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

Methods for realization of an immittance whose argument is nearly constant at < 1, over an extended frequency range, are discussed. In terms of the generalized complex frequency variable , these immittances are proportional to , and as such they are approximations of Riemann-Louville fractional operators. First, we present a method which is applicable only for the special case . This is based on the continued fraction expansion (CFE) of the irrational driving-point function of a uniform distributed RC (U ) 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 between -1 and +1. One is based on the CFE of ; the two signs result in two different circuits which approximate 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 versus the number of elements, where and , 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 networks which can approximate 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.