Radio-Frequency Propagation on Polyphase LinesL. 0. Barthold, Senior Member IEEE

Suminary: Component analysis is widely used in the solution may be written in terms of 3-phase voltages, as indicatedof power frequency problems and is also adaptable to radio- in equation 1:frequency (r-f) problems on multiconductor overhead lines.The appropriate component sets (modes) for radio frequency are vO 1 1 1 Vaa funietion of the line geometry. Matrix methods can utilize 1these components in the solution of r-f propagation problems 1 = 3 1 a a Vb (1)involving complex systems and including effects of transposition, V2 1 a' a vcgrounid wires, intermediate faults, and general impedanceterminations. or in more concise nomenclature

I92= 9a Vab (2)The past several vears have witnessed a rapid development ofthe radio noise analysis techniques initially undertaken in where a is a transformation matrix relating the symmetrical1955 by Dr. G. E. Adams of the General Electric Company. components of voltage to the phase voltages themselves.In both the United States and Europe, application of this work The symmetrical component transformation is power in-has been largely limited to the appraisal of radio-interference variant; i.e., the sum of the powers in each of the networkscharacteristics of extra-high-voltage transmission lines, al- is equal to the total phase power. Symmetrical componentsthough the propagation analysis used in this solution is equally are convenient to 60-cycle solutions because each set of com-adaptable to carrier-current problems. ponents, considered by itself, comprises a balanced 3-phaseA number of authors have analyzed r-f problems on multi- circuit which, in turn, may be reduced to a single-phase line-

conductor systems in terms of symmetrical components or to-neutral equivalent. This equivalent is usually assumedidentification of signals as either line-to-line or line-to- to have no mutual coupling to similar equivalents representingground.''1 Modal (eigenvalue) analyses of multiconductor other sequence networks.telephone lines were made as early as 1927.3 More recently, In seeking a corollary advantage in r-f analyses, one appar-a discussion of modal propagation on unbalanced high-voltage ent difference is that the scope of the problem is restrictedlines was presented in which the development was in terms to the line (or lines) and line terminations. With this reduc-of electromagnetic field quantities and the emphasis was on the tion in scope, three questions develop:identity and characteristics of the components themselves,rather than their application.4 This paper examines the fore- 1. What is the nature of conyeniently analyzed equiva-going multiconductor power line propagation theory in terms lent circuits in terms of radio frequencies?of circuit quantities, with particular emphasis on practical 2. What is the number and description of these equiva-methods of solution. lents?The material presented in this paper is not intended as an 3. How may such noninteracting networks be used to

application guide but as a suggested means of making a more represent terminations, transpositions, and faults as theyrigorous analysis of the carrier current application problem. affect carrier performance?Several important aspects of the problem, such as intermod- These questions will be discussed in the balance of thisulation distortion introduced by fundamental frequency paper.corona and the variations in carrier attenuation due toweather conditions and conductor icing, have been intention- The Nature of Modal Networksally bypassed. Although some field evidence exists to justify The r-f energization of one conductor of a long poly-the type of component analysis discussed in the paper, the phase line with respect to ground shows the ratio of voltagemethods of solution are generally applicable to other com- to current is different at the point of application than at var-ponent systems as well. In the interest of conciseness, ele- ious points along the line. Alternatively considered, thementary matrix notation will therefore be used throughout attenuation of the applied signal in decibels per mile appearsthis paper. to vary with length. Apparently the most fundamentalThe Use of Components in Electrical Problems component network would be one in which voltage and cur-

In a variety of engineering problems, it is convenient to rent maintained a constant ratio and attenuation was con-resolve the total problem into component parts; each of these stant, for example, a 2-conductor line in free space. For aparts., in its interconnection with the others, is treated more semi-infinite line of this sort, simple line equations apply andsimply than the whole. Power frequency analyses often V(x, t)= V(0, t)e-y (3)make use of symmetrical components. The zero, positiveand negative sequence components of voltage for example Appendix I discusses in detail the derivation of modal

