POWER TRANSMISSION AND DISTRIBUTION …

1 Tom Penick 05/10/99 Page 1 of 11 POWER TRANSMISSION AND DISTRIBUTION ee368 INDEX attentuation and buck voltage per unit rod voltage per unit and Mengele wave ratio ' of two to a single LOSSLESS LINEMost TRANSMISSION lines fall into this category. Theformulas are simplified since 0 R and +=V0 ZBeAezz =I = (gamma) propagationconstant = attenuation constant = phase constant [rad/m]A = voltsB = voltsZ0 = characteristic or surgeimpedance [ ]when LR << and CG << then:CLGLCR22+= and LC = dZdRRS + =sinhcosh0 IVVddZRRS + =coshsinh0 IVICLZ=0 SHORT TRANSMISSION LINEA TRANSMISSION line is considered short when there is asmall angular variation, z << 1 LINE BASICSL ocations along thetransmission line aretraditionally referencedfrom the receiving = -dz= 0 VSdVRBAR+=V)(10 BAzR =Id = length [m]VS = voltage (complex) at thesending end [v]VR = voltage (complex) at thereceiving end [v]IS = source current (complex)[A]IR = load current (complex)[A]z = distance from the receivingend (a negative value orzero) [m] = (gamma) propagationconstant.

2 Refer to"ALPHA, BETA, GAMMA"on page + =sinhcosh0 IVVddzRRS + =coshsinh0 IVITELEGRAPHER S EQUATIONSfor a lossless linetvLCzv2222 = tiLCzv2222 = L = inductance [H/m]C = capacitance [F/m]v = voltage [V]z = distance along line [m]t = time [s]Tom Penick 05/10/99 Page 2 of 11 EQUIVALENT CIRCUIT FOR ATRANSMISSION LINES ourcevLG22 RiiRL22 LoadCv + dvi + dii + diKVL:()02222= + +++ + + dtdidzLidzRdvvdtdidzLidzRv()()0=++dvdtdi dzLidzR()()dtdidzLidzRdv+= + =dtdiLRidzdvPhasor form: ()IVLjRz + = KCL:()()()0=++++++ diidvvdtddzCdvvdzGi()()dvvdtddzCdvvdzGdi +++= dtdvCGvdzdi =Phasor form: ()VICjGz + = 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]WAVELENGTHfvvLCpp= = = = 222 = wavelength [m] = phase constant [ ] = frequency [ ]f = frequency [Hz]vp = velocity of propagation( 108 for a conductor inair) [m/s]TWO-PORT SYSTEMRVSSII2-PORTRV = RRSS ddzdZdIVIV)cosh()sinh(1)sinh()cosh(00 This matrix equation is equivalent to:RRSdZdIVV + =)sinh()cosh(0 andRRSddzIVI + =)cosh()sinh(10 This can also be expressed: + += 220ddRddRSeezeeIVV ++ = 22100ddRddRSeeZeezIVITHE PI EQUIVALENT MODELyyVSSRySRRV002tanh)sinh(1)cosh(zddz dyySR = ==)sinh(10dzySR =OPEN-CIRCUIT COAXIAL LINE~dV tsin = RRSSdZdjdjZdIVIV cossinsincos00In this case, rr = Penick 05/10/99 Page 3 of 11 VELOCITY OF PROPAGATION vpThe speed at which a wave travels down the line.

3 Fora TRANSMISSION line in air, this is near the speed oflight, c = 108 m/sLCvp1=vp = velocity of propagation [m/s]L = inductance [H/m]C = capacitance [F/m]SURGE IMPEDANCE orCHARACTERISTIC IMPEDANCEThe cable materials and the arrangement of theconductors determine the surge impedance. It hasnothing to do with + +=0Z0 = 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, GAMMA when LR << andCG << then:CLGLCR22+= and LC = + = + += j))((CjGLjRzzBeAe +=V0 ZBeAezz =IBAVR+=)(10 BAZIR = = attenuationconstant = phase constant[ ] = (gamma)propagationconstantZ0 = surge impedance(has nothing to dowith resistance) [ ]A = voltsB = voltsVR = voltage at thereceiving endIR = current at thereceiving endCAPACITANCE PER UNIT LENGTHS ingle line in air(capacitance decreaseswith height):rhC2ln20 =Two conductors in abundle:ababiaaiDrDDCln20 =Coaxial cable.

