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

The GMD Method for Inductance Calculation Applied to Conductors with Skin Effect

The GMD method (geometric mean distance) to calculate inductance offers undoubted advantages over other methods. But so far it seemed to be limited to the case where the current is uniformly distributed over the cross section of the conductor, i.e. to DC (direct current). In this paper, the definition of the GMD is extended to include cases of nonuniform distribution observed at higher frequencies as the result of skin effect. An exact relation between the GMD and the internal inductance per unit length for infinitely long conductors of circularly symmetric cross section is derived. It enables much simpler derivations of Maxwell’s analytical expressions for the GMD of circular and annular disks than were known before. Its salient application, however, is the derivation of exact expressions for the GMD of infinitely long round wires and tubular conductors with skin effect. These expressions are then used to verify the consistency of the extended definition of the GMD. Further, approximate formulae for the GMD of round wires with skin effect based on elementary functions are discussed. Total inductances calculated with the help of the derived formulae for the GMD with and without skin effect are compared to measurement results from the literature. For conductors of square cross section, an analytical approximation for the GMD with skin effect based on elementary functions is presented. It is shown that it allows to calculate the total inductance of such conductors for frequencies from DC up to 25 GHz to a precision of better than 1 %

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

The GMD Method for Inductance Calculation Applied to Conductors with Skin Effect

The GMD method (geometric mean distance) to calculate inductance offers undoubted advantages over other methods. But so far it seemed to be limited to the case where the current is uniformly distributed over the cross section of the conductor, i.e. to DC (direct current). In this paper, the definition of the GMD is extended to include cases of nonuniform distribution observed at higher frequencies as the result of skin effect. An exact relation between the GMD and the internal inductance per unit length for infinitely long conductors of circularly symmetric cross section is derived. It enables much simpler derivations of Maxwell’s analytical expressions for the GMD of circular and annular disks than were known before. Its salient application, however, is the derivation of exact expressions for the GMD of infinitely long round wires and tubular conductors with skin effect. These expressions are then used to verify the consistency of the extended definition of the GMD. Further, approximate formulae for the GMD of round wires with skin effect based on elementary functions are discussed. Total inductances calculated with the help of the derived formulae for the GMD with and without skin effect are compared to measurement results from the literature. For conductors of square cross section, an analytical approximation for the GMD with skin effect based on elementary functions is presented. It is shown that it allows to calculate the total inductance of such conductors for frequencies from DC up to 25 GHz to a precision of better than 1 %

Drift waves and magnetic field oscillations in cylindrical plasmas

A general investigation of linear drift-wave phenomena in cylindrically bounded plasmas, immersed in a magnetic field without shear and curvature, is performed within the two-fluid hydrodynamical approximation, taking into account electron-temperature oscillations and inhomogeneous radial distributions of the undisturbed electron density and temperature. For plasmas in which the electron temperature strongly exceeds the ion temperature the problem is reduced to an ordinary complex second-order differential equation describing the radial distribution of the oscillating electric potential. It is shown that the presence of electron-temperature oscillations (which must always exist in order to satisfy electron-energy conservation) and of radial gradients in the undisturbed electron temperature (which must always exist owing to cooling of the plasma at the boundary) leads to an important modification of the theory of drift waves in cylindrical plasmas (with regard to their stability and the radial distribution of the oscillating quantities) compared with previous papers in which these phenomena were disregarded. A numerical program for solving the corresponding complex-eigenvalue problem has been derived that allows a realistic calculation of all the quantities pertaining to drift-wave phenomena. It has been applied, in particular, to the calculation of the radial distribution of the oscillating coherent magnetic fields accompanying the coherent drift waves. The numerical results prove to be in good agreement with experiments performed with a helium plasm

Comparative Study of the Accuracy of Analytical Inductance Formulae for Square Planar Spiral Inductors

In the design of radio frequency (RF) microelectronic integrated circuits (IC’s) and of antennas for short-wave radio frequency identification (RFID) and telemetry systems, planar spiral coils are important components. Many approximate analytical formulae for calculating the inductance of such coils can be found in the literature. They can simplify the problem of designing inductors to a predefined inductance considerably. But the error statistics given by different authors cannot be compared because they are based on different or unknown domains of definition. Hence, it is not possible to decide which formula is best in a given case by merely studying the literature. This paper compares the maximum relative errors of six of some of the most cited formulae in the literature. To all formulae, the same domains of definition are applied. Each of them spans all four dimensions of the parameter space. Precise inductances are obtained numerically with the help of the free scientific and industrial standard software FastHenry2 and used as reference values to calculate the errors of the formulae. It has been found that the alleged maximum errors reported by some authors are far too optimistic. Only two formulae feature small enough errors to be useful in circuit design. The method and the domains of definition applied in the present study may also prove useful for the assessment of future formulae

