Restrictions on Linear Phase, Low-Pass Networks with Prescribed Parasitic Loads

In this paper integral restrictions on the time delay of linear phase, low-pass matching networks are derived. These networks operate between a resistive generator and a load involving parasitic elements. The restrictions relate the minimum group delay, size of the parasitic elements, and order of the matching network (number of elements of the system) with the level of the overall gain. This paper is limited to the case in which the load has a zero of transmission at infinity. The restrictions involve the Appell integral. In the case of unit gain at dc, this integral becomes the Riemann Zeta function. The restrictions are derived by assuming a Gaussian amplitude characteristic. The synthesis procedure is based on the fact that the Gaussian amplitudes are asymptotic forms of Bessel polynomials which were investigated by Thomson and Storch.