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

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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.

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Circuit Theory, IEEE Transactions on  (Volume:14 ,  Issue: 3 )