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The data on which Medhurst's semi-empirical self-capacitance formula is based are re-analysed in a way that takes the permittivity of the coil-former into account. The updated formula is compared with theories attributing self-capacitance to the capacitance between adjacent turns, and also with transmission-line theories. The inter-turn capacitance approach is found to have no predictive power. Transmission-line behaviour is corroborated by measurements using an induction loop and a receiving antenna, and by visualising the electric field using a gas discharge tube. In-circuit solenoid self-capacitance determinations show long-coil asymptotic behaviour corresponding to a wave propagating along the helical conductor with a phase-velocity governed by the local refractive index (i.e., v = c if the medium is air). This is consistent with measurements of transformer phase error vs. frequency, which indicate a constant time delay. These observations are at odds with the fact that a long solenoid in free space will exhibit helical propagation with a frequency-dependent phase velocity > c. The implication is that unmodified helical-waveguide theories are not appropriate for the prediction of self-capacitance, but they remain applicable in principle to open-circuit systems, such as Tesla coils, helical resonators and loaded vertical antennas, despite poor agreement with actual measurements. A semi-empirical method is given for predicting the first self-resonance frequencies of free coils by treating the coil as a helical transmission-line terminated by its own axial-field and fringe-field capacitances.

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... An accurate formula of the inner capacitance of a single layer RF coil, derived from Medhurst data and rearranged by Knight [23], can be expressed as: ...

... 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 [23,[31][32][33]. The coil self-capacitance obtained from the Medhurst formula was 2.7 pF. ...

... Our parallel equivalent self-capacitance formula is derived from the lumped element and the Miller theories-recent findings about the self-resonance frequency (SRF) for coils also gives credit to the lumped element theory. The Grandi-Kazimerzuc-Massarini-Reggiani (GKMR) theory about self-capacitance, widely used in the works of Knight [23][24][25], considers all turns to be in series connection. Although this is true locally for the separation materials (i.e., air, conductor insulation and coil-formers), the electric field of the coil behaves more similar to the transmission-line theory-the coil is essentially an antenna. ...

In this paper, a modular electromagnetic transducer that achieves the optimal transfer of energy from the electric and/or magnetic fields is proposed. Both the magnetic field resonance coupling and the influence of the electric field near the copper transducers of the printed circuit board and inside the FR4-type epoxy material are considered. In our printed arrays of flat transducers, we consider face-to-face capacitances for the study of resonance coupling. Because the space between coil turns is almost double the plate thickness, the coplanar capacitance can be ignored for frequencies under 2 MHz. A radio frequency (RF) transmitter and transducer were built to demonstrate the increased energy transfer efficiency when using both electric and magnetic fields in the near-field region. The transversal leakage flux coupling of a long RF coil was more efficient than a simple axial magnetic field coupling when using pancake transceiver coils. The optimal configuration having one long coil at the base and two or more flat coils as capacitor plates near coil ends generated the highest tandem of magnetic and electrical fields. A power regression tool was used to convert and simplify the transducer current and voltage variation with distance. In this regard, the current change corresponded to magnetic field variation and the voltage change to the electric field variation. New formulas for estimating the near-field region and the self-capacitance of the RF transformer coil are proposed; the optimal function in the frequency domain for a given transducer distance was defined by simulation.

... The prevailing approach in the literature is to analyze MC systems under magnetoquasistatic (MQS) assumption which neglects the far-field terms [1], [2], [4]- [12], [14]- [20], [24], [26], [27], [31], as well as high-frequency (HF) effects in the coil transducers (e.g., skin and proximity effects [33], radiation losses, and the coil's self-resonance frequency (SRF) [34]- [36]). This is justified by the relatively low frequency used, in which the NF components are dominant. ...

... However, prediction models for the SRF of a given coil are not well established and apply mostly to air coils in free space (see Appendix C). Based on recent transmission-line approach, Knight [36] suggests that the first resonance of an air-coil in free space occurs when the total wire length l w equals half the wavelength, i.e., l w = λ/2. Using the general expression of λ [46], this may also be applied to a dielectric medium but not to a lossy medium. ...

... For the coaxial orientation k tr = 1, and for the parallel orientation k tr = 0.5. Alternatively, using spherical coordinateŝ n R X = a r R Xêr + a θ R Xêθ + a φ R Xêφ (36) and k tr ≡ (a r R X cos θ + a θ R X 2 sin θ). ...

