pwrtransanddistr
TRANSCRIPT
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Tom Penick [email protected] www.teicontrols.com/notes 05/10/99 Page 1 of 11
POWER TRANSMISSION AND DISTRIBUTION EE368
INDEXattentuation constant ..........3boost and buck transformers9bundle
capacitance ....................3radius ............................8
calculus............................11
capacitanceinline voltage drop .........5pos./neg. sequence .........3zero sequence.................3
capacitance matrix .............5capacitance per unit length.3coaxial line ........................2
capacitance ....................3complex depth.................... 4constants..........................11corona..............................10cross product.................... 10
dot product.......................10electric field.......................6electric shock.....................5flashover..........................10flux....................................7flux linking........................7
geomagnetic storm .............7geometry..........................10ground rod resistance .........9impedance
characteristic..................3surge..............................3
inductanceinline voltage drop .........5pos./neg. sequence .........4zero sequence.................4
inductance per unit length ..4interference........................6
lightning and transients ......9lossless line........................1magnetic flux.....................7Markt Mengele method ......6natural log........................11noise..................................6
phase constant.................... 3p-matrix.............................8positive sequence ...............4propagation constant ..........3reflection coefficient
current...........................9voltage...........................9
standing wave ratio SWR ... 4step voltage........................5substation...........................9telegraphers' equations .......1transformers.......................9
transit time.........................5transmission coefficient......9transmission line................1traveling waves..................4trickle pulse.....................10trigonometric identities ....11
velocity of propagation....... 3voltageacross insulators.............6between two points ........5due to a single line......... 5peak...............................5rms................................5step................................5
voltage regulator ................9wavelength.........................
THE LOSSLESS LINE
Most transmission lines fall into this category. The
formulas are simplified since 0R and G .
L2
C
L2
zz BeAe +=V
0Z
BeAe zz =I
= (gamma) propagationconstant
= attenuation constant= phase constant [rad/m]A= volts
B= volts
Z0= characteristic or surge
impedance []
when LR
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EQUIVALENT CIRCUIT FOR ATRANSMISSION LINE
Source
vL
G22
R
i
i R L2 2
Load
C v + dv
i + di
i + di
KVL:
( ) 02222
=
+
+++
+
+
dt
didz
Lidz
Rdvv
dt
didz
Lidz
Rv
( ) ( ) 0=++ dvdt
didzLidzR
( ) ( )dt
didzLidzRdv +=
+=
dt
diLRi
dz
dv
Phasor form: ( )IV
LjRz +=
KCL:
( ) ( ) ( ) 0=++++++ diidvvdt
ddzCdvvdzGi
( ) ( )dvvdt
ddzCdvvdzGdi +++=
dt
dvCGv
dz
di=
Phasor form: ( )VI
CjG
z
+=
R= resistance [/m]L= inductance [H/m]G= conductance [v/m]C= capacitance [F/m]
i= current, amps [A]z= distance [m]V= V phasorI= I phasor
= phase angle [rad]
WAVELENGTH
f
vv
LC
pp ==
=
=
2
22= wavelength [m]= phase constant [rad./m]= frequency [rad./s]f= frequency [Hz]vp= velocity of propagation
(2.998108
for a conductor in
air) [m/s]
TWO-PORT SYSTEM
R
VS
SI I
2-PORT RV
=
R
R
S
S
ddz
dZd
I
V
I
V
)cosh()sinh(1
)sinh()cosh(
0
0
This matrix equation is equivalent to:
RRSdZd IVV += )sinh()cosh( 0 and
RRSdd
zIVI += )cosh()sinh(
1
0
This can also be expressed:
+
+=
22 0dd
R
dd
RS
eez
eeIVV
++
=
221
00
dd
R
dd
RSeeZee
zIVI
THE PI EQUIVALENT MODEL
y yVS S R
ySR
RV
00
2tanh
)sinh(1)cosh(
z
d
dz
dyy
SR
=
==
)sinh(1
0 dzy
SR =
OPEN-CIRCUIT COAXIAL LINE
~d
V tsin
=
R
R
S
S
dZ
dj
djZd
I
V
I
V
cossin
sincos
0
0
In this case, rr=91033.3
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VELOCITY OF PROPAGATION vpThe speed at which a wave travels down the line. Fora transmission line in air, this is near the speed oflight, c =2.998 10
8m/s
LCv
p
1=
vp= velocity of propagation [m/s]L= inductance [H/m]C= capacitance [F/m]
SURGE IMPEDANCE orCHARACTERISTIC IMPEDANCE
The cable materials and the arrangement of theconductors determine the surge impedance. It hasnothing to do with resistance.
