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GT ECE 6390 - Space and Line Current Radiation

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Lecture Notes ANT2: Space and Line Current RadiationIn this lecture, we study the general case of radiation from z-directed spatial currents. The far-field radiation equations that result from this treatment form some of the foundational principles offield radiation equations that result from this treatment form some of the foundational principles of all antenna engineering. In fact, after this lecture, a student should be able to look at most types of antennas and, regardless of type or construction specifics, be able to infer the basic radiation pattern from the size and shape.In the later section of the talk, we simplify the analysis to include the special (but very important) case of the general wire antenna. Concentrating on results for the half-wave dipole, we demonstrate how a radiator more realistic than the ideal Hertzian dipole operates. We close with ademonstrate how a radiator more realistic than the ideal Hertzian dipole operates. We close with a thorough summary of the most common types of wire antennas and their radiation and electrical parameters.Page 1Lecture Notes ANT2: Space and Line Current RadiationWe start with the z-component of the vector magnetic potential’s Greens formula, which was derived in ANT2 This timeharmonic equation relates a zcomponent of current density (Jz) toderived in ANT2. This time-harmonic equation relates a z-component of current density (Jz) to the z-component of Az. Note that in a general relationship, we would have 3 separable equations –one for z-components, one for y-components, and one for z-components. This divide-and-conquer approach is what makes the vector-magnetic-potential method much more straight-forward than other techniques for solving radiation problems.Note that the integration occurs over 3dimensions (x’ y’ z’) which are not to be confused with theNote that the integration occurs over 3-dimensions (x’,y’,z’), which are not to be confused with the point of observation (x,y,z). The integral is sliding around the mass of z-directed current, picking up the radiative contributions of each amplitude and phase of infinitessimal current elements. In this way, we view the spatial current distribution as simply the superposition of numerous Hertzian dipole elements.Page 2Lecture Notes ANT2: Space and Line Current RadiationNow we can simplify this integral if we assume that the point of observation is a significant distance from the spatial distribution of current which should be roughly centered in the origin ofdistance from the spatial distribution of current, which should be roughly centered in the origin of our problem. Thus, for large r, we can make some simplifications to the integral in the previous page. Exactly what constitutes a large r will become evident after we make the approximations.For amplitude terms, the magnitude of the difference between observation and integration vectors can be approximated as the distance from observation to the origin – or simply r. Amplitude terms in general tend to be insensitive to slight variations in this termin general tend to be insensitive to slight variations in this term.Phase terms, however, are much more sensitive to approximation, largely because it’s the modulus-2pi of the distance that contributes to a phase term – not the absolute, cumulative value of the observation distance. Making the same r approximation that we did in amplitude would be catastrophic, erasing all of the proper phase behavior that is critical to synthesizing a radiation tt I t d k th i ti th t d f i t th tpattern. Instead, we make the approximation that rays drawn from any point on the current distribution to the point of observation are parallel. This is true as long as the point of observation is greater than D^2/. Thus, when we talk about current distributions, there are actually two conditions that we need to define the far field. First, we must be greater than 1-wavelength away from the antenna because we used simplified far-field expressions in our superposition formula. But we also need to make sure that the observation distance is D^2/which can be a much more stringent condition, particularly for electrically large antennas such as satellite dish antennas.Page 3gpyygLecture Notes ANT2: Space and Line Current RadiationHere is the reduced formula for vector magnetic potential once all of the far-field approximations are inserted Here we have also expanded out the unit vectorr that points toward the observer inare inserted. Here we have also expanded out the unit vector-r that points toward the observer in terms of azimuth and elevation angles. When this is inserted into the expanded 3D integral, one of the more remarkable principles of antenna engineering becomes evident.The radiation pattern of an antenna is effectively the Fourier transform of the spatial distribution of currents. As such, the pattern follows all of basic rules of a 3D Fourier Transform. Larger current distributions tend to result in smaller patterns (smaller half-power-beamwidths). Smaller current distributions tend to result in broader patterns (very small radiators are always near-distributions tend to result in broader patterns (very small radiators are always nearomnidirectional in their patterns). Expanding the dimension of the current distribution in only one direction of space will only reduce the radiation pattern width in the corresponding plane. These basic principles allow antenna engineers to shape their radiation patterns by squeezing and stretching their radiative element in various dimensions.One further consequence of this relationship: it is impossible to design an antenna pattern with an extended null region without resorting to an infinite current distribution in space. Why? In a Fourier relationship, a function in one domain with finite support (a non-singular region over which a function has value of zero) *must* have infinite support in its transformed domain (non-zero across the entire domain excepting singular points). This is a mathematical property, and since the physics follows this principle, a realistic antenna with finite support of Jz in the space domain *necessarily* has infinite support in the pattern domain. Synthesizing near-zero backlobes under these conditions is one of the most common and challenging tasks in antenna engineering.Page 4Lecture Notes ANT2: Space and Line Current RadiationNow, once we have made the calculation for


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