495

American Telephone and Telegraph Co., “Measurement of transients at the central office termination of a subscriber loop,” Contribution to Study Group V of the International Telegraph and Telephone Consultative Committee (CCITT), Com V, no. 34, Aug. 1982. “Measurement of transients at the subscriber termination of a telephone loop,” Contribution to Study Group V of the International Telegraph and Telephone Consultative Committee (CCITT), Com V, no. 35, Nov. 1983. R. Pomponi, H. Kijima, M. Parente, E. Popp, W. Scott, and A. Zeddam, Surge Voltages and Currents Measured on Telecommunications Lines. EMC Zurich, 1993, 126R6. A. Papoulis, Probability, Random Variables, and Stochastic Processes. New York: McGraw-Hill, 2nd ed. GR-974-CORE, Generic Requirements for Telecommunications Line Protector Units (TLPU’s), Bellcore, Nov. 1994, Issue 1. Standard for Gas Tube Surge Arresters on Wire Line Telephone Circuits, ANSI C62.61-1985. ANSMEEE Std. 820-1984, “IEEE standard teleuhone loop uerfomance

-h -A A h

a

Fig. I. Geometry of the problem

flowing on cylindrical antenna, two hollow cylinders separated by an infinitesimal gap A (Fig. 1).

In general this problem is formulated [2], [3] as an integral equation of the first kind with an unknown constant in the “source” term: this is the well-known equation written by HallCn, namely, for IzI 5 h

\‘”“, I

jc- I - --- J -- - Federal Communications Commission, 47 CFR 68.308. where k is the wavenumber, C is the characteristic impedance of the

medium surrounding the antenna, V is the voltage across the gap. It is worth noting that the constant C has to be determined by means of the boundary condition I ( f h ) = 0, which can be imposed only after the functional form of the current is known. Besides the kernel of HallCn’s equation is discontinuous and difficult to evaluate, being defined as A Hybrid Procedure to Solve HallCn’s Problem

G. Miano, L. Verolino, and V. G. Vaccaro

Ahtract- The knowledge of the electromagnetic fields in the neigh- borhood of an antenna needs the accurate evaluation of the current distribution on it. This is a subject which deserves a particular attention mainly for sensors. It is called Hallen’s problem the one relevant to the current distribution on a cylindrical antenna. We have already shown [l] that this problem can be formulated as a Fredholm integral equation of the second kind with a continuous kernel, and that this integral equation can be solved by a transformation into a linear system of algebraic equations. Even if this solution has a number of doubtless improvements with respect to previous approaches [2], [3], however it does not explicit the logarithmic singularity of the current due to the infinite capacitance of the infinitesimal gap. As a consequence the above expansion requires more and more terms to obtain an assigned precision of the solution, the closer we are to the gap. In this paper, we show that it is possible to extract the singular part of the current, and to obtain a reasonable precision with a finite number of terms regardless of the distance from the gap. The method seems to be suitable for thick dipole antennas. The procedure has been defined hybrid because we first resort to a finite number of steps of the iterative solution, and then the nth integral equation is solved by the Bubnov-Galerkin projection method.

I. INTRODUCTION This paper mainly deals with the solution of HallCn’s integral

equation. This equation relates the unknown current distribution on the antenna to the voltage impressed at the gap. An accurate solution of this problem can be of valuable interest, among others, for sensors. Our particular aim is to compute the current distribution

Manuscript received April 1, 1995; revised April 4, 1996. G. Miano and L. Verolino are with the Universitk degli Studi di Napoli

“Federico 11,” Dipartimento di Ingegneria Elettrica and INFN Sezione di Napoli, 1-80125 Napoli, Italy.

V. G. Vaccaro is with the Universith degli Studi di Napoli “Federico 11,” Dipartimento di Scienze Fisiche and INFN Sezione di Napoli, 1-80125 Napoli, Italy.

Publisher Item Identifier S 0018-9375(96)06144-3.

. KO a u2 - k 2 du

where IO (z) and KO (x) are modified Bessel functions of order zero. As is well known, Fredholm equations of the first kind may lead

to an ill conditioned numerical formulation; all these considerations illustrate how really difficult it is to carry out a numerical procedure and to formulate a robust numerical algorithm to solve this integral equation.