CjG
LjRZ
++
=0
Z0= surge impedance (hasnothing to do with resistance)
[]R= resistance [/m]L= inductance [H/m]G= conductance [v/m]
C= capacitance [F/m]= frequency [radians/sec.]
ALPHA, BETA, GAMMAwhen LR
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INDUCTANCE PER UNIT LENGTH
Single line in air(inductance increases
with height):
r
hL
2ln
20
=
Coaxial cable:
i
r
r
rL 00 ln
2=
0= (mu) Permeability constant410-7 [H/m, Tm/A]
r= (mu) relative permeability, avalue near 1 for many materials
h= height of transmission line [m]r= radius of the conductor [m]r0= outer radius of a coaxial
conductor [m]ri= inner radius of a coaxial
conductor [m]
POS./NEG.-SEQUENCE INDUCTANCE
3-phase positive or negative sequence inductance:
+
++
=
/
/0/ ln2 GMR
GMDL where the
geometric mean distance between conductors is:
3/ bcacab DDDGMD =+and the geometric mean radius is:
3/ cba rrrGMR =+ (see note)
Note: Apply a multiplier of4/1
e to the physicalradius of each conductor.
ZERO-SEQUENCE INDUCTANCE
3-phase zero-sequence inductance:
0
000 ln2
3GMR
GMDL
= where the
geometric mean distance between conductors is:
90 ccicbicaibcibbibaiaciabiaai DDDDDDDDDGMD =
and the geometric mean radius is:
90 cbcabcbaacabcba DDDDDDrrrGMR = (see note)
Note: Apply a multiplier of4/1
e to the physicalradius of each conductor.
dc COMPLEX DEPTHComplex depth is an adustment to the actual depth ofthe conductor image, used when calculatinginductance.
The value of dc added to theabove-ground height of theconductors gives thedistance to effective earth.
In other words, instead ofDaai = 2h, we now haveDaai = 2h + 2dc.
( ) +=
01
1
fjd
c
f= frequency [Hz]0= (mu) Permeability
constant 410-7 [H/m,Tm/A]
= constant, 0.01 forlimestone[(-m)-1]
effective earth
ai
h conductorimage
dc
dc
real earth
h
conductor
a
POSITIVE SEQUENCE
VB
VC
VA
Va= Vag 0Vb= Vag -120Vc= Vag 120
STANDING WAVE RATIO
the ratio of peak voltage to minimum voltage:
+
==11
))((MIN
))((MAXSWR
rms
rms
zV
zV
where is the magnitude of the reflection coefficient
TRAVELING WAVES
( ) ( ) ( )44344214434421
-Vwavereverse
2
Vwaveforward
1, LCztFLCztFztv ++=+
( ) ( ) ( )44 344 214434421-Iwavereverse
0
2
Iwaveforward
0
1, LCztz
FLCzt
z
Fzti +=
+
Forward traveling wave:
ze Reverse traveling wave:
ze +
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TRANSIT TIME
p
dv
dt =
td= 1-way transit time [s]d= length of transmission line [m]vp= velocity of propagation [m/s]
VOLTAGE DROP ACROSS INLINEELEMENTS
Series inductance: )( Ljv = I
Series capacitance:Cj
v
=I
PEAK, RMS, and VOLTAGE TO GROUND
2rmsabpeakab VV =3
)3( = abagV
V
VOLTAGE BETWEEN TWO POINTSband cDUE TO A CHARGED LINE a
aibac
aicaba
bcDD
DDqV ln
2 0=
qa= CV= unit charge on the line [c/m]0= Permittivity of free space
8.8510-12
[F/m]Dab= distance from line ato point b[m]Dcai= distance from point cto the
image of line a[m]
a
ai
cb
Dab
D
DbaiDcai
ac
VOLTAGE TO GROUND AT POINT bDUE TO TRANSMISSION LINE a
a
ab
bai
abg
r
h
D
D
VV2
ln
ln
=
Dbai= distance from point b to theimage of line a[m]
Dab= distance from line ato point b[m]r= radius of conductor a[m]
Dab
b
h
a
Dbai
ai
CAPACITANCE MATRIX
The upper and lower triangles of the capacitancematrix are equal.
