traveling wave

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Traveling Wave Transient Overvoltages

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Page 1: Traveling Wave

Traveling Wave

Transient Overvoltages

Page 2: Traveling Wave

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

Page 3: Traveling Wave

Impulse Voltage Waveform

Page 4: Traveling Wave

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

Page 5: Traveling Wave

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)]

Page 6: Traveling Wave

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

9

7

/2ln1018

/2ln1021

rh

rh

LCv

Page 7: Traveling Wave

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

Page 8: Traveling Wave

2.3 Superposition of Forward and Backward-Traveling Wave

Page 9: 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

Page 10: Traveling Wave

3.1 Line Termination in Resistance

fc

cb

cf

fc

vZR

ZRv

R

ZRv

vZR

i

iRv

2

2

Rbf

bfR

c

bb

c

ff

PPPR

vv

R

vP

Z

vP

Z

vP

22

2

2

Page 11: Traveling Wave

3.2 Line Termination in Impedance (Z)

c

f

fc

fc

ZZ

Z

vv

vZZ

Zv

iZZ

i

2

2

2

c

c

fb

fc

cb

cf

ZZ

ZZ

vv

vZZ

ZZv

vR

ZZv

2

Page 12: Traveling Wave

• 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.

cR

cRR

cs

css ZZ

ZZ

ZZ

ZZ

;

Page 13: Traveling Wave
Page 14: Traveling Wave

• 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

Page 15: Traveling Wave

• 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

Page 16: Traveling Wave
Page 17: Traveling Wave

3.5 Termination Through Capacitor

)21()(

2)(

)1(2)(

:

/1

112

/1

/12

1

12

/1

)/1(2)(

/

/

/

CZtfb

CZt

c

f

CZtf

cf

c

cf

sc

ff

c

c

c

c

evtv

eZ

vti

evtv

So

CZssv

CZs

CZ

s

v

CZs

v

s

v

CsZ

Cssv

f

c

c

vv

CsZ

Cs

ZZ

Z

/1

)/1(2

2

Page 18: Traveling Wave
Page 19: Traveling Wave

3.6 Termination Through Inductor

)12()(

)()()(

)1(2

)(

2)(

)/(

)/(

)/(

tLZfb

fb

tLZ

c

f

tLZf

c

c

c

evtv

tvtvtv

eZ

vti

evtv

Page 20: Traveling Wave

4. Junction of Two Line

2

1

1

c

c

bb

c

ff

Z

vi

Z

vi

Z

vi

vZ

Zv

Z

v

Z

v

Z

v

iii

vvv

c

cf

cc

b

c

f

bf

bf

2

1

211

12

Page 21: Traveling Wave

1

2

2

2

1

2

c

bb

c

c

ff

Z

vP

Z

vP

Z

vP

fcc

ccb

fcc

ccb

fcc

c

fcc

c

iZZ

ZZi

vZZ

ZZv

iZZ

Zi

vZZ

Zv

21

21

21

12

21

1

21

2

2

2

Page 22: Traveling Wave
Page 23: Traveling Wave

5. Junction of Several Line

2/

2

2/

2

22/

2

21

21

1

2

21

cc

ff

fcc

c

c

cc

f

ZZ

vi

iZZ

Zv

Z

ZZ

vv

Example:

Page 24: Traveling Wave
Page 25: Traveling Wave

6. Bewley Lattice Diagram

Page 26: Traveling Wave