65 1

Abstract-The problem of pulsed antennas has two comple- mentary parts: i ) analysis of the radiation field when the driving voltage is given and ii) synthesis of the driven voltage when the radiation field is given. In this paper a number of heuristic procedures are presented, relating to the computation of transient radiation from elementary sources, coaxial apertures, infinitely long cylindrical antennas, ~ t e cylindrical antennas, and loop antennas. Comparison with available rigorous solutions and experiments is also provided.

I. IKTRODUCTION AND DISCUSSION O F PRINCIPAL RESULTS

In early 1969, Prof, T. T. Wu stated: “It is worth em- phasizing t.hat, our knowledge about. the transient, response of antennas is very mea.ger indeed. Any progress in this rat,her neglected field is certainly going to be of tremendous value” [l]. Although a few new papers have appeared on t,his subject since then, the above st,at,ement is probably still valid.

Most, of the papers concerned with transient, currents on and transient, radiation from ant,ennas use solut,ions of frequency-domain integral equa.t,ions which are then trans- formed into the t,ime-doma.in by Fourier techniques. These papers cover the t,ransient, beha.vior of t.he infinite cylin- drical antenna [2-)[7], t,he finit,e cylindrical antenna driven by a coaxial line [SI, or by a gap t.ype excit,at,ion [SI, [lo], the conical antenna [ll], the aperture antenna. [l2>[14], t.he loop ant,enna [E] , [lS], and the loaded ant.enna [17]. In a few pa,pers, an experimental verification of the theory is provided, wit,h reference to t.he radiated and received fields [lS], [19] or t.o t,he input response of t,he antennas [20], [21]. Also, the effects of thin-wire approximations and of source-excitation modeling ca.n be found in t.he literature Dl], [32]. Other approaches that have been used are t.he singularity expa.nsion method [Z], [ZS], the solution of the t,ransient current, integral equa- t,ion directly in time-domain [24>[29], and an almost rigorous application of the Wiener-Hopf t,echnique [30].

A general charact.eristic of most. of the referenced material is the emphasis on eit.her t,he extensive use of numerica.1 met.hods or the presentation of the solution in a.n integra.1 form tha.t must be eva.luated numerically or at least, asymptotically [7]. A somewhat different ap- proa.ch has been suggest,ed [33] which aims at the general properties (if any) of transient radiat,ion. In this paper, this approach is formalized and further developed.

Manuscript received December 20, 1973; revised April 8, 1974.

Pasadena, Calif., on leave from the Department of Electrical Engi- G. Franceschetti was u7it.h the California Institute of Technology,

neering, University of Kaples, Kaples, Italy.

Pasadena, Calif. 91109. C. H. Papas is with the California Institute of Technology,

In Section 11, the transient radiat,ion from elementary electric or magnetic sources w-it.h prescribed current dis- tributions is analyzed by using a formal Fourier inversion of known steady-state results. The source environment is assumed not. only isotropic a.nd homogeneous, but also nondispersive, since completely different results are ob- tained when dispersion is taken into account, [34>[37]. This assumption will apply t,hroughout, this paper. The a.nalysis shows that the transient radiated power is ex- pressed in terms of t,he l/r components of the field only, when the antenna excitation is confined in time. These component.s in the case of an electric dipole are propor- t,ional to the the-derivative of the current f ( t * ) and to the second he-derivative of the input. voltage V ( t * ) , a.t the retarded time t* = t - ./c. In t,he case of a mag- netic dipole, the l / r fields a.re proportional to j ( t * ) , f T ( t * ) , where I ( t ) ,V(t) are, respectively, the current along, and the voltage across t.he small current loop.

In Section 111, a deeper analysis of transient, radiation from the element.ary sources is presented. For the electric dipole, a model of t.wo opposite charges, which collapse one upon the other, is assumed. For the magnetic dipole, a. pulsed current, a.long a circular loop is considered. In each case, it is shown that the far-field waveform is not proportional to the first or second time deriva.tives of V ( r ) or I ( t ) , when t.he time resolution is great,er than h/c, h being a characteristic linear dimension of t.he ant.enna, and c the light, ve1ocit.y. In this same scale, t.he radiation due to an assumed pulse of current. appears in the form of two pulses (electric dipole), or a cont,inuous radiated waveform with square-root singularities (magnetic dipole). However, when a time scale larger t,han h/c is considered, it is shown that. a proper limiting process leads to the for- mal results of Sect.ion 11. From a physica.1 standpoint, the process can be understood as a contraction of t.he physical dimensions of the mdiators, or equivalent,ly, as a limit,ed t,ime-resolution of the field measuring devices.

Also t,he more complex radiating structures that will be considered here are subject to t.he above time-resolution property. A4ccordingly, a macroscopic and a. microscopic a.pproa.ch are defined for the transient radiation, with respect to a time-resolution of order h./c. In all cases the t,ransition from one time-scale t.0 the ot,her is developed in detail.

In Sect.ion IV, an ana.lysis of transient. radiation from coaxial apert,ures is developed. In the macroscopic ap- proach, advantage is taken of the equivalence bebeen the aperture field and a distributed magnetic loop [38]. The far-field is shown to be proportional t.0 P ( t * ) , V ( t ) being the voltage across the aperture. In t,he microscopic ap-

IEEE TRXVSACTIONS ON ~ T E N N . ~ AND PROPAGATIOX, SEPTENBER 1974

proach, the far-field is computed as due to the sudden reflection of the current a t the open end along the edges of the conductors of the coaxia,l 'aperture. The radiated waveform is characterized by four square-root singular- ities, where the incident voltage is a. pulse and the coaxial aperture is circular. These results are in agreement with a quasi-rigorous solution of the problem [30].