=
cg
bg
ag
ccbcac
bcbbab
acabaa
c
b
a
V
V
V
CCC
CCC
CCC
q
q
q
a
aa
r
hC 2
ln2 0=
ab
abi
ab
D
DC
ln2 0=
ELECTRIC SHOCK
Danzeil's Electro-cution Formula:
At
I165.0
=
Perception level: 1 mALet go level: 10-20 mADeath level: 100 mABody resistance: 1k
STEP VOLTAGE
Step voltage is the potential perunit length across the surface ofthe earth. This can be a shockhazard in the case of a lightningstrike or large fault (short circuit).
For the flag pole:
=21
12
2 rrrrI
V
For the ground rod:
( )
( )21
12ln2 rlr
rlr
l
IV
++
=
V= step voltage (as shown)[V] = constant, 0.01 for limestone
[(-m)-1]l= length of the ground rod [m]
flagpole
groundrod
l
V
V
V
2r
r1
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VOLTAGE ACROSS INSULATORSDUE TO CAPACITANCE
z
nVV
gn
=sinhsinh
Cc/== capacitance ratioc= capacitance between
insulator and arm [F]C= capacitance acrossinsulator [F]
Vn= voltage between arm andinsulator unit n[V]
Vg= line voltage [V]
n= integer value denoting aparticular insulator unit.n= 1 is the unit attachedat the tower arm
z = total no. of insulator units
c
c
c
c
C
CONDUCTOR
C
C
TOWER ARM
c
c
C
C
C
ELECTRIC FIELDPerception level: 10 kV/m rmsAnnoyance level (sparks): 15-20 kV/m rmsDesign limit (peak): 2200 kV/m or 22 kV/cmCritical (air breakdown): 3000 kV/m or 30 kV/cm
r
l
rr
qaE)
02=
Breakdown in dry air:3/2
4599.17
3000
+=
T
PEBK
Er= radial electric field[V/m]
ql= line charge [C/m]r= conductor radius [m]r= radial unit vector
EBK= breakdown [V/m]P=atmospheric pressure
[in. Hg]T=temperature [F]Electric field for a bundle of 2:
)2(2
2/
)(2
2/
00 xrA
q
xr
qE ll
+++
=
CENTER OF CHARGE
ONDUCTORS
A
x
r
x= displacement of thecenter of charge fromthe center of the
conductor [m]A= bundle radius,
measured from centerof bundle to center of
conductor [m]
MARKT MENGELE METHOD
for computing average maximum peak bundlegradientused in noise calculations
1. Treat each phase bundle as asingle equivalent conductorwith radius:
( ) NNeq NrAr/11=
2. Find the CNxN matrix. Kron reduce it to C3x3. Selectthe phase bundle with the maximum diagonal C term
(this is usually the inside bundle). Put Vmax on it, (-Vmax/2) on the other two bundles, and compute the
peak bundle line charge ql peak.
