Planar Elliptical Inductor Design for Wireless Implantable Medical Devices

Wireless implantable medical devices (WIMDs) have seen unprecedented progress in the past three decades. WIMDs help clinicians in better-understanding diseases and enhance medical treatment by allowing for remote data collection and delivering tailored patient care. The wireless connectivity range between the external reader and the implanted device is considered one of the key design parameters in WIMD technology. One of the common modes of communication in battery-free WIMDs is inductive coupling, where the power and data between the reader and the implanted device are transmitted via magnetically coupled inductors. The design and shape of these inductors depend on the requirements of the application. Several studies have reported models of standard planar inductors such as circular, square, hexagonal, and octagonal in medical applications. However, for applications, constrained by narrow implantable locations, elliptical planar inductors may perform better than standard-shaped planar inductors. The aim of this study is to develop a numerical model for elliptical inductors. This model allows for the calculation of the inductance of the elliptical planar inductor and its parasitic components, which are key design parameters for the development of WIMDs powered by inductive coupling. An area transformation technique is used to transform and derive elliptical inductor formulas from standard circular inductor formulas. The proposed model is validated for various combinations of the number of turns, trace width, trace separation, and different inner and outer diameters of the elliptical planar inductor. For a thorough experimental validation of the proposed numerical model, more than 75 elliptical planar inductors were fabricated, measured, and compared with the numerical output of the proposed model. The mean error between the measured inductor parameters and numerical estimates using the proposed model is <5%, with a standard deviation of <3.18%. The proposed model provides an accurate analytical method for estimating and optimizing elliptical planar inductor parameters using a combination of current sheet expression and area transformation techniques. An elliptical planar inductor integrated with a sensing element can be used as a wireless implant to monitor the physiological signal from narrow implantation sites.


Introduction
Wireless implantable medical devices (WIMDs) have gained significant attention due to their suitability for home monitoring and diagnostic surveillance of various devices, including pacemakers, cardiac defibrillators, insulin pumps, and neurostimulators [1,2]. These devices can also enable the monitoring of bone repair and joint stress, which are typically evaluated by modeling [3]. WIMDs have enabled remote patient monitoring and the delivery of personalized care [4,5]. The wireless linkage distance between the WIMD and the external reader device poses a significant challenge while designing and theorem to transform the design parameters, such as inner diameters (minor and major) and outer diameters (minor and major) of the elliptical planar inductor into the inner and outer diameters of a circular planar inductor. The proposed model was numerically evaluated using MATLAB and was tested for more than 75 inductors while varying the trace width, trace separation, number turn, and major-to-minor axis ratio. In this study, more than 75 elliptical planar inductors were designed, fabricated, and measured. The proposed model was also experimentally evaluated using fabricated elliptical inductors with different design parameters. The results of the proposed model were compared and validated with the experimental results. The error was found to be less than 5% with a standard deviation of 3.18%, which is comparable with the existing literature.

Materials and Methods
The elliptical planar inductor shape is novel, and there is no existing expression to calculate the inductance and other parasitic parameters. An ellipse can be represented using an equivalent circular shape using the simple area transformation mathematical theorem. Therefore, using this theorem, all the design parameters (inner and outer major and minor diameters) of the elliptical planar inductor can be transformed into the design parameters (inner and outer diameters) of a circular planar inductor. Figure 1 represents the overall research approach, where it can be seen that the proposed numerical model is validated against the experimental results.
This study aims to develop a numerical model of elliptical planar inductors that is computationally simple and provides inductor parameters that can be verified experimentally. The proposed model will allow for the computation of key inductor parameters such as inductance and other parasitic components. Elliptical planar inductors can be useful in various medical implantable applications where symmetrical shapes cannot be used due to narrow implantation site constraints. This has been achieved using the existing geometrical theorem to transform the design parameters, such as inner diameters (minor and major) and outer diameters (minor and major) of the elliptical planar inductor into the inner and outer diameters of a circular planar inductor. The proposed model was numerically evaluated using MATLAB and was tested for more than 75 inductors while varying the trace width, trace separation, number turn, and major-to-minor axis ratio. In this study, more than 75 elliptical planar inductors were designed, fabricated, and measured. The proposed model was also experimentally evaluated using fabricated elliptical inductors with different design parameters. The results of the proposed model were compared and validated with the experimental results. The error was found to be less than 5% with a standard deviation of 3.18%, which is comparable with the existing literature.

Materials and Methods
The elliptical planar inductor shape is novel, and there is no existing expression to calculate the inductance and other parasitic parameters. An ellipse can be represented using an equivalent circular shape using the simple area transformation mathematical theorem. Therefore, using this theorem, all the design parameters (inner and outer major and minor diameters) of the elliptical planar inductor can be transformed into the design parameters (inner and outer diameters) of a circular planar inductor. Figure 1 represents the overall research approach, where it can be seen that the proposed numerical model is validated against the experimental results.

Planar Inductor Design
As discussed in the introduction, the inductance of the planar inductor depends on the geometrical parameters of the inductor design, which include inner diameter ( ), outer diameter ( ), number of turns ( ), trace width ( ), and trace separation ( ). The inner or outer diameter of the planar inductor can be calculated from Equation (1) [46].

Planar Inductor Design
As discussed in the introduction, the inductance of the planar inductor depends on the geometrical parameters of the inductor design, which include inner diameter (d in ), outer diameter (d out ), number of turns (N), trace width (W), and trace separation (S). The inner or outer diameter of the planar inductor can be calculated from Equation (1) [46].
Here is the average diameter of the planar inductor and represents the fill ratio of the inductor, which is an indicator of how hollow the inductor is; a smaller corresponds to a hollower inductor and and are approximately similar. From Equations (1) and (2), it is evident that the inductance of the planar inductor depends on the inductor's geometrical parameters. From Figure 2, it can be observed that all the planar designs are symmetrical in shape. The constants listed in Table 1 are specifically computed for symmetrical shapes. If the shape of the inductor becomes asymmetrical or semi-symmetrical, then these constants cannot be used directly. The elliptical planar inductor is a semi-symmetrical shape; therefore, the inductance of the elliptical inductor cannot be directly calculated from Equation (2) in combination with the constants given in Table 1. In the current study, an area transformation concept is used to calculate the inductance of an elliptical planar inductor using a simple mathematical translation technique.

