Propagation of Surges
The basic differential equations for voltage and current in a distributed constant line are as follows:
and these equations represent travelling waves. The solution for the voltage may be expressed in the form,
that is one wave travels in the positive direction of x and the other in the negative. Also, it may be shown that because dv/dx = —L(di/dt), the solution for current is
In more physical terms, if a voltage is injected into a line (Figure 10.16) a corresponding current i will flow, and if conditions over a length dx are considered, the flux set up between the go and return wires,
where L is the inductance per unit length.
As the induced back e.m.f., —dF/dt, is equal to the applied voltage v
where U is the wave velocity
Also, charge is stored in the capacitance over dx, that is.
From (10.9) and (10.10),
Substituting for U in (10.10),
where Z0 is the characteristic or surge impedance.
For single-circuit three-phase overhead lines (conductors not bundled) Z0 lies in the range 400-600 V. For overhead lines, U = 3 x 108m/s, that is the speed of light, and for cables
where er is usually from 3 to 3.5, and = 1.
From equations (10.9) and (10.10),
The incident travelling waves of v,- and ii, when they arrive at a junction or discontinuity, produce a reflected current ir and a reflected voltage vr which travel
Figure 10.17 Application of voltage to unenergized loss-free line on open circuit at far end. (a) Distribution of voltage, (b) Distribution of current. Voltage source is an effective short circuit
back along the line. The incident and reflected components of voltage and current are governed by the surge impedance Z0, so that
In the general case of a line of surge impedance Z0 terminated in Z (Figure 10.17), the total voltage at Z is v = v; + vr and the total current is i = i; + ir.
where a is the reflection coefficient equal to ——=° )
M V— + — oj
From (10.12) and (10.11), the total voltage and current is given by:
From equations (10.13) and (10.14), if Z ! 1, v = 2v, and i = 0. This is shown in Figure 10.17. As shown in the first plot, a surge of vi travels towards the open circuit end of the line. This creates a current surge of i = Vi/—0. When this surge reaches the open circuit end, a reflected voltage surge of Vi and a current surge of —i are created and they start travelling back down the line. Therefore the total voltage surge is now 2vi and the current is zero as shown in the second plot. At the voltage source, an effective short circuit, a voltage wave of — v, is reflected down the line. The reflected waves will travel back and forth along the line, setting up, in turn, further reflected waves at the ends, and this process will continue indefinitely unless the waves are attenuated because of resistance and corona.
From equations (10.13) and (10.14), if Z = Z0 (matched line), a = 0, that is there is no reflection. If Z > Z0, then vr is positive and ir is negative, but if Z < Z0, vr is negative and ir is positive.
Summarizing, at an open circuit the reflected voltage is equal to the incident voltage and this wave, along with a current (—i), travels back along the line; note that at the open circuit the total current is zero. Conversely, at a short circuit the reflected voltage wave is (—v,) in magnitude and the current reflected is (ii), giving a total voltage at the short circuit of zero and a total current of 2ii. For other termination arrangements, Thevenin's theorem may be applied to analyze the circuit. The voltage across the termination when it is open-circuited is seen to be 2vi and the equivalent impedance looking in from the open-circuited termination is Z0; the termination is then connected across the terminals of the Thevenin equivalent circuit (Figure 10.18).
Consider two lines of different surge impedance in series. It is necessary to determine the voltage across the junction between them (Figure 10.19). From (10.13):
Figure 10.18 Analysis of travelling waves-use of Thevenin equivalent circuit, (a) System, (b) Equivalent circuit
Figure 10.19 Analysis of conditions at the junction of two lines or cables of different surge impedance
The wave entering the line Z is the refracted wave and b is the refraction coefficient, that is the proportion of the incident voltage proceeding along the second line (Zi). From (10.14), with vi = Z0b
From equations (10.15) and (10.16), the equivalent circuit shown in Figure 10.19(b) can be obtained.
When several lines are joined to the line on which the surge originates (Figure 10.20), the treatment is similar, for example if there are three lines having
Figure 10.20 Junction of several lines, (a) System, (b) Equivalent circuit
Figure 10.21 Surge set up by fault clearance, (a) Equal and opposite current (7) injected in fault path, (b) Equivalent circuit, (c) Voltage and current waves set up at a point of fault with direction of travel
equal surge impedances (Zi) then
An important practical case is that of the clearance of a fault at the junction of two lines and the surges produced. The equivalent circuits are shown in Figure 10.21; the fault clearance is simulated by the insertion of an equal and opposite current (I) at the point of the fault. From the equivalent circuit, the magnitude of the resulting voltage surges (v)
and the currents entering the lines are
The directions are as shown in Figure 10.21(c).