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Wireless power transfer (WPT) is an emerging technology that enables the wireless transfer of electrical energy from power supplies to electrical equipment. It has been widely used in electric vehicles, mobile phones, household appliances, medical devices, and other fields. However, the frequency splitting and bifurcation phenomena existing in WPT systems are the fundamental obstacles and challenges that affect the effective operation of WPT systems. In this paper, a method based on the fractional‐order circuit is proposed to simultaneously suppress the frequency splitting and bifurcation phenomena by changing the order of fractional‐order capacitor. By replacing the compensation capacitor in the transmitter of the traditional WPT system with a fractional‐order capacitor, a fractional‐order WPT system is formed. Then, using fractional calculus and circuit theory, the mathematical model of the proposed WPT system containing a fractional‐order capacitor is established, and the frequency splitting and bifurcation phenomena are analyzed. The theoretical results show that the frequency splitting and bifurcation phenomena are suppressed only by adjusting the order of fractional‐order capacitor, and the output power of the original resonant frequency is improved. Finally, the experimental prototype is implemented to validate the theoretical results. [ABSTRACT FROM AUTHOR]

As the core component of the all-metal micro resonant gyroscope, the structural parameters and form and position errors of the resonator significantly influence its vibration characteristics, and consequently, the accuracy of the gyroscope. By establishing the finite element model of an ideal hemispherical resonator and optimizing the meshing method, we refined the frequency difference to 0.1 Hz, enhancing the accuracy of the simulation model. Through finite element simulation, we examined the impact of various structural parameters and processing errors on the natural frequencies of each mode. We analyzed how form and position errors, including shell thickness error, central axis error, equatorial plane error, and edge rectangular tooth position error, affect the frequency splitting of the resonator. We provided optimization suggestions for the structural parameters, ensuring frequency splitting variations of less than 1 Hz. Theoretical modeling and simulation analysis indicated that the primary factors influencing the vibration modes and frequency splitting are the rectangular tooth structure and shell thickness. Following the optimized parameters, the frequency splitting of the All-Metal Micro Resonant Hemisphere was reduced by an order of magnitude to 14 Hz, demonstrating that these optimized conditions can significantly enhance the resonator's performance. [ABSTRACT FROM AUTHOR]

The parameter variations of magnetically coupled resonant wireless power transmission (MCR‐WPT) system cause frequency splitting and resonant frequency shift, which lead to the system detuning and reduce the transmission efficiency. To solve the problems, a frequency tracking control method based on the phase difference between the primary‐side voltage and the secondary‐side current (Frequency Tracking Control‐the primary‐side voltage and the secondary‐side current, FTC‐PS) in this paper. According to the characteristics that the phase difference between the voltage of transmitting side and the current of receiving side is not affected by frequency splitting, the phase difference is controlled at 90°, so that the system can achieve the maximum transmission efficiency under the condition of frequency splitting and resonant frequency shift. The experimental results show that the method can make the system always work at the natural resonant frequency when the parameters change, and the transmission efficiency is not affected by the frequency splitting phenomenon. © 2024 Institute of Electrical Engineers of Japan and Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

When two resonators of coupled silicon resonators are identical and the gain on one side is equal to the loss on the other side, a parity-time (PT) symmetric-coupled silicon resonator is formed. As non-Hermitian systems, the PT-symmetric systems have exhibited many special properties and interesting phenomena. This paper proposes the strain-induced frequency splitting in PT symmetry-coupled silicon resonators. The frequency splitting of the PT system caused by strain perturbations is derived and simulated. Theory and simulation both indicate that the PT system is more sensitive to strain perturbation near the exceptional point (EP) point. Then, a feedback circuit is designed to achieve the negative damping required for PT symmetry. Based on a simple silicon-on-insulator (SOI) process, the silicon resonator chip is successfully fabricated. After that, the PT-symmetric-coupled silicon resonators are successfully constructed, and the frequency splitting phenomenon caused by strain is observed experimentally. [ABSTRACT FROM AUTHOR]

A twist sensor based on the frequency splitting effect of fiber Bragg grating laser is proposed and experimentally demonstrated. Laser longitudinal mode was split into two frequencies under the action of birefringence induced by fiber twist and by monitoring the periodical beat frequency response, the twist angle could be deduced. The experimental results show that the relationship between the fiber twist angle and the beat frequency is linear (linearity is more than 99%) when the twist angle is in the range of 10°–55°. Moreover, the wavelength drift caused by the temperature does not affect this linearity. Since the sensor is based on frequency rather than on the wavelength, its sensitivity is not limited to the resolution of the spectrometer, and has been improved to 31.59 MHz/rad (0.55 M/degree). [ABSTRACT FROM AUTHOR]

Reduced-order dynamic models of wireless power transfer (WPT) systems are desired to simplify the analysis and design of power control, phase synchronization, and maximum efficiency tracking. The reduced-order dynamic phasor model is a good choice because of its straightforward physical meaning and concise mathematical formula. However, the model relies on the assumption of loose coupling and loses accuracy when the coupling becomes stronger. In this paper, a model reduction method with split frequency matching is proposed to improve model accuracy under relatively strong coupling conditions, which is suitable for most short-distance WPT applications, such as wireless electrical vehicle charging. Split frequency matching is achieved through a pair of conjugate equivalent mutual inductances, which are derived from the asymmetry characteristics of the full-order dynamic phasor model in the positive and negative frequency domains. The proposed model retains the advantages of the existing model while significantly improving the accuracy under strong coupling conditions. Its characteristics are verified by comparing the experimental results and model predictions under both large step changes and small-signal perturbations. [ABSTRACT FROM AUTHOR]

