Measurement Techniques, Vol. 37, No. 1, 1994

THEORY OF THE INTERACTION BETWEEN A PULSED ELECTROMAGNETIC

F I E L D AND A T W O - E L E C T R O D E

HORN ANTENNA

S. A. Podosenov, A. A. Sokolov, and S. V. Al'betkov

UDC 631.3.018.7

A system of differential equations describing the symmetrical excitation of a horn antenna with an interelectrode dielectric layer in the form of a pyramid by a pulsed electromagnetic TEM-wave is obtained.

Measuring transducers in the form of sections of striplines are used to measure the strength of short electromagnetic-field pulses. Transducers of this kind reproduce pulses, without appreciable distortion, over a range equal to the time taken for a double transit of the signal along the line. However, a stripline has low sensitivity. Increasing the interelectrode gap does not solve the problem because of the difficulties that arise in matching it to the recording device. To record small signals it is best to use a two-electrode horn antenna, since it has a higher sensitivity than a stripline and can be matched in a natural way to the recorder. Unlike a stripline, a horn antenna distorts the shape of the signal. However, if the transfer characteristic of the transducer is known, the form of the input signal can easily be established from the form of the output signal. The problem therefore arises of calculating the transfer characteristic of a horn antenna. A rigorous analytic solution of the problem presents considerable mathematical difficulties. Even in the case of a thin linear antenna with a sinusoidal signal, the expression for the current leads to an integrodifferential equation, obtained for the first time in [1] and independently in [2]. A large number of papers have been published on the modification of these equations and their numerical solution, among which the most well-known are the reviews [3, 4]. For a horn antenna with a small aperture angle a comparatively simple method of analysis is possible based on a model of the telegraph equations, where the external electromagnetic field acts as the source of excitation. The problems involved in exciting transmission lines by pulsed electromagnetic fields have been widely discussed in the literature (see, for example, [5-10] and the bibliography given in them). In [11, 12] we developed a theory of the interaction between electromagnetic pulses and stripline pickups on a dielectric substrate and carried out experimental investigations. In this paper we give an electrodynamic derivation of the telegraph equations when there is an alternating external electromagnetic field present for a horn antenna with a small aperture angle. It is shown that the introduction between the generatrices of the horn antenna of a dielectric layer leads to the occurrence in the excitation equation of an additional magnetic source, distributed along the antenna, which considerably changes the voltage and current distribution pattern in the antenna. We will consider a transmission line, which is the horn antenna, formed by two diverging triangular thin linear plates, situated symmetrically about the yOz plane (see Fig. 1). The space between the plates is filled with a medium with relative permittivity ei. Arbitrary loads Z1 and 2 2 are connected to the ends of the line. We will assume that the shape of the conductors ensures that the characteristic impedance of the line is constant along the length S. The gap 2h ,~ l, where I is the length of the arm of the antenna. Half the aperture angle /~ and the angle between the generatrices ~ of the antenna are also small. All this ensures that it is possible to excite a scattered TEM-mode in the line considered [13] and justifies the telegraph-equations approximation. The line considered contains, of course, higher modes also (leakage waves). They disturb the form of the signal and attenuate along the direction of propagation [13].

Translated from lzmeritel'naya Tekhnika, No. 1, pp. 26-28, January, 1994. 0543-1972/94/3701-0045512.50

9 1994 Plenum Publishing Corporation

45

Electrode 1

c~

l I

]-. aj

F

I

Y

9

I' /'/

az

lir

\

Electrode 2

Fig, 1

Suppose a plane electromagnetic wave propagates along the Oz axis of the line so that the electric-field vector E' is directed along the Ox axis. The plane of the antem~a qczc4q is perpendicular to the yOz plane, bisects it and makes an angle with the a with the Oz axis. Our purpose is to derive equations for the excitation of the antenna when an electromagnetic wave is incident on it. The initial equation is Maxwell's equation where r o t ' E t ~ - O - B t / Ot '

(1)

~t is the total electric-field vector, and fit is the total magnetic-induction vector. Using Stokes' theorem for the contour ala2asa4 (see Fig. 1), we obtain r

