This dissertation is concerned with the synthesis of linear passive electrical and mechanical networks. The main objective is to gain a better understanding of minimal realisations within the simplest non-trivial class of networks of restricted complexity--the networks of the so-called "Ladenheim catalogue"--and thence establish more general results in the field of passive network synthesis. Practical motivation for this work stems from the recent invention of the inerter mechanical device, which completes the analogy between electrical and mechanical networks.

A full derivation of the Ladenheim catalogue is first presented, i.e. the set of all electrical networks with at most two energy storage elements (inductors or capacitors) and at most three resistors. Formal classification tools are introduced, which greatly simplify the task of analysing the networks in the catalogue and help make the procedure as systematic as possible.

Realisability conditions are thus derived for all the networks in the catalogue, i.e. a rigorous characterisation of the behaviours which are physically realisable by such networks. This allows the structure within the catalogue to be revealed and a number of outstanding questions to be settled, e.g. regarding the network equivalences which exist within the catalogue. A new definition of "generic" network is introduced, that is a network which fully exploits the degrees of freedom offered by the number of elements in the network itself. It is then formally proven that all the networks in the Ladenheim catalogue are generic, and that they form the complete set of generic electrical networks with at most two energy storage elements.

Finally, a necessary and sufficient condition is provided to efficiently test the genericity of any given network, and it is further shown that any positive-real function can be realised by a generic network.