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Surface integral equation (SIE) methods are of great interest for the efficient electromagnetic modeling of various devices, from integrated circuits to antenna arrays. Existing acceleration algorithms for SIEs, such as the adaptive integral method (AIM), enable the fast approximation of interactions between well-separated mesh elements. Nearby interactions involve the singularity of the kernel, and must instead be computed accurately with direct integration at each frequency of interest, which can be computationally expensive. We propose a novel algorithm for reducing the cost-per-frequency of near-region computations for both homogeneous and layered background media. In the proposed extended AIM (AIMx), the SIE operators are decomposed into a frequency-independent term containing the singularity of the kernel, and a nonsingular frequency-dependent term. Direct integration is only required for the frequency-independent term, and can be reused at each frequency, leading to significantly faster frequency sweeps. The frequency-dependent term is captured with good accuracy via fast Fourier transform (FFT)-based acceleration even in the near region, as confirmed with an error analysis. The accuracy and efficiency of the proposed method are demonstrated through numerical examples drawn from several applications, and CPU times are significantly reduced by factors ranging from three to 16., 15 pages, 18 figures, accepted for publication in the IEEE Transactions on Antennas and Propagation

The knowledge of the steady-state security region is essential for secure operation of the integrated energy system (IES). The two-time-scale feature of the IES is explored to facilitate construction of its exact steady-state security region. Conditions allowing decomposition of the IES into fast and slow subsystems are established. A complete characterization of the IES steady-state security region via the stable equilibrium manifold of the fast subsystem and the slow subsystem is derived and theoretically proven. This characterization can significantly facilitate construction of the IES steady-state security region (SSR) and is derived under the AC power flow model and gas flow model without any model approximation and simplification. Numerical studies are conducted to validate the derived characterization of the SSR and to illustrate the advantages of the proposed characterization in terms of robustness and reduced computational burden. Numerical studies show that exploring the fast and slow subsystems can lead to fast computation of feasible solutions with a 20-fold speed up in calculation time, as compared with the direct integration of the original IES system. Hence, the proposed two-time-scale method appears to be fast and yet accurate.

This article is primarily concerned with finite-time convergence (FTC) and predefined-time convergence (PTC) design for a class of general zeroing neural network (ZNN) by constructing different activation functions (AFs). Based on the limit comparison test for improper integrals, some useful theoretical criteria are proposed to determine whether a nonlinear-activated ZNN model has FTC, PTC, or not. This novel method can avoid the valuation loss of the zoom method and the unsolvable barrier of the direct integration method that are widely used in the previous ZNN design. According to these convergence criteria, some instructive corollaries are derived to design valuable AFs to make ZNN models with FTC or PTC more easily. By taking a matrix-inversion ZNN model, some commonly used AFs are used to verify the usability of the criteria. In addition, some new AFs are constructed to further design some better ZNN models with superior FTC or PTC. Finally, convergence types of the ZNN model based on different AFs are visualized in numerical experiments.

To characterize the particle mass flow rate of gas–solid two-phase flow, a model of particle mass flow rate based on the comprehensive application of transferred current and induced current signals from an electrostatic sensor is proposed in this article. Compared with the characterization model of the particle mass flow rate using the transferred current signal or induced current signal separately, the proposed model can make full use of the characteristics of the two signals. The induced and transferred current signals are extracted from the electrostatic sensor based on the time-domain waveform characteristic method and the direct integration function method, respectively. According to the sensitivity of the time-domain characteristics of the transferred current signal (the radial motion of particles) and the macroscopic stability of the frequency-domain characteristics of the induced current signal (the axial motion of particles), the new signal characteristic value based on the two signals is used to establish a comprehensive signal characterization model (CSCM) of the particle mass flow rate. The coefficients in the model are fit using the Levenberg–Marquardt (LM) method. The experimental results show that the proposed model has higher accuracy and stability than the separate signal model under the experimental conditions in this article.