Magnetic communication systems are most often analyzed assuming magneto-quasistatic conditions, which neglect full-field terms and high-frequency effects in the transmitting and receiving coils. Such approximations may lead to non-optimal designs in terms of operating frequency, size, and coil orientation. This paper presents an optimal design approach for maximizing the channel capacity, using both magneto-quasistatic and fullwave analysis, while incorporating high-frequency effects, such as skin and proximity effects, radiation losses, and the selfresonance of coils. For a given medium and required transmission distance, the optimal operating frequency is such, for which the receiver is located be in the radiative near-field, and not in the reactive near-field. The optimal power allocation and the resulting channel capacity were obtained using a "water-filling" algorithm. The high-frequency effects reduced the signal-to-noise ratio, and limited operating frequency and coil size. This is especially true for short-distance transmission through low-loss media, where the optimal signal frequency is relatively high. In addition, full-wave analysis significantly improved potential data rates compared to the typical magneto-quasistatic approach. This improvement was achieved due to a higher operating frequency, and sometimes a change of mutual orientation from coaxial to parallel. Electromagnetic simulations validated the primary effects presented here.

... The CC of MCS is relatively small due to the low frequencies involved [4], the rapid decay with distance which results in low SNR, and the narrow bandwidth dictated by the resonance operation mode [15]. A common approach to overcome this drawback is to deploy many passive relay coils between TX and RX in the form of a network [27], [6]- [13], [17], [53], [55]-[58] ("magnetic waveguide"). However, this increases the complexity and price of the system. ...

... However, this refers only to the central frequency (fopt in our case). Based on [27] we choose the limit of the self-resonance frequency (SRF) of the TX coil. The SRF determines the maximum usable frequency of an inductor before it switches to capacitive reactance. ...

... The SRF determines the maximum usable frequency of an inductor before it switches to capacitive reactance. According to the extensive analytical and experimental work of [27], the first SRF of coil in free space occurs when the (18) where S.F stands for "Safety margin factor." This result, however, applies only to free space and may not be necessarily valid when the coil is embedded in a lossy medium, or winded over a permeable core. ...

... Este efecto fue estudiado por Nagaoka (Nagaoka, 1909), que presentó un factor de corrección tabulado para bobinas monocapa con núcleo de aire de longitud finita. Knight (Knight, 2010) reformuló el coeficiente tabulado para obtener la ecuación de Nagaoka. El desarrollo posterior del coeficiente de Nagaoka para bobinas cortas con piezas en su interior dio lugar al coeficiente de 41 Capítulo 2. Estado del arte y marco teórico Nagaoka efectivo, en el que se tiene en cuenta la fracción de bobina ocupada por la pieza (Kennedy et al., 2011a). ...

... En una segunda parte del estudio realizado, Kennedy et al. (Kennedy et al., 2011b) investigaron los métodos de diseño presentados anteriormente para el calentamiento por inducción de probetas de aluminio. Los autores compararon la potencia medida experimentalmente con los valores calculados mediante métodos analíticos, concluyendo que los resultados más precisos los proporciona el modelo desarrollado a partir del trabajo de Davies (Davies, 1990) incluyendo el coeficiente de corrección de Nagaoka para bobinas cortas (Knight, 2010). Sin embargo, se estudiaron materiales lineales o casi lineales y no analizaron la precisión de los modelos para materiales no lineales. ...

The induction hardening process is a surface hardening technique that is increasingly used in industry due to the advantages it offers over other conventional heat treatments. It is typically used on critical components that are subjected to high loads and high-pressure contacts, which require an elevated surface hardness. Although the industry’s interest in this heat treatment in increasing, the definition of the most important process parameters is generally limited to the technicians’ know-how and to previous experiences, increasing the associated costs and the time-to-market, since the design of the process is normally carried out by means of trial-and-error procedures. The simulation of induction hardening is highly complex and computationally expensive because of the numerous interactions between physical fields. In the literature review, it has been observed that the application of induction hardening in complex industrial components is limited by the lack of numerical models capable of predicting the consequences of induction hardening.
Numerical simulation is therefore key in the development of the induction hardening process and its effective implementation in modern industry. Furthermore, the study of the implications that induction hardening has on the in-service behavior of hardened components is a task to which engineers and scientists have devoted their attention in recent years, although it is not yet fully resolved.
In this doctoral thesis, the simulation of the induction hardening process is addressed in order to solve the limitations found in the existing literature. A numerical model to efficiently simulate the induction heating phase of ferromagnetic materials has been developed, which couples the electromagnetic and thermal fields through a semi-analytical model.This model has been experimentally validated on low alloy 42CrMo4 steel cylinders, obtaining more accurate and 80% faster results than using other commercial software. Additionally, a coupled multiphysics model has been developed to simulate the second phase of the induction hardening process. This model couples thermal, mechanical and microstructural physics and has been experimentally validated in terms of prediction of microstructure, hardness and residual stress generation. The developed model, unlike other commercial software, allows to evaluate the impact of the various models used to describe different phenomena occurring during quenching. In this thesis, the impact of Transformation Induced Plasticity (TRIP) on the low alloy 42CrMo4 has been investigated, concluding that computational models should include this effect to improve residual stress predictions.
Finally, the developed models have been combined with experimental techniques to investigate the influence of induction hardening residual stresses on rolling contact fatigue (RCF) behavior. In this study, a computational methodology has been developed to incorporate residual stresses in RCF life analyses and the influence of residual stresses has been studied numerically and experimentally using a modified three-ball-on-rod test. It has been observed that compressive residual stresses at the hardened case extend the life of the component and modify the depth at which the most critical damage occurs. For the numerical analysis, the Dang Van multiaxial criterion has been used and three critical shear stress quantities (Tresca, orthogonal shear and octahedral shear) have been compared in terms of life prediction and critical damage location. Numerical and experimental results indicate that the orthogonal shear quantity predicts more accurate results. The contributions made in this doctoral thesis are expected to reduce the current
gap between the simplified models generally developed in the literature and the industrial cases of higher complexity, allowing empirical trial-and-error procedures to be considerably reduced and increasing the control over the resulting material characteristics obtained with the process. With this change in the paradigm, it is expected that, at an industrial level, greater economical, temporal and energetical efficiency can be achieved, also reducing the number of defectives.

