Calculation and analysis of thermal impedance of microelectronic structures from analytical models

This article deals with thermal impedances of microelectronic components that are useful in Simulation Programs with Integrated Circuit Emphasis (SPICE). In devices like heterojunction bipolar transistors, the active regions thicknesses are often much smaller than the substrates thicknesses. The devices can then be thermally assimilated to heat densities located on top of solid media. In addition to that, when the other dimensions of the heat sources are also much smaller than the substrates dimensions, it is reasonable to consider that the substrate is semi-infinite. First, the expression of the thermal impedance Z of a circular shape heat source centered on top of a half space is presented. For this purpose, the integral transform technique has been used to solve the tri-dimensional heat conduction equation in the frequency domain. The original expression is explicit, exact and allows obtaining results very quickly. After that, the case of a circular heat source on top of a cylinder is treated. A complete analysis of the substrate dimensions influence on the thermal impedance is done. It is based on the impedance decomposition into the one-dimensional impedance and the spreading impedance. By comparing these impedances with that obtained for the heat source on top of the semi-infinite medium, the threshold pulsation at which the thermal impedance of the finite medium differs from the thermal impedance of the half space is extracted. Moreover the geometrical criteria resulting in an error of less than 2% between the spreading impedance of the finite medium and the semi-infinite one are extracted. When these criteria are observed the impedance can be calculated using two perfectly known impedances: the spreading impedance of the semi-infinite medium and the one-dimensional impedance. The results are plotted on the Nyquist diagram, providing a compact representation. Finally the assumption of a circular shape heat source to approximate the thermal impedance of a square shape heat source is validated by evaluating the associated error. The calculation times have been compared to confirm the interest of using this hypothesis.     

Analysis of the microstrip disk antenna element

The circular microstrip antenna element is formed by a radiating disk closely spaced above a ground plane. It is modeled as a cylindrical cavity with magnetic walls which can be resonant in the transverse magnetic (TM) modes. The far fields and the radiation conductances for different mode structures have been calculated assuming a magnetic line current flowing along the perimeter of the disk. The directivity of a disk antenna excited in the dominant mode is between 4.8 dB and 9.9 dB, depending on the size. Losses, due to imperfect supporting dielectrics and to the finite conductivity of the conductors, have been derived by means of a perturbation technique. Graphs are given for design purposes showing the input impedance, theQfactor, and the radiation efficiency at resonance for different modes and thicknesses. The air-filled microstrip antenna has the highest efficiency and the broadest bandwidth at a given resonant frequency.     

Radiation on Losses Due to Variations of Radius Dielectric or Optical Fibers

The total loss of the HE/sub 11/ mode to the radiation field of a finite dielectric rod with small amplitude surface irregularities is considered, and a simple approximate analytic expression for radiation due to sinusoidal roughness is presented. It is shown that radiation occurs only when the frequency of surface roughness /spl Omega/ is in the range /spl beta/-k/sub 2/相似文献   

Scattering Coefficients for Wall Impedance Changes in Waveguides

The Wiener-Hopf tectilque is used to obtain an exact solution to a two-dimensional scattering problem. In the problem solved, an incident TE/sub 10/ mode, traveling from z= -/spl infin/ in the positive z direction, is confined by infinite bounding planes; these planes have infinite conductivity for z<0 and an impedance Z/sub 1/, for z>0. The scattering from the junction at z=0 gives rise to reflection and transmission coefficients that are exactly determined. An approximate solution for the reflection coefficients is also given when the TE/sub 10/ mode is incident from the opposite direction. Finally, a table is presented which lists some transmission and reflection coefficients for rectangular and circular waveguides with discontinuities in the wall impedances.     

