On the Realization of a Constant-Argument Immittance or Fractional Operator

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Abstract
Methods for realization of an immittance whose argument is nearly constant at\lambda \pi/2, |\lambda|< 1, over an extended frequency range, are discussed. In terms of the generalized complex frequency variables, these immittances are proportional tos^{\lambda}, and as such they are approximations of Riemann-Louville fractional operators. First, we present a method which is applicable only for the special case|\lambda| = \frac{1}{2}. This is based on the continued fraction expansion (CFE) of the irrational driving-point function of a uniform distributed RC (U\overline{RC}) network; the results are compared with those of earlier workers using lattice networks and rational function approximations. Next we discuss two methods applicable for any value of\lambdabetween -1 and +1. One is based on the CFE of(1 + s^{\pm 1})\pm\lambda; the two signs result in two different circuits which approximates^{-\lambda}at low and high frequencies, respectively. The other method uses elliptic functions and results in an equiripple approximation of the constant-argument characteristic. In each method, the extent of approximation obtained by using a certain number of elements is determined by use of a digital computer. The results are given in the form of curves of\omega_2/ \omega_1versus the number of elements, where\omega_2and\omega_1, denote the upper and lower ends, respectively, of the frequency band over which the argument is constant to within a certain tolerance. From the lumped element networks, we derive some\overline{RC}networks which can approximates_{\lambda}more effectively than the lumped networks. The distributed structures can be fabricated in microminiature form using thin-film techniques, and should be more attractive from considerations of cost, size, and reliability.

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