Network Synthesis Filters - Driving Point Impedance

Driving Point Impedance

The driving point impedance is a mathematical representation of the input impedance of a filter in the frequency domain using one of a number of notations such as Laplace transform (s-domain) or Fourier transform (jω-domain). Treating it as a one-port network, the expression is expanded using continued fraction or partial fraction expansions. The resulting expansion is transformed into a network (usually a ladder network) of electrical elements. Taking an output from the end of this network, so realised, will transform it into a two-port network filter with the desired transfer function.

Not every possible mathematical function for driving point impedance can be realised using real electrical components. Wilhelm Cauer (following on from R. M. Foster) did much of the early work on what mathematical functions could be realised and in which filter topologies. The ubiquitous ladder topology of filter design is named after Cauer.

There are a number of canonical forms of driving point impedance that can be used to express all (except the simplest) realisable impedances. The most well known ones are;

• Cauer's first form of driving point impedance consists of a ladder of shunt capacitors and series inductors and is most useful for low-pass filters.
• Cauer's second form of driving point impedance consists of a ladder of series capacitors and shunt inductors and is most useful for high-pass filters.
• Foster's first form of driving point impedance consists of parallel connected LC resonators and is most useful for band-pass filters.
• Foster's second form of driving point impedance consists of series connected LC anti-resonators and is most useful for band-stop filters.

Further theoretical work on realizable filters in terms of a given rational function as transfer function was done by Otto Brune in 1931 and Richard Duffin with Raoul Bott in 1949. The work was summarized in 2010 by John H. Hubbard.