The properties of small size, low noise, high performance and no wear-out have made the hemispherical resonator gyroscope a good choice for high-value space missions. To enhance the precision of the hemispherical resonator gyroscope for use in tasks with large angular velocities and angular accelerations, this paper investigates the standing wave precession of a non-ideal hemispherical resonator under nonlinear high-intensity dynamic conditions. Based on the thin shell theory of elasticity, a dynamic model of a hemispherical resonator is established by using Lagrange's second kind equation. Then, the dynamic model is equivalently transformed into a simple harmonic vibration model of a point mass in two-dimensional space, which is analyzed using a method of averaging that separates the slow variables from the fast variables. The results reveal that taking the nonlinear terms about the square of the angular velocity and the angular acceleration in the dynamic equation into account can weaken the influence of the 4th harmonic component of a mass defect on standing wave drift, and the extent of this weakening effect varies with the dimensions of the mass defects, which is very important for steering the development of the high-precision hemispherical resonator gyroscope. [ABSTRACT FROM AUTHOR]

Abstract This paper focuses on the long time behavior of the solutions to the Cauchy problem of the three-dimensional compressible magneto-micropolar fluids. More precisely, we aim to establish the optimal rates of temporal decay for the highest-order spatial derivatives of the global strong solutions by the method of decomposing frequency. Our result can be regarded as the further investigation of the one in (Wei, Guo and Li in J. Differ. Equ. 263:2457–2480, 2017), in which the authors only provided the optimal rates of temporal decay for the lower-order spatial derivatives of the perturbations of both the velocity and the micro-rotational velocity.

As the core component of the all-metal micro resonant gyroscope, the structural parameters and form and position errors of the resonator significantly influence its vibration characteristics, and consequently, the accuracy of the gyroscope. By establishing the finite element model of an ideal hemispherical resonator and optimizing the meshing method, we refined the frequency difference to 0.1 Hz, enhancing the accuracy of the simulation model. Through finite element simulation, we examined the impact of various structural parameters and processing errors on the natural frequencies of each mode. We analyzed how form and position errors, including shell thickness error, central axis error, equatorial plane error, and edge rectangular tooth position error, affect the frequency splitting of the resonator. We provided optimization suggestions for the structural parameters, ensuring frequency splitting variations of less than 1 Hz. Theoretical modeling and simulation analysis indicated that the primary factors influencing the vibration modes and frequency splitting are the rectangular tooth structure and shell thickness. Following the optimized parameters, the frequency splitting of the All-Metal Micro Resonant Hemisphere was reduced by an order of magnitude to 14 Hz, demonstrating that these optimized conditions can significantly enhance the resonator’s performance.

When two resonators of coupled silicon resonators are identical and the gain on one side is equal to the loss on the other side, a parity-time (PT) symmetric-coupled silicon resonator is formed. As non-Hermitian systems, the PT-symmetric systems have exhibited many special properties and interesting phenomena. This paper proposes the strain-induced frequency splitting in PT symmetry-coupled silicon resonators. The frequency splitting of the PT system caused by strain perturbations is derived and simulated. Theory and simulation both indicate that the PT system is more sensitive to strain perturbation near the exceptional point (EP) point. Then, a feedback circuit is designed to achieve the negative damping required for PT symmetry. Based on a simple silicon-on-insulator (SOI) process, the silicon resonator chip is successfully fabricated. After that, the PT-symmetric-coupled silicon resonators are successfully constructed, and the frequency splitting phenomenon caused by strain is observed experimentally.

Microwave resonance sensing has attracted significant interest due to its promising potential in developing ultracompact integrated sensors. For biomedical sensing applications, it is required to realize high sensitivity to tiny volumes of solution using such long microwave wavelengths. Here, a chiral microwave plasmonic resonator, which can induce exceptional point (EP) state based on the coalescence of non‐Hermitian dipole eigenmodes, is proposed. The intrinsically geometric asymmetry leads to strong asymmetry of non‐Hermitian Hamiltonian and strong coupling between the two non‐degenerate dipoles, hence resulting in significantly enhanced frequency splitting signals. Benefiting from the subwavelength dipole resonances, the proposed resonator has advantages of the EP state's high sensitivity to tiny targets and strong plasmonic wavelength compression. Its sensing performance is experimentally validated by metal scatterer detection and glucose solution measurement. The minimum detectable metal scatterer features a diameter of λ0/280 and a volume of 2.9 × 10−9λ03 (λ0 is free‐space wavelength). The solution volume under detection features a volume of 4.4 × 10−7λ03 and the reporting limit for glucose reaches 0.26 µmol. These results envision a feasible route for trace‐amount biomedical sensing at the microwave frequencies. [ABSTRACT FROM AUTHOR]

The MEMS gyroscope is one of the basic units of inertial navigation, whose performance and accuracy is noteworthy. Because of the limitations of processing technology and other factors, the relative manufacturing error of MEMS gyroscopes is usually large. Errors directly lead to a frequency mismatch of resonant structures and consequently restrict the performance improvement of the gyroscope. This study proposes a mechanical trimming technique combining the addition and removal of gold in a ring MEMS gyroscope. Firstly, the analysis of the gyroscope dynamics and error model and trimming theory provides theoretical guidance for the trimming process. Secondly, the method of adjusting the mass is investigated, and the ablation threshold of femtosecond laser parameters on gold is analyzed, which provides the process with parameters for the trimming experiment. Finally, the frequency trimming process is conducted in three steps, including the addition of gold spheres and the removal of gold spheres and gold film, which are applicable to the trimming process at different rates of frequency split. The results shows that the proposed method can reduce the frequency split of the gyroscope from 4.36 to 0.017 Hz. [ABSTRACT FROM AUTHOR]

