Showing papers on "Dielectric loss published in 1974"
TL;DR: In this article, a nematic material in which e∥ exhibits a dielectric loss in the audio-frequency range is reported, and the activation energy for the loss is 0.91 eV.
Abstract: A new nematic material in which e∥ exhibits a dielectric loss in the audio‐frequency range is reported. At 25 °C Δ e=(e∥−e⊥) reverses sign at 2.5 kHz and can be as positive as 6.2 at low frequencies and as negative as −2.2 at high frequencies; the activation energy for the loss is 0.91 eV. The availability of a dielectric anisotropy of either sign depending on the frequency of the applied field allows the electrical orientation of the material with its optic axis either parallel or perpendicular to the electrodes. The possibilities for improving the voltage threshold and dynamic response of electro‐optical effects by means of a two‐frequency addressing scheme are demonstrated.
114 citations
TL;DR: In this article, the authors examined the factors that affect the dielectric response of food materials in an attempt to establish a predictive model and to obtain a more basic understanding of dielectrics properties.
Abstract: The factors which affect the dielectric response of food materials are examined in an attempt to establish a predictive model and to obtain a more basic understanding of dielectric properties. Detailed dielectric measurements were performed on rehydrated non-fat dry milk using the precision slotted line technique, at temperatures of 25, 35, 45, and 55°C and at frequencies of 300, 1000, and 3000 MHz. Solute-solute and solute-solvent interaction were found to reduce the dielectric loss factors to levels substantially below those predicted by chemical composition alone; therefore, the dielectric loss cannot be predicted from a linear, additive model based on composition. The results of dielectric measurements of solid food products, beef and turkey were reported. Conductivities of beef juice obtained from cooking were measured and the dielectric losses at 5 to 65°C and 300, 915, and 2450 MHz calculated based on the measured conductivities.
65 citations
TL;DR: In this article, an explicit mechanism for the anomalous increase in dielectric constant and loss at low frequencies and high temperatures for poly(vinylidene fluoride) containing ionic impurities was described.
Abstract: An explicit mechanism is described for the anomalous increase in dielectric constant and dielectric loss at low frequencies and high temperatures for poly(vinylidene fluoride) containing ionic impurities. Relations are proposed for the ionic contributions, ei″ and ei″, to the dielectric constant and dielectric loss:
where v0 and D0 are the concentration and the diffusion coefficient of the mobile ions at infinite temperature, q is the charge of an ion (in cgs electrostatic units), l is the distance between electrodes, k is the Boltzmann constant, T is the absolute temperature, Ed is the apparent activation energy for diffusion of the ions, and W is the dissociation energy of the ionic impurities. From the slopes of curves of log eT′ versus 1/T and log e″T versus 1/T for poly(vinylidene fluoride), energies Ed = 34 kcal/mole and W = 342 kcal/mole were obtained.
59 citations
TL;DR: In this paper, a comparison between cavity and open-resonator measurements of permittivity and dielectric loss angle at 35 GHz was made, using one set of samples of unsintered polytetrafluoroethylene measured in both systems and two sets of similar samples, one set measured in each system, of a high density polyethylene (Rigidex 2000), cut from the same rod.
Abstract: A comparison has been made between cavity and open-resonator measurements of permittivity ?' and dielectric loss angle ? at 35 GHz. Two comparisons were made, the first using one set of samples of unsintered polytetrafluoroethylene measured in both systems and the second using two sets of similar samples, one set measured in each system, of a high density polyethylene (Rigidex 2000), cut from the same rod. The measured values of ?' and ? obtained by the two systems agreed within the limits of the experimental errors. The standard deviations obtained for ?' and ? were 0.0021 and 2.1 ?rad, respectively. A discussion of the uncertainties associated with each measured parameter is given.
51 citations
TL;DR: In this article, the effect of magnetic loss on the electromagnetic response of a body has been studied in a wide range of frequencies from 125 to 5000 hz for a variety of samples, with varying amounts of magnetite dispersed in a nonmagnetic matrix.
