new method for calculating pulse radiation from an antenna with a reflector

48 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 39, NO. 1, FEBRUARY 1997

New Method for Calculating Pulse Radiationfrom an AntennaWith a Reflector

Oleg V. Mikheev, Stanislav A. Podosenov,Member, IEEE, Konstantin Y. Sakharov,Alexander A. Sokolov,Member, IEEE, Yanis G. Svekis,Member, IEEE, and Vladimir A. Turkin

Abstract—A simple method for calculating the pulse radiationof an antenna with a reflector is proposed both in the near and farzones. The method is based on a substitution of the radiation fieldfrom a parabolic mirror by the radiation field from an exciting V-antenna reflected from the mirror. An experimental investigationof the system radiation field and a comparison with the theoreticalresuts have been performed.

I. INTRODUCTION

T HE PROBLEM of a directed radiation of short electro-magnetic pulses is of importance for solving the applied

problems of electromagnetic compatibility. The theory of pulseradiation of a reflex antenna with small angular divergence hasfirst been proposed in papers by Baum and Farr [1]–[4]. In[2] a formula has been obtained to calculate the electric fieldintensity on the reflector axis at the distancefromit at the instant .

(1)

Here , where is an input impedance of theV-antenna irradiating the parabolic reflector,is the velocityof light, is a wave impedance of vacuum.

The antenna is excited by an arbitrary time-dependentvoltage pulse from the V-antenna. The excitation pointof the V-antenna is located at the reflector focus with a focallength . The distance between the antenna ends is equal tothe mirror diameter .

The formula contains a term with a derivative of thegenerator voltage at the instant of the signal arrival at thepoint of observation with respect to the time, related to theradiation from the reflector, and the term with a difference oftwo voltages corresponding to the radiation from the excitingantenna. The field intensity decreases in inverse proportion tothe distance from the emitter.

The analysis of (1) shows that the term with a derivativein the near zone does not reflect the problem essence. As weshall show below, while applying a step signal to the input,the amplitude of the signal being emitted by the mirror isindependent of the distance and constant in magnitude at anyfinite distance from the mirror. There changes only a pulse

Manuscript received May 7, 1996; revised November 15, 1996.The authors are with the All-Russian Research Institute for Optophysical

Measurements, Moscow 119361, Russia.Publisher Item Identifier S 0018-9375(97)01782-1.

duration which decreases in inverse proportion to the squaredistance from the mirror.

Starting from the aforesaid, we introduce a definition of thenear zone for pulsed fields.

The distance from the source for which the duration of theemitted pulse from a step signal is equal to the front durationfrom a quasistep one is called a boundary distance of the nearzone .

The physical meaning of is evident. At the distancesfrom the source less than the quasistep pulse does notvary in magnitude. Beyond the boundary of the near zone theamplitude will be diminished due to adding the fronts fromthe pulses of opposite polarity. As will be shown below, thiscritical distance is determined by the formula

(1a)

where is a quasistep pulse front duration. From the formulait follows that, as the front duration vanishes, the bound-ary between the zones tends to infinity. For example, for

.In paper [5] the pulse radiation from a parabolic antenna

with a reflector has been investigated experimentally. How-ever, as distinct from (1), the calculated relation presentedin the paper for determining the electric field intensity at theobservation point is independent of the output voltage valuebut depends on the value of the current in the reflector. Theabsence of a formula for the antenna input impedance doesnot allow absolute values of fields to be calculated. Thereforepaper [5] is appropriate for determining only time-dependencesof emitted pulses and their relative amplitudes.

II. M ETHOD FOR CALCULATING THE RADIATION

FIELD OF AN ANTENNA WITH A REFLECTOR

The present paper gives a new method of calculating theradiation field of an antenna with a reflector. Consider theessence of the method.

The mirror image method is used in the antenna theory tocalculate the radiation field of sources located over a perfectlyconducted plane. This method lies in the total field from asource over the conducted plane being a sum of the sourceradiation field in a free space and the radiation field from itsmirror image.

Apply the mirror image method not to a flat surface but to aparabolic mirror, having reflected a V-antenna from it. Sincean excitation point of the V-antenna is located at the parabolic

0018–9375/97$10.00 1997 IEEE

MIKHEEV et al.: NEW METHOD FOR CALCULATING PULSE RADIATION 49

Fig. 1. Modes of a V-antenna with a reflector.

Fig. 2. Geometry of symmetric parts with a facture.

reflector focus, the excitation point image in the mirror will bevirtual and lies at infinity beyond the mirror. The ends of theexciting antenna and its mirror image will coincide with eachother and lie on the mirror generatrix. The distance betweenthe ends is equal to the mirror diameter.

