IEEE Transactions on Power Apparatus and Systems, Vol.PAS-98,No.l Jan/Feb 1979PROPAGATION ON POLYPEASE LOSSY POWER LINES;

A NEW PARAM4ETER SENSITIVITY MODEL

S. Cristina, Member, IEEE M. D'Amore, Member, IEEE

University of RomeRome, Italy

ABSTRACT

The mathematical model of the transmission syst.emtakes into account the dependence of the line and earthparameters on the frequency. The solution of the propa-

gation matrix equations for a polyphase lossy power

line, is obtained by applying modal analysis. In orderto define the effects of parameter variations on modalquantities,a new sensitivity model is presented. The algorithm for the calculation of the modal sensitivityfunctions is suitable for easy computer implementation.The sensitivity analysis of the modal attenuation con-

stants is carried out for horizontal lines with groundwires: the normalized sensitivity functions presentedare of general validity. The computation of the niodalattenuation constants, carried out with the "exact" andthe "approximate" method, leads to values whichprove tobe in closer agreement with experimental results alreadyknown.

INTRODUCTION

The computation of the switching transients, of thepropagation of power-line carrier signals, of the radiointerference level, concerns a considerably wide range

of frequencies. An accurate mathematical model of thetransmission system constituted by an overhead poly-phase lossy line with ground wires, must take into ac-

count parameter variations with frequencv: skin effectphenomenon must be included in the phase conductors,ground wires and earth. In spite of the numerous studiescarried out, the evaluation of the high-frequency con-

ductor resistance and the correct utilization ofCarson's earth correction termsstill seemuncertain. Thenecessity of defining a mathematical model withconstantparameters for a homogeneous line requires the assign-ment of average values to conductor height and to thespecific quantities of the earth. On the other hand theheight of each phase bundle above theground planevariesconsiderably along the line. Some theoretical and experimental studies have shown that the influence of thisdistributed non-uniformity may be seen both as 'a sharpresonance increasing modes attenuation and disturbancesof modal impedances in the vicinity of certain resonance

frequencies' . Furthermore the difficulty of evaluatingresistivity is well-known since it varies considerablywith the nature and conditions of the ground. Among theelectric and geometric parameters which describe thetransmission system, there can be distinguished, therefore, the ones that are constant and known with a goodaccuracy from the ones affected by errors of evaluationand/or variable along the line.

The propagation of the high-frequency voltages andcurrents on polyphase systems can be studied byapplying the modal analysis method: thephase quantitiesare transformed into non-interacting moda quantities

F 78 086-1. A paper recamended and approved bythe IEEE Power System Comunications Ccanmittee of theIEEE Power Engineering Society for presentaticn at theTIEE PES Winter Meeting, New York,, NY, January 29 -

February 3, 1978. Manuscript submitted September 6,1977; made available for printing November 4, 1977.

which propagate according to a law depending on themodal propagation constants which have been the objectof numerous theoreticaland experimental studies. Howe-ver the analytical correlations between each transmissionsystem parameter and the modal quantities are not yetknown. It is clearly seen, finally, how the reliabilityof the mathematical model conditions accuracy in the calculation of the modal quantities and therefore of thesolution of the propagation equations.

Purpose of the paper

This study sets out to:- improve the transmission line mathematical model andthe algorithm to calculate the modal quantities, alreadydefined in a preceding paper2;- define a new model of parameter sensitivity, whichtakes into account the series-impedance and shunt-admittance matrices, the propagation matrix and its cigenvalues;the modal propagation constants;- carry out the sensitivity analysis of the modal at-tenuation constants for horizontal lines with groundwires, and to define normalized sensitivity functions ofgeneral validity;- apply the defined calculation methods to the case of a

practical power line for which the experimental values ofthe modal attenuation constants are known.

1. TRANSMISSION LINE MATHEMATICAL MODEL

The geometrical paranetersLet the transmission system be constituted by p,

three-phase lines with pgground wires. The total num-

ber of conductors is N= N + pg, N-=3.p1 being the num-

ber of the phases. Each phase bundle has an average distance hi from the ground, equal to the maximum heightminus two thirds of the sag measured in the middle ofthe span. Distances di;, Dii, Aij are indicated inFig.l.The bundle conductor oi radius rbi is made up of nisubconductors of radius ri, resistivity pi, permeabilitypi and surface state coefficient mi. If si is the uni-form distance between the subconductors, the equivalentradius of the bundle is equal to: -1

r 1 ~~n.-l n.(1) ri Iniri _.5 si(sin n. ) }

1~~~~~~~

The coeff icients of matrix[A] of ordergarding the geometry of the system, are:

112 ^1

(N, N) , re-

-1A.j = ln (D.id..)

The eZectricaZ LoarconetersIn matrix [A]l- the pg rows and columns relating to

the ground wires are discarded, to have the matrix [F]of order (N,N ):

(3) [F]= [r]in which u nd e r s c r i p t 'r'ind icates the matr ix reduc t ionfrom order (N, N) to order (N, N) .

The matrix of the shunt-susceptance is:

(4) [B] = 2Tr w[ F]co being the free space permeability and w the angularvelocity of the phase voltage. Therefore, neglecting theconductance of the air path, the shunt-admittancematrixof order (N, N), is:

(5) [']= j [B]in which the superscript point indicates complex quan-

0018-9510/79/0100-0035$00.75 1979 IEEE

35

36

Fig. 1. Schematic of conductors.

tity.The

cludesby the

series-impedance matrix of order (N, N),which inboth earth and conductor skin effect, is definedfollowing equation:

(6) [7]= [R] + [Re] +j{[X']+[X"]+[Xe]}

Let the diagonal matrix of the conductor resistancesbe:

(7a) [R] = diag {R1 R2 ... Ri...RN}where the generic coefficient Ri can be calculated bymeans of an expression3 which takes into account theskin effect and the proximity effect of the n8i outerstrands of radius rsi:

(7b) Ri= K n. r (2+n.)]-1Si i Si Si i

as Si= Pip1fTr and Ksi is the stranding factor forwhich an average value equal to 2.25 has been computed.Expression (7b) guarantees good accuracy for fre-quenciesabove4 kHz. The uncertainty in evaluating theresistance of the conductor, prompted consideration ofthree other formulae, alternatives to (7b), shown in Ap-pendix 1. For the purpose of establishing thedifferencesof the results that can be obtained by applying the fouravailable expressions of Ri, four different aluminium-steel conductors were considered with resistivityp= 2.83-10-8 Sn and permeability 11= po= 4tr-10-7Hm1, 19steel strands with a radius of 1.05 mm with p=2010-8SOmand p= 103 po. The values of r, ns, rs are indicated inFig. 2. For each conductor the ratio Rac*Rjl was calcu-lated by means of the formulae presented, varying thefrequency from 5 kHz to 1 MHz. Fig.2 shows the curves

obtained by applying (7b) and (43) of Appendix 1: thevalues obtained by means of eq. (41)and eq. (42) for whicha range of stranding factor is indicated, coincide with

AA..