Abstract: Relaxation processes are important in the study of electromagnetic prospecting techniques. These processes can be the result of magnetic fields due to eddy currents in electrical conductors or due to displacement currents from dielectric processes or magnetic fields associated with direct magnetic loss phenomena. In the past, most attention has been focused on eddy-current losses since this is the dominant effect. Dielectric losses also may be of importance, particularly at high frequencies, where the conductivity is extremely low or where the dielectric properties become anomalously high. In general, however, little attention has been given to magnetic losses and the effect they might have on the electromagnetic response parameter. Magnetic minerals do have a measurable magnetic loss due to various mechanisms. This effect has been measured in a frequency range from 125 to 5000 hz for a variety of samples. In synthetic samples with varying amounts of magnetite dispersed in a nonmagnetic matrix the effect is linearly proportional to the magnetite content. The magnetic loss causes a distinct, but small, change in the electromagnetic response of a body. The effect can be detected, provided the peak frequency of magnetic loss is lower than the peak frequency of electrical loss. The most diagnostic feature is the reversed phase relationship from the electrical response (for constant geometries).
41 citations
TL;DR: In this article, the authors measured the dielectric constant, the total ac loss tangent, and the dc conductivity of the Apollo 15 lunar soil sample in ultrahigh vacuum from room temperature to 827 C for the frequency spectrum from 100 Hz through 1 MHz.
Abstract: Electrical property measurements have been made on an Apollo 15 lunar soil sample in ultrahigh vacuum from room temperature to 827 C for the frequency spectrum from 100 Hz through 1 MHz. The dielectric constant, the total ac loss tangent, and the dc conductivity were measured. The dc conductivity showed no thermal hysteresis, but an irreversible (in vacuum) thermal effect was found in the dielectric loss tangent on heating above 700 C and during the subsequent cooling. This appears to be related to several effects associated with lunar glass above 700 C. The sample also showed characteristic low-frequency dispersion in the dielectric constant with increasing temperature, presumably due to Maxwell-Wagner intergranular effects. The dielectric properties may be fitted to a model involving a Cole-Cole frequency distribution that is relatively temperature-independent below 200 C and follows a Boltzmann temperature distribution with an activation energy of 2.5 eV above 200 C. The dc conductivity is fitted by an exponential temperature distribution and becomes the dominant loss above 700 C.
41 citations
TL;DR: In this article, the authors used the liquid-immersion method to measure the dielectric properties of magnetic recording tape and found that the accuracy of the final result depends mainly on the validity of the model.
Abstract: Dielectric and magnetic properties are measured by assuming an electrical model for the specimen and its electrodes or windings; the accuracy of the final result depends mainly on the validity of the model. Distributed parameters may have to be assumed for frequencies above 10 MHz. Measurements on dielectrics may be of two different capacitances for specimen in and out, or of two capacitances adjusted to be nearly equal. British and American practices are different in allowing for edge effects in a disk specimen. The liquid-immersion method is of limited use in measuring dielectric loss. For frequencies up to 1 MHz, bridge methods are recommended; at higher frequencies, resonance methods. For frequencies at which distributed parameters must be assumed, there are coaxial line and various waveguide and resonator methods. For 10-5 Hz to 20 GHz there are time-domain methods of somewhat less sensitivity. Magnetic properties are not usually reproducible enough to justify accuracy better than a few parts in 1000. Bridge methods exist for both low- and high-flux densities; at high fluxes there is also the wattmeter method. A special method is used for measuring the properties of magnetic recording tape.
41 citations
Patent•
01 Jul 1974TL;DR: In this paper, the authors describe a low loss dielectric material, which is made up of BaO and TiO 2 and corresponds closely in composition to the compound Ba 2 Ti 9 O 20.
Abstract: Devices are described which utilize a low loss dielectric material. This dielectric material is unique in that it combines low dielectric loss with a high and temperature-stable dielectric constant. The material is made up of BaO and TiO 2 and corresponds closely in composition to the compound Ba 2 Ti 9 O 20 . Devices utilizing this material are useful at a variety of frequencies, but are particularly advantageous in the microwave region above 0.5 GHz.
35 citations
TL;DR: Several series of glasses with the general formula x R 2 O · (1 − x )R′ O · P 2 O 5 were prepared and the dielectric loss and internal friction data were taken.
Abstract: Several series of glasses with the general formula x R 2 O · (1 − x )R′ O · P 2 O 5 were prepared and the dielectric loss and internal friction data were taken. The dielectric losses of alkali metaphosphate glasses are greatly reduced by the substitution of alkaline-earth oxide for alkali oxide. Some of the investigated series show a weak minimum in the dielectric loss. The internal friction measurements reveal a high temperature peak, in addition to the single alkali peak and the intermediate temperature peak commonly observed in phosphate glasses. The origin of this high temperature peak is discussed.