To circumvent such problems, we will adopt a method which has been proved quite fruitful in similar cases of scattering problems [4]. We transform [l] HallCn’s problem into a Fredholm integral equation of the second kind according to the following steps.

a) We describe the problem in a cylindrical coordinate system; it is worth noting that on the surface matching the contour of the antenna (infinite cylinder) two different boundary conditions have to be imposed: the tangential component of the electric field has to vanish on the metallic surface of the antenna, and the current has to be zero outside the tube.

P 1

b) Using the following spectral representation of the current

I ( z ) = 47~a F ( w ) cos (wz) dw 1- boundary conditions give two coupled integral equations arising in complementary regions.

c) Representing the unknown as

F ( w ) = - v ( t ) J o ( w t ) d t j k V -.c-0 sh 0

one of the two integral equations is automatically verified, while the other one will give a Fredholm integral equation of the second kind.

0018-9375/96$05.00 0 1996 IEEE

496 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 38, NO. 3, AUGUST 1996

The system of dual integral equations, introducing the representa- tion of the unknown mentioned in c), is reduced to a single Fredholm integral equation of the second kind

(1)

where z E [0, h], N ( t , z ) is a continuous and symmetric kernel and is given as

00

N ( t , z ) = 1 [T(w) - w]Jo(wz)Jo(wt)dw (2)

and T(w) is defined by

T ( w ) = 2a(w2 - k2)Io

The current I ( z ) , object of our study, is related directly to the auxiliary function p(t) [ l ] by a transformation of the Abel type [SI

(3)

Moreover, (3) immediately shows that the current vanishes at the edges, namely, I ( h ) = 0.

From the kernel and free term properties it follows that p(t) must be a continuous function. Now, because if z + Of (1) states that p(0) = 1, we can approximate [6] formula (3) in order to have the current for small z as

(4)

obtaining the prescribed logarithmic behavior for the singularity [7]. Equation (1) has been solved [l] by the following series:

M

where

is a complete set in L2(0, h ) and Pk(z) are Legendre polynomials. This expansion is proved to be quite suitable for the manipulation of the integral operator, but it cannot uniformly converge to 1 for z + 0. As a consequence it cannot give explicitly the logarithmic singularity of the current I ( z ) for z + 0.

In this paper we shall find a representation of the current on the antenna in which we can recognize two terms:

a) a term which accounts for the infinite capacitance at the gap; b) another term which takes into account the radiative contribution

not suffering for the aforesaid logarithmic pathology. It is shown that (1) can be solved resorting to an hybrid procedure

consisting in a finite number of steps of iterative solution and then to the solution of the integral equation resulting from the final step. This new integral equation has an extremely regular solution which can be found by means of the weighted residuals method.

11. THE HYBRID PROCEDURE Allowing for (l), we may rewrite it as

where L is a linear integral operator. It is well known that an approximate solution of (7) can be found setting it equal to a function which satisfies the following recursive equation:

with the initial condition = f . Furthermore the distance A$(") ( t ) between the "true" solution and the approximated one satisfies the following integral equation:

Ag'"' = L(")[f] + L[A4(")] (9)

and A$(') = y. The hybrid procedure consists in setting the exact solution of (7) as the sum

(10)

where $(") ( t ) is found from recursive equation (8) and A$(")( t ) is found by solving the integral equation (9) by means of the Bubnov-Galerkin method, or other numerical methods.

We have seen that in our case, in order to obtain the advantages of this method, it is sufficient to set the index n equal to 1. For sake of simplicity this index will be dropped. It is easy to verify that (9) becomes

y ( z ) = $ ( n ) ( z ) + A$("'(.)

A$(.) = - z N ( t , z ) d t 1" - z lh N ( t , z ) A $ ( t ) d t (11)

and that (8) becomes

$(.) = 1. (12)

Since the kernel z N ( t , 2) is regular, A$(z) is at least linearly vanishing for z = 0. Resorting to (3), the current I(.) can be expressed as

In (13), the logarithmic singularity, accounting for the infinite ca- pacitance at the gap, has been extracted. Furthermore because of the vanishing properties of A$(.) around zero, the integral will converge for any value of z , in particular for z = 0. This means that the input admittance of the antenna can be well represented by the contribution of the infinite static capacitance plus a radiative term given by the integral in square brackets. A good estimation of the input admittance for an antenna with a small but finite gap A, is obtained by setting in (14) z = A, namely,

y ,=

Now we are ready to transform (1 1) into an algebraic system of linear equations which is easier to treat numerically. There are many ways of reformulating a Fredholm integral equations into a system of equations [8]; in this paper we use the weighted residual method [9].