25 50 100 200 400 600 800

FREQUENCY - KHZ

Fig. 2. VaZues of ratio Ra cR1 for different steeZ-aZuminiwn conductors, computeYby eq .(7b) ( and eq. (43)(-7)

the ones derived from (7b) when the value 1.55 and 1.19respectively is fixed for Ks. From examination of thecurves presented, it can be noted that the values ob-tained with (7b) and (43) are increasingly differen-tiated with increase of the frequency: this divergency,which, moreover, is not considerable, is justified bythe consideration that eq. (43),which uses Bessel's formulae, does not take into account the proximity effectof the outer strands. It is considered opportune, therefore, to give preference to (7b) both because of theaccuracy shown as compared with the experimental valuesand because of the simplicity of the calculation.

Let the matrix of the internal reactances of eachconductor be:

(8a) [X']= diag {X1 X2 Xi Xk}

The generic coefficient can be calculated by meansof the expression:

4 bei (Ci) *bei '(Ci)+ber (C i) *ber ' ( i)

i i lbei'(Ci)12+lber'((i)I2in which ber, bei, ber', bei' are the Besselfunctions, real and imaginary, and their derivatives, with Ci= ri(w pi {i)l/2. The curves of Xi,calculated as a function of the frequency by means of(8b) for each of the four conductors previously con-

sidered, differ by some per cent from the values of Riobtained with (43), to a decreasing extent with the increase of f. It is therefore possible to accept theequality Ri= Xi for frequencies higher than 5 kHz. Onthe other hand the reasons have been set forth why itis opportune to calculate Ri by means of (7b), which istherefore used also to evaluate Xi. This approximationis also justified by the values of the coefficients of[X'], considerably less than those of matrices [X"] and[Xe] which prevail in the formation of the reactancepresent in expression (6) of [fj.

The reactance matrix due to physical geometry of theconductors relative to the earth plane, is equal to:

(9) [X"]= f [A]

po being the free space permeability.Matrices [Re], [Xe], which make up the contribution

of the earth-return path, have the following form:

(10) [Re]= 2le f [P], [Xe] = 2 Pe f [Q]Pe being the permeability of theground. The coefficientsof matrices [P] and[Q]are calculated by means of theCarson's isymptotic or general expressions4, shown inAppendix 2. The two types of formulae areused accordingto the value assumed by parameter rij=Di (2rfvePe)

The conclusions of a study carried out in this con-

nection suggest using the general expressions for rijbetween 0 and 10 and the asymptotic ones for rij greaterthan 10. These indications differ from the ones givenby Carson. For rij between 5 and 1O, the values obtainedby means of the two types of formulae almost coincide.However, this diff erence aff ects considerably the valuesof the modal attenuation Constants. In particular, on

using the general formulae the modal constant of thelowest attenuation assumes higher values, which are generally nearer to the experimental ones.

The series-impedance matrix [Z] of reduced order(N,N) is obtained from [1]-l by elimination of the pgrows and columns relating to the ground-wires-. Assuming

(lla) [H] = r

the following is obtained:

(llb) [Z] = [H1

The modal parcanetersThe propagation equations along a system of parallel

conductors, are expressed with matrix notation:

imageofXith conducbri image of

rj 7jthconductor

(12) [2] [P1 2' rtI](12) ~ 2 24 z dz

in which [I] and [I] are the column vectors of the phasevoltages and currents:

( 13) [V] = [v1v2 .**VN] t [I]= [i1i2...*iN]t'[P] is the propagation matrix

(14) [PI = *[Z]IYand underscript 't' indicates the transposed matrix.

In order to transform each of eqs, (12) into a

system of N equations describing the propagation of Nnon-interacting modal quantities included in the columnvectors:

(15) [V] ()V2).VN0[°=:1I2 ()

we take

(16) [V=M*[V [I=[][°

For the transformation square matrices [M], [N], the following normalized expressions are assumedl:

[t]=*21 *2 2 * * *2N []=*21 .2 2 * 2N

t1 M2 * MNN NNN NN2 N@NN

in which the generic coefficients are equal to:

V(k) *(k)

Mj k = +(k) Njk I(k)V1 1

This normalization requires that number one shouldnot be assigned to a conductor placed in the symmetryplane of the system.

Taking eqs. (16) into account, eqs. (12)become:

(17) d2[ [ ] i- = [A]. [i0]

since(18) [A] [M p M=[]*[]Z iag{l2 N

where J1X2. . N are the eigenvalues of [P and of.P3t.Two eigenvectors constituting the i-th columns of [MJ andof [N] are associated with each eigenvalue Xi.

For a ref lection-free transmission line, the solutionof (17) leads to the following equations:

(19) [°o]z= [1$] [V°]

37

[.][z °0] [. °1

in which there appear the vectors of themodal quantitiescalculated at distance z from the reference point z= 0,and the modal transition matrix:

(20a) [bz= diag{exp(-m(l)z)exp(4m(2)z). exp(m(N)z)}

The propagation constant of the generic mode (k), isdefined by the equation:(20b) ;(k)= {> 1/2 (k) (k)(k)b k (k)in which aL(k) is the attenuation constant and (k) thephase coefficient from which the modal propagation vel-ocity is obtained:

(21) v (k)= w~(k)

Finally, the characteristic-impedance matrix is

( 22a) [z ] = [t41 * Pa2 [Ml * [Z

where r0()() *NLem]= diag{m m .... .mand the modal impedance matrix can be evaluated by thefollowin-g-equation:(22b) [Z ]= I] [ c]' [ ]The modes are numbered in the order of increasing modalimpedances. Therefore the phase quantities, solutionsof differential equations (12), can be calculated withthe expressions:

2 = Ml,¢ z * M]1 IV]

[I]z- NJ ¢ ( z[N] .IO [N]-[Ef" 1 z [M] [P ]

where [Y°]= [°]l

2. PARAMETER SENSITIVITY MODELThe analysis of parameter sensitivity will Ecarried

out for frequencies above 5kHz, for which matrix [X'2 ofthe internal reactances is considered as coinciding with LR]The parcaneters vector

The matrices present inthe expressions (5) of[Y] and (6)of [Z], depend on the parameters of the system accordingto the fqllowing notations:

R]= [X'=fl(rhKsApipenef) [Xe=f4(h,Aij e ,f)tRe] f3(h,Ltij Pe,Pe,f) rXe=f4(h9,Aij PeyPe9f)

[B]= f5-(r,n,ssh, .fTable I. Expressions of the sensitivity functions of matrices [R]=[X'], [X"], [Rely [Xe, [B].