P

Here F is the area of the surface between the dashed lines, ~" is the closed contour a~a2asa4, and s is the unit vector normal to the surface F. Integration of the left-hand side of (2) over the contour ~" gives (/a a~

E~aZ=ut(s+ds)= 0td{s,~) 0"--7~ ds,i ut(sd); a~

a~

(3)

O~

Ias elet=-el/,. Hence it follows that ~ Udl = [ Os

46

(4)

where Els t and E2s t are the projections of the total electric-field vector on the axial lines of the electrodes 1 and 2, respectively, u t is the total voltage between the electrodes, which will represent in the form of a sum [9] I~t~s,t)=u~(s.t)+ui(s,t) ,

(5)

u s is the scattering voltage, and u i is the voltage induced in the electrodes by the excitation field. By definition, the quamity u i can be expressed by the equation a,

,?= .~; ~z)ctT.

(6)

at

The external field in the dielectric layer -fi(0/ is related to the external field of the surrounding medium -fi(e)i. This connection, which is governed by the choice of the model, can be represented in tensor notation as follows:

where summation is carried out over repeated subscripts n. The components of the tensor B~n depend, in general form, on the electric and magnetic properties of the layer and the surrounding medium, on the geometry of the dielectric, and on the spectral characteristics of the external field. We will assume that the characteristic wavelength X of the external field is large compared with the dimensions q c a and 21sin@/2 (see Fig. 1), which agrees with the known representation of quasistationarity. In this case X may also be less than the length of the arm 1. However, as was pointed out in [14], by dividing 1 into small sections, we can successfully use the quasistationarity condition in this case also. In the approximate method, which is based on the quasistationarity representation, the fields are calculated in the same way as for stationary processes. The quasistationarity condition, in particular, enables one to find the components of the tensor Bin by considering the polarization of the dielectric wedge in an electrostatic field. To f'md the components of the symmetrical tensor B~ a derivation similar to that in [15] for the polarization of a dielectric ellipse in an external uniform electric field is used. Like any symmetrical second-rank tensor, Bkn has a principal axis in which it is diagonal. Consequently, if we orientate the body so that the external field e)t is parallel to one of the principal --ql* . axes, the field E(i)~ will also be parallel to ~(e)~. For the wedge shown in Fig. 1, the axes of the coordinates x, y, z coincide with the direction of the principal axes e~, ~2, ~'s- Hence, the integral (6), taking expressions (7) and the result obtained in [15] into account, one can be represented in the form 'li=~11

--h--ssinf5 .f e{e}xdx ;

~ ~[l-'-n(l) (~(i) /e{e)"-1)] -1 , where rt(1) is the depolarization coefficient of the wedge in the direction of the Ox axis, the specific values of which will be introduced later. Assuming that the external field in the uniform medium e(e) when there is no wedge satisfies Maxwell's equation (1) with the replacement-fit._, -fi(e)i and "~t..., "if(e)', integrating it over the area bounded by the contour ala4c2cla a, and using Stokes' theorem, we have c3s =~xl ~s

Eic}2sd~~- h E(e)21xdx+

s

i d ~+ 3"E,~,aT")=fllx I Ei,)l~(s,t ) + ~9 E,,),~ --Eie)2~fs t)--COS~ O h~ssin[~

(9)

]

where the integration over the contour is carried out in an anticlockwise direction, and E(e)tsi and E(e)2si are the projections of the external electric field on the axial lines clc 3 and c2c4, respectively. We convert the right-hand side of (2) to the form 0 {'

-- -37- # (-~e'-;)dF~-c~

0 hq-ssin~

yf

~

-h--ssin[3

(-~i.--d)dx-

(10)

47

0

- f6T #l (~'.~)aF, in which we have represented the total magnetic-induction vector in the form of a sum B~--~--B%E~ ,

(11)

where ffs is the magnetic induction due to the currents in the antenna. In view of the fact that the angle ~b is small and the excitation of the antenna by the external field is symmetrical, there will be no in-phase currents in it, while the antiphase currents will be equal, i.e., i 2 = - i 1 = i. It follows from Ohm's law that