Area Transformation Technique to Model Elliptical Planar Inductor
In this paper, an area transformation technique is used to transform the area of an elliptical shape into a circular shape [47]. In this approach, the inductance of the elliptical planar inductor ( ) is estimated by translating the area of the ellipse ( ) into the area of the circle ( ) by using mathematical transformation formulas [47]. Using this technique, the design parameters of the elliptical planar inductors are transformed into a circular planar inductor, and all the circular planar inductor formulas can be reused. Figure 3 represents the schematic flow of the transformation of the ellipse area into the area of the circle. Figure 3a shows an ellipse with a minor radius ( ) and major radius ( ), which can be transformed into a circle with a radius ( ) using Equations (3)-(5).
Equations (6) and (7) represent the radius ( ) and diameter ( ) of the translated circle.  Here d avg is the average diameter of the planar inductor and τ represents the fill ratio of the inductor, which is an indicator of how hollow the inductor is; a smaller τ corresponds to a hollower inductor and d out and d in are approximately similar. From Equations (1) and (2), it is evident that the inductance of the planar inductor depends on the inductor's geometrical parameters. From Figure 2, it can be observed that all the planar designs are symmetrical in shape. The constants listed in Table 1 are specifically computed for symmetrical shapes. If the shape of the inductor becomes asymmetrical or semisymmetrical, then these constants cannot be used directly. The elliptical planar inductor is a semi-symmetrical shape; therefore, the inductance of the elliptical inductor cannot be directly calculated from Equation (2) in combination with the constants given in Table 1. In the current study, an area transformation concept is used to calculate the inductance of an elliptical planar inductor using a simple mathematical translation technique.

Area Transformation Technique to Model Elliptical Planar Inductor
In this paper, an area transformation technique is used to transform the area of an elliptical shape into a circular shape [47]. In this approach, the inductance of the elliptical planar inductor (L ellipse ) is estimated by translating the area of the ellipse (A ellipse ) into the area of the circle (A circle ) by using mathematical transformation formulas [47]. Using this technique, the design parameters of the elliptical planar inductors are transformed into a circular planar inductor, and all the circular planar inductor formulas can be reused. Figure 3 represents the schematic flow of the transformation of the ellipse area into the area of the circle. Figure 3a shows an ellipse with a minor radius (r min ) and major radius r maj , which can be transformed into a circle with a radius (r) using Equations (3)-(5). Equations (6) and (7) represent the radius (r) and diameter (d c ) of the translated circle.
while A ellipse = π r min r maj (4) substituting Equations (4) and (5) into Equation (3), π r 2 = π r min r maj r = r min r maj (6) d c = 2 r min r maj (7) Bioengineering 2023, 10, x FOR PEER REVIEW 6 of 29 (a) (b) Similarly, the elliptical planar inductor in Figure 3b can also be translated into a circular planar inductor by using Equation (7). Therefore, the inner diameter ( _ ) and outer diameters ( _ ) of the translated circular planar inductor in terms of elliptical inner diameter ( _ , ) and outer diameter ( _ , _ ) can be given as follows in Equations (8) and (9): similarly, After transforming an elliptical planar inductor into a circular planar inductor, which is a standard shape of a planar inductor, all formulas of a circular planar inductor can be used to estimate the inductance of an equivalent planar elliptical inductor through Equation (2). The inductance of the elliptical planar inductor is given by Equation (10). Moreover, the parasitic components can also be calculated using the standard formulas of the circular planar inductor. Similarly, the elliptical planar inductor in Figure 3b can also be translated into a circular planar inductor by using Equation (7). Therefore, the inner diameter (d in_c ) and outer diameters (d out_c ) of the translated circular planar inductor in terms of elliptical inner diameter d in_min , d in maj and outer diameter (d out_min , d out_maj ) can be given as follows in Equations (8) and (9): similarly, After transforming an elliptical planar inductor into a circular planar inductor, which is a standard shape of a planar inductor, all formulas of a circular planar inductor can be used to estimate the inductance of an equivalent planar elliptical inductor through Equation (2). The inductance of the elliptical planar inductor is given by Equation (10). Moreover, the parasitic components can also be calculated using the standard formulas of the circular planar inductor.

Parasitic Components
After transforming the area of the ellipse into the circular inductor, the values of the parasitic components for the elliptical planar inductor can be calculated using the existing circular inductor expressions. The lumped model of the planar inductor is shown  Table 1.

Parasitic Components
After transforming the area of the ellipse into the circular inductor, the values of the parasitic components for the elliptical planar inductor can be calculated using the existing circular inductor expressions. The lumped model of the planar inductor is shown in Figure 4a, consisting of parasitic resistance ( ), parasitic capacitance ( ), and an ideal inductor ( ).
(a) (b) (c) The total conductor length of the planar elliptical planar ( ) can be calculated by Equation (11) after achieving the newer inner diameter ( _ ) and outer diameter ( _ ) using Equations (8) and (9).
The quality factor of the inductor plays a critical role in the inductor's performance, and the quality factor is hugely impacted by the parasitic resistance ( ). The value of is dependent on the direct current resistance ( ) and the alternating current resistance ( ). Equation (12) is used to compute .
Equation (13) is used to compute the DC component of the resistance ( ).
= (13) where is the total length of the elliptical spiral conductor, is the trace thickness, is the trace width, and is the resistivity of the conductor.
The AC resistance ( ) of the planar inductor is frequency dependent and becomes significantly higher than the DC resistance ( ) at higher frequencies due to the skin and proximity effects [48]. Therefore, the AC resistance ( ) can be computed from Equation (14).
At higher frequencies, the alternating current flows through the outer area of the conductor rather than flowing through the complete cross-sectional area of the conductor; this effect is called the skin effect. Due to a reduction in the effective cross-sectional area, the current flow faces more resistance, which is known as the skin effect resistance ( ).
In Figure 4b, the red area shows the skin depth ( ) through which current flows, and the black area represents the area with no current flow. Equation (15) is used to compute the resistance due to the skin effect ( ) [15]. The total conductor length of the planar elliptical planar (l ellipse ) can be calculated by Equation (11) after achieving the newer inner diameter (d in_c ) and outer diameter (d out_c ) using Equations (8) and (9).
The quality factor of the inductor plays a critical role in the inductor's performance, and the quality factor is hugely impacted by the parasitic resistance (R parasitic ). The value of R parasitic is dependent on the direct current resistance (R DC ) and the alternating current resistance (R AC ). Equation (12) is used to compute R parasitic .
Equation (13) is used to compute the DC component of the resistance (R DC ).
where l ellipse is the total length of the elliptical spiral conductor, t is the trace thickness, W is the trace width, and ρ is the resistivity of the conductor. The AC resistance (R AC ) of the planar inductor is frequency dependent and becomes significantly higher than the DC resistance (R DC ) at higher frequencies due to the skin and proximity effects [48]. Therefore, the AC resistance (R AC ) can be computed from Equation (14).
At higher frequencies, the alternating current flows through the outer area of the conductor rather than flowing through the complete cross-sectional area of the conductor; this effect is called the skin effect. Due to a reduction in the effective cross-sectional area, the current flow faces more resistance, which is known as the skin effect resistance (R skin ). In Figure 4b, the red area shows the skin depth (δ) through which current flows, and the black area represents the area with no current flow. Equation (15) is used to compute the resistance due to the skin effect (R skin ) [15]. here µ o is the permeability constant, and its value is 4π × 10 −7 H/m, µ r is the relative permeability of the conductor, and f is the operational frequency.
Similar to the skin effect, the proximity effect also becomes significant at higher frequencies. At a specific frequency (crowding frequency ( f critical )), the magnetic field of the nearby turns of the planar inductor becomes significantly high and causes a nonuniform flow through the traces. This nonuniform distribution of current results in increased resistance which is known as proximity resistance (R proximity ) and can be computed using Equation (16) [49].
The parasitic capacitance (C parasitic ) is one of the significant parasitic components and can limit the functionality of the inductor. The parasitic capacitance for the planar inductors can be computed from Equation (17) [50,51]. Total parasitic capacitance is a combined effect of capacitances between the nearby metallic traces due to the air gap between traces and underlying substrate material (polyimide).
The contributing factors of parasitic capacitances are α = 0.9 and β = 0.1. The parasitic capacitance due to the air gap (C air ) and parasitic capacitance due to the underlying substrate (C substrate ) are shown in Figure 4c. The relative permittivities of the substrate material and air are expressed as substrate and air , respectively.
Once the parasitic capacitance is computed, then the self-resonance frequency ( f SRF ) of the planar elliptical inductor can easily be computed using the self-inductance of the elliptical inductor and parasitic capacitance. The f SRF is very critical while designing an inductor for specific applications with a wide range of operational frequencies. Above its self-resonance frequency, an inductor works more like a capacitor than an inductor. The f SRF can be computed using Equation (18) [51].

