This thesis is concerned with the synthesis of passive networks, motivated by the recent invention of a new mechanical component, the inerter, which establishes a direct analogy between mechanical and electrical networks. We investigate the minimum numbers of inductors, capacitors and resistors required to synthesise a given impedance, with a particular focus on transformerless network synthesis. The conclusions of this thesis are relevant to the design of compact and cost-effective mechanical and electrical networks for a broad range of applications.

In Part 1, we unify the Laplace-domain and phasor approach to the analysis of transformerless networks, using the framework of the behavioural approach. We show that the autonomous part of any driving-point trajectory of a transformerless network decays to zero as time passes. We then consider the trajectories of a transformerless network, which describe the permissible currents and voltages in the elements and at the driving-point terminals. We show that the autonomous part of any trajectory of a transformerless network is bounded into the future, but need not decay to zero. We then show that the value of the network's impedance at a particular point in the closed right half plane can be determined by finding a special type of network trajectory.

In Part 2, we establish lower bounds on the numbers of inductors and capacitors required to realise a given impedance. These lower bounds are expressed in terms of the extended Cauchy index for the impedance, a property defined in that part. Explicit algebraic conditions are also stated in terms of a Sylvester and a Bezoutian matrix. The lower bounds are generalised to multi-port networks. Also, a connection is established with continued fraction expansions, with implications for network synthesis.

In Part 3, we first present four procedures for the realisation of a general impedance with a transformerless network. These include two known procedures, the Bott-Duffin procedure and the Reza-Pantell-Fialkow-Gerst simplification, and two new procedures. We then show that the networks produced by the Bott-Duffin procedure, and one of our new alternatives, contain the least possible number of reactive elements (inductors and capacitors) and resistors, for the realisation of a certain type of impedance (called a biquadratic minimum function), among all series-parallel networks. Moreover, we show that these procedures produce the only series-parallel networks which contain exactly six reactive elements and two resistors and realise a biquadratic minimum function. We further show that the networks produced by the Reza-Pantell-Fialkow-Gerst simplification, and the second of our new alternatives, contain the least possible number of reactive elements and resistors for the realisation of almost all biquadratic minimum functions among the class of transformerless networks. We group the networks obtained by these two procedures into two quartets, and we show that these are the only quartets of transformerless networks which contain exactly five reactive elements and two resistors and realise all of the biquadratic minimum functions. Finally, we investigate the minimum number of reactive elements required to realise certain impedances, of greater complexity than the biquadratic minimum function, with series-parallel networks.