_____L1?1 EII

' e][ ela[B]

x~ [s j=aih. EX,] [r1ax]_a][s. Ee s1Lxi |i [sax] 2 a S3iaS1~D X 1l ax 34iaxJ axi 5iJ ax i

x r [W 1] [R] i f [W21] | 0 0 M[ W5 11 *B].x2=Ks EW1[wl2][R] 0 0 0

x3=p [W13]. [R] 0 0 0 0

x4=-} [W14w1j [R] 0 0 0 0

x5= n [W 5].[R] | Di f [W25] 0 0 -[F]X[W5] [B]x6= s PO f[W26] 0 0 [B]

|x7=h | OP0

f IW27] 2pef [J371 -211 ef [W47] | [FXpW5 7] .[B]

x8=Aij | 00

f [W28] -2|ief[W38] -2|Pef[W48] -[F]-[W 8] * [B]

x9= Pe °__efp_el_3_ | 9fp[Wef] 2f1e[W49]X'O1 1e 0 0 2f [[P]-[W39]] 2f[[P]-[W4] 0

(2f) [R] | f1 [XI] 2PeI[P -[W39] | 2ipe[[P]{W491] f 1 [B]

38having considered P0 and co invariant quantities.The parameters column vector, of order Np- 11, has theform:

(24) [Xp]=[r KsppunshAij pe 11e f]tThe parcaneter sensitivtty functions

The sensitivity functions of [Y]and [Z]with regardto the generic parameter xi, present in the vector indicated in (24), are:

(25) 1x] a= ' - a [zi i= l-NpTaking into account the equations (5),(6),(llb), to

calculate (25) it is necessary to obtain the followingpartial derivatives:

(26) [s1i~I=~[Rl = M~.1 [2~-= ' il-aX. ax.1i i= 1-N

(26) [iax 5pax S}x

the first four matrices being of order (N,N) and thelast of reduced order (N,N). The calculation of (26), carried out for each of the eleven parameters considered,led to the expressions shown in Table I: the coeff icientsof the matrices presented have the form indicated in Appendix 2.

The following expressions are therefore obtained:

(27) [S*y] = Xs L1*[Z] II][L] i= 1-Npin which matrix [Dxi] of order (N,N) is equal to:

(28) [X]= [ Ilb1 i-§xwhere

(29) Cs~~[zx]=axiTaking into account (14), the sensitivityfunctions

of the propagation matrix are obtained:

(30) [P] = [Z ][]{Y i= 1-NThese lxi1-li i xiI

These sensitivity matrices can be used to carry out thesensitivities of N ei§envalues Xk of P. The following,in fact, are :dbtained:OX1 *Pk{S.* k*N]} M0k k= 1-Nx3j xs Lk

in which [skI' [Nk] are the eigenvectors respectively oft and Pt, associated with the eigenvalue Xk,and the as-terisk indicates the scalar product of two vectors.

Taking (20b) into account, the sensitivity functionsof the modal propagation constants m(k) are:

(32a) m(= (2 (k) ) 1AXk k= 1-N

xi x

Then, too, the following expressions are true:

(32b) *m(k) = sa(k) +jS (k) k= 1-N

Comparing (32a) with (32b), the sensitivities ofthe attenuation constants and of the phase coefficientsare defined as follows:

(33a) Sa = Ret(2r(k)1Xk k=1-Nxi Xi

(33b) S (k = Im{(2;(k))1l.gXk} k= 1-N

Finally for modal propagation velocities we havethe expressions:

v(k) v k =(k)(33c) 5v 5~xi (k) x.

In order to make the N definite sensitivities,con-cerning the same modal quantity, comparable with oneanother, the (33 a,b,c,) are normalized to obtain thenew expressions:

-3a a-(k (k) S k)a

(34b) v(k)= x(i S(k)

(34c) s()x sv (k

k= 1-N

Taking (33c) into account, (34c) can also rewritten inthe form: (k) g(k)

x. x.1 1

Let ixPo be the vector including the nominal valuesof the parameters of the reference transmission system:the (34 a,b,c,),calculated for this vector,constitute ameasure of the infinitesimal variations of the modalquantities due to an infinitesimal variationof each parameter. Therefore it is possible to establish to wyhatparameter each modal quantity is most sensitive, byevaluating the normalized coefficients of the jacobian:

-Y(1) S(3 . s()

[ SY(N) Sy(N) SY(N)(35) Lxi x2 xll

in which (k) coincides with one of the quantities a(k)B(k), v(kT relative to the k-th mode.

Suppose a finite variation is attributed to the parameters of the reference system: let the new configura-tion be decribed by vector lXplj. The normalized vector,including the parameter variations, is defined:(36) [AXX]=I[^ Ax2 .I tllttas the generic coefficient ARi= (xi1-x. ).x.i.

The consequent normalized variations of the modalattenuation constants and of the phase coefficients areincluded in the following vectors:

(7)[hA-]= [tA^(1) ,A_(2) ^A(k) A(N)]t(37b)~ ~~~A[°] ^() f(2) A(k) -f(N)]t

as ( k) A (k)= (k) (k)) (k)

The vectors (37a,,b)can be calculated by expandingeach modal quantity in Taylor's series at [Xpj. To first-order approximation, the following expressions aretherefore obtained:

(384)[A-ao]= [30]*[AX I.1p(38b) [A^aO=EJS] *[AX]

in which the jacobians [Ja],[jP] have the form indicatedin (35).

The comparison between the values of (37 a,b ) ob-tained by applying the method described in section 1 andformulae (38 a,b), makes it possible to establish themaximum admissible variations A i for each parameter.The approximated eqs. (38 a,b) are particularly usefulwhen values ack), a,Qk)relative to the reference system,have been obtained experimentally. Furthermore, takinginto account the variations due to uncertainties ofevaluation of the parameters such as for example Ks,h,Pe, the corresponding errors of the modal quantities arecalculated by means of (38 a,b).

