Alternating currents in conductors: circular cross-section

This paper presents a method of testing sheet materials at elevated temperatures using direct resistance heating. The problems associated with the use of direct resistance heating are discussed by referring to investigations on the heating characteristics of short, rectangular sheet blanks. Experimental results show that elevated-temperature tensile tests at considerably fast strain rates can be carried out using direct resistance heating. The major drawback is shown to be the non-uniformity of temperature of the test specimen at its ends due to heat transfer from the specimen to the unheated jaws. Ways of minimizing this non-uniformity are discussed and a suggestion is made for further reduction by indirect resistance heating of jaws using an independent power supply.

This chapter deals with some specific applications of induction and direct resistance heating, that have been investigated also by the authors of this book. When available, analytical solutions for these ‘special’ applications are provided. Section 4.1 is about the resistance heating of ferromagnetic workpieces of rectangular cross-section, Sect. 4.2 describes the direct resistance heating of bended conductors. The transverse flux heating is a technology that is applied for treating thin slabs and stripes with high electrical efficiency and is the topic of the Sect. 4.3. Planar coils, often named pancake inductors, are described in the Sect. 4.4, while the effects due to the finite length of inductor are discussed in Sect. 4.5. Induction hardening of workpieces of complex geometry is one of the most important applications of induction heating, single and dual frequency processes for contour hardening are presented in Sect. 4.6. The innovative process of through heating by rotating a nonferrous billet inside a DC field produced by superconductive coils is detailed in Sect. 4.7, while in Sect. 4.8 the induction heating is produced by rotation of permanent magnets. Section 4.9 deals with the induction heating of hollow workpieces heated by internal inductors and Sect. 4.10 describes the forces acting in induction heating applications.

A method is presented for solving for the electromagnetic field in a multilayer circular cylindrical structure, excited by a concentric coaxial helical winding. The method is based on a double transmission line technique and it uses the concept of E-mode and H-mode lines. The use of Bessel functions is completely obviated by employing micro-T circuits. Supporting experimental evidence is also presented.

Thermal gradient effects in direct resistance heated metal matrix composites

An oscillating infinite series involving product of Bessel function J o(x) and an oscillating infinite series involving trigonometric function sin(x) were evaluated and computed numerically in [1] and [2] respectively. In this paper, an oscillating infinite series involving product of exponential, Bessel and trigonometric functions is evaluated. The series is transformed first into the sum of two infinite integrals by using contour integration and then the infinite integral with oscillating integrand is transformed through some identities into a finite integral containing modified Bessel function K 1(x). Finally, theset two integrals are evaluated numerically without any computational difficulties at all.

We first consider a summation procedure for some trigonometric series in terms of the Riemann zeta and related functions. In some cases these series can be brought in closed form, which means that the infinite series are represented by finite sums. Afterwards, we show some applications of our results to the summation of series involving Bessel or Struve functions. Further, relying on results from the previous sections, we obtain sums of series involving a Bessel or Struve integral. These series are also represented as series in terms of the Riemann zeta and related functions of reciprocal powers and can be brought in closed form in certain cases as well. By replacing the function appearing in a Bessel and Struve integral with particular functions, we find sums of new series.

