Determination of System Voltages Produced by Travelling Surges
In the previous section the basic laws of surge behaviour were discussed. The calculation of the voltages set up at any node or busbar in a system at a given instant in time is, however, much more complex than the previous section would suggest. When any surge reaches a discontinuity its reflected waves travel back and are, in turn, reflected so that each generation of waves sets up further waves which coexist with them in the system.
To describe completely the events at any node entails, therefore, an involved book-keeping exercise. Although many mathematical techniques are available and, in fact, used, the graphical method due to Bewley (1961) indicates clearly the physical changes occurring in time, and this method will be explained in some detail.
Bewley Lattice Diagram
This is a graphical method of determining the voltages at any point in a transmission system and is an effective way of illustrating the multiple reflections which take place. Two axes are established: a horizontal one scaled in distance along the system, and a vertical one scaled in time. Lines indicating the passage of surges are drawn such that their slopes give the times corresponding to distances travelled. At each point of change in impedance the reflected and transmitted waves are obtained by multiplying the incidence-wave magnitude by the appropriate reflection and refraction coefficients a and ft. The method is best illustrated by Example 10.3.
A loss-free system comprising a long overhead line (Zj) in series with a cable (Z2) will be considered. Typically, Zj is 500 V and Z2 is 50 V.
Referring to Figure 10.25, the following coefficients apply:
Line-to-cable reflection coefficient,
Figure 10.25 Bewley lattice diagram-analysis of long overhead line and cable in series, (a) Position of voltage surges at various instants over first complete cycle of events, that is up to second reflected wave travelling back along line, (b) Lattice diagram
Line-to-cable refraction coefficient, Cable-to-line reflection coefficient,
Cable-to-line refraction coefficient,
As the line is long, reflections at its sending end will be neglected. The remote end of the cable is considered to be open-circuited, giving an a of 1 and a b of zero at that point.
When the incident wave v, (see Figure 10.25) originating in the line reaches the junction, a reflected component travels back along the line (ajv;), and the refracted or transmitted wave (biv,-) traverses the cable and is reflected from the open- circuited end back to the junction (1 x b1v,). This wave then produces a reflected wave back through the cable (1 x b1a2vi) and a transmitted wave (1 x b1b2v,) through the line. The process continues and the waves multiply as indicated in Figure 10.25(b). The total voltage at a point P in the cable at a given time (t) will be the sum of the voltages at P up to time t, that is b1vi(2 + 2a2 + 2a2), and the voltage at infinite time will be 2b1vi(1 + a2 + a2 + a2 + a|+ ....).
The voltages at other points are similarly obtained. The time scale may be determined from a knowledge of length and surge velocity; for the line the latter is of the order of 300 m/ms and for the cable 150 m/ms. For a surge 50 ms in duration and a cable 300 m in length there will be 25 cable lengths traversed and the terminal voltage will approach 2v,. If the graph of voltage at the cable open-circuited end is plotted against time, an exponential rise curve will be obtained similar to that obtained for a capacitor.
The above treatment applies to a rectangular surge waveform, but may be modified readily to account for a waveform of the type illustrated in Figure 10.1. In this case the voltage change with time must also be allowed for and the process is more complicated.