Conformal Transformation Technique Research Articles

Properties of high-gradient magnetogravimetric systems--I. Single rods and gratings with oval and circular cross sections

The properties of two-dimensional rod systems are examined, use being made of the complex potential in conjunction with the conformal-transformation technique. The concept of prescribed trajectories whereby stronger, effective, levitational forces can be obtained is introduced. It is shown that the high-gradient magnetogravimetric-separation system is amenable to scale-up since sufficient levitational force, in terms of the equivalent specific gravity, is generated. Examples of gratings with circular and different oval cross sections, which are subject to the magnetizing field in two alternative directions, are discussed.

Four-point sheet resistance correction factors for thin rectangular samples

A conformal transformation technique is employed to determine the geometrical correction factor associated with four-point sheet resistance measurements of thin rectangular samples. Results are presented which are useful in identifying sample geometries for which the correction factor is insensitive to the placement of the four electrodes on the interior of the sample. The correction factors for corner-contacted rectangular samples (van der Pauw resistors) are shown to be highly sensitive to small variations in the length-to-width ratio, such as might be experienced when fabricating microelectronic structures. Experimental data which serve to confirm the validity of the analytical solution are provided for several cases of practical interest.

Elimination of current concentration due to Hall effect by variable resistive electrodes

Presents a theoretical investigation on elimination of current distortion caused by the Hall effect in Faraday-type MHD power generators. Taking as an example an MHD generator with a pair of electrodes opposed, and applying the conformal transformation technique to the generator channel, it was shown that the excessive current concentration at the electrode edges due to Hall effect can be completely eliminated by adopting suitably variable resistive electrodes, and therefore, the current distribution in the generator channel is not influenced by the Hall effect. This method is based on the principle that the induced Hall field is exactly compensated by the potential gradient produced by the current flow in the resistive electrodes.

ConformalTransformations in the Presence of Field Components along a Third Axis — Part I

The presence of field components along a third axis does not prevent the application of planar conformal transformation techniques. Some complications are introduced in that the characteristics of the media are altered in the model, but in many cases this poses no major difficulties.

Seepage through Opening in Cutoff Wall Under Weir

A solution to the problem of seepage through an opening in a cutoff wall under an impervious structure lying on the ground level is presented with the use of the conformal transformation technique. The cutoff extends from the base of the structure to the impervious layer, which is overlaid by an isotropic and homogeneous porous stratum. The solution has been found to agree with established solutions for a partial cutoff wall and for a partial cutoff wall under a weir sitting on the ground level. Graphical presentation of the solution which takes into account the geometry, has been made for ready use of designers.

SEEPAGE CHARACTERISTICS OF FOUNDATIONS WITH A DOWNSTREAM CRACK

The seepage characteristics of foundations with a downstream crack is analysed using conformal transformation technique. Two cases are investigated: a downstream vertical crack in a finite depth of pervious stratum and a downstream inclined crack of finite depth in a pervious stratum of infinite depth. Experimental verification of the flow lines with an electrolytic tank model shows satisfactory agreement with the theory. It is shown that a downstream foundation crack, irrespective of its length and inclination, has negligible effect on the seepage characteristics when the crack is at a distance equal to or greater than the depth of the pervious stratum from the downstream end of the hydraulic structure.

Fringe and permeance factors for segmented-rotor reluctance machines

The field distribution in reluctance machines with profiled segmented rotors is analysed using numerical and analytical conformal-transformation techniques. Results, necessary for the sound design of such machines, are presented in terms of the permeance factors applicable to the regions of the field between segments and the fringe factors applicable to the fluxes passing to the stator.

