# Modular Electromagnetic Transducer for Optimized Energy Transfer via Electric and/or Magnetic Fields

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{−2}, the safety limit for human exposure, operating at 2–100 GHz. Between 50–100 MHz, even class E or D amplifiers are difficult to control and rising switching losses are a big issue when the operating voltage is higher than 100 V. Under 27–50 MHz, switching losses can be minimized by using silicon carbide (SiC) semiconductors with minimum parasitic capacitance.

^{2}) and capacitive (61 × 61 cm

^{2}) system for wireless power transfer with inductive and capacitive (LC)-compensated topology at resonance was developed. It transferred 3 kW at a 1 MHz switching frequency and a 15 cm air gap with an efficiency of 94.5% [4]. Based on this, our work proposes coupling electrostatic and magnetic fields to optimize wireless power transfer by constructing a class-E fast-gate-controlled resonant Royer oscillator, to obtain resonant frequencies in the 1–10 MHz interval. A Royer circuit is a circuit with two metal-oxide-semiconductor field effect transistors (MOSFETs), with zero voltage switching (ZVS) tandem control that allows a significant reduction in switching losses. Thus, the transistors can be used at frequencies of 1–5 MHz and primary coil voltages over 120 V, without significant temperature spikes. ZVS control is realized by a TPS2814P fast-gate driver and by twin inductive “kickback”-type signals; these are produced by two ultra-fast Schottky suppressor diodes (or rectifiers) with a fast response of 50–75 ns (UF5404, UF5408), and with the help of a voltage divider designed for the maximum supply voltage of 120 V

_{dc}. Gate MOSFET signals are attenuated to 5–8 V by the voltage divider and used in tandem (pins 1 and 3 as input, and 5 and 7 as output on the transistor gate) by the TPS2814P circuit (see the system with a TPS2814P gate driver presented in the paper). The selected frequencies are a compromise between longitudinal wave attenuation for shorter distances (up to several meters in the near field zone) and increased wave directivity and propagation along the ground or other conducting surfaces.

## 2. Materials and Methods

#### 2.1. Methods of Design and Manufacturing RF Transducers

^{−1}, from low audio frequencies up to very high frequencies (VHF). A spatially resolved electric field measurement above the surface of an RF stripline was demonstrated. The method from [21] employs a feedback coil placed near the receiver in the sensor, to partially cancel the errors introduced by the magnetic properties of the soil. A soil sensitivity metric was introduced to quantify the effects of the soil, and this metric was used to optimize a circuit for driving the feedback coil. A similar method, presented in [21], can be used for a flat coil transducer array to detect the magnetic fields. A PCB-printed feedback coil can be mounted in front of each flat coil transducer to quantify the effects of the surroundings.

#### 2.2. Resonant Air Transformer Design

_{m}) is weak: under 0.5 (in most cases reaching 0.2). If we want to increase the secondary voltage, the obvious option is to increase the ratio between secondary and primary inductance until we reach the desired frequency. As a secondary option, the parallel circuit quality factor should be slightly adjusted to increase the current and to reach the desired resonant frequency. For optimum energy transfer efficiency, both options, i.e., good magnetic and electric coupling, must be considered.

_{m}= 1 and the ratio between voltages U depends only on the inductance L and capacitance C ratios:

^{2}cross-section and polyvinyl chloride (PVC) or polytetrafluoroethylene (PTFE) insulation. The wire can withstand an amperage of 1–3 A without overheating at 1.53 MHz and 1.9 MHz (five turns). If we want to decrease all dielectric losses, the ideal insulator is PTFE, which can be used for windings and PCB plates.

_{sec,C}). To reach 1.9 MHz, the secondary inductance was decreased to 0.25 mH and self-capacitance to 14–15 pF. From these experimental measurements and trial and error tests to achieve the desired resonance frequency, we observed that self-capacitance increased with the higher number of turns. This led us to the conclusion that a coefficient must be introduced in the internal capacitance formula or at least a part of this capacitance must be regarded in parallel.

_{2}):

_{2}). The PCB plate that was used had U-shaped aluminum cooling profiles. At 1.86 MHz, the total resonant secondary capacitance is 29–30 pF (14 pF self-capacitance); at 1.53 MHz, we have a higher number of turns (230) and the internal coil capacitance increases to 18 pF, for a 34 pF summed resonant capacitance (C

_{sec}). For further resonance frequency adjustment, we prefer to change the primary number of turns (±1 or 2 turns) and to cut the connection wires to the desired length until we reach exactly the same secondary resonance frequency value. It is the easiest way because we use a lower wire length (1–2 m) and fewer turns (5–10 turns):

^{−5}digits error). The Nagaoka coefficient decreases with the coil shape factor, $u={D}_{c}/{H}_{c}$. This signifies the increase in leakage flux as the coil becomes shorter: coil length or height H

_{c}becomes smaller when compared to the coil diameter D

_{c}. As the secondary coil becomes longer than its diameter, the dispersion flux is minimized and the magnetic-coupling efficiency is increased.