2 ' ~~~networks for an earth of infinite conductivity. There are, inPaper 63-202, recommnended by the AIEE Power System Com-munications Committee and approv-ed by the AIEE TechnicalOperat;ions Department for presentation at the IEEE Winter Table I. Modal Impedances and Phase Coefficients for aGenerawl Meeting, New York, N. Y., January 27-February 1, 1963.Manuscript submaitted October 26, 1962; made available for printing Sample Single-Circuit Horizontal 345-Ky LineDeebr18 92 Mode l Mode 2 Mode 3

L. 0. B$ARTHOLD is with the General Electric Company, Schenectady, Surge impedance 351 396 546N. Y. ~~~~~~~~~~~~~~~Phasea coefficient 0.454 -0.707 0.542

The aulthor 'wishes to acknowledge assistance from Dr. M. H. Hesse Phase b coefficient -0.767 0 0.642concerning matrix techniques as applied to network solutions. Phase c coefficient 0.454 0.707 0.542

JULY 1964 Barthold-Rad jo-Frequlency Propagation on Polyphase Lines 665

PHASE total attenuation into conductor and ground components in

IMPEDANCE a IMPEDANCE the same ratio as calculated conductor and ground compo-

--U-g1DITCuOQY IN- t c UIuITY t--- nents. It is interesting to note that the earth's reducedinfluence on attenuation at lower frequencies reduces the

->/ // variation in attenuation between the various line-to-line(A) modes of propagation.

There are corresponding differences in phase constant dfor each of the modal networks. Phase constants are also

