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Showing papers on "Dipole antenna published in 1987"


Journal ArticleDOI
TL;DR: In this article, the authors used reciprocity and a transmission line model to determine the radiation properties of printed circuit antennas (PCA's) in a multilayered material configuration.
Abstract: Reciprocity and a transmission line model are used to determine the radiation properties of printed circuit antennas (PCA's) in a multilayered material configuration. It is demonstrated that extremely high directive gain may result at any scan angle, with practical materials, if the thickness of the substrate and multiple superstrate layers is chosen properly. This model is also used to analyze the radiation characteristics of printed circuit antennas in inhomogeneous substrates.

326 citations


Journal ArticleDOI
TL;DR: The pattern averaging gain (PAG) method to estimate the average gain of mobile antennas in a multipath propagation environment is proposed and the antenna configuration with the PIFA element mounted on the lateral side of the radio case is found to be most suitable for portable radio units.
Abstract: The pattern averaging gain (PAG) method to estimate the average gain of mobile antennas in a multipath propagation environment is proposed. By using this method and a wire-grid model, the radiation characteristics of the planar inverted F antenna (PIFA) mounted on a portable radio case is analyzed. In particular, the variation of the antenna gain with the radio case dimensions and inclination angle of the radio case during operation is clarified. Also, the effect on antenna patterns of the operator holding the portable radio is experimentally investigated. Based on this analysis, the antenna configuration with the PIFA element mounted on the lateral side of the radio case is found to be most suitable for portable radio units. In addition, an appropriate selection of the radio case dimensions is found to result in further improvements in the antenna bandwidth.

239 citations


Journal ArticleDOI
TL;DR: In this paper, the radiation pattern of a bow-tie antenna of finite length and infinitesimal thickness, placed on a lossless dielectric substrate, is analyzed based on a representation of the current density on the metal surface of the antenna as a sum of an imposed quasistatic term and a set of current modes with unknown amplitudes.
Abstract: A new formulation is discussed for the rigorous calculation of the radiation pattern of a bow-tie antenna of finite length and infinitesimal thickness, placed on a lossless dielectric substrate. The analysis is based on a representation of the current density on the metal surface of the antenna as a sum of an imposed (quasistatic) term and a set of current modes with unknown amplitudes. Free-space fields that are expressed in terms of continuous spectra of symmetrized plane waves are matched to the current modes using the method of moments. The resulting set of equations are solved for the unknown current amplitudes. The calculations show that for increasing bow length the antenna impedance spirals rapidly to a value predicted by transmission line theory. The theory also shows that the E -plane pattern of a two wavelength, 60\deg bow-tie antenna is dominated by low-loss current modes propagating at the dielectric wavenumber. As the bow tie narrows, the loss of the modes increases, and the dominant wavenumber tends to the quasistatic value. Pattern measurements made at 94 GHz are shown to agree well with theoretical predictions. Measurements for a long-wire antenna, a linear array of bow-tie elements, and a log-periodic antenna are also presented.

173 citations


Patent
George J. Monser1
31 Jul 1987
TL;DR: In this paper, a vertical monopole antenna is coupled to a first feed on a ground plane conductor and a dipole antenna consisting of a plurality of horizontal dipole antennas is employed to produce a predetermined (such as omni-directional) radiation pattern about the first feed.
Abstract: An antenna system comprising a vertical monopole antenna coupled to a first feed on a ground plane conductor and a dipole antenna comprising a plurality of horizontal dipole antenna elements coupled to a corresponding plurality of feeds on the ground plane conductor, the plurality of dipole antenna elements being disposed about the first feed. The vertical monopole antenna comprises a plurality of monopole antenna elements, the plurality of monopole antenna elements and the plurality of dipole antenna elements being alternately radially disposed about the first feed. With such arrangement, the plurality of vertical monopole elements produce a vertically polarized beam having a predetermined (such as omni-directional) radiation pattern about the first feed, and the plurality of horizontal dipole antenna elements produce a horizontally polarized beam having a predetermined (such as omni-directional) beam pattern about the first feed; that is, the antennas are provided with substantially coincident phase centers. The monopole antenna elements do not substantially adversely affect (i.e. shadow) the omni-directional beam pattern produced by dipole antenna elements, and the dipole antenna elements do not substantially adversely affect the omni-directional beam pattern produced by the monopole antenna elements. Thus, the vertically polarized antenna elements may be disposed in close proximity to the horizontally polarized antenna elements, with no substantial degradation of the omni-directional beam patterns of either antenna, thereby allowing the total size of the antenna system to be reduced.

