Passive Analogue Filter Development - Realisability and Equivalence

Realisability and Equivalence

Realisability (that is, which functions are realisable as real impedance networks) and equivalence (which networks equivalently have the same function) are two important questions in network synthesis. Following an analogy with Lagrangian mechanics, Cauer formed the matrix equation,

where ,, and are the nxn matrices of, respectively, impedance, resistance, inductance and elastance of an n-mesh network and s is the complex frequency operator . Here, and have associated energies corresponding to the kinetic, potential and dissipative heat energies, respectively, in a mechanical system and the already known results from mechanics could be applied here. Cauer determined the driving point impedance by the method of Lagrange multipliers;

where a₁₁ is the complement of the element A₁₁ to which the one-port is to be connected. From stability theory Cauer found that, and must all be positive-definite matrices for Z_p(s) to be realisable if ideal transformers are not excluded. Realisability is only otherwise restricted by practical limitations on topology. This work is also partly due to Otto Brune (1931), who worked with Cauer in the US prior to Cauer returning to Germany. A well known condition for realisability of a one-port rational impedance due to Cauer (1929) is that it must be a function of s that is analytic in the right halfplane (σ>0), have a positive real part in the right halfplane and take on real values on the real axis. This follows from the Poisson integral representation of these functions. Brune coined the term positive-real for this class of function and proved that it was a necessary and sufficient condition (Cauer had only proved it to be necessary) and they extended the work to LC multiports. A theorem due to Sidney Darlington states that any positive-real function Z(s) can be realised as a lossless two-port terminated in a positive resistor R. No resistors within the network are necessary to realise the specified response.

As for equivalence, Cauer found that the group of real affine transformations,

where,

is invariant in Z_p(s), that is, all the transformed networks are equivalents of the original.