_ -MODE(I) z amenable to calculation, but no experimental evidence existsto confirm these calculations. Modal phase constants (or

~~~4 <z MODE (2) < z --- velocity) will be the subject of special tests at Project EHV._ 0o o o - ~~~A perspective of the extent to which differences in phase con-

_r z a:_stants between modes influence the answers on a 100-mile_ _ - M ( line, for example, can be achieved by realizing that a difference

in phase constant of 0.3% between two modes would result in(B) a difference in phase of two components of a 100-ke sinusoidal

voltage of 1 radian at the remote terminals. Experience in-Fig. 1. Conceptual diagram of a 3-phase transmission line dicates differences between mode phase constants may be of

and the corresponding uncoupled modal networks the order assumed previously.

A-Three-phase transmission line with r-f discontinuities Methods of SolutionB-Uncoupled modal networks with interconnection at each

point of r-f discontinuity Any attempt to represent impedance discontinuities asspecific interconnections of modal networks at the point ofdiscontinuity is extremely difficult. Since several simultane-ous discontinuities are likely to be involved in any one prob-lem, a direct algebraic solution of the modal networks in

general, as many modal networks as there are conductors. which discontinuities are described mathematically, ratherReasonable assumptions permit bundled conductors to be con- than as connections of circuit elements, proves to be the mostsidered as a single equivalent conductor and ground wires to expedient procedure.be treated as a minor modification to the main conductor The example line shown in Fig. 3 will help to illustrate thematrix. solution procedure. The line of Fig. 3(A) contains a generalAn example solution for modal impedances and phase co- receiving impedance matrix and a general sending impedance

efficients for a single-circuit horizontal 345-kv line is shown in matrix. Figs. 3(B) and 3(C) show simple examples of theTable I. sending- and receiving-end impedance matrices; two phasesModal networks will be numbered in order of increasing see an r-f termination represented by the station capacitance-

modal surge impedance. (Previous papers have not been to-ground, the coupled phase is isolated from the station by aconsistent in this regard.) For a single-circuit horizontal line, trap (open circuit), and the line-side of the trap is connectedmodes 1, 2, and 3 correspond roughly to the a, 3, and 0 com- capacitively to carrier equipment. An r-f signal of knownponents of 60-cycle analyses. magnitude is coupled to the line at the sending end, and the

Fig. 1 shows a conceptual diagram of an actual 3-phase magnitude of the received signal is the unknown quantity.transmission line and the corresponding modal component Fig. 4 shows the designation of modal voltages and currentsnetworks, each consisting of an idealized 2-conductor trans- in the network of Fig. 3.mission line. The line of Fig. 1 (A) is assumed to be of a con-stant configuration between discontinuities, such as transposi- THE METHOD OF ABCD CONSTANTStions, faults, taps, terminals, etc. The corresponding modal ABCD constants are commonly used to solve simple 2-networks of Fig. 1 (B) are uncoupled, except at each point of terminal networks. The constants are defined such thatdiscontinuity.Each mode is characterized by its own propagation con-

stant. Modal attenuation constants have been the subjectof a number of field-test programs in the United States andEurope,5', and although efforts have been concentrated in MODAL ESTIMATEDthe area of 1,000 kc data exist for frequencies down to 10 ON TENUATION ATTENUATIONI ~~~ ~~~dwnt 0FIGURATION MCODE FROM TESTS p I100 KCkc. An approximate method for the calculation of modal (db/MILE) 1db/MILE)attenuation constants has also been published in the litera- a b c (I) O. 20O* 0.035ture.7 (2) O.T73 0.036

Fig. 2 shows values of modal attenuation for a single-cir- (3) 6.03 0.728cuit 345-ky line of horizontal configuration and a double-cir- a d 0.033cuit vertical-configuration line, based on test measurements b e (2) 0.1 8t 0 .035at 840 kc and 1,000 kc, respectively. Also shown in Fig. 2 . .1(3) O.23t 0.036are estimated attenuations for the same lines at 100 kc. c 15 ) 0.323 0.0o37This estimate is based on the assumptions that the conductor ~ <^f<x (61 3.50t 0.55portion of r-f losses varies as the square root of frequency and

. . . . ., X~~~~~~~~~~~AVERAGE OF MEASUREMENTS t AVERAGE Of MEASUREMENTSthat thle earth component of attenuation vraries dilrectly with AT 840 KC AT 1000 KCfrequency, as suggested by unpublished work of J. Clade andC. Gary of Eleetricite de France. The 100-ke attenuation Fig. 2. Approximate attenuation constants for two examplevalues shown in Fig. 2 are based on a subdivision of measured 345-ky configurations

