Schwartz-Christoffel Transformation Research Articles

A multi-dimensional Child–Langmuir law for any diode geometry

While prior theoretical studies of multi-dimensional space-charge limited current (SCLC) assumed emission from a small patch on infinite electrodes, none have considered emission from an entire finite electrode. In this paper, we apply variational calculus (VC) and conformal mapping, which have previously been used to derive analytic solutions for SCLC density (SCLCD) for nonplanar one-dimensional geometries, to obtain mathematical relationships for any multi-dimensional macroscopic diode with finite cathode and anode. We first derive a universal mathematical relationship between space-charge limited potential and vacuum potential for any diode and apply this technique to determine SCLCD for an eccentric spherical diode. We then apply VC and the Schwartz–Christoffel transformation to derive an exact equation for SCLCD in a general two-dimensional planar geometry with emission from a finite emitter. Particle-in-cell simulations using VSim agreed within 4%–13% for a range of ratios of emitter width to gap distance using the thinnest electrodes practical for the memory constraints of our hardware, with the difference partially attributed to the theory's assumption of infinitesimally thin electrodes. After generalizing this approach to determine SCLCD for any orthogonal diode as a function of only the vacuum capacitance and vacuum potential, we derive an analytical formulation of the three-dimensional Child–Langmuir law for finite parallel rectangular and disk geometries. These results demonstrate the utility for calculating SCLCD for any diode geometry using vacuum capacitance and vacuum potential, which are readily obtainable for many diode geometries, to guide experiment and simulation development.

Analytical Modeling of Eccentric PM-Inset Machines With a Slotless Armature

In this paper, by solving the Maxwell's equations, an analytical model is developed for the prediction of the air gap flux density in the eccentric permanent magnet (PM)-inset machines at no-load and on-load conditions. Since in this model neither Schwartz–Christoffel transformation nor perturbation analysis is used, it is fast and accurate. In the modeling process, the stator of the eccentric machine is replaced with an equivalent surface current (ESC), which is injected at the stator bore. Distribution of the ESC is unknown and must be selected, such that the resultant tangential field satisfies the boundary condition at the stator bore contour. To apply this condition, the flux density of the PMs, armature current, and the ESC are obtained by applying the subdomain analysis and using a conformal transformation. The armature current and ESC are considered as a surface current sheet at the stator bore. The back electromotive force and inductances of the eccentric machine are obtained in the proposed model. In addition, using the Maxwell stress tensor, the unbalanced magnetic force, and the electromagnetic torque are computed. Finally, the model is verified by means of finite element analysis.

Analysis and Application of MEMS Microshield Line in Millimeter Filters

This paper introduces the conformal mapping method for analyzing the quais-TEM characteristic of MEMS microshield line in detail. The conformal mapping method is an accurately method based on the Schwartz-Christoffel transformation which could invert the complex structure of the microshield line to the conventional plate capacitance. In this paper, the characteristic impedance Z0 and the effect permittivity eeff of the microshield line in wide dimension range are obtained. Making use of the simulation result, three filters are designed in millimeter frequency. Furthermore, the S-parameter of the filter indicates the filter designed by the line has good performance, low insertion loss and good out-band suppression.

Characteristics and Modeling of a Nonplanar Nonrectangular Metal Oxide Semiconductor Field Effect Transistor for Charge Sensing in the Si Micro-Fluidic Channel

In this work, a nonplanar, nonrectangular metal-oxide-semiconductor field effect transistor (MOSFET) with an asymmetrical channel structure for sensing charge in the Si micro-fluidic channel was fabricated, and the electrical characteristics of the fabricated three-dimensional (3-D) MOSFET were measured. The device was formed in the convex corner of a Si micro-fluidic channel using tetramethyl ammonium hydroxide (TMAH) anistropic etching solution, so that it would be suitable for combination with a micro-fluidic system. We approximated the nonplanar, nonrectangular 3-D MOSFET to a two-dimensional rectangular structure using the Schwartz-Christoffel transformation. The LEVEL1 device parameters of the 3-D MOSFET were extracted from the measured electrical device characteristics and were used in a simulation program with integrated circuit emphasis (SPICE) simulation. The measured and simulated results for the 3-D MOSFET were compared and found to show good agreement. We also investigated the feasibility of the proposed 3-D MOSFET as a charge sensor for detecting charged biomolecules.

