# Traveling Wave

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```					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/m C=1/[18 x 109 ln(2h/r)] F/m • v
 2 10 ln 2h / r   1    18 109 ln 2h / r    LC   
7 1/ 2

   

1

• In the cable :  = 1/(LC) = 3 x 108 K K=dielectric constant (2.5 to 4.0)

m/s

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=vf/Zc - vb/Zc and v + iZc= 2vf so vf = ½ (v+iZc) and

I(x,t)=If + Ib iZc=vf – vb v=2vf=iZc vb = ½ (v+iZc) or vb= vf-iZc

3.1 Line Termination in Resistance
v  iR 2 i vf R  Zc R  Zc 2R R  Zc vb  vf R  Zc vf 

Pf 

v2 f Zc

2 vb Pb  Zc

v f R R Pf  Pb  PR PR 

2

v 

 vb 

2

3.2 Line Termination in Impedance (Z)

2 i if Z  Zc 2Z v vf Z  Zc v  v f 2Z  Z  Zc

Z  Zc vf  v 2R Z  Zc vb  vf Z  Zc vb  v f Z  Zc  Z  Zc

• 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. •
Z s  Zc Z R  Zc s  ; R  Z s  Zc Z R  Zc

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 end v=vf+vb=2vf • Voltage at the open end is twice the forward voltage wave

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 end v=if+ib=2if • Current at the open end is twice the forward current wave

3.5 Termination Through Capacitor
2Z  Z  Zc 2(1 / Cs )  Z c  1 / Cs v  v f
2(1 / Cs ) v f 2v f 1 v( s )   Z c  1 / Cs s s Z c Cs  1 1 / Z cC 1 1   2v f  s s  1 / Z cC s s  1 / Z cC So : v(t )  2v f (1  e t / Z cC ) i (t )  2v f Zc e t / Z c C 2v f

vb (t )  v f (1  2e t / Z cC )

3.6 Termination Through Inductor

v(t )  2v f e i (t )  2v f Zc

( Z c / L )t

(1  e

( Z c / L )t

)

vb (t )  v(t )  v f (t ) vb (t )  v f (2e
( Z c / L ) t

 1)

4. Junction of Two Line
if  vf Z c1
v f  vb  v i f  ib  i vb v   Z c1 Z c1 Z c 2  Z c1  v 2v f   1   Z  c2   vf

vb ib  Z c1 v i Zc2

2Z c 2 v vf Z c1  Z c 2 2 Z c1 i if Z c1  Z c 2 Z c 2  Z c1 vb  vf Z c1  Z c 2 Z c1  Z c 2 ib  if Z c1  Z c 2

Pf 

v

2 f

Z c1
2

v P Zc2 v Pb  Z c1
2 b

5. Junction of Several Line
Example:

Zc2 v Z c1  Z c 2 / 2 2 2 Z c1 v if Z c1  Z c 2 / 2 if  2v f Z c1  Z c 2 / 2

2v f

6. Bewley Lattice Diagram

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