#### 2.3. New Self-Capacitance and Secondary Coil 3D Electrical Field Model

_{r}.

_{sec}is the inductance of the secondary air core transformer, C

_{sec,C}the inner capacitance of the parallel lumped element of the coil, C

_{t}the coil series self-capacitance, N = N

_{s}the number of turns and D

_{i}= d

_{i}is the insulated winding diameter. Assuming the Miller model for lumped capacitor and lumped element theory [11,26], an equivalent parallel capacitance of three times the coil self-capacitance should cause a coil to resonate at half of its self-resonance frequency. If we take only one third of the coil (L

_{sec}/3) and make it resonate with the previous equivalent parallel capacitance C

_{parallel}, we will obtain the same self-resonance frequency, but now the self-capacitance C

_{t}= 3 C

_{sec,C}= C

_{parallel}. So, from this equivalence we can see that all turn capacitances are now added in parallel for one third of the coil (L

_{sec}/3). Of course, this assumption can only be true if we consider a uniform distribution of inductance and inner capacitance, i.e., all these three parts are equal. The formula below is applied when we have two different dielectrics (mediums), insulation (i) and air (a) between the circular conductors (turns):

_{B}= D

_{c}is the inner coil diameter, not to be confused with d

_{c}, the winding conductor diameter, considered without insulation. C

_{a}is air capacitance between three turns, C

_{i}is the insulation capacitance of one circular wire, ε

_{r}is the relative insulation permittivity and ε

_{0}is the vacuum permittivity.

_{B}is the inner diameter of the coil, d

_{i}is the conductor insulation diameter, and g is the insulation thickness (2g is the space between turns when insulation is not considered). All the turns of the coil are wrapped tightly around each other, so the insulation diameter is the sum between conductor’s diameter d

_{c}and the 2g space between turns. The insulation capacitance of one circular wire can be expressed as:

_{B}is much bigger than the conductor insulation thickness g, D

_{B}>> 2g. So, the final logarithmic terms can be eliminated from Equation (8), to give:

_{c}, then this capacitance contribution can be neglected, and Equation (11) becomes:

_{r}is 2.7–4, p is little over 1, at maximum 1.2. For other dielectrics, such as PVC, ɛ

_{r}is 2.2–2.4; for thicker 0.7 mm insulation and a conductor over 1 mm, p increases up to 1.6.

_{x}, the variable electrical field distance between two turns in air. The minimum distance covered by the electrical field is the space between turns 2g, even though there is no insulation and the maximum covered distance is d

_{i}, the outside conductors’ diameter plus two times the insulation thickness.

_{c}or D

_{B}is much bigger than the height H

_{c}, u >> 1, the electrical field will concentrate only around this three-turn toroid. For longer coils, the H

_{c}height increases and the cumulated electric field encloses a larger elliptic torus (elliptical Gauss–Kummer series for the perimeter) with the same 3D symmetrical view. The self-capacitance, Equation (18) can be successfully applied to any coil having 0.80 < u < 10.

_{i}≈ d

_{c}, the first self-capacitance formula should also be considered:

_{i}, and the power factor m also appears inside the logarithmic term.

_{S}= 1, because we consider this distribution for a single lumped element (or a single three-turn model). Additionally, from Miller transmission-line theory [11,26], with uniform distribution of inductance and self-capacitance, our parallel self-capacitance C

_{sec,C}can be regarded as three equal capacitances mounted in series: ${C}_{\mathrm{sec},C}={\scriptscriptstyle \frac{1}{3}}{C}_{t}$.

_{i}<< H

_{C}, thus:

_{SRF}of a coil can be rapidly estimated by using the G(u) function from Figure 2 or Equation (24).

_{e}for longer or shorter coils. The k

_{e}factor is related to the electric field behavior for longer or shorter coils and is also related to the number of turns. In the special case of very small air RF coils, the power factor m is 1, because charges are very close to the coil center and d

_{i}≈ d

_{c}since the insulation thickness is very small. For large coils, with heights and diameters larger than 1 or 2 cm, charges are far away from the center, and m is 1.5. Here, we can see a resemblance to the electric dipole. When dipole charges are close to the calculated reference point, the power factor is 3/2; when dipole charges are far away from the reference point, the power factor is 3.