In Section V the radiat,ion coming from the gap of a linear antenna is considered. In the macroscopic approach, the far-field turns out t o be proportional to p(t*). In the microscopic approach, the radiated waveform ha.s a square- root singularity when the applied volt.age is a step. This last result is consistent vith the early-t,ime behavior of the far-field radiated by an insnitely long cylindrical ant,enna

In Section VI, an approximate method is developed for computing transient radiation from linear antennas and loops large with respect to GT, T being the pulse width. The approximation involved is the use of transmission line analysis for modeling currents and voltages. This results in a sinusoidal current distribution. It is therefore ex- pected tha.t the results of the analysis are valid in the limit of vanishingly smdl wire radius [39].

For the linear antenna, the far-field turns out to be radiated from the input terminals and t.he end-tips, as qualitatively suggested in previous analyses [ll], [13] and experimentally confirmed [18]. For the loop antenna, the far-field appears in the form of a continuous waveform, with square-root singularities (when the applied voltage is a Dirac pulse). For each observationa.1 position there are two flash points on the loop from which most. of the radiation appears t o be emanating.

In Section VII, conclusions and recommendations are made. The suggested recipes and techniques of analysis are simple and rather general and allow the handling of a large variety of radiators of practical interest. However, the dispersion of the pulsed waveforms propagating a.long the radiating structures is not taken into account. On the other hand, this dispersion d produce a. continuous radiation, which is generally small and gives the 1at.e-time response of the radiating structure r/].

c77.

11: FORMAL SOLUTION FOR TRANSIENT RADIATION FROM ELEMENTARY SOURCES Suppose we have an infinitesimal oscillating electric

dipole of moment p e = po exp [-kt] in a homogeneous isotropic environment. The dipole is located at the origin of a spherical coordinate system (~,6,+) and is oriented parallel to the direction 6 = 0. The components of the dipole's electromagnetic transformed field are given by the well-known expressions

where P = (p/t) and c = 1/ ( ~ p ) ~ / ~ .

From (1) the transient fields can be easily obtained for the case of a nondispersive environment, when the dipole moment varies arbitra,rily in time, i.e., p e = p e ( t ) . By taking the Fourier transform of (1)

and doing the same for the ot,her field components, it is easily obtained

where t* = t - r /e is the retarded time, and a dot means derivation with respect to time.

The corresponding expressions for the magnetic dipole are immedia.tely obtained by duality ( H + E, E -+ - H, b + I/{, P e 4 - p m ) * Thus

The r-component of the Popting vector is

S,e(t) = EB(t)H,(t) Srm(t) = --E,(t)He(t) (5)

for the electric and the magnetic dipole, respectively. The radiated energy density a t (r,t?) is

B*(w) = P Po exp [ k r / c l sin e [y - + 51 (lb) When the dipole excitation has a f 3 t e duration the 4nr bracketed terms inside the large parentheses are equal to

FRkNCESCHETTI -4ND PAPAS: PULSED .4WTEl’lXAS 653

zero, and (6) transform as

Equations (7) show that the total radiated energy is -q

dependent only on the l / r components of the field accord- ingly identified as the “radiative” terms.

It is interesting to calculate in which regions of space the radiative terms a,re the domina,nt field components. Reference is made to a Gaussian pulse of electric moment L t

V

T being the effective width of the pulse (distance between inflection points). Substituting (8) in (3) one obtains

Fig. 1. Relevant to physical model for computing transient radia- tion from electric dipole.

. s ine[b@ - 4) - 4,,+ cT ‘* frT>I - (9b) In (10) and (11) C and L are the static capacitance and inductance of the dipole and the loop, respectively, l ( t ) and V ( t ) the current along and the voltage across the

H 6 ( t ) = .4W(cT)2 sin e [ (8 c;) - 41 - 4 $ $1 . (9c) radiators, and the quasi-static relations

It follows from (9) that the radiative terms are always (dipole) I ( t ) = CV(t) (loop) v(t) = - ~ l ( t )

654 I E E E TRANSACTIONS ON ANTENNAS AND PROPAGATION, SEPTENBER 1974

Formal espressions of current density in the time and frequency domains are the follonkg :

J(r, t) = -eZp6(z)6(y)6(z + vt)

1 = -ezq6(s)6(y)6 It1 5 -

V

J(7,t) = 0,

A

J(r ,o) = -ezq6(2)6(y) exp

J(r ,w) = 0, I z I > 1 (14) , , *.

e, being a unit vector in the z direction. The transformed radiative fields associated with (14) are

R ( w ) = -i{ exp [iwr/c] sin [ ( w l / c ) (7 + cos e) ] 2ar 4

?I + COS e sin e

A Be ( w ) H + ( w ) = - C

V b ?) = - .

I I I * t -T I

I +:

By Fourier transforming (15) into the time-domain, the Fig. 2. Relevant to physical model for computing transient r a c k result

. . . tion from magnetic dipole.

E d t ) = b - q sine flows in a circular loop of radius R 1oca.ted in the plane

4~ ?I + COS e (2,Y).