3. SENSITIVITY ANALYSIS OF MODAL ATTENUATION CONSTANTS

Base-case sgstemIn a first utilization of the model def ined in

sect. 2, the sensitivity analysis of the modal quantityof greatest interest, that is the attenuation constant,is carried out, for the case of horizontal transmissionlines with ground wires.

The base-case system is constituted by a power linealready considered in the preliminary studies carried

1'1J' + IS3 i1i +j { [SI 'J + IS2 'J + IS4 Ji 1

out in the Italian 1000 kV Project6, having the geo-metric configuration and the parameters ind7icated inFig. 3. The attenuation constants a,(k)and the respect-ive normalized sensitivity functions (k), indicated in(34 a), were calculated, varying the friquency. InTableII there appear the sensitivities values for each of theeleven parameters present in (24), calculated for theindicated frequency values fj, with j=l-5. In particularfor the parameter X7= h- the sensitivities were calcylated with regard to the he6ght of the central phase Sx7aof the, lateral nhases, S*7b, of the three phases,The analysis of the values obtained makes it possible toestablish a comparison among the sensitivities of eachmode with regard to all the parameters, and among thesensitivities of the three modes with regard to eachparameter. Let {gj(k) }fl and {tR(k)}f2 1e the values forthe first two frequencies fl,f2 and {.kJ}f12 the ar-ithmetic mean of these values: the calculations carriedout suggest conpidering value {Sak)flforfLf<(f..f)/2and value {SX }1fl2 for (f2-f1) 2<f<f2. Going on ina similar way for the successive frequency intervals, thecomplete spectrum of each sensitivity function is ob-tained.

Approximate caZcuZation of the sensitivity functions.

If the transmission system has different parametersfrom the ones considered in the base-case, it is necessary to calculate the new sensitivities by means of thealgorithm described in sect. 2. However, it is possibleto develo ) an approximate calculation byutilizing thevalues {SR3 )}bc presented in Tab. II for thebase-case,multiplied by two suitable factors vl,v2 evaluated onthe basis of the results obtained in the numerous applications carried out. The variation of a parameter as

39

Fig. 3 . Base-case transmission Zine configuration

compared with the base-case determines, thtrfore, new

values of all the sensitivity functions SgiK :itwas sujpposed that these normalized variations are limited only toparameters r, Ks, n., h, Aij' peand have the values indicated in Tab. III. If only one parameter is modifiedthe new twelve sensitivity functions for each tode are

obtained by multiplying the corresponding {SR, hb by

the values of vl, relatile to each parameter xi, indi--ated in Tab. ITI and by the values of v2, indicated inTab. IV, which depends on the frequency considered. Ifthe normalized variations of parameters fall between twovalues indicated in Tab. III, it is necessary to takeinto account that vl varies linearly in each interval.If various parameters are modified at the same time,

T.a a(k)Table II. Values of modal attenuation constants normalized sensitivitiest3 1bc for base-case 8ystem.

|xl=r x =K X3=Q x4=y xS=n x6s x =h x =,A XSQe Xig=e x,=f

|Mode f kHz | 5a(k) | Sk)a (| a k-a k a(k)S| x ak -a |k a | .a(k) | a0(k) | ..a(k) | 0(k)

5 _0.853 0.898 0.446 _Q520 0.438 1.090 0.239 _0.102 0.312 _0.028 0.092 0.579

50 _0.531 0.572 0.279 _0.184 0.438 0592 1.408 _0.791 1.676 _0.184 0.592 0.898

250 _0.210 0.252 0.123 0.127 0.439 _1.210 1.318 _1.704 2.762 _0.169 0.911 1.043

500 _0.134 0.176 0.087 0.200 0.439 -1.632 0.816 _2.164 3.014 _0.056 0.876 0.968

1000 Q.0.090 0.133 0.065 0.243 0.440 |1.916 0.687 |2.255 2.895 0.113 |0752 0.821

L5 | 0.129 0.214 0.101 0.111 |Q367 0 0.285 0844 Q685 |_Q241 0.750 1.134

50 .0.095 0.156 0.023 0.269 0.370 0 . D0.426 |-0.426 1.248 | 0.172 0966 1.094

2 | 250 |0.052 OD99 0.010 0l296 0Q373 0 .1.109 .1.109 1.215 0.076 0.808 0.874

500 _0.041 0.084 0.008 Q300 0.373 0 _1.234 _1.234 1.120 0.152 0.751 0.805

1000 Q0.033 0.074 QX007 0.304 0.375 0 _1.423 1.423 1.092 0.269 0.648 0.694

5 0.017 0.210 0.003 0.197 0.239 _0.023 _0D59 _Q099 0.106 0.066 0.567 0.829

50 0.021 0.190 0.002 0.219 0.262 -0.164 _Q33 1 _0.510 0.087 0.167 0.543 0.738

13 L 250 0.022 0.137 0.001 0Q230 Q0274 |0.276 |_0494 |0.786 0.027 0Q288 0.512 0.643

500 0a023 0.122 0.001 0.233 0.277 |_0.312 |0.542 0.871 0.007 |0340 0.490 |0599

1000 0.023 0.111 0.001 |0236 0.280 |.0.339 | 0.567 |.0.925 0.016 0.367 0.486 0.577

1.''"M- n=6

* * , r = 15.75 mm

15.2 m rb=450 mm

9L.80m 19.80m

3rw = 14.70 mm36.9m g

1~~~Qe=250 Dm

iz=4Jle107 Hn'

h=23.43 m

.a(k)Table III. VaZues of sensitivities {Sx. eec crrectionfactorv, for parameter normaZized deviationsfrom base-case.