3. Assuming equal charge division, calculate the averagemaximum bundle gradient and the average maximumpeak bundle gradient.
rN
qE
peakl
avg
0max 2
=
( )
+=
A
rNEE avgpeakavg 11max
N= number of conductors in the bundle
r= conductor radius (not the bundle radius) [m]A= bundle radius, measured from center of bundle to
center of conductor [m]
NOISE AND INTERFERENCE
TRANSMISSION LINE NOISE
Transmission line noise is caused by corona. It has a120 Hz base frequency and is the effect of positiveand negative ions moving back and forth. Theattenuation is 3 dB per doubling distance from theline. This is a slow attenuation due to the length ofthe line. In the 1000 kV range, sound is a limitingfactor.
pressurereferencenPa20pressuresound
log20(dB)levelsound 10=
Noise per phase:
0log4.11log55loglog120 ANDdNKgAN +++=AN= some kind of noise [dB or dBA?, meaning above
level of perception]g= peak surface gradient by Markt Mengele method
[kV/cm]d= dunno [m]
D= wire to listener distance [m]AN0= some other kind of noise [dB]
RADIO INTERFERENCE
Corona discharge occurs only on very highvoltage lines.
Gap discharge usually indicates a physicalproblem, can occur on distribution lines.
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GEOMAGNETIC STORM
Low frequency flux, almost DC. Occurs in east/westlines in polar areas.
ELECTRIC FIELD and CAPACITANCE
in a coaxial conductor
** r
q
r
l
r 02=E
i
o
r
l
r
rrr
l
r
rrr
r
rq
drr
q
drV
o
i
o
i
ln2
12
0
0
=
=
=
=
=E
i
o
rl
r
rV
qC
ln
2 0==
Er= radial electric field [V/m]
ql= line charge [C/m]0= Permittivity of free space
8.8510-12
[F/m]r= relative permittivity [constant]r= radial distance [m]ri= inner conductor radius [m]ro= outer conductor radius [m]V= voltage between inner and
outer conductor [V]C= capacitance per meter [F/m]Emax= maximum electric field
(near center conductor) [m]
i
o
iir
l
r
rr
V
r
qE
ln2 0max ==
**When using this formula to find the height above ground
at which breakdown occurs, ris the conductor radius, notthe height, because breakdown begins at the surface ofthe conductor.
MAGNETIC FLUX
MAGNETIC FLUX
I
NL
=
rH
= 2
1
HB =
= sS dsB
L= inductance [H]N= number of turns
I= current [A]H= magnetic field intensity (direction
by right-hand rule) [A/m]B= magnetic flux density [W/m2]= permiability of free space 410-7
r= relative permittivity [constant]r= radial distance [m]S= amount of magnetic flux passing
through a surface [H/m
2
]
FLUX LINKING 2 conductors
The amount of flux linking two wires is the amount offlux passing between them. This applies to twoconductors of equal radius carrying equal current inopposite directions.
eq
eqrD
rx
rD
rx r
rDIdx
x
Idx
x
I eq
eq
eq
eq
=
+
=
=
=ln
22000
eq
lrDL ln0=
rrereq 7788.04/1 ==
D= distance between two conductors(center to center) [m]
I= current [A]x= a distance alongD[m]= amount of magnetic flux passing
through a surface [H/m2]
Ll= inductance per meter between 2conductors [H/m]
req= equivalent radius [m]
magneticflux
FLUX LINKING 1 conductor above earth
The amount of flux linking a single wire above earth isthe total flux passing between the conductor and theground, summing the contributions by the conductorand by its image.
eq
eqrh
hx
h
rx r
rhIdx
xdx
x
I eq
eq
=
+
=
==
2ln
211
20
20
NOTE: Use theequivalent radius,
4/1
=rer
eq
.
eq
eq
lr
rhL
=2
ln2
0
conductor image
magneticflux
ground
h
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FLUX LINKING TO GROUND
multiple conductors above earth
The diagram andformula concern theflux linking forconductor a due toconductor b only.
a
b
ai ci
cmagneticflux
ground
conductors
biconductor
images
ab
abibD
Dr
bD
Dr
bba
D
DI
r
drI
r
drI abi
gbi
bg
ba
ln222000
=
+
= ==
P MATRIX
The relationship between three conductors in air is:
=
c
b
a
cccbca
bcbbba
acabaa
cg
bg
ag
q
q
q
PPP
PPP
PPP
V
V
V
021
where Vag is the voltage from conductor ato ground,
where
a
aaiaa
rDP ln= withDaai being the distance
between conductor a and its image, and ra being the
radius of conductor a.
where
ab
abi
abD
DP ln= withDabi being the distance
between conductor a and the image of conductor b,
andDab being the distance between conductors a and
b. The remaining P terms follow this pattern.The expression can also be written:
abcabcabc QPV02
1=
Relationships involving capacitance are:
abcabcCP = *102 and abcabcabc QVC =
The term Pabc-1* means a matrix in which the inverse
of the individual members has been carried out.