A Parallel-Plate Waveguide Approach to Microminiaturized, Planar Transmission Lines for Integrated Circuits

The parallel-plate waveguide with a two-layer loading medium, a conducting semiconductor substrate, and a relatively thin dielectric layer approximates the interconnections in many integrated systems if the fringing fields are ignored. The fundamental mode of this structure is an E mode which is a surface wave. Its propagation behavior is analyzed in this paper and the equations are evaluated by highly accurate numerical methods. The semiconducting substrate is characterized by its dielectric constant and conductivity. A critical conductivity /spl sigma//sub min/ exists and is related to the cross sectional and material parameters. If the substrate conductivity is given by /spl sigma//sub min/ then the attenuation constant of the line is a minimum. The same value of conductivity yields minimum phase distortion at maximum bandwidth. If the conductivity is larger than /spl sigma//sub min/ the substrate acts as a poor conductor with associated skin effect; if it is smaller, lossy dielectric behavior results. Analysis shows that it is appropriate to subdivide the frequency range into three intervals. The lowest-frequency interval is characterized by propagation which resembles diffusion. This is caused by the loss in the dielectric layer. The next frequency range extends to some upper frequency which is determined by substrate conductivity and the cross-sectional dimensions. In this interval, the phase velocity of the fundamental mode is controlled by the ratio of dielectric to semiconductor thickness, which, if typical interconnections are considered, implies a very low velocity. This property indicates that the structure can serve as a delay line. Further increases in frequency result in higher phase velocities. Skin effect and dielectric loss behavior describe the propagation in this third interval.     

Conservation of complex power technique for waveguide junctionswith finite wall conductivity

Scattering at the junction of two waveguides with finite wall conductivity is rigorously treated using E-field mode matching and the conservation of complex power technique. At the transverse junction discontinuity between the two waveguides, the complex power absorbed by the junction wall is taken into account along with the usual transfer of complex power from one guide to the other. This leads to a generalized form of the scattering matrix [S] of the lossy junction which incorporates the surface impedance Z_m of the transverse metallic wall, assumed to be a good conductor. The specific case of a copper transverse diaphragm with centered circular iris in an X-band guide is considered and the equivalent TE₁₀ shunt admittance is computed. Numerical results are also given for lossy X-band cavity resonators with circular coupling holes     

On optimal current patterns for electrical impedance tomography

We develop a statistical criterion for optimal patterns in planar circular electrical impedance tomography. These patterns minimize the total variance of the estimation for the resistance or conductance matrix. It is shown that trigonometric patterns (Isaacson, 1986), originally derived from the concept of distinguishability, are a special case of our optimal statistical patterns. New optimal random patterns are introduced. Recovering the electrical properties of the measured body is greatly simplified when optimal patterns are used. The Neumann-to-Dirichlet map and the optimal patterns are derived for a homogeneous medium with an arbitrary distribution of the electrodes on the periphery. As a special case, optimal patterns are developed for a practical EIT system with a finite number of electrodes. For a general nonhomogeneous medium, with no a priori restriction, the optimal patterns for the resistance and conductance matrix are the same. However, for a homogeneous medium, the best current pattern is the worst voltage pattern and vice versa. We study the effect of the number and the width of the electrodes on the estimate of resistivity and conductivity in a homogeneous medium. We confirm experimentally that the optimal patterns produce minimum conductivity variance in a homogeneous medium. Our statistical model is able to discriminate between a homogenous agar phantom and one with a 2 mm air hole with error probability (p-value) 1/1000.     

The effect of radial radiation transport on intensity characteristics and oscillation frequency of homogeneously broadened lasers

In the present treatment of performance characteristics of laser systems, having homogeneous line broadening, a radial gain profile is taken into account. It will be shown that such a profile, which may be induced by the laser beam itself, changes the curvature of the phase front, so that additional radiation transport in the radial direction takes place. It turns out that for many practical systems the radial radiation transport can be comparable to the radiation gained directly from the medium and therefore may not be neglected. Furthermore, it will be shown how the additional radial radiation transport affects the oscillating frequency. Conditions are deduced in which mode competition leads to the survival of the mode having highest radiation intensity. This is not the one closest to the line center as found in systems where radial radiation transport is not taken into account. Treating this problem we approximate medium parameters by a quadratic profile so that the modes have a Gaussian structure. The oscillation frequency, determined on one side by the dispersion of the active medium and on the other side by the spontaneous decay of the upper level, is discussed. The frequency shift is not very sensitive to changes of mirror reflectivity but increases considerably with increasing thermal effects. Numerical calculations show, for instance, that for a semiconfocal system a frequency shift close to 0.5 in units of normalized frequency can be predicted.     

