Regular positive-real functions and passive networks comprising two reactive elements

The purpose of this paper is to formalise the concept of a regular positive-real function and to demonstrate its usefulness in the classification of low-complexity two-terminal networks. We will develop a series of lemmas characterising the basic properties of regularity. We will give a simple proof that all five-element series-parallel networks with two reactive elements are regular, and we will show that there are only two bridge-networks with two reactive elements which may be non-regular. We will discuss the class of two-terminal networks with two reactive elements and an arbitrary number of resistive elements and highlight a theorem of Reichert. The results are motivated by an approach to passive mechanical control which makes use of the inerter device.