... However, induction heating of nonlinear ferromagnetic materials was not studied, as well as the applicability of the proposed model for multi-turn coils. In their follow-up work [6], the authors investigate the previously presented design methods for induction heating of aluminum billets, comparing the experimentally measured power with calculated values using these methods, concluding that the most accurate results are provided by the model developed from Davies [7] including the Nagaoka correction coefficient for short coils [8]. However, linear or nearly-linear materials were studied and the accuracy of the models for nonlinear materials was not discussed. ...

... The presented semi-analytical approach is divided into several steps, see workflow in Fig. 4. First, the governing electromagnetic equations are evaluated numerically by commercial finite element software ANSYS Maxwell, Release 2019R1 [24]. Solving a time-harmonic electromagnetic FE analysis using equation (8) and the linearized magnetic permeability shown in equation (9) enables us to determine the fields generated by the inductor at room temperature according to the input current, the geometry of the billet and the inductor, the coil-workpiece configuration and the surrounding media. The magnetic field in the workpiece at room temperature is extracted from this analysis, referred as H T0 in the flowchart. ...

The numerical simulation of the induction heating process can be computationally expensive, especially if ferromagnetic materials are studied. There are several analytical models that describe the electromagnetic phenomena. However, these are very limited by the geometry of the coil and the workpiece. Thus, the usual method for computing more complex systems is to use the finite element method to solve the set of equations in the multiphysical system, but this easily becomes very time consuming. This paper deals with the problem of solving a coupled electromagnetic - thermal problem with higher computational efficiency. For this purpose, a semi-analytical modeling strategy is proposed, that is based on an initial finite element computation, followed by the use of analytical electromagnetic equations to solve the coupled electromagnetic-thermal problem. The usage of the simplified model is restricted to simple geometrical features such as flat or curved surfaces with great curvature to skin depth ratio. Numerical and experimental validation of the model show an average error between 0.9% and 4.1% in the prediction of the temperature evolution, reaching a greater accuracy than other analyzed commercial softwares. A 3D case of a double-row large size ball bearing is also presented, fully validating the proposed approach in terms of computational time and accuracy for complex industrial cases.

... where the numerical pre-factors are the empirical coefficients 20 . Assuming the shield and the coil as coaxial cylinders, the shield-to-coil (s-c) capacitance is calculated using the relation, where τ is the winding pitch of the coil and the numerical term arises from unit conversion of the formula, as given in ref. ...

... where p is the integer number of RF wavelength λ RF that gets accommodated within x. Figure 4(a) shows numerically estimated values of C ex for our ion-trap and calculated φ following the Eqn. (20) as a function of the machining inaccuracy. Even though there is no direct relation, the capacity factor in our ion-trap geometry is α c = C ex /φ = 1.9, where C ex and φ are in fF and milli-degree, respectively. ...