Based on the perturbation method, the task of determining the frequency spectrum splitting of elastic thin-walled shell, the geometry of which slightly differs from the axial symmetry, is solved. A mathematical model of the shell elastic element with arbitrarily small nonaxisymmetric geometry parameter errors and a software-algorithmic complex of numerical calculation of its natural frequency splitting are developed. With the help of computer analytics the perturbed differential-matrix operators for a thin-walled shell with arbitrary small deviations from the axial symmetry of the mid-surface shape are constructed. The results of calculations of specific shell elastic elements are presented. [ABSTRACT FROM AUTHOR]

The properties of small size, low noise, high performance and no wear-out have made the hemispherical resonator gyroscope a good choice for high-value space missions. To enhance the precision of the hemispherical resonator gyroscope for use in tasks with large angular velocities and angular accelerations, this paper investigates the standing wave precession of a non-ideal hemispherical resonator under nonlinear high-intensity dynamic conditions. Based on the thin shell theory of elasticity, a dynamic model of a hemispherical resonator is established by using Lagrange’s second kind equation. Then, the dynamic model is equivalently transformed into a simple harmonic vibration model of a point mass in two-dimensional space, which is analyzed using a method of averaging that separates the slow variables from the fast variables. The results reveal that taking the nonlinear terms about the square of the angular velocity and the angular acceleration in the dynamic equation into account can weaken the influence of the 4th harmonic component of a mass defect on standing wave drift, and the extent of this weakening effect varies with the dimensions of the mass defects, which is very important for steering the development of the high-precision hemispherical resonator gyroscope.

As an inertial sensor with excellent performance, the hemispherical resonator gyro is widely used in aerospace, weapon navigation and other fields due to its advantages of high precision, high reliability, and long life. Due to the uneven distributions of material properties and mass of the resonator in the circumferential direction, the frequencies of the two 4-antinodes vibration modes (operational mode) of resonator in different directions are different, which is called frequency splitting. Frequency splitting is the main error source affecting the accuracy of the hemispherical resonator gyro and must be suppressed. The frequency splitting is related to the structure of the resonator. For the planar-electrode-type hemispherical resonator gyro, in order to suppress the frequency splitting from the structure, improve the accuracy of the hemispherical resonator gyro, and determine and optimize the equivalent bottom angle parameters of the hemispherical resonator, this paper starts from the thin shell theory, and the 4-antinodes vibration mode and waveform precession model of the hemispherical resonator are researched. The effect of the equivalent bottom angle on the 4-antinodes vibration mode frequency value under different boundary conditions is theoretically analyzed and simulated. The simulation results show that the equivalent bottom angle affects the 4-antinodes vibration mode of the hemispherical resonator through radial constraints. The hemispherical resonator with mid-surface radius  R = 15   mm  and shell thickness  h = 1   mm  is the optimization object, and the stem diameter D and fillet radius  R 1  are experimental factors, with the 4-antinodes vibration mode frequency value and mass sensitivity factor as the response indexes. The central composite design is carried out to optimize the equivalent bottom angle parameters. The optimized structural parameters are: stem diameter  D = 7   mm , fillet radii  R 1 = 1   mm ,  R 2 = 0.8   mm . The simulation results show that the 4-antinodes vibration mode frequency value is 5441.761 Hz, and the mass sensitivity factor is 3.91 Hz/mg, which meets the working and excitation requirements wonderfully. This research will provide guidance and reference for improving the accuracy of the hemispherical resonator gyro. [ABSTRACT FROM AUTHOR]

This paper presents a system for simultaneous wireless power and data transfer (SWPDT) over a single inductive link. We design the frequency-splitting inductive link to address the trade-off between high power transfer efficiency (PTE) and high data rate (DR). An active rectifier is designed for high power conversion efficiency. Forward data communication is based on frequency-shift keying (FSK) modulation to support high DR transmission while delivering uninterrupted power. Moreover, we proposed low-power FSK demodulation circuits to further improve the energy efficiency of the data demodulation. The system is implemented with 180nm CMOS process and occupies an area of 0.836mm2. The post-layout simulation results show that the overall PTE (including the resonant tanks and the active rectifier) of the SWPDT system can be up to 76.5% while delivering 40mW power to the load. Meanwhile, a data rate of 1.11Mbps can be achieved. The power consumption for data demodulation is $69.1~\mu \text{W}$ , showing an energy efficiency of 62.2pJ/bit. The proposed SWPDT system based on a single inductive link shows potential for implantable biomedical applications.

Wireless power transfer has gradually become a promising way for the charging of electric vehicles (EVs) and the design of a high efficiency and high stability charging system is inevitable. Inductive power transfer (IPT) systems, as one of the most important method to realize wireless power transfer, usually work under weak coupling conditions and obtain high power transmission efficiency through resonance compensation circuits. In the case of reduced power transmission distance, the IPT system will enter an over-coupled state, causing the system output power to change drastically and the system to detune, and frequency bifurcation and frequency splitting will also occur. This paper first theoretically deduces the root cause of the output power variation of several typical topological IPT systems under over-coupled conditions, and then analyzes their frequency bifurcation and frequency splitting characteristics by numerical analysis methods, which can provide a reference for the design of IPT systems. Finally, the theory of this paper is verified by simulation and a 2 kW test prototype.

