Traveling Wave
Transient Overvoltages
1.Introduction
• Transient Phenomenon : – Aperiodic function of time– Short duration
• Example :Voltage & Current Surge :(The current surge are made up of charging or discharging capacitive currents that introduced by the change in voltages across the shunt capacitances of the transmission system)
– Lightning Surge– Switching Surge
Impulse Voltage Waveform
2.Traveling Wave• Disturbance represented by
closing or opening the switch S.• If Switch S closed, the line
suddenly connected to the source.
• The whole line is not energized instantaneously.
• Processed :– When Switch S closed
– The first capacitor becomes charged immediately
– Because of the first series inductor (acts as open circuit), the second capacitor is delayed
• This gradual buildup of voltage over the line conductor can be regarded as a voltage wave is traveling from one end to the other end
Voltage & Current Function
• vf=v1(x-t)
• vb=v2(x+t)
= 1/(LC)
• v(x,t)=vf + vb
• vf=Zcif• vb=Zcib
• Zc=(L/C)½
• If=vf/Zc
• Ib=vb/Zc
• I(x,t)=If + Ib
• I(x,t)=(C/L) ½ [v1(x-t) -v2(x+t)]
2.1 Velocity of Surge Propagation
• In the air = 300 000 km/s = 1/(LC) m/s• Inductance single conductor Overhead Line (assuming
zero ground resistivity) :L=2 x 10-7 ln (2h/r) H/mC=1/[18 x 109 ln(2h/r)] F/m
•
• In the cable : = 1/(LC) = 3 x 108 K m/sK=dielectric constant (2.5 to 4.0)
12/1
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2.2 Surge Power Input & Energy Storage
• P=vi Watt
• Ws= ½ Cv2 ; Wm= ½ Li2
• W=Ws+Wm = 2 Ws = 2 Wm = Cv2 = Li2
• P=W = Li2 /(LC) = i2 Zc = v2 / Zc
2.3 Superposition of Forward and Backward-Traveling Wave
3. Effects of Line Termination
• Assuming vf, if,vb and ib are the instantaneous voltage and current. Hence the instantaneous voltage and current at the point discontinuity are :
• v(x,t)=vf + vb and I(x,t)=If + Ib
• I=vf/Zc - vb/Zc and iZc=vf – vb
• v + iZc= 2vf so v=2vf=iZc
• vf = ½ (v+iZc) and vb = ½ (v+iZc) or vb= vf-iZc
3.1 Line Termination in Resistance
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3.2 Line Termination in Impedance (Z)
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• Line is terminated with its characteristic impedance :– Z=Zc
=0, no reflection (infinitely long)
• Z>Zc
– vb is positive
– Ib is negative
– Reflected surges increased voltage and reduced current
• Z<Zc
– vb is negative
– Ib is positive
– Reflected surges reduced voltage and increased current
• Zs and ZR are defined as the sending-end and receiving end.
•
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css ZZ
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;
• Boundary condition for current i=0
• Therefore if=-ib• Vb=Zcib=Zif=vf
• Thus total voltage at the receiving endv=vf+vb=2vf
• Voltage at the open end is twice the forward voltage wave
3.3 Open-Circuit Line Termination
• Boundary condition for current v=0
• Therefore vf=-vb
• If=vf/Zc=-(vb/Zc)=ib• Thus total voltage at the receiving end
v=if+ib=2if• Current at the open end is twice the
forward current wave
3.4 Short Circuit Line Termination
3.5 Termination Through Capacitor
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3.6 Termination Through Inductor
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4. Junction of Two Line
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5. Junction of Several Line
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Example:
6. Bewley Lattice Diagram