Capacitive, inductive and resistive loads of an ion-trap system, which can be modelled as LCR circuits, are important to know for building a high accuracy experiment. Accurate estimation of these loads is necessary for delivering the desired radio frequency (RF) signal to an ion trap via an RF resonator. Of particular relevance to the trapped ion optical atomic clock, determination of these loads lead to accurate evaluation of the Black-Body Radiation (BBR) shift resulting from the inaccurate machining of the ion-trap itself. We have identified different sources of these loads and estimated their values using analytical and finite element analysis methods, which are found to be well in agreement with the experimentally measured values. For our trap geometry, we obtained values of the effective inductive, capacitive and resistive loads as: 3.1 μH, 3.71 (1) μH, 3.68 (6) μH; 50.4 pF, 51.4 (7) pF, 40.7 (2) pF; and 1.373 Ω, 1.273 (3) Ω, 1.183 (9) Ω by using analytical, numerical and experimental methods, respectively. The BBR shift induced by the excess capacitive load arising due to machining inaccuracy in the RF carrying parts has been accurately estimated, which results to a fractional frequency shift of 6.6 × 10−17 for an RF of 1 kV at 2π × 15 MHz and with ±10 μm machining inaccuracy. This needs to be incorporated into the total systematic uncertainty budget of a frequency standard as it is about one order of magnitude higher than the present precision of the trapped ion optical clocks.

... David W. Knight develops an empirical model called HTL'//CEF [31] to simulate the nominal helical velocity factor u hx /c in a free solenoid coil. The nominal helical velocity factor is then used to calculate the SRF of the free solenoid coil. ...

... For solenoid coil SRF calculation, David W. Knight developed an empirical model called HTL'||C EF in [31] to simulate the nominal helical velocity factor u hx /c in a free solenoid coil based on the nominal helical velocity factor data. ...

This paper presents a design methodology for a 3-coil magnetic resonance wireless power transfer (WPT) system with a long transfer distance (up to 20 cm) and a small implanted receiver (RX) (2mm in diameter) at a specific frequency. The methodology aims to find out the optimal value of dimensional parameters (i.e., coil diameter, gap interval and coil turn number) of the transmitter (TX) coils to maximize the magnetic field strength at the target distance while keeping the coil self-resonant frequency (SRF) twice of the target operational frequency. Firstly, the circuit model of the TX circuits is developed, which include a single-turn coupling coil and a multi-turn primary coil. Secondly, the co-dependences between the dimensional parameters are analyzed, which shows the dominant factors and secondary factors of each dimensional parameter, and how the optimal values of dimensional parameters are changed by these factors. Based on the analysis, design flow of the TX circuit is proposed to decide the optimal values of dimensional parameters given the transfer distance, source voltage, operational frequency and wire diameter. Using the design flow, optimal values of dimensional parameters are predicted for 20-cm transfer distance and 16-cm transfer distance and are verified with finite element analysis (FEA) software COMSOL Multiphysics. With the optimal TX design, the power received by a 2-mm ferrite core solenoid RX is calculated. At 20 cm transfer distance, up to 4.3 mW can be achieved in air, and 0.8 mW can be achieved in conductive human tissue.

... Secondary winding inductance of TRT [9]: ...

... The scheme of single-wire electric power transmission system.III. CALCULATION OF PARAMETERS AND SPICE MODELINGSelf capacitance of a Tesla transformer secodary winding (L2) can be calculated[9] by formula (was supported by the student research project SGS-2018-023. l -coil length, D -coil diameter, ε0 -permittivity of free space medium. ...

... where the numerical pre-factors are the empirical coefficients 20 . Assuming the shield and the coil as coaxial cylinders, the shield-to-coil (s-c) capacitance is calculated using the relation, where τ is the winding pitch of the coil and the numerical term arises from unit conversion of the formula, as given in ref. ...

... where p is the integer number of RF wavelength λ RF that gets accommodated within x. Figure 4(a) shows numerically estimated values of C ex for our ion-trap and calculated φ following the Eqn. (20) as a function of the machining inaccuracy. Even though there is no direct relation, the capacity factor in our ion-trap geometry is α c = C ex /φ = 1.9, where C ex and φ are in fF and milli-degree, respectively. ...

Capacitive, inductive and resistive loads of an ion-trap system, which can be modelled as LCR circuits, are important to know for building a high accuracy experiment. Accurate estimation of these loads is necessary for delivering the desired radio frequency (RF) signal to an ion trap via an RF resonator. Of particular relevance to the trapped ion optical atomic clock, determination of these loads lead to accurate evaluation of the Black-Body Radiation (BBR) shift resulting from the inaccurate machining of the ion-trap itself. We have identified different sources of these loads and estimated their values using analytical and finite element analysis methods, which are found to be well in agreement with the experimentally measured values. For our trap geometry, we obtained values of the effective inductive, capacitive and resistive loads as: 3.1 μH, 3.71 (1) μH, 3.68 (6) μH; 50.4 pF, 51.4 (7) pF, 40.7 (2) pF; and 1.373 Ω, 1.273 (3) Ω, 1.183 (9) Ω by using analytical, numerical and experimental methods, respectively. The BBR shift induced by the excess capacitive load arising due to machining inaccuracy in the RF carrying parts has been accurately estimated, which results to a fractional frequency shift of 6.6 × 10−17 for an RF of 1 kV at 2π × 15 MHz and with ±10 μm machining inaccuracy. This needs to be incorporated into the total systematic uncertainty budget of a frequency standard as it is about one order of magnitude higher than the present precision of the trapped ion optical clocks.