In order to solve the problems of frequency splitting and low efficiency in wireless power transfer system, the capacitive coupled wireless power transfer system with double-side LCLC compensation topology is taken as the research object in this paper. The effect laws of variable parameters on the system output are analyzed by system modeling. The relationship between system efficiency and the important parameters is derived by the using circuit theory, and the system efficiency is analyzed in detail by using MATLAB simulation software. The experiment of wireless power transfer with electronic components is conducted in this paper. The existence of frequency splitting and frequency offset is proved by experiments, and the influence of frequency and coupling capacitance on the output efficiency of the system is verified. The results show that the frequency splitting of the system can be avoided by adjusting the capacitance of the coupling mechanism below the frequency split point. Fixing the relative position of the coupling plates, the system efficiency can be improved by increasing the frequency of the system to the efficiency key point. [ABSTRACT FROM AUTHOR]

In manufacturing, batch size is decreasing, while the product range is increasing. In these conditions, it is important to ensure continual operation of the available production equipment. Downtime of the equipment may be due to maintenance. Two categories may be distinguished: planned maintenance and sudden unplanned maintenance. Sudden maintenance is due to the unforeseen incapacity of production equipment; planned maintenance is scheduled on statistical principles to prevent breakdown. Quantitative analysis of temporary production losses due to downtime permits assessment of the scope for decreasing downtime. One possibility is active diagnostics of the performance of critical components so as to establish when maintenance is required. Theoretical and experimental principles are proposed for diagnostics of the equipment without shutdown and informational support of maintenance planning. [ABSTRACT FROM AUTHOR]

The tire acoustic cavity resonance noise (TACRN) is known to contribute to audible noise in the passenger compartment of a vehicle. In order to reduce TACRN effectively, its mechanism needs to be grasped better. In this paper, the calculation formulas of tire acoustic cavity resonance frequency for four different conditions such as static unloaded tire, static loaded tire, rotating unloaded tire, and rotating loaded tire are analyzed and verified by the simulation and experiment. In particular, the resonance frequency formulas of static loaded tire introducing inflation pressure and rotating loaded tire are proposed and verified, respectively, in this paper. And the influence of tire inflation pressure, load, and running velocity on splitting frequency are studied. Some new findings are described and discussed; for example, the first-order resonance frequency may split into four resonance frequencies in most cases, and may split into three resonance frequencies in certain cases when a loaded tire is rotating. And the existing conditions for three and four resonance frequencies are also discussed. [ABSTRACT FROM AUTHOR]

The MEMS gyroscope is one of the basic units of inertial navigation, whose performance and accuracy is noteworthy. Because of the limitations of processing technology and other factors, the relative manufacturing error of MEMS gyroscopes is usually large. Errors directly lead to a frequency mismatch of resonant structures and consequently restrict the performance improvement of the gyroscope. This study proposes a mechanical trimming technique combining the addition and removal of gold in a ring MEMS gyroscope. Firstly, the analysis of the gyroscope dynamics and error model and trimming theory provides theoretical guidance for the trimming process. Secondly, the method of adjusting the mass is investigated, and the ablation threshold of femtosecond laser parameters on gold is analyzed, which provides the process with parameters for the trimming experiment. Finally, the frequency trimming process is conducted in three steps, including the addition of gold spheres and the removal of gold spheres and gold film, which are applicable to the trimming process at different rates of frequency split. The results shows that the proposed method can reduce the frequency split of the gyroscope from 4.36 to 0.017 Hz.

As an inertial sensor with excellent performance, the hemispherical resonator gyro is widely used in aerospace, weapon navigation and other fields due to its advantages of high precision, high reliability, and long life. Due to the uneven distributions of material properties and mass of the resonator in the circumferential direction, the frequencies of the two 4-antinodes vibration modes (operational mode) of resonator in different directions are different, which is called frequency splitting. Frequency splitting is the main error source affecting the accuracy of the hemispherical resonator gyro and must be suppressed. The frequency splitting is related to the structure of the resonator. For the planar-electrode-type hemispherical resonator gyro, in order to suppress the frequency splitting from the structure, improve the accuracy of the hemispherical resonator gyro, and determine and optimize the equivalent bottom angle parameters of the hemispherical resonator, this paper starts from the thin shell theory, and the 4-antinodes vibration mode and waveform precession model of the hemispherical resonator are researched. The effect of the equivalent bottom angle on the 4-antinodes vibration mode frequency value under different boundary conditions is theoretically analyzed and simulated. The simulation results show that the equivalent bottom angle affects the 4-antinodes vibration mode of the hemispherical resonator through radial constraints. The hemispherical resonator with mid-surface radius R=15 mm and shell thickness h=1 mm is the optimization object, and the stem diameter D and fillet radius R1 are experimental factors, with the 4-antinodes vibration mode frequency value and mass sensitivity factor as the response indexes. The central composite design is carried out to optimize the equivalent bottom angle parameters. The optimized structural parameters are: stem diameter D=7 mm, fillet radii R1=1 mm, R2=0.8 mm. The simulation results show that the 4-antinodes vibration mode frequency value is 5441.761 Hz, and the mass sensitivity factor is 3.91 Hz/mg, which meets the working and excitation requirements wonderfully. This research will provide guidance and reference for improving the accuracy of the hemispherical resonator gyro.