In order to predict the performance of designed electric machines, and also based on which the high efficiency of the drive systems can be obtained, the accurate calculation of the iron loss is necessary. Taking modeling accuracy versus modeling complexity into account, this article proposes a general and accurate iron loss calculation method considering harmonics based on the loss surface (LS) hysteresis model and finite-element method (FEM). Compared with the previous triangular wave excitation, this article uses the data under sinusoidal wave excitation to establish the LS model, which is more convenient for engineering application. A specimen is tested for different flux densities with harmonics, the extensive results verify the modeling method. Due to a powerful hysteresis model, the iron loss calculation can be realized by the direct integration of hysteresis loops. A separated iron loss calculation package is developed, and it is combined with the commercial finite-element software to compute the iron loss distribution. The method that combines the LS model and FEM is proved feasible by comparing the calculated iron loss with the measured one for the Epstein frame.

Layer-by-layer manufacturing of composite mechanisms allows fast and cost-effective fabrication of customized robots in millimeter and centimeter scales which is promising for research fields that rely on frequent and numerous physical iterations. Due to the limited number of components that can be directly integrated in composite structures, however, often an assembly step is necessary which diminishes the benefits of this manufacturing method. Inspired by the Japanese craft of cutting (kiri-) paper (-gami), Kirigami, we introduce quasi-2D and highly stretchable functional Kirigami layers for direct integration into the composite mechanisms. Depending on the material and geometrical design; functional Kirigami layers can perform as flat springs, stretchable electronics, sensors or actuators. These components will facilitate the design and manufacturing of composite robots for different applications. To illustrate the effectiveness of these components, we designed and realized a foldable composite inchworm robot with three Kirigami layers serving as actuator, sensor and contact pad with directional friction. We elaborate on the working principle of each layer and report on their combined performance in the robot.

We present our computationally efficient approach to modeling tunable graphene-based active metadevices, where the integral multivariate surface conductivity is reformulated in the time and frequency domains with physically interpretable and fast-to-compute integration-free terms. The derivation is built on an expansion to power series of $ {z}= -\text {exp}(-\mu /k_{B}T)$ that converges very fast for non-zero chemical potential values. The model reveals interesting decomposition of graphene response into a universal constant term plus a damped oscillator (digamma functions in the frequency domain) plus non-oscillating correction terms for near-zero potentials. The number of terms in that series is analyzed theoretically for a given accuracy. In practice, only a few series terms are required, making our approach very efficient for simulation of active metasurfaces compared with direct integration of Kubo’s formulas. A simple performance test comparing run times with our code versus the numerical integration of the original Kubo’s formulas demonstrates a speedup exceeding 103. The proposed models can be critical for the initial ellipsometric characterization of graphene and advanced global optimization of graphene-controlled metadevices.

High-density, point-of-load (PoL) power conversion and power delivery are required to continue scaling electronic systems with increased functionality, more bandwidth, and smaller sizes. To meet the demands of these highly complex and miniaturized electronic systems, new 3-D integration schemes of advanced passive and active components are needed to enable the next-generation power distribution networks (PDNs). In addition, shorter interconnect lengths are required to provide low losses and better transient response in switch-based power conversion systems. Tantalum capacitors have the potential to provide some of the highest volumetric densities of any current capacitor technology, but are generally bulky components with low frequency stability, limiting their use in future 3-D power systems. In this paper, a process for the embedding and integration of ultrathin, high-density tantalum capacitors with improved frequency stability is demonstrated. The 5-V capacitors show a density of $1~\mu \text{F}$ /mm2 at 1 MHz with only a 100- $\mu \text{m}$ thickness, and are shown to be capable of direct integration on silicon for short interconnect length of $ . The electrical performance of the capacitors is tested after integration and shows that they retain high capacitance density and low equivalent series resistance (ESR), while also providing low leakage currents. The combination of low loss, high volumetric density, and 3-D integration capability make the capacitors an ideal candidate for next-generation power modules.

The coupling between hysteresis and linear dynamics remains a challenge for modeling and control of piezo-actuated nanopositioning stages. A decoupling identification approach is proposed in this paper. The system is first modeled by a Bouc–Wen hysteresis model cascading a linear dynamic part. Then, a proposed direct integration method with high computational efficiency is used for the identification of the Bouc–Wen model. With the identified hysteresis parameters, a harmonic linearization method is subsequently used to estimate the actual input to the linear dynamics. Experiment results on a two degree-of-freedom nanopositioning stage demonstrate that both the rate-dependence and amplitude-dependence resulted from the coupling between the hysteresis and linear dynamics can be well predicted by the proposed approach. Compared with existing methods, the proposed approach shows advantages in terms of both model accuracy and identification efficiency.