... La bobina secundaria presenta una capacitancia a tierra, además de la capacitancia que hay entre las espiras que la conforman. Para determinar la capacitancia producida por la bobina secundaria, usamos el modelo de un solenoide cilíndrico, la denominada formula de Medhurst, la cual nos da la capacitancia por unidad de longitud 2 [5]: Donde = 1.27 corresponde al radio del núcleo, y ℎ2 = 5.8 a la altura de la bobina secundaria, 2=1.32 La capacitancia real de la bobina secundaria, sin colocarle ninguna carga en su punta que esta al aire, es la siguiente: ...

Se realiza un estudio tanto teórico como experimental de los circuitos resonantes aplicados a la transmisión y recepción de energía eléctrica inalámbrica. Se construye una bobina de Tesla de estado sólido para lo cual, se calcula la frecuencia de resonancia y el voltaje máximo que se genera al tener una carga en la punta de la bobina secundaria. Mediante un receptor de ondas electromagnéticas diseñado para resonar a la misma frecuencia de la bobina de Tesla, se mide el voltaje emitido por el secundario. Mediante un microcontrolador PSoC, se despliega en una pantalla LCD para medir el alcance, la frecuencia, la potencia y el voltaje efectivo del secundario.

... To wit, there is no accurate, generalized (pseudo-)analytical solution for the self-capacitance of a coil but it is an on-going topic [34][35][36]. For a single-layer coil in free-space, but connected to a circuit, a self-capacitance arises due to the stored axial electric field of a wave propagating along a helical wire [37] and inter-turn capacitance from separated charges of differing potential due to nonzero resistance along the wire [12]. Separate coil layers also affect the self-capacitance. ...

... A resonant MM can in general be described as a RLC circuit where the R, L and C are the inductance, capacitance and resistance, respectively, of an equivalent circuit model of the structure. The resonance frequency of such a circuit would then be f L C 1 2p = , where the inductance L and the capacitance C of a spiral are approximated with the expressions valid for concentric coils [53] and, respectively, for a solenoid [54]: ...

We use finite element simulations in both the frequency and the time-domain to study the terahertz resonance characteristics of a metamaterial (MM) comprising a spiral connected to a straight arm. The MM acts as a RLC circuit whose resonance frequency can be precisely tuned by varying the characteristic geometrical parameters of the spiral: inner and outer radius, width and number of turns. We provide a simple analytical model that uses these geometrical parameters as input to give accurate estimates of the resonance frequency. Finite element simulations show that linearly polarized terahertz radiation efficiently couples to the MM thanks to the straight arm, inducing a current in the spiral, which in turn induces a resonant magnetic field enhancement at the center of the spiral. We observe a large (approximately 40 times) and uniform (over an area of ~10 μm²) enhancement of the magnetic field for narrowband terahertz radiation with frequency matching the resonance frequency of the MM. When a broadband, single-cycle terahertz pulse propagates towards the MM, the peak magnetic field of the resulting band-passed waveform still maintains a six-fold enhancement compared to the peak impinging field. Using existing laser-based terahertz sources, our MM design allows to generate magnetic fields of the order of 2 T over a time scale of several picoseconds, enabling the investigation of nonlinear ultrafast spin dynamics in table-top experiments. Furthermore, our MM can be implemented to generate intense near-field narrowband, multi-cycle electromagnetic fields to study generic ultrafast resonant terahertz dynamics in condensed matter.

... A resonant MM can in general be described as a RLC circuit where the R, L and C are the inductance, capacitance and resistance, respectively, of an equivalent circuit model of the structure. The resonance frequency of such a circuit would then be f = 1/2π √ LC, where the inductance, the capacitance of a spiral are approximated with the expressions valid for concentric coils for L [53], and for a solenoid for C [54]: ...

We use finite element simulations in both the frequency and the time-domain to study the terahertz resonance characteristics of a metamaterial (MM) comprising a spiral connected to a straight arm. The MM acts as a RLC circuit whose resonance frequency can be precisely tuned by varying the characteristic geometrical parameters of the spiral: inner and outer radius, width and number of turns. We provide a simple analytical model that uses these geometrical parameters as input to give accurate estimates of the resonance frequency. Finite element simulations show that linearly polarized terahertz radiation efficiently couples to the MM thanks to the straight arm, inducing a current in the spiral, which in turn induces a resonant magnetic field enhancement at the center of the spiral. We observe a large (approximately 20 times) and uniform (over an area of $\sim 10~\mu m^{2}$) enhancement of the magnetic field for narrowband terahertz radiation with frequency matching the resonance frequency of the MM. When a broadband, single-cycle terahertz pulse propagates towards the metamaterial, the peak magnetic field of the resulting band-passed waveform still maintains a 6-fold enhancement compared to the peak impinging field. Using existing laser-based terahertz sources, our metamaterial design allows to generate magnetic fields of the order of 2 T over a time scale of several picoseconds, enabling the investigation of non-linear ultrafast spin dynamics in table-top experiments. Furthermore, our MM can be implemented to generate intense near-field narrowband, multi-cycle electromagnetic fields to study generic ultrafast resonant terahertz dynamics in condensed matter.