123 citations


Journal ArticleDOI
TL;DR: In this paper, a microwave printed planar log-periodic dipole array (LPDA) antenna analysis and design procedure is presented together with the description of a planar feeding technique.
Abstract: A microwave printed planar log-periodic dipole array (LPDA) antenna analysis and design procedure is presented together with the description of a planar feeding technique. The effects of the dielectric substrate on the antenna design are also discussed and taken into account. A numerical technique using the finite element method is employed to obtain the characteristic impedance and effective dielectric constant of the transmission lines involved, i.e., an unbalanced stripline and a balanced line having only two parallel strips. A 4-8 GHz printed LPDA antenna was manufactured and tested, showing very good radiation pattern and voltage standing-wave ratio (VSWR) values over the entire frequency range.

92 citations


Journal ArticleDOI
15 Jun 1987
TL;DR: In this article, a printed wire antenna of arbitrary configuration is analyzed and the electric field tangential to the wire is derived using the current expanded by piecewise sinusoidal functions.
Abstract: A printed wire antenna of arbitrary configuration is analyzed. The electric field tangential to the wire is derived using the current expanded by piecewise sinusoidal functions. These functions are also used to form the impedance matrix elements. Use of the stationary phase method leads to a simple expression for the radiation field. Numerical analyses based on the present formulation yield radiation characteristics of a zigzag dipole antenna, a loop antenna, and a round spiral antenna. >

88 citations


Patent
08 Jan 1987
TL;DR: A laminar antenna as mentioned in this paper includes a conductive ground plane (102), a first dielectric lamina (106), conductive exciter lamina, a second Dielectric Lamina (114), and conductive radiator lamina.
Abstract: A laminar antenna includes a conductive ground plane (102), a first dielectric lamina (106), a conductive exciter lamina (108), a second dielectric lamina (114), and a conductive radiator lamina (116). The radiator partially overlaps the exciter and the amount of overlap determines the input impedance of the antenna. The laminar antenna can be positioned within the wall of a plastic radio housing (302). Multi-radiator wideband and duplex embodiments of the antenna are also described. In another embodiment, the ground plane extends above the radio housing while the radiator and dielectric laminae wrap around the extended portion of the ground plane.

79 citations


Journal ArticleDOI
TL;DR: In this article, a nearly distortionless antenna beam pattern and 180° steering were demonstrated in a linear seven-element array with a Bragg cell and an offset and radar frequency of 3.2 GHz.
Abstract: Microwave signals for all array elements in a phared array microwave antenna are simultaneously generated by spatial sampling of a moving sinusoidal interference pattern formed between two frequency-offset injection-locked laser diodes. The offset and radar frequency of 3.2 GHz were generated by a Bragg cell. A nearly distortionless antenna beam pattern and 180° steering are demonstrated in a linear seven-element array.

64 citations


Patent
03 Aug 1987
TL;DR: In this paper, an improved element for use in an electrically steered antenna array is disclosed comprising a dipole, an integral balun and a 180° phase shift bit, which utilizes printed circuit techniques throughout using an unbalanced microstrip for connection to electrical circuitry.
Abstract: An improved element for use in an electrically steered antenna array is disclosed comprising a dipole, an integral balun and a 180° phase shift bit. The arrangement utilizes printed circuit techniques throughout using an unbalanced microstrip for connection to electrical circuitry, a balun for transitioning from unbalanced microstrip to a balanced dipole antenna and includes a low loss 180° phase shift bit formed by the use of a branched feed network including two diodes whose conductive states determine the sense of antenna excitation, and produce the equivalent of a 180° phase shift bit.

48 citations


Patent
14 Dec 1987
TL;DR: In this paper, an integrated circuit antenna structure for transmitting or receiving millimeter and/or submillimeter wave radiation having an antenna relatively unimpaired by the antenna mounting arrangement is disclosed.
Abstract: An integrated circuit antenna structure for transmitting or receiving millimeter and/or submillimeter wave radiation having an antenna relatively unimpaired by the antenna mounting arrangment is disclosed herein. The antenna structure of the present invention includes a horn disposed on a substrate for focusing electromagnetic energy with respect to an antenna. The antenna is suspended relative to the horn to receive or transmit the electromagnetic energy focused thereby.