Formal expressions of the current in the time and fre-

cos e)] quency domains are the following:

I ( t ) = Io[U(t + r ) - U(t - r ) ] (18) (16a) sin wr

f ( w ) = ZIO - . (19)

1 c The transformed radiative fields a.ssociated with (19) are E9 &(t) = - t* = t - - r w

is obta.ined. Equations (16) clearly show that the radiation = - RIo - sin (w7)J1

emanates from t,he end points of the configuration at times rc - Z/v (acceleration) and +E/v (deceleration), respectively (see the diagram of Fig. 1). Accordingly, the radiated field is not proportional to the time derivative of the current J1 (2) being the Bessel funct,ion of first kind. (see (10) ) , when a time scale less than 21/v is considered. By Fourier transforming (20) into the time-domain, the

(10) -macroscopic approach-is readily accomplished by letting 21/v approach zero. One obtains &( t ) = - ~ [He+ + He-]

e.$ ( w ) = -{Be W b )

The transition from (16) -microscopic approach-to result

Io 2m sin e ( 2 1 4

2ql 6( t * + 2) - 6 ( t * - x ) Ee(t) = lim [-- sin0 E+(t) = -{Ha (21b)

211t-o 4m-c 2s N a f ( t ) = f

t* f 7

Po 1 [ ( (R/c) sin 8) - (t* f r ) 2]1/2 ’ = { -sin ei(t*), z = - (7 + COS e) (17a)

h r c C R I t + T I 5 -sin e (22a) H,(t) = TEe C

thus recovering (lo), since I ( t ) 2E = - q6 ( t ) 21 = - po6 ( t ) .

Fig. 2. A constant current Io is suddenly switched on a.t time t = - r and switched off a.t time t = +r. The current is obtained.

For the case of a magnetic dipole, reference is made to He*(t*) = 0, R I t f r I > -sine (22b) C

FRAKCESCHE'ITI .4XD PAPAS: PULSED ANTENSAS 655

Fig. 3. Relevant to transition from continuous radiation t,o equiva- lent pulses.

Equations ( 2 2 ) clearly show that the antenna radiates at. t.imes t = -7, t = 7. For each direction of observation there is continuous radiation for I t - r [ 5 R sin O/c, characterized by square-root singularies (see the diagram of Fig. 2). The radiation emanates mainly from points AB' (flash points) a.t t.he intersections of t.he loop with the pla.ne containing the direction of obsermtion and the z axis. No radiation emanates from point,s BB' (black points), due t,o a cancellation effect. Accordingly, the radiated field is not proport,ional to the second time derivative of the current (see (11)), when a time scale less than R / c is considered.

The transhion from (2 1) -microscopic approach-to (11) -macroscopic approach-is acconlplished by letting R/c a.pproach zero.

Reference is ma.de to the radiation which takes place at

for the actual radiated waveform some equivalent pulses (see Fig. 3). Reasonable requirements a.re: i) the strength of each pulse must be equal to the time integral of the act,ual waveform and ii) the time allocation of each pulse must divide the actual waveform int.0 part,s of equal area. When these requirements are fulfilled, (21) and ( 2 2 ) transform into

t = - r. For R + 0, it seems quite reasonable to substitute

thus recovering (11) , since I ( t ) = U ( t + r ) , apart from the fact.or 3/7r = 0.995 = 1 (depending upon the s0mewha.t arbitrary identification between actual radiated wave- forms and equivalent pulses).

Fig. 4. Relevant. to transient radiation from coaxial apertures.

IV. TRANSIENT RADIATION FROM COAXIAL APERTURES ,,

In this section, the transient radiation from coaxial apertures will be considered. Reference is made to Fig. 4 where the radiation from an open ended coaxid cable of circular cross section is considered. However, the results whieh will be present,ed can'readily be generalized to apply to radiating a.pertures of different geometry.

First, the macroscopic approach will be exploited. Let V ( t ) be the incident volta,ge at the open end o f the coaxial cable. Under the assumption of perfect open-circuit con- ditions, the electric field in the aperture has only a p component given by

and t,he magnetic field is zero. By using the equivalence theorem, the radiated field can

be computed as due to an equivalent magnetic current density with only a component, Jw ( t ) = E,(t) . A z- directed electric dipole of moment given by [38]

is therefore associated to the aperture. In (25) A = a(b2 - a2) a.nd is the aperture area, and Zo = [l In b/a]/2a and is the characteristic impedance of the coaxial cable.

When A is small compared to ( c T ) ~ , the higher order moments associated with the a.perture are negligible, and the radiated field can be obtaihed from the results of Sction 11. Hence

A V( t * ) Ee(t) = Zo 2a 3arc2 sin e (26a.j

Accordingly, the radiated field is proportional to the second t,he-derivative of the incident voltage.

As a preliminary to the study of the radiation from the microscopic point,of view a model similar to that of Section I1 for the electric dipole is considered. A charge q at

666 lEEE TRtiNSACTIONS ON INTEXNAS AND PROPAGATION, SEPTEUBER 1974

position z = - E is suddenly accelerated at t = - z/v and is then made t.0 travel at constant velocity u along the positive direction of the z axis. At t = 0 and z = 0, the charge is suddenly reflected and is again made to travel a t consta.nt velocity v along the negative direction of t,he z axis. At t = z/v and z = -1 the cha.rge is suddenly stopped.

The radiat.ed field can be comput,ed by following the same procedure as in Section 111. Only the results are given in the following. The field is radiated a.t t,imes t = -l/v, 0, Z/v, from points z = --I, 0, I, respectively. Particularly, the field radiated at t = 0, z = 0 (reflection of t,he charge) is given by

Ee ( 0 C

5 V H,(t) = - , ?I=-. (27b)

Turning to the coaxia.1 struct,ure of Fig. 4, let I ( t ) =

q6 ( t - z / c ) be the current propagating along the cable. The pulsed linear charge densities on the inner and outer conductor will be q / 2 ~ a and -q/2abJ respectively. These pulsed charges will apparently be propagated with velocity c along the cable. Accordingly, the radiated field can be computed via (27) (n-it,h 9 = 1) and superposition. Hence

- [ [ ' 6 ( t - :) d+ - ('6 (1 - :) d$] (28)

where r&b are the distances of the point (?-,e) from the inner and outer rims of the cable, respectively,

r, 'V r - a sin e COS 4 rb = 1' - b sin e COS 4. (29)

In order to compute the integrals which appear in (29) it is convenient to use the Fourier transform

= d+ I+& 6 (t* + - sin e COS b

0 -00 c

= exp[i:r] [d+exp[- @ b s i n e c o s + C I

and similarly for t-he other integral By Fourier inverting (30) in the time-domain and substituting in (29) , the result

is obt,ained. In (31) the functions Eamb(l*) are given by

B ( t * ) = 0, X C

[ t* I > -sin 8.