Inline phase-change switch (IPCS) devices were fabricated on top of an AlN heat spreading layer deposited on a thick SiO2 layer. The thick SiO2 emulates the inter-layer dielectric thermal properties that prevent direct integration of IPCS devices at the end of a back-end-of-line CMOS process. The AlN acts as a heat spreading layer, enabling the proper thermal quench rate despite the low thermal conductivity of the oxide underneath. Multiple AlN thicknesses were examined and shown to properly quench the device. Use of this heat spreading layer experimentally demonstrates a substrate agnostic monolithic integration scheme, where IPCS devices can be integrated at the end of any microfabrication process—regardless of the substrate or materials underneath. This is attractive for many CMOS applications, where wafer real-estate is expensive, or in III–V applications, where the wafer thermal properties are poor and vary among different technologies.

Various numerical tools have been developed to predict the level of aeolian vibrations for a damped span of transmission-line conductors. In part I of this study, nonlinear time history models of two types of transmission-line dampers were developed. In this paper, a model for the complete conductor-damper system is presented. When combined with empirical equations for wind power input and conductor self-damping using the Energy Balance Principle, the direct integration time history model proposed allows the prediction of the vibration amplitudes expected on a damped span. The amplitudes predicted by the model compare well to experimental data sets available in the literature. Since the damper is modelled from its mechanical properties of geometry, mass, stiffness, and damping, the optimization of a conductor-damper system can be done easily with this model, without additional experimental tests. A sensitivity analysis is conducted to demonstrate the capabilities of the model.

We study the effects of random roughness on wave propagation in a parallel plate metallic waveguide with finite conductivity. The rough surface is three dimensional (3D) with roughness heights varying in both horizontal directions. Integral equations are obtained from the extinction theorem formulated with layered medium Green's function. The second order small perturbation method is then applied to solve the integral equations. A closed form expression for the coherent wave is derived, which is expressed in terms of a three-fold Sommerfeld type integral due to the waveguide structure. Approximate methods are applied to calculate the Sommerfeld integral. The results based on the approximations are shown to agree with that of direct integration. The coherent wave enhancement factors of absorption are computed. Results of the present 3D roughness case are compared with the previous results of two dimensional (2D) roughness. The results for waveguides are also compared with the results obtained for a plane wave incident on a metal surface with 3D roughness. Results are illustrated for 3D roughness with a variety of power spectra. It is shown that enhancement factors of 3D roughness are larger than for 2D roughness. We also show that enhancement factors for a waveguide are larger than that of the plane wave case. Comparisons are also made between the theoretical results and measurements.

Sommerfeld-integrals (SIs) are ubiquitous in the analysis of problems involving antennas and scatterers embedded in planar multilayered media. It is well known that the oscillating and slowly decaying nature of their integrands makes the numerical evaluation of the SI real-axis tail segment a very time consuming and computationally expensive task. Therefore, SI tails have to be specially treated. In this paper we compare two recently developed techniques for their efficient numerical evaluation. First, a partition-extrapolation method, in which the integration-then-summation procedure is combined with a new version of the weighted averages (WA) extrapolation technique, is summarized. The previous variants of WA technique are also discussed. Then, a review of double-exponential (DE) quadrature formulas for direct integration of the SI tails is presented. The efficient way of implementing the algorithms, their pros and cons, as well as comparisons of their performance are discussed in detail.

A low-power silicon photonic disk resonator is monolithically and two-dimensionally integrated with complementary metal-oxide-semiconductor (CMOS) electronics. The results show direct integration with established CMOS manufacturing lines without changing either the CMOS or the photonic process stack. Comparable results for each integration method show subpicojoule per bit modulation and the potential for further reductions. The analysis reveals that the process of integration does not limit the performance of photonic or electronic components, and clear steps to use smaller drivers to achieve sub-50 fJ/bit at 10 Gb/s operation are available.