Analytical Approximation for the Inductance of Circularly Cylindrical Two-Wire Transmission Lines with Proximity Effect

The paper describes a simple analytical approximation for the inductance of two-wire transmission lines of circularly cylindrical wires with proximity effect. It yields precise results up to very high frequencies, and also at all interaxial distances between the wires above some limit. Its accuracy is established by comparison to numerical computations and to measurements. It is shown that the proximity effect cannot be neglected unless the interaxial distance between the wires amounts to at least four wire diameters. Further, images of the current distribution in various situations are discussed

insights by genetic characterization

Background Giardia duodenalis is a common flagellated protozoan parasite that infects the small intestine of a wide range of vertebrate hosts. This study aimed to determine whether tracing of G. duodenalis isolates by current genetic typing tools is possible using an exemplary set of samples from infected cattle, buffalo and children from the Ismailia province, Egypt. Method A total of 804 fecal samples from ruminant animals was collected from 191 herds and 165 samples from diarrheal children below the age of 10 years. Parasites were detected in these samples using the copro-antigen RIDA®QUICK test and by real-time PCR. Samples were then genetically characterized based on the triosephosphate isomerase, glutamate dehydrogenase and β-giardin genes. Results The prevalence of G. duodenalis was 53% in ruminants and 21% in symptomatic children and infection was not positively correlated with diarrheal symptoms. Sequence typing analysis confirmed predominance of B-type sequences (>67%) in humans and E-type sequences (>81%) in ruminants over A-type sequences. For 39 samples the complete sequence information of the three marker gene fragments could be derived. Integration of the concatenated sequence information of the three marker gene fragments with the spatial data of the respective sample revealed that identical or near identical (only up to 1 out of 1358 bp different) concatenated sequencing types were spatially related in 4 out of 5 cases. Conclusion The risk of zoonotic infection emanating from ruminants even in high prevalence areas is negligible. Genetic characterization indicated a predominant anthropogenic cycle of infection within the pediatric population studied. Integration of sequence typing data with information on geographic origins of samples allows parasite sub- population tracing using current typing tools

Epidemiology of Giardia duodenalis infection in ruminant livestock and children in the Ismailia province of Egypt: insights by genetic characterization

Background: Giardia duodenalis is a common flagellated protozoan parasite that infects the small intestine of a wide range of vertebrate hosts. This study aimed to determine whether tracing of G. duodenalis isolates by current genetic typing tools is possible using an exemplary set of samples from infected cattle, buffalo and children from the Ismailia province, Egypt. Method: A total of 804 fecal samples from ruminant animals was collected from 191 herds and 165 samples from diarrheal children below the age of 10 years. Parasites were detected in these samples using the copro-antigen RIDA®QUICK test and by real-time PCR. Samples were then genetically characterized based on the triosephosphate isomerase, glutamate dehydrogenase and β-giardin genes. Results: The prevalence of G. duodenalis was 53% in ruminants and 21% in symptomatic children and infection was not positively correlated with diarrheal symptoms. Sequence typing analysis confirmed predominance of B-type sequences (>67%) in humans and E-type sequences (>81%) in ruminants over A-type sequences. For 39 samples the complete sequence information of the three marker gene fragments could be derived. Integration of the concatenated sequence information of the three marker gene fragments with the spatial data of the respective sample revealed that identical or near identical (only up to 1 out of 1358 bp different) concatenated sequencing types were spatially related in 4 out of 5 cases. Conclusion: The risk of zoonotic infection emanating from ruminants even in high prevalence areas is negligible. Genetic characterization indicated a predominant anthropogenic cycle of infection within the pediatric population studied. Integration of sequence typing data with information on geographic origins of samples allows parasite sub-population tracing using current typing tools

Materials with Colossal Dielectric Constant: Do They Exist?

Experimental evidence is provided that colossal dielectric constants, epsilon >= 1000, sometimes reported to exist in a broad temperature range, can often be explained by Maxwell-Wagner type contributions of depletion layers at the interface between sample and contacts, or at grain boundaries. We demonstrate this on a variety of different materials. We speculate that the largest intrinsic dielectric constant observed so far in non-ferroelectric materials is of order 100.Comment: 3 figure