Biconical Antennas with Unequal Cone Angles

The problem of radiation and reception of electromagnetic waves associated with a spherically capped biconical antenna having unequal cone angles 1 and 2 is investigated. Both cones that comprise a bicone are excited symmetrically at the apices by a voltage source so that the only higher order modes are TM. A variational expression for the terminal admittance is derived. Under the wide-angle approximation, expressions for the radiated field, the effective height, and the terminal admittance are obtained. In addition, limiting values of these quantities are derived for electrically small and electrically large wideangle bicones. The results for arbitrary cone angles are new and subsume results that appear in the existing literature as special cases such as where 1 = 2 or 2 = =2. Moreover, the approximations of this paper are more accurate than many in the literature. It is argued that the radiation pattern of an electrically small cone is proportional to sin , which is similar to that of a short dipole; whereas the pattern behaves like 1= sin for electrically large cones. The parameter is the angle from the bicone’s axis of symmetry to the observation direction. Consequently, the direction of maximum radiation changes with exciting frequency for a bicone of fixed length. Although most of the analyses are presented in the frequency-domain, time-domain responses of bicones are discussed for some special cases that are similar to situations considered by Harrison and Williams. In particular, the time-domain radiated field and the received voltage are shown to depend on the input’s passband and on the match between the source and the bicone.

Biconical Antennas with Unequal Cone Angles Surendra N. Samaddar, Life Senior Member, IEEE, and Eric L. Mokole, Member, IEEE Abstract-The problem of radiation and reception of electromagnetic waves associated with a spherically capped biconical antenna having unequal cone angles 1 and 2 is investigated.Both cones that comprise a bicone are excited symmetrically at the apices by a voltage source so that the only higher order modes are TM.A variational expression for the terminal admittance is derived.Under the wide-angle approximation, expressions for the radiated field, the effective height, and the terminal admittance are obtained.In addition, limiting values of these quantities are derived for electrically small and electrically large wideangle bicones.The results for arbitrary cone angles are new and subsume results that appear in the existing literature as special cases such as where 1 = 2 or 2 = =2.Moreover, the approximations of this paper are more accurate than many in the literature.It is argued that the radiation pattern of an electrically small cone is proportional to sin , which is similar to that of a short dipole; whereas the pattern behaves like 1= sin for electrically large cones.The parameter is the angle from the bicone's axis of symmetry to the observation direction.Consequently, the direction of maximum radiation changes with exciting frequency for a bicone of fixed length.Although most of the analyses are presented in the frequency-domain, time-domain responses of bicones are discussed for some special cases that are similar to situations considered by Harrison and Williams.In particular, the time-domain radiated field and the received voltage are shown to depend on the input's passband and on the match between the source and the bicone.

I. INTRODUCTION
I N this paper, the biconical antenna analyses provided by Schelkunoff [1], Smith [2], Tai [3]- [5], Papas and King [6], [7], and Sandler and King [8] are generalized by considering axially symmetric bicones having unequal cone angles.The geometry of the antenna configuration is shown in Fig. 1, and are spherical coordinates.The common axis of the two cones is oriented along the axis and the cone angles and satisfy and .Relative to , the lateral surfaces of the upper and lower cones correspond to and , respectively.By a proper choice of and , the exit aperture of the antenna can be adjusted so that the radiated power will be directed in a desired direction, which is one of the motivations behind this investigation.Furthermore, knowledge of biconical antenna characteristics is helpful in understanding why TEM horns, V-antennas, triangular plates, and bow-tie antennas are very wide band.
The cones are excited symmetrically at the apices so that only TEM and TM modes are generated.The field components can then be expressed in terms of a scalar Manuscript received October 10, 1996; revised September 25, 1997.This work was supported by the Naval Research Laboratory and the Office of Naval Research.
The authors are with Naval Research Laboratory, Washington, DC 20375 USA.
Publisher Item Identifier S 0018-926X(98)01486-0. function [1]- [5], which is equivalent to the radial component of the vector potential and is azimuthally invariant.Formal expressions for the field components are presented first in each of the regions and , where is the length of each cone and corresponds to the radius of the sphere in Fig. 1.On using these field components, a variational expression for the terminal admittance is derived in Sections II-A and II-B.When , this variational expression agrees with Tai's result [5].
Also of interest are the wide-band and ultrawide-band behaviors of biconical antennas, which have application in surveillance and communications.Papas and King [6] demonstrated that both the input resistance and reactance of wideangle bicones having cone angles exceeding 40 are very slowly varying functions of frequency for very wide frequency ranges.Furthermore, they showed that the higher-order TM modes in the antenna region can be neglected for wide-angle bicones.Under this wide-angle approximation, general results for a bicone's effective length, input impedance, terminal admittance, and radiated field are derived (Section II-C).These results are further analyzed for the limiting cases of electrically small and large wide-angle cones.Radiation patterns for two specific wide-angle bicones with U.S. Government work not protected by U.S. copyright.
inches are discussed to contrast the behavior of equal cone angles versus that of unequal cone angles: 1) and 2) and .In Section III, the transient responses of wide-angle biconical antennas for some special cases, similar to the situations considered by Harrison and Williams [9], are studied.

