Travelling wave

1. ‫الموجات‬
‫المسافرة‬ Travelling waves
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Power system 3
Lecture 1
Dr. Zakieldeen Mohammed Eltayeb Elhassan
zakideenzain@yahoo.com
Karary University –College of Engineering
Department of Electrical and Computer Engineering

2. Introduction
Open End Line
Short circuited Line
Line Terminated through R,C
Reflection and Refraction
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4. We have to find relation between the voltage and current waves travelling
over the transmission lines and their velocity of propagation:
-Electrostatic flux is associated with the voltage wave:
We get
The charge between the conductors of the line up to a distance x is given by:
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5. -Electromagnetic flux is associated with the current wave:
Also we can find electromagnetic flux linkages:
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Rate of change in flux linkages link around the conductor:
Dividing equation (5) by (3), we get:

6. Now, multiplying equations (3) with (5), we get:
L&C for overhead line are:
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7. Substituting these values in equation (7), the velocity of propagation of the wave:
This means the velocity of propagation of the travelling
waves over the overhead transmission lines
The velocity of light. !
=
•In actual practice because of losses in line a velocity of approximately =
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8. Impedance well be infinite as the line is open-ended
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9. Electromagnetic energy stored by the element dx equal to
Electrostatic energy in the element dx equal to :
At open end the current is zero
Electromagnetic energy vanishes and is transformed into electrostatic energy
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10. This means the potential of the open end is increased by Vvolts.
The total potential of the open end when the wave reaches this end =V+V=2V
Refracted wave ‫المنكسرة‬ ‫=الموجة‬ Incident wave ‫العابرة‬ ‫الموجة‬ + Reflected wave ‫الموجة‬
‫المنعكسة‬
•Incident wave is the wave that starts travelling over the line when the switch S is
closed.
•Reflected wave is the wave that reaches the open end and potential rise by V.
•Refracted wave is actual voltage at the open end.
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12. The voltage at the shorted end is zero
Electrostatic energy vanishes and is transformed into electromagnetic energy
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13. Let the change in current be i then we get:
•This is means the current is increase by I amperes.
•The total current at the shorted end when the current wave reaches the end is
=I+I=2I
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14. Variation of voltage and current in a short ended line
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16. Assume V″ and I″ are the refracted voltage and current waves into the resistor R
when the incident waves V and I reach the resistance R.
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17. Similarly substituting for V″ in terms of (V + V′), equation (8) becomes:
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18. From above equations we can get the following coefficient:
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βi=
βv=
αi=
αv=

19. In case of open circuit
Coefficient of refraction for current waves
Coefficient of refraction for voltage waves
Coefficient of reflection for current waves
Coefficient of reflection for voltage waves
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20. In case of short circuit
Coefficient of refraction for current waves
Coefficient of refraction for voltage waves
Coefficient of reflection for current waves
Coefficient of reflection for voltage waves
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21. In case of line is terminated through a resistance
The coefficient of refraction is unity whereas the coefficient of reflection is zero
That is means when a transmission line is terminated through a resistance
equal to its surge impedance the wave does not suffer reflection and the wave
will enter fully into the resistance.
After substituting we get that :
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22. Line Connected to a Cable:
The refracted voltage:
-The impedance of the overhead line is approximately 400Ω.
-The impedance of the cable is approximately 40 Ω.
That is means that the voltage entering the cable is about
of the incident voltage V
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23. We can find out the voltage across the capacitor i.e., the refracted voltage. using
equation (9):
After take Laplace transformation we get:
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24. V″(s) is not a travelling wave but it is the voltage across the capacitor C.
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25. Substituting in equation (13) the values of currents we get:
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26. Substituting for
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27. Problems
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1. Two Long transmission lines A and B are connected together with cable (C) 1.5
km long. The lines have capacitance of 10 pF/m and inductance 1.6 µH/m and
the cable has capacitance 89 pF/m and inductance 0.5µH/m. A rectangular
voltage wave of magnitude 10 kV and of long duration travels along line A
towards the cable C. Find the following:
a. Magnitude of the second voltage step occurring at the junction of the cable
and line B.
b. What will be the voltage at the junction of line A and the cable after 20 µsec?
c.Stored energy in cable in 20 µsec.
d.Incident voltage in junction B at 30 µsec.

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.2
‫محاثته‬ ‫نقل‬ ‫خط‬
1.6 mH/km
‫وسعته‬
0.01µF/km
‫طوله‬ ‫كيبل‬ ‫مع‬ ‫موصل‬
2 km
‫المميزة‬ ‫ممانعتة‬ ‫و‬
70 Ω
‫نبضية‬ ‫جهد‬ ‫موجة‬ ‫الخط‬ ‫هذا‬ ‫علي‬ ‫وتسافر‬
/µsec kV
𝑣 = 250 𝑒−0.06𝑡
− 𝑒−0.9𝑡
‫سرعتها‬ ‫كانت‬ ‫اذا‬
‫االتي‬ ‫أوجد‬ ‫النقل‬ ‫خط‬ ‫على‬ ‫سرعتها‬ ‫نصف‬ ‫الكيبل‬ ‫علي‬
:
‫الكيبل‬ ‫نهاية‬ ‫الي‬ ‫الموجة‬ ‫لوصول‬ ‫الالزم‬ ‫الزمن‬ ‫كم‬ ‫أ‬
-
.
‫الجهد‬ ‫ب‬
-
‫مرور‬ ‫بعد‬ ‫للكيبل‬ ‫العابر‬
8µsec
.
‫مرور‬ ‫بعد‬ ‫الكيبل‬ ‫في‬ ‫الطاقة‬ ‫ج‬
-
10µsec
.
‫عند‬ ‫والكيبل‬ ‫الخط‬ ‫بين‬ ‫اإللتقاء‬ ‫نقطة‬ ‫عند‬ ‫المنعكس‬ ‫الجهد‬ ‫قيمة‬ ‫د‬
-
t= 32 µsec
.
‫أعاله‬ ‫للموجة‬ ‫المقدمة‬ ‫زمن‬ ‫علي‬ ‫الحصول‬ ‫كيفية‬ ‫ناقش‬ ‫ه‬
-
.
.3
‫قدره‬ ‫جهد‬
70 kV
‫قدرها‬ ‫مميزة‬ ‫ممانعة‬ ‫له‬ ‫نقل‬ ‫خط‬ ‫في‬ ‫ارسل‬
350 Ω
‫من‬ ‫مكون‬ ‫توازي‬ ‫فرع‬ ‫نهايته‬ ‫عند‬ ‫و‬
‫مقاومة‬
100 Ω
‫ومكثفة‬
0.08µF
‫االتي‬ ‫أحسب‬
:
‫ا‬ ‫القيمة‬ ‫وماهي‬ ‫الزمن‬ ‫بداللة‬ ‫التوازي‬ ‫فرع‬ ‫الى‬ ‫يعبر‬ ‫الذي‬ ‫الدفعي‬ ‫للجهد‬ ‫تعبيرا‬ ‫أ‬
-
‫له؟‬ ‫لقصوى‬
‫المقاومة‬ ‫في‬ ‫يمر‬ ‫الذي‬ ‫الدفعي‬ ‫التيار‬ ‫قيمة‬ ‫ب‬
-
100 Ω
.
‫للجهد‬ ‫الفعال‬ ‫الزمن‬ ‫قيمة‬ ‫ج‬
-
)
‫الموجة‬ ‫مقدمة‬ ‫زمن‬ ‫أي‬
. (

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