In an elliptical magnetic dot two vortices and a flower state configuration can be accommodated at remanence. We show that a two-fold degeneracy in the position of the flower state can be geometrically introduced and exploited for solid state data storage. Switching between two, energetically-equivalent, magnetic configurations is obtained by injecting a perpendicular electric current. Switching currents of the order of ~105 A/cm2 and high thermal stability are demonstrated by micromagnetic simulations. This system can be used as a bit-cell for magnetic solid state data storage. The low value of the switching current and the high thermal stability allow direct integration with the current complementary metal oxide semiconductor technology.

A direct integration algorithm, based on double exponential-type quadrature rules, is presented for the efficient computation of the Sommerfeld integral tails, arising in the evaluation of multilayered Green's functions. The proposed scheme maintains the error controllable nature of the so-called partition-extrapolation methods, often used to tackle this problem, whereas it requires substantially reduced computational time. Moreover, the proposed method is very easy to implement, since the associated weights and abscissas can be precomputed. The overall behavior of the proposed method both in terms of accuracy and efficiency is demonstrated through a series of representative numerical experiments, where compared with one of the most proven methods available in the literature.

The mode quality of the output of free-electron lasers (FELs) is near the diffraction limit. In this paper, we analyze the optical mode quality in FELs using the M 2 parameter, which is a measure of the size and divergence of the optical beam. We calculate M 2 in two ways: 1) by a direct integration over the transverse mode structure and 2) by allowing the mode to expand beyond the wiggler and analyzing the divergence. A numerical analysis is conducted using the MEDUSA simulation code that shows that M 2, as expected, is near unity at saturation but increases with the higher order mode content if the interaction proceeds past saturation.

Multirate simulation of electric networks exhibiting a wide variety of time constants decreases the simulation runtimes by exploiting the property of circuit latency. The fundamental idea is to use different integration steps for different network subsystems, according to the requirements of accuracy of each subsystem. Programs that exploit circuit latency are usually based on relaxation methods that require iterations among the different subsystems of the original networks. These methods lack the numerical robustness of direct implicit integration methods used in circuit simulators and are not adequate for real-time simulation due to their non-uniform solution times. This paper proposes circuit latency exploitation without iterations making use of the concept of interlinked Multi-Area Thevenin Equivalents (MATE). Results are presented showing the efficiency and accuracy of the method

The Generalized Cobweb Method is presented as a technique to improve the accuracy and speed of calculations of the magnetization of hard magnetic materials that can be successfully described by means of the Preisach model. These calculations are usually carried out by a direct integration of the Preisach probability distribution over a certain area. However, this integration can be replaced by a faster and more accurate procedure based on the definition of a grid on the Preisach plane. The paper gives the steps to create these grids for a general probability distribution, and some details on how to use them to calculate the magnetization of the simulated system.

We describe the packaging and testing of a two-dimensional array parallel-optics module with 36 channels with each channel operating up to 3.3 Gb/s. This represents the first commercial module based on direct integration of vertical-cavity surface-emitting lasers (VCSELs) onto silicon very large scale integration (VLSI) circuits using a hybrid optoelectronic-VLSI technology. The module eliminates wire bonds between the driver/receiver chips and the corresponding VCSELs thereby reducing crosstalk and power dissipation, simplifying packaging and alignment, and simultaneously improving bandwidth and total link jitter performance.

10 pages, 9 figures, 1 table., Domains of existence of the main resonances in directly modulated semiconductor lasers are obtained by application of quasi-conservative theory. The predictions are compared with numerical results coming from a direct integration of the model equations and with experimental observations reported by other groups. In both cases we find a qualitative good agreement. We consider a model that contains explicitly the gain saturation and spontaneous emission terms. We find that the spontaneous emission strongly modifies the qualitative behavior of the instabilities boundaries, while the gain saturation leads to a simple quantitative shift of boundaries. We also find that modulation of the pump or loss produce equivalent results if the respective modulation amplitudes are conveniently rescaled., This work was supported by DGES (Spain) under Projects PB97-0141-C02-01, BMF2000-1108, and TIC99-0645-C55-02. The work of S. I. Turovets was supported by a fellowship from MEC, Spain.

This paper examines the most popular advances in the FDTD algorithm development, as applied to electromagnetic wave propagation in linearly dispersive media. Several methodologies are presented, and comparisons are made in order to demonstrate the strengths and weaknesses of the various approaches. In particular, direct-integration and recursive-convolution schemes, associated with wave propagation in a cold plasma, Debye dielectrics, and Lorentz materials, are considered.