This work constructs a circuit model, forms a resonant frequency expression of the WPT system, and investigates a novel single side hybrid frequency adjusting strategy. Firstly, by the tracking strategy of minimizing the reflection coefficient, a novel hybrid frequency adjusting control strategy is exerted in the transmitting side by uniting the Bang-Bang control method and the fuzzy PID control technique. Secondly, the receiver is permitted to move to nearly any location within the system's working scope, and the system still reaches a near constant efficiency and a maximum load power. Finally, the simulation results verify that the proposed hybrid adjusting method can be adopted to track the resonant frequency more rapidly and more accurately. [ABSTRACT FROM AUTHOR]

In order to solve the problems of frequency splitting and low efficiency in wireless power transfer system, the electrical-field coupled wireless power transfer system with double-side LCLC compensation topology is taken as the research object in this study. The effect laws of variable parameters on the system output are analyzed by system modeling. The relationship between system efficiency and the important parameters is derived using the circuit theory, and the system efficiency is analyzed in detail using MATLAB simulation software. The correctness of simulation results are verified by experiments. The results show that adjusting the capacitance value of the coupling mechanism below the frequency division point can avoid frequency splitting in the system, while fixing the relative position of the coupling plates can improve the system efficiency by increasing the frequency of the system to the critical point of efficiency, and the best efficiency of the system can reach 89.4%. [ABSTRACT FROM AUTHOR]

This paper presents an embroidered textile frequency-splitting microwave sensor based on a pair of identical stepped-impedance resonators (SIRs) loading a microstrip transmission line. The sensor is implemented by means of conductive threads. The sensing region of the proposed structure is the capacitive square patch of one of the SIRs. If such region is kept unaltered, the structure is symmetric, and the frequency response (transmission coefficient) exhibits a single transmission zero. However, if symmetry is broken (e.g., through liquid absorption in the sensing region), the frequency response of the proposed sensor exhibits two transmission zeros (frequency splitting). The difference (in frequency and magnitude) between such zeros (or notches) is intimately related to the dielectric properties of the absorbed liquids to be sensed $/$ detected. The proposed sensing structure is applied to the detection of deionized (DI) water absorption, and to the quantification of the number of DI water drops. The maximum measured sensitivity is found to be 2.70 MHz $/ \mu \text{l}$ and 0.03 dB $/ \mu \text{l}$ for the incremental frequency and incremental magnitude of the notches. [ABSTRACT FROM AUTHOR]

Using frequency splitting, two energy management strategies (EMS) based on Haar wavelet decomposition and Fourier analysis for fuel cell hybrid vehicle (FCHV) are proposed to manage efficiently the power flow between components. The paper aims to discuss the performances of the proposed EMS in terms of dynamic behavior, robustness operation, real time application and fuel economy. For apply this methodology, two EMS approaches are elaborated and successfully tested for parallel Fuel Cell/UC: conventional approach using Fourier Transform analysis (FT) and Wavelet analysis approach allowing natural frequency splitting. Finally, and to evaluate the performance and relevance of the developed approach, a comparison analysis were conducted. The simulation results exhibit the effectiveness of both strategies. Indeed, Wavelet analysis leads to better results in terms of energy flow and dynamic behavior, excellent robustness and stability of system, as well as energy economy improvement. A very relevant strategy is proposed based on Wavelet analysis using digital filtering techniques, which enables a natural frequency splitting to ensure the best global performances. In addition, the approach remains simple and suitable for real time operation. [ABSTRACT FROM AUTHOR]

• Frequency split of a ring resonator with two harmonics of density defect was calculated using the perturbation method. • Frequency split grows if the condition p ± m = 2 n is met (p , m - defect harmonics numbers, n - wave number). • The obtained results were validated by FEA, there is a good agreement between numerical and analytical results. • Frequency split does not depend on the angle between harmonics for density defects. • As frequency split depends on a combination of first and third harmonics, their balancing influence frequency splitting. A ring resonator with two harmonics of density defect is considered. Harmonics are of arbitrary numbers m and p. Using an analytical implementation of perturbation techniques for eigenvalue problem, the second-order frequency mismatch approximation is found for different combinations of defect harmonics numbers. It is shown that for combination of numbers conforming the rule | p ± m | = 2 n (where n is a working mode number), frequency split is proportional to the product of amplitudes of defect harmonics. More specifically, it is shown that harmonics with numbers 1 and 3 considered individually yield no frequency split, while simultaneous emergence of these two harmonics do yield a high rise of frequency mismatch. From analysis, we derived that for two harmonics of density inhomogeneity, frequency split does not depend on the angle between harmonics. The same results are obtained numerically by finite element analysis of a ring resonator with two harmonics of density defect. Also, the eigenvalue problem for a ring resonator with two different defects (density and radius) is considered using FEM. We show that the dependence of the frequency mismatch on the angle between harmonics is only observed for certain combinations of harmonics. [ABSTRACT FROM AUTHOR]

Mutual inductance highly determines the performance of dual-coil magnetic coupled resonance wireless power transfer (MCR-WPT). Due to the frequency splitting, the system gives inferior energy transfer efficiency in overcoupled state, where the two coils are close with each other and a large mutual inductance is presented. This raises the need to finely tune the mutual inductance for frequency splitting elimination and consequently performance optimization. Meanwhile, with the built MCR-WPT system, the mutual inductance remains constant given a fixed distance between the coils, which limits its potential application. In this article, the resonant coils with magnetoelectric composites core consisting of Metglas/lead zirconate titanate/Metglas were adopted to tune the mutual inductance. A large tunability of 56.5% in the low-frequency range and a tunability of 16.6% in the high-frequency range were achieved. The tunability of the mutual inductance indicates its great potential and provides a new route to eliminate the frequency splitting for high efficient MCR-WPT. [ABSTRACT FROM AUTHOR]