A Semi-Infinite Array of Parallel Metallic Plates of Finite Thickness for Microwave Systems

An array of parallel metallic plates of finite thickness are useful in microwave lenses. The effect of finite thickness in the idealized situation of a semi-infinite array of perfect conductivity, is treated theoretically and experimentally for normal incidence of a uniform plane wave on the plane interface separating the medium from free space. The theoretical discussion involves the approximate variational method and a procedure is given for estimating the order of magnitude of the error in the final result. It is shown that it can be advantageous to use plates of finite thickness since the reflection from the interface can be reduced from that existing for infinitely thin plates.     

An improvement on analytic accuracy of circular groove guide

In this paper, an Nth order approximate eigenequation of circular groove guide is presented. Using the sixth order approximate results, some propagation properties of the TE₁₁ mode of circular groove guide are recalculated. The results are more accurate than before and in good agreement with the measurements     

A waveguide polarization controller

In this paper a novel waveguide polarizer is introduced that does not require rotary joints and the frequency of operation can easily be adjusted by a few set screws. In this method the degenerate eigenvalues of a circular waveguide are separated by deforming the waveguide cross section slightly. In order to generate a desired polarization, the orientation angle of the deformation point with respect to the polarization of the incident wave can be adjusted using a rotary roller mechanism concentric with the circular waveguide. Analysis of the problem based on the finite element method and an approximate analytical method is given. A prototype model at 34.5 GHz is built and tested. Experimental results shows excellent agreement with the theoretical prediction     

Mode Conversion in Tapered Waveguides At and Near Cutoff

The coupling coefficient between the TE/sub 11/ mode and the TM/sub 11/ mode in tapered circular waveguides is derived, and at cutoff frequency it tends to approach an infinity of the order of 0/sup -1/4/. It is surprising to discover that the corresponding coupling coefficient between the TE/sub 10/ mode and the TM/sub 12/ mode in tapered rectangular waveguides approaches instead a zero of the order of 0/sup 1/4/ at cutoff frequency. Accordingly, for the modes concerned, the choice of using circular or square waveguides as tapers for transition at and near cutoff frequency is significant in reducing mode conversion level. At and near cutoff frequency a "synthesized" square taper is better in that it is shorter than a "synthesized" circular taper for the same mode conversion levels. On the other hand, for frequencies far away from cutoff the choice is insignificant. Design procedures for "synthesized" waveguide tapers at and near cutoff are presented, and the results of measurements are in agreement with the theoretical calculations.     

Low-frequency radio transmission in a circular tunnel containing a wire conductor near the wall

The general theory for transmission in a circular tunnel containing a thin axial conductor is employed to calculate the attenuation rate of the propagating mode. The remarkable property is that the attenuation rate is approximately pro-portional to frequency and it does not depend critically on the wall conductivity for typical conditions.     