F) T ? ^s A n [ AhLtij (x162) ^

E a -0.302_+0.25 1+0.51 -0.13 F+0.26 1+0.45 _3 1 +2 +6 +10 IPj.211+0.io1o0-43-05 1-0.34 1-0.251+0.25 i0.881+3.oo1.333 0.824 0.695 Q.886 1.214 1.357

1.282

1.293

0.268

0.984

0.907

0.903

0.905

0.899

0.905

0.905

0.873

0.788

1.081

1.364

0.912

Q987

1.000

0.991

0.991

Q992

0.987

Q991

0.997

Q995

1.007

1.357

Q943

0.989

1.000

0.997

0.997

0.963

1.000

0.998

1.000

0.841 0.730

0.846 0.724

1.425 1.740

1.009 1.018

1.051 1.088

1.053 1.092

1.052 1.090

1.056 1.096

1.053 1.089

1.053 1.091

1.027 1.047

1.269 1.588

0.960 0.939

0.764 0.636

1.054 1.095

1.008 1.016

1.000 1.000

0.727 1.007

1.004 1.007

1.003 1.006

1.000 1.013

1.004 1.007

1.002 1.003

1.045 1.045

1.000 1.000

0.786 0.643

1.039 1.065

1.007 1.011

1.000 1.000

1.002 1.002

1.000 1.001

1.074 1.074

1.000 1.000

1.000 1.000

1.000 1.000

Q901

0.902

1.197

1.000

1.128

1.010

1.034

1.032

1.036

1.034

1.017

0.885

0.919

06845

1.010

1.000

1.000

1.046

1.046

1.016

1.000

1.011

1.005

1.045

0.920

0.857

1.004

1.000

1.047

1.052

1.050

1.185

1.007

1.010

1.006

1.179

1.187

0.646

1.298

1.309

0.409

1.000 1.000

0.787 0.659

0.973 0.950

0.937 0.896

0.941 0.901

0.935 0.893

0.937 0.896

0.967 0.946

1.192 1.308

1.131 1.212

1.182 1.364

0.986 0.973

1.000 1.000

1.000 1.000

0.917 0862

0.917 0.862

0.970 0.950

0).987 0.987

0.979 0.965

0994 0.990

1.000 1.000

1.139 1.219

1.143 1.357

0.996 0.991

1.000 0.996

0.917 0.862

0.906 0.846

0.908 0.847

0.704 0.519

0.990

0.980

0.991

0.979

0.969

0.986

1.571

1.587

1.610

0.800

0.654

0.804

0.802

0.802

0.806

0.799

0.801

0.896

0.885

1.192

1.909

0.223

0.670

1.000

0.979

0.979

0.998

0.987

0.980

0.991

1.773

1.022

1.928

0.335

0.701

1.007

0.987

0.987

0.219

1.007

1.002

0.995

0.805

0806

0.805

2.071

1.153

1.066

1.065

1.066

1.060

1.065

1.066

1.035

1.058

0.949

0.718

1.247

1.137

1.000

1.005

1.005

0.995

1.000

1.005

1.003

Q818

1.000

0.714

1.196

1.113

0.996

1.006

1.005

1.40 7

0.997

0.998

1.002

0.581

0.583

0.569

3.480

1.392

1.143

1.139

1.141

1.121

1.1421.141

1.074

1.057

0.949

0.482

1.571

1.337

1.000

1.010

1.010

0.984

1.000

1.008

1.005

0.591

0.993

0.486

1.430

1.266

Q.989

1.016

1.012

2.000

0.993

0.994

1.005

0.452

0.456

OA447

4A472

1.589

1.188

1.178

1.183

1.149

1,183

1.182

1.094

1.115

0.879

0.364

1.801

1.496

1.000

1.013

1.013

0.972

1.000

1.010

1.008

0.445

0.985

0.364

1.587

1.380

Q.982

1.024

1.019

2.481

0.990

0.990

1.008

0.148

0.298

0.309

2.386

1.011

1.004

0.100

0.981

1.013

2.260

1.420

1.290

0.105

0.212

0.445

1.081

1.038

1.000,

1.025

1.025

0.454

0.554

1.209

1.172

1.182

0.358

0.643

1.122

1.117

1.004

0.914

0.942

0.926

0.892

1.117

1.118

0.490

0.583

0.593

1.803

1.002

1.005

0.468

1.370

1.038

1.580

1.222

1.144

0.127

OA65

0.673

1.034

1.011

1.000

1.13 3

1.133

OA93

0.382

1.119

1.085

1.091

0.613

0.786

1.043

1.040

1.101

1.022

1.0 33

0.980

0.969

1.070

1.056

1.248 2.238 3.548 2.600 2.076 0,457 0.795 1.524

2.036

2.081

1.102

0.995

0.442.

3.041

0.799

0.988

0.750

0.542

0.720

5.173

3.828

1.273

0.997

1.019

1.000

0.803

0.803

0.152

1.118

0.585

0.830

1.000

1.873

1.200

0.996

1.000

1.043

0.593

0.684

0.456

0.906

0.770

0.908

3.163

3.154

0.10

1.220

1.010

3.076

0.110

0.260

0.225

0.254

0.613

1.404

1.293

3.000

0.990

1.129

1.000

0.868

0.868

0.605

0.286

1.036

1.046

0.864

1.139

0.929

0.865

0.869

1.072

0.961

0.964

6.667

0.917

1.021

1.023

2.349

2.317

0.205

1.120

1.015

228)

0.400

0.584

0.509

0.530

0.757

1.173

1.14 1

1.818

1.007

1.067

1.000

0.914

0.913

0.575

OA50

0.998

1.038

0.955

1.102

0.929

0.922

0.920

1.112

1.004

1.005

6.148

0.941

1.014

1.017

1.901 0.544

1.878 0.56

0.500 1.709

1.077 0.948

1.050 1.045

1.885 0.423

0.642 1.120

0.743 1.050

0.675 1.089

0.685 1.144

0.838 1.070

1.096 .0.942

1.101 0.929

1 A55 0.709

1.010 0.993

1.043 0.976

1.000 1.000

0.941 1.027

0.941 1.027

0.551 0.407

0.605 1.447

1.027 0.965

1.030 0.966

0.955 1.045

1,073 0.934

0.929 1.000

0.948 1.039

0.945 1.040

1.123 1.138

1.012 0.967

0.991 0.992

5.741 6.000

0.955 1.045

1.010 0.990

1.012 0.988

1.020

0.967

0.701

0.470

0.510

0.503

1.168

0.871

1.532

0.591

0.644

0.135

1.333

1.000

0.716

0.469

1.000

0.934

0.934

0.737

1.000

0.621

1.686

0.955

1.409

1.929

0.730

0.474

0.808

0.663

0.699

0.700

1.111

0.764

0.813

1.202

1.203

0.614

0.998

0.612

1.310

0.982

0.980

0.793

0.905

0.942

1.577

1.374

1.091

0.993

1.000

1.000

0.803

0.803

0.436

1.171

0.986

0.965

1.000

1.212

1.071

0.991

0.993

1.145

0.933

0.935

1.148

1.000

0.961

0.978

1.417

1.398

0.228

0.995

0.770

1A93

0.738

0.850

1.645

0.996

1.054

1.212

1.121

1.000

0.990

0.995

1.000

0.751

0.751

0.740

.O000

1.274

1.263

0.950

0.949

0.664

0.974

0.967

0.975

0.975

0.938

0.700

0.622

1.096

1.176

40

1

2

3

r

Ks

n

s

hahb

hc

AijOe

/Le

r

Ks

n

S

ha

hb

hc

aijQe

Pe

Ks

n

s

ha

hb

hc

AiiQe

/le

41Table IV. Values of sensitivi-ties {?=(k)l correction faotor v2.

xi bc

each function Sxj results from the arithmetic,meanofthe partial values obtained for the variation of eachparameter.