This paper presents a frequency-splitting-based wireless power and data transfer IC that simultaneously delivers power and forward data over a single inductive link. For data transmission, frequency-shift keying (FSK) is utilized because the FSK modulation scheme supports continuous wireless power transmission without disruption of the carrier amplitude. Moreover, the link that manifests the frequency-splitting characteristic due to a close distance between coupled coils provides wide bandwidth for data delivery without degrading the quality factors of the coils. It results in large power delivery, high data rate, and high power transfer efficiency. The presented IC fabricated in a 180-nm BCD process simultaneously achieves up-to-115-mW wireless power delivery to the load and 2.5-Mb/s downlink data rate over the single inductive link. The measured overall power efficiency from the DC power supply at the transmitter module to the load at the receiver module reaches 56.7 $\%$ at its maximum, and the bit error rate is lower than 10 $^{-6}$ at 2.5 Mb/s. As a result, the figure of merit (FoM) for data transmission is enhanced by 2 times, and the FoM for power delivery is improved by 38.7 times compared to prior state-of-the-arts using a single inductive link. [ABSTRACT FROM AUTHOR]

At present, magnetically coupled resonant (MCR) wireless power transfer (WPT) has been widely used in many practical applications. However, the traditional symmetrical coils structure has limited the development of WPT technology, such as the mining and medical field, where the coils structure is always asymmetrical because of their special environment requirements. In the study a novel method was presented for the analysis of the frequency characteristics of the MCR‐WPT system with asymmetric coils structure. The concept of the quality factor ratio of the coils with asymmetrical coils structure was proposed, and then the mathematical relationship between the quality factor ratio and the output power was deduced. In addition, the effects of the quality factor ratio, generalized coupling factor, generalized mismatching factor and frequency splitting factor on the output power and transmission efficiency were studied respectively. The simulation results not only demonstrate the regular pattern between the frequency splitting, the quality factor ratio and the frequency splitting factor, but also validate the condition of the output power achieving the local maximum value. Besides, the results show the relationship between the coupling factor and the transmission efficiency. Finally, an experimental platform was built to verify the theoretical analysis. And the conclusions of the study not only provide theoretical and practical basis for the future study in WPT technology but also can provide beneficial references for the design of WPT system in particular applications. © 2021 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

In practical engineering, the frequency splitting of Hemispherical Resonator Gyro (HRG) caused by uneven mass distribution seriously affects the precision of HRG. So, the inherent frequency is an important parameter of micro-Hemispherical Resonator Gyro (m-HRG). In the processing of hemispherical resonator, there are some morphological errors and internal defects in the hemispherical resonator, which affect the inherent frequency and the working mode of m-HRG, and reduce the precision and performance of m-HRG. In order to improve the precision and performance of m-HRG, the partial differential equation of the hemispherical resonator is solved, and the three-dimensional model using ANSYS software accurately reflected the actual shape is established in this paper. Then, the mode of hemispherical resonator in ideal state and uneven mass distribution state are simulated and analyzed. The frequency splitting mechanism of the hemispherical resonator is determined by calculation and demonstration, and the frequency splitting of the hemispherical resonator is suppressed by partial mass elimination. The results show that the absolute balance of energy can ensure the high-quality factor and the minimum frequency splitting of the hemispherical resonator. Therefore, during the processing of hemispherical resonator, the balance of mass should be achieved as much as possible to avoid various surface damage, internal defects and uneven mass distribution to guarantee the high-quality factor Q and minimum frequency splitting of hemispherical resonator.

A dual-band sensor based on a planar rectangular cavity loaded with pairs of improved planar resonator is used to measure the permittivity difference of liquids with a small volume ($2.5~\mu \text{l}$). The planar cavity is connected by two microstrip transmission lines, constituting a two-port device with two branches. A pair of interdigital capacitor (IDC)-based split ring resonator (SRR) is coupled to the middle part of each branch. The distance between the IDC–SRRs is large, and their coupling can be neglected. Thus, two resonance frequencies are produced (4.15 and 9.18 GHz). One frequency is generated from the resonance of the two IDC-based SRRs (4.15 GHz), whereas the other results from the coupling of the rectangular cavity and the two IDC-based SRRs (9.18 GHz). Frequency splitting is used to detect changes in the dielectric characteristics of the liquids. If the two branches (IDC-based SRRs) of the proposed sensor are placed on the same liquid, only two resonance frequencies are obtained. When two different liquids are loaded by the sensor, the resonance frequency will be split into four. A planar rectangular cavity is used instead of a splitter configuration to motivate the two IDC–SRRs. This novel design affords a dual-band miniaturized sensor. [ABSTRACT FROM AUTHOR]

The magnetically coupled resonant wireless power transfer (MCR-WPT) system for implantable medical devices has gradually become one of the research hotspots. The transmission characteristics of the system in weak-coupling regions are of research significance. The mathematical model of the system's transmission characteristics is established by equivalent circuit theory. Several key factors that affect the transmission characteristics of the system are obtained, to include the self-inductance of the coil, the mutual inductance between the coils, and the load resistance. A magnetically coupled resonant wireless power transfer system is designed. A method of coil design is proposed. The transmission efficiency of the system in the weak-coupling region is improved by optimizing the design of the coil. Simulation and experimental results exhibited good consistency. The results show that the transmission performance of the system in the weak-coupling area is improved. This work provides guidance and practical significance for the research of the MCR-WPT system with implantable medical equipment.