4 IrrrC00ln2 = 0 = Permittivity of free 10-12 [F/m] r = relative permittivity[constant]h = height of TRANSMISSION line[m]r = radius of the conductor [m]Daai = distance from conductor ato its image [m]Dabi = distance from conductor ato the image of conductor b[m]Dab = distance from conductor ato conductor b [m]r0 = outer radius of a coaxialconductor [m]ri = inner radius of a coaxialconductor [m] CAPACITANCE3-phase positive or negative sequence capacitance: + + + =//0/ln2 GMRGMDC where thegeometric mean distance between conductors is:3/bcacabDDDGMD= +and the geometric mean radius is:3/cbarrrGMR= +ZERO-SEQUENCE CAPACITANCE3-phase zero-sequence capacitance:0000ln23 GMRGMDC =where thegeometric mean distance between conductors is:90ccicbicaibcibbibaiaciabiaaiDDDDDDDD DGMD=and the geometric mean radius is:90cbcabcbaacabcbaDDDDDDrrrGMR=Tom Penick 05/10/99 Page 4 of 11 INDUCTANCE PER UNIT LENGTHS ingle line in air(inductance increaseswith height):rhL2ln20 =Coaxial cable:irrrL00ln2 = 0 = (mu) Permeability constant4 10-7 [H/m, T m/A] r = (mu) relative permeability, avalue near 1 for many materialsh = height of TRANSMISSION line [m]r = radius of the conductor [m]r0 = outer radius of a coaxialconductor [m]ri = inner radius of a coaxialconductor [m] INDUCTANCE3-phase positive or negative sequence inductance: + + + =//0/ln2 GMRGMDL where thegeometric mean distance between conductors is:3/bcacabDDDGMD= +and the geometric mean radius is:3/cbarrrGMR= + (see note)Note: Apply a multiplier of 4/1 e to the physicalradius of each INDUCTANCE3-phase zero-sequence inductance:0000ln23 GMRGMDL =where thegeometric mean distance between conductors is:90ccicbicaibcibbibaiaciabiaaiDDDDDDDD DGMD=and the geometric mean radius is.

5 90cbcabcbaacabcbaDDDDDDrrrGMR= (see note)Note: Apply a multiplier of 4/1 e to the physicalradius of each COMPLEX DEPTHC omplex depth is an adustment to the actual depth ofthe conductor image, used when value of dc added to theabove-ground height of theconductors gives thedistance to effective other words, instead ofDaai = 2h, we now haveDaai = 2h + 2dc.() +=011fjdcf = frequency [Hz] 0 = (mu) Permeabilityconstant 4 10-7 [H/m,T m/A] = constant, forlimestone [( -m)-1]effective earth aihconductorimagedcdcreal earthhconductoraPOSITIVE SEQUENCEVBVCVAVa = Vag 0Vb = Vag -120Vc = Vag 120 STANDING WAVE RATIOthe ratio of peak voltage to minimum voltage: +==11))((MIN))((MAXSWR rmsrmszVzVwhere is the magnitude of the reflection coefficientTRAVELING WAVES()()()44344214434421-V wavereverse2V waveforward1,LCztFLCztFztv++ =+()()()44344214434421-I wavereverse02I waveforward01,LCztzFLCztzFzti+ =+Forward traveling wave:ze Reverse traveling wave:ze +Tom Penick 05/10/99 Page 5 of 11 TRANSIT TIME pdvdt=td = 1-way transit time [s]d = length of TRANSMISSION line [m]vp = velocity of propagation [m/s]VOLTAGE DROP ACROSS INLINEELEMENTSS eries inductance: )(Ljv =ISeries capacitance.

6 Cjv =IPEAK, RMS, and VOLTAGE TO GROUND2rmsabpeakabVV=3)3( =abagVVVOLTAGE BETWEEN TWO POINTSb and c DUE TO A CHARGED LINE aaibacaicababcDDDDqVln20 =qa = CV = unit charge on the line [c/m] 0 = Permittivity of free 10-12 [F/m]Dab = distance from line a to point b [m]Dcai = distance from point c to theimage of line a [m]aaicbDabDDbaiDcaiacVOLTAGE TO GROUND AT POINT bDUE TO TRANSMISSION LINE aaabbaiabgrhDDVV2lnln=Dbai = distance from point b to theimage of line a [m]Dab = distance from line a to point b [m]r = radius of conductor a [m]DabbhaDbaiaiCAPACITANCE MATRIXThe upper and lower triangles of the capacitancematrix are equal. = cgbgagccbcacbcbbabacabaacbaVVVCCCCCCCCC qqqaaarhC2ln20 =ababiabDDCln20 =ELECTRIC SHOCKD anzeil's Electro-cution level: 1 mALet go level: 10-20 mADeath level: 100 mABody resistance: 1k STEP VOLTAGEStep 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).

