ELEC-E8409 HIGH VOLTAGE ENGINEERING Exercise 3 ELEC-E8409 HIGH VOLTAGE ENGINEERING

Question 1 A step wave of amplitude u is applied to the open line illustrated in the figure. Study voltage as a function of time at points A, B and C. s m v l Z j / 10 300 450 400 6 = W k 150 40 A B C u .

Transmission coefficient: Reflection coefficient: Ground Bare Conductor Cable Open End Z = 0 Z = 400 Ω Z = 40 Ω Z =∞ τ1 τ2 τ3 τ4 ρ1 ρ2 ρ3 ρ4 A B C (τ2, ρ2): Z1 = 400 Ω, Z2 = 40 Ω (τ1, ρ1): Z1 = 400 Ω, Z2 = 0 (τ4, ρ4): Z1 = 40 Ω, Z2 = ∞ (τ3, ρ3): Z1 = 40 Ω, Z2 = 400 Ω

Propagation of waveform: B C 10 9 8 7 6 t [µs] 5 4 3 2 1 τ2 τ3 ρ1 ρ2 ρ3 ρ4 τ4

Uc 0.96 (7.5µs) Ub 0.78 (7.5µs) 0.66 (5.5µs) 0.66 (6.5µs) 0.36 (3.5µs) 1.2 Ua 0.2 0.4 0.6 0.8 1 1.2 2 3 4 5 6 7 8 9 10 Time [µs] Uc 0.96 (7.5µs) 1.0 Ub 0.8 0.78 (7.5µs) 0.66 (5.5µs) 0.66 (6.5µs) Voltage 0.6 0.36 (3.5µs) 0.4 0.33 (4.5µs) 0.2 0.18 (1.5µs) 0.0 1 2 3 4 5 6 7 8 9 10 Time [µs]

Question 2 A current transformer is linked to a 20-kV bare conductor line. The primary coil inductance is 100 µH. Wave impedance of the line is 500 Ω. A step wave of amplitude 110 kV arrives at the current transformer. Show, using equations and approximate drawings, both of the transformer’s 20-kV terminals against ground and also the voltage between the terminals.

û =110 kV 1 2 L = 100 µH Z = 500 Ω 1 2 Z1 Z2 u2 u1 u12 2û Easy Solution Lecture slides:

Derive the equations using Laplace transformation: 1 2 Z1 Z2 u2 u1 u12 2û Derive the equations using Laplace transformation: Z sL + Z 2û s u1 u1 Laplace Laplace

Derive the equations using Laplace transformation: 1 2 Z1 Z2 u2 u1 u12 2û Derive the equations using Laplace transformation: sL + Z Z 2û s u2 u2 Laplace

Derive the equations using Laplace transformation: 1 2 Z1 Z2 u2 u1 u12 2û Derive the equations using Laplace transformation: Z sL 2û s u12 u12 Laplace

250 Z1 Z2 u2 u1 u12 2û 200 150 Voltage [kV] 100 50 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Time [µs]

Question 3 An overhead line receives a square impulse of amplitude 100 kV and length 2 km at point A. Point A is connected to a spark gap parallel with a 0.01 µF capacitor. The inception voltage of the gap is 50 kV. Draw the reflecting waveform at point A and define the waveform parameters.

Equivalent Circuit s = 2000 m û = 100 kV Z1 = 500 Ω Z2 = 500 Ω A C = 0.01 µF A Z1 Z2 u2 2û Equivalent Circuit

Inception time (moment of ignition) Inception voltage of spark gap = 50 kV: Inception time (moment of ignition) Duration of impulse (v = speed of light c)

U1 U2 Voltage [kV] t [µs] Ur = U2 – U1 120 100 80 60 40 20 1 2 3 4 5 6 U2 = 100(1 – e -t/τ ) 60 50 kV U1 40 U2 20 t = 1.73 µs Voltage [kV] t [µs] 1 2 3 4 5 6 7 8 9 10 -20 -40 -60 -80 Ur = U2 – U1 -100 -120

Derive the equations using Laplace transformation: Z1 Z2 u2 2û Derive the equations using Laplace transformation: A Z sC 1 2û s Z u2 u1 Laplace