666 Barthold-Rad jo-Frequcency Propagation onb Polyphase Linles JULY 1964

S E F T U R THE TRANSMISSION LINE_ 1. 'a > a Let us consider one of the three line sections of Fig. 3(A)

Z s c R ABCD constants for any of the line sections may be writtenin matrix form, either on a phase basis or a modal basis. Since

fby definition, there is no intercoupling between modes, themodal ABCD matrices contain zeros for all off-diagonal terms.

/777 77~ 77 ""'"7'""7"'"' From fundamental transmission line equations, it can be(A) shown that

- a-- --a |___ cosh -Y(l)l 0 0

r.. b-- -b Amf. 2Arf_b-- _c -TRA0 1 9t = | 1) cosh-y2ly 0TRAP TRA

STATION CARRIER CARRIER STATIONCAPACI- CURRENT CURRENT CAPACI-TANCE TERMINAL TERMINAL TANCE (10)

_,__| Z() sinh y(l)lt 0 0

(B) (C) Bm = 0 Z(2) sinh ,(2)l 0

Fig.3. Phasediagramofanexample3-phasetransmissionline 0 0 Z(3) sinh -Y(8)l

A-Three-conductor transmission line with two intermediate (11)discontinuities

B-Example transmitting terminal Y|(1 sinh Y(l)l 0 0C-Example receiving terminal Cm = 0 y(2) sinh (El 0

0 0 Y(3) sinh y(S)l(12)

VI = A V2+B12

1I= CVI2+DI2 (4) Dm <Am(This method can be generalized for use with matrix quantities. Then, at point U of Fig. 3(A), the voltage and current areThe over-all problem of Fig. 3(A) may then be described asfolloNvs: Vmu Avm +± Bm i'tR

vms= + mR ImU = Cm Vn + D 2m1 (14)

s C Vin t+ D |inE (5) Substituting equation 8 into equation 14 gives

SirLce the vinR and i,'R matrices are related by a known im- nU 4 _+ B YnRVRpedance matrix, the ABCD constants of equation 5 permit ___-;sending-end current and voltage to be expressed in terms c= Cm + D vmE (15)of receiving-end voltage. It will be shown how the over-allABCD constants can be determined from a progressive solu-tion of the modal network equations.

THE RECEIVING TERMINALIn terms of phase quantities, the voltage and current at the IFME 3 mT m]

receiving terminal are related as is indicated by the following I__IequalIlon: Zms MR

Vplt ZR, pE (6) V

Recognizing that

VMR 1 VPR (A

tmE Al 'pR (7) [9

and substituting into equation 6 results in a modal terminal Fig. 4. Modal quantities for [Ii

equation the network of Fig. 3 p

Vmg 1 = ZmBX ~~~~~(8)A-Modal voltage and cur- >m >mXViR mtm rent designation for network mS

xvhere of Fig. 3(A) em____ ~~~~~~~~~~B-Modal voltage and cur-|

zme =~M-1Mj 9 rent designation for network Z\X/Rv/m\/S

JULY 1964 Barthold-Radio-Frequ>enc?y Propagation> on Polyphase Li/nes 667

TRANSPOSITIONS

In passing through the transposition of Fig. 3(A) from left to b -K 2K -Kright, the conductors are seen to change position. In a modalconcept it will be least confusing if a, b, and c are considered K K I2Kas referring to geometric positions in space. In this sense, *the voltage on either side of the transposition can be equated b K 0 0as follows: C 0 K 0

VaT = Vl~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~j ~~~~~~~~~0 0K

VbT Vu (16) b i K iVT VaU C Ypf1 =

At point T, the modal voltages may be written in terms ofthe phase voltages: a K -K *

C Yf -K K 0VT(l)M Ma(l) |M&l)JMC(') VaT | C0

VT(2) = Ma(2) Mb(2) MC(2) VbT ( 17)_I'I_) * K > YmOX WHERE YmOX IS THE LARGEST TERMIV()Ill()MDIIMC3||VCT | IN THE Ymatrix SEEN LOOKING EITHER DIRECTIONFROM THE FAULT.

If equation 16 is substituted into equation 17, it can then beshown that Fig. 5. Matrix representation of some example fault types

VT0() | Ma(l) Mb(i) |M,(l) VbUVT(2) MaAa(2) Mtb(2) j7IC(2) Vtcu (18)

VT(3) M1fa(3) Mb(3) Mc(3) Vau FAULTS

The voltages on either side of the short-circuit shown inAn expression for the phase voltages at Uin terms of themode Fig. 3(A) are equal; i.e.,voltages at that point can be written as F

VbU Nb(l) Nbh(2) Nb(3) VU; 2VimE = VmF (23)VcU = N C()Nc(2) Nc(3) VUu(2. (19) The current at point E is merely the sum of the fault currentVaU Na(1~N-(2) I

a' VI) I I Iand the current at F:|Vau |ATa(l) Na(2) Nxa(3) u(3)1imE Y| j= < VmF + ZmF (24)Note that the elements of the N matrix in equation 19 have

shifted upward one row to express the phase voltages in the Both vmF and imF matrices are known in terms of the vm,same order as they occur in equation 18. It will now be con- matrix; the Yinf matrix is known from the fault. Appar-venient to define a row cycling matrix C. such that ently the Yinf matrix will contain elements equal to infinityfor short circuits. There are several ways of preventing0 1 0 Na(') Na(2 Naa(3' Nb(') Vb(2)| Nb(3| multiplication by infinity; the simplest is to substitute a finite

o 1 ATb(l) |b(2) AT(|3) Nc A |'2 Nc(3) number K into the Ymf matrix and then assign a value to Kmuch larger than the surge admittances of the connected sys-1 0 NC(I) Nc(2) 0N,c(3) Na"() IA1a(2) Na(3) tem. Fig. 5 shows several common faults and the correspond-

(20) ing Y,jf matrix. Converting to a modal admittance matrix,Then equation 18 may be written as f = YMf N (25)VinT = 1M1{ Cr N Vin (21) Fig. 5 and the foregoing analysis consider only shunt faults,

