Useful Network Theory
A lumped-constant circuit, provided that it is passive, linear and bilateral, can be represented by the four-terminal network shown in the diagram in Figure 2.27. The complex parameters A, B, C and D describe the network in terms of the sending- and receiving-end voltages and currents as follows:
and it can be readily shown that AD—BC = 1.
Figure 2.27 Representation of a four-terminal (two-port) network
A, B, C, and D may be obtained by measurement and certain physical interpretations can be made, as follows:
1. Receiving-end short-circuited:
2. Receiving-end open-circuited:
Expressions for the constants can be found by complex (that is, magnitude and angle) measurements carried out solely at the sending end with the receiving end open and short-circuited.
Often it is useful to have a single four-terminal network for two or more items in series or parallel, for example, a line and two transformers in series.
It is shown in most texts on circuit theory that the generalized constants for the combined network, A0, B0, C0, and D0 for the two networks (1) and (2) in cascade are as follows:
For two four-terminal networks in parallel it can be shown that the parameters of the equivalent single four-terminal network are:
The delta network connected between the three terminals A, B and C of Figure 2.28 can be replaced by a star network such that the impedance measured between the terminals is unchanged. The equivalent impedances can be found as follows:
Impedance between terminals AB with C open-circuited is given by Similarly and
From these three equations ZOA, ZOB and ZOc, the three unknowns, can be determined as
A star-connected system can be replaced by an equivalent delta connection if the elements of the new network have the following values (Figure 2.28):
Figure 2.28 Star-delta, delta-star transformation