34 citations
01 Oct 1974
TL;DR: In this article, a method has been developed for precise non-destructive measurement of the dielectric constant and losses of slab-like samples, such as microstrip substrates for instance.
Abstract: A method has been developed for a precise non-destructive measurement of the dielectric constant and losses of slab-like samples, such as microstrip substrates for instance. The test setup basically consists of an open-ended rectangular waveguide whose flange is placed in contact with one side of the dielectric material, the other one being backed by a metal plate. The measured amplitude and phase of the reflection coefficient with respect to a short-circuit in the aperture plane are related to the real and imaginary parts of the permittivity by means of computer generated charts.
29 citations
TL;DR: In this article, a non-destructive method for measuring the dielectric constant and the loss factor of a slab-type material using a microstrip cavity was developed, which uses a simple and rapid substitution procedure, yields accurate results and has several advantages over currently available techniques.
Abstract: A new nondestructive method has been developed for measuring the dielectric constant and the loss factor of a slab-type material using a microstrip cavity. The method, which uses a simple and rapid substitution procedure, yields accurate results and has a number of advantages over currently available techniques. Experimental details and the theoretical basis are explained and experimental data are presented.
IBM1
TL;DR: In this paper, the effects of crystallinity and orientation on secondary dielectric losses were considered, and related to the structural entities involved in these motions, and a stress relaxation peak was found in oriented polycarbonate.
Abstract: Polycarbonate has three secondary dielectric losses in the temperature range between –200°C and the glass transition. The effects of crystallinity and orientation on these secondary losses are considered, and related to the structural entities involved in these motions. In addition, a stress relaxation peak was found in oriented polycarbonate.
TL;DR: In this paper, a dc conduction mechanism is proposed based on electronic hopping favored by the micro-Brownian motions responsible for dielectric losses, and the experimental results are discussed and interpreted in terms of the proposed mechanism.
Abstract: Dc conductivity measurements were performed as a function of temperature on unplasticized poly(vinyl chloride) and on PVC plasticized with various amounts of dioctylphthalate. The conductivity curves consist of two or three straight-line segments denoted I, II, and III with increasing of temperature. The intersection of segments I and II occurs at the glass-transition temperature Tg. The slope in region I is independent of the DOP concentration, while the slope in region II decreases slowly with an increase in the amount of DOP. No dependence of the conductivity on the molecular weight was found. From the conductivity curves, activation energies were evaluated below and above Tg. These satisfactorily coincide with those determined by dielectric loss or by electrical transient phenomena. A dc conduction mechanism is proposed based on electronic hopping favored by the micro-Brownian motions responsible for dielectric losses. These motions involve smaller chain lengths below than above Tg. The experimental results are discussed and interpreted in terms of the proposed mechanism.
TL;DR: The low frequency (102 to 105 Hz) dielectric relaxational behaviour of di-n-butyl phthalate, tri-o-tolyl phosphate and tri-2-chloroethyl phosphate has been studied in this paper.
Abstract: The low frequency (102 to 105 Hz) dielectric relaxational behaviour of di-n-butyl phthalate, tri-o-tolyl phosphate and tri-2-chloroethyl phosphate has been studied and one loss process, due to Brownian motions of the molecules, is observed in di-n-butyl phthalate and tri-o-tolyl phosphate. Results for tri-2-chloroethyl phosphate show evidence of bimodal behaviour.These dielectric loss data are compared with the viscoelastic retardation data at 30 MHz obtained previously by Barlow and Erginsav for the same samples of these materials. For di-n-butyl phthalate and tri-o-tolyl phosphate the Argand diagrams for dielectric relaxation and viscoelastic retardation are very similar for each liquid, suggesting a common mechanism for both processes.The viscoelastic retardation results are fitted by the empirical Davidson–Cole equation for these two liquids whilst the dielectric relaxation results are fitted better by calculated curves based upon the Williams–Watts empirical relaxation function. It is shown that the defect diffusion stress relaxation functions of Phillips, Barlow and Lamb can be fitted by the Williams–Watts empirical relaxation function with a suitable choice of parameters: these predict, however, a broader dielectric loss curve than is found in practice.