35 citations


Patent
23 Nov 1987
TL;DR: An antenna array for receiving dual polarized electromagnetic waves, comprised of a first thin-film printed circuit rectenna having a plurality of linear half-wavelength dipole antennae oriented in a first direction for receiving a first component of the dual polarized waves, and a second thin-filtered printed circuit Rectenna parallel to the first rectenna, having a multiplicative linear halfwavelength antennae orientated in a second direction to receive the second orthogonal components of the electromagnetic waves as discussed by the authors.
Abstract: An antenna array for receiving dual polarized electromagnetic waves, comprised of a first thin-film printed circuit rectenna having a plurality of linear half-wavelength dipole antennae oriented in a first direction for receiving a first component of the dual polarized waves, and a second thin-film printed circuit rectenna parallel to the first rectenna, having a plurality of linear half-wavelength dipole antennae oriented in a second direction for receiving the second orthogonal component of the electromagnetic waves. A reflector screen is disposed parallel and behind the second rectenna, for reflecting incident electromagnetic waves transmitted through the first and second rectennae back to the first and second rectennae for reception thereby. The dipole antenna of the first rectenna are disposed in a predetermined pattern in relation to the second rectenna dipole antennae, the first and second rectennae are separated by one of either substantially zero distance or by a multiple half-wavelength distance, and the second rectenna and reflector screen are separated by a predetermined distance to effect substantial cancellation of transmission line shielding effects and mutual coupling, resulting in high efficiency signal reception.

Journal ArticleDOI
M. Weiner1
TL;DR: In this paper, the input impedance and directive gain of a monopole element at the center of a circular ground plane in free space are summarized for arbitrary element length and element radius with ground plane radii of zero to more than two wavelengths.
Abstract: The input impedance and directive gain of a monopole element at the center of a circular ground plane in free space are summarized for arbitrary element length and element radius with ground plane radii of zero to more than two wavelengths. Numerical results are obtained by utilizing various models and are compared with measurements. The models include a ground plane of zero extent, the integral equation method, method of moments, oblate spheroidal wave functions, and the method of moments combined with the geometric theory of diffraction (GTD).

Patent
23 Dec 1987
TL;DR: In this paper, a unitary, multiband and dual polarity mobile antenna for vehicles and the like is described, which includes a television antenna (B) having a horizontally polarized radiation pattern and a 2-way communications antenna (C) having vertically polarized radiation patterns.
Abstract: A unitary, multiband and dual polarity mobile antenna (A) is disclosed for vehicles and the like which includes a television antenna (B) having a horizontally polarized radiation pattern and a 2-way communications antenna (C) having a vertically polarized radiation pattern. A null current point (50) is created on communications antenna (C) by a phasing coil (34) which is intersected by a horizontal plane (52) in which television antenna (B) lies. This reduces cross-coupling, cross-polarization, and interference between the two antenna during operation. Horizontal television antenna (B) includes a J-folded, generally boomerang shaped, curved antenna configured to have a radiating element (20) and a half radiating element (22). These elements together form a third radiating element providing operation at three resonant frequencies.

Patent
20 Apr 1987
TL;DR: An antenna for mounting on a nonconductive surface such as a windshield of an automobile coupled by a capacitor to housing on the inside of the resonant plate is described in this article.
Abstract: An antenna for mounting on a non-conductive surface such as a windshield of an automobile coupled by a capacitor to housing on the inside of the resonant plate. The antenna has a jack for receiving a plug from a car radio and provides for interchangeability of the cable between the radio receiver and the jack. The circuit allows for fine tuning to the resonant frequency of the antenna.