Fabrication
To validate the proposed elliptical inductor model, different elliptical planar inductors were made using a wet-etching method. In the first fabrication stage, a LaserJet printer (HPM553, HP Technology, Dublin, Ireland) was used to print the inductor mask directly onto a 50 µm thick single-sided copper-coated polyimide film (Flexible Isolating Circuit 50 Microns-Coppered 35 Microns-1 Side, CIF, Buc, France). As the next step, these coppercoated polyimide films with printed masks were attached to the plastic stand of the etching machine (PA104 Heated Bubble Etch tank, Fortex, UK). The etchant was made by combining sodium persulphate (Na 2 S 2 O 8 ) and deionized water in a 1:5 ratio. The etching was carried out inside a transparent acrylic tank with a diaphragm air pump attached to microporous tubing to produce tiny air bubbles that would help the etching process. To increase the speed of the etching process, the etchant's temperature was set at 42 • C by using a suspended glass heater dipped inside the machine tank. The overall etching process was completed within 20-25 min. In the next step, this patterned flexible PCB was removed from the etching tank and washed with hot water. The ink particles from the patterned inductor designs were removed using an acetone bath. In the final step of the fabrication, flexible multithread copper wires were soldered on the terminal points of the inductors for electrical connections. The stepwise fabrication process is shown in Figure 5.
process was completed within 20-25 min. In the next step, this patterned flexible PCB was removed from the etching tank and washed with hot water. The ink particles from the patterned inductor designs were removed using an acetone bath. In the final step of the fabrication, flexible multithread copper wires were soldered on the terminal points of the inductors for electrical connections. The stepwise fabrication process is shown in Figure  5.

Device Validation
To validate the proposed model, a Keysight E4990A impedance analyzer (Keysight Technologies Inc., CA, USA) was used for the measurements of fabricated elliptical inductors. Before the measurements, the impedance analyzer was calibrated using the standard test fixture 16047E (Keysight Technologies Inc., CA, USA) (open and short calibration). The fabricated elliptical planar inductors were connected in this test fixture, as shown in Figure 6, and the inductance ( ) and the real ( ) and imaginary ( ) parts of the impedance were measured for a range of frequencies between 20 Hz and 120 MHz (the full-scale measurement range of the E4990A). However, to validate the proposed model, the inductance measurements reported in the tables were taken at 1 MHz frequency as the equipment error is only 0.1% at 1 MHz, and it became 5-10% when the measurement frequency is >100 MHz [52].

Results
To validate and assess the estimation accuracy of the proposed elliptical model, different sets of elliptical planar inductors were fabricated. Considering the significant impact of parasitic components in varying geometries with different trace widths, trace separations, minor and major diameters, and the number of turns, a large number of inductors (N = 75) were fabricated. In one set, the major-to-minor ratio ( ) between the inner minor and inner major diameters ( _ to _ ) was kept fixed, and the trace width (W) and trace separation (S) were varied. Similarly, for another set of fabricated inductors, the ratio between the outer minor and outer major diameters ( _ to _ ) was varied while keeping the trace width and trace separation constant.

Device Validation
To validate the proposed model, a Keysight E4990A impedance analyzer (Keysight Technologies Inc., CA, USA) was used for the measurements of fabricated elliptical inductors. Before the measurements, the impedance analyzer was calibrated using the standard test fixture 16047E (Keysight Technologies Inc., CA, USA) (open and short calibration). The fabricated elliptical planar inductors were connected in this test fixture, as shown in Figure 6, and the inductance (L ellipse ) and the real R ellipse and imaginary (X ellipse ) parts of the impedance were measured for a range of frequencies between 20 Hz and 120 MHz (the full-scale measurement range of the E4990A). However, to validate the proposed model, the inductance measurements reported in the tables were taken at 1 MHz frequency as the equipment error is only 0.1% at 1 MHz, and it became 5-10% when the measurement frequency is >100 MHz [52].
process was completed within 20-25 min. In the next step, this patterned flexible PCB was removed from the etching tank and washed with hot water. The ink particles from the patterned inductor designs were removed using an acetone bath. In the final step of the fabrication, flexible multithread copper wires were soldered on the terminal points of the inductors for electrical connections. The stepwise fabrication process is shown in Figure  5.