An approach is given to analyze the modal coupling of open-ended waveguides with arbitrary cross sections located in a conducting screen. The presented theory enables the determination of reflection characteristics of a single waveguide, as well as the analysis of mutual coupling between elements in waveguide antenna arrays. The field inside each waveguide is expressed as a sum of the transverse-electric and transverse-magnetic modes and expressions for the mutual admittances of modes excited at the aperture are obtained using a direct integration method. From these expressions, the mode reflection and conversion coefficients are determined. Computed and measured results are presented. Furthermore, this approach has been used to design a new type of horn antenna with high return loss and equal radiation patterns in the two principle planes.

An integrated structure is demonstrated as a refractive index sensor. The structure consists of a liquid-filled elliptical microchannel embedded in silica glass and integrated with waveguides. The microchannel features entry points that are open to the top surface of the device and distinct from the optical input. The structure allows light to couple from a solid-core input waveguide to the liquid-core waveguide formed by the microchannel, and back to a solid-core output waveguide. Bimodal interference allows the structure to be sensitive to the refractive index of the liquid, with a full beat corresponding to a refractive index change of /spl sim/10/sup -4/. The structure allows the direct integration of optical fluids with silica waveguides for sensing and optical processing applications.

The generation and propagation properties of transient leaky Rayleigh waves are characterized by a two-dimensional finite element model. The displacement vector is used as the primary variable for the solid medium and a potential scalar, which is a replacement for the pressure, is taken as the fundamental variable for the fluid medium. The coupled system of finite element equations are solved in the time domain by direct integration through the central difference scheme. Three configurations are considered: the conversion of a Rayleigh surface wave into a leaky Rayleigh wave, a focused beam probing a fluid/solid interface at the Rayleigh angle, and the interaction of a defocused wave with the interface. The wave velocity in the fluid (water) is lower than the Rayleigh wave velocity in the solid (aluminum). The wave propagation profile in each case is predicted by the model. The finite element model proves to be an effective tool for surface acoustic device design and ultrasonic NDE.

The cryogenic performance of a high-speed GaInAs/InP p-i-n photodiode, with graded bandgap layers at the heterostructure interfaces, was investigated for the first time. DC measurements show that the dark current of the diode decreases sharply as the temperature decreases from 300 to 200 K. A factor of 1000 in dark current reduction was found for this photodiode, when it was cooled from room temperature to about 150 K. Similar modulation bandwidths were found for this device for temperatures between 9 and 300 K, with a bandwidth greater than 20 GHz. No degradation was found in performance at cryogenic temperature compared to room temperature. This enables direct integration of high-speed photodiodes with superconductive and other cryogenic electronics.

This paper employed the spectral domain method of moment for the calculation of the current distribution and inductance of the spiral inductor on silicon multilayer structures. We used FFT (fast Fourier transform) in the calculation of the infinite 2-dimensional integrals and "divided FFT" in the case of the very small site analysis model. Using this algorithm, the computation time can be reduced considerably compared with the direct integration. The computed inductance was compared with that measured to show the validation of the formulation.

A direct integration procedure for far-zone vector potentials of thin circular loop antennas has been known for many years. This method is general in the sense that it leads to simple integrals which have closed form solutions for most commonly assumed loop current distributions. However, a comparable integration technique has not been available for evaluating the more complicated near-zone vector potentials. This paper introduces a systematic approach for the exact integration of general near-zone vector potentials associated with current-carrying circular loop antennas. A particular example is considered where this new integration technique is used to find exact solutions to the vector potential and electromagnetic field integrals for loops with a Fourier cosine-series expansion of the current. The observation is made that degenerate forms of these exact representations lead to simplified expressions for the important special cases of a uniform and cosinusoidal current loop. Two equivalent forms of exact series expansions are derived for the uniform current vector potential and field integrals. It is shown that the familiar small-loop approximations, as well as the classical far-field expressions, may be obtained as limiting cases of the more general exact series representations for the uniform current loop obtained in this paper. Convenient asymptotic far-field expansions are derived for the loop with a cosinusoidal current distribution. Finally, the far-field analysis for the cosinusoidal loop is generalized to loops having an arbitrary current represented by a Fourier cosine series.