TL;DR: In this paper, the dielectric losses of aluminum/organic monolayer/aluminum structures were studied between −170 and +50°C within the frequency range 100 Hz to 50 kHz.
Abstract: Dielectric losses of aluminum/organic monolayer/aluminum structures were studied between −170 and +50°C within the frequency range 100 Hz to 50 kHz. They are due largely to oxide layers on the electrode surfaces and partly to movements of dipoles linked to the organic molecules. At a measurement frequency of 1 kHz, absorption maxima appear at −100 and −30°C and are attributed to dipole movements in the amorphous and crystalline regions of the layers, respectively.
TL;DR: In this paper, the measurement and theory of dielectric relaxation in glass are discussed, and it is shown that contacting electrodes such as Ag paste or Au sputtered film can lead to anomalously high Dielectric losses and high capacitance, but the use of stainless steel electrodes reduces these losses and capacitances substantially.
Abstract: The measurement and theory of dielectric relaxation in glass are discussed. It is shown that contacting electrodes such as Ag paste or Au sputtered film can lead to anomalously high dielectric losses and high capacitance, but the use of stainless steel electrodes reduces these losses and capacitances substantially. The height of the loss peak was proportional to the sample thickness for a sodium silicate glass. Broad, asymmetric loss peaks were found for different glasses. The results are compared to equations derived from a space-charge model. These equations predict the thickness dependence found and also the shapes of some of the loss peaks. Predictions of the frequency of maximum loss are in fair agreement with experimental data.
TL;DR: In this paper, a model to account for the electrical relaxation behavior of glasses in which a vacancy mechanism prevails is presented. But the model assumes that the vacancies existing in glass a priori are constant in number and usually are in excess over those generated by thermal motion in the sense of Frenkel's defect theory.
Abstract: This paper presents a model to account for the electrical relaxation behaviour of glasses in which a vacancy mechanism prevails. The vacancies existing in glass a priori are constant in number and usually are in excess over those generated by thermal motion in the sense of Frenkel's defect theory. The pathways of vacancies form a net system with junctions and impasses. The displacement of vacancies in the impasses is deemed especially responsible for the relaxation process observed when a glass is subjected to an external electric field. According to the model (i) the dispersion of permittivity is inversely proportional to the temperature, and (ii) as the temperature is varied the curves of the ratios of the permittivity and dielectric loss, versus the dispersion, as a function of the logarithm of angular frequency, remain constant in shape. It follows from the model that a certain proportion of alkali ions in glass cannot be replaced either in an ion exchange process or by electrolysis if the temperature is not very high.
TL;DR: In this article, the influence of the presence of a DC field on the dielectric properties of MoO3 has been observed by measuring the permittivity and loss tangent of polycrystalline specimens in the range of frequencies 002-100 Hz and temperatures 20-210°C using AC voltage, both with and without superimposed DC voltage.
Abstract: The influence of the presence of a DC field on the dielectric properties of MoO3 has been observed by measuring the permittivity and loss tangent of polycrystalline specimens of MoO3 in the range of frequencies 002-100 Hz and in the range of temperatures 20-210°C using AC voltage, both with and without superimposed DC voltage.
01 Apr 1974
TL;DR: In this paper, a search was made for new cationic conducting phases in alkali-tantalate and niobate systems, and phase equilibrium diagrams were constructed for the six binary systems Nb2O5-LiNbO3, Nb 2 O5-NaNb O3, nb 2O5 NbO5 KTaOO3 and Ta2O 5-KTaO3.
Abstract: A search was made for new cationic conducting phases in alkali-tantalate and niobate systems. The phase equilibrium diagrams were constructed for the six binary systems Nb2O5-LiNbO3, Nb2O5-NaNbO3, Nb2O5-KNbO3, Ta2O5-NaTaO3, Ta2O5-LiTaO3, and Ta2O5-KTaO3. Various other binary and ternary systems were also examined. Pellets of nineteen phases were evaluated (by the sponsoring agency) by dielectric loss measurements. Attempts were made to grow large crystals of eight different phases. The system Ta2O5-KTaO3 contains at least three phases which showed peaks in dielectric loss vs. temperature. All three contain structures related to the tungsten bronzes with alkali ions in non-stoichiometric crystallographic positions.