An extension of the finite hankel transform and applications

The point-spread function (PSF) is used in optics for design and assessment of the imaging capabilities of an optical system. It is therefore of vital importance that this PSF can be calculated fast and accurately. In the past 12 years, the Extended Nijboer-Zernike (ENZ) approach has been developed for the purpose of semi-analytic evaluation of the PSF, for circularly symmetric optical systems, in the focal region. In the earliest ENZ-years, the Debye approximation of the diffraction integral, by which the PSF is given, was considered for the very basic situation of a low-NA optical system and relatively small defocus values, so that a scalar treatment was allowed with a focal factor comprising a quadratic function in the exponential. At present, the ENZ-method allows calculation of the PSF in low- and high-NA cases, in scalar form and for vector fields (including polarization), for large wave-front aberrations, including amplitude non-uniformities, using a quasi-spherical phase focal factor in a virtually unlimited focal range around the focal plane, and no limitations in the off-axis direction. Additionally, the application range of the method has been broadened and generalized to the calculation of aerial images of extended objects by including the finite distance of the object to the entrance pupil. Also imaging into a multi-layer is now possible by accounting for both forward and backward propagation in the layers. In the advanced ENZ-approach, the generalized, complex-valued pupil function is developed into a series of Zernike circle polynomials, with exponential azimuthal dependence (having cosine/sine azimuthal dependence as special cases). For each Zernike term, the diffraction integral reduces after azimuthal integration to an integral that can be expressed as an infinite double series involving spherical Bessel functions, accounting for the parameters of the optical system and the defocus value, and Jinc functions comprising the radial off-axis value. The contribution of the present paper is the formulation of truncation rules for these double series expressions, with a general rule valid for all circle polynomials at the same time, and a dedicated rule that takes into account the degree and the azimuthal order of the involved circle polynomials to significantly reduce computational cost in specific cases. The truncation rules are based on effective bounds and asymptotics (of the Debye type) for the mentioned spherical Bessel functions and Jinc functions, and show feasibility of computation of practically all diffraction integrals that one encounters in the ENZ-practice. Thus it can be said that the advanced ENZ-theory is more or less completed from the computational point of view by the achievements of the present paper.

By means of elliptical functions the rectangular line with inner central conductor of circular cylindrical shape is transformed into a coaxial line with circular outer conductor and nearly circular inner conductor, then by employing a finite number of terms of circular cylindrical harnomics we can fit the boundary conditions at the outer conductor and at a finite number of points at the inner conductor, thus the problem of the rectangular line with inner central circular conductor is solved. Similarly, by means of trigonometrical functions, the trough line with central inner conductor can be handled, i.e., it is transformed into a wire of nearly circular cross-section parallel to and in front of a grounded plane, then by means of the bipolar coordinate transformation, the problem of this trough line can be solved by using a finite number of rectangular harnomics. When the distance between the axis of the inner conductor and the bottom of the trough tends to infinity, the results obtained in this paper go into that of the well-known slab line.

We considered the problem of estimating the error in the solution of the wave equation recorded using infinite series Fourier-Bessel. The algorithm that adjusts the number of elements in a partial sum of infinite series, based on the assessment of the series balance. The application of the algorithm made it possible, without loss of accuracy, to substantially reduce the number of summable elements of the series in the numerical simulation of the light pulse propagation in a circular cross-section.

Warm Forming of AA7075-T6 with Direct Electrical Resistance Heating

VHF transequatorial propagation via three dimensional waveguides

The problem of sound scattering by an infinitely long penetrable and impenetrable cylinder suspended over a realistic impedance ground is investigated. The analytical approach using the image source method in the scattering of sound involves expressing the total sound field at any receiver point (over a locally reacting ground) as the sum total of the direct field, the ground reflection from the source, scattered field by the actual cylinder and finally by its image. The exact solutions can then be expressed as an infinite series, containing Bessel and Hankel functions of increasing order. The coefficients of the scattered field are determined by matching the desired boundary condition (for a rigid circular cylinder, for example, the normal component of velocity is zero on the boundary). Although the preceding approach is commonly used in theoretical treatments of sound scattering, to the best of our knowledge this is the first time it has been attempted numerically with coupling to the Parabolic Equation (PE) method. Presented is a Two-dimensional case of sound scattering of PE generated acoustic field by an impenetrable (soft or rigid) and penetrable cylindrical obstacle over a finite impedance ground in a non-refractive atmosphere.

In this work we apply the classical Plana summation formula to derive integral representations for certain infinite series that their general terms containing Gamma, q q -Gamma, theta, Bessel, and Hurwitz zeta functions.

Study on critical current of a flexible quasi-isotropic HTS conductor with high engineering current density