_{c}of 156 mm, a height H

_{c}of 43 mm and a self-resonance frequency (SRF) at 18.9 MHz, a self-capacitance of 10.5 pF was calculated instead of 9.33 pF according to the results from past SRF measurements performed at Applied Scientific Instrumentation, registered as a ham radio AF7NX [30]. For a coil with 7.2 turns, 6.73 µH inductance, 123 mm in diameter, and 49 mm in height, a self-resonance frequency of 22.6 MHz was determined. For this coil we have estimated a self-capacitance of 7.306 pF, instead of 7.369 pF according to the SRF measurements. For this, the Medhurst formula for short coils is out of range (18–20 pF). Another interesting measurement involved a 17-turn coil made from a cable with almost double relative permittivity (4 instead of 2.3). For this calculation we had to modify the power factor to 1.2 instead of 1.5. This 17-turn coil had 31.1 µH inductance, 103 mm in diameter, 49 mm in height and a SRF of 12.86 MHz.

_{c}= 35.2 cm, D

_{c}= 10.235 cm, d

_{i}= 0.31 mm, n = 1136 and H

_{c}= 48.5 cm, D

_{c}= 6.08 cm, d

_{i}= 0.31 mm, and n = 1515 were constructed and their self-resonance frequency (SRF) was investigated by de Miranda et al. [34]. For these Tesla coils (last rows in Table 2), the power factor is maximum (1.2–1.3) and k

_{e}is calculated by a fast Gauss–Kummer converging series. This series was considered because some experiments suggested that the self-capacitance could be 4/π, 27% higher than the predicted one. This is true only for coils with a higher number of turns.

#### 2.4. Updated Theoretical Near-Field Limits for Wireless Power Transmission

^{3}), the reactive electric or magnetic field part (1/r

^{2}) (in the boundary region around the antenna both fields are still contributing separately), and we have the final radiative electric field part where electric and magnetic fields are in phase and closely interlinked (1/r) [35]. Additionally, the total electric field has two components, one component is along the radius or distance r and the other is an angular component θ:

_{0}is the free space impedance, can be rewritten by expressing the E power degree in absolute values:

_{rad}is equivalent to P

_{T.}In terms of irradiance, ${P}_{R}/{A}_{R}={\scriptscriptstyle \frac{1}{2}}c{\epsilon}_{0}{E}^{2}\left(r,\theta \right)$, the total received power from the E field can be expressed as a modified Friis formula for the near-field case, where $\left({P}_{R}\le {P}_{T},0\le \eta \le 1\right)$ [36]:

_{m}= 8 m at 1.5 MHz and r

_{m}= 3 m at 4 MHz for the highest efficiency, can be described using the above Equation (32). For a frequency over 10 MHz, a maximum mounting distance of 1 m should be considered to obtain the maximum efficiency.

_{mloop}= 8…14.928 is the minimum function value to obtain the maximum mounting distance for an omnidirectional D

_{T}= D

_{R}= 1 loop. The maximum positioning distance ${r}_{mloop}$ will be ${r}_{mloop-1}$. Formulas (29)–(32) are only applicable for single loops or short coils. This theoretical limit lies between $\lambda /6\pi $ and $\lambda /10\pi $, thus this mounting distance limit should be considered for any transducer coil. For a real, short dipole $dL\ll \frac{\lambda}{4},{A}_{T}=\frac{{\lambda}^{2}{D}_{T}}{4\pi},{A}_{R}=\frac{{\lambda}^{2}{D}_{R}}{4\pi},{P}_{T}={R}_{r}{I}^{2},{R}_{r}=20{\pi}^{2}{\left(\frac{dl}{\lambda}\right)}^{2}\cong \frac{1}{2}{Z}_{0}{\left(\frac{dl}{\lambda}\right)}^{2}$, the flat coil case can also be extended as a short dipole array. From this formula, we extract the maximum usable distance r

_{m}between the transmitter and transducer, for the near-field case, depending on the coil positioning angle, frequency and directivity (surface) of the coil:

_{m}= 17 m at 1.5 MHz and r

_{m}= 9 m at 4 MHz, for best efficiency, are described by the formula:

_{m}has the minimum function value, different from 0, we should obtain the maximum mounting distance r

_{m}. Both the electric dipole and the current loop limits can be calculated by visually representing each function graph and by identifying the minimum inflexion points of f(θ,X

_{m}). All 3rd-order electrical dipole solutions are identified visually when the function f(θ,X

_{m}) = 0. After all the dipole solutions are extracted, we observe that we can further interpolate the results by using Equation (35).

