Optimal Taper Research Articles

Velocity tapering in microwave amplifiers

The low-to-moderate RF conversion efficiencies characteristic of kinetic energy conversion devices may be enhanced by appropriately tapering the RF circuit phase velocity. The nonlinear interaction equations have been solved in closed form for an HKB (hard-kernel-bunch) model to obtain an optimum taper form. Several velocity taper forms have been investigated in experimental amplifiers and the power output, gain, efficiency, and phase shift characteristics of these amplifiers are discussed in detail. Conversion efficiencies up to 50 percent and near constant output power for a 20-dB variation in RF drive power have been realized in near optimally tapered circuit amplifiers. Further improvements realized include operation at a single voltage for maximum gain and power output and a nearly linear phase shift characteristic for a ±10 percent voltage modulation.

The transient response of tapered transmission lines

Tapered transmission lines are frequently used in communications engineering to transform voltages and impedances. Much has been done in the steady-state analysis <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> ‾ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">12</sup> of these devices. The need remains, however, for a theory of the optimum taper based upon transient considerations because of intrinsic limitations of any steady-state theory. Steady-state treatments are sufficient for a-c applications, but are difficult to apply meaningfully to transient phenomena. Very little has been done in the realm of tapered-transmission line transient analysis. An analysis for exponential tapers has appeared in the literature, <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">13</sup> ‾ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">16</sup> but there has been no attempt to find the optimum taper from transient considerations.

The optimum transmission-line pulse transformer

Tapered transmission lines have been found to be useful for pulse-transformer service.1,2 It is the object of this paper to develop a new theory for the optimization of the taper which is based upon the transient rather than the steady-state response of the transmission-line pulse transformer. A criterion for the optimum taper will be established and then applied in a special case in which both ends of the line are nominally matched. Finally, the case of general terminations at both the input and output will be discussed.

The Optimum Tapered Transmission Line Matching Section

The tapered transmission line matching section is analyzed by treating it as a high pass filter. The ideal reflection coefficient characteristic for the taper that gives the smallest pass band tolerance for a given cutoff frequency and vice versa is derived from the expression for the reflection coefficient of the optimum designed n section quarter-wave transformer by taking the limit as n tends to infinity. By neglecting the square of the reflection coefficient in the differential equation for the reflection coefficient on the taper, a synthesis procedure is derived for obtaining an optimum taper. The procedure is similar to that used in designing an optimum line source distribution in antenna theory. Conditions are derived for the maximum allowable pass band reflection and change in impedance level for which the theory remains accurate. For a maimum reflection coefficient of 0.1 in the pass band the theory remains accurate for all frequencies above the cutoff value provided the change in impedance level does not exceed 7.5. This optimum taper is compared with the well-known exponential and Gaussian tapers and is found to be 13.9 per cent and 27 per cent shorter respectively for the same cutoff frequency and pass band tolerance.

A Transmission Line Taper of Improved Design

The theory of the design of optimal cascaded transformer arrangements can be extended to the design of continuous transmission-line tapers. Convenient relationships have been obtained from which the characteristic impedance contour for an optimal transmission-line taper can be found. The performance of the Dolph-Tchebycheff transmission-line taper treated here is optimum in the sense that it has minim reflection coefficient magnitude in the pass band for a specified length of taper, and, likewise, for a specified maximum magnitude reflection coefficient in the pass band, the Dolph-Tchebycheff taper has minimum length. A sample design has been carried out for the purposes of illustration, and its performance has been compared with that of other tapers. In addition, a table of values of a transcendental function used in the design of these tapers is given.

On the optimum illumination taper for the objective of a microwave aerial

On the optimum illumination taper for the objective of a microwave aerial

On the optimum illumination taper for the objective of a microwave aerial

The Fourier-transform description of the radiation pattern from an aperture is used to obtain curves relating the overall gain of an idealized aerial system, comprising an objective and waveguide flare, with the size and illumination of the aperture of the latter. For rectangular objectives the dimensions of the aperture of the flare may be varied independently in the two planes of symmetry and the overall gain obtained by multiplying together the gain factors for the distributions across the objective in these two planes. In the case of the circular objective this no longer applies, but curves are drawn showing the variation of the gain of the system as the size of a symmetrical feed is varied by equal amounts in the two planes of symmetry. It is shown that maximum overall gain is obtained with a rectangular objective when the size of the feed is adjusted to give a primary pattern such that the illumination intensity at the edges of the objective is 8½ dB below that at the centre. The corresponding optimum taper for circular objectives is 11½ dB.Primary radiation-patterns are drawn for “constant,” “cosine” and “double-cosine” illuminations of the feed aperture. The variations of secondary beam-widths and first side-lobe levels with feed size for rectangular and circular objectives are discussed.

Impedance matching by tapered transmission lines

Expressions are found for the impedance deviations with change of wavelength at the input of a tapered section of transmission line, when the far end of the tapered section is joined to a coaxial line, which in turn is terminated by its characteristic impedance. It is found that the added resistance and reactance are least when the length of the tapered section is approximately an integral multiple of half wavelengths. If both the outer and inner conductors are tapered, there is an optimum taper which will give unity standing-wave ratio (s.w.r.) for half-wavelength sections, and a s.w.r. only slightly different from unity for other lengths. If it is desired to keep the diameter of one conductor the same on each side of the junction, the best method is to taper both conductors in opposite ways for a half-wavelength and then the same way. This produces either a bulge on the inner conductor or a constriction in the outer conductor.

Impedance matching by tapered transmission lines

Expressions are found for the impedance deviations with change of wavelength at the input of a tapered section of transmission line, when the far end of the tapered section is joined to a coaxial line, which in turn is terminated by its characteristic impedance. It is found that the added resistance and reactance are least when the length of the tapered section is approximately an integral multiple of half wavelengths. If both the outer and inner conductors are tapered, there is an optimum taper which will give unity standing-wave ratio (s.w.r.) for half-wavelength sections, and a s.w.r. only slightly different from unity for other lengths. If it is desired to keep the diameter of one conductor the same on each side of the junction, the best method is to taper both conductors in opposite ways for a half-wavelength and then the same way. This produces either a bulge on the inner conductor or a constriction in the outer conductor.