The various representations are shown in Figure 10.37. A switching operation changes the topology of the network and hence the [Y] matrix. If the [Y] matrix is formed with all the switches open, then the closure of a switch is obtained by the adding together of the two rows and columns of [Y] and the associated rows of [i].
Figure 10.36 The equivalent circuit of Figure 10.35
Another area where switching is used is to account for non-linear C — i characteristics of transformers, reactors, and so on. The representation is shown in Figure 10.38(a) and (b), in which
Figure 10.37 Representation of switch
where bt = incremental inductance.
If C is outside of the limits of segment K, the operation is switched to either k — 1 or k + 1. This changes the [Y] matrix.
Because of the random nature of certain events, for example switching time or lightning incidence, Monte Carlo (statistical) methods are sometimes used. Further information can be obtained from the references at the end of this book.
Lines and cables would require a large number of n circuits for accurate representation. An alternative would be the use of the travelling-wave theory.
Consider Figure 10.39,
where U = speed of propagation and Z0 = characteristic impedance. At node k, where
Figure 10.40 Equivalent circuit for single-phase line
The equivalent circuit is shown in Figure 10.40. An advantage of this method is that the two ends of the line are decoupled. The value of Ik depends on the current and voltage at the other end of the line t seconds previously, for example if t = 0.36 ms and At = 100 ms, storage of four previous times is required.
Detailed models for synchronous machines and h.v.d.c. converter systems are given in the references.