WITH GROUND CONDUCTORS:
=
w
v
c
b
a
wwwvwcwbwa
vwvvvcvbva
cwcvcccbca
bwbvbcbbba
awavacabaa
wg
vg
cg
bg
ag
q
q
q
q
q
PPPPP
PPPPP
PPPPP
PPPPP
PPPPP
V
V
V
V
V
02
1
Since the grounds vand whave zero potential, thematrix dimension 5 can be reduced to 3.
EQUIVALENT BUNDLE RADIUS
( ) NNeq NrAr/11=
req= equivalent bundle radius [m]N= number of conductors in the
bundle
r= conductor radius [m]A= bundle radius, measured from
center of bundle to center of
conductor [m]
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LIGHTNING AND TRANSIENTS
LIGHTNING
Lightning is essentially a currentsouce. The model at rightapproximates the waveform of a
1.5 50 strike, meaning 1.5 srise time, 50 kA peak. Fall time isassumed 10 the rise time.
On distribution lines, chancesare greater that lightning damagewill be from a ground return stroke,so the ground wire is placed belowthe current-carrying conductors.
With transmission lines, theground wire(s) is placed above thecurrent-carrying conductors,providing a shadow angleofprotection.
t
50kA
01.5 s
16.5 s
I
ground
ground
GROUND ROD RESISTANCE
= 1
2ln
21
a
h
hR
R= ground rod resistance []= conductivity of earth
[1/( m)]h= rod depth [m]a= rod radius [m]
V REFLECTION COEFFICIENT OF
VOLTAGEThe reflection coefficient is the factor by which thevoltage is multiplied to find the voltage of thereflected wave. A voltage is reflected when it reachesa discontinuity in the line, such as a load, tap, orconnection to a line of different characteristicimpedance. The reflection coefficient for an open-ended line is 1 and for a short it is 1.
0
0
ZZ
ZZ
L
L
V +
=ZL= load impedance []Z0= characteristic impedance of the
line []
V TRANSMISSION COEFFICIENT OFVOLTAGE
0
2
1
ZZ
Z
L
L
VV
+=
+= V= reflection coefficient of voltageZL= load impedance []Z0= characteristic impedance of the
line []
I REFLECTION COEFFICIENT OFCURRENT
( )
0
0
ZZ
ZZ
I
I
L
L
VI
+
=
==+
I+= forward-traveling current wave
[A]I
-= reverse-traveling current wave
[A]V= reflection coefficient of
voltage
ZL= load impedance []Z0= characteristic impedance of
the line []
TRANSFORMERS
BOOST AND BUCK
These two transformer types are the basis for theregulating transformer below.
LineBoost
Buck
Line
VOLTAGE REGULATOR
The voltage regulator is used at substations tomaintain a near constant output voltage under varyingload conditions. The transformer can boost or buckincrementally and employs motor-driven contacts.
Line
RegulatedLoad
A substation will typically have two of these units.Substation capacity is limited to 75% load. If onetransformer should fail, the remaining unit can handle150% of its rated load for 2 hoursenough time toswitch loads to other substations.
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GENERAL
MISCELLANEOUSCorona is the ionization of air; it is a source of radio
interference, power loss, hissing, crackling noise, andvisible blue light. Not a catastrophic arc. The effect isintensified when moisture is in the air.
A flashover is an arc.
Sparking occurs with poor, dirty connections.A trickle pulse is thousands of pulses per half-cycle,
responsible for interference in the AM band.
Reasons for using multiple conductors: Most current flowsin the outer of the conductor, especially at higherfrequencies. The use of multiple conductors reduces theelectromagnetic field, when arranged in a circular pattern,but even two conductors side-by-side has a dramaticeffect.
