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

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 variable
s
, these immittances are proportional to
s^{\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
\lambda
between -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 approximate
s^{-\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_1
versus the number of elements, where
\omega_2
and
\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 approximate
s_{\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.