The Rogowski coil (RC) is emerging as a robust current measurement solution for various kinds of applications. Due to its endorsed electrical performance and flexible geometrical design, the RC is widely used in a variety of power system measurements. The ability of measuring a wide range of frequencies and signal amplitudes makes the RC a favorable sensor for assessing normal as well as faulty system operations. Researchers dealing with specific measuring or monitoring solutions confine the focus towards a certain design approach for the development of RC sensors. In order to provide a wider perspective, this paper presents a comprehensive overview of RCs considering different perspectives including: design, construction, modelling, and ongoing advances in RC designs and applications. The applications are mainly discussed based on the operation of the power components and characteristics of the measured signals. Further exploration of RC sensors can greatly contribute in increasing the measurement capability and reliability of power grids.

In this paper, we present the establishment of a magnetic flux density standard in the national calibration laboratory in Croatia. A single-layer Helmholtz coil was constructed as a magnetic field source at frequencies from 25 Hz to 100 kHz to calibrate magnetic field meters up to field limits for occupational exposure. First, the calibration error margin was determined for probes up to 100 cm² of the cross-sectional area with respect to the predefined calibration volume inside the coil. Next, the equivalent electrical model of the assembled standard was parameterized as a multiple resonance circuit to estimate the correction of the reference field with frequency. Finally, the calculated correction was experimentally verified using a single turn induction loop with an amplifier. The resulting uncertainty budget of the standard confirms the laboratory's capability to calibrate environmental magnetometers with the best relative uncertainty of 0.8 %.

This article proposed an underwater simultaneous wireless power transfer and data transfer system for the autonomous underwater vehicle. The double-sided
LLCC
compensation topology is proposed to effectively suppress the interference to the data channel at switching instant. With the increase of the size and turns of the coupling coils, the self-resonance frequency gradually decreases. However, the frequency of data carriers is usually much lower than the self-resonant frequency, which will reduce the data rate or improve the bit error rate at a high data rate. Moreover, the high voltage generated by coupling coils causes high voltage stress on the data channel. To deal with these two issues, the last turn of each coupling coil is used to transfer data in the proposed system. The two transceivers are, respectively, connected with the last turn of the two coupling coils, which excites the entire coils to self-resonate and generates two resonant frequencies. The two resonant frequencies are utilized to realize high-speed full-duplex communication. And the voltage stress of the data channel is effectively reduced. A 518-W full-duplex prototype is built, and the maximum data rate is 700 kb/s, which verifies the feasibility of the proposed system.

ТНE PURPOSE. The objective of this study is to investigate the possibilities of longer distance resonant energy transmission applying Wireless Power Transfer (WPT) in the MHz frequency range. The planned final purpose is the energy to be transferred to all types (aerial and terrestrial) of electric vehicles (EV), mainly for the battery charging at a larger distance, compared to the normal dis-tances of WPT in use at this moment. The key to this type of High Frequency (HF) WPT system is the strong resonant inductive coupling. METHODS. This project is based on the HF power oscillations generating equipment, which original function is to generate several kW of power at MHz frequency for welding of acrylic or other plastic details. RESULTS. As a first step, the equipment was modified to supply HF power for the WPT transmitter coil, instead of supplying power to the soldering plates. The operation frequency is defined by the factory, and it is made now regulable between 8 and 14 MHz by introducing a vacuum variable capacitor. The internal powerful oscillator is based on the electronic vacuum tube ITL 5-1, a military type, capable to deliver up to 3.5 kW active power at the output. The original output had a coaxial form for supplying finally the capacitive load of the dielectric welder. This had to be reworked and a resonant loop, i.e., a capacitively compensated transmitting coil, is now connected. The intended application of this HF system is to charge the batteries of a public transport EV, possibly during its periodic stops, while the passengers will enter and leave. CONCLUSION. The applied frequency is relatively high and the distances are larger, this system still uses the magnetic field as the energy transporter, i.e., it is a near field transmission, a non-radiating system, and is expected not to produce adverse effects on the human being’s health, or to achieve a safe protection from the field.