For the position of four coils of traditional four-coil magnetically coupled resonant wireless power transmission system is relatively fixed and cannot be changed at will, which leads to limitation of its use, a new four-coil structure was proposed: power coil and load coil are connected with the lumped capacitor separately to form power resonant circuit and load resonant circuit, the power coil can produce magnetic resonance with transmitting coil, the load coil can produce magnetic resonance with receiving coil. At the same time, the relative position of the four coils can be changed, the distance between the four coils can be adjusted appropriately, which not only can increase transmission distance of wireless electric energy, but also can improve frequency splitting, so as to maximize output power and transmission efficiency of the system. The effects of different coupling coefficients between the four coils on transmission efficiency and output power of the system were analyzed by simulation and experiment. The results show that the larger the coupling coefficient between the power coil and the transmitting coil is, the higher the transmission efficiency of the system will be, with the increase of the coupling coefficient, the transmission efficiency increases first and then decreases. The output power and transmission efficiency decrease obviously with the increase of the transmission distance between the load coil and the receiving coil. For different transmission distances, adjusting the distance between the load coil and the receiving coil enables the system to obtain the maximum transmission efficiency and output power under the same load.

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The acoustic cavity resonance inside the tire–wheel assembly is known to contribute to audible noise in the passenger compartment of vehicles. To obtain control methods of tire acoustic cavity resonance, its characteristics and producing mechanism need to be clarified first. In this article, the finite element model of a tire coupled with acoustic medium in the tire cavity is constructed. The Euler method is introduced to study the modal characteristics of tire cavity under the influence of tire inflation pressure, load, and tire rotation velocity. Frequency splitting phenomena under four separate conditions (stationary tire without load, stationary tire with load, rotating tire without load, and rotating tire with load) are simulated and analyzed. The slope change of the resonance frequency as a function of rotation speed is found to be close to the reciprocal of tire radius which can be explained by a model of wave propagation in a ring-shaped channel with moving media inside the ring. The obtained function of the slope change can help determine the frequency variation range under different vehicle velocity, structure load, and tire inflation pressure, which can then help to control the cavity resonance energy and provide a more comfortable driving experience. [ABSTRACT FROM AUTHOR]

One of the major challenges in inductive wireless power transfer (WPT) systems is that the optimal frequency of operation may shift predominantly due to coupling variation as a result of so-called frequency splitting phenomenon. When frequency splitting occurs, two additional resonance frequencies split from the coupler’s resonance frequency. Maximum power levels are observed at these split resonance frequencies; however, these frequencies are strongly-dependent on the coupling coefficient, hence the distance and alignment between the couplers. In addition to that, peak power values at these frequencies can be different from each other due to small impedance differences between the primary and secondary side resonant couplers, forming a local and a global maximum. Therefore, the WPT system should adaptively operate at the correct frequency for achieving maximum power transfer. In this paper, a metaheuristic Particle Swarm Optimization (PSO) based frequency tracking algorithm is proposed for use in WPT systems. The WPT system employs multi sub-coil flux pipe couplers, a full-bridge inverter which is driven by TMS320F28069 controller card and is suitable for high power charging applications. The control algorithm can accurately find the global maximum power point in case of frequency splitting with asymmetric peaks. The proposed frequency tracking algorithm operates only at the primary side based on measurement of the input power level. Therefore, no additional communication link is needed between the primary and the secondary side. Effectiveness of the proposed control method is validated by performing experiments under three different misalignment scenarios and compared to the traditional Perturb and Observe algorithm.

An equivalent circuit model of three-coil magnetic coupling resonant wireless power transmission system was built, and formulas of system transmission power and transmission efficiency were deduced. Coupling factors relation with the maximum system transmission power was obtained under critical coupling state, and relationship expression between coupling factor and frequency splitting point was given. Frequency characteristics of the system were analyzed. The theoretical analysis results show that there is frequency splitting phenomenon in the system under overcoupling state with three frequency splitting points; the system only resonates at inherent resonant frequency under critical coupling state and undercoupling state; whether there is frequency splitting phenomenon or not, the system resonates at inherent resonant frequency, and transmission power of the system reaches the maximum. The simulation and experimental results verify correctness of the theoretical analysis.

One of the major challenges in inductive wireless power transfer (WPT) systems is that the optimal frequency of operation may shift predominantly due to coupling variation as a result of so-called frequency splitting phenomenon. When frequency splitting occurs, two additional resonance frequencies split from the coupler's resonance frequency. Maximum power levels are observed at these split resonance frequencies; however, these frequencies are strongly-dependent on the coupling coefficient, hence the distance and alignment between the couplers. In addition to that, peak power values at these frequencies can be different from each other due to small impedance differences between the primary and secondary side resonant couplers, forming a local and a global maximum. Therefore, the WPT system should adaptively operate at the correct frequency for achieving maximum power transfer. In this paper, a metaheuristic Particle Swarm Optimization (PSO) based frequency tracking algorithm is proposed for use in WPT systems. The WPT system employs multi sub-coil flux pipe couplers, a full-bridge inverter which is driven by TMS320F28069 controller card and is suitable for high power charging applications. The control algorithm can accurately find the global maximum power point in case of frequency splitting with asymmetric peaks. The proposed frequency tracking algorithm operates only at the primary side based on measurement of the input power level. Therefore, no additional communication link is needed between the primary and the secondary side. Effectiveness of the proposed control method is validated by performing experiments under three different misalignment scenarios and compared to the traditional Perturb and Observe algorithm. [ABSTRACT FROM AUTHOR]