(32b)

A plot of the radiated field is given in Fig. 5. The preced- ing results a.re in complete agreement with those of a4 exact a.nalysis carried out by using WienepHopf tech- niques [30] when pulses such that cT << b axe considered. This last restriction does not apply in the proximity of the forward direction.

Equations (31) and (32) clearly show that, for each direction of observa,tion, there is a continuous radiation for I t* I 5 b sin 8/c characterized by square-root singuhrities. The radiat,ion emanates mainly from p0int.s AA', BB' (flash points). Accordingly, t.he radiated field is not proportional to t,he second time-derivative of the incident voltage V ( t ) = Zoq6 ( t ) = P6(t) (see (26) ), xhen a time sca.le less t.han b / c is considered.

Also in this ca.se the transition from (31)-microscopic approach-to (26)-macroscopic a.pproach-can be ac- conlplished by substituting for the actual radiated wave- form equivalent pulses (see Fig. 5 ) as already done in Section 111. Then b / c and a/c are required to approach zero. One obtains (see Fig. 5)

.[(X - Y) 6 ( t * - x ) - 6(t* - y)

X - Y

6 ( t * + y) - 6 ( t * + x ) - (X - Y)

X - Y -1

Ee(t) = 5--4,rsine ' [Eb(t*) - Ea(t*)] (31a) th-us recovering (26), since V ( t ) = TS'g(t)

FRAXCESCHEWITI AND P.4P.G: PVLSED ANTENNAS

~ -, Diroc pulses

Di roc pu ISOS

Fig. 5. Plot of field radiated by coaxial circular aperture and transition from cont.inuous radiation to equivalent. pulses. Sote that. radiated field is superposit.ion of positive and negat.ive plots of figure.

1,'. TRASSIEST RADIATIOK FRO31 BNTENKA GAP

In order to compute t,he transient radiation from the gap, an infinitely long cylindrical antenna of circular cross sect,ion is considered (see Fig. 6). The antenna is excited at. a gap of thickness 2c! by a step voltage T ' ( t ) = 170U(f). Let. E(z,tj = & ( z ) V ( t ) e z be the electric field across the gap. The radiated field can be comput,ed by using the equivalence principle [3S]. The gap is short-circuited and a n1agnet)ic current, density

Jm(z , t ) = E o ( z ) 6 ( p - a ) L,'(t)e, (34)

is assumed to flow around t,he antenna, where the gap was located. When t,he gap t.hickness is vanishingly small, t,he magnetic current, distribution (34 j can be substibuted by a ring of magnet,ic current.

1, = lz d P rd J , (z , t j dz = - voc(t). ~ 3 5 )

On the other hand, the radiation from a loop of elect,ric current was conlput,ed in Section I11 ( (21) and (22) ) . These results can be easily accommoda,ted for the case at hand by using duality (see Section 11). For each observa- t.ion point only one half of t.he magnetic ring will contribute t.o t.he radiat,ion, the other half part being screened by t.he metallic cylinder. Accordingly, only t,he single flash point, A vd.l be present, (see Fig. 6). Furthermore, t,he radiated field will be doubled, due to the image of the nmgnetic ring on t.he metallic conduct.or.

The resulting radiated field is given by

VO ( a / c ) sin 0 - t* Eo(t) = ~ (36a)

pr sin e [t* ( (2u./c) sin 6 - t * ) jl'?

Fig. 6. Relevant to transient. radiat.ion from gap of cylindrical antenna.

and is zero for f. < 0, tY > (2a sin O)/c. Note that, the distances are measured from t,he flash point A .

When t* = t - r j c can be neglected wit,h respect, to (2a sin e) /c, (36) coincide wit.h t.hose describing t,he early- time behavior of t,he radiated field of an infinitely long Cylindrical a.ntenna given by Latham a.nd Lee [7].' When the assumption of infinitesimal gap is relaxed and the gap is allowed to be of finite dimensions, t,he results given by Pyne and Tesche [32 ] for hhe early-time behavior of the field can be obtained.

Equa.t,ions (36) give the field radiat,ed by the gap, and previous analysis proves t.hat this field coincides wit,h t,he early-time behavior of the field. Obviously, the launching of the currents onto t,he ant,enna will produce other fields, which add to t,he fields (36). These furt,her terms will be given in Section 61.

The transit.ion from (36) -microscopic approach-to t,he macroscopic formulation is readily obt.a.ined by using t,he techniques presented in Sections I11 and IV. The field n-aveform (36) is substituted by an equivalent pulse, and t.hen a/c is made to approach zero. The field radiated by an applied voltage V ( t j can be obtained by differentiating (36) and using convolution. Hence

VI. ELEMENTARY THEORY OF RADIATION FRO31 LINEAR ANTENNAS L I D CIRCVLAR

LOOPS An approxin1at.e analysis of transient. radiation from

linear antennas and loops of arbit,rary dinlensions is

S o w that t.he voltage spplletl to the gap is - T - o C ( 1 ) in [i].