In this work, theoretical and experimental results of solid-state resonant circuits for the transmission and reception of wireless electrical energy, for applications in mobile devices are presented. Analytical expressions are found to calculate the voltage range as a function of the distance between the emitter and the load, as well as the current at the front end of an electromagnetic wave receiver. These expressions show the parameters to be varied to achieve a greater range in the transmission of wireless electrical energy. The transmitted voltage and current are measured by an electromagnetic wave receiver and compared with theoretical values, finding an excellent correspondence between the two.

This article introduces a kilowatt-scale large air-gap capacitive wireless power transfer (WPT) system for electric vehicle (EV) charging that achieves high-power transfer density and high efficiency. High-power transfer density is achieved by operating at a multi-MHz frequency (13.56 MHz), and by utilizing innovatively designed matching networks that enable effective power transfer by providing gain and reactive compensation while absorbing the parasitics present in the EV charging environment. High efficiency is achieved through the use of new interleaved-foil air-core coupled inductors in the matching networks. Interleaved-foil inductors provide a better tradeoff between quality factor, size, and self-resonant frequency compared to conventional solenoidal inductors, making them suitable for compactly and efficiently processing kilowatt-scale power at multi-MHz frequencies. Two variants of the interleaved-foil concept are presented: a semi-toroidal interleaved foil (STIF) inductor and a toroidal interleaved-foil (TIF) inductor. The superior performance of interleaved-foil inductors is demonstrated using analytical formulations, which are validated through finite-element analysis and measurements. Compared to the highest-quality-factor solenoidal inductor, the STIF inductor is shown to achieve 32% smaller box volume while having only 3% lower measured quality factor, while the TIF inductor is shown to achieve an even better tradeoff with 27% smaller box volume and 30% higher measured quality factor. A 13.56-MHz, 12-cm air-gap prototype capacitive WPT system utilizing TIF inductors with a quality factor of 2055 in its matching networks is designed, built, and tested. This system achieves record-breaking performance for a capacitive EV charging system, with an efficiency of 94.7% while transferring 3.75 kW using 22-cm-diameter coupling plates, corresponding to a power transfer density of 49.4 kW/m
<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup>
. This TIF-inductor-based prototype is also shown to outperform a second high-performance prototype utilizing STIF inductors.

It is well known that the Maxwell equations predict the behavior of the electromagnetic field very well. However, they predict only one wave equation while there are significant differences between the "near" and "far" fields and various anomalies have been observed involving the detection of super luminous signals in experiments with electrically short coaxial cables, optical fibers as well as other methods.
We show that the mathematical Laplace operator defines a complete set of vector fields consisting of two potential fields and two fields of force, which form a Helmholtz decomposition of any given vector field F. We found that neither in Maxwell’s equations nor in fluid dynamics vector theory this result has been recognized, which causes the potential fields to not be uniquely defined and also makes the Navier-Stokes equations unnecessarily complicated and introduces undesirable redundancy as well. We show that equivalents to both the Maxwell equations as well as the Navier-Stokes equations can be directly derived from a single diffusion equation describing Newton’s second law in 3D. We found that the diffusion constant ν in this equation has the same value as the speed of light squared, but has a unit of measurement in meters squared per second thus uncovering problems with time derivatives in current theories, showing amongst others that the mass-energy equivalence principle is untenable. Finally, we show that the diffusion equation we found can be divided by mass density ρ, resulting in a velocity diffusion equation that only has units of measurement in meters and seconds, thus decoupling the dynamics of the medium from it’s substance, mass density ρ. This reveals the quantized nature of spacetime itself, whereby the quantum circulation constant ν is found to govern the dynamics of physical reality, leading to the conclusion that at the fundamental quantum level only dynamic viscous forces exist while static elastic forces are an illusion created by problems with a number of time derivatives in current theories.
With our equivalents for the Maxwell equations three types of wave phenomena can be described, including super luminous longitudinal sound-like waves that can explain the mentioned anomalies. This paper contributes to the growing body of work revisiting Maxwell’s equations by deriving all of the fields from a single equation, so the result is known to be mathematically consistent and free of singularities and uniquely defines the potential fields thus eliminating gauge freedom. Unlike Maxwell’s equations, which are the result of the entanglement of Faraday's circuit level law with the more fundamental medium arguably creating most of the problems in current theoretical physics, these revisions describe the three different electromagnetic waves observed in practice and so enable a better mathematical representation.
Keywords: Classical Electrodynamics, Superfluid medium, Fluid Dynamics, Theoretical Physics, Vector Calculus.