Device Validation
To validate the proposed model, a Keysight E4990A impedance analyzer (Keysight Technologies Inc., CA, USA) was used for the measurements of fabricated elliptical inductors. Before the measurements, the impedance analyzer was calibrated using the standard test fixture 16047E (Keysight Technologies Inc., CA, USA) (open and short calibration). The fabricated elliptical planar inductors were connected in this test fixture, as shown in Figure 6, and the inductance ( ) and the real ( ) and imaginary ( ) parts of the impedance were measured for a range of frequencies between 20 Hz and 120 MHz (the full-scale measurement range of the E4990A). However, to validate the proposed model, the inductance measurements reported in the tables were taken at 1 MHz frequency as the equipment error is only 0.1% at 1 MHz, and it became 5-10% when the measurement frequency is >100 MHz [52].

Results
To validate and assess the estimation accuracy of the proposed elliptical model, different sets of elliptical planar inductors were fabricated. Considering the significant impact of parasitic components in varying geometries with different trace widths, trace separations, minor and major diameters, and the number of turns, a large number of inductors (N = 75) were fabricated. In one set, the major-to-minor ratio ( ) between the inner minor and inner major diameters ( _ to _ ) was kept fixed, and the trace width (W) and trace separation (S) were varied. Similarly, for another set of fabricated inductors, the ratio between the outer minor and outer major diameters ( _ to _ ) was varied while keeping the trace width and trace separation constant.

Results
To validate and assess the estimation accuracy of the proposed elliptical model, different sets of elliptical planar inductors were fabricated. Considering the significant impact of parasitic components in varying geometries with different trace widths, trace separations, minor and major diameters, and the number of turns, a large number of inductors (n = 75) were fabricated. In one set, the major-to-minor ratio (R) between the inner minor and inner major diameters (d in_maj to d in_min ) was kept fixed, and the trace width (W) and trace separation (S) were varied. Similarly, for another set of fabricated inductors, the ratio between the outer minor and outer major diameters (d in_maj to d in_min ) was varied while keeping the trace width and trace separation constant.

Estimation Results for Varying Trace Separation (S) and Trace Width (W) While Keeping the Ratio (R) between the Inner Minor Diameter to Inner Major Diameter Constant
This section details the comparison between the numerical results of the proposed model and the measured results of the fabricated elliptical planar inductors. In this comparison, the ratio between the inner minor and inner major diameters was kept fixed at 3, while the trace width and trace separation were varied between 200 µm and 600 µm for 10-and 5-turn elliptical inductors. The outer minor and outer major diameters were dependent on the combinations of trace width and trace separation. Tables 2 and 3 show the calculated and measured results, respectively, for the 10-turn elliptical inductor. The ratio between the outer minor and outer major diameters was kept at 3. Figure 7 represents the fabricated elliptical inductors with a trace width of 600 µm and trace separation varying from 200 µm to 600 µm, while the major-to-minor ratio (R) and N are kept constant at 3 and 10, respectively.   Table 3. Measured inductances of elliptical planar inductors for different combinations of trace separation (S) and trace width (W) while major-to-minor ratio (R) = 3 and number of turns (N) = 10. This section details the comparison between the numerical results of the proposed model and the measured results of the fabricated elliptical planar inductors. In this comparison, the ratio between the inner minor and inner major diameters was kept fixed at 3, while the trace width and trace separation were varied between 200 µm and 600 µm for 10-and 5-turn elliptical inductors. The outer minor and outer major diameters were dependent on the combinations of trace width and trace separation. Tables 2 and 3 show the calculated and measured results, respectively, for the 10-turn elliptical inductor. The ratio between the outer minor and outer major diameters was kept at 3. Figure 7 represents the fabricated elliptical inductors with a trace width of 600 μm and trace separation varying from 200 μm to 600 μm, while the major-to-minor ratio (R) and N are kept constant at 3 and 10, respectively.   Table 3. Measured inductances of elliptical planar inductors for different combinations of trace separation (S) and trace width (W) while major-to-minor ratio (R) = 3 and number of turns (N) = 10.   The percentage error, as given in Equation (19), is an evaluation metric commonly used to compare estimated and experimental measurements [23,27]. The percentage error between the numerical inductance of the proposed model and measured inductance values of the elliptical planar inductor is computed using Equation (19) and given in Table 4. A maximum error value of 6.42% was found when the trace separation and trace width were 600 µm, while the minimum error was found to be 0.08% when the trace separation (S) and trace width (W) were 500 µm and 200 µm, respectively. However, the average error was 2.47%, with a standard deviation of 1.8% for the different combinations of trace separation (S) and width (W).  The measured inductances (black lines) and proposed model inductances (red lines) with color bar graphs for various trace separations and trace width are represented graphically in Figure 8. It is evident from Figure 8 that the proposed model inductance and measured inductance values are approximately the same, with a percentage difference of less than 5% between the values. This difference is primarily associated with fabrication inaccuracies and measurement errors.