Et2s=ir~+L 2 "O'(

}

E~s= - (irx_}_L1 3-/Oi ) '

(12)

where r 1 and r z are the resistances per unit length, and L l and L 2 are the internal inductances per unit length. The quantity J (ffs.'~)dF in (I0) is the magnetic flux through the hatched contour alaza3a 4 due to the currents flowing F

in the antenna. If we neglect the effect of the far currents on the flux, which is a fundamental requirement when deriving the telegraph equations [16], it follows from the local nature of the magnetic field that

( BS.~)dF=L3ids ,

(13)

where L 3 is the external inductance per unit length, which depends on the geometrical shape of the antenna over the transverse cross section. From (4), (5), and (9)-(13) we have

Ous Ot Os q-L ~ -}-tR--~xx(E(e)2s--Elelis}-h+siinf~ .

_

t

--(i--~)cos~ ~

i

(-~i.~)dx;

(14)

L-Lx-~-L2q-L3; R-rl-]-r ~. To obtain the second equation of the transmission line, which is a consequence of the equation of continuity, we will use the result obtained in [16] or [9], replacing Vs by u' in it. If the medium is not conducting, we obtain

Oi

Ons

o-7 + g a T =0,

(15)

where C is the capacitance per unit length. Thus, the system of equations (14) and (15) is the system required, and describes the excitation of a horn antenna by an external electromagnetic field. As 3 --" 0 and ~b --, 0 the conical line becomes a two-wire line. Comparing the system of equations (14) and (15) with the similar system (33) and (32) from [9], it can be shown that in (33) there is no magnetic field in the source. However, this does not contradict the result we have obtained since the transmission line in [9] was considered to be in an isotropic medium, whereas in our case the conductors of line are separated by a dielectric layer. For the case of a uniform medium (e0) = e(e)), as follows from (8), 311 = 1, which leads to disappearance of the magnetic source in (14) and is identical with the results obtained in [5, 9].

REFERENCES 1. 2. 3. 4. 5.

48

E. Hellen, Nova Acta Uppsala, 11, No. 1 (1938). M.l.eontovich and M. Levin, Zh. Tekh. Fiz., 14, 481 (1941). R. Mittra (ed.), Computational Methods in Electrodynamics [Russian translation], Mir, Moscow (1977). S . L . Bennet, Proc. IEEE, 66, No. 3, 35 (1978). D. V. Razevig, Atmospheric Voltage Surges in Electrical Transmission Lines [in Russian], Gosenergoizdat, Moscow-Leningrad (1959).

6. 7, 8. 9. 10. 11. 12. 13. 14. 15. 16.

C.D. Taylor, R. S. Satterwhite, and C. W. Harrison, IEEE Trans. Antennas and Propagation, AP-13, No. 4, 987 (1965). C.R. Paul, IEEE Trans. Electromagnetic Compatibility, 18, No. 4 (1976). E.F. Vens, The Effect of Electromagnetic Fields on Screened Cables [in Russian], Radio i Svyaz, Moscow (1982). A.K. Agrawal, H. J. Price, and S. H. Gurbaxani, IEEE Trans. Electromagnetic Compatibility, 22, No. 2, 119 (1980). Y. Kami and R. Sato, IEEE Trans. Electromagnetic Compatibility, 30, No. 4, 457 (1988). S.V. Al'betkov et al., Proceedings of the 14th All-Union Conference "High-Speed Photography, Photonics and Metrology of Rapidly Occurring Processes," VNI/OFI, Moscow (1989), p. 216. S.A. Podosenov and A. A. Sokolov, Izmer. Tekh. No. 1, 41 (1991). K.E. Baum, Proc. IEEE, 64, No. 11, 53 (1976). A. Sommerfeld, Electrodynamics [Russian translation], IL, Moscow (1958). L.D. Landau and E. M. Lifshits, Electrodynamics of Continuous Media [in Russian], Nauka, Moscow (1982). L.A. Vainshtein, Electrodynamics of Continuous Media [in Russian], Nauka, Moscow (1982).

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