The Exponential Transmission Line*

Abstract

The theory of the exponential transmission line is developed. It is found to be a high pass, impedance transforming filter. The cutoff frequency depends upon the rate of taper.The deviation of the exponential line from an ideal impedance transformer may be decreased by an order of magnitude by shunting the low impedance end with an inductance and inserting a capacitance in series with the high impedance end. The magnitudes of these reactances are equal to the impedance level at their respective ends of the line at the cutoff frequency.For a two-to-one impedance transformer the line is 0.0551 wavelengths long at the cutoff frequency. For a four-to-one impedance transformer the line is 0.1102 wave-lengths long at the cutoff frequency, etc. , The results have been verified experimentally. Practical lines 50 meters and 15 meters long have been constructed which transform from 600 to 300 ohms over the frequency range from 4 to 30 me. with deviations from the ideal that are small compared with the deviations from the ideal of commercial transmission lines, either two-wire or concentric.When an exponential line is used as a dissipative load of known impedance instead of a uniform line it is possible to approach more nearly the ideal of constant heat dissipation per unit length. This makes it possible to use a shorter line.T H E exponential line may be defined as an ordinary transmission line in which the spacing between the conductors (or conductor size) is not constant but varies in such a way that the distributed inductance and capacitance vary exponentially with the distance along the line. That is, the impedance ratio for two points a fixed distance apart IS independent of the position of these two points along the line. A disturbance is propagated down an exponential transmission line in the same manner as it would be down a uniform line with the additional effect that the voltage is increased by the square root of the change in impedance level and the current is decreased by the reciprocal of this quantity.The exponential line has the properties of a high pass impedance transforming filter. The cutoff frequency depends upon the rate of • Presented before joint meeting of U.

Adaptive transient solution of nonuniform multiconductor transmission lines using wavelets

“…In order to quantify the degree of adaptivity and its dependence on the cutoff threshold , we performed a study on a nonuniform line whose solution is known analytically, the exponential line [4]. This line is characterized by exponentially tapered per-unit-length inductance and capacitance where the parameter controls the rate of taper and are the nominal per unit length inductance and capacitance at the…”

Tapered radio frequency transmission lines

“…The earliest work found in the literature is the classic article by Burrows, of Bell Labs (1938) [8]. His study includes the analysis and conclusions presented here.…”

Magnetic tuning characteristics of an He–20Ne laser in a transverse magnetic field

“…The resulting circuit is, however, difficult to analyse because it incorporates a non-uniform transmission line. Although an extensive list of references is available on the analysis of lossless tapered lines (Kaufman 1955), exact solutions have been obtained in some special cases only (Burrows 1938, Wheeler 1939, Sugai 1960, 1961. The non-uniform R C line presents a more difficult task because the differential equation for voltage (as also for current) involves an imaginary term.…”