but a similar method may be used to represent series faultsSimilarly, (open conductors).

Once again the ABCD line equations of the form of equa-inmT = 1 C )_N i|mu (22) tion 14 may be used to translate the solution at point E topoint S, giving values of the vms and i;S matrices in termsFrom the foregoing it is apparent that premultiplication of of the receiving-end voltage matrix VmR.

equation 15 by the matrix product MlCrN would give anexpression for the VmT and imT matrices in terms of the receiv- SENDING TERMINALing-end voltage matrix Vinn. For the example termination shown in Fig. 3(B),The ABCD constants for the section of line between pointsF and T are identical in nature to those used between points 9m = Yfl2 ms +- 9m (26)U and R. Since at this point in the solution both the VmT and1inT matrices are known (in terms of the Vmn matrix), an ex- and thatpression of the form shown in equation 14 will give the VmF and1mF matrices in terms of the Vi,?matrix. ITmx j= < emx VnS j(7

668 Barthold-RadiopFrequency PropagXation on Pol?yphase Lines JULY 1964

Equating equations 26 and 27 and collecting terms results in Equation 35 applies to a specific set of phase quantities, andsince equation 32 is perfectly general, it too must be valid forYmc+ Y75x VMS m emx - (28) these special values:

Suppose that the progressive solution of the circuit of Fig. V) - Zv7 J (36)3(A) up to point S yields the numerical matrices If equation 36 is subtracted from equation 35

VMS = P8 VMR (29) ZpZ z| in=n O (37)

mSi QSR VmS (30) Matrix theory shows that if the i,(nl) matrix in equation 37exists, then the determinant of the Z matrix in the same equation

Sub'stitution of equations 29 and 30 into 28 shows that: must equal zero. The determinant of this 3-by-3 matrix willyield a cubic equation whose solution will have three roots;

|UmI) = YmC+imX PSR +; Q s mx em these roots will be real for the assumption of a perfectly conduct-ing earth. Apparently there are three independent sets of(31) voltage and current which satisfy equation 35. These sets

have been identified as "modes" of propagation. The modalThus, the receiving-terminal voltage is expressed as a impedances are Z(1', Z(2), and Z(3) where, for the sake of con-

function of the sending-terminal excitation matrix and the sistent nomenclature, the modes will be numbered in order ofincreasing impedance magnitude.system impedance matrices. The solution of such problems Now that the modal impedances have been identified, they

is a laborious, but not overly complicated, hand calculation. may be substituted individually into equation 37 to yield theWith the aid of a digital computer capable of complex matrix ratio of currents peculiar to each mode, ia(f)1i,(7) and i8(?n)/ij(7mulltiplication and inversion, the problem can be reduced to for n =1, 2, and 3.anominalamount of hand calculation. For the study of The ratio of phase voltages in any of the modes is the samea nominal amount of hand calculation. For the study of as the ratio of phase currents in that mode, as is apparent from

condLitions such as variations in coupling methods and fault equation 35. This solution may be recognized as a straight-types, it is possible to use again the solution for those por- forward eigenvalue analysis in which the modal impedances aretions of the system not changed between cases. eigenroots and the corresponding mode voltages (or currents)

are eigenvectors; additional discussion of the eigenvalue problemConclusions is available.8

In developing a modal-network concept analogous to the1. -A method has been discussed for resolution of r-f problems symmetrical component networks of power-frequency analyses,on polyphase lines into modal (eigenvalue) networks, each it will be useful to define a transformation matrix M such thatconsisting of an idealized uncoupled 2-conductor line.2. Some of the constants characterizing these networks are VM M (38)calculable with fair accuracy; all of them require further tests Equation 38 is analogous to equation 2. Solving equation 38and theoretical work. for the phase voltages shows that3. Techniques are presented foi the closed solution of r-f -1 ()problems by modal networks including the effects of internet- ___ =v__ (V)work coupling at impedance discontinuities represented bytranspositions, faults, and terminals.

Vp = A VM| (401Appendix I

THE DERIVATION OF MODAL NETWORKS whereThe following derivation will be specifically concerned with a AN M -

3-phase, 3-conductor line. The extension of the argumentsN

developed to lines using bundled conductors, double circuits, If equation 40 is expanded,and ground wires will be discussed separately.At; the terminals of a semi-infinite 3-conductor line, the

voltage-current relationship may be expressed as Va Na(11A|a182 Na131 V(l1Vb()=Arhi A AT5(3 v(2i (41 )