The performance of randomized and conventional waveform for airborne radars in cluttered environments is compared. We compute the clutter effects on randomized waveform by direct integration of the cross-ambiguity function with the unfolded clutter density function. It is concluded that, when available system resources are equalized, randomized waveform cannot compete with conventional waveforms in high clutter situations, though their performance is superior in low or zero clutter situations. The properties of specific spread spectrum codes that would yield superior performance are investigated as are other considerations with differential impact on conventional and randomized waveforms. >

The dispersion errors associated with various frequency-dependent FDTD methods are considered herein. Particularly, we provide a rigorous error analysis of both direct integration and recursive type schemes for two media models: the one-pole Debye and the two-pole Lorentz. The error equations are cast in terms of a dispersion relation that shows explicitly the errors associated with numerically induced dispersion and dissipation. From the dispersion relation, plots are provided that typify the errors of each method. In general, all methods have about the same propagation characteristics; the differences, however, are seen in the attenuation plots. To validate the claims herein, data obtained from FDTD scattering simulations (both 1-D and 3-D geometries) are also given. >

A novel numerical method for the standard propagation equations of Raman amplifiers with multiple pumps is presented and derived based on the four-step Adams-Bashforth method. Our proposed multistep method can increase the accuracy and stability in designing Raman amplifiers compared with the one-step method. Output signal, pump, and backscattering pump powers have been numerically simulated and compared by utilizing our multistep method and one-step method. The results show that our method can reduce the computing time over three orders of magnitude with excellent accuracy by comparison with the direct integration method for the Raman amplifier propagation equations.

In the quasi-TEM limit a circuit model is developed to predict the voltages and currents induced on a multilayered microstrip structure illuminated by a low-frequency uniform plane wave. The analysis is based on a low-frequency series expansion rather than on the direct integration of Maxwell's equations. >

Circularly short bent, graded-index dielectric slab waveguides are mathematically modeled using both a direct integration method and a source-type integral equation method. The former method is easy to implement, requiring only small amounts of computer time. The latter method has the potential to be extended to a rigorous model for bent channel waveguides. Both homogeneous and inhomogeneous slab waveguide layers are considered. The mathematical models are validated through comparison of the numerical results with results obtained by the analytical resonance conditions. Numerous numerical examples are given. >

A numerical algorithm simulating the electrodynamics of many-particle systems is presented. This algorithm requires solution of the 3-D Poission equation for charge conservation using the boundary integral equation method, and integration of the dynamical equations of motion for discrete interacting particles. Motion of the particles is due to both Coulomb forces and Stokes's drag. Particles suffer elastic collisions with walls and each other. A scatter-gather method together with the use of influence coefficient matrices allowed two-order-of-magnitude speedups in computation compared to direct integration methods. The simulation is implemented within a user interface on a minisupercomputer to enable interactive steering of solution parameters and dynamic visualization of results. Sample solutions for xerographic development of image pixels are discussed. >

A photonic switching device with very large gain and high sensitivity has been developed by the vertical and direct integration of a heterojunction phototransistor and a laser diode. The device switches on with very low input power of approximately 10 nW and emits output power of approximately 4 mW under continuous-wave conditions at room temperature. The minimum energy for switching on is estimated to be as low as 80 fJ. The internal optical feedback of the device is quantitatively discussed to interpret the low-power operation for the switch-on. >

For the first time, we derive efficient modeling equations for the average power analysis of Raman amplifiers (RAs) from the standard propagation equations. Applications of these equations to the numerical analysis of practical RA-based systems show a reduction in computation time of over two orders of magnitude compared with the direct integration approach based on ordinary coupled differential equations, while reproducing all the essential system performances precisely. In addition to enhanced computational efficiency, the derived equations also give deeper insights into the detailed dynamics of RAs.

Expressions are presented for estimating the finite-range gain of pyramidal horns with improved accuracy. These are obtained by avoiding the conventional path length approximation in estimating the gain reduction factors, by a direct integration of the aperture field in the frequency range from 18 to 26.5 GHz, these expressions compare with available measured data on a pyramidal horn with an accuracy of better than /spl plusmn/0.15 dB, which is an improvement of about the same order over the earlier formulas.

