Schwarz-Christoffel Research Articles

Analytical design and experiment of radial inductive displacement sensor with full-bridge structure for magnetic bearings

In magnetic bearing systems, the air gap between the stator and the rotor is generally measured with an eddy current displacement sensor. However, the inductive displacement sensor (IDS) is a preferred choice due to its structural consistency with magnetic bearings, which contributes to a compact system design. While most of the reported IDS research studies are based on three-dimensional finite element method (3D FEM) simulation and the IDS of such studies is designed with a half-bridge structure, in this paper, a higher sensitivity IDS with a full-bridge structure is proposed. To optimize the response of the sensor, an accurate theoretical analysis method is presented for the sensor based on the Schwarz-Christoffel transformation, which is used for compensating fringe effects of sensor inductance. Compared to traditional 3D FEM simulation, fast sensor design and optimization can be achieved with the proposed method. Experimental results agree well with theoretical predictions. The sensitivity of the proposed sensor is nearly 15.5 mV/μm, and the displacement resolution is better than 1 µm.

Stress intensity factor analysis of kinked and hole crack in an infinite plate using numerical conformal mapping

The stress intensity factors (SIFs) of the kinked crack emanating from a hole in an infinite plate are calculated by Muskhelishvili’s formalism. A numerical Schwarz–Christoffel (S-C) conformal mapping method is proposed to calculate the mapping function which transforms the relatively complex configuration of the crack onto a unit circle. The application of this method to the cases of kinked crack and hole crack problems shows good accuracy of SIFs compared to the results of other researches. The performance of the application to the hole crack with multiple kinks shows the potential of this method in the analysis of crack with complex configuration.

Three-Dimensional Channel Potential Model of a Triple Gate MOSFET based on Conformal Mapping Technique

In this paper, an analytical channel potential model of an experimental nano-scaled triple gate MOSFET has been developed that is valid in the subthreshold regime. The potential model is derived by applying the Schwarz–Christoffel transformation method and conformal mapping techniques. The model is derived by superposing the influence of top gate on the potential barrier with the 2D potential solution, which is obtained by considering the presence of only two side gates. From that, the threshold voltage model of the device is also evaluated. To validate our model, the result has been compared with numerical simulation data from 3D TCAD Sentaurus that shows good accuracy for channel length (as short as 30 nm) considered for our device.

An Affine Model of a Riemann Surface Associated to a Schwarz–Christoffel Mapping

In this paper, we construct an affine model of a Riemann surface with a flat Riemannian metric associated to a Schwarz–Christoffel mapping of the upper half plane onto a rational triangle. We explain the relation between the geodesics on this Riemann surface and billiard motions in a regular stellated n-gon in the complex plane.

Hybrid Analytical Modeling of Air-Gap Magnetic Field in Asymmetric-Stator-Pole Flux Reversal Permanent Magnet Machine Considering Slotting Effect

This article proposes an accurate and computationally efficient hybrid analytical (HA) model for the air-gap magnetic field computation of asymmetric-stator-pole flux reversal permanent magnet machine, which can provide a thorough insight into the working mechanism by taking the slotting effects into account. The proposed HA model combines the subdomain method and Schwarz–Christoffel (SC) transformation, which can account for the stator and rotor slotting effects, respectively. The subdomain method is first employed to predict the air-gap magnetic field considering the rotor slotting effect by converting the machine geometry to a slotless stator model. Afterward, a complex relative permeance function is obtained by the SC transformation, in which the stator slotting effect is taken into account. Finally, the complete air-gap magnetic field distribution of the machine can be efficiently predicted by the proposed HA analytical model. The effectiveness of the proposed HA model is verified by both finite-element analyses and experimental results.

About the robustness of Schottky conjecture when quasi-one-dimensional stages are present

The Schwarz–Christoffel transformation is used to analytically evaluate the field enhancement factor close to the apex of two-stage conducting structures consisting of a vertical line centered on the top of an isosceles trapezoidal protrusion on an infinite line. This way the validity of the Schottky conjecture (SC) is assessed for different ratios involving the dimensions of the system, which is expected to model quasi-one-dimensional structures, such as nanotubes, on the top of multistage structures used in different scientific and technological applications. The results obtained in this work suggest that the SC remains a good approximation beyond the region in which it is usually expected to be valid.