_{R}= D

_{T}= 1, both antennas are omnidirectional with r

_{m}= 0.12 λ. For a short dipole flat coil case D

_{R}= D

_{T}= 1.5, r

_{m}= 0.14 λ; for a two flat coil dipoles array D

_{R}= D

_{T}= 2…3, r

_{m}= 0.2 λ; for eight or more dipoles D

_{R}= D

_{T}= 12, r

_{m}≥ λ. These limitations from Equation (35), only apply to where the near field is located. A similar approach was presented by Schantz, in experiments with antennas to prove their gain variation with distance [37]. Here, we extract the maximum recommended distance r

_{m}where the flat coil should be mounted. The gain or directivity D

_{T}is replaced by an effective coil or aperture diameter D

_{b}and frequency f.

_{R}, defined as the solid angle 4π/Ω

_{A}(steradians), i.e., about a 2.7π solid angle for a short dipole. The average transmitted power in the near field per unit solid angle is P

_{T}/4π; as the distance from the transmitter is increasing, we need to consider increasing the number of resonating coils or the total transducer surface.

_{p}coefficient (the same when using a power regression approximation) can be determined by using the logarithmic transformation:

## 3. Results

#### Wireless Power Transfer Measurements

- (i).
- The secondary long coil of the resonant air transformer was coupled by using the axially magnetic field with a single double-sided flat coil transducer, C0 configuration;
- (ii).
- The secondary long coil of the resonant air transformer together with one flat pancake coil as the capacitor plate was coupled by using both the magnetic and electric fields, with the same flat double-sided coil-capacitor plate mounted as the receiver or transducer, C1 configuration, with a diode bridge comprising 1N5817 rectifier diodes;
- (iii).
- The secondary long coil of the resonant air transformer was coupled by using only the magnetic field with four double-sided transducer plates mounted near and behind one another, (see Figure 5 (top)), C2 configuration, with SB1100 rectifier diodes;
- (iv).
- The secondary long coil of the resonant air transformer together with two flat pancake coils connected as capacitor plates were coupled by using both the magnetic and electric fields with the same four transducer plates with double-faced pancake coils, regarded also as the capacitors, C3 configuration, with SB1100 rectifier diodes. The flat pancake coils of the resonant air transformer have both sides connected in parallel to be used as capacitors plates, thus eliminating their internal capacitance.

_{0}was chosen to be 1 m, so the following expressions can be derived:

_{estim}). Since the estimated electric field variation with distance is close to the proposed power regression expressions, by using Equation (38), these pancake coils can be used as electric and magnetic field transducers.

_{estim}of the 3.2 order power was estimated at 1–5 MHz frequencies and a 300 mm maximum transducer positioning distance (Section 2.3). If we look at the measured voltage graph, it seems that the voltage drops faster the higher the distance (25–80 mm range), from 126 V to 50 V, when we use only the magnetic field coupling at 1.53 MHz resonance frequency (C2 configuration). When we connect two pancake plates to both secondary wire ends, we have the C3 configuration where both the magnetic and electric fields are enclosing the same area and the voltage is kept almost constant, from 127 V at 25 mm down to 122 V at 80 mm.

## 4. Discussion

_{e}flattening factor $\left(1\le {k}_{e}\le 4/\pi \right)$ and the power factor m $\left(1\le m\le 1.5\right)$. If the dielectric constant is not known, m should be considered as 1 for very small insulation thicknesses, or as 1.5 for usual dielectrics, such as PVC, and for thicker insulations. In the special cases of very small air RF coils, the power factor m is 1, because charges are very close to the coil center and d

_{i}≈ d

_{c}as the insulation thickness is very small. For large coils, with height and diameters higher than 1 or 2 cm, electric charges is further from the center, thus m should be considered as 1.5.

_{R}= D

_{T}= 1, both the electrical dipole transceivers are omnidirectional, at a maximum transducer mounting distance of 0.12 λ. For omnidirectional current loops, the maximum transducer positioning distance was 0.0562 λ.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Calculation of the Radiation Resistance for a Current Loop

_{rad}received at the terminals and the radiation resistance R

_{rad}is:

**Figure A1.**A circular loop carrying a current I and the magnetic flux passing through the surface S

_{Γ}(

**left**); the magnetic flux, Φ, is out of phase with the current by the angle α, which increases with frequency (

**right**) [42].

_{e}, obtained for the filiform coil, is:

_{e}:

**Figure A2.**A circular loop of radius a, lying in the x–y plane, centered at the origin, and carrying a current I.

^{′}are the distances considered from the origin point to point P.