Coil inductance and capacitor capacitance depend on overall dimensions, structure, and ambient factors. They do not vary with frequency. Reactive component impedance is determined by inductance or capacitance respectively, if active resistance is not considered. This is true for the frequencies which are significantly lower than the self-resonant frequency of the component. Parasitic parameters contribution increases on approaching the self-resonant frequency. Therefore, the componentʼs actual inductance and actual capacitance on operating frequency are defined. They are provided by manufacturers and differ from the nominal values. The actual values provide more accurate impedance of components near the considered frequency. Significant deviation from the considered frequency can cause impedance mismatch even more than the nominal values can provide. Frequency response of the high-frequency circuits such as analog filters and impedance match networks are determined by components impedance, not the nominal values. Thus, calculated values must be close to the actual values. The purpose of this article is to justify actual values application instead of nominal values.

The electrical properties of RF coils are deduced from the boundary value solution of Maxwell's equations. In the 1890s Tesla concluded, on the basis of experiment, that very large voltages were attainable on helical resonators through the mechanism of wave interference and standing wave phenomena. For such structures lumped element circuit analysis fails because its inherent presuppositions are inadequate. In the limit, as the frequency is lowered, the mode distribution becomes uniform and the RF solution passes to conventional lumped circuit elements. This passage is demonstrated analytically. Experimental measurements are employed to support Tesla's assertions.

The natural, undamped modes of propagation in a concentric line having a circular outer conductor and a tape‐helix inner conductor are investigated. The investigation is primarily concerned with a study of the propagation characteristics of the modes and, by making suitable approximations, closed expressions for the propagation factors of a few of the lower modes are obtained. Numerical results are then presented for a system having typical geometrical parameters, and generalizations of a qualitative nature are indicated. In the course of the work, the natural‐mode field expressions are derived.

By modeling a wire-wound coil as an anisotropically conducting
cylindrical boundary, one may start from Maxwell's equations and deduce
the structure's resonant behavior. Not only can the propagation factor
and characteristic impedance be determined for such a helically disposed
surface waveguide, but also its resonances, "self-capacitance"
(so-called), and its voltage magnification by standing waves. Further,
the Tesla coil passes to a conventional lumped element inductor as the
helix is electrically shortened

Propagation of a wave along a helix

- Twt J Beam-Type
- Pierce

Theory of the Beam-Type TWT. J R Pierce. Proc. IRE. Feb. 1947. p111-123. See Appendix B, p121-123,
"Propagation of a wave along a helix", which gives Schelkunoff's derivation of propagation parameters for the
Ollendorff helix.

Correction Feb. 1955, p148

37 Coaxial Line with Helical Inner Conductor. W Sichak. Proc. IRE. Aug. 1954. p1315-1319. Correction Feb. 1955,
p148. Reprinted in: Electrical Communication 32(1), March 1955. p62-67.

40 Fields and Waves in Communication Electronics. [cited earlier] Section 9.9: Surface guiding

40 Fields and Waves in Communication Electronics. [cited earlier] Section 9.9: Surface guiding.
Technical Journal. Vol. 25(2), April 1946. See fig. 22.

The idealixed helix and other slow-wave structures

- Waves Fields
- Communication In
- Electronics
- J Ramo
- T Whinnery
- Van Duzer
- Wiley

35 Fields and Waves in Communication Electronics. S Ramo, J R Whinnery and T Van Duzer. Wiley 1994. ISBN 0-
471-58551-3. Section 9.8: The idealixed helix and other slow-wave structures.

36 Some wave properties of helical conductors

- Bryant
- Elec
- Comm

36 Some wave properties of helical conductors, J H Bryant, Elec. Comm. 31(1) 1954. Considers the free coil case
and also the effects of inner and outer coaxial conducting cylinders.

- H Poritsky
- P Abetti
- R Jerrard

38 Field Theory of Wave Propagation ALong Coils. H Poritsky, P A Abetti, R P Jerrard. Power Apparatus and
Systems, Part III, Trans. of the AIEE. 72(2), Oct. 1953. p930-939.

Power Apparatus and Systems, Part III

- H Poritsky
- R P Abetti
- Jerrard

Field Theory of Wave Propagation ALong Coils. H Poritsky, P A Abetti, R P Jerrard. Power Apparatus and
Systems, Part III, Trans. of the AIEE. 72(2), Oct. 1953. p930-939.

A Radar History of World War II. Louis Brown. 1999. Taylor and Francis

- Military Technical
- Imperatives

Technical and Military Imperatives: A Radar History of World War II. Louis Brown. 1999. Taylor and Francis.
ISBN13: 978-0-7503-0659-1. See Ch. 4. Resonant Magnetron: p153, p409.

Lower modes of a concentric line having a helical inner conductor. L Stark

Lower modes of a concentric line having a helical inner conductor. L Stark, J. Appl. Phys. 25(9), 1954. p1155-1162.