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

2. 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 .

3. 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

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

5. 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]

6. 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.

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

8. 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

9. 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

10. 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

11. 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]

12. 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.

13. 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

14. 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)

15. 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

16. 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