01 Nov 1974
TL;DR: The theory of the barrel resonator with a central rod of dielectric material is developed in this paper, and the equations for resonance are derived and it is shown how this configuration can be used for the measurement of permittivity.
Abstract: The theory of the barrel resonator with a central rod of dielectric material is developed. The equations for resonance are derived and it is shown how this configuration can be used for the measurement of permittivity.
TL;DR: In this article, a study of copper impurity centers in NH 4 Cl by dielectric loss and electrical conductivity measurements is reported, where a theory is given for the reorientations of impurity-vacancy dipoles in a CsCl-type lattice by means of jumps of constituent point defects.
TL;DR: In this article, an accurate method for the measurement of the dielectric constant of liquids and solids is described and the guide wavelength is measured for two different thicknesses of the material through a characteristic equation which can be solved graphically or numerically.
Abstract: An accurate method is described for the measurement of the dielectric constant of liquids and solids. The dielectric material partially loads a slotted rectangular waveguide and the guide wavelength is measured for two different thicknesses of the dielectric. The guide wavelengths are related to the dielectric constant of the material through a characteristic equation which can be solved graphically or numerically. Some experimental results are obtained and found to be in close agreement with the values of the dielectric constant available in the literature.
TL;DR: In this article, the authors compared the dielectric properties of terrestrial and lunar materials as a function of density, temperature, and frequency, and concluded that density prevails over temperature as the controlling factor of dielectrics permittivity in the lunar regolith.
Abstract: The dielectric properties of lunar fines 74241,2 in the audio-frequency range under lunarlike conditions are presented. Results suggest that volatiles are released during storage and transport of the lunar sample. Apparently, subsequent adsorption of volatiles on the sample surface alters its dielectric response. The assumed volatile influence disappears after evacuation. A comparison of the dielectric properties of lunar and terrestrial materials as a function of density, temperature, and frequency indicates that if the terrestrial sample analyzed were completely devoid of atmospheric moisture, it would present dielectric losses smaller than those of the lunar sample. It is concluded that density prevails over temperature as the controlling factor of dielectric permittivity in the lunar regolith and that dielectric losses vary slowly with depth.
TL;DR: In this article, the relative complex permittivity components /spl epsilon/sub r/' and loss tangent, and conductivity from measurements data on materials in coaxial, rectangular, or cylindrical waveguides were computed.
Abstract: Computation of the relative complex permittivity components /spl epsilon//sub r/' and /spl epsilon//sub r/'' loss tangent, and conductivity from measurements data on materials in coaxial, rectangular, or cylindrical waveguides.
TL;DR: The Curie-Weiss-like behavior of the loss tangent of ceramic barium titanate is explained in terms of interaction of the soft-mode at the grain boundaries as mentioned in this paper.
Abstract: The Curie-Weiss-like behaviour of the loss tangent of ceramic barium titanate is shown to depend on the surface area of the grains and is explained in terms of interaction of the soft-mode at the grain boundaries. A relation tan δ = 0.004 D −1 (T - T 0)−1 is obtained for the region just above T c with D, the mean grain diameter, in cm and T in kelvins. The region over which this effect is dominant is limited by conduction loss introduced when oxygen is lost during specimen preparation.
TL;DR: In this paper, the Hartshorn-Ward Resonance Substitution (RS) method is used for the determination of dielectric properties of material in sheet form, and a modified form of Q meter is described for use in the frequency range 1-100 MHz where application of the resonance substitution method permits dielectoric loss measurements to be made with a precision of better than five microradians.
Abstract: There are three methods for the determination of the dielectric properties of material in sheet form: the Hartshorn-Ward 'resonance-substitution' technique (BS 2067), the Lynch method using a transformer bridge (BS 4542) and the two fluid immersion method. These methods may be used at frequencies up to 1 MHz, and, in this range, it is shown that the two fluid immersion method offers advantages for permittivity measurement but is not suitable for determination of tan ?. A modified form of Q meter is then described for use in the frequency range 1-100 MHz where application of the resonance substitution method permits dielectric loss measurements to be made with a precision of better than five microradians. Measurements at microwave frequencies are made by resonant cavity techniques and comparison is made between the value of permittivity determined at 1 MHz by a cross checkable two fluid method, and those calculated from the high frequency values of permittivity and dielectric loss.
Patent•
11 Feb 1974
TL;DR: In this article, a method for checking the quality of a piezoelectric material, wherein the frequency of the dielectric loss angle tangent maximum was estimated from the temperature-frequency dependence of the Dielectric Loss Angle tangent of the material, was found.