658 IEEE TIUSSACTIOKS ON ANTENKAS AND PROPAGATIOX, SEPTEMBER 1974

possible by using transmission line techniques for modeling currents and voltages. As far as the cylindrical ant,enna is concerned, the preceding procedure has already been used [30], [43], [44]. Hereafter it is presented, in a slightly dif- ferent form appropriate to previous analysis, for complete- h

E& t)

T i

h

ness and for comparison with experimental results. A cylindrical antenna of height 2h and dia,meter 2a is

considered (see Fig. 7). If the assumption is made that the antenna is very thin, i.e., D = 2 In [2h/a] >> 1, the current, distribution in the frequency domain reduces t.0 C391

f ( z ) = - vin 2~ sin [(w/c) (h - I z i)] .m i COS [wh/c]

(35)

where Pi, is the transformed volta.ge at the antenna ter- minals. Equation (35) is formally equivalent to that of the current distribut.ion along a.n open ended transmission line of length h, characteristic impeda.nce Zo = p / 2 n and fed by an ideal voltage generat.or of zero internal im- pedance. When the internal inlpeda.nce of t.he generator is different from zero and equa.1 t.0 aZo(real), (3s) can be imnlediat,ely generalized by using the preceding transmis- sion line analogy. Hence

where is the transformed voltage of the generator.

seen t.0 be The t,ransformed fields associated with (39) are readily

COS [ (w/~)h] - COS [ (w/c)h COS e] 1 + r exp [2i(w/c)h]

* {COS [wh/c] - COS [ ( w / c ) h COS e]) m -xn ( - r)n exp [2niwh/c] 0

* e, H = - r where r = (1 - a ) / ( 1 + a) is the reflect,ion coefficient.

By Fourier transforming (40) the radiated fields in the time-domain a.re obtained. For a Dirac pulse excitation V ( t ) = Vo6 ( t ) , P = Vo and the radiated fields are given

Fig. i. Plot of field radiated by linear antenna fed by voltage puke. Input. reflection coefficient. r is assumed to be close to unit.?.

( h + ?I cos e))]

Equations (41) shorn that the transient radiation of the antenna consists of a chain of Dirac pulses, each being a replica of the applied signal (save for the amplitude and eventually the sign). The first pulse (first term inside the la.rge parenthesis of (41a) ) represents the radiateion due to the launching of t,he current along the a.nt,enna at time t = 0. The second series in (41) represenh the radiation from the upper and lo--er tips of the antenna due to the charges associated to the current and going back a.nd forth along the antenna.

The first series in (41) represents the radiation due to the currents relaunched toward the generator. These currents are absorbed in the internal resist,ance of the generator itself. When the antenna is short-circuited at. its terminals I’ = 1, these last terms disappear.

For an arbitrary applied voltage V ( t ) , the radia.t,ed field is easily obtained from (41) by using convolution. h sketch

FRANCESCHE'ITI AXD PAPAS: PULSED -4NTlCNNAS 659

t'

Fig. 8. Relevant to transient radiatiou from circular loop.

of the transient. field radiated by t,he antenna when the applied volt,age is a rectangular pulse is given in Fig. 7.

Previous result.s can be easily applied to t,he case of a monopole antenna on an infinite perfectly conducting ground plane. The radiation from the lower tip of the antmema disappears, but the transient pulses re0ected from t.he ground plane must be t,aken into account. In this way, a very satisfactory agreement with the experimental result,s given in [lS] is ob ta ind2

The transient radiation from a circular loop can be obt,ained folloming similar lines. The loop of ra.dius R is fed by a voltage generator of internal impedance azo (see Fig. 8). The transformed radiative field due to a t,raveling- wave current. la exp [isx] along the loop is given by [40]

80 = E1 COS 0 En * (-i)"nJ,( ( w / c ) R sin 0 ) (2 - n2) ( w / c ) R sin 0

- [s sin n+ - in cos n+]

4 e,x2 H z -

where f

(42a)

(42b)

(42C)

El = if - (exp [?2?rs] - 1) exp IoRw 2m-c

Expressions (42) are the generalization of the field radiated by a loop with a. t,raveling-mave current Io exp [i7nx], rrz integer. Letting s = m + E and t.hen .$ approach zero, t.he conventional results [41] are recovered.

nately, possible due to the lack of quantitative data in [18]. No checking relative to the amplihde of t.he fields is, unf0rt.u-

Now, t,he transfornled current a.long the loop is assumed to be coincident with t,hat of a transmission line of length TR, characteristic impedance 2, (real) and short-circuit,ed at its end. Hence

f = I , cos C(w/c)R(a - I x 113 cos C(4C)Rl

P cos [(W/C)R(T - 1 x i ) ] - _ - 20 a cos [ ( w / c ) R ] - i sin [ ( w / c ) R ] (44)

\\-here is the t,ransformed volt,a.ge of t.he generator. Expression (44) can be recast, in t,erms of two traveling-

Rave currents moving in the positive and negat,ive x angular coordinate directions. By algebraic manipulation

f = P exp [ i ( w / c ) ~ x l + exp [ i ( w / c ) ~ ( 2 n - x) 1

(45) From (45) and (42) with s = wR/c, and using super-

position, the transformed field radiated by the loop can be computed. Hence

Zo(a + 1) 1 - r exp [ 2 ~ i ( w / c ) R ]

2 cos e ( - i ) W n ( (u / c )R sin 0) E0 = Eo - C n sin n+ (46b) sine (wR/c)' - n2

where

(47)

The solut,ion for the transient. radiat.ed field is obtained by Fourier transforming (46). In t,he case of a. Dirac pulse excitation, V ( t ) = VoS(t), and thetransform corresponds to the field radiated by a point charge going around in the loop. This transform ca.n be comput,ed in t,erms of k n o w functions and convolution integrals [45].