The example described in sect. 4 will clarify the

method of calculation proposed.

Approximate calculation of modal quantitiesThe modal attenuation constant.s and the phase coef-

ficients can be calculated by means of the approximatedformulae (38 a,b). For this purpose the maximum norma-

lized variations of the parameters indicated in (36)were defined, taking into account the results obtainedin the numerous tests carried out by modifying the par

ameters- values indicated for the base-case. In absol-ute value the following is obtained:

IAr|l=0V15 IAkZ 1=0,50 1|A1=0,25 lAPv|=0.25lAfii=0.50 |A-S 1=0.25 1ARI=0.15 AA..I 1=0.15

A^e I030A11 1=0.30 jAf1=0.20

4. APPLICATIONS TO A PRACTICAL POW4ER LINE

A 500 kV horizontal transmission line7 with groun:dwires was considered: the geometric config uration aildthe parameters of the transmission system are indicated in Fig. 4.

By means of the algorithm described in sect. 1, theattenuation constants of the first and second modefor whichthe experimental values are known7, were ca.lculated,. for

frequencies ranging between 50 kHz and 400 kHz. The re

sults obtained are shown in Fig.5. The attenuation con

stants were found.also by utilizing the approximate formula:(k) (k) (La (k)(39) {ac } {cr } -t-(--- )f. 1(f -f. l) k= 1,2

fi fj-1 3f fj-1 j j-1

haying assumed for f0= 50 kHz,{fa( I) {cra(2) coinciding with the experimental values indicated in Fig. 5 .Further on are shown some values of the partial derivatives of a (l), ac(2) with regard to the frequency, cal-

culated by means of the

f kHz |

a(1) -lof afxlO

a (2) -loafc -x

Npm1Hz

Npm1Hz

method

0.187

1.462

I described in sect. 2:

100 200 1 250 1 380

0.169

1.212

0.132

0.927

0.126

0.846

0.110

0.705

In the computation iterative process a constant increase Af= f -fj -1= 5kHz -was f ixed. The re-sults obtainedfor c(l) ,a(2 appear in Fig. 5.

Fig. 4. Configuration of a 500 kV horizontaZ transmis-sion- Zine7.

prmeterh ifi |r | K Q | n | | ha hb hc Qe ||

E f kH

1.06244 1.10360 1.10067 1.59408 1 1.23855 0.91344 1.37015 1.27787 1.03734 1.10349 1.09941

50 0.96207 0.98421 0.98078 1.90401 1 1.50896 0.89404 1.09468 1.08336 0.92821 0.99235 1.04984

1 250 1 1 1 1 1 1 1 1 1 1 1 1

500 1.11007 1.06622 1.05536 1.30093 1 1.10975 1.26814 0.97768 0.97236 1.16359 1.01643 1.00840

1000 1.10321 1.03704 1.04606 1.51765 1 1.11926 1.24938 1.02363 0.98844 1.07892 1.04000 1.02003

5 1.03241 1.19796 0.99810 1.03063 0.99813 1 3.46714 1.05001 1.07538 1.27100 1.00798 0.97000

50 0.99497 1.01552 0.99000 0.93210 0.99675 1 0.85090 0.90876 0.98917 1.37813 0.99336 0.98000

250 1 1 1 1 1 1 1 1 1 1 1 1

500 0.99035 1.00781 0.93742 0.91205 1.00061 1 1.04419 1.01272 1.01403 1.21266 0.99281 0.95000

1000 0.97229 1.01254 0.92845 0.92894 1.00420 1 1.05469 1.01743 1.00763 1.23897 1.00158 0.99000

5 0.96987 1.01366 1.06371 0.99506 0.99670 2.73350 0.98686 1.29406 0.53831 0.98521 1.00943 0.99833

50 0.96670 1.00178 1.06042 0.99882 0.99424 1.09416 0.98914 1.02570 0.59635 0.97249 1.00214 0.99722

250 1 1 1 1 1 1 1 1 1 1 1 1

500 0.97482 1.00157 1.01073 1.00215 0.99835 0.98456 1.00123 0.99614 2.58448 0.99486 1.02367 1.01759

1000 0.98880 1.00355 1.04117 1.00051 1.00227 0.96592 1.00329 0.99526 0.70922 1.02352 1.00501 1.00129

42

20 304z

30F +

z + +tMode2

ooU

0 2-

z

0

0

20 100 180 280 340 420FREQUENCY - kHz

Fig. 5. Values of modal attenuation constants for a500 kV transmission line7.measured7- computed by 'exact' method- -- -

computed by 'approxcimate' method:+ exact sensitivitieso sensitivities from base-case values

Thetained

(40)

partial derivative present in (39) canbeob-also by means of the expression:

3a(k) a(k) _a(k) a(k)k=2= s = s k=1, 2

_f f f fa(k)

deriving from (34a).Sensitivity function Sf was calculated by means of the approximate method described insect. 3. The parameters of the line under considerationdiffer from those of the base-case system in agreementwith the following values:

A-r=-0.134, An=-2, Ah=-0.343, AA. .=-0.384, A$ =-0.880.3-9 a(k) e

These variations makesensitivities Sf with k=1,2,different from the reference values {Sf }bc. Factorsvi calculated for the variation of each parameter, are:

Mode A? An Ata Aij AQe1 0.943 0.931 1.232 0.721 0.6442 0.998 0.994 1.137 1.040 0.686

Having fixed the frequency, the value of v2 is seen inTab. IV and value of {S kk)1bc inTab.II. The products {Skf}bc-Vl-V2 corresponding to the five va-lues of v1 are therefore obtained. The arithmetic meanof these products is rked out in order to have thesensitivity sought Sf . The calculation, repeated forthe five frequencies defined in the base-case, led tothe following values:

50 [ so250 _ 500 I 1000 l

0.842 0.932 0.872 0.748

1.041 0.848 0.742 0.667

Following the method set forth in sect. 3, the normalized sensitivities were calculated also for frequencies lying between two consecutive values indicatedabove. The calculation of the derivatives was then carried out by means of (40): the values obtained are incloser agreement with those above presented. The at-tenuation constants, the values of which are shown inFig. 5, were calculated again with (39), having assuned,at every calculation step, a f requency increase(f 1-f j_1 )=0.2 fj1l.

