Formula For Mutual Inductance Research Articles

Calculation and regulation of mutual inductance of wireless charging coils with metal shielding materials based on multiple intelligent optimization algorithms

SummaryIn wireless charging systems for electric vehicles, fluctuations in coil mutual inductance can affect both system output power and efficiency. This paper introduces a coil mutual inductance model that accounts for metal shielding material. The formula for coil mutual inductance is derived using Maxwell's equations and boundary conditions. Additionally, the paper presents a tactic for adapting the mutual inductance with the aid of an intelligent optimization algorithm, which is utilized for steadying the mutual inductance of the transmission coil. Lastly, a model of the transmission coil is developed to authenticate the practicality of the mutual inductance equation and the adaptation scheme. Measurement and calculation results indicate that the difference between the calculated value of the mutual inductance formula and the actual mutual inductance value of the coil, following deflection and offset, is below 5%. Moreover, the algorithm, along with the employed adaptation strategy, consistently demonstrates effectiveness in maintaining the mutual inductance value within a predefined range. The temporal efficiency of the algorithm, as measured by the time taken to determine this coordinate position, is consistently under 1 s. This ensures that the algorithm exhibits a rapid response time, a crucial attribute in ensuring its practical utility.

Simulation and Theoretical Study on the Mutual Inductance of the B-Dot Used for High-Power Transmission Lines

For the B-dot used for high-power transmission lines, due to its special structure, the current method to obtain its mutual inductance is calibration experiments and there lacks an effective method to estimate its mutual inductance at the design stage. In this paper, the mutual inductance of this kind of B-dot is studied by simulation and theoretical analysis. Through simulation, the frequency response of the mutual inductance is obtained and it is found that the mutual inductance at high frequency which is the main working frequency range of the B-dot is lowest and stable. Through theoretical analysis, the approximate calculation formula of mutual inductance at any frequency is derived. Using the experimental results in previous reference, the theoretical results and the simulation results are verified. The influencing factors of the B-dot’s mutual inductance at high frequency are analyzed. The results show that the number of wire loops, the inner radius of wire loops and the radius of hole are positively related to the mutual inductance, and the height of the center of the wire loops is negatively related to the mutual inductance. Among these four factors, increasing the inner radius of wire loops can improve the mutual inductance more significantly. The work in this paper reveals the physical meaning of the mutual inductance change and provides guidance for the design of the B-dot.

A Concise Analytical Formula of Mutual Inductance for Circular Rogowski Coil Revealing Spatial Distribution Characteristics of Mutual Inductance Deviation

Mutual inductance between primary wire and circular Rogowski coil varies with the position of primary conductor, the deviation of mutual inductance referred to center of circle causes the measurement error. Usually, the mutual inductance of Rogowski coil can be calculated by the summation of each turn of coil, however the spatial distribution characteristics of mutual inductance deviation has not been revealed. A concise analytical formula of mutual inductance is derived based on the summation form and verified by experiment in this paper, which leads to three characteristics of mutual inductance deviation. The first is the mapping relation of mutual inductance deviation between inside and outside Rogowski coil which established connection between positional accuracy and anti-interference ability. The second is the zero deviation line of mutual inductance deviation given by simple equation, which describes the position of primary wire where the measurement error vanishes. The third is approximate function on turns number, inner and outer radius of Rogowski coil and measurement error, which is easy to calculate and simplifies the design progress of Rogowski coil. Furthermore the formula and characteristics of mutual inductance can be generalized from uniform Rogowski coil and perpendicular primary wire to discontinuous PCB RC, primary conductor with cross section, and inclined primary wire. The mapping relation, zero deviation line and improved formula of mutual inductance deviation are discussed.

Modeling and optimization of pavement scale-model for magnetically coupled resonant in wireless power transmission systems

Electric vehicles (EVs) can achieve energy conservation and sustainable development. However, the development has been limited by mileage and battery size. The application of magnetically coupling resonant (MCR) wireless power transmission (WPT) can be used to solve the drawbacks of the EVs. In this study, the parameters of the primary coils and the secondary coils were selected and optimized by using COMSOL Multiphysics finite element software. The influence of the primary coils on the mutual inductance of the secondary coils were conducted to adopt the modes of non-superposition, 1/3 superposition and 1/2 superposition, respectively. The mutual inductance formulas of coils were derived in different superposition modes, the calculation results were in well coincided with the simulation. The pavement scale-model based on MCR wireless power transmission was established by simulation and calculation results to determine the coil parameters and the optimum superposition mode. The influence of the secondary coil on the wireless charging efficiency was explored in static and different moving speeds. The results showed that speed of 40–60 cm/s and spacing of 28 mm were deemed as the optimal charging efficiency when considering the requirements of movement state of the EV and coil embedment.

