Theory of exponentially tapered RC transmission lines for phase-shift oscillators

Tapered RC transmission lines are useful in phase-shift oscillators, because they permit (i) a great reduction in the required gain of the active device (vacuum tube or transistor) and (ii) generation of frequencies up to a few megacycles per second. They are particularly useful for microminiature oscillator circuits because of their small size and the possibility of integration with the active device, which is a transistor. In the paper, a general, exponentially tapered RC line is analysed to find expressions for the current and voltage at any point of the line. These are used to deduce an equation giving the frequency of phase reversal and an expression for the attenuation at this frequency, for the possible combinations of generator and load impedances that are of practical interest. In general, the frequency-determining equation is transcendental and highly involved. An approximation is developed in the paper for small-taper parameters; using this approximation, it has been shown that, for a particular frequency, there exists an optimum combination of load impedance and taper parameter for which the attenuation is a minimum. Each special case is illustrated by a numerical example, and the results are verified by experiment.