CONCLUSION

The reliability of the mathematical model of thetransmission line and of the algorithm for the compu-tation of the modal quantities, is proved by the good

accuracy of the analytical results compared with the experimental ones.

The parameter sensitivity model proposes a compu-tation method that is very suitable for a computer im-plementation. It makes it possible to obtain the sensi-tivity functions of the modal quantities for a line ofgeneral configuration; they can be utilized both duringdesigning and to ascertain the effects due to errors ofevaluation of somieparameters. The sensitivity functionsappear in the formulae for the approximate calculationof modal quantities.

The sensitivity analysis of the modal attenuation constants carried out for a horizontal line with ground wires, led tothe evaluation of normalized sensitivity functions, comparabewith one another, for eleven line paranmeters . These sensitivities, multiplied by suitable factors, cai be utilized also forsystem with diff erent paraneters, and therefore must be consi-dered of general validity for horizontal lines withground wires. The modal attenuation constants computedwith the 'exact' and the 'approximate' method in theapplication carried out for a practical power line, showexcellent agreement with the experimental results.

APPENDIX 1

Three calculation formulae of high-frequency con-ductor resistance are considered:

-1 1/2(41) R.= K U(n r )1 Si i i i~~

withS.i P.1'. f ir and 1.25<K < 1.80i 1, il,i -5s-

(42) R.=K . -R- 10- [KI ( i f ) +K2] Q-m

w-here R. is the conductor resistance measured at powerfrequency, pi the relative permeability, Sali the alu-minium-section in cm2, K1=0.354, K2= 0.2, 0.25dKE<2.75

ber(Ci)bei' (Ci)-bei(Ci)ber' tCi) P i(43) R.=0.5r. 12 2

11|bei' (4 ) j21ber' (G i) 12 wr2i '

in which ber, bei, ber', bei' are the Bessel functions,real and ima in7ry, and their derivatives, with

(WVriUipT ) 2.

APPENDIX 2

Matrices [W1l], with j= 1-5, [W211, [W251, [W26]order (N,N), are iagonal. The generic coeff icientwith i= 1-N, has the following expressions:

W I r -(n r -l1 -1

LW14] IW151 [W2611

1 -11

1Wi [05pi | -n 2(ni-1)5(nisi)_ ~~~~~~[W25].

ofWi,

-2 Si lnjW. -n. {l-lnj -I+2(na-lI -11 31 2n:'risin (7Tnj-l) tg(inn )1

Matrices [W51], [W55], [W56 , of order (N,N, resultrespectively from [W21], [W25j , LW26] by discarding the Pgrows and columns relating to theground wires. Thematrix[W27] w7as defined with regard to the height of each phasebundle. For a three-phase line with two ground wires,the following matrices are true:

afl a,2 a83 a4 a15821 0 0 0 0

a31 0 0 0 0

a8l 0 0 0 0

a5 0 0 0 0j

0 b, 0 0 0

b2 b22 b23 b24 b2b

0 b32 0 0 0

b42 0 0 0

L0 b52 °0 0

43Table V. Expressions of coeffi&c-ients skd

43

0 0 C13

0 0 C23

C31 C32 C33

0 0 C43

0 0 C53

0 00 0

C34 C.350 00 0

the coefficients have the following expressions:

The coefficients of matrix 1W281 are:

W..= 0 ~~-2 _-2W W..= A.. (D.. -d..).11 ' 1j ]J ij ij

Matrices [W57], EW58] are obtaindd respectively by reducing matrices [W27], [W28].

The matrices which take into account the earth-returnpath are:

a[- n] j= 7-11EW3j]x='xj [W4j] ax 7

Katrices [P] and [Q] are calculated by means of theCarson's asymptotic or general expressions4.Asymptotic Formulae.

ij = k Pk QiiP 3Pin which:

Cos-ij cos2 co's31 3cos55pi =

112P2 P32 112 3 %= 211252 ..r. r.j rjwit h

r*j = 2112 D11 , jj=arcsin4iD;j=.^j)We take:

- sinkj tgOe.1- 2152

2 ri, in i tgipi 2!12 Ij

qk= Pk tg2oi j

sin 2Aji tg4.22r?

3 sin5o. tg1.12112 r5

k = 1-4 .

Matrices [W37, [W47] have the form indicated in (44)The coefficients of matrices [W3j],[W4j], with j= 7-9,are the following:

vvij(hi+hj) [p1- pl+2(p2+ P2)+3(p3-p;Y+ 5(p4- p4)]4 Djj2[p1+q1+ 2(p2- q2)+ 3 (p3+ q3) + 5 (p4+ q4)]

0.5 p,+ p2+1.5 p3 +2.5p4

(h +hj)EDji[P1- p -3(p3- p) + 5(p4-7P4)]A4j Dij2[p1 + q1- 3 (p3+ q3) + 5 (P4+ q4)]

0.5 (P1-3P3+5 P4)

=0.5 1 s = 16,=o-24S = 0 5 y1,2= o.8941o'Qvg 13-=o..64 25 621= .~t

S22=-0.694-10P3 S33. 0.116610 / S31=0.833'1062 S32=-0.347-10i34

-.l4 -1-3S33= 0.116- l% s4, =0.833-10 L2 S42 =-0.347-10 p4 s43=0.116 10 I*

71h(0.5

r; 2h ii (0.5 h = 1-6rj)cos(2ho~1 ,h=1=O5 j) I(2O6

go= 0.333ri. cog1o =32-0.635 10 3r. cos51d 0r,,0.102-164ru C0891%.11 I II ii I2 I

I1 1.25 si, 022.1.958 812 (123= 2.366 8S13

31=0.22210 r1 cos3 a32S-O0.4O1 r cos7 3=0.712 10 rrosnI.1131 = 0I 32- j 3307118G41=1666 831 042= 2183 032 43.2.521 933

(Y" = all a.2= a.2 C1'3 = 063=1.25 821 2=2=1.958 822 (=2.386 823___2 ______2=_ 1 (933 (23

31 = 1631 a32= a832 (=533

..4j=1.66" S4, Cr4|2= 2.183 S42 943=2.521 843

The sensitivity matrices which appear in Table I,referto asymptotic formulae of coefficients P.. Qi.jGeneral Formulae

Pij (1-a3 ) + ' log 2+ "2 ij +

i 4 2 y rj 2 2 2 2

1 - a log 2 '1Ai U + "2 Yr 22 8 24 24 Y r~~~~~~~~~~~ij

where 3uk- Ij skj

1

3

a'li 'kj1=. k= 1-4

in which coefficients skj' Okj have the exressionspresented in Table V. Unfortunately the partial derivatives of Pij, Qij are nOt shown in the paper for lackof space.