Full-wave analysis of coplanar waveguides for LiNbO3optical modulators by the mode-matching method considering nonidealconductors on etched buffer layers

Rigorous analysis of traveling-wave coplanar waveguide electrodes for LiNbO₃ optical modulator applications is presented by using an extended full-wave mode-matching method. The microwave propagation characteristics under the composite influence of substrate anisotropy, uniform or etched buffer layers, finite electrode thickness and conductivity, and metallization undercutting are accurately assessed by employing a network equivalent formulation. Variations of the coplanar waveguide microwave effective index and the characteristic impedance at low frequencies due to finite electrode conductivity are illustrated, and are important even though the mode is quasi-TEM in nature. The effect of etching the SiO₂ buffer layer is shown to be one possible method for lowering the microwave effective index while keeping the conductor loss at a fixed level     

Analysis of Circular Chiral Dielectric Waveguides

The general eigenequations and the field expressions for circular waveguides containing chiral material in both the core and the cladding are given in a simple formulation. The results indicate that there are two types of field distribution which is related to the operating wavelength and chiral admittance. When the chiral admittance is larger than a critical value, the fields of one kind of circularly polarized wave in the core become exponentially-like damping in the transverse direction. Based on the eigenequations, the characteristics of a circular waveguide with chiral medium filled in the core are investigated. The relation between the cutoff frequency and the chiral admittance, the dispersion curves for some modes with different angular mode number and the transverse distribution of fields are presented. Some features different from the ordinary circular waveguide are resulted.     

Resonant Frequency of Open-Ended Cylindrical Cavity

The TE/sub 011/ mode of oscillation in an open-ended circular cylindrical microwave cavity is analyzed. The cavity consists of a circular waveguide that is terminated at each end with a thin cylindrical partition coaxial with the circular waveguide. The resonant frequency of the cavity is computed by using Laplace transform and Wiener-Hopf techniques. Numerical values for the resonant frequency are presented.     

An equivalent transmission-line model containing dispersion forhigh-speed digital lines-with an FDTD implementation

The increase in processor speeds in the last few years has created a growing need for the accurate characterization of waveform propagation on lossy printed-circuit-board (PCB) transmission lines. Due to the dispersive nature of pulse propagation on lossy transmission lines, approximations of the classic transmission-line model can fail in this application (i.e., lossless or DC losses approximations). This paper shows how an equivalent transmission-line model can be used to analyze dispersive transmission lines for high-speed digital applications. The equivalent-circuit elements of this transmission-line model incorporate the frequency dependence of the per unit length impedance and admittance caused by the finite conductivity of the conductors as well as the dielectric losses. We show that these equivalent circuit elements can be readily implemented into finite-difference time-domain (FDTD) transmission-line codes, and we present such a FDTD implementation. S-parameters and pulsed waveforms for a circular wire, coplanar waveguides (CPW) and microstrip lines are shown. Finally, we present approximate expressions for analytically obtaining the resistance and inductance per length of a microstrip line     

Scattering by finite, open cylinders using approximate boundaryconditions

Simplified integral equation formulations for electromagnetic scattering from finite, open, circular cylinders are developed by invoking approximate boundary conditions. The approximate calculations are compared to calculational results of formulations based on exact boundary conditions. It is shown that the approximations work quite well despite the curved edges of the cylinders     

Scattering by a rotating circular cylinder with finite conductivity

The effect of motion on the signal scattered by a rotating circular cylinder with finite conductivity is investigated. The problem is solved by means of the "instantaneous rest-frame" hypothesis. The analysis shows that a surface current must be taken into account to calculate the jump in the tangential magnetic field at the cylindrical surface. This holds even in the case of finite conductivity. For a perfectly conducting cylinder, the influence of the motion on the fields is negligible. This is shown by considering the limit of high but finitesigma.     

On the determination of an optimum microwave diathermy frequency for a direct contact applicator

An approximate theoretical analysis is presented for determining the relative heating in a two layer fat-muscle medium due to a dielectric loaded dipole-corner reflector applicator in direct contact with the finite fat layer. The results indicate that convenient applicator sizes, equivalent to those now used at 2450 Mc/s, operating at a frequency in the neighborhood of 750 Mc/s provide the maximum ratio of relative heat per unit volume in the muscle as compared to that in the fat. Experimental data taken with the aid of an electrical model of the fat-muscle layers shows reasonable agreement with the results of the approximate analysis.