Measured Inductance (μH) of Fabricated Elliptical Inductors (Number of Turns (N) = 10 Major-to-Minor Ratio (R) = 3)
The percentage error, as given in Equation (19), is an evaluation metric commonly used to compare estimated and experimental measurements [23,27]. The percentage error between the numerical inductance of the proposed model and measured inductance values of the elliptical planar inductor is computed using Equation (19) and given in Table 4. A maximum error value of 6.42% was found when the trace separation and trace width were 600 μm, while the minimum error was found to be 0.08% when the trace separation (S) and trace width (W) were 500 μm and 200 μm, respectively. However, the average error was 2.47%, with a standard deviation of 1.8% for the different combinations of trace separation (S) and width (W).   Figure 8. It is evident from Figure 8 that the proposed model inductance and measured inductance values are approximately the same, with a percentage difference of less than 5% between the values. This difference is primarily associated with fabrication inaccuracies and measurement errors. To evaluate the impact of the number of turns of the elliptical inductor on varying trace separation and trace width, five-turn elliptical inductors were investigated. The fabricated elliptical inductors with five turns are shown in Figure 9. For a fair comparison To evaluate the impact of the number of turns of the elliptical inductor on varying trace separation and trace width, five-turn elliptical inductors were investigated. The fabricated elliptical inductors with five turns are shown in Figure 9. For a fair comparison with previous results, trace separation and trace width combinations were kept the same as in the last test setup. The calculated and measured inductance values are tabulated in Tables 5 and 6, respectively. The maximum value of the calculated inductance was found for a trace width and trace separation of 200 µm, and the minimum inductance value was seen for a trace width of 600 µm and trace separation of 400 µm.
with previous results, trace separation and trace width combinations were kept the same as in the last test setup. The calculated and measured inductance values are tabulated in Tables 5 and 6, respectively. The maximum value of the calculated inductance was found for a trace width and trace separation of 200 μm, and the minimum inductance value was seen for a trace width of 600 μm and trace separation of 400 μm. S = 200 μm S = 300 μm S = 400 μm S = 500 μm S = 600 μm     The measured inductance of elliptical planar inductors is tabulated in Table 5. It can be observed from Table 6 that the measured inductance values are lower than the calculated inductance values. The measured inductance was higher due to the additional copper leads soldered with the fabricated inductor for electrical connections with an impedance analyzer. The maximum measured inductance is 1 μH, while the minimum measured inductance is observed to be 0.833 μH for a trace separation of 600 μm and a trace width of 500 μm.  The measured inductance of elliptical planar inductors is tabulated in Table 5. It can be observed from Table 6 that the measured inductance values are lower than the calculated inductance values. The measured inductance was higher due to the additional copper leads soldered with the fabricated inductor for electrical connections with an impedance analyzer. The maximum measured inductance is 1 µH, while the minimum measured inductance is observed to be 0.833 µH for a trace separation of 600 µm and a trace width of 500 µm.
The percentage error between the numerical inductance of the proposed model and the measured inductance values of the elliptical planar inductor is given in Table 7. It can be observed from Table 7 that the maximum error was observed to be 9.93% for a trace separation of 300 µm and a trace width of 500 µm. The minimum error was observed to be 0.59% for a trace separation and trace width of 200 µm. Moreover, the average error was 4.85%, with a standard deviation of 3.18% for the different combinations of trace separation and trace width for a fixed number of turns (5) and fixed major-to-minor ratio (3). Further, it can be observed from Table 7 that the average percentage error is slightly higher for 5-turn elliptical inductors compared to 10-turn elliptical inductors. This error is higher as fewer-turn inductors are more prone to variability in fabrication and measurement phases than large-turn inductors. Table 7. The percentage error between the numerically calculated and measured inductances of elliptical planar inductors after using the proposed model for different combinations of trace separation (S) and trace width (W) while major-to-minor ratio (R) = 3 and number of turns (N) = 5. The measured inductances (black lines) and numerical inductances of the proposed model (red lines) with color bar graphs for various trace separations and trace width have been represented graphically in Figure 10. It is evident from Figure 10 that the numerical inductance of the proposed model and measured inductance values are approximately the same, with a percentage difference of less than 8% between the values.

Test Parameters
The percentage error between the numerical inductance of the proposed model and the measured inductance values of the elliptical planar inductor is given in Table 7. It can be observed from Table 7 that the maximum error was observed to be 9.93% for a trace separation of 300 μm and a trace width of 500 μm. The minimum error was observed to be 0.59% for a trace separation and trace width of 200 μm. Moreover, the average error was 4.85%, with a standard deviation of 3.18% for the different combinations of trace separation and trace width for a fixed number of turns (5) and fixed major-to-minor ratio (3). Further, it can be observed from Table 7 that the average percentage error is slightly higher for 5-turn elliptical inductors compared to 10-turn elliptical inductors. This error is higher as fewer-turn inductors are more prone to variability in fabrication and measurement phases than large-turn inductors. Table 7. The percentage error between the numerically calculated and measured inductances of elliptical planar inductors after using the proposed model for different combinations of trace separation (S) and trace width (W) while major-to-minor ratio (R) = 3 and number of turns (N) = 5.

Test Parameters
Trace The measured inductances (black lines) and numerical inductances of the proposed model (red lines) with color bar graphs for various trace separations and trace width have been represented graphically in Figure 10. It is evident from Figure 10 that the numerical inductance of the proposed model and measured inductance values are approximately the same, with a percentage difference of less than 8% between the values.

Estimation Results for Varying Ratios of Inner Minor Diameter to the Inner Major Diameter between 1 to 5 While Keeping Trace Separation and Width Fixed at 200 µm
To investigate the effect of the varying major-to-minor ratio between one and five, elliptical inductors of different sizes were fabricated and tested, as shown in Figure 11. In all planar inductors, the number of turns, trace width, and trace separation were kept fixed at 10, 200 µm, and 200 µm, respectively. The inner minor diameter was set to 5 mm, while the inner major diameter was varied from 5 to 25 mm to achieve major-to-minor ratio (R) values of between one and five. The numerical inductance (L cal ) is calculated using the proposed model mentioned in Section 2.1, whereas the measured inductance (L meas ) is measured using the impedance analyzer. The numerical inductance of the proposed model and measured inductance values are listed in Table 8. To calculate the difference between the numerical inductance of the proposed model and measured inductance values, the absolute percentage error has been calculated and is tabulated in Table 8.
To investigate the effect of the varying major-to-minor ratio between one and five, elliptical inductors of different sizes were fabricated and tested, as shown in Figure 11. In all planar inductors, the number of turns, trace width, and trace separation were kept fixed at 10, 200 μm, and 200 μm, respectively. The inner minor diameter was set to 5 mm, while the inner major diameter was varied from 5 to 25 mm to achieve major-to-minor ratio (R) values of between one and five. The numerical inductance ( ) is calculated using the proposed model mentioned in Section 2.1, whereas the measured inductance ( ) is measured using the impedance analyzer. The numerical inductance of the proposed model and measured inductance values are listed in Table 8. To calculate the difference between the numerical inductance of the proposed model and measured inductance values, the absolute percentage error has been calculated and is tabulated in Table 8.  It can be observed from Table 8 that the maximum error was observed to be 6.38% for the major-to-minor ratio of 5, and the minimum error was observed to be 0.45% for the major-to-minor ratio of 2.5. Moreover, the average error between the calculated and measured inductances was observed to be 3.61%, with a standard deviation of 2.11%. The major  It can be observed from Table 8 that the maximum error was observed to be 6.38% for the major-to-minor ratio of 5, and the minimum error was observed to be 0.45% for the major-to-minor ratio of 2.5. Moreover, the average error between the calculated and measured inductances was observed to be 3.61%, with a standard deviation of 2.11%. The major contributors to this error are the fabrication, measurement, and methodology to estimate the inductance of the planar elliptical inductors.
To validate the proposed model, the inductance of the elliptical inductor was calculated for a frequency range of 20 Hz to 120 MHz. The measured inductance of all elliptical inductors was recorded using an impedance Analyzer. Figure 12a,b show the numerical inductance of the proposed model and the measured inductance of the elliptical inductors, respectively. It can be observed from Figure 12 that, for all combinations of the major-to-minor ratio (1 to 5), the trends in the measured and the calculated inductance values look similar. These similar trends between the calculated and measured values validate the proposed model. Moreover, the percentage difference between the calculated and measured inductance values has been calculated for the observed frequency range, and the values are tabulated in Table 8.
contributors to this error are the fabrication, measurement, and methodology to estimate the inductance of the planar elliptical inductors.
To validate the proposed model, the inductance of the elliptical inductor was calculated for a frequency range of 20 Hz to 120 MHz. The measured inductance of all elliptical inductors was recorded using an impedance Analyzer. Figure 12a,b show the numerical inductance of the proposed model and the measured inductance of the elliptical inductors, respectively. It can be observed from Figure 12 that, for all combinations of the major-tominor ratio (1 to 5), the trends in the measured and the calculated inductance values look similar. These similar trends between the calculated and measured values validate the proposed model. Moreover, the percentage difference between the calculated and measured inductance values has been calculated for the observed frequency range, and the values are tabulated in Table 8. The self-resonance frequency ( ) is an important performance metric to analyze the behavior of an inductor as the parasitic capacitance dominates at frequencies higher than the self-resonance frequency. Thus, while designing an inductor for higher frequencies, it is not enough to choose the correct inductance but also essential to use an inductor with a self-resonance frequency substantially lower than the . Therefore, to analyze The self-resonance frequency ( f SRF ) is an important performance metric to analyze the behavior of an inductor as the parasitic capacitance dominates at frequencies higher than the self-resonance frequency. Thus, while designing an inductor for higher frequencies, it is not enough to choose the correct inductance but also essential to use an inductor with a self-resonance frequency substantially lower than the f SRF . Therefore, to analyze the proposed numerical model and measured elliptical planar inductors, this study has compared f SRF values calculated using our model with experimental data. It can be noted from Table 9 that the maximum deviation between calculated ( f SRF_cal ) and measured self-resonance frequency ( f SRF_meas ) values is observed to be 6.72% for a trace separation and width of 200 µm and a major-to-minor ratio of 4.5. The N/A in this Table 9 represents that the percentage error is not available as the self-resonance frequency was higher than the frequency range (>120 MHz) of the impedance analyzer. The impedance (real and imaginary) of the elliptical planar inductor calculated from the proposed model was compared with measured values from the impedance analyzer over the frequency range of 20 Hz to 120 MHz. The calculated and measured impedance results are shown in Figures 12b and 13a. In both Figure 13a,b, the real part of the impedance is shown with solid lines, whereas the imaginary part is shown with dotted lines. It can be observed from Figures 12b and 13a that the real and imaginary impendence profiles for both the calculated and measured results of the elliptical inductors are similar.