An alternative method for field analysis in inhomogeneous domains

PurposeThis paper aims to solve inhomogeneous dielectric problems by matching boundary conditions at the interfaces among homogeneous subdomains. The capabilities of Hilbert transform computations are deeply investigated in the case of limited numbers of samples, and a refined model is presented by means of investigating accuracies in a case study with three subdomains.Design/methodology/approachThe accuracies, refined by Richardson extrapolation to zero error, are compared to finite element (FEM) and finite difference methods. The boundary matching procedures can be easily applied to the results of a previous Schwarz–Christoffel (SC) conformal mapping stage in SC + BC procedures, to cope with field singularities or with open boundary problems.FindingsThe proposed field computations are of general interest both for electrostatic and magnetostatic field analysis and optimization. They can be useful as comparison tools for FEM results or when severe field singularities can impair the accuracies of other methods.Research limitations/implicationsThis static field methodology, of course, can be used to analyse transverse electro magnetic (TEM) or quasi-TEM propagation modes. It is possible that, in some case, these may make a contribution to the analysis of axis symmetrical problems.Originality/valueThe most relevant result is the possible introduction of SC + BC computations as a standard tool for solving inhomogeneous dielectric field problems.

Generalized Design Technique of Ultra-Wideband Transitions for Quasi-TEM Planar Transmission Lines Based on Analytical Models

A generalized design technique of ultra-wideband planar transitions using EM-based analytical models is presented in this paper. Among various planar transmission lines, each transmission line has unique advantages over other line types, and a microwave component in a system can be designed to perform much better if it is implemented on a certain type of transmission line. Therefore, low-loss and high-performance transitions between transmission lines, with easy integrability with the main circuit board, are needed. The proposed transition design technique uses the Schwarz-Christoffel transformation, which is one of conformal mapping methods, and can be applied to design any planar transition between a pair combination of planar transmission lines based on TEM or quasi-TEM waves. To optimally match the characteristic line impedance and smoothly transform the electromagnetic field distribution between the planar transmission lines, entire cross-sections through the planar transition should be analyzed with proper models. In this paper, the cross-sections of planar transitions are categorized into 4 cross-sectional models, where each cross-section is divided into multiple regions for the analysis to obtain line capacitance. Therefore, for the 4 cross-sectional models, 7 types of basis structures are identified to obtain their capacitances by applying conformal mapping. By adding capacitances of a combination of the 7 analysis types, the total line capacitance, thus the characteristic line impedance, of any cross-sectional model of the planar transition can be obtained. The characteristic line impedance of each cross-sectional model calculated with the proposed analytical formulas is compared with the 3D EM calculations, and it is found that the deviated values are mostly well under 6%. This proposed technique enables to design various planar transitions efficiently and quickly for the maximum performance without parameter tuning trials, by providing optimal impedance matching and smooth field transformation up to mm-wave frequencies.

Parasitic Capacitance Analysis of Three-Independent-Gate Field-Effect Transistors

Three-Independent-Gate Field-Effect Transistors (TIGFETs) are a promising alternative technology that aims to replace or complement CMOS at advanced technology nodes. In this paper, we extracted the parasitic and intrinsic capacitances of a silicon-nanowire TIGFET device using three-dimensional numerical simulations in an attempt to accurately compare its capacitances and, consequently, circuit-level performances to CMOS at comparable nodes. Analytical models of the parasitic capacitances of a TIGFET transistor were derived using techniques such as the equivalent Schwarz-Christoffel transformation and standard cylindrical capacitors and show close agreement with numerical simulations. The maximum capacitance of a TIGFET transistor is $2\times $ larger than for a 15 nm CMOS High Performance (HP) device due to the TIGFET’s two additional gated contacts, but this is countered by its ability for multiple modes of operation which reduces the effective switching capacitance per device. A TIGFET transistor sees, on average, only a 30% increase in total capacitance compared to a CMOS HP device. Additionally, the TIGFET’s increased device functionality can be used to modify the circuit-level architecture of a TIGFET-based design to mitigate the performance impact of its larger device-level capacitance. This combination of a TIGFET’s (1) multiple modes of operation, and (2) circuit-level architecture lead to enhanced system performance. In particular, we show that at the 15 nm technology node TIGFET technology has 18% lower energy-delay product for a fan-out of 4 and higher when using 1-bit full-adder logic circuit than for the equivalent node CMOS HP.

Research on a Novel Transverse Flux Permanent Magnet Motor With Hybrid Stator Core and Disk-Type Rotor for Industrial Robot Applications

Traditional surface permanent magnet synchronous motor (SPMSM) used as a servo motor limits the miniaturization and performance improvement of industrial robot, due to its large overall volume and heavy weight of the rotor. In this article, a novel transverse flux permanent magnet motor with a hybrid stator core and disk-type rotor (HSDR-TFPM) is proposed to adapt to the industrial robot well. Moreover, a better space design by which the usual components used in industrial robot is considered. Considering the computing time and accuracy, an analysis method based on Schwarz-Christoffel mapping is employed to analyze the magnetic field of HSDR-TFPM and conduct research on motor parameters. In order to prevent the axial movement of rotor, the detailed mechanical design is introduced and the strength of motor is analyzed further. In the end, the analysis results and performance of the proposed HSDR-TFPM are validated and compared with a conventional servo SPMSM.