Unilateral Route Method to Estimate Practical Mutual Inductance for Multi-Coil WPT System

Multi-coil WPT systems require mutual inductance information between coils to increase the power transmission efficiency. However, in the high frequency (HF) bands such as 6.78 MHz and 13.56 MHz, the presence of surrounding coils changes the value of the mutual inductance between the two coils due to the parasitic element effect of the coils. These parasitic effects make it harder to estimate the mutual inductance among three or more coils. In contrast to ideal mutual inductance, which has a constant value regardless of frequency and surrounding coils, we define the practical mutual inductance as the mutual inductance varied by parasitic elements. In this paper, a new method is presented to estimate the practical mutual inductance between multiple coils in the HF band. The proposed method simply configures the expression of practical mutual inductance formula because only one of two bilateral dependent voltage sources generated by mutual inductance is considered. For several coils placed along the same axis, the practical mutual inductances between coils were measured with respect to the distance between them to validate the proposed method. The practical mutual inductance obtained from the proposed method was consistent with the simulated and measured values in HF band.

Three‐dimensional rotatable omnidirectional MCR WPT systems

Recently, Wi-Fi wireless power transfer (WPT) has attracted increasing attention. In this study, a rotatable three-dimensional (3D) transmitter (Tx) is first designed which only needs to be powered by a single source without current amplitude or phase control. Mutual inductance formulas of the Tx and a receiver (Rx) during the Tx's 3D rotation are derived based on Biot-Savart law and rotating matrix, and the maximum coupling areas are obtained. A method of achieving equal coupling of Rxs with the Tx which simultaneously charges two Rxs is proposed. A real rotatable 3D Tx and an omnidirectional magnetically coupled resonant (MCR) WPT prototype are fabricated, and the theoretical analysis is verified by simulation and experiment. Experiment displays that the maximum or desired coupling of single or two Rxs in 3D space can be obtained by calculated rotation combinations. The maximum power transfer efficiency (PTE) of the system can reach 90.2% with the transmission distance of 1.5 times the radius of Tx. Finally, the PTE can be maintained at 73% with the transmission distance varying from 18 to 27 cm by a novel method. The proposed 3D technique could be applied to many fields including Wi-Fi WPT home devices, remote sensing and some mechanical systems.

Inductance Calculation of Multilayer Circular Printed Spiral Coils

In the past few years, implanted biomedical devices have been widely used in healthy and medical applications and wireless power transfer (WPT) techniques used in these devices have been become a research hot topic. The printed spiral coils (PSCs), which have characteristics of small volume, low cost, high precision and high-integration, are better suited for WPT system. It is noteworthy that mutual inductance value of PSCs is a significant parameter which can influence the transfer efficiency of WPT system. Considering the geometry of multilayer circular PSCs and the distance between two adjacent layers, the previous methods are not suitable for calculating the mutual inductance of multilayer circular PSCs. In this study, an approximate calculation formula of mutual inductance between two single-layer circular PSCs is presented based on the existing research. On this basis, the self-inductance values of double-layer circular PSCs are calculated by MATLAB, the calculation results are verified by the ANSYS Q3D simulation and measurement results.

Miniaturization of SCMR Systems Using Multilayer Resonators

Multilayer elements are used to miniaturize the size of highly efficient wireless power transfer systems based on the Strongly Coupled Magnetic Resonance (SCMR). Specifically, inspired by miniaturization techniques used in antennas, multilayer resonators in conformal SCMR systems (instead of single-layer resonators traditionally used) are tightly stacked to significantly shrink the size of CSCMR systems. An analytical methodology is developed, based on Kirchhoff's law and Maxwell's mutual inductance formula using elliptic integrals, which can predict the system's resonance. Excellent agreement is shown between analytical, simulated and measured results. Based on our results the proposed multilayer CSCMR systems can operate at significantly lower frequencies than traditional CSCMR systems, while maintaining significantly smaller size as well as providing significantly higher efficiency and transmission range than traditional CSCMR systems.