~~~~~~~~~~~~~~(32) 41_AJ) AT (2 A~3 c1(3)

where vp and i, matrices are column matrices in which the phasevoltage and current are considered as sinusoidal functions of The modal voltages v(l), V(2), and v(3 [,are abstract quantitiestime, and the Zp matrix is a square surge impedance matrix. which represent the magnitude of each of the modal components.If a perfect ground plane is assumed, then the self- and mutual The columns of the N matrix apparently represent the sameelements of the Z4 matrix are of the form ratio of magnitudes as do the modal-voltage ratios determined

2hi from a solution of equation 37. In fact, a workable'definitionZi0= 60 loge 2 (33) of V(1), V(2., and V13) would result from assigning values

N_ (fl) - a(8i= 6401oge d(34) VC(8

N (n2)_ &Equation 32 is completely general. Suppose the hypothesis bi v~Ct)iS made that, for a particular set of values of phase quantities, _f)-1(2

IlV (e) =Z8 8 (35) Rather than proceed with the definitions as suggested bywhere Z(4i is a constant instead of a matrix, of modal networks, to imposve thenretrcinofs powmaer inaricance

JUTLY 1964 Barthold-Radio-Frequency Propagation on Polyphase Lines 669

on the 111 matrix, i.e., to stipulate that the sum of powers in each M- Nmodal network be equal to the total phase power. If the com-plex power is defined as

Thus, to invert either M or N matrices, it is sufficient to inter-S |- P + J Q (43) change the rows and columns.

GROUND WIRESthen the phase power isthen_the phase power is

The original Zp matrix for a 3-conductor line with two ground71* t , . * wires for example, would be a 5-by-5 matrix. If a zero valuei _____-p I___=__pl_Zp__ p (44) is assigned to the ground wire voltages, then an equivalent Zpmatrix may be defined which is once again 3 by 3 in dimension.where the prime denotes the transpose and the asterisk the This is apparent if one considers that, for two ground wirescomplex conjugate of a matrix. x and y,

Similarly,

S1-t jm ' n1 = Zm *~ I * (45) *vx= 0 = iaZza+i±Zxb+icZxc+ ixZ x+5yZxyI Sm = V=7(5y==iaZy+ibZyb+i,ZycQ ixZyx+iyZyy (52)Then, recognizing that Equation 52 may be solved for ix and iy in terms of ia, ib, and i,.__ _____ ____ The resulting values of ix and iy may be substituted into the

iN iMl phase-voltage equations corresponding to equation 52. Collect-ing terms results in an equation of the form

1 I' I

0P= ~rnIm N (46) Vp = Zpeq (53)and substituting into equation 44, the phase power matrix Sp where the influence of the ground wires is included irn the modifiedand inode power matrix Sm may be equated, with the result that Z matrix.