_{r}cannot be neglected, and the Poynting’s vector S can be generalized as:

_{near}for the near-field case:

_{rad}. Since, in relation to the distance, the component ${\underset{\_}{H}}_{r}$ varies by 1/r2, it means that in the wave area only the component ${\underset{\_}{H}}_{\theta}$ has appreciable values. At great distances from the loop, the components ${\underset{\_}{E}}_{\varphi}$ and ${\underset{\_}{H}}_{\theta}$ form a plane electromagnetic wave that propagates along the unit vector ${\overline{u}}_{r}$. The energy flux is given by Poynting’s vector S:

_{rad}is:

_{rad}:

_{eff}is the ferromagnetic material relative permeability placed in the vicinity of the coil (µ

_{eff}= 1 for air or vacuum), λ is the wavelength and a is the loop radius. If the loop antenna or transducer has N turns, then the radiation resistance increases with a factor of N

^{2}, because the radiated power increases as I

^{2}.

_{rad}for far-field case, see Equations (A1) and (A22). The energy flux or Poynting’s vector S is determined for both the near- and far-field regions.

## References

- He, J.; Han, M.; Wen, K.; Liu, C.; Zhang, W.; Liu, Y.; Su, X.; Zhang, C.; Liang, C. Absorption-dominated electromagnetic interference shielding assembled composites based on modular design with infrared camouflage and response switching. Compos. Sci. Technol.
**2023**, 231, 109799. [Google Scholar] [CrossRef] - Aqeel, M.J.; Rosdiadee, N.; Sadik, K.G.; Haider, M.J.; Mahamod, I. Opportunities and Challenges for Near-Field Wireless Power Transfer: A Review. Energies
**2017**, 10, 1022. [Google Scholar] [CrossRef] - Seung, H.L.; Kyung, P.Y.; Myung, Y.K. 6.78-MHz, 50-W Wireless Power Supply Over a 60-cm Distance Using a GaN-Based Full-Bridge Inverter. Energies
**2019**, 12, 371. [Google Scholar] [CrossRef] [Green Version] - Lu, F.; Zhang, H.; Hofmann, H.; Mi, C.C. An Inductive and Capacitive Combined Wireless Power Transfer System with LC-Compensated Topology. IEEE Trans. Power Electron.
**2016**, 31, 8471–8482. [Google Scholar] [CrossRef] - Hui, S.Y.R.; Zhong, W.; Lee, C.K. A Critical Review of Recent Progress in Mid-Range Wireless Power Transfer. IEEE Trans. Power Electron.
**2014**, 29, 4500–4511. [Google Scholar] [CrossRef] [Green Version] - Grayson, Z.; Juan, M.R.D. Single-Turn Air-Core Coils for High-Frequency Inductive Wireless Power Transfer. IEEE Trans. Power Electron.
**2020**, 35, 2917–2932. [Google Scholar] - Minnaert, B.; Stevens, N. Maximizing the Power Transfer for a Mixed Inductive and Capacitive Wireless Power Transfer System. In Proceedings of the IEEE Wireless Power Transfer Conference (WPTC), Montreal, QC, Canada, 3–7 June 2018. [Google Scholar] [CrossRef]
- Abdelatty, O.; Wang, X.; Mortazawi, A. Position-Insensitive Wireless Power Transfer Based on Nonlinear Resonant Circuits. IEEE Trans. Microw. Theory Tech.
**2019**, 67, 3844–3854. [Google Scholar] [CrossRef] - Vincent, D.; Sang, P.H.; Williamson, S.S. Feasibility Study of Hybrid Inductive and Capacitive Wireless Power Transfer for Future Transportation. In Proceedings of the IEEE Transportation Electrification Conference and Expo (ITEC), Chicago, IL, USA, 22–24 June 2017; pp. 229–233. [Google Scholar] [CrossRef]
- Kung, M.L.; Lin, K.H. Enhanced Analysis and Design Method of Dual-Band Coil Module for Near-Field Wireless Power Transfer Systems. IEEE Trans. Microw. Theory Tech.
**2015**, 63, 821–832. [Google Scholar] [CrossRef] - Miller, J.M. Electrical oscillations in antennas and inductance coils. Proc. Inst. Radio Eng.
**1919**, 7, 299–326. [Google Scholar] [CrossRef] [Green Version] - Mariscotti, A. Measuring the Stray Capacitance of Solenoids with a Transmitting and a Receiving Coil. Metrol. Meas. Syst. J.
**2011**, XVIII, 47–56. [Google Scholar] [CrossRef] [Green Version] - Zhang, Z.; Pang, H.; Georgiadis, A.; Cecati, C. Wireless Power Transfer—An Overview. IEEE Trans. Ind. Electron.
**2019**, 66, 1044–1057. [Google Scholar] [CrossRef] - Garnica, J.; Chinga, R.A.; Lin, J. Wireless Power Transmission: From Far Field to Near Field. Proc. IEEE
**2013**, 101, 1321–1331. [Google Scholar] [CrossRef] - Leyh, G.E.; Kennan, M.D. Efficient Wireless Transmission of Power Using Resonators with Coupled Electric Fields. In Proceedings of the 40th North American Power Symposium 2008, Calgary, AB, Canada, 28–30 September 2008. ISBN 978-1-4244-4283-6. [Google Scholar] [CrossRef] [Green Version]
- Yoon, I.J.; Ling, H. Investigation of Near-Field Wireless Power Transfer Under Multiple Transmitters. IEEE Antennas Wirel. Propag. Lett.
