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Imperfections in fractally scaled transmission lines

Models of fractally scaled, discrete-component, transmission lines consisting of resistors and capacitors have been investigated experimentally and computationally over extended frequency ranges. Agreement between the predicted performances of the lines and those measured has been found. In particular, a fractional power-law dispersion with frequency, which is a characteristic of scaling, has been observed in the form (, where p is defined by a fixed ratio of the logarithms of the scaling parameters. The effect of localized imperfections on the scaling has been examined by introducing errors into the scaling of the lines and determining the standard deviation of the exponent resulting from these defects. Both for increases and for decreases in magnitude, either of the resistive or of the capacitive elements, the limiting behaviour has been found to be characterized by a square-root dependence of the standard deviation introduced by the imperfection on the magnitude of the relative change in the component from the scaled value. The errors introduced into the dispersions are discussed in terms of the commonly observed fractional power-law response with frequency and the significance of variations from perfect scaling behaviour is discussed.