed with KiCad based on the GA results in Table A6 and was fabricated in three samples. Since the optimization has been performed for k = 0.08, we must adjust the coupling factor on this value before any experimental measurement. Distance between the primary and secondary coils is the only adjustable parameter to tune the coupling factor.
One way to estimate the coupling factor is measuring the mutual inductance (M) and calculating the coupling factor by using Equation (2). Mutual inductance (M) can be calculated by Equation (18) and measuring the secondary coil open circuit voltage.
Accessing to the primary current (I Primary ) and zeroing the secondary current are not feasible due to the primary and secondary coil intrinsic capacitors (C P and C S in Figure 2). The effect of intrinsic capacitors on the primary and secondary current decreases by decreasing the working frequency but it also attenuates the secondary voltage which makes it immeasurable due to a poor SNR. Thus, we adopted a practical method for measuring the mutual inductance [18]. In this method, an LCR analyzer (Agilent 4294A, 40 Hz-110 MHz) measures the equivalent selfinductance of aiding and opposing connection of primary and secondary coils and the mutual inductance (M) is calculated by: where L aid and L opp are the measured self-inductances in aiding and opposing connection of two coils, respectively. Figure 9 shows the measurements results by using Equation (19) and the measured self-inductances for three pairs of PSCs. As mentioned on Figure 9, all three PSC pairs are in good agreement for k = 0.08 ± 0.003 at 16-mm distance. The result is in complete agreement with the simulation results with FEA tool, COMSOL (COMSOL Inc., Burlington, MA) [19]. To the best knowledge of the authors, there is no single accurate expression in the literature to calculate the coupling factor between two rectangular PSCs. We have used the equation that proposed by Grover [20], generalized by Babic et al. [21], and modified by [22] which have been adopted by some researches [23]. However, the result is not consistent with Figure 9 which is also verified by the circuitry simulation (of secondary voltage). Therefore, according to experimental results, we adjusted the distance between primary and secondary PSCs on 16 mm to obtain k = 0.08. Generally, coupling factor of PSCs is smaller than the traditional coils. While k is less than 0.1 at few millimetres distance in PSCs [24], it can reach to tenths at the centimetres distance range in the traditional wired coil [25].

| Measuring the sensitivity
Since the fabricated PSCs do not show the exact calculated equivalent components (Table A6 ), we must find the new optimum C 1 , C 2 , and C in according to the fabricated PSC equivalent circuit. The measurement results show the equivalent coil series resistor does not match with Equation (9). Several reasons can lead to this discrepancy, including fabrication materials (ρ), w, s, t (conductor thickness) tolerances, and inaccuracy in Equation (9). However, the optimization theoretical results with measured resistors are in complete agreement with the measured and simulated sensitivity. Table A7 shows the measured components, new optimum capacitors, and the maximum sensitivity by the GA method. The sensitivity decline and the major changes in the optimum capacitors are thoroughly due to the coils' series resistors. Figure 10 shows the experimental setup. The voltage across the primary coil is measured by an oscilloscope (Agilent, Model DSO-X 2024A, Probe Tektronix TPP0200) with the equivalent R OSC = 10 MΩ and C OSC = 11.5 pF. Figure 11 shows the schematic of the circuit under the test. The optimum capacitors (see Table A7) fine-tuned by a series trimmer with signal generator and parallel trimmers with primary and secondary coils. We measured the primary voltage for six transducer resistances from 935 to 1235 Ω and calculated the sensitivity by using Equation (20) and the results are shown in Figure 12.
R i and R j are two consecutive resistors out of six resistors.
It is worth to mention that the sensitivity in Table A7 is the normalized sensitivity in Equation (8) that shows the primary voltage change due to 1 Ω transducer change for 1 V input voltage (V OSC ). Thus, according to Figure 11 with 10 V input voltage, the sensitivity in Table A7 is comparable with Figure 12 by considering a factor of 10.

| Simulation and theoretical results
The sensitivity is measured by varying the transducer resistance around its nominal value, that is, 1 KΩ, and substitute its F I G U R E 7 Genetic algorithm flowchart 174 -NOROOZI AND MORSHED associate voltage in Equation (20). The final circuit shown in Figure 11, has been used for simulation, measurement, and theoretical calculation. LTspice (Linear Technology, Milpitas, CA) was employed to simulate the voltage change in response to the transducer resistance variation. A MATALB code developed to compute Equations (2)-(8) as the theoretical sensitivity for different transducer resistances. According to Table A7, the expected sensitivity for 1 KΩ transducer resistance, is 7.9 mƱ (the average sensitivity for three fabricated boards) which is in good match with simulation and measurement values. Figure 13 shows the average results of measurement, simulation, and theoretical for three PSC pairs. As this figure shows, the simulation, calculation, and measurement results are in good agreement with the maximum 8.8% difference between the calculation and measurement at R S = 1 kΩ. According to Figure 11, the sensitivity has the descending trend which is reasonable.

| DISCUSSION AND CONCLUSION
Previously, we had designed an optimum PSC pair to maximize the sensitivity by adopting and applying an iterative method for maximum power transfer. First we defined the sensitivity as a function of circuit components and the coil profiles. Then we used two more methods, GA and analytical, in addition to the iterative method, to find the optimum coil profiles at 1 kΩ transducer resistance. For coil profile optimization, we first found the best impedance matching (C in ) and tuning capacitors (C 1 and C 2 ) for the defined selfinductance ranges. Although the maximum sensitivity resulted from all three methods are in the same range, since the GA is faster, more reliable, easier to apply, and has the highest sensitivity, its result was adopted for fabrication. theoretical results, respectively. More discussion on coil selfinductance range and the susceptibility to the components' tolerances is provided in the supplementary document. In this study, the coupling factor has been taken as a constant value, independent of coil profiles, whereas it is affected by the PSC physical characteristics. The effect of coil profiles on the coupling factor should be introduced in the optimization process that requires further study. In addition, to design a practical wearable sensor based on the optimized PSC pair, the effect of the primary and secondary movements (misalignment) that affects the coupling factor should be also considered in the PSC design. The effects of coil profiles and coil misalignment on the coupling factor are two parameters that should be studied in the coil design and optimization process in future work.

ACKNOWLEDGEMENT
This material is based on work supported by the National Science Foundation under Grant No. 1637250.

APPENDIX TA B L E (A1)
Maximum sensitivity and optimum capacitors for two ranges of primary and secondary self-inductances (R P = 1.1 Ω, R S = 5Ω, R T = 1 KΩ, k = 0.08) Note. The appendix provides more information from our analysis for interested readers.

NOROOZI AND MORSHED
-177 TA B L E A5 The analytical resultant sensitivity optimization for the PSC primary and secondary profiles in Table A4. Each design shows the feasible L P and L S with minimum error to the optimum values in