PwrTransAndDistr

8/6/2019 PwrTransAndDistr

1/11

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


2/11


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


3/11


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


4/11


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 +


5/11


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


6/11


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.


7/11


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


8/11


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]


9/11


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.


10/11


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+


11/11


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

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