Wideband Modeling of Integrated Power Passive Structures: The Series Resonator

This paper presents a higher order frequency plane model for integrated series resonant module (ISRM). The cause for the occurring of the high frequency impedance peak is identified as the parallel resonance between the winding inductance and its inter-winding capacitance. The inter-winding capacitance can be calculated from a transmission-line-based lumped model using Schwartz-Christoffel transformation. The simulation results using the proposed model correlate the small-signal test results very well. The proposed higher order impedance model will help to evaluate the high frequency behavior of an ISRM or to minimize the high frequency parasitics. It can also be easily implemented in the design-oriented algorithm to facilitate the design and optimization of an ISRM. More simulation and experimental results will be included in the final paper. The application of this model to more complicated structures as well as some practical design issues will also be discussed.

Schwartz-Christoffel Panel Method

The classical Schwartz-Christoffel transformation can be extended to airfoils by a modern computational approach, which otherwise would be limited to calculating potential flows around simple geometric shapes. A numerical method is proposed to determine the Schwartz-Christoffel transformation from a unit circle to paneled general aviation airfoils. The present calculations agree with the theoretical predictions, and they are also compared to experimental results.

Effect of head field rise time in perpendicular recording

An analytic recording model for the transition parameter and the nonlinear transition shift (NLTS) with the rise-time effect has been developed. In this model, the Schwartz-Christoffel transformation is used to calculate the combined image field from the single-pole writer and the soft underlayer. It is shown that both medium velocity and finite head-field rise-time increase the transition parameter. The NLTS for a dibit pattern increases with linear density, but decreases with medium velocity. A large transition parameter arises because a finite rise time reduces the record field-gradient as the bit time is reduced. For a dibit, the demagnetizing field from the first bit increases the transition interval of the dibit, while the write-time difference of the second bit and the reference bit decreases the interval.

垂直磁気記録における磁化反転幅，ＮＬＴＳの解析モデル

A new analytic perpendicular-recording model for the transition parameter and nonlinear transition shift (NLTS) has been developed. In this model, the Schwartz-Christoffel transformation is used to calculate the demagnetizing field in the medium, combining the effects of the single-pole writer and the soft underlayer. The Westmijze field turned sideways is used to calculate the head field. Both the head field and the demagnetizing field are averaged over the thickness of the recording medium. The Williams-Comstock approach is utilized to determine the transition parameter. Track-edge effects are not included. S/N dependence on the density and the transition width of the isolated pulse (T50) that are calculated from our model agree well with published data. When the write pole image is considered, the calculated NLTS decreases (e. g. from 15% to 10%), but the transition parameter does not change significantly.

GUIDED AND UNGUIDED INTERFACIAL SOLITARY WAVES

We deal with progressing solitary waves at the interface of two superimposed fluids of different densities. In the case of a two-fluid system bounded above and below by rigid walls, we refer to the wave as guided. If the top wall is absent, that is, the top fluid has its free surface exposed to air, the wave is unguided. The problem is formulated by using a generalized Schwartz-Christoffel transformation technique which results in a system of nonlinear integro-differential equations for the interfacial angle θ i , free surface angle θ s and a connection equation for the jump in the potential across the interface. Numerical solutions for the system are presented for a range of Froude numbers showing the effect of density and depth ratios

Similarity of sharp-edged porous media

The dynamic interaction between a rigid porous structure (porosity φ) and its saturating fluid is studied. From the microscopic conservation laws and constitutive relations, macroscopic equations are derived. An averaging technique proposed and discussed by authors like Lévy, Auriault and Burridge and Keller is used. The macroscopic equations are studied in the high-frequency limit. The high-frequency behaviour is characterized by the tortuosity parameter α ∝ and by an effective pore radius Λ, denned previously by Johnson et al. The low frequency behaviour is characterized by the ratio of the steady-state permeability K 0 and the porosity φ. A similarity parameter M = 8α ∝K 0 φΛ 2 (in the two-dimensional case M = 12α ∝K 0 φΛ 2 ) is defined, which is approximately equal to 1 for configurations that have smooth microscopic geometries. For sharp-edged pore geometries, however, M is no longer found equal to one. Numerical computations are performed using a Schwartz-Christoffel transformation.

Approximate predictions of gravity flows over irregular topography for large Froude number

Approximate predictions are reported for two-dimensional, steady, incompressible flows over backward-facing bottom, of small slope, at high Froude number. Schwartz-Christoffel transformation is used to map the region, in the complex potential-plane, onto the upper half-plane. The Hilbert transformation as well as the perturbation technique are used as a basis for the approximate solution of the problem for large Froude number, and small slope of the bottom. General equations, in integral form, for any order of approximation are obtained. Solution up to first-order approximation is discussed and illustrated.

Schwartz-Christoffel transformation approach to the electrostatic field of a charged conducting closed cylinder with some symmetries

Classical solutions to the charged conducting closed cylinder are external transformations of the closed boundary to a circle, or the real axis of another plane. By making use of some symmetries, if any, of the field, we can change this external into an internal problem by the Schwartz-Christoffel transformation, resulting in easier and better solutions.