An approach to the finite element method applied to the solution of electromagnetic fields problems is presented. This methodology is suitable for teaching electrical engineering students at undergraduate level, because the problem formulation is based solely on the direct integration of Maxwell's equations and the approach is only valid for first-order elements, thereby avoiding the use of an excessively complex mathematical treatment.

A crucial part of so-called ?strapdown? techniques for inertial guidance systems is the generation of the matrix of direction cosines relating the body axes to the reference axes. This is generally done by direct integration of a set of differential equations having the body angular velocities as inputs. This paper is devoted to an error analysis of two commonly used integration schemes, namely, direction cosines and quaternions. Scale, skew, and drift errors are defined, and the susceptibility of the integration schemes to these types of error is examined. It is concluded that the quaternion scheme offers an advantage because it intrinsically yields zero skew error. The paper is presented in two parts, for convenience.

Two computer programs have been developed for the rigorous performance analysis of circular waveguide tapers. They are based on the direct integration of the coupled wave equations and on a mode-matching procedure applied to a step-ladder equivalent of the taper, respectively. Close agreement has been obtained between the predictions of these two approaches. The computer programs are utilized to examine the usefulness of existing taper design procedures. It has been found that a Dolph-Chebychev design is actually capable of predicting the level of the first unwanted mode, provided that the operational frequency is far above its cutoff. The spurious mode excitation can still be improved by taking into account the reconversion of power from the unwanted into the wanted modes. It has been verified by the computer analysis that the above design procedures are applicable even for frequencies near cutoff of the dominant spurious mode; however, that is at the expense of a certain degradation of the input reflection of the incident mode. A further improvement of the performance is possible only by direct computer optimization of the taper. >

An approach is developed for obtaining closed-form expressions for variance estimators of frequency stability as functions of several parameters. It is based on a direct integration method and therefore expressions for autocorrelation functions of phase and frequency instabilities are not needed. The method is particularly useful when the transfer function approach of J. Rutman (1974) and E. Boileau (1978) is used, according to which the integral expressions of the variance estimate are chosen a priori. Using this method, the integrals can be evaluated without the need of calculating the corresponding sample of the variance. The relationship between this approach and the structure function characterization of frequency stability developed by W.C. Lindsey and C.M. Chie (1976) is also presented. >

As a consequence of the spatial phase distribution in the resonators of cesium-beam time and frequency standards, the generated frequency depends on the specific path of the atomic beam. A change of the position of the atomic beam source may result in a beam displacement normal to the beam direction. For a deflection system consisting of a combination of quadrupole and hexapole magnets for two-dimensional beam deflection, the displacement of the center of mass of the beam resulting from a misalignment of the beam source is computed. To this end, the distribution of the beam intensity on the collector is first determined. It is shown that for the cesium-beam time and frequency standard CSI of the Physikalisch-Technische Bundesanstalt (PIB), the uncertainty of the position of the center of mass of the beam entails a contribution to the uncertainty of the standard of less than 1 X 10-15. The amount of the displacement of the center of mass of the beam can be determined from the decrease of the beam flux on the collector caused by an adjustment of the beam source.

The transient response of a 0.4 by 0.6 cm rectangular bore rail accelerator was analyzed using a three-dimensional finite-element code. Results are presented for the case of a 210-kA current input and 5-cm arc length. Typically, the copper rail deflected to a peak value of 0.08 mm in compression and then oscillated at an amplitude of 0.02 mm. Simultaneously the insulating side wall of glass fabric base, epoxy resin laminate (G-10) was compressed to a peak value of 0.13 mm and rebounded to a steady state in extension. Projectile pinch or blow-by due to the rail (or side-wall) extension or compression, respectively, can be identified by examining the time history of the rail (or side-wall) displacement. For the case presented, the effect of blow-by was most significant at the side wall characterized by mm-size displacement in compression. Dynamic stress calculations indicate that the G-10 supporting material behind the rail is subjected to over 21 MPa - at which the G-10 could fail if the laminate was not carefully oriented. Results for a polycarbonate resin (Lexan) side wall show much larger displacements and stresses than for G-10. Therefore the tradeoff between the transparency of Lexan and the mechanical strength of G-10 for side-wall material is obvious. Displacement calculations from the modal method are smaller than the results from the direct integration method by almost an order of nagnitude, because the high-frequency effect is neglected. However, this effect is significant and dominating for a highly-impulsive, wave-propagation problem such as the rail accelerator structure.