We start with the expansion of the unknown A$(z) in the functional space ~ ' ( 0 , h )

,=l

where the functions cpn(z) are defined by relation (6). We point out that expansion (15) converges uniformly, and the expansion coefficient E, goes to 0 as n goes to infinity.

In order to determine the unknowns E,, we first plug (15) into integral (11) and then we project the equation onto the set { cpm ( z ) / z } . By using the orthogonality property [6]

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 38, NO. 3, AUGUST 1996 497

-480

-4,85

(1 l), after some algebraic manipulations, can be transformed into the following system of algebraic equations (n = 1, 2, 3, .. .)

j kh=x/2 I

j z/h=O.l -

L ............................ :. ............................ j .................................. : .... / h/a=60..-

....... ...; . . . . . . . . . . . . . -4 ................................... ..................... .-

M M

-4,75 ' 8 ' I ' I I b I I , I I I I

............................................................ .-

........ i.. . . . . . . . i .......................... ~

i (a) I -5,oo " I ' " " " I ' " " I I '

where the matrix coefficients are defined by the following integrals:

The solution of system (16) requires the inversion of the symmetric and infinite matrix D + B where B = diag(1, 1/3, 1/5, ...). However, it is necessary to truncate the system of equations to finite order and invert the finite matrix AN = DN + BN using a digital computer. We have verified, by numerical experiments, that the inversion procedure of the matrix AN is well-behaved and stable. This property holds because our discrete system comes from a Fredholm integral equation of the second kind with the eigenvalues of the kernel different from 1 . Now, because of these properties, we are sure that the approximate solutions converge to the exact solution for N + m in L2(0 , h ) [lo].

A possible acceleration of the slowly convergent integrals (17) can be obtained using higher order terms of the following asymptotic expansion [l 11:

as it has been shown in [l]. Finally, it is possible to derive the current distribution I ( z )

directly, without employing intermediate formulas. Using expansion (15) and relation (14), we obtain the following series expansion of Chebyshev's polynomials [6]:

+ log h+yl. From (18) it is immediately verified that the current vanishes at the end of the antenna I ( h ) = 0.

111. NUMERICAL RESULTS

It is evident that much of the theory developed has been constructed expressly for the purpose of facilitating numerical computations. That is why, in this numerical section, we shall reproduce some classical results given in the literature as examples to demonstrate the accuracy of the proposed method, and we shall compare the old method of solution [ 1 j with the new hybrid procedure.

First of all, it is necessary to say the following words about the numerical algorithms used [ 121:

9 An adaptive Gaussian quadrature routine has been used to perform numerical integration (relative error less than The inverse of the coefficients matrix has been computed by triangular factorization with row interchanges. Bessel functions of large order have been computed by the Miller's algorithm [12j, while Bessel functions of order 0 and 1 have been approximated by rational functions or truncated Chebyshev series.

........ &==I2 h/a=60 4

0 5 10 15 20 N

Fig. 2. Old and new procedure for the real part are coincident.

-4,75

-4,80

-4,85

-4,90

-4,95

-5,OO

Imaginary part (mAN)

0 20 40 60 80 100 120 N

(b) Fig. 3. imaginary part.

(a) New procedure for the imaginary part. (b) Old procedure for the

We started by making a graph of the "admittance" I ( z ) / V versus the order of the solution ( N ) for a given z / h to study the convergence of the proposed method. A few terms are necessary to stabilize the real part (Fig. 2) for both cases (old and new procedure). This is not so for the imaginary part. The new procedure gives a quite satisfying convergence with the same order N of solution as the real part [Fig. 3(a)]; in contrast, the old procedure, even with a much larger order of solution, is not able to give the same accuracy [Fig. 3(b)].

498

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IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 38, NO. 3, AUGUST 1996

j fi=n: I

j h/a=60: ............................................................................................. - 1

3

2

1

0

-1

-2

-3

-4

("1 ! ' ' ' l ~ ' ' l ' ' ' I ~ " -

j N=5 I .......................................