Patent
29 Sep 1987
TL;DR: In this article, the performance of an electronic system utilizing a phased array antenna as its principal radiator into a predetermined sector of space is tested using an enclosure which bounds a miniature anechoic chamber located in the near-field region of the antenna array and lined with material absorbing free space electromagnetic radiation.
Abstract: An arrangement for testing the performance of an electronic system utilizing a phased array antenna as its principal radiator into a predetermined sector of space includes, as one of its main components, an enclosure which bounds a miniature anechoic chamber located in the near-field region of the antenna array during the performance of a testing operation and lined with material that absorbs free space electromagnetic radiation. The enclosure has an opening that is aligned with the phased array antenna the performance of which is being tested. Two dual-polarized testing horn antennas are situated at a wall of the enclosure that is opposite to the wall provided with the opening. Electrical signals received from the horn antennas and representative of the near-field electromagnetic radiation emitted by the phased array antenna are supplied to an evaluating circuit which gives indication of the performance of the electronic system of the phased array antenna. A control circuit is employed for energizing the testing antennas in any desired manner to issue electromagnetic radiation toward the phased array antenna being tested. A polarimeter is used to accurately determine the polarization capabilities of the antenna over a wide bandwidth.

Patent
01 Sep 1987
TL;DR: In this paper, a conformal switched beam array antenna (10) is provided which includes a plurality of antenna elements (14) that are mounted onto a curved (16) or a planar surface (28) and that are arranged into a pluralityof arrays (22).
Abstract: A conformal switched beam array antenna (10) is provided which includes a plurality of antenna elements (14) that are mounted onto a curved (16) or a planar surface (28) and that are arranged into a plurality of arrays (22). The antenna elements (14) in each array (22) are arranged in a hexagonal array with one antenna element (14) in the center of each hexagonal array space. A plurality of multiple-throw switches (34), a RF power divider or power combiner (42), and a digital programmer (46) cooperate to select antenna elements (14) in the direction of a given target, and also cooperate to reduce the required number of phase shifters (40) and amplifiers (38 or 58) to a fraction of the total number of antenna elements (14).

Patent
Lee A. Meadows1
03 Sep 1987
TL;DR: In this article, a radio frequency transponder/repeater is provided where an array of receiving antenna elements is coupled to the array of transmitting antenna elements to provide a directional antenna system.
Abstract: A radio frequency transponder/repeater is provided wherein an array of receiving antenna elements is coupled to an array of transmitting antenna elements to provide a directional antenna system and including additionally an internal radio frequency source which is also coupled to the array of transmitting antenna elements. With such arrangement, an internally generated signal provided by the radio frequency source may be transmitted using the transmitting array of the Van Atta System.

Patent
14 Dec 1987
TL;DR: In this article, a foreshortened log-periodic antenna with various configurations of dipole elements and a tapered feedline is described. But the antenna is not considered in this paper.
Abstract: A foreshortened log-periodic antenna which includes variously configured dipole elements and a tapered feedline is disclosed. The dipole elements are arranged in four regions. A first region, located nearest to a feed end of the antenna, includes solid triangular dipoles having constant base-to-height ratios. A second region, adjacent to the first region, includes solid triangular dipoles having decreasing base-to-height ratios. A third region, located adjacent to the second region, includes a single linear dipole. A fourth region, located adjacent to the third region and nearest to a low frequency end of the antenna, includes foreshortened dipoles. The feedline may be implemented either in coaxial or microstrip form so that a lower characteristic impedance is pressed at the feed end of the antenna than at the low frequency end of the antenna. Specifically, for a coaxial feedline having individual conductors separated by a distance "d" at the feed end of the antenna, the individual conductors taper relative to each other so that they are separated by a distance of 2d-5d at the low frequency end of the antenna.

Journal ArticleDOI
TL;DR: In this paper, transient electromagnetic fields are produced when breakers, disconnectors, and switches of various types are operated under varying load conditions, and discharge current pulse is assumed during switching operations, which can act as a dipole antenna.
Abstract: Transient electromagnetic fields are produced when breakers, disconnectors, and switches of various types are operated under varying load conditions. For analysis purposes, a discharge current pulse is assumed during switching operations, which can act as a dipole antenna. The transient electromagnetic fields of this antenna can be calculated for near-and far-field conditions.

Journal ArticleDOI
TL;DR: In this paper, the static electric field distribution in the gap between two perfectly conducting semi-infinite cylinders is obtained in terms of a Fourier-Bessel eigenfunction series, which is transformed into a rapidly convergent series of ultrasphetical polynomials whose weighting function explicitly satisfies the Meixner edge condition.
Abstract: The static electric field distribution in the gap between two solid perfectly conducting semi-infinite cylinders is obtained in terms of a Fourier-Bessel eigenfunction series. For dipole antennas whose cylinder diameter 2a and gap length 2d are both much less than the operating wavelength \lambda , this field can serve as the quasistatic excitation field in the gap of the dipole. However, the Fourier-Bessel series is slowly convergent. It is transformed into a rapidly convergent series of ultrasphetical polynomials whose weighting function explicitly satisfies the Meixner edge condition. Numerical results are presented graphically for both the axial electric field on the gap surface and the associated potential distribution. Gap ratios of d/a from 0.01 to 10.0 are considered and it is shown that as d/a \rightarrow 0 the solution approaches the two-dimensional solution obtainable by conformal mapping.