II. ANALYSIS
In this section, formal expressions of the electric-and magnetic-field components for the antenna region and its complement are expanded in terms of series involving Bessel , Legendre , and Hankel functions.The unknown coefficients of these expansions and the terminal admittance of the bicone at are determined.The expression for is recast in a variational form, which is evaluated to obtain a series representation of .At this point, the expressions for the fields, the input impedance , the effective height , and are simplified by using a wide-angle approximation for the cones.These results are reduced further by applying approximations for two special cases: electrically small and electrically large wide-angle cones.Within the antenna region and , the components of the electric and magnetic fields are given by (1a) where denotes differentiation of with respect to its argument and and are constants that must be determined.Furthermore, by [ where is the derivative of with respect to its argument, are unknown constants, and (4d)

A. Determination of , , and
Representations for the unknown coefficients and and the terminal admittance of ( 1) and ( 4) are determined by applying the continuity conditions at and the orthogonality relations for and .In particular, since and are continuous at for The manipulations in the remainder of this section simplify the coupling among the unknowns and completely uncouples them under the wide-angle cone assumption, so that each unknown is expressed in terms of known quantities.
Integrating both sides of ( 6) with respect to from to and applying the boundary conditions yield The function is introduced to account for the situation when the cone with angle is replaced with an infinite perfect conductor at the plane (Fig. 2) that occurs in [6]- [9].After applying the orthogonality of on to (5), one obtains (11a) The quantity is given by (11b) Next, substituting (7b) for and (11a) for in (9) results in (12) Fig. 2. Cross-sectional view of spherically capped conical antenna in rectangular (x; y; z) and spherical (r; ; ) coordinates above an infinite conducting ground plane (xy-plane).The cone angle is 1 , the altitude is h 1 , and a is both the cone's slant height and the hemisphere's radius.The hemsiphere is indicated by the dotted curve.
which involves only the unknowns .Finally, replacement of in (7a) by the right side of ( 12) provides an expression for the terminal admittance (13) in terms of only.Hence, (11a), (12), and (13) represent the coupling among and more simply than ( 5) and ( 6).As mentioned in the introduction, when TM waves or complementary waves in the antenna region are negligible can be neglected, which is accurate for wide-angle cones according to [2] and [6].Letting in (9) implies the following approximate value of : (14) Subject to the wide-angle cone condition , which may be called the zero-order approximation, the expression for the terminal admittance becomes (15) (10a) Equation ( 15) can also be obtained from ( 13) by neglecting the term consisting of the triple sum and then replacing with (7b) and ( 14).