Estimation Results for Varying Ratios of Inner Minor Diameter to the Inner Major Diameter between 1 to 5 While Keeping Trace Separation and Width Fixed at 300 μm
To validate the impact of varying trace width and separation for varying major-tominor ratios of one to five, the trace width and separation were set to 300 μm while the number of turns was kept fixed at 10. The inner minor diameter was set to 10 mm, while the inner major diameter was varied from 10 to 50 mm to achieve major-to-minor ratio

Estimation Results for Varying Ratios of Inner Minor Diameter to the Inner Major Diameter between 1 to 5 While Keeping Trace Separation and Width Fixed at 300 µm
To validate the impact of varying trace width and separation for varying major-tominor ratios of one to five, the trace width and separation were set to 300 µm while the number of turns was kept fixed at 10. The inner minor diameter was set to 10 mm, while the inner major diameter was varied from 10 to 50 mm to achieve major-to-minor ratio values of between one and five. The calculated and measured inductance values are listed in Table 10. To compute the difference between the calculated and measured inductance values, the absolute percentage error has been calculated and is tabulated in Table 10. Table 10. Key results of elliptical planar inductors for varying major-to-minor ratios when trace separation (S) and trace width (W) were kept constant at 300 µm. The maximum error was observed to be 7.85% for a major-to-minor ratio of five, whereas the minimum error was observed to be 0.28% for a major-to-minor ratio of three. The average error between the calculated and measured values was observed to be 4.19%, with a standard deviation of 2.39%. As explained previously, the errors arise mainly from the fabrication, measurement, and methodology of estimating the inductance of the planar elliptical inductors.
To validate the proposed model, the inductance of the elliptical inductor was computed for a frequency range of 20 Hz to 120 MHz. The measured inductance of all elliptical inductors was recorded using an impedance analyzer. Figure 14a,b show the calculated and measured inductance of the elliptical inductors, respectively. It can be observed from Figure 14, that for all combinations of major-to-minor ratios (one to five), the trends in the measured and the calculated inductance values look similar. One apparent outlier in Figure 14b may have been affected by interference during measurement. However, the trend remains consistent with other data points.
It is evident from Table 11 that the maximum deviation between the calculated ( f SRF_cal ) and measured self-resonance frequency ( f SRF_meas ) values is 9.55% for the trace separation and width of 300 µm and a major-to-minor ratio of two, while the average error was 4.88% with a standard deviation of 3%. Table 11. Self-resonance frequency of elliptical planar inductors for varying major-to-minor ratios when trace separation (S) and trace width (W) were kept constant at 300 µm.   Figure 14. Inductance of elliptical planar inductors for the full range of frequencies from 20 Hz to 120 MHz with varying major-to-minor ratios while trace separation (S) and trace width (W) were kept constant at 300 μm (a) Computed response using proposed model (b) Measured response from the impedance analyzer.
It is evident from Table 11 that the maximum deviation between the calculated ( _ ) and measured self-resonance frequency ( _ ) values is 9.55% for the trace separation and width of 300 μm and a major-to-minor ratio of two, while the average error was 4.88% with a standard deviation of 3%. Table 11. Self-resonance frequency of elliptical planar inductors for varying major-to-minor ratios when trace separation (S) and trace width (W) were kept constant at 300 μm.  To further validate the proposed model, a comparison between the numerical impedance of the proposed model and the measured impedance for the different major-to- Figure 14. Inductance of elliptical planar inductors for the full range of frequencies from 20 Hz to 120 MHz with varying major-to-minor ratios while trace separation (S) and trace width (W) were kept constant at 300 µm (a) Computed response using proposed model (b) Measured response from the impedance analyzer.
To further validate the proposed model, a comparison between the numerical impedance of the proposed model and the measured impedance for the different major-to-minor ratios is made. During this analysis, the trace width and trace separation were kept constant at 300 µm. The calculated and measured impedance results for a frequency range of 20 Hz to 120 MHz are shown below in Figure 15a,b. The real part of the impedance is shown with solid lines, whereas dotted lines represent the imaginary part of the impedance. The results show that the profiles of real and imaginary components are similar for both the calculated and the measured results. minor ratios is made. During this analysis, the trace width and trace separation were kept constant at 300 μm. The calculated and measured impedance results for a frequency range of 20 Hz to 120 MHz are shown below in Figure 15a,b. The real part of the impedance is shown with solid lines, whereas dotted lines represent the imaginary part of the impedance. The results show that the profiles of real and imaginary components are similar for both the calculated and the measured results.
(a) (b) Figure 15. Impedance (Real (solid lines) and Imaginary (dotted lines) components) of elliptical planar inductors for the full range of frequencies from 20 Hz to 120 MHz with varying major-tominor ratios while trace separation (S) and trace width (W) were kept constant at 300 μm (a) Computed response using proposed model (b) Measured response from the impedance analyzer.