**2011**, 10, 662–665. [Google Scholar] [CrossRef] - Park, J.; Tak, Y.; Kim, Y.; Kim, Y.; Nam, S. Investigation of Adaptive Matching Methods for Near-Field Wireless Power Transfer. IEEE Trans. Antennas Propag.
**2011**, 59, 1769–1773. [Google Scholar] [CrossRef] - Besic, H.; Kainz, A.; Kahr, M.; Hortschitz, W.; Keplinger, F. Electric Field Sensor with Stabilized Interferometric Readout. In Proceedings of the 21st International Conference on Solid-State Sensors, Actuators and Microsystems (Transducers), Online, 20–24 June 2021; pp. 1311–1314, ISBN 978-0-7381-2562-6. [Google Scholar] [CrossRef]
- Luo, W.; Ge, J.; Liu, H.; Wang, G.; Bai, B.; Yuan, Z.; Zhu, J.; Zhang, H.; Dong, H. An Orientation Sensitivity Suppression Method for an Overhauser Sensor Based on a Solenoid Coil. IEEE Sens. J.
**2021**, 21, 7793–7807. [Google Scholar] [CrossRef] - Toney, J.E.; Pollick, A.; Retz, J.; Sriram, S. Noncontact Electro-Optic Near Field Probe for Surface Electric Field Profiling. In Proceedings of the IEEE SENSORS Conference 2016, Orlando, FL, USA, 9 January 2017. [Google Scholar] [CrossRef]
- Waymond, R.S. Magnetic Feedback Amplifier for Electromagnetic Induction Sensors. In Proceedings of the IEEE International Geoscience and Remote Sensing Symposium IGARSS 2015, Milan, Italy, 26–31 July 2015. [Google Scholar] [CrossRef]
- Lundin, R.A. Handbook Formula for the Inductance of a Single-Layer Circular Coil. Proc. IEEE
**1985**, 73, 1428–1429. [Google Scholar] [CrossRef] [Green Version] - Knight, D.W. The self-resonance and self-capacitance of solenoid coils: Applicable theory, models and calculation methods. Tech. Rep.
**2016**. [Google Scholar] [CrossRef] - Grandi, G.; Kazimerzuc, M.K.; Massarini, A.; Reggiani, U. Stray Capacitances of Single-Layer Solenoid Air-Core Inductors. IEEE Trans. Ind. Appl.
**1999**, 35, 1162–1168. [Google Scholar] [CrossRef] [Green Version] - Grandi, G.; Kazimerzuc, M.K.; Massarini, A.; Reggiani, U. Stray capacitance of single-layer aircore inductors for high-frequency applications. IEEE Ind. Appl. Conf.
**1996**, 3, 1384–1388. [Google Scholar] - Chute, F.S.; Vermeulen, F.E. On the self-capacitance of solenoidal coils. Can. Electr. Eng. J.
**1982**, 7, 31–37, ISSN 0700-9216. [Google Scholar] [CrossRef] - Medhurst, R.G. HF resistance and self capacitance of single layer solenoids—Part 1. Wirel. Eng.
**1947**, 24, 35–43. [Google Scholar] - Medhurst, R.G. HF resistance and self capacitance of single layer solenoids—Part 2. Wirel. Eng.
**1947**, 24, 80–92. [Google Scholar] - Kyocera, A.V.X. AL Series Air Core RF Inductors. Available online: https://ro.mouser.com/pdfDocs/alaircore.pdf (accessed on 1 March 2022).
- Rondeau, G. Designing Self Resonant Coils for Antenna Traps and Chokes. Available online: https://squashpractice.com/2021/01/10/designing-self-resonant-coils-for-antenna-traps-and-chokes/ (accessed on 10 January 2021).
- Pettit, A. Solenoid Self-Resonance Measurements by Alex Pettit, KK4VB. Available online: https://g3ynh.info/zdocs/magnetics/appendix/kk4vb_srf.html (accessed on 1 March 2022).
- Saini, D.K.; Ayachit, A.; Kazimierczuk, M.K. Design and characterisation of single-layer solenoid air-core inductors. IET Circuits Devices Syst.
**2019**, 13, 211–218. [Google Scholar] [CrossRef] - Russell, A. Problems in connection with two parallel electrified cylindrical conductors. J. Inst. Electr. Eng.
**1926**, 64, 238–242. [Google Scholar] [CrossRef] - Miranda, C.M.; Pichorim, S.F. Self-resonant frequencies of air-core single-layer solenoid coils calculated by a simple method. Electr. Eng.
**2014**, 97, 57–64. [Google Scholar] [CrossRef] - Moorey, C.L.; Holderbaum, W. A Study on the Wireless Power Transfer Efficiency of Electrically Small, Perfectly Conducting Electric and Magnetic Dipoles. Prog. Electromagn. Res. C
**2017**, 77, 111–121. [Google Scholar] [CrossRef] [Green Version] - Shaw, J.A. Radiometry and the Friis transmission equation. Am. J. Phys.
**2013**, 81, 33–37. [Google Scholar] [CrossRef] [Green Version] - Schantz, H. A Near-field propagation law & a novel fundamental limit to antenna gain versus size. In Proceedings of the IEEE Antennas and Propagation Society International Symposium 2005, Washington, DC, USA, 3–8 July 2005; Volume 3A, pp. 237–240. [Google Scholar]
- Lu, F.; Zhang, H.; Hofmann, H.; Mi, C.C. A Double-Sided LCLC-Compensated Capacitive Power Transfer System for Electric Vehicle Charging. IEEE Trans. Power Electron.
**2015**, 30, 6011–6014. [Google Scholar] [CrossRef] - Lope, I.; Carretero, C.; Acero, J. First self-resonant frequency of power inductors based on approximated corrected stray capacitances. IET Power Electron.
**2021**, 14, 257–267. [Google Scholar] [CrossRef] - Guan, Y.; Wang, Y.; Wang, W.; Xu, D. A High-Frequency CLCL Converter Based on Leakage Inductance and Variable Width Winding Planar Magnetics. IEEE Trans. Ind. Electron.
**2018**, 65, 280–290. [Google Scholar] [CrossRef] - Darie, E.; Pîslaru-Dănescu, L. Distribution of magnetic field and exposure level around of overhead power lines. Electroteh. Electron. Autom.
**2020**, 68, 59–65. [Google Scholar] [CrossRef] - Mocanu, C.I. Electromagnetic Field Theory (Teoria Câmpului Electromagnetic); Publishing House Editura Didactică și Pedagogică: Bucharest, Romania, 1984. [Google Scholar]