Fig. 4. Equation (18) without logarithmic singularity.

1 ...... ..............

......................... ..... 0 -1 . . . . ................

-2

Fig. 5. Both terms of (18)

We could not test, because of limitation of CPU elaboration time, at which order we could get the same accuracy as the new procedure.

Calculations have been performed for half wavelength antenna ( k h = 7r) with h / u = 60. Graphs of the current deprived of the logarithmic contribution are shown in Fig. 4 for N = 5; this is meant to be the radiative contribution. It is interesting to underline the stationary behavior of this part of the current in the vicinity of z = 0. The total current distribution, as given by (18), is drawn in Fig. 5 for the same order of solution. The comparison between the behavior of those two distributions validates the assumption that it is possible to calculate the input admittance of an antenna with a finite gap A from the value of the total current at z = A. The current distribution, as calculated by the old procedure, is drawn in Fig. 6 for an order of solution N = 20. It is quite apparent that it cannot account of the logarithmic solution and that it cannot reproduce a smooth behavior as in Fig. 5. It was not possible to reach an order of solution N to have the same distribution as drawn in Fig. 5 .

IV. CONCLUSIONS

We have shown that the results of the method for solving HallCn's problem already adopted in [ 11 are suitable for remarkable improve- ments using a hybrid procedure. At least two orders of magnitude have been gained in the truncation order of the matrices, regardless on how thick are the antennas; this can be of particular interest for dipole sensors.

From the behavior of the current without "the static capacitance contribution" we learn that both the real and the imaginary part are stationary around z = 0. This means that the input admittance

...................... .......................... ........................... ......................... I h/a=60.1 i.

Fig. 6. Current distribution (old procedure).

measurements of equal antennas, having different gaps, must coincide provided that the susceptance of the "static capacitance" is correctly subtracted according to (15).

The accuracy obtained and the possibility of examining a wide range of frequencies encourage us to apply the method to other canon- ical problems in order to have nonambiguous responses in testing some proposed approximations. Therefore our work will continue in the near future, applying the method to Pocklington's equation (finite gap) and to an array of antennas: this will be performed not only to provide numerical values for the input impedance as a function of the frequency, but it will, at the same time, ascertain the validity of certain asymptotic expansions.

It is worth noting that the method explained can be extended to the case of a cylinder of finite conductivity. We hope also that the extension to a two-dimensional version of dual integral equations, appearing in diffraction by rectangular obstacles, will be developed using a double Neumann series.

REFERENCES

[l] G. Miano, V. G. Vaccaro, and L. Verolino, "A new method of solution of Halltn's problem," J. Math. Phys., vol. 36, no. 8, Aug. 1995.

[2] G. Franceschetti, "Campi elettromagnetici," Boringhieri, Torino, Italy, 1991.

[3] R. W. P. King, The Theory of Linear Antennas. Cambridge, MA: Harvard University Press, 1956.

[4] G. DBme, L. Palumbo, V. G. Vaccaro, and L. Verolino, "Longitudinal coupling impedance of a circular iris," I1 Nuovo Cimento, vol. A-104, p. 1241, 1991.

[SI I. N. Sneddon, The Use of Integral Transforms. New York: McGraw- Hill, 1972.

[6] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products. New York: Academic, 1980.

[7] J. Meixner, "The behavior of electromagnetic fields at edges," IEEE Antennas Propagat., vol. AP-20, p. 442, 1972.

[8] I. N. Sneddon, Mixed Boundary Value Problems in Potential Theory. Amsterdam, The Netherlands: North-Holland, 1966.

[9] W. F. Ames, Numerical Methods for Partial Differential Equations. New York: Academic, 1977.

[lo] L. V. Kantorovich and G. P. Akilov, Functional Analysis in Normed Spaces. New York: Pergamon, 1964.

[ll] G. N. Watson, A Treatise on the Theory of Bessel Functions. Cam- bridge, England: Cambridge University Press, 1966, 2nd ed.

[12] B. P. Flannery, W. H. Press, S. A. Teukolsky, and W. T. Vetter- ling, Numerical Recipes: The Art of Scientific Computing. Cambridge, England: Cambridge University Press, 1986.