N AT_ =1 (47) BUNDLED CONDUCTORSIt is apparent that a complete phase impedance matrix of aSince the N matrix contains only real numbers, the conditions single-circuit line containing m conductors per bundle would befor power invariance reduce to a square matrix 3m in dimension. Since the conductors of any[~ 1, given phase have the same potential, the m equations pertaining

N |\T =1 (48) to any phase may be combined to reduce the order of the matrixto an equivalent 3 by 3. An alternate method is to replace each

Any choice of values for the elements in the columns of the conductor bundle by its geometric mean equivalent and proceedN matri2i:, which assures that N,(nz) Nb (n), and NVc) have ratios with the calculation of an equivalent 3-by-3 impedance matrix.the same as the ratios produced by a solution of equation 37, The latter method, though less accurate, is sufficient for mostwill result in an N matrix which, when premultiplied by its own purposes.transpose, will produce a square matrix with off-diagonal terms DouBLECIRCUIT LINESof zero.The requirement of concern, then, is that the diagonal of this The extension of the foregoing arguments to a 6-conductorproduct be unity. The terms of the diagonal, from upper left system is generally apparent. For a 6-conductor line, equationto lower right, are equated to unity. 37 would be a sixth-order equation with six real roots. In aI,ATa(1)'+ Nr (1) 2+ I'V (1)' solution for the elements of the N matrix, six values of A(n) wouldresult from the solution of six equations of the form of equation1=Ta(i2)'+N7V (02+Ne (i)2 50. The mode-voltage terms in equation 50 would consist of

An (3)2+ NT (3)2 V (3) 2 the square of five ratios plus a value of unity assigned to one1 = Na(')',VT(i)l±Nc( (49) arbitrarily chosen conductor.Since the columns of the N matrix in equation 41 correspond in In general, there will be as many modes as there are conductors,ratios to the mode voltage (or current), equation 49 can be bundled conductors, and ground wires capable of being eliminatedrewritten as by the tecbnique described.

1=A(1)2[(lV:7-(1) (v ci) + j Appendix IL Nomenclature

(2) Vta(2)8 2 V702)- 2 -y propagation constantI=~A(2)2(V:)2~)+ ( +:)±i 3 paecntnV\V6(2)/ V,(2)/ J a = attenuation constant17a (3)2v (3) 2 ~~~=phase constant,

1AO~)'[ Va3)\ (V1(3)\v = phasor voltageVc )] + 1(h)/ + j] (50) it=phasor currentZ=impedance or surge impedance

where AM"8 are constants. By solving equation 50, the magni- Y=admittance or surge admittancetudes of these constants can be found; once they are known p = phase-quantity subscriptthe "normalized" N-matrix terms may be evaluated. m=modal-quantity subscript

(n) = mode-number superscriptN (n)_ (n) Va_ ) M=modal transformation matrixaT - V. (U) N =inverse of Mc~~~~~~~~~~~~~~A=height above ground

N_X_=__no r = conductor radiusATb?i)- D1y = distance between conductor i and the image of conductor]jdy=distance between conductors i and j

NCn-(.C (51)

It is both interesting and useful to note that for the normalized RfrneNV matrix RfrneI I I I, ~~~~~~~~~~1.PROPAGATION OF HIGH-FREQUENCY WAVES ALONG A SYM-

IAT I= |Ml METRICAL THREE-PHASE LINE, A. Chevallier. Revu>e Generale deL'Electricitt, Paris, France, vrol. 54, Jan. 1945, pp. 25-32.

670 Barthold.-Radio-Frequencjy Propagation on Polypha*e LinesI JULY 1964

2. EXPERIMENTAL VALUES OF THE CHARACTERISTICS DETERMINING VILLE POWER ADMINISTRATION MCNARY-ROSS 345-KV LINE, G. E eTHE BEHAVIOR OF HIGH-VOLTAGE POWER NETWORKS AT THE FRE- Adams, M. G. Poland, T. W. Liao, F. J. Trebby. Ibid., June 1959,QUENCIES USED BY CARRIER CURRENT COMMUNICATION CIRCUITS, pp. 380-88.E. Alsleben. CIGRE, Paris, France, vol. III, 1958. 6. RADIO NOISE ATTENUATION AND FIELD FACTOR MEASUREMENTS3. PROPAGATION OF PERIODIC CURRENTS OVER A SYSTEM OF ON THE AMERICAN ELECTRIC POWER CORPORATION BREED-OLIVE3.PROPAGANTION OPEIDCCRETOVRASTMOF 345-Ky LINE, L. 0. Barthold, J. 3. LaForest, R. H. Schlomann,PARALLEL WIRES, J. R. Carson, R. S. Hoyt. Bell System Technical F.3. Trebby. Ibid., vBol.79, June 1960, pp. 303-09.Journal, New York, N. Y., vol. 6, July 1927, pp. 495-545.