Thus, instead of the V-antenna-reflector system we obtainedtwo V-antennae with different expansion angles. Thereby, theantenna expansion angle will vanish for the antenna replacingthe reflector. A superposition of the fields from these antennagives the sought-for field (Fig. 1).

In [6]–[8] a theory of nonstationary radiation from travelingwave wire antenna has been elaborated. Apply the resultsof the theory to the calculation of our antenna. From [9] itfollows that it will have a very narrow directional pattern.On the basis of [8] it is easy to calculate the field at anarbitrary point of space for any instant, any wire shape,and any time-dependence of the exciting voltage pulse. Thebasic formulae of the theory proved most easily obtainable

by a direct calculation of the electromagnetic field tensorcomprising both the electric and magnetic fields [10], [11].

Fig. 2 depicts symmetric sections of the antenna with abreak formed by intersection of two rectilinear sections. Thebeginning of the rectilinear section is at a distance, mea-sured along the wire, the end is at a distanceand thebreak is at a distance from the antenna excitation point.The contributions to the electric and magnetic fieldsfrom the selected parts of the antenna are determined by theexpressions obtained in [8]

(2)


Fig. 3. Coordinates of the points of measuring the electric field.

(3)

From formula (3) we shall find the electric field of anantenna with a reflector having replaced the parabolic mirrorby the V-antenna “reflected” from it (Fig. 1). The fact that thesignal in the “reflected” antenna propagates with the velocitymore than that of light is unimportant, i.e., the signal velocityin it is a phase one which may take any value. Outside theantenna the signal propagation velocity coincides with that oflight in vacuo. The total electric field is represented asa sum of the fields , where corresponds to theexciting antenna field and —to the reflector field. The fieldfrom the exciting antenna we present in the form

(4)

where is an excitation current, is an exciting antennaarm length (Fig. 1). Suppose that the current is not reflectedfrom the antenna ends.

The field from the reflector can be obtained similarlytaking account of the limiting transition , where is anangle between the generatrices of the antenna “reflected” from

the reflector. Then in the phase expressions for the currentswe allow for the signal, reflected from the antenna reflector,falling on the focal plane of the mirror, with a delay by thevalue from the initial excitation instant.

From (4) we obtain a value of the field from the excitingantenna on the mirror axis. From the problem symmetry itfollows that one field component will be nonzero. For itwe find

(5)

where is a distance from the antenna ends to the pointof observation, is an angle between the vector and themirror axis, is a slope of the exciting antenna (Fig. 1).

We represent the total field on the mirror axis at anarbitrary distance from the focus in the form

(6)

For a parabolic mirror the formula is valid

(7)

In the far zone , whereis a distance from the reflector expansion plane to the point

of observation. From the analysis of (6) it follows that thefield from the exciting antenna falls in inverse proportion


to the distance, and the field reflected from the mirror isproportional to the difference of two voltages phase shifted.While exciting the antenna by a step voltage, the amplitude ofthe field reflected from the mirror remains unchanged alongthe mirror axis, with the pulse duration decreasing as thedistance increases. From (6) it follows that the pulse duration,depending on the distancefrom the mirror expansion planeto the point of observation, can be representented in the form

(8)

The same result can be obtained from paper [3] as well.In the far zone, according to (1a), the pulse transforms into arectangular one with the duration

(9)

From (9) and the definition of the near zone we find (1a).For the signal of an arbitrary form in the far zone we find

(10)

which coincides (to within the choice of the origin of a timescale) with the result [2] obtained by another method andapplicable, from our viewpoint, only away from the emitter,i.e., at . (Note that in the expressions for phases offormulae (10) and (1) there are different values displaced inphase by . This means that we measured the time fromthe instant of switching on the generator, and the authors ofpaper [2] did it from the instant of the signal arrival at thepoint of observation).

In the near zone a differentiation of the signal does notoccur, and instead of (10) one should use (6) applicable bothnear and away from the mirror as well.

In [8] an expression has been found for the radiation patternof a V-antenna in the equatorial and meridional planes whenthe field observation point lies at a finite distance from theantenna excitation point and the antenna arms are infinite.In particular, in the equatorial plane, an expression has beenfound for an infinite V-antenna, with the expansion angle,to be in the form

(11)

where is an equatorial angle being measured from theantenna plane, is a distance from the excitation point to theobservation one. As applied to our case, (11) is representablein the form

(12)

where

(13)

Using the result of [8], we find the expression for the field inmeridional plane

(14)

where

(15)

Apply (11)–(15) for our case, namely, when the fieldobservation point lies at an infinitely large distance from theimaginary source (the distance from the mirror to the to theobservation point is finite and may take any value). Thus, weshould like to investigate the field at a finite distance from themirror axis. The solution to the problem to be sought reducesto finding the limits of (13)–(15) at .It’s evident that , and the angle isconventionally introduced instead of, with beingvalid. Introduce designations

(16)

Equalities (16) are evident from the geometry, withandbeing displacements from the mirror axis along the corre-

sponding coordinate axes setting the points at which we shouldlike to find the field.