**Figure 1.**Coil antenna as a transmission line, with self-capacitance of the turns regarded in parallel; (

**a**) long coil elliptic torus model with three long turns or three series C

_{t}capacitors; (

**b**) short coil toroid model with three turns or three series capacitors.

**Figure 2.**Coil geometry dimensionless function (factor) derived from the Lundin inductance formula and the Medhurst coil self-capacitance formula; here ${\epsilon}_{m}=1$.

**Figure 3.**Coil capacitance dimensionless function (factor), comparison between the Medhurst formula and our theoretical logarithmic function; (

**a**) 0 < u < 1, m = 1 and (

**b**) u > 0, m = 1.5.

**Figure 4.**(

**a**) Absolute power degree for voltage (+1 for the E field) function of the frequency spectrum (1–10 MHz) and an E field angle, in radians; (

**b**) absolute power degree for voltage (+1 for the E electrical field) function of the frequency spectrum (1–10 MHz) and a transducer distance in m; (

**c**) absolute power degree for voltage (+1 for the E field) at a fixed 0.3 m distance, function of the frequency spectrum (1–5 MHz) and an E field angle in radians.

**Figure 5.**Photograph of the wireless power system for the C3 configuration, pancake coils at 30 mm distance (

**top**), schematic of the system with a TPS2814P gate driver, a Royer circuit with ZVS, a resonant air transformer and two, four or more double-sided pancake coil transducers (

**bottom**), where E is the coupling electric field and M is the coupling magnetic field (or mutual inductance).

**Figure 6.**(

**a**) Measured DC current for one pancake plate (C1 configuration) and a rectifier made from four 1N5817 diodes. (

**b**) Measured DC current for four transducer plates, a long RF coil case (C2 configuration) or both an RF coil and pancake plates (C3 configuration).