Estimation Results for Varying Ratios of Inner Minor Diameter to the Inner Major Diameter between 1 to 5 While Keeping Trace Separation and Width Fixed at 400 μm
To further analyze the impact of varying trace width and separation for varying major-to-minor ratios of one to five, the trace width and separation were set to 400 μm while the number of turns was kept at 12. The inner minor diameter was set to 12 mm, while the inner major diameter was varied from 12 to 60 mm to achieve major-to-minor ratio values of between one and five. To compute the difference between the calculated and measured inductance values, the absolute percentage error has been calculated and is tabulated in Table 12.

Estimation Results for Varying Ratios of Inner Minor Diameter to the Inner Major Diameter between 1 to 5 While Keeping Trace Separation and Width Fixed at 400 µm
To further analyze the impact of varying trace width and separation for varying major-tominor ratios of one to five, the trace width and separation were set to 400 µm while the number of turns was kept at 12. The inner minor diameter was set to 12 mm, while the inner major diameter was varied from 12 to 60 mm to achieve major-to-minor ratio values of between one and five. To compute the difference between the calculated and measured inductance values, the absolute percentage error has been calculated and is tabulated in Table 12. Table 12 shows that the maximum error was 5.31% for a major-to-minor ratio of five, whereas the minimum error was 0.03% for a major-to-minor ratio of three. The average error between the calculated and measured values was observed to be 2.22%, with a standard deviation of 1.82%. As stated earlier, the errors arise mainly from the fabrication, measurement, and methodology of estimating the inductance of planar elliptical inductors. For further validation, the inductance of the elliptical inductor was computed and measured for a frequency range of 20 Hz to 120 MHz. Figure 16a,b show the calculated and measured inductance of elliptical inductors, respectively. It can be observed from Figure 16 that for all combinations of major-to-minor ratio values (one to five), the trends in the measured and the calculated inductance results look similar. As previously discussed, the self-resonance frequency is a critical parameter when designing an inductor. Thus, the measured and calculated self-resonance were compared and listed in Table 13, and a maximum deviation of 7.89% was noticed for a major-tominor ratio of 2.5. However, the average error was 4.17%, with a standard deviation of 2.67%. Figure 16. Inductance of elliptical planar inductors for the full range of frequencies from 20 Hz to 120 MHz with varying major-to-minor ratios while trace separation (S) and trace width (W) were kept constant at 400 µm (a) Computed response using proposed model (b) Measured response from the impedance analyzer. As previously discussed, the self-resonance frequency is a critical parameter when designing an inductor. Thus, the measured and calculated self-resonance were compared and listed in Table 13, and a maximum deviation of 7.89% was noticed for a major-to-minor ratio of 2.5. However, the average error was 4.17%, with a standard deviation of 2.67%. Table 13. Self-resonance frequency of elliptical planar inductors for varying major-to-minor ratios when trace separation (S) and trace width (W) were kept constant at 400 µm. The real and imaginary impedance components are shown in Figure 17a,b when the trace width and trace separation were kept constant at 400 µm for varying major-to-minor ratio values of between one and five. It is evident from Figure 17a,b that there was a similar response for the calculated and measured impedances of elliptical inductors. The real and imaginary impedance components are shown in Figure 17a,b when the trace width and trace separation were kept constant at 400 μm for varying major-to-minor ratio values of between one and five. It is evident from Figure 17a,b that there was a similar response for the calculated and measured impedances of elliptical inductors.

Discussion
In this study, a numerical model of a planar elliptical inductor has been presented that uses an area transformation technique to estimate inductor parameters from a circular model. During the transformation, the minor and major inner and minor and major outer diameters of the elliptical planar inductor were transformed into the inner diameter and outer diameter of the circular planar inductor. After the transformation, the new inner