Patent
05 Jan 1987
TL;DR: In this article, an orbit antenna for a nuclear magnetic resonance (NMR) imaging apparatus is symmetrized by joining together two concentric conductive loops in series, and the antenna is frequency-tuned by a capacitor placed in parallel between the non-interconnected ends of the loops.
Abstract: An orbit antenna for a nuclear magnetic resonance (NMR) imaging apparatus is symmetrized by joining together two concentric conductive loops in series. The antenna is frequency-tuned by a capacitor placed in parallel between the non-interconnected ends of the loops. A receiving circuit having a high input impedance is connected between the midpoint of the loops and one of the capacitor terminals. It is thus shown that the antenna is not unbalanced and endows the received signal with all the advantages of symmetrized antennas.

Proceedings ArticleDOI
01 Oct 1987
TL;DR: In this paper, the dispersion behavior of metal-strip-loaded dielectric image guide leaky-wave antennas is analyzed and completely characterized, based on a novel multimode transverse equivalent network.
Abstract: The dispersion behavior of metal-strip-loaded dielectric image guide leaky-wave antennas is carefully analyzed and completely characterized. Many measurements have been made on this antenna type but until now no direct practical theory was available from which such antennas could be systematically designed. The theory is based on a novel multimode transverse equivalent network that is simple in form and easy to use. Finally, it is shown how the dispersion behavior and the antenna performance can be effectively controlled by properly adjusting the structural parameters.

Journal ArticleDOI
TL;DR: In this paper, the radiation impedance of a dipole printed on periodic dielectric slabs standing over a ground plane in an infinite phased array is calculated by means of modal expansions in terms of longitudinal section electric (LSE) and longitudinal section magnetic (LSM) modes.
Abstract: The radiation impedance of a dipole printed on periodic dielectric slabs standing over a ground plane in an infinite phased array is calculated. The calculation is made by means of modal expansions in terms of longitudinal section electric (LSE) and longitudinal section magnetic (LSM) modes. Derivations of these hybrid space modes and development of the modal expansions are presented first. The impedance of the dipole element is then calculated by a method based on the complex power relations with the aid of Poynting's theorem. The results calculated by the present theory in the limiting case of no dielectric slab loading have been compared with the results calculated by the method of ordinary plane wave expansions. A comparison is also made between results calculated by the theory and the experimental data obtained from waveguide simulators or test array. These results are shown to be in good agreement. An outline of steps to formulate the solution of the current distribution on dipole is also given.

DOI
01 Feb 1987
TL;DR: In this paper, the real-axis path of integration is deformed to a contour, which is the steepest-descent path for the exponential function present in the integrand.
Abstract: A novel approach is described to the numerical evaluation of Sommerfeld integrals, which offers a compromise between efficiency and simplicity. In this method, the real-axis path of integration is deformed to a contour, which is the steepest-descent path for the exponential function present in the integrand. This new contour also avoids the singularities of the integrand which occur near the original path. The proposed technique is applicable if the radial displacement of the source and field points is electrically small and is most efficient if this displacement is smaller than the vertical separation between these points. Such conditions prevail, for example, in the analysis by means of the integral equation technique of a thin, vertical wire antenna, which penetrates the interface between two media (the so-called ground stake antenna)