Having calculated the limits, we find

(17)

(18)

Equation (17) gives the distribution for the field on theequatorial plane shifted, relative to the mirror axis away fromthe antenna plane, by the value. Assuming , weobtain the contribution to the field on the mirror axis, whichcorresponds to the first term in the second square brackets of(6). The noncoincidence of signs is natural, as the components

and are opposite in sign. Since we are interested in achange of the leading edge, we do not take account of thesignal from the antenna ends arriving at the observation pointwith a delay and being determined by the second term in thesection square brackets of (6). If one assumes , then weobtain the value of the field at the point , which istwice less than that on the axis.

Equation (18) gives the distribution of the field in themeridional plane coinciding with the antenna plane. Forwe obtain the field on the axis, which is in agreement with(6). However, for , i.e., for (18) diverges.The reason for divergence is that (18) makes a contributionfrom an infinite V-antenna to the field, and the equalitycorresponds to the field on one of the antenna arms. Sincein theory the antenna is asummed to be infinitely thin, thenthe field on it tends to infinity. Really, the observation pointunder consideration lies on not the antenna arm, but outsideit, namely, on the line being a continuation of the arm. Fromgeometrical consideration it follows that for finding the field inthis case it is necessary to take also into account the field fromthe antenna end, which arrives in this case simultaneously with


Fig. 4. Comparison of the experimental results with the calculated ones at points 1 through 10,calculated, measured.

the field from the exciation point but due to (6) is of oppositesign. One may show that the field from the antenna end hasthe form

(19)

which on adding to (18) gives

(20)

Thus, the divergence vanishes with regard to the field of thenearest end.

The method being proposed allows one to calculate the fieldinside the central spot taking into account the arrival of thesignal from the ends of the exciting and “reflected” antennae.Omitting simple but rather cumbersome transformations, wepresent the general expression for fnding the-component ofthe toal electric field for the points having arbitrary coordinates

(Fig. 3)(21)

In particular, for we obtain the formula for thefield on the antenna axis reducing to (6).


Fig. 4. (Continued.) Comparison of the experimental results with the calculated ones at points 1 through 10,calculated, measured.

III. COMPARISON OFTHEORY WITH EXPERIMENT

An objective of the experimental investigation was to verify(21). The research set comprised emitting and measuringsystems.The emitting one was shaped as a parabolic reflectorwith the diameter 0.9 m and ratio shaped as a parabolicreflector with the diameter 0.9 m and ratio .The V-antenna with served as an irradiationfor it. The exciting generator produced at the input of the V-antenna step pulse with the amplitude 9.9 V, risetime 80 ps andduration 50 ns. The measuring unit consisted of a widebanddigital oscillograph with the risetime 30 ps and a stripline-like transducer, with the theory of the latter being presentedin [12], [13]. The transducer had a transient response risetime75 ps, its duration 4.7 ns, conversion ratio 0.67 (mV/(V/m)),and overall dimensions 500 60 8 mm.

To determine a dependence of the-field intensity distribu-tion inside the central spot, the measurements were performedat the points located on the geometrical axis of the reflectorat different distances from the expansion surface and at thepoints shifted from the reflector axis. Fig. 3 presents ten pointsat which the field measurements have been performed. Points1, 4, 7 lay on the antenna axis at the distances of and

from the expansion surface. Points 2, 5, 8 and 3, 6, 9 layat the corresponding distances from the expansion surface butwere shifted along the axes or by the value . Point10 had coordinates . Fig. 4 presentsthe results measured and calculated by (21) at selected points,where are experimental values, are calculated ones.

As seen from the figures, the experimental results arein good agreement with the theoretical ones. A noticeablediscrepancy between theory and experiment in Fig. 4 was dueto the pulse decay along the transducer of field away from theexciting antenna. As increases, the effect diminishes.

The measured duration of pulses coincides with the dura-tion calculated by (8) to within the measurement error. Thepresence of bursts on the ends of pulses on the experimentalcurves and their absence on the corresponding calculatedones is related to matching the V-antenna with the reflector.Theoretically, such an agreement has been assumed. Theperformed measurements were related mainly to the near zonewhere Baum and Farr’s formula (1) is not valid. In the farzone (1) is in agreement with the experimental data.