**Figure 7.**(

**a**) Measured DC voltage only for a long RF coil (C2 configuration) and for both an RF coil and all pancake plates. (

**b**) Transmitter and transducer plates at a 150 mm distance, producing 0.7 W to power 14 LEDs.

**Figure 8.**(

**a**) Measured DC power for the C1 configuration, one long RF coil, one flat pancake coil as the transmitter and one pancake plate as the transducer. (

**b**) Measured DC power for four transducers in the C2 and C3 wireless configurations, (C2 configuration) a long RF coil as the transmitter and (C3 configuration) both an RF coil and two pancake plates as the transmitters.

**Table 1.**RF inductors measured self-capacitance compared to the calculated Medhurst and our logarithmic formulas.

AVX RF Inductors | Part. No. AL02390N0 | Part. No. AL023246N | Part. No. AL023422N | Part. No. AL12B43N0 | Part. No. AL01627N0 | Part. No. AL016100N |
---|---|---|---|---|---|---|

Turns No. | 9 | 15 | 18 | 10 | 5 | 9 |

C_{measSRF} [pF] | 0.2166 | 0.2194 | 0.2058 | 0.2610 | 0.1287 | 0.1759 |

C_{calcMED} [pF] | 0.3026 | 0.3026 | 0.3026 | 0.3130 | 0.3514 | 0.3514 |

C_{calcLOG} [pF] | 0.2205 | 0.2271 | 0.2295 | 0.1941 | 0.1636 | 0.1714 |

**Table 2.**Tesla coils and RF cable inductors measured self-capacitance compared to the calculated Medhurst and our logarithmic formulas.

SRF Tested Inductors | 7.6 µH, u = 3.62 | 6.73 µH, u = 2.51 | 31.1 µH, u = 2.10 | 215 µH, u = 0.79 | 16.4 mH, u = 0.125 | 33.6 mH, u = 0.290 |
---|---|---|---|---|---|---|

Turns No. | 6.2 | 7.2 | 17 | 80 | 1515 | 1136 |

C_{measSRF} [pF] | 9.330 | 7.369 | 4.924 | 2.272 | 6.824 | 6.329 |

C_{calcMED} [pF] | 19.501 | 18.443 | 14.67 | 2.700 | 6.532 | 6.274 |

C_{calcLOG} [pF] | 10.515 | 7.306 | 5.027 | 2.018 | 6.967 | 6.619 |

RF Coil and Transducers Configuration | Distance (mm) | DC Current (mA) | DC Voltage (V) | DC Power (W) |
---|---|---|---|---|

C1 | 25 | 80.0 | 34 | 2.72 |

C2 | 25 | 164.0 | 126 | 20.66 |

C3 | 25 | 195.5 | 127 | 24.83 |

C1 | 155 | 7.1 | 12.4 | 0.09 |

C2 | 155 | 8.1 | 18.5 | 0.15 |

C3 | 155 | 11.0 | 27 | 0.30 |

C1 | 295 | 1.52 | 3.5 | 0.0053 |

C2 | 295 | 0.8 | 7.63 | 0.0061 |

C3 | 295 | 2.0 | 5.27 | 0.0105 |

SRF Transducers Tests | I_{0} (A) | bp-Power Degree | U_{0} (V) | bp-Power Degree |
---|---|---|---|---|

C1 (E+H, 1Rec) | 0.000328 | 1.689 | 0.4825 | 1.6965 |

C2 (H, 4Rec) | 0.000100 | 2.300 | 1.4936 | 1.3466 |

C3 (E+H, 4Rec) | 0.000200 | 2.180 | 0.3740 | 2.2245 |

E_{meas} (H_{meas}) | - | 3.180 | - | 3.2245 |

E_{estim} (H_{estim}) | - | 3.200 | - | 3.2000 |

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## Share and Cite

**MDPI and ACS Style**

Zărnescu, G.-C.; Pîslaru-Dănescu, L.; Tiliakos, A.
Modular Electromagnetic Transducer for Optimized Energy Transfer via Electric and/or Magnetic Fields. *Sensors* **2023**, *23*, 1291.
https://doi.org/10.3390/s23031291

**AMA Style**

Zărnescu G-C, Pîslaru-Dănescu L, Tiliakos A.
Modular Electromagnetic Transducer for Optimized Energy Transfer via Electric and/or Magnetic Fields. *Sensors*. 2023; 23(3):1291.
https://doi.org/10.3390/s23031291

**Chicago/Turabian Style**

Zărnescu, George-Claudiu, Lucian Pîslaru-Dănescu, and Athanasios Tiliakos.
2023. "Modular Electromagnetic Transducer for Optimized Energy Transfer via Electric and/or Magnetic Fields" *Sensors* 23, no. 3: 1291.
https://doi.org/10.3390/s23031291