Equivalent Impedance Transforms - 2-terminal, n-element, 3-element-kind Networks

2-terminal, n-element, 3-element-kind Networks

Simple networks with just a few elements can be dealt with by formulating the network equations "by hand" with the application of simple network theorems such as Kirchhoff's laws. Equivalence is proved between two networks by directly comparing the two sets of equations and equating coefficients. For large networks more powerful techniques are required. A common approach is to start by expressing the network of impedances as a matrix. This approach is only good for rational networks. Any network that includes distributed elements, such as a transmission line, cannot be represented by a finite matrix. Generally, an n-mesh network requires an nxn matrix to represent it. For instance the matrix for a 3-mesh network might look like;

The entries of the matrix are chosen so that the matrix forms a system of linear equations in the mesh voltages and currents (as defined for mesh analysis);

The example diagram in Figure 1, for instance, can be represented as an impedance matrix by;

and the associated system of linear equations are,

In the most general case, each branch, Z_p, of the network may be made up of three elements so that,

where L, R and C represent inductance, resistance, and capacitance respectively and s is the complex frequency operator .

This is the conventional way of representing a general impedance but for the purposes of this article it is mathematically more convenient to deal with elastance, D, the inverse of capacitance, C. In those terms the general branch impedance can be represented by,

Likewise, each entry of the impedance matrix can consist of the sum of three elements. Consequently, the matrix can be decomposed into three nxn matrices, one for each of the three element kinds;

It is desired that the matrix represent an impedance, Z(s). For this purpose, the loop of one of the meshes is cut and Z(s) is the impedance measured between the points so cut. It is conventional to assume the external connection port is in mesh 1, and is therefore connected across matrix entry Z₁₁, although it would be perfectly possible to formulate this with connections to any desired nodes. In the following discussion Z(s) taken across Z₁₁ is assumed. Z(s) may be calculated from by;

where z₁₁ is the complement of Z₁₁ and |Z| is the determinant of .

For the example network above;

and,

This result is easily verified to be correct by the more direct method of resistors in series and parallel. However, such methods rapidly become tedious and cumbersome with the growth of the size and complexity of the network under analysis.

The entries of, and cannot be set arbitrarily. For to be able to realise the impedance Z(s) then, and must all be positive-definite matrices. Even then, the realisation of Z(s) will, in general, contain ideal transformers within the network. Finding only those transforms that do not require mutual inductances or ideal transformers is a more difficult task. Similarly, if starting from the "other end" and specifying an expression for Z(s), this again cannot be done arbitrarily. To be realisable as a rational impedance, Z(s) must be positive-real. The positive-real (PR) condition is both necessary and sufficient but there may be practical reasons for rejecting some topologies.

A general impedance transform for finding equivalent rational one-ports from a given instance of is due to Wilhelm Cauer. The group of real affine transformations,

where,

is invariant in Z(s). That is, all the transformed networks are equivalents according to the definition given here. If the Z(s) for the initial given matrix is realisable, that is, it meets the PR condition, then all the transformed networks produced by this transformation will also meet the PR condition.