Summary: To eliminate frequency splitting in wireless power transfer, a novel spindle‐shaped coil‐based wireless power transfer system is introduced and designed. Based on the equivalent circuit model, the transmission coefficient and the input impedance are calculated. A comparative analysis is performed between the traditional and the proposed systems. It is found that the frequency splitting is suppressed by utilizing spindle‐shaped transmitting and receiving coils in the proposed system because of the larger mutual inductance between the source coil (load coil) and transmitting coil (receiving coil) and the smaller mutual inductance between the transmitting and the receiving coils in comparison with the traditional system. In the overcoupled region, the transmission coefficient of the proposed system almost remains unchanged and keeps at a high value. The input impedance is characterized with high amplitude and small argument, which is beneficial for efficient power transfer. Experimental setup is established to validate the performance of the proposed wireless power transfer system. The results indicate that the proposed system is superior to the traditional one in eliminating the frequency splitting, which is well consistent with the simulation analysis. [ABSTRACT FROM AUTHOR]

Wireless power transmission technology avoids the problem of towed wires in the process of using electric energy, and increases the flexibility of using electricity. It is a hot research topic at present. In order to improve the system performance, a field-circuit coupling algorithm is proposed to analyze the system performance, with the help of the concept of supercomputing. Frequency splitting is a phenomenon in wireless power transfer (WPT) system when the coupling distance is less than the splitting point, the load power changes from a single-peak curve to a double-peak curve driven by two non-intrinsic resonant frequencies. Asymptotic coupled mode theory (CMT) method is used to analyse the frequency splitting phenomena in WPT system. It provides detailed information about interaction of field strength under different coupling states through coupled solution of FEM and CMT. Over coupling, critical coupling and under coupling are three typical states classified by frequency splitting. Experimental results are acquired by two helical resonators. The overall system reaches the critical coupling state when resonators space 1.5 m and the total power on the load is 110 W. Therefore, it is an efficient way to forecast transmission characteristics by using this method. [ABSTRACT FROM AUTHOR]

• Frequency splitting is the equivalent form of the defective mass on the low-frequency axis. • Removal of defective mass on the low-frequency axis eliminates frequency splitting. • Frequency splitting and rigid-axis azimuth can be recognized based on standing-wave oscillation effects and spectral analysis. • An ion beam trimming process for frequency splitting is designed and optimized. • The designed trimming process is stable and efficient, which is significant in engineering applications. This paper studies a high-precision etching process based on focused ion beam static trimming of the defective mass of fused silica hemispherical resonators to eliminate frequency splitting. This trimming method effectively overcomes the shortcomings of inefficiency and poor robustness, and the accuracy is seriously dependent on the stability of the trimming equipment in the traditional approach based on the harmonic form of trimming. The vibration model of an imperfect hemispherical resonator is firstly established by equating structural defects into point mass defective units, revealing that the frequency splitting is a function of the size of the equivalent defect mass and that the angular position of the stiffness axis is determined by the spatial distribution of the defect mass, which further elucidates that the frequency splitting of the resonator is the equivalent form of the defect mass on its low-frequency axis. To validate the accuracy of the present method, a simulation model of the imperfect resonator is established, and the results show that the theoretical model is in good agreement with the finite element method. Then, a high-precision nonlinear parameter identification model containing frequency splitting and low-frequency axis azimuth parameters is established based on the standing wave oscillation effect and spectral analysis, and the frequency splitting and low-frequency axis azimuth are identified by analyzing the vibration function of the hemispherical resonator. Finally, based on the parameter identification results and the ion beam removal function analysis, an ion beam trimming process is designed to eliminate frequency splitting by trimming the defective mass on the low-frequency axis. The results show that the frequency splitting after mass trimming is better than 0.001 Hz, significantly improving the hemispherical resonator's performance. [Display omitted] [ABSTRACT FROM AUTHOR]

Due to the effects of splitting frequency and cross coupling, the resonant frequency of the WPT system usually deviates from the given frequency band, and the system operating at the given frequency band suffers a very low output power. Ensuring that electric vehicle wireless power transfer (EV-WPT) systems operate at a resonant state is the prerequisite for efficient energy transfer. For this purpose, a novel design method by manipulating the eigenstate parameters is proposed in this paper. The proposed system can make a EV-WPT system with arbitrary coil successfully to resonate at any given bands, not just a single band. Therefore, the method designed in this article cannot only eliminate the problem of low power caused by frequency deviation, but also realize the application requirements of multiple frequency bands. Firstly, this article establishes an accurate state space model of an n-coil fully coupled EV-WPT system, and after that, the analytical current response on each circuit is derived. Based on that, a detailed frequency spectrum analysis is presented, along with several essential spectrum parameters’ derivations, including center frequencies and bandwidths. Then, with the center frequency and bandwidth as the design indexes, a novel methodology of designing to make EV-WPT systems achieve resonant-state at arbitrary given bands is derived. Finally, simulation and experimental verification are carried out. Simulation and experimental results show that whether it is a single-band or multi-band system, the accuracy of the value under designed resonant frequency is less than 0.01, which can effectively eliminate the frequency deviation phenomenon and obtain the maximum power output at the given frequency band.