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Traveling Wave

Traveling Wave. Transient Overvoltages. 1.Introduction. Transient Phenomenon : Aperiodic function of time Short duration

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Traveling Wave

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  1. Traveling Wave Transient Overvoltages

  2. 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

  3. Impulse Voltage Waveform

  4. 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

  5. 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)] Voltage & Current Function

  6. 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)

  7. 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

  8. 2.3 Superposition of Forward and Backward-Traveling Wave

  9. 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

  10. 3.1 Line Termination in Resistance

  11. 3.2 Line Termination in Impedance (Z)

  12. 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.

  13. 3.3 Open-Circuit Line Termination • 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

  14. 3.4 Short Circuit Line Termination • Boundary condition for current v=0 • Therefore vf=-vb • If=vf/Zc=-(vb/Zc)=ib • Thus total voltage at the receiving endv=if+ib=2if • Current at the open end is twice the forward current wave

  15. 3.5 Termination Through Capacitor

  16. 3.6 Termination Through Inductor

  17. 4. Junction of Two Line

  18. 5. Junction of Several Line Example:

  19. 6. Bewley Lattice Diagram

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