Compact Modeling of Short-Channel Effects in Staggered Organic Thin-Film Transistors

This article introduces analytical compact models of short-channel effects in staggered organic thin-film transistors (TFTs). The effects of subthreshold-swing degradation, threshold-voltage roll-off, and drain-induced barrier lowering (DIBL) on the static current-voltage characteristics of staggered TFTs are extracted from an analytical potential solution of the 2-D problem of the staggered geometry. This solution is derived by using the Schwarz-Christoffel transformation that leads to a complex mapping function linking the staggered geometry to an equivalent in another coordinate system for which an analytical potential solution exists. The technology CAD (TCAD) Software Sentaurus is used to verify the compact models. Finally, the closed-form and physics-based equations are incorporated into an existing compact current model and verified by measurements on staggered organic TFTs with channel lengths as small as 0.4 μm fabricated on flexible plastic substrates by stencil lithography.

Analysis of End-Effect Force and Weakening Method for Permanent Magnet Toroidal Motor

In the research of torque ripple of permanent magnet toroidal motor (PMTM), the planet gears are affected by the end-effect force during its rotation due to the special structure of the central worm stator. The existence of the end-effect force will cause vibration and noise, which will lead to poor operation accuracy of the planet gears. In view of the above problems, the formation mechanism of the torque ripple is analyzed. We point out that the sudden change in the magnetic induction intensity at the end of the central worm stator is the main source of the end-effect force. The Schwarz-Christoffel transformation method is used to analyze the magnetic field distribution at the end of the stator. The mathematical model of the end-effect force of PMTM is derived by applying Maxwell stress tensor method. The influence of the structural parameters on the end-effect force of PMTM is analyzed by finite-element simulation. The results show that the output fluctuation of PMTM can be effectively reduced by choosing the optimal wrap angle of the central worm stator by calculating. In addition, choosing the appropriate polar arc coefficient of the planet gears can further reduce the influence of the end-effect force. These research results can provide important reference for the structural optimization of PMTM.

Mitigation of cogging torque in transverse‐flux permanent‐magnet machines with flux concentrators by step skewing of stator pole

The objective of this article is to study how the stator step skewing method can lead to the reduction of the cogging torque of a transverse-flux permanent-magnet (TFPM) machine with flux concentrators. Three structures are analysed and compared in terms of their influence of the skewed displacement on the cogging torque using the three-dimensional finite element method (3D-FEM). Thereafter, to validate the FEM results, cogging torque is calculated by applying a Schwarz–Christoffel (SC) conformal mapping. To apply this transformation, the 3D TFPM generator structure with axially magnetised permanent magnets (PMs) is converted into a 2D structure with radially magnetised PMs and the cogging torque of the machine is predicted using both the analytical method and 3D-FEM. The accuracy of the approach is demonstrated by the adequate agreement between the results obtained through this SC mapping and those of the 3D-FEM. In addition, two 100 W prototyped TFPM machines are designed, simulated, manufactured and tested to validate the effects of the step-skewed stator yoke on the predicted cogging torque and back-EMF.

Cogging Torque Computation and Optimization in Dual‐Stator Axial Flux Permanent Magnet Machines

Combining Schwarz–Christoffel transformation, magnetic field reconstruction method, and Maxwell stress tensor method, the calculation model of cogging torque of a dual‐stator axial flux permanent magnet machine is established. The end effect and slotting effect are both been taken into account. The results are verified by finite element method. And with the proposed method, calculation time can be significantly reduced compared with finite element method. Three measures to reduce the cogging torque of dual‐stator axial flux permanent magnet machines are discussed based on which the cogging torque is optimized by genetic algorithm. © 2020 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC.

Schwarz–Christoffel accessory parameter for quadrilaterals via isomonodromy

We develop the recent proposal by the authors to exploit the isomonodromic tau function defined by Jimbo, Miwa and Ueno (JMU) to solve the accessory parameter problem in conformal mapping theory. We focus here on mappings of Schwarz–Christoffel type: in particular, the mapping from the upper half plane to a four-sided polygon where the sides are all straight lines. We show that one can obtain the relevant accessory parameters—the pre-image of the polygonal vertices—via a special ‘zero curvature limit’ in which the radius of curvature of some of the edges tends to zero. We apply the procedure to rectangular domains where the JMU tau function is given by a ratio of Riemann theta functions, known as the Picard solution, and take the zero curvature limit to recover the accessory parameter obtained by Nehari using quite different methods. We then turn to trapezoids, deriving new asymptotic formulas for the accessory parameters in the limit of large and small aspect ratios. Our work lends a new geometrical perspective to problems of isomonodromy that we believe provides theoretical insight, while also showing how classical problems in conformal mapping can benefit from new ideas emerging from isomonodromic deformation theory.

