Realization of Variable Active Networks

The synthesis of variable active networks is investigated. The networks studied have a frequency or time response that is a function of an independent variable, which is assumed to vary sufficiently slowly with time that the systems can be considered to be time-invariant. The only variable elements considered are controlled sources. The sufficiency of fixed passive and variable active elements is demonstrated for transfer functions that can be realized as transfer functions of lumped passive networks (for any fixed value of the variable parameter). These functions are assumed to be rational insfor fixedx, the independent variable, but arbitrary functions ofx. It is shown that the fixed passive network need only be an RC network with, at most,m + 3ports, wheremis the degree of the denominator polynomial ins. The only other requirement of the RC network is that the numerators of the transfer impedances from the input port to the remainingm + 2ports be linearly independent polynomials ins. In addition, it is shown that the number of controlled sources can be reduced from2m + 2in a variety of situations by a particular choice of the RC network.