7. THE CALCULATION OF ATTENUATION CONSTANTS FOR RADIO4. WAVE PRO0PAGATION ALONG UNBALANCE:D HIGH-VOLTAGE NOISE ANALYSIS OF OVERHEAD LINES, G. E. Adams, L. 0. Barthold.TRANSMISSION LINES, G. E. Adams. AIEE Transactions, pt. III Ibid., P. 97s81.(Power Apparatus and Systems), vol. 78, Aug. 1959, pp. 639-47.

8. APPLIED ANALYSIS (book), C. Lanczos. Prentice-Hall, Inc.,5. RADIO NOISE PROPAGATION AND ATTENUATION TESTS ON BONNE- Englewood Cliffs, N. J., 1956.

Expansion Joint for Pipe-Type Cable LinesN. B. Timpe, Senior Member IEEE

Summary: The Philadelphia Electric Company has developed tion, the fourth method would also require large and costlyan expansion joint which provides flexibility, simplification, and anchors to retain the pipe in a stressed condition. The hori-savings when installing pipe-type cable lines on bridges andaerial structures. Pipe installed on a bridge or on an aerialstructure requires compensation for expansion and contraction 150 tons for a 7-inch pipe.resulting from temperature changes. The temperature range Expansion bends were previously used by the Philadelphiawhich the pipe will experience is determined by ambient condi- Electric Company for lines installed across bridges. Troughstions and operating losses of the line.

were installed at the end of the bridge to hold the expansionbends. In order to force the pipe expansion to occur in the

Methods of Providing for Expansion bend, anchors were installed on the pipe at each end of anExpansion and contraction of the pipe can be provided for bv extensive system of track and rollers, located on the bridge andone of the following methods: < in the trough. The rollers and track guided the pipe expan-

sion and contraction into the bend. The space required for a1. An expansion joint. trough in a congested area was difficult to provide, and the2. An expansion bend. dead air space between the pipe and the wall of the trough3. A completely rigid structure which prevents expansion or created a high thermal resistance, resulting in higher cable tein-contraction of the pipe. peratures. In addition, the troughs became filled wvith dirt4. Prestressed pipe installed so that expansion or contraction and required considlerable maintenance.will subtract or add to the stresses.

It generally is not permissible to transmit forces of the magni- Expansion Joint Methodtude required to the bridge or aerial structure by the third and The 138-kv high-pressure oil pipe-type cable line betweenfourth methods without extensive reinforcement. In addi- the Delaware Generating Station and the Juniata substation

crosses the main line of the Pennsylvania Railroad on the "G"Paper 63-923, recommended by the IEEE Insulated Conductors Street bridge. Two expansion joints and unidirectionalCommittee and approved by the IEEE Technical Operations Com- anchors were installed on this pipe line bridge crossing, result-mittee for presentation at the IEEE Summer General Meeting and ing in a savings of 70,000 dollars over the expansion bend con-Nuclear Radiation Effects Conference, Toronto, Ont., Canada,June 16-21, 1963. MIanuscript submitted October 8, 1962; made struction previously used. Fig. 1 shows a section of one endavailable for printing February 9, 1963. of the bridge, including the location of an expansion joint andN. B. TIMPE is with the Philadelphia Electric Company, Phila- one of the unidirectional anchors.delphia, Pa.

PENS] EPANSION JOIN ANCHOR MANHOLEINSULATOR

I~~~~~~~~~~~~~~~~~~~ ~RSE GAKE IN

Fig. 1. High-pressure pipe-type cable installation on "G"'OTRCSNStreet bridge Fig. 2. Expansion joint

JULY 1964 Timpe-Expansion Joint for Pipe- Type Cable Lines 671'