Patent
19 May 1987
TL;DR: In this article, the authors proposed a multi-path processing unit to prevent the transmission band width from being made broader than the signal band by radiating a modulation wave from the 1st transmission antenna, retarding the modulation wave by one time slot or over from the 2nd transmission antenna and providing a multihop processing unit extracting a main wave component included in a received wave to the receiver side.
Abstract: PURPOSE: To prevent the transmission band width from being made broader than the signal band by radiating a modulation wave from the 1st transmission antenna, radiating a delayed wave retarding the modulation wave by one time slot or over from the 2nd transmission antenna and providing a multi-path processing unit extracting a main wave component included in a received wave to the receiver side. CONSTITUTION: A transmission code series is inputted to a modulator 9, the modulation output is radiated directly from a transmission antenna 10, converted into a delay modulation wave retarded by one time slot or over (Td) by a delay circuit 11 and radiated from a transmission antenna 12. A signal wave received by a reception antenna 13 is the synthesis of the signal from the transmission antenna 10 and the signal from the transmission antenna 12, and the transmission antennas 10, 12 are located with a proper distance so that the correlation between both the waves is nearly 0 at the receiving point. The received synthesis wave is amplified by a receiver 14, detected by a detector 15 and the main wave is emphasized and extracted by a waveform equalizer 16 as a multi-path processing unit and the component other than the main wave is cancelled. COPYRIGHT: (C)1988,JPO&Japio

Journal ArticleDOI
01 Aug 1987
TL;DR: In this paper, a new microwave antenna design with greatly enhanced heating at the tip for interstitial hyperthermia was presented, and temperature distribution patterns surrounding the new balun-fed folded dipole antenna were measured in a saline phantom under transient and steady-state conditions.
Abstract: This letter presents a new microwave antenna design with greatly enhanced heating at the tip for interstitial hyperthermia. Temperature distribution patterns surrounding the new balun-fed folded dipole antenna are measured in a saline phantom under transient and steady-state conditions.

Journal ArticleDOI
TL;DR: In this article, the effect of mutual coupling on the performance of a least mean square (LMS) adaptive array using dipole elements is analyzed in consideration of the signal bandwidth, and the results show that the effect in the broad-band signal cases is much greater than that in the narrow-band cases, particularly when few antenna elements are used.
Abstract: The effect of mutual coupling on the performance of a least mean square (LMS) adaptive array using dipole elements is analyzed in consideration of the signal bandwidth. The purpose here is to illustrate quantitatively the significance of the effect of mutual coupling. The results show that the effect in the broad-band signal cases is much greater than that in the narrow-band cases, particularly when few antenna elements are used.

Journal ArticleDOI
TL;DR: In this article, a new technique of synthesis of near-field amplitude and phase patterns of linear, planar, of volume arrays of finite size or arrays located on a planar contour of a finite size is presented.
Abstract: A new technique of synthesis of near-field (NF) amplitude and phase patterns of linear, planar, of volume arrays of finite size or arrays located on a planar contour of finite size is presented The array could consist of point dipoles or directive elements The criterion for prescribing the NF (amplitude and phase) pattern information in the synthesis problem for unique determination of array excitation currents is also stated The proposed near-field synthesis technique is based on the potential integral solution of source currents, Nyquist sampling of the near-field data and the technique of linear least square approximation (LLSA) The NF pattern synthesis technique is illustrated to synthesize a variety of NF patterns with a number of array configurations Application of the proposed NF pattern synthesis technique to minimize distortion in far-field patterns of arrays mounted on a conducting platform and to realize array antennas with low sidelobes in the near and far field is also presented

Patent
28 Sep 1987
TL;DR: Close tolerance waveguide assemblies for use in antenna arrays are disclosed in this paper, in particular the waveguide assembly is configured along the zero electrical current lines in the antenna array, where each such protruding member has two formed or unformed ends.
Abstract: Close tolerance waveguide assemblies for use in antenna arrays is disclosed. In particular, the waveguide assemblies are configured along the zero electrical current lines in the antenna array. Such waveguides are particularly useful in antenna arrays for radar systems operating at the microwave and millimeter-wave frequencies. In the invention, antenna array assemblies, such as rectangular or ridged waveguide assemblies, are configured by combining plates which are formed into a plurality of equal length members protruding from and perpendicularly disposed to a structural member, wherein each such protruding member has two formed or unformed ends.

Journal ArticleDOI
TL;DR: In this paper, a general method for exact integration of vector potentials of thin dipole antennas characterized by various current distributions is developed, independent of the usual farfield restrictions involving dipole length, observation point distance, and wavelength.
Abstract: A general method for exact integration of vector potentials of thin dipole antennas characterized by various current distributions is developed. Such solutions are shown to be independent of the usual farfield restrictions involving dipole length, observation point distance, and wavelength. Their convergence is rapid in the induction and near-field regions.