7 For the flag pole: = 21122rrrrIVFor the ground rod:()()2112ln2rlrrlrlIV++ = V = step voltage (as shown) [V] = constant, for limestone[( -m)-1]l = length of the ground rod [m]flagpolegroundrodl V VV 2rr1 Tom Penick 05/10/99 Page 6 of 11 VOLTAGE ACROSS INSULATORSDUE TO CAPACITANCEznVVgn =sinhsinhCc/= = capacitance ratioc = capacitance betweeninsulator 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 = 1 is the unit attachedat the tower armz = total no. of insulator unitsccccCCONDUCTOR CCTOWER ARMccCCCELECTRIC FIELDP erception 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/cmrlrrqaE)02 =Breakdown in dry air:3 +=TPEBKEr = radial electric field[V/m]ql = line charge [C/m]r = conductor radius [m] r = radial unit vectorEBK = breakdown [V/m]P = atmospheric pressure[in.]

8 Hg]T = temperature [ F]Electric field for a bundle of 2:)2(22/)(22/00xrAqxrqEll++ + =CENTER OF CHARGE CONDUCTORSAxrx = displacement of thecenter of charge fromthe center of theconductor [m]A = bundle radius,measured from centerof bundle to center ofconductor [m]MARKT MENGELE METHODfor computing average maximum peak bundlegradient used in noise each phase bundle as asingle equivalent conductorwith radius:()NNeqNrAr/11 = 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 thepeak bundle line charge ql equal charge division, calculate the averagemaximum bundle gradient and the average maximumpeak bundle =() +=ArNEEavgpeakavg11maxN = number of conductors in the bundler = conductor radius (not the bundle radius) [m]A = bundle radius, measured from center of bundle tocenter of conductor [m]NOISE AND INTERFERENCETRANSMISSION LINE NOISET ransmission line noise is caused by corona.

9 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 reference nPa 20pressure soundlog20(dB) level sound10=Noise per + ++=AN = some kind of noise [dB or dBA?, meaning abovelevel 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 INTERFERENCEC orona discharge occurs only on very highvoltage discharge usually indicates a physicalproblem, can occur on DISTRIBUTION Penick 05/10/99 Page 7 of 11 GEOMAGNETIC STORMLow frequency flux, almost DC. Occurs in east/westlines in polar FIELD and CAPACITANCEin a coaxial conductor** rqrlr 02=EiorlrrrrlrrrrrrqdrrqdrVoioiln21200 === ==EiorlrrVqCln20 ==Er = radial electric field [V/m]ql = line charge [C/m] 0 = Permittivity of free 10-12 [F/m] r = relative permittivity [constant]r = radial distance [m]ri = inner conductor radius [m]ro = outer conductor radius [m]V = voltage between inner andouter conductor [V]C = capacitance per meter [F/m]Emax = maximum electric field(near center conductor) [m]ioiirlrrrVrqEln20max== **When using this formula to find the height above groundat which breakdown occurs, r is the conductor radius, notthe height, because breakdown begins at the surface ofthe FLUXMAGNETIC FLUXINL =rH = 21HB = = sSdsB L = inductance [H]N = number of turnsI = current [A]H = magnetic field intensity (directionby right-hand rule)

10 [A/m]B = magnetic flux density [W/m2] = permiability of free space 4 10-7 r = relative permittivity [constant]r = radial distance [m] S = amount of magnetic flux passingthrough a surface [H/m2]FLUX LINKING 2 conductorsThe amount of flux linking two wires is the amount offlux passing between them. This applies to twoconductors of equal radius carrying equal current inopposite = + = = =ln22000eqlrDLln0 = D = distance between two conductors(center to center) [m]I = current [A]x = a distance along D [m] = amount of magnetic flux passingthrough a surface [H/m2]Ll = inductance per meter between 2conductors [H/m]req = equivalent radius [m]magneticfluxFLUX LINKING 1 conductor above earthThe 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 = + = ==2ln2112020 NOTE: Use theequivalent radius,4/1 = =2ln20conductor imagemagneticfluxground hTom Penick 05/10/99 Page 8 of 11 FLUX LINKING TO GROUND multiple conductors above earthThe diagram andformula concern theflux linking forconductor a due toconductor b conductors biconductorimagesababibDDrbDDrbbaDDIrdrI rdrIabigbibgbaln222000 = + = ==P MATRIXThe relationship between three conductors in air is: = cbacccbcabcbbbaacabaacgbgagqqqPPPPPPPPPV VV021where Vag is the voltage from conductor a to ground,where aaaiaarDPln=with Daai being the distancebetween conductor a and its image, and ra being theradius of conductor ababiabDDPln=with Dabi being the distancebetween conductor a and the image of conductor b,and Dab being the distance between conductors a andb.