Mutual inductance calculation for coils with misalignment in wireless power transfer

In wireless power transfer (WPT) systems, the energy transmission channel is established by the coupling relationship between transmitting coil and receiving coil, and the coupling strength is usually measured by coupling coefficient. Therefore, it is necessary to calculate the mutual inductance between transmitting coil and receiving coil when describing their coupling strength. Here, filamentary circular coils and cylindrical helix coils are taken as the object of study, and the formulas for mutual inductance of two coils with misalignment are derived based on Neumann's formula. In addition, the influence of multiple factors, such as axial distance, lateral distance and angle, on mutual inductance is analysed, and the calculated results are verified by experiments. The research results of this paper can be used to estimate the mutual inductance between the couplers in WPT systems, especially when there is misalignment.

Dual-Side Independent Switched Capacitor Control for Wireless Power Transfer with Coplanar Coils

The transfer coils of traditional Magnetic Coupling Resonant Wireless Power Transfer (MCR-WPT) systems are generally arranged coaxially. Coplanar coils can be an alternative scheme that can save more space in some applications, such as mobile phone wireless charging. However, the inductances of coplanar coils are sensitive to foreign objects, which leads to a reduction in transfer efficiency and output power. A MCR-WPT system with coplanar coils and its control strategy are proposed in this paper. First, the characteristic of the mutual inductance and the magnetic field distribution between the coplanar coils are analyzed. A formula to calculate mutual inductance between the coplanar coils is proposed in this part. Secondly, the effect of inductance offset and frequency detuning on transfer efficiency and output power are analyzed. Then, the control strategy to eliminate frequency detuning is proposed. The proposed method implements switched capacitor to take place of constant compensation capacitor. The equivalent capacitance of switched capacitor is adjusted when the frequency detuning occurred. Thus, the inherent frequency of the resonant tank tracks the source frequency all the time. Since the switched capacitor of each side is controlled based on the quantity of their own, the control process is independent and does not require wireless communication. The complexity and the cost of the system are reduced. At the end of the article, the veracity of mutual inductance formula and the effectiveness of the proposed control strategy are verified by experiments. The experimental coils are placed in in an environment full of interference. The inductances of the coils are reduced from 224 μH to about 214 μH. The transfer efficiency and output power of MCR-WPT system with closed-loop control are higher than the one without control. At a distance of 5 cm from edge to edge, the transfer efficiency is 76.37% under the proposed control and 72.21% under no control. The output power is 285.66 W under the proposed control and 271.37 W under no control.

Modeling circular inductors coupled to a semi-infinite magnetic medium considering the proximity effect

This paper presents a simple 2D electromagnetic model that solves the high-frequency current distribution, considering the proximity effect, in a planar spiral coil above a linear conductive or non-conductive semi-infinite magnetic medium. The conductive regions are divided into axisymmetric elements in which the current is assumed constant. The current flowing in each element depends on its complex impedance and is computed by Kirchhoff’s circuit law. To take the effect of the magnetic medium into account, a new set of mutual inductance formulas are presented. Those formulas are expressed in terms of elliptic integrals and the fast-converging arithmetic–geometric mean iteration of Gauss. The geometric mean distance method is used to deal with elements of arbitrarily shaped cross-section. Elliptic integrals are also used to express the magnetic flux density. The current distribution, the magnetic field and the equivalent impedance computed with the multifilament model agree well with the results obtained using commercial finite element software.

Modeling and Optimization of Magnetically Coupled Resonant Wireless Power Transfer System With Varying Spatial Scales

Previous work reveals that the magnetically coupled resonant (MCR) wireless power transfer (WPT) technology is efficient and practical for mid-range wireless energy transmission, able to handle nontrivial amount of power. This paper presented an equivalent analytical model for MCR WPT system to incorporate spatial misalignments between the transmitting coils and receiving coils. The mutual inductance formulas were derived when receiving coils are laterally, angularly or generally misaligned from transmitting coils. The relationship among the output power, transmission efficiency and the mutual inductance, load resistance were analyzed in detail. Experiments had also been carried out to facilitate quantitative comparison and validate the theoretical analysis.