Discussion
In this study, a numerical model of a planar elliptical inductor has been presented that uses an area transformation technique to estimate inductor parameters from a circular model. During the transformation, the minor and major inner and minor and major outer diameters of the elliptical planar inductor were transformed into the inner diameter and outer diameter of the circular planar inductor. After the transformation, the new inner diameter and outer diameter were used for further calculations of the inductance, impedance, self-resonance frequency, and other parameters of the elliptical planar inductor.
To validate the proposed model, several elliptical planar inductors were fabricated. The measured and proposed numerical model results were compared to assess the accuracy of the proposed model for planar elliptical inductors.
The estimated inductor parameters using the proposed model showed an excellent match with the measured values from a large batch of fabricated inductors. The proposed model was validated for robustness using different combinations of trace widths, trace separation, and other geometrical features. The trace width and trace separation were varied between 200 µm to 600 µm. In the first step, the trace separation and trace width were varied for 10-and 5-turn inductors while the major-to-minor axis ratio was kept constant at three. Figure 18 represents the boxplot of percentage error between the measured and calculated inductances using the proposed model for the elliptical inductor when trace width and separation were varied between 200 µm and 600 µm. The ratio between the major and minor inner diameters was kept fixed at three while the number of turns was set to 5 and 10. From Figure 18a, it is clear that the median and variation of the percentage error were higher for N = 5 compared to the percentage error for N = 10. Table 14 shows the average percentage error and standard deviation of all measurement scenarios. From Table 14, it is evident that in all cases, the average percentage error was less than 5%, and the maximum measured standard deviation was 3.18%. The estimated inductor parameters using the proposed model showed an excellent match with the measured values from a large batch of fabricated inductors. The proposed model was validated for robustness using different combinations of trace widths, trace separation, and other geometrical features. The trace width and trace separation were varied between 200 μm to 600 μm. In the first step, the trace separation and trace width were varied for 10-and 5-turn inductors while the major-to-minor axis ratio was kept constant at three. Figure 18 represents the boxplot of percentage error between the measured and calculated inductances using the proposed model for the elliptical inductor when trace width and separation were varied between 200 μm and 600 μm. The ratio between the major and minor inner diameters was kept fixed at three while the number of turns was set to 5 and 10. From Figure 18a, it is clear that the median and variation of the percentage error were higher for N = 5 compared to the percentage error for N = 10. Table 14 shows the average percentage error and standard deviation of all measurement scenarios. From Table 14, it is evident that in all cases, the average percentage error was less than 5%, and the maximum measured standard deviation was 3.18%.  In this investigation, it was observed that the average error of inductance for the measured and proposed numerical model was approximately two times higher for inductors with fewer turns (N = 5) than that for 10 turns. The standard deviation was found to be approximately two times higher for 5-turn inductors than for 10-turn inductors. This higher average error and standard deviation of percentage error in smaller inductors were because smaller inductors are more prone to show variation during the fabrication, measurement, and designing process. A very small error due to calibration or measurement  In this investigation, it was observed that the average error of inductance for the measured and proposed numerical model was approximately two times higher for inductors with fewer turns (N = 5) than that for 10 turns. The standard deviation was found to be approximately two times higher for 5-turn inductors than for 10-turn inductors. This higher average error and standard deviation of percentage error in smaller inductors were because smaller inductors are more prone to show variation during the fabrication, measurement, and designing process. A very small error due to calibration or measurement will cause a higher percentage error for small inductors than for large inductors.
In the next set of experiments, the major-to-minor ratio was varied between one and five while the trace separation and trace width were kept at 200 µm, 300 µm, and 400 µm. Figure 18b represents the boxplots of percentage errors for this set of inductors. Table 14 shows percentage errors of 3.61%, 4.19%, and 2.22% when the trace width and separation were 200 µm, 300 µm, and 400 µm, respectively. Moreover, the standard deviation was found to be 2.11%, 2.39%, and 1.82% when trace width and separation were 200 µm, 300 µm, and 400 µm, respectively. This investigation observed that the average error and standard deviation were smaller when the trace width and separation were kept at 400 µm. From all sets of experiments, it can be seen clearly that the maximum error was observed to be 4.85%, and the minimum average error was observed to be 2.22%, while the overall average error was 3.47%. The maximum and minimum standard deviations were 3.18% and 1.80%, respectively, while the average standard error was 2.26%.
For comparison with existing methods, a summary of the state-of-the-art approaches is presented in Table 15. The limitations associated with each approach are also listed in Table 15. It can be observed that some of the expressions are very complex and demand high computational power and resources to evaluate the inductance of planar inductors. Most of the approaches listed in Table 15 are only suitable for symmetrical planar inductor computation. Moreover, the accuracy is also dependent on the design parameters such as trace width, trace separation, number of turns, and inner and outer diameters. The fourth column in Table 15 shows the absolute percentage error between the inductance values computed using the expression and finite element simulator. These errors tend to increase when compared with the experimental data, as experimental results may also vary due to variations in the fabrication process and measurement setup. The percentage error reported in this study between the proposed numerical model and experimental results is under 5%. This is a collective error due to the fabrication, measurement, and estimation error of the model of the planar elliptical inductor. Using a similar area transformation technique, the inductance and other parasitic components can be computed for other semi-symmetrical shapes without performing high-intensity computational power and complex mathematical modeling. The proposed model is computationally simple as shown in Equations (8)- (10). In terms of computational complexity, it takes only 5.7 ms on average to compute the inductance of a single planar inductor using the proposed model and MATLAB 2020b running on a desktop computer (Processor (Intel Core i5 CPU at 1.60 GHz 2.11 GHz), RAM 8 GB, etc.).
Overall, a small error and small standard deviation between the experimental and numerical results have been observed; however, the main limitation of this work was the variation in the fabrication process, especially when there are fine traces in the design of the elliptical planar inductor. As mentioned earlier, smaller inductors are more prone to error as the parasitic effect due to measurement setup can also change the actual values. This variation can cause an overall increase in the error between the experimental and numerical results. The proposed model in this study is based on the current sheet expression for the planar inductor model. The current sheet expression shows a 2-3% error when the trace width and separation of the planar inductor are relatively similar. This error becomes 8% when the trace separation is less than or equal to three times the trace width, which could also limit the accuracy of the proposed model. Another limitation is the impedance analyzer frequency range of 20 Hz to 120 MHz, as a smaller inductor shows the response in the higher frequencies >120 MHz; also, the measurement error increases when the measurement frequency is >100 MHz. Table 15. Comparison of state-of-the-art approaches for computing the inductance of planar inductors.

Methodology/Expression/Simulator Limitations % Error
GroverExpression (L gmd2 ) [37] L gmd2 = L sel f + M + − M − L sel f = 0.002l ln 2l W+t + 0.50049 + W+t 3l L sel f is the self-inductance of the single current-carrying electrode, W and t are the trace thickness and width, and l is the length of the conductor, |M+| = |M−| = 2lQQ = ln l gmd + 1 + l Where gmd is the geometric mean distance between two conductors and can be computed using the below equation. P is the pitch of the coil. gmd = ln(P) − Wheeler Expression (L wh ) [40] L wh = N 2 r 2 8r+11∆ Here r is the radius of the coil, ∆ = (dout −d in ) 2 Only accurate for circular solenoid coils Error increases with the increase in trace width

Conclusions
In this study, a numerical model is developed to calculate the inductance of the elliptical planar inductor and its parasitic components. An area transformation technique from an elliptical to a circular shape was used to adapt the circular planar inductor formulas. The proposed numerical model was validated for various combinations of the number of turns, trace width, trace separation, and different inner and outer diameters of the elliptical planar inductor. For the validation, a large batch of elliptical planar inductors (n = 75) were designed, fabricated, and measured to assess the estimation accuracy and robustness of the model. The overall average error between the measured and proposed numerical model results was less than 5%, with a standard deviation of less than 3.18%. The main factors for a higher variation in the measured results were the limitations in the fabrication process, as masks were directly printed on the flexible copper-coated sheets using a LaserJet printer, which has a lower resolution on this type of print media. Nevertheless, an excellent match of inductor parameters between the model estimates and the measured values suggests that the proposed model is a good candidate for modeling and designing elliptical planar inductors.
In future studies, the planar inductors can be fabricated using laser technology to achieve less variation and high accuracy. Using this approach, elliptical planar inductors can be designed, optimized, and fabricated for several applications, including implantable devices. In the future, these elliptical planar inductors can be integrated with passive sensing capacitive elements to realize LC wireless sensors. The elliptical inductors can be folded into a compact shape, making them suitable for a catheter delivery system to remote and narrow implantation sites. The proposed approach of the area transformation technique can be used to compute the inductance and other parasitic components for other semi-symmetrical shapes without performing high-intensity computational power and complex mathematical modeling.