Abstract: A method for checking the quality of a piezoelectric material, wherein the frequency of the dielectric loss angle tangent maximum, from which the quality of the piezoelectric material is estimated, is found from the temperature-frequency dependence of the dielectric loss angle tangent of the piezoelectric material.
TL;DR: In this paper, the authors measured the dielectric constant and dielectral loss at 0.22, 1.25, 3.22 and 10.00 cm on benzene, hexafluorobenzene, an equimolar mixture of benzene and hexaphlorobenzenes, and hexane, dodecane, and octadecane.
Abstract: Measurements of dielectric constant and dielectric loss at 0.22, 1.25, 3.22, and 10.00 cm have been made upon benzene, hexafluorobenzene, an equimolar mixture of benzene and hexafluorobenzene, a dilute solution of hexachlorobenzene in benzene, and hexane, dodecane, and octadecane. The very small sizes of the dielectric losses observed in this frequency region, where loss due to the rotational orientation of dipolar molecules of this size should be at or near a maximum, indicates the absence of any appreciable permanent molecular dipole moment. The increase of the loss values with increasing frequency in this region indicates that, as already established for benzene, they form the low‐frequency tail of a very‐far‐infrared absorption band for each liquid. The absorption presumably arises from the small temporary dipole moments induced by molecular collisions. It is found that the absorption coefficients are increased by the presence of mutually opposing and compensating permanent dipoles and of π electrons in the molecules.
TL;DR: In this article, the additive properties of the Na2O-Fe2O3-P2O5 system were studied over the temperature range 20°-180°C.
Abstract: The electrical conduction and dielectric relaxation of some glasses in the system Na2O-Fe2O3-P2O5 were studied over the temperature range 20°-180°C. This system contains Na2O which confers an ionic conductivity on glass and Fe2O3 an electronic conductivity.The conductivity of this kind of glass may be expected to have an additive property from which it can be calculated by the addition of ionic and electronic conduction. In this study the experimental results were discussed in view of the additive property. Samples used in this experiment were prepared by melting under different conditions, i.e. in the air and in a reducing atmosphere. The ratio (Fe2+/Fe2++Fe3+) in the glass was determined by chemical analysis. The conductivity was observed to vary greatly with the melting temperature. The values of log σ increase with increasing Fe2+ ion in the glasses of compositions 40Fe2O3-60P2O5, 10Na2O-30Fe2O3-60P2O5, 20Na2O-20Fe2O3-60P2O5; but on the contrary in the galss containing relatively small amount of Fe2O3, 30Na2O-10Fe2O3-60P2O5, the log σ decreases with increasing Fe2+ ion. This can be considered from the fact that the mechanism of hopping theory holds for the glass containing a large amount of Fe2O3 and does not hold for the glass with small amount of Fe2O3 content. The d.c. conductivity (σ) of glasses of the formula xNa2O-(40-x)Fe2O360P2O5 was shown by replacing Fe2O3 with Na2O to have a minimum value at the composition near x=20 at temperature above 100°C, but at temperature below 100°C the x-value showing the minimum conductivity was seemed to shift toward its higher value. The activation energies ΔHdc and ΔHac were found to be grouped into two parts, the higher values in the compositions near (40Na2O-60P2O5) in which ionic conduction predominate and the lower values in the compositions near (40Fe2O3-60P2O5) in which electronic conduction predominate.The dielectric relaxation was observed by changing frequency and temperature. The values of dielectric constant e′ and dielectric absorption e″ were measured in glasses melted by various conditions.The normalized dielectric losses at various temperature was shown by plotting e″/e″max against log (f/fmax) but it was independent of temperature. The relationship between σ and the frequency at which the dielectric loss becomes maximum, fmax, was found to follow the equation proposed by H. Namikawa.
01 Oct 1974
TL;DR: In this paper, a numerical solution for any ratio of dielectric thickness to height of guide is found, but only when the loss is small, and the numerical results agree with some obtained by previous authors.
Abstract: The. problem is to calculate the dielectric loss in a waveguide partially filled across its smaller dimension. A numerical solution can be found which applies for any ratio of dielectric thickness to height of guide, but only when the dielectric loss is small. Numerical results agree with some obtained by previous authors.