A general solution for the field of polygonal electromagnet

A general solution for the field of a two-dimensional polygonal electromagnet is presented. The solution is applicable to magnetic recording head models of finite size and arbitrary polygonal configuration. It uses a combination of analytic and numerical techniques. Analytic procedures are based on mixed boundary conditions and a succession of conformal mappings starting with the Schwartz-Christoffel (S-C) transformation. A concept of an idealized winding allows any magnetostatic problem to be reduced to an electrostatic problem of two equipotential shims on the plane of conformal mapping. The result of the analytic solution is a parametric formula for the field vector that can be easily obtained for any model by an inspection of its configuration. This formula contains S-C constants and parameters of unknown values. These difficulties are resolved by numerical procedures.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Novel embedded capacitive sensors for monitoring the cure of polyimide in multichip modules

Efficient and cost-effective manufacturing techniques can be aided by built-in microsensors. In order to monitor the curing of polyimide, an important component of multichip modules two types of embedded capacitive sensors have been developed. One is composed of parallel plates, the other is formed by interdigitated electrodes. The capacitors fabricated on multiple UltradelR 4212 polyimide layers probe the dielectric constant and thickness of the polyimide. Capacitance and loss factor of the sensors are measured after curing the polyinide over the temperature range 100–350 °C, as a function of frequency from 100 kHz to 1 MHz. The interdigitated capacitance decreases during the curing cycle to a stable value indicating that the cure has taken place. Curing time-temperature measurements indicate that a temperature is critical for complete curing. The dielectric constant is calculated from the measured interdigitated capacitance using Schwartz–Christoffel transformation. The film thickness is determined from the parallel plate capacitance. The interdigitated capacitance increases by 4.5% due to humidity, however, when baked at 350 °C the capacitance returns to its initial value. Interdigitated capacitance changes by 10.8% with extra polyimide coating layers.

Magnetic field of an asymmetric ring head with an underlayer

The fringe field of an asymmetric ring head with an underlayer is obtained using conformal mapping techniques. The Schwartz-Christoffel transformation is used to determine the magnetic field distribution around the head poles, and an analytical field equation is obtained. It is found that an asymmetric ring head with an underlayer is highly suitable for perpendicular recording.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Exact field calculations for asymmetric ring heads

An exact solution for the fields of an unconventional ring head with an inclined gap is derived. The Schwartz-Christoffel transformation is used to determine the distribution on the fringe field components in the gap vicinity, and an analytical field equation is obtained. The transcendental equation of the mapping function is solved using Newton's method.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Saturation effects on read/write characteristics of asymmetric ring head

Asymmetric head field profiles, flux line distribution, and the corresponding spectral response functions, etc. have been calculated for a linear core with constant permeability and a nonlinear core with B-H loop based on finite-element analysis. The inclination angles ranging from 0 degrees to 75 degrees have been considered. It is shown that beyond some inclination angles the field intensity based on a Schwartz-Christoffel (S-C) transformation calculation is generally overestimated, the signal level too high by up to one to two orders of magnitude, compared to those of the nonlinear results depending on sharpness of the head considered. Numerically, the error incurred in the head field calculation in the linear core is less than 10% up to 45 degrees inclinations, while the corresponding S-C results are generally >or=45% for the magnetomotive force equal to 0.112 A-T considered. >

Conformal mapping calculation of railgun skin inductance

The common rail arrangement consisting of two long, parallel, rectangular rails is discussed. The authors calculate the inductance per unit length L' in the short flight time limit where the skin depth is much smaller than any rail dimensions, the current is all on the rail surface, and the magnetic field does not penetrate the rails. The solution given is based on the Schwartz-Christoffel transformation that maps the boundaries of the problem into a simpler shape. >

Approximate solution of a flow over a ramp for large Froude number

An approximate method is presented to solve the problem of steady free-surface flow of an ideal fluid over a semi-infinite ramp in the bottom. Schwartz-Christoffel transformation is used to map the region of flow, in the complex potential-plane, onto the upper half-plane. The Hilbert transformation as well as the perturbation technique are used as a basis for the approximate solution of the problem for large Froude number and small inclination angle of the ramp. General equations, in integral form, for any order of approximation are obtained. Solution up to first-order approximation is discussed and illustrated. Elevation of the free-surface for different ramp heights, different inclination angles of the ramp and different Froude numbers are plotted. An approximate formula of maximum elevation of the free-surface in terms of the ramp heights and its inclination angle is found.

An Integrated Experiment in Field Theory

The various means available for the solution of field problems are reviewed and the solutions for a simple problem, that of the fringing field between the pole and armature of an electric machine, obtained using three methods; direct field plotting, Schwartz-Christoffel transformation and finite elements are presented.