مرکز منطقه ای اطلاع رساني علوم و فناوري - On the Realization of a Constant-Argument Immittance or Fractional Operator

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 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.