VII. CONCLUSIONS AND RECOMMENDATIONS

In the preceding sections the t.ransient radiation from a number of radiating st.ructures has been considered: element.ary electric and magnetic dipoles, coaxial apertures, infinitely long cylindrical antennas, cylindrical antennas, and loops of finite dimensions. It has been shown that two

- . -

660 IEEE TRANSACTIONS ON ANTENNAS ATTD PROPAGATIOK, SEPTEMBER 1974

approaches-the microscopic and the macroscopic-are possible, depending on the time-scale which is being con- sidered. The procedure for recovering one result from the other has also been exploited.

The cases considered cover most practical applica,tions, and the theory developed seem to be simple, sound, and apt to an engineering development. The only limitation is the neglecting of the dispersion of the pulsed currents along the radiating structures.

In the case of coaxial apertures, this corresponds to neglecting higher order modes of excitation. It. is reason- able t o surmise that this is acceptable provided t.hat, the transverse dimensions of the aperture are small compared to cT, T being the equivalent durat,ion of the applied pulse. This has been rigorously proved by using a fre- quency doma@ analysis for the circular coaxial aperture [30]. .

In the case of cylindrica.1 antenna.s and loops, t,he approximation corresponds t o neglecting the dispersion of the current pulse injected in the radiating structure. The deformation of the pulses increases as time elapses, so t.hat the deformation becomes more pronounced on “long” radiating structures. Therefore, the analysis pre- sented is a.cceptable provided that the radiator dimensions are not large compared to GT, or, in any case, if reference is made to t,he early-time behavior of the fields.

A better underst.anding of the pulse deformation a.long the radiators is therefore a necessary condition for extend- ing the validity of the results presented in this paper. “A wise man once said that science is no good because it generates ten new problems for every problem it solves. A wiser man answered him by saying that science is good because it uncovers ten new problems for every problem it solves” [42].

ACKNOWUDGMENT The authors are indebted t.o R. Stratton for t.he careful

typing of the ma.nuscript.

REFERENCES [ l ] T. T. Wu, “1nt.roduction to linear antennas,” in Antenm

Theory, R. E. Collin and F. J. Zucker, E&. h’ew York: McGran~-Hill, 1969, vol. 1, ch. 8, p. 350.

121 P. 0. Brundell, “Transient electromagnetic waves around a cylindrical t.ransmitting antenna,’’ Em&xson Tech., vol. 16,

131 T. T. Wu, “Transient response of a dipole antenna,” J . Math. no. 1, pp. 137-162, 1960.

(41 S P. Morgan, Transient response of a dipole ant.enna,” J . Phys., vol. 2, n9; 6, pp. 892-894, 1961.

151 0. Einarsson, The s,t,ep voltage current response of an infinite Math. Phys., vyl. 3, no. 3, pp. 56.1-565, 1962.

conducting cylinder, Trans. Roy. Inst. Techml. Stockholm, ’ vol. 191, 1962.

[6] C. W. Harrison and R. W. P. King, “On the transient response of an infinite cylindrical antenna,” IEEE Trans. Antennas Propagat. (Commun.), vol. AP-15, pp. 301-302, Mar. 1967.

[7] R. W. Latham and I(. 5. H. Lee, “Transient propert.ies of an infinite cylindrical antenna,” Radio Sei., vol. 5, no. 1. pp. 715- 723, 1970.

181 T. T. Wu and R. W. P. King, “TFmsient response of linear antennas driven by a coaxial line, IEEE Tram. Antennas

191 R. J. Palciauskas and R. E. Beam, Transient fields of thin Propagat., vol. AP-11, pp. 17-23, Jan.‘?963.

cylindrical antennas,” ’ IEEE Trans. Antennas Propagat.

[IO] A. M. Abo-Zena and R. E. Beam, “Electromagnetic fields at (&mmun.), vol. AP-18, pp. 276-278, Mar. 1970.

points near pulseexcited linear antenna,” IEEE Trans. An-

tennas Propagat. (Commun.), voI. AP-19, pp. 129-131, Jan. 1971. C. W . Harrison and C. 5. Williams, “Transients in wide angle

AP-13, pp. 236-246, Mar. 1965. conical antennas,” IEEE Trans. Antennas Prqmgat., vol.

C . Polk, “Transient behavior of aperture antennas,” PTOC. IRE,

antenna pattern characterist,ics for arbit.rary time signals,” D. K. Cheng and F. I. Tseng, Transient and steady-state

492-493, July 1964. IEEE Trans. Antennas Propagat. (Commun.), vol. AP-12, pp.

flush-mounted coaxial aperture,” IEEE Trans. Ekhmna.gn. D. C. Chang and C. W. Harrison, “On 6he pulse response of a

Cmpat. , vol. EMC-13, pp. 14-18, Feb. 1971.

a circular loop antenna, IEEE Trans. Antennas Propagat. A. M. Abo-Zena and R. F,. Beam, “Transient radiation field of

(Commun.), vol. AP-20, pp. 380-383, May 1972. . A . R. Dion, “Transmission of step funct.ions by loop antennas,” IEEE Trans. Antennas Propagat. (Commun.), vol. AP-18, pp. 389-392, May 1970. D. L. Sengupta and P. P. Liu, “Waveforms radiated by con- tinuously loaded linear antennas excited by Gaussian pulse,” presented at Spring URSI Meet.ing, Washington, D. C., 1972. H. J. Schmitt, C. W . Harrison, and C. 5. Williams, Jr., “Calcu- lated and experimental response of thin cylindrical antennas to pulse excitation,” IEEE Trans. Antennas Propagat., vol. AP-14, pp. 120-127, Mar. 1966.