A method is presented for the computation of the far field radiation patterns of paraboloid reflector antennas by using a modified stationary phase approximation to eliminate one integration. This method is applicable to open cassegrain, offset paraboloid and horn reflector antennas. For symmetrical paraboloid antennas the modified approximation reduces to the exact expression obtained by direct integration. The errors introduced by the stationary phase and modified stationary phase approximations are investigated. Specifically the far field of an open cassegrain with a 128 wavelength aperture diameter is computed by the approximate method up to 20 degrees off-axis. The difference between these radiation patterns and those computed by double integration, is less than a few tenths of a dB up to 1.0 degree, and less than a few hundredths of a dB at larger angles off-axis. In order to estimate the computational advantage of this approximation, the number of points required for integration of an oscillatory function by Simpson's rule is also examined and it is determined that at least 6 points per cycle are necessary to obtain 4 decimal accuracy. For fewer points the error is appreciable.

A method of focusing a long hollow electron beam is described in which beam space-charge fields are balanced against uniform radial electric fields and periodic radial and longitudinal electric fields. There is no magnetic field associated with this focusing method. Elimination of the usual large axial magnetic-field requirement of "confined flow" focusing allows the tube system employing the electrostatically focused hollow beam to be lighter in weight, more compact, and to enjoy an enhancement of over-all efficiency. A physical description of the focusing mechanism is given, followed by a mathematical treatment which yields conditions for stable beam flow. Expressions are derived for electrode voltages required to focus a given hollow beam, as well as for velocity variation and beam deflection. A comparison of beam "stiffness" is made with a conventional "confined flow" beam. Beam stability is investigated under various "overfocusing" conditions and imperfect entrance conditions. Finally, this focusing method is compared with other purely electrostatic methods. An experimental program was carried out which demonstrated the usefulness of this type of focusing. Results indicate that good beam definition and transmission coefficients can be obtained with some insensitivity to entrance conditions.

The Gilbert equation is solved numerically to obtain detailed structures of two-dimensional vertical Bloch lines. To avoid situations in which magnetization aligns parallel to the polar axis, the computing region is divided into three subregions, and two kinds of coordinates systems are used depending on the orientation of the magnetization in each subregion. The instability associated to the equation is suppressed completely by discretizing the equation according to the backward Euler method. The paper formulates the method of calculation, describes the derived structure of a plane-like wall with periodically-spaced VBL's, and makes comparison with the results of simulations based on various modelings developed so far.

A novel technique for direct integration of PbS-Si heterojunction IR detectors and read-out circuitry using planar Si technology has been developed and implemented. X-Y address read-out mode of a detector array has been demonstrated. Injection efficiency of close to unity has been obtained under optimum bias condition at room temperature.

A stream function or vector potential for that density vector is used to develop a finite element code for calculating induced currents in thin conducting plates. While two-dimensional, the code includes self field effects for harmonic fields for skin depths on the order of the plate thickness and larger. The solution of the resulting integro-differential equation, using a Galerkin method, leads to a complex, nonsymmetric, fully populated global matrix. In addition to the current density, the code also calculates induced temperatures due to Joule heating and the magnetic forces on the plate. The results are compared to infrared measurements of induced currents in rectangular plates. Extension of the code to transient problems using both fast Fourier transform and direct integration methods is in progress.

An exact expression for the characteristic impedance of two infinite cones of elliptic cross section is found through direct integration of the transverse electromagnetic (TEM) electric and magnetic fields in conical coordinates. Two special cases, the infinite circular biconical antenna and the infinite "bowtie," agree with previously known results.

Starting from the radiation characteristics of a differential current element, the vector potential produced by a single-turn circular loop antenna is formulated in terms of a general loop current by a direct integration process. Evaluation of the resulting integrals for a limited number of specific loop current distributions leads to generalized expressions for both the standing-wave and traveling-wave cases. Radiation fields can be found from the integrated vector potential expressions. This technique represents only a slight modification of the one used by Sherman, but it extends his work in such a way that the results are equivalent to those obtained by Knudsen in a somewhat different manner.