REFERENCES

(1) M.C.Perz, "Propagation Analysis of HF Currents andVoltages on Lossy PowerLines" IEEE Trans., P.A.S.,Vol. 92, pp. 2032-2043, Noyember/December, 1973.

(2) S.Cristina, M.D'Amore, "EHVAC PolyphaseLossyPowJerLines Corona Performance Computation", A 77 560-6 pre-sented at IEEE PES Summer Meeting, Mexico City, July17-22,1977.(3) R.H.Galloway, W.B. Shorrocks, L.M. Wedepohl,"Calculation of Electrical Parameters For Short and Long Polyphase Transmission Lines", Proc.IEE, Vol. 111, No.12,December, 1964.

(4) J.R.Carson, "Wave propagation in ovexhead wireswith ground return , Bell Sys. Tech. J. , 1926, Vol.5,pp. 539-554.

(5) Faddeev, D.K., V.N. Faddeeva,"ComputationalMethodsof Linear Algebra", San Francisco, Calif.,Freeman,1963,pp. 228-229.

(6) L.Paris, F.Reggiani, M.Sforzini, M.Valtorta, "TheItalian 1000 kV Project",IEEE Canadian Communication andPower Conference, Montreal, November 1974.

(7) B,Bozoki, D.E, Jones, "Power Line Carrier CouplingInvestigations on a 500 kV Line", IEEE Transactions on

P.A.S., pp. 197-200, March, 1965.

[w27]

(44)where

[w37

[w31[W471[Ww

[INJI

L.

44

Savenro Cristina (M'76) was born in Montevago,Italy, on October 18, 1948. He received a doctordegree in electrical engineering, in 1974, fromthe University of Rome, Rome, Italy.

In 1974 he was appointed Researcher at theInstitute of Electrical Engineering, University ofRome. His main interests are corona effects andpropagation's analysis on EHV transmissionlines.

Dr. Cristina is a member of AEI.

Marcello D'Amore (M'73) was born inMesagne, Italy, on May 4, 1942. He received adoctor degree in electrical engineering. in 1966,and the post graduate degree in nuclear engineer-ing, in 1970, from the University of Rome,Rme, Italy.

In 1969 he joined the National Committeeof Nuclear Energy where he performed the postgraduate thesis on modern optimal control ofCirene prototype nuclear plant. In 1970 he wasappointed Assistant at the Institute of Elec-

trical Engineering, University of Rome. In 1974 he became anAssistant-Professor of Basic Electrical Engineering at the University ofRome. His main interests are basic electrical engineering, corona effectson HV transmission lines, computer application to modern analysis oftransients on power systems.

Dr. D'Amore is a member of the Italian Electrical and ElectronicSociety.

Discussion

R. G. Wasley (Drexel University, Philadelphia, PA): The authors pre-sent the results of a sensitivity study related to the modal characteristicsof multiconductor transmission lines. An interesting feature of thepaper is the generation of an approximate method for providing modalattenuations from sensitivity values which depend upon partial deriva-tives with respect to frequency. If such approximate methods could beextended to system performance calculations, especially those systemswith transposed line sections, then a valuable tool might be found forthe more expedient assessment of various system configurations and, inparticular, a method which might quickly indicate which of the possiblecoupling arrangements should be avoided because of sensitivity tomodal cancellation effects.

In keeping with these comments, I would suggest that a sensitivitystudy for a complete power line carrier channel would be morebeneficial than a mode by mode sensitivity study. In practice, it is theoverall system performance which must be assessed and this depends, ineach case, on the modes generated and their interaction at the receivingstation. Thus far, the sensitivity study of the authors does not addressthis interactive behavior and, in consequence, the results obtained aredifficult to interpret and apply.

In regard to the approximate evaluation, which was applied ap-parently to aerial mode attenuation calculations only, can the authorsindicate whether a similar calculation was made for the ground modeand compared with an accurate calculation, since no correspondingmeasurements were available? If so, I would appreciate the authorscomments as to the possibility of calculating the ground mode attenua-tion by the approximate formula. Have the authors considered approx-imate formulas for mode velocities?

Manuscript received February 16, 1978.

S. Cristina and M. D'Amore: The authors wish to thank Dr. Wasley forthe appreciations and the interesting questions presented in his discus-sion.

The approximate formulae for the calculation of the modal at-tenuation constants, which aroused Dr. Wasley's interest, are one ofthe possible applications of the sensitivity model. In a number of studiescarried out in the past, the attempt was made to show empirically the in-fluence of some parameters on modal quantities: the formulaepresented in the paper, make it possible to carry out the completeanalysis of parameter sensitivity.

In the second phase of the research the authors proposed to definea parameter sensitivity algorithm of voltage attenuation for a completepower line carrier channel for various coupling arrangements withtransposed line sections. This study, which Dr. Wasley too considereddesirable, uses the mathematical model defined in the first phase of theresearch and, in particular, the sensitivity functions of the modal pro-pagation constants.

The attenuation constant of the third mode, the ground-mode, wasnot presented in Fig. 5 since no experimental values were available.

The a(3) curves were calculated with exact and approximate for-mulae as functions of the frequency: a comparison of the two profilesproves the validity of the results obtained with the approximatemethod.

In Table 11 are shown the normalized sensitivity functions of modalattenuation constants only; actually the sensitivities of the modal phasePI" coefficients, by means of which the sensitivities of velocities v(k) areobtained, were also calculated. The partial derivative of v'*' with regardto the frequency, is defined as follows:(44) dv(k)/df = 1/'(3( [2n - v'*' 61P'3/df] k = 1-3This sensitivity function is present in the approximate formula:(45) [v()Jf, = [v(kfJ_, + [dv(k6/dff,-, * (f,- f,, k = I -3that was applied for the transmission line having the parameters in-dicated in Fig. 4, with satisfactory results.

Manuscript received May 12, 1978.