ant.enna,” IEEE Trans. Antennas Propagat. (Commun.), vol. D. Lamensdorf, “The transient response of the coaxial cone

R. W. P. King and H. J. Schmitt., “The transient. response of linear antennas and loops,” IRE Trans. Antennas Propagat., vol. AP-IO, pp. 222-228, May 1962. H. J. Schmitt, “Experiment,al observation of t,he t,ransient re- sponse of linear ant.ennas and loops,” IEEE Trans. A n t a n a s Propagat. (Commun.), vol. AP-11, pp.. 509-510, July 1963. C. E. Baum, “Electromagnetic translent. interaction nith ob- jects with emphasis on finite size objects, and some aspects of transient pulse production,” presented at Spring URSI Meet- ing, Washington, D. C., 1972. F. M. Tesche, “On t.he analysis of scatt,ering and antenna prob- lems using the singularity expansion technique,” IEEE Trans. Antennas Propagat., vol. AP-21, pp. 53-62, Jan. 1973. C. L. Bennett. and J. Martine, “-4 space-time integral equation solution for currents on wire structures with arbitrary excit.a- tion,” present:: at. Fall URSI Meet.ing, Columbus, Ohio, 1970. A. J. Poggio, The space-time domain magnetic vector pot.en- tial integral- equation,” IEEE Tra.ns. Antennas Propagal.

C. L. Bennett and A. M. Auckenthaler. Transient and time- (Commun.), vol. AP-19, pp. 702-704, Seyf. 1971.

V O ~ . 48, pp. 1281-1288, July 1960.‘[

AP-18, pp. 799-802, XOV. 1970.

domain solutions for antennas and scatterers,” IEEE Int. Cow.

E. P. Sayre and R. F. Harrington, “Time-domain radiation and D i g . , pp. 624-4525, 1971.

scattering by thin wires.” Avpl . Sci. Res., vol. 26, no. 6. vv. 413-444,-19?2. E. K. Miller, A. J. Poggio, and G. J. Burke, “An integro- ditferential equat,ion technique for the time-domain analysis of thin mire struct.ure, 1, the numerical method,” J . Comput. Phys., vol. 12, no. 1, pp. 24-48, 1973. T. I(. Lui and K. K. Mei, “A time-domain integral equation solution for linear antennas and scatterers,” Radio Sei., vol. 8, nos. 8 9 , pp. 797;804, 1973. N. L. Broome, Transient radiation from coaxial waveguide and cylindrical monopole ant.ennas,” Caltech Antenna Lab., Pasadena, Calif., Rep. 67, 1973. F. M. Tesche, “The effect. of the t.hin wire approximation and t.he source gap model on the high-frequency integral equation solution of radiating antennas,” IEEE Trans. Antennas Propagat. (Commun.), vol. AP-20, pp. 210-211, Mar. 1972. 2. L. Pyne and F. M. Tesche, “Calculation of the early time radiated electric field from linear antenna nith a finite source gap,” IEEE Trans. Antennas Propagat., vol. AP-21, pp. 740- 743, 1973. C. H. Papas, “Pulsed antennas,” in Yerevan Lectures on Electro- magnetic Thmry:, pp. 64-71, 1972. P. I. Richards, Transients in conducting media,” IRE Trans.

J. R. Wait, “Propagation of electromagpetic pulses in a ho- Antennas Propagat., vol. AP-6, pp. 158-182, Apr. 1958.

mogeneous conducting earth,” Appl. Sn’. Res. Sect. B, vol. 8, no. 3, pp. 213-253, 1960. -. ‘‘Exact. field of Dulsed diDole in a homogeneous cold

. ._ . * _

[37] J. R. Wait and D. A. Hill, “Transient. signals from a buried plasma,” J. Appl. Phys.; vol. 41, i o . 7, 288&2884,-1970.

magnet,ic dipole,” J. Appl. Phys., vol. 42, no. 10, pp. 3866- 3869, 1971.

[38] R. F. Harrington, Tim-Ha.rmonic Ekctromagnetic Fields. New York: McGraw-Hill, 1961, ch. 3, pp. 110-113.

FR.%XCESCHETTI AND PAPAS: PULSED ANTENNAS 661

[39] J. D. Kraus, Antennas. New Pork: hlcGraa-Hill, 1950, ch. 9, pp. 235-247 and 247-249.

[40] S. M . Prasad and B. N. Das, “A circular loop antenna wit.h traveling-wave current. distrlbution,” ZEEE Trans. Antennas Propagat. (Commun.), vol. $P-18, no. 2, pp. 278-280, Mar. 1970.

[-I11 R. W . P. King, “The loop antenna for t.ransmission and recep- tion,” in Antenna Theory, R.. E. Collin and F. J. Zucker, Eds.

[42] C. H. Papas, “Pulsed antennas,” in Yereuan Lectures on Electro- New Pork: hIcGran=Hill, 1969, vol. 1, ch. 11, pp. 458482.

[43] V. C. Martins, “A met,hod for sketching the transient electro- magnetic Theory, p. 79, 1970.

magnetic response radiating from a dipole-monopole antenna excit.ed by a general source impedance and driving function,”

[44] B. X. Krasil’nikov and G. I. Makarov, “Nonstatimary Processes in 1968 Znt. Yynzp. on Antennas and Propagation, Bost.on, Mass.

i n Linear Oscilhtors,” Vestnik Leningradskogo Cniversitet.a,

1453 G. Franceschetti and C. H. Papas, “Pulsed ant.ennas,” Caltech 16, 27-50, 1958.

hntenna Lab., Pasadena, Calif., Rep. 70, 1973.