IV. CONCLUSION

The proposed calculation method permits one to determine,to sufficient accuracy, the radiation field of an anternna witha reflector at any distance from the reflector. The use of themethod being proposed allows one to find simply the solutionto the problems whose calculation by other known methodsrequires considerable efforts.

As distinct to [2] and [5], the field measurements were per-formed not over a conducting surface but in a free space, whichpermits the elaborated equipment to be used for measuring thefield of real emitters.

REFERENCES

[1] E. G. Farr and C. E. Baum, “Impulse radiating antennas,”Int. Symp.Electromag. Environ. Consequenes, Book of Abstracts. Bordeaux,France, May 30–June 4, 1994.

[2] E. G. Farr, C. E. Baum, and C. J. Buchenauer, “Impulse radiating an-tennas,”Ultra Wideband/Short-Pulse Electromagnetics 2. New York:Plenum, 1995, Pt. II, pp. 159–170.

[3] E. G. Farr and C. E. Baum, “The radiation pattern of reflector impulseradiating antennas: Early-time response,”Sensor and Simulation Note358, June 1993.

[4] E. G. Farr and G. D. Sower, “Design principles of half impulse radiatingantennas,”Sensor and Simulation Note 390, Dec. 1995.


[5] Hao-Ming Shen, “Experimental study of electromagnetic missiles,”Microwave Particle Beam Sources Propagat., vol. 873, pp. 338–346,1988.

[6] S. A. Podosenov and A. A. Sokolov, “Calculations for nonstation-ary wire emitters in the electromagnetic compatibility problems,”Metrologiya, no. 1, pp. 17–25, 1994 (in Russian).

[7] , “Nonstationary radiation of a V-antenna and a linear source,”Metrologiya, no. 1, pp. 26–35, 1994 (in Russian).

[8] S. A. Podosenov, Y. G. Svekis, and A. A. Sokolov, “Transient radiationof traveling waves by wire antennas,”IEEE Trans. Electromag. Compat.,vol. 37, pp. 367–383, Aug. 1995.

[9] C. E. Baum and E. G. Farr, “Impulse radiating antennas,”UltraWideband/Short-Pulse Electromagnetics, H. L. Bertoni et al., Eds.New York: Plenum, 1993, pp. 139–147.

[10] L. D. Landau and E. M. Lifshitz,Field Theory. Moscow, Russia:Nauka, 1973 (in Russian).

[11] J. A. Stratton, Electromagnetic Theory. New York: McGraw-Hill,1941.

[12] S. A. Podosenov and A. A. Sokolov, “Linear two-wire transmissionline coupling to an external electromagnetic field, Part I: Theory,”IEEETrans. Electromag. Compat., vol. 37, pp. 559–566. Nov. 1995.

[13] S. A. Podosenov, K. Y. Sakharov, Y. G. Svekis, and A. A. Sokolov,“Linear two-wire transmission line coupling to an external electromag-netic field, part II: Specific cases, experiment,”IEEE Trans. Electromag.Compat., vol. 37, pp. 566–574, Nov. 1995.

Oleg V. Mikheev graduated in 1992 from theMoscow Engineering Physical Institute and special-ized in electrophysical plants.

Since 1992, he has been working at the laboratoryfor electromagnetic compatibility in the All-RussianResearch Institute of Optophysical Measurements,State Standards of Russia (VNIIOFI). Presently, heis working in the field of short EM-pulse radiationand measurement.

Stanislav A. Podosenov, for a photograph and biography, see p. 10 of thisTRANSACTIONS.

Konstantin Y. Sakharov graduated in 1981 fromthe Moscow Engineering Physical Institute and spe-cialized in electrophysical plants. He received thePh.D. degree in measurement and generation of thepulse EM-field in 1987.

Since 1980, he has been working at the labora-tory for electromagnetic compatibility in VNIIOFI.Presently, he is working in the field of EM-pulsesradiation and measurement.

Alexander A. Sokolov, for a photograph and biography, see p. 10 of thisTRANSACTIONS.

Yanis G. Svekis graduated in 1985 from theMoscow Engineering Physical Institute andspecialized in electrophysical plants.

Since 1985, he has been working at laboratoryfor electromagnetic compatibility in VNIIOFI.Presently, he is working on problems of short EM-pulse radiation and measurement and elaboratingEM-field sensors of radio-frequency range.

Vladimir A. Turkin graduated in 1989 from theMoscow Engineering Physical Institute and special-ized in electrophysical plants.

Since 1988, he has been working at VNIIOFI.Presently, he is working in the field of short EM-pulse radiation and measurement.

new method for calculating pulse radiation from an antenna with a reflector

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