Improved Formulae for the Inductance of Straight Wires

The best analytical formulae for the self-inductance of rectangular coils of circular cross section available in the literature were derived from formulae for the partial inductance of straight wires, which, in turn, are based on the well-known formula for the mutual inductance of parallel current filaments, and on the exact value of the geometric mean distance (GMD) for integrating the mutual inductance formula over the cross section of the wire. But in this way, only one term of the mutual inductance formula is integrated, whereas it contains also other terms. In the formulae found in the literature, these other terms are either completely neglected, or their integral is only coarsely approximated. We prove that these other terms can be accurately integrated by using the arithmetic mean distance (AMD) and the arithmetic mean square distance (AMSD) of the wire cross section. We present general formulae for the partial and mutual inductance of straight wires of any cross section and for any frequency based on the use of the GMD, AMD, and AMSD. Since partial inductance of single wires cannot be measured, the errors of the analytical approximations are computed with the help of exact computations of the six-dimensional integral defining induction. These are obtained by means of a coordinate transformation that reduces the six-dimensional integral to a three-dimensional one, which is then solved numerically. We give examples of an application of our analytical formulae to the calculation of the inductance of short-circuited two-wire lines. The new formulae show a substantial improvement in accuracy for short wires.

New Formulas for Mutual Inductance and Axial Magnetic Force Between Magnetically Coupled Coils: Thick Circular Coil of the Rectangular Cross-Section-Thin Disk Coil (Pancake)

This paper deals with novel formulas for calculating the mutual inductance and the magnetic force between two coaxial coils in air comprising a circular coil of the rectangular cross section and a thin disk coil (pancake). The mutual inductance and the magnetic force are expressed over the complete elliptical integrals of the first and second kinds, Heuman's lambda function, and three terms that must be solved numerically. All singular cases have been resolved analytically. We also give another approach as comparative method that is based on the filament method where all conductors are approximated by the set of Maxwell's coils. The expressions are obtained over the complete elliptical integrals of the first and second kinds. These new consistent expressions are accurate and simple for useful applications. All results obtained by different approaches are in excellent agreement.

헬리컬 코일을 이용하는 자기 공진형 무선 전력 전송 시스템에서 새로운 상호 인덕턴스의 계산식 제안

본 논문에서는 헬리컬 코일을 이용하는 공진형 무선 전력 전송 시스템에서 새로운 상호 인덕턴스 계산식을 이용한 해석적 계산을 제안한다. 상호 인덕턴스 계산 과정 중에 중요한 변수의 look-up table을 수식화 하였다. 송 수신이 대칭인 구조에서 두 헬리컬 코일 간의 거리를 53 mm부터 500 mm까지 10 mm 간격으로 거리를 증가시키며, 공진 주파수와 삽입 손실에 대한 계산 결과와 실험 결과를 비교하였다. 실험 결과와 계산 결과의 공진 주파는 평균적으로 5.63 %의 차이를 보였다. 삽입 손실은 실험 결과와 290 mm에서 0.33 dB의 가장 작은 차이를 보였고, 평균적으로 2.25 dB의 차이를 보였다. 발룬을 사용하지 않았을 때의 실험 결과가 계산 결과와 더 근접한 결과를 보이는 것을 확인했다. In this paper, analytical calculations using a novel mutual inductance formula for a resonant wireless power transmission system using helical coils. The look-up table of critical variables during the mutual inductance calculation process was formulated. The calculation results for resonant frequency and insertion loss were compared with experimental results when the distance between the two helical coils in a structure where the transmission and reception is symmetrical was varied with 10 mm increments from 53 mm to 500 mm. On average, the resonant frequency showed a difference of 5.63 % between the experimental results and the calculation results. The insertion loss had an average difference 2.25 dB where the smallest difference of 0.33 dB occurred with 290 mm. It was found that the experimental results without using a balun were in greater agreement with the calculation results.

Correction to “New Formulas for Mutual Inductance and Axial Magnetic Force between a Thin Wall Solenoid and a Thick Circular Coil of Rectangular Cross-Section” [Aug 11 2034-2044

Equation (6) in the above titled paper (ibid., vol. 47, no. 8, pp. 2034-2044, Aug. 2011), was presented incorrectly. The corrected equation is presented here.