Additiom.1 References

1461 K. Franz. “Behavior of diuole imuedance with time exuonential ~1

ercitat.ion of dipole,” Ar&. Elek Ubertragung, vol. i4, no. 4, pp. 167-168, 1960.

1471 H. J. Frankena. “Transient Dhenomena associated with Som- merfield’s horizontal dipole troblem,” Appl. Sci. Res. See. B, vol. 8, no. 4, pp. 357-358, 1960.

[48] A. T. de Hoop- and H. J. Frankena, “Radiation of pulse gen- erated by a vertical dipole above a plane nonconducting earth,” Appl. Sci. Res. Sec. B, vol. 8, no. 4, pp. 36!3-375, 1960.

[49] D. %I. Bolle, “The radiation pat.tern of long t,hin antennas for shortrpulse _excit.at,ion,” IRE Trans. Antanas Propagat., vol. AP-10, pp. ,877788, Sept. 1962.

1501 C. R. Burrom, “Transient response in an imperfect dielectric,”

May 1963. IEEE Trans. Antennas Propagat., vol. AP-11, pp. 286-296,

(511 C. W. Harrison, “Physical limitations on the measurement of

magnetic probes,” ZEEE Trans. Antennas Propagat., vol. t.ransient fields in air and dissipat,ive media using electric and

AP-12, pp. 53G533, Sept,. 1964. (521 E. A. Lewis, J. E. Rasmussen and J. R. Stahmann, “Wave-

forms and relat.ive phase stability of tram3ient.s radiated from a helicopter-supported antenna wire,” ZEEE Trans. Anfennus Propagat., vol. BP-13, pp. 257-261, Mar. 1965.

[53] D. E. hlereucther, “Transient. pulse transmission using im- pedance bonded cylindrical antennas,” Ph.D. dissertat,ion, Univ. Microfilms (from University of New Mexico, Albu- querque) order No. 68-13077, 1968.

[54] E. P. Saurce and R. F. Harrington, “Transient response of

st.raight rrire scatterers and ant.ennas,” in 1968 Znt. Symp. on

[55] C. L. Bennett and W. H. Weeks, “Electromagnet,ic pulse re- Antennas a.nd Propagation. Boston, Mass., pp. 160-164.

and Propagation, Boston, Mass., pp. 176183. sponse of cylindrical scatterers,” in 1968 Znt. Symp. on Antennas

[56] R. H. Ott, “The time domain backscat.ter from dipole antennas illuminated at broadside,” Air Force Cambridge Research

[57] R. J. Palciauskas, Transient fields of thin cylindrical an- Labs., Rep. 68-0808,‘11968.

wr-estern, Evanston, Ill.) order no. 69-2906, 1968. tennas,” Ph.D. dissert.ation, Univ. Microfilms (from Xorth-

(581 R. H. Ott, “The time domain backscatter from dipole antennas illuminated at broadside,” Sci. Rep. B-2 ,$eo., (AD688209), 1968.

[59] C. D. Taylor and T. H. Shumpert, Electromagnet.ic pulse generat.ion by an impedance loaded dipole ant.enna,” ZEEE Trans. Antennas Propagat. (Commun.), vol. AP-18, pp. 110-

1601 J. R. Wait and K. P. Spies, “Transient magnetic field of a 112, Jan. 1970.

Antennus Propagat. (Commun.), vol. ,4P-18, pp. 714-716, Sept.. pulsed electrlc dlpole in a dissipative medium,” ZEEE Trans.

1970. [61] C. h9. Knop, “On transient radiation from a log-periodic dipole

array,” ZEEE Trans. Antennas Propagat. (Commun.), vol. AP-18, pp. 8Oi-808, Wov. 1950.

[62] A. 31. Abo-Zena and R. E. Beam, “Transient radiation field of a pulse excited linear antenna parallel to a conduct.ing plane,” in 1970 Int. S y m p . on Antennas and Propagation, Columbus, Ohio, pp. 31G317.

1631 -. “E1ect.romaenet.ic fields at Doints near uulse-excited linear . . antenna,” ZEEE -Trans. Anten& Propagai. ~ ( C o G u n . ) , vol.

[64] Chen-to Tai and H;,E. Foster, “Processes of transient radia- AP-19, pp. 129-131, Jan. 1971.

t,ion and recept.ion. in 1971 Inter. Sump. on Antennas and Propagation, Sundai, Japan, pp. 1-2.

!651 A. M. AbwZena and B. E. Beam, “Transient. radiation fields of a pulse-excited loop ant.enna parallel to a conducting plane,” in 1971 Znt. Symp. on Antennas and Propagation, Sundai, Japan, PP. ,

[66] D. L. R aldelick, “Response of pulsed dipoles in conduct.ors,” in 1971 Int. S y m p . on Anten.nas and Propagation, Sundai, Japan, PP. 54.

(671 Y. K. Lin and K. K. hlei, “A line domain integral equation solution for thin wire dipoles,” in 1971 Znt. Sytnp. on Antennas and Propagation, Sundai, Japan, pp. 13-14.

(681 S. F. Fralick, “Radiation of electromagnetics Raves with Walch-function time variation. Preliminary results,” in 1972 ZEEE Znt. Conj. on Communications, Philadelphia, Pa., pp. 38-164.

[69] V. P. Pyat.i, R. W . Graff, and L. J. Ricardi, “Comments on a transform solution for the fields from a current. element, Trans. Anten.nas Propagal. (Commun.), vol. AP-21, p. 132, Jan. 1973.

~.