Surge impedance loading means that 0zzLOAD = , i.e. theimpedance of the load is the same as the characteristicimpedance of the line. Under this condition, no wavereflection occurs.
Charge on a conductor. The only way a conductor can havea charge is by direct contact. No charge does not meanno voltage.
GRAPHING TERMINOLOGYWithx being the horizontal axis and y the vertical, we have
a graph of y versus x or y as a function of x. The x-axisrepresents the independent variable and the y-axisrepresents the dependent variable, so that when a graphis used to illustrate data, the data of regular interval (often
this is time) is plotted on the x-axis and the correspondingdata is dependent on those values and is plotted on the y-axis.
MILLER THEOREMThe circuit at left may be replaced by the circuit at right.One of the resistances in the circuit at right will actually bean admittance.
2vv1
R
12 /vvK=
Rv1 R1 2v2
K
RR
=
11
K
RR
/112 =
For an admittance Ywe have:
v1 2v
Y
12 /vvK=
v1 1 2Y Y 2v
)1(1 KYY =)/11(2 KYY =
DOT PRODUCTThe dot product is a scalar value.
( (zzyyxxzyxzyx BABABABBBAAA ++=++++= zyxzyxBA
ABcos = BABA
0 =yx , 1 =xx
( ) yzyx BBBB =++= yzyxyB
B
A
AB
Projection of Balong :
( )aaB
B
B
The dot product is commutative and distributive:
ABBA = ( ) CABACBA +=+
CROSS PRODUCT
( ) ( ) ( )xyyxzxxzyzzyzyxzyx
BABABABABABA
BBBAAA
++=
++++=
zyx
zyxzyxBA
ABsin = BAnBA
where n is the unit vector normal toboth A and B (thumb of right-hand rule).
BAAB =The cross product is distributive:
( ) CABACBA +=+
n
AB
A
B
NABLA, DEL OR GRAD OPERATOR
zyx +
+
zyx
GEOMETRY
SPHERE
Area 24 rA =
Volume 3
34
rV =
ELLIPSEArea ABA =Circumference
22
22ba
L+
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TRIGONOMETRIC IDENTITIES
+= j
)sin()cosh()cos()sinh()sinh( ddjddd +=
)sin()sinh()cos()cosh()cosh( ddjddd +=
)cos()cosh()sin()sinh(
2tanh
dd
djdd
++
=
)sinh(1)cosh(
2tanh
d
dd
=
L+
=!3)(
)()sin(3d
dd
L+
=!2)(
1)cos(2d
d
L+
+=!3)(
)()sinh(3d
dd
L++=!2)(1)cosh(
2dd
CONSTANTS
Avogadros number
[molecules/mole]231002.6 =
AN
Boltzmanns constant231038.1 =k J/K
51062.8 = eV/K
Elementary charge191060.1 =q C
Electron mass 310 1011.9 =m kg
Permittivity of free space12
0 1085.8= F/m
Permeability constant (mu)7
0 104= [H/m]
Plancks constant341063.6 =h J-s
151014.4 = eV-sRydberg constant 678,109=R cm-1
kT @ room temperature 0259.0=kT eVSpeed of light
810998.2 =c m/s1 (angstrom) 10-8 cm = 10-10m1 m (micron) 10-4cm1 nm = 10 = 10-7cm
1 eV = 1.6 10-19J
1 V = 1 J/C 1 N/C = 1 V/m 1 J = 1 Nm = 1 CV
CALCULUS OPERATIONS
= drV rr E
Edz
vd= ?
dt
diLtv
L=)(
00
1)( idv
Lti
t
L +=
00
1)( vdi
Ctv
t
C +=
dt
dvCti
C =)(
Vr= voltage [V]Er= radial electric field [m]iL= current in an inductor [A]iC= current in a capacitor [A]vL= voltage across an inductor [V]vC= voltage across a capacitor [V]
PROPERTIES OF THE NATURAL LOG
ABBA lnlnln =+
B
ABA lnlnln =
A
BBA lnln = AeBA B ==lnABA Be =ln
AeA =ln