# Problems

- 10.1 A 345 kV, 60 Hz system has a fault current of 40 kA. The capacitance of a busbar to which a circuit breaker is connected is 25 000 pF. Calculate the surge impedance of the busbar and the frequency of the restriking (recovery) voltage on opening.
- (Answer: 727 V, 8760 Hz)
- 10.2 A highly capacitive circuit of capacitance per phase 100 pF is disconnected by circuit breaker, the source inductance being 1 mH. The breaker gap breaks down when the voltage across it reaches twice the system peak line-to-neu- tral voltage of 38 kV. Calculate the current flowing with the breakdown, and its frequency, and compare it with the normal charging current of the circuit. (Answer: 24 kA, 503 Hz; note I = 2V
_{p}/Zq)

- 10.3 A 10 kV, 64.5 mm
^{2}cable has a fault 9.6 km from a circuit breaker on the supply side of it. Calculate the frequency of the restriking voltage and the maximum voltage of the surge after two cycles of the transient. The cable parameters are (per km), capacitance per phase = 1.14 pF, resistance = 5.37 V, inductance per phase = 1.72 mH. The fault resistance is 6 V. - (Answer: 374 Hz; 16kV)
- 10.4 A 132 kV circuit breaker interrupts the fault current flowing into a symmetrical three-phase-to-earth fault at current zero. The fault infeed is 2500 MVA and the shunt capacitance, C, on the source side is 0.03 pF. The system frequency is 50 Hz. Calculate the maximum voltage across the circuit breaker and the restriking-voltage frequency. If the fault current is prematurely chopped at 50 A, estimate the maximum voltage across the circuit breaker on the first current chop.
- (Answer: 215.5kV; 6.17kHz; 43kV)
- 10.5 An overhead line of surge impedance 500 V is connected to a cable of surge impedance 50 V. Determine the energy reflected back to the line as a percentage of incident energy.
- (Answer: reflected energy =— 0.67 x incident surge energy)
- 10.6 A cable of inductance 0.188 mH per phase and capacitance per phase of 0.4 pF is connected to a line of inductance of 0.94 mH per phase and capacitance 0.0075 pF per phase. All quantities are per km. A surge of 1 p.u. magnitude travels along the cable towards the line. Determine the voltage set up at the junction of the line and cable.
- (Answer: 1.88 p.u)
- 10.7 A long overhead line has a surge impedance of 500 V and an effective resistance at the frequency of the surge of 7 V/km. If a surge of magnitude 500 kV enters the line at a certain point, calculate the magnitude of this surge after it has traversed 100 km and calculate the resistive power loss of the wave over this distance. The wave velocity is 3 x 10
^{5}km/s. - (Answer: 250 kV; 375 MW)
- 10.8 A rectangular surge of 1 p.u. magnitude strikes an earth (ground) wire at the centre of its span between two towers of effective resistance to ground of 200 and 50 V. The ground wire has a surge impedance of 500 V. Determine the voltages transmitted beyond the towers to the earth wires outside the span.
- (Answer: 0.44v; from 200 V tower and 0.17v; from 50 V tower)
- 10.9 A system consists of the following elements in series: a long line of surge impedance 500 ft, a cable (Z
_{0}of 50 V), a short line (Z_{0}of 500 V), a cable (Z_{0}of 50 V), a long line (Z_{0}of 500 V). A surge takes 1 ps to traverse each cable (they are of equal length) and 0.5 ps to traverse the short line connecting the cables. The short line is half the length of each cable. Determine, by means of a

**Figure 10.41 **Solution of Problem 10.9

lattice diagram, the p.u. voltage of the junction of the cable and the long line if the surge originates in the remote long line.

- (Answer: see Figure 10.41)
- 10.10 A 3 p.h., 50 Hz, 11 kV star-connected generator, with its star point earthed, is connected via a circuit breaker to a busbar. There is no load connected to the busbar. The capacitance to earth on the generator side terminals of the circuit breaker is 0.007 mF per phase. A three-phase-to-earth short circuit occurs at the busbar with a symmetrical subtransient fault current of 5000 A. The fault is then cleared by the circuit breaker. Assume interruption at current zero. (a) Sketch the voltage across the circuit breaker terminals of the first phase to clear. (b) Neglecting damping, calculate the peak value of the transient recovery voltage of this phase. (c) Determine the time to this peak voltage and hence the average rate of rise of recovery voltage.
- (Answer: (b) 17.96 kV; (c) 16.7 ms, 1.075 kV/ms)
- (From
*Engineering Council Examination, 1996)*

10.11 A very long transmission line AB is joined to an underground cable BC of length 5 km. At end C, the cable is connected to a transmission line CD of 15 km length. The transmission line is open-circuit at D.

The cable has a surge impedance of 50 V and the velocity of wave propagation in the cable is 150 x 10^{6}m/s. The transmission lines each have a surge impedance of 500 V. A voltage step of magnitude 500 kV is applied at A and travels along AB to the junction B with the cable.

Use a lattice diagram to determine the voltage at:

a. D shortly after the surge has reached D;

b. D at a time 210 ms after the surge first reaches B;

c. B at a time 210 ms after the surge first reaches B.

Sketch the voltage at B over these 210 ms.

- (Answer: (a) 330 kV, (b) 823 kV and (c) 413 kV)
- (From
*Engineering Council Examination, 1995)*