Application of the Schwarz-Christoffel Transformation to the Solution of Harmonic Dirichlet Problems in Electrostatics

In this paper, a purely conformal mapping method for efficiently solving harmonic Dirichlet problems of electrostatic in domains free of charge and with charge whose boundaries have inconvenient geometries consisting of straight-line segments is presented. The method which uses the inverse of an appropriately determined Schwarz-Christoffel transformation as the mapping function, was applied to harmonic Dirichlet problems in an infinite strip and infinite sector and the solution or electrostatic potential for the problem obtained for each case. Furthermore, the equipotential lines of the electric field were also obtained in order to show the features of the solution and the field analysed accordingly. The electric field intensity was also analysed to show its variation in the field. This method could therefore be a suitable alternative method for solving Laplace's equation in two dimensional electrostatic problems.

MOSFETs’ Electrical Performance in the 160-nm BCD Technology Process With the Diamond Layout Shape

This article introduces an innovative approach that describes the drain–source current improvements of MOS transistors. It is based on the geometrical modification of MOSFET’s channel from a rectangular layout shape (RLS) into a diamond layout shape (DLS). In this way, the drain–source current enhancement is increased up to 11% for the DLS MOS transistors with an effective aspect ratio ( ${W}/\textit {L)}_{\mathrm {eff}}$ equal to 2.0 and an angle $\alpha $ set to 80°. Moreover, we present the comparison of 3-D TCAD simulations data, analytical model data based on Schwarz–Christoffel transformation (SCT), and measurement data given by measurement of the MOS transistors fabricated in the Bipolar-CMOS-DMOS (BCD) 160-nm technology process. For this purpose, there have been fabricated 1124 samples, which were proportionally divided into RLS MOSFETs and DLS MOSFETs with the angles $\alpha $ equal to 120°, 100°, and 80°. For all studied aspect ratios, the presented model has an excellent analytic description in comparison with the 3-D TCAD simulation results with an error lower than 3%. So, it proves the quality of the analytical model based on the SCT approach and it is the recommended approach to use also for modeling other MOSFET gate layout shapes.

Electromagnetic Force Analysis of Eccentric Axial Flux Permanent Magnet Machines

Based on Schwarz–Christoffel mapping, this paper presents a fast analytical method to analyze the electromagnetic force of eccentric axial flux permanent magnet machines, considering static, dynamic, and mixed eccentricities. A quasi-3D model of an axial flux permanent magnet machine is established, and the magnetic field is obtained by Schwarz–Christoffel mapping. The electromagnetic force density is obtained by the Maxwell stress tensor method, and the electromagnetic force density is used to characterize the variation of electromagnetic force. The distribution law of electromagnetic force is investigated. The calculated results are verified by the finite element method, which has shown that the method in this paper can be widely used in the analysis of axial flux permanent magnet machines.

Large-size expansion for triangular Wilson loops in confining gauge theories

The asymptotic behavior of Wilson loops in the large-size limit (L → ∞) in confining gauge theories with area law is controlled by effective string theory (EST). The L−2 term of the large-size expansion for the logarithm of Wilson loop appears within EST as a 2-loop correction. Ultraviolet divergences of this 2-loop correction for polygonal contours can be renormalized using an analytical regularization constructed in terms of Schwarz-Christoffel mapping. In the case of triangular Wilson loops this method leads to a simple final expression for the L−2 term.

Analysis of a Tubular Linear Permanent Magnet Oscillator With Auxiliary Teeth Configuration for Energy Conversion System

The tubular linear permanent magnet oscillator (TLPMO) has been widely used in the industrial field as an energy conversion machine. Detent force is an obvious drawback of TLPMO, and one of the efficient ways to minimize this phenomenon is by adding auxiliary teeth. In this article, a magnetic field reconstruction (MFRC) method is proposed to analyze the TLPMO comprehensively, in which the auxiliary-teeth, slot, and end effects are included. First, the subdomain conversion model is used to consider the end effects of the slotless magnetic field under no-load condition. Thereafter, the two-step Schwarz–Christoffel transformation method is launched to account for auxiliary-teeth and slot effects. Subsequently, the MFRC technology is performed to integrate the above-mentioned results, and the forces are obtained. Finally, the accuracy of the presented method is validated by comparing the results from the FEA model and experiment of the TLPMO prototype. The main contribution of this article is presenting a hybrid method that has a good tradeoff between the computation time and accuracy. The analytical results using the MFRC method closely agree with that of the FEA model, and it has less than 7% deviation compared with the experimental measurement. Hence, it can be regarded as an efficient tool in the initial design procedures of TLPMO.