Extrapolation of Nucleon Form Factors. II

The magnetic isoscalar form factor is fitted with two poles: the $\ensuremath{\omega}$ and the $\ensuremath{\varphi}$ resonance. The magnetic isovector form factor ${G}_{\mathrm{MV}}$ for spacelike and timelike momentum transfers is fitted using the conformal-transformation techniques of Levinger, Peierls, and Wang, and these techniques are tested using artificial data. If we do not assume a contribution from the $\ensuremath{\rho}$ resonance, we find a spectral function with a broad peak some 100 MeV below the position of the $\ensuremath{\rho}$, and with a negative dip around 1200 MeV. If we assume that the $\ensuremath{\rho}$ resonance contributes additively at its known position and width, but with an adjustable coefficient for its strength, we argue that the $\ensuremath{\rho}$ contributes about 90% of the static moment. The complete spectral function shows a shoulder around 500 MeV, and again a negative dip around 1200 MeV. If we assume that ${G}_{\mathrm{MV}}$ is a product of the form factor for the $\ensuremath{\rho}$ and an adjustable function, we find that the spectral function has a high peak near the $\ensuremath{\rho}$ resonance, and a marked negative dip at 900 MeV. The $\ensuremath{\rho}$ again contributes some 90% of the static moment. The different phenomenonological fits are in semiquantitative agreement with each other and with the recent field-theoretical calculations of Furuichi et al.

The analysis of the magnetic field variation in a magnet with a stepped gap

The gap of a permanent magnet has been stepped to provide a large value of BdB/dx which enables a sensitive susceptibility balance to be made. An analysis of the field variation using a conformal transformation technique is given. It shows that the maximum value of BdB/dx obtained in the region of the step is considerably larger than that produced by the infinite step at the edge of the pole pieces. An experimental determination of the maximum value of BdB/dx gives a result in satisfactory agreement with the predicted one and the discrepancy between the two is briefly discussed. Finally, a brief analysis is given of a double stepped system for the production of a uniform value of BdB/dx over an extended region.

Synthesis of Multiple Resistance Networks from Single Resistive Films

A design procedure is described whereby multiple resistance networks can be synthesized from a single layer, monolithic, resistive structure. The method is illustrated by detailing the design of attenuators operating on an image-impedance basis. The single sheet geometry used is a rectangle with conducting tabs placed on its boundary. Conformal transformation techniques are applied to facilitate the computation of the required open-circuit resistance values which characterize a given rectangular geometry. Two SchwarzChristoffel transformations and a bilinear transformation permit the rapid design of a monolithic resistive network meeting prescribed specifications. The mapping of critical points is diagrammed to help the reader clearly visualize the transformations involved. Specific analytical examples are carried out, and curves are presented showing the dependence of network characteristics upon the parameters of sheet resistivity, geometric ratios, and position and size of the conducting tabs. The synthesis of two pads is described to illustrate graphical techniques involved. Both attenuators are designed to operate between 75 \Omega characteristic impedances, and to have 2 db of insertion loss. The first pad is realized by holding tab size constant and deriving the necessary rectangular geometry and resistivity. In the second case, the resistivity is held constant while the required geometry is determined. In general, alternative physical embodiments which are electrically equivalent afford a desirable flexibility from the viewpoint of network fabrication. The above synthesis techniques have been substantiated by the construction and measurement of laboratory models. Experimental data are given which represent the results obtained using tantalum film designs. Excellent correlation with theory is noted. Possible advantages which monolithic single-sheet networks offer over the discrete multiple resistor circuits which they replace are reduction of the number of components, better HF performance, ease of manufacture, and higher reliability due to the need for fewer interconnections. The single-sheet resistive components are notable for their compatibility with current thin-film fabrication techniques.

A Solution to the Approximation Problem for RC Low-Pass Filters

This paper presents a method for investigating the transfer functions of RC low-pass networks. The procedure involves the use of the concept of potential analogy and the technique of conformal transformation. The poles and zeros are arranged in a certain symmetrical fashion in the transformed plane and on this plane the frequency characteristic may be written in closed form. The frequency characteristics obtained by this method have equal-ripple variation inside the pass band. By mathematical manipulations, the several quantities that are essential in describing the quality of each frequency characteristic; i. e., the pass-band tolerance, the stopband-passband ratio and the stop-band attenuation, may be expressed in terms of several geometrical quantities in the transformed plane and the parameters of the transformation function. Thus, the design of this group of network functions may be accomplished by adjusting the geometric dimensions in the transformed plane and the transformation functions. The transformation used here is a function that involves elliptic functions. Some design charts which are helpful in the selections of these values are given.