New Formulas for Mutual Inductance and Axial Magnetic Force Between a Thin Wall Solenoid and a Thick Circular Coil of Rectangular Cross-Section

This paper presents new analytic formulas for determining the mutual inductance and the axial magnetic force between two coaxial coils in air, namely a thick circular coil with rectangular cross-section and a thin wall solenoid. The mutual inductance and the magnetic force are expressed as complete elliptical integrals of the first and second kind, Heuman's Lambda function and one well-behaved integral that must be solved numerically. All possible singular cases are automatically handled by the proposed formulas. The results of the work presented here have been verified by the filament method and previously published data. The new formulas provide a substantially simple alternative over previously published approaches, which involve either numerical techniques (finite element method, boundary element method, method of moments) or other semianalytic or analytic approaches.

High precision computation of solenoid magnetic fields by Garrett's methods

For magnetic field calculations of solenoids two methods, due to M.W. Garrett (1967), are developed. The first one consists of the expansion of the magnetic field in a zonal harmonic series in the central region of solenoid and in outer space. For both regions the expansion coefficients are given either by simple recurrence relations, or by explicit expressions. In particular, this method is suitable for high accuracy calculations of the field within windings. In the second method, the field is expressed in terms of incomplete elliptic integrals of the first and second kinds computed by use of raising Landen transformations. Given the high precision and high computation rate of this method, it has advantages in analyzing field problems for windings consisting of many radial sections carrying different currents, as occurs during the quench of superconducting multisectioned magnets. Also, the general formulas for mutual inductance and axial force between a pair of coaxial solenoids are given.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The Ampere‐Neumann Electrodynamics of Metallic Conductors

AbstractThis is a review of the old electrodynamics which prevailed during the first half of the 165‐year history of electromagnetism. Amperes principal achievement was the deduction of his empirical force law from experiments with several current balances. Faraday then discovered electro‐magnetic induction. This prompted F. E. Neumann to work out a quantitative explanation of induction based on Amperes force law. It involved the concept of the electrodynamic potential which, as we know now, is the same entity as magnetic energy. With the newtonian principle of virtual work, Neumann found his potential yielded the correct mechanical forces on metallic current circuits. Neumanns theory contains a physical quantity which today is called the magnetic vector potential and treated as a mathematical contrivance.Neumanns mutual inductance formula has become a powerful tool of inductance calculations. Maxwell made a major contribution to the Ampere‐Neumann electrodynamics by developing the mean‐geometric‐distance method for calculating the inductance of conductors of finite cross‐sections. This became particularly useful after Sommerfeld solved Neumanns double integral for parallel, straight wires. Maxwell built all of Neumanns mathematical theory into his field equations but the lingo changed. Electrodynamic potential became kinetic energy of the field; conductor element interactions became flux linkage; and so on. Maxwells equations do not contain a magnetic force law. He believed both Amperes law and the law currently in use, which was first suggested by Grassmann in 1845, were compatible with field theory. Lorentz later found that the motion of charges in vacuum obeyed only Grassmanns law and not Amperes. From then onward the old electrodynamics fell into disuse and field theory has reigned supremely ever since.Recent developments have shown the conflict between Amperes and Grassmanns law to be related to the nature, of the electric current. Conduction currents in metals obey Amperes law and convection currents in vacuum obey Grassmanns law. Both laws agree on the reaction forces between closed metallic circuits, because the relativistic contribution from the Grassmann law then integrates to zero. This fact appears to have mislead Lorentz in believing that the drifting electron in vacuum is magnetically equivalent to the current element of metals.An examination of the long debate concerning the validity of Newtons third law of motion in electromagnetism proves the Ampere‐Neumann electro‐dynamics to be valid for metallic circuits while the theory of special relativity and field momentum conservation are required for convecting charges in vacuum. This conclusion is strongly supported by experimental evidence. It demands a change in the concept of the metallic current element.

The Prediction of Current Distribution in Induction Heating Installations

This paper describes a coupled equivalent circuit approach for predicting the current distribution in three-dimensional geometries having axial symmetry. Specimens of arbitrary cross section can be handled by this numerical method so long as they have circular symmetry about the coil axis. Several self-inductance and mutual inductance formulas are investigated, and the merits of these formulas for calculating the parameters for the equivalent circuit are compared. Experimental and theoretical results are compared for a variety of geometries. Current density probes were used to obtain the experimental results.