B. Evaluation of
The representation of is first recast in terms of the unknown aperture field , which is then expanded in a series involving the sequence .Finally, is derived by requiring that give a stationary value of .With the aid of (8a) and (8b), express and as follows in terms of the aperture field at : Substituting ( 3), (16a), and (16b) for , and , respectively, into (6) Additional information is now used to derive an alternate representation for the coefficient preceding the summand in (32b) and to obtain under the wide-angle approximation.To these ends, equate the expressions in (24) and (26) at and use (2) and (30).After manipulation, one obtains the relation for the terminal admittance.Equations ( 32) and (34) provide expressions for the field, the effective height, and the terminal admittance under the wideangle approximation.In the next two sections, these results are simplified for electrically small and electrically large wide-angle bicones.
1) Small-Cone Approximation : In this case, consider of (33b) first.Through (27a), express in terms of the Bessel functions of the first and second kinds and substitute the standard asymptotic approximations of and for the small argument [10].In the resulting asymptotic approximation of , one eliminates terms with for and expands the remaining expression in a series involving powers of with the binomial expansion.Finally, retaining only terms of the expansion through inclusive leads to the approximation (35a) Next, substitute (35a) into (33b) to get (35b) Now an approximation of is obtained.One can argue that (36a) Moreover, substituting (35b) into (34a) and expanding the trigonometric functions in Maclaurin series yield (36b) By taking only the first term of each series and excluding terms of order and higher, the approximations for and can be simplified further to where and .The approximations in (38) are accurate provided is not near and is not near zero; otherwise, more than one term in the series for of (35b) is needed to obtain reasonable accuracy.For example, the second and third terms are not negligible compared to the first term for .According to (37) and (38b), the radiation pattern of an electrically small wide-angle bicone is approximately proportional to and, consequently, is similar to the pattern of a short dipole, which agrees with an observation made in [8].
Another parameter of interest is the bicone's driving impedance of (29), which, by (34a) and (35b), is approximately (39a), shown at the bottom of the page, where the arguments of , , and are suppressed.Under the caveat on accuracy pertaining to (38a) and under the assumption that is negligible for , a simpler approximation of is (39b) As expected, the input impedance of an electrically small bicone behaves like a capacitive impedance , where the equivalent capacitance is computed by making the obvious identification in either of (39).
An identity between two evaluationsand -of the zero-order approximation of the terminal admittance is now established.When either or , the boundary conditions impose the constraint that (39a) the index of the summation in (15) runs over the odd natural numbers only.Since for odd (40a) However, by (2a), which implies that (40b) Note that (40) and relations based on it do not depend on the small-cone approximation.In summary, observe that the radiated field and the input impedance for the electrically small wide-angle bicone depend explicitly on the cone angles and .Moreover, upon setting , , and , in (38b) and in (39b) reduce to and (41) which are [9, Eq. ( 10)] and [11, Eq. ( 22)], respectively.
2) Large-Cone Approximation : In the highfrequency region, expressions for and must be established to derive results for and .The analysis of this section extends the approach of [9, Appendixes B and C] to obtain the desired results for the more general situation of arbitrary and .In particular, an expression for is obtained by using identities involving the Legendre polynomials.
According to [12] (42a)  This limit implies that the behavior of the field for approaches as the frequency increases without bound and that the directions of maximum radiation approach and .Consequently, maximum radiation of the electrically large wide-angle bicone does not occur at broadside .Furthermore, with (29) and (45c), one can easily show that the bicone's input impedance has the limit (51) Thus, for high frequencies the input impedance is essentially constant.Hence, the electrically large wide-angle bicone is a broadband and possibly an ultrawide-band antenna.
3) Examples: Two special cases are considered for in: 1) and 2) and .The relative field pattern associated with the -plane radiation pattern [7] (52) is plotted for various values of in both cases.In case 1), as mentioned previously, only the odd terms of the series are present because .Hence, on re-indexing each sum in (52) by setting and by allowing to run from zero to , (52) becomes [8, Eq. ( 6)].Truncate each series of the re-indexed version of (52) at terms and denote the resulting fraction as .The approximate pattern is plotted in Fig. 3 for five frequencies ranging from to .The correspondence between and frequency for each plot is shown in Table I.One high frequency and one low frequency are chosen to represent situations when and , respectively; while the other three frequencies are selected for comparison to [8, Fig. 5(c)], as well as for providing nominal values in the transition between and .The pattern has azimuthal symmetry and symmetry about .The former means that the pattern may be graphed in two dimensions and the latter means that one may restrict to to gain a complete understanding of the behavior of .As one can observe [Fig.3(a)], the pattern at low frequencies is similar to that of the short dipole as [8] notes.In fact, on utilizing the low-frequency approximation in (38b), one finds that , which is the short-dipole pattern.The graph of is coincident with that of Fig. 3(a) when they are  overlayed.Even the pattern for the transitional value of is only slightly distinguishible from the short dipole's pattern.Therefore, the approximation for is excellent for and is good for .As increases from 1 to 10.640 through the transitional region between electrically small and electrically large bicones [Fig.3(b)-(d)], the peak radiation remains at broadside; however, the radiation decreases for , and for each between 0 and 18 the radiation level exceeds that of the short dipole with a local maximum appearing at some intermediate angle. .This value of is picked because each pattern is accurate for and because Harrison used this value for calculating tables in [11].
In contrast, to obtain a reasonably accurate pattern for , at least 61 terms must be used [Fig.3(e)].Fig. 3(f) plots for 30 terms and is provided as a comparison to Fig. 3(e).Clearly, the detail is missing in Fig. 3(f).The pattern in Fig. 3(e) is considerably less smooth than the patterns for small and transitional values of .In particular, many smaller amplitude lobes appear for .The effect is less pronounced between 30 and 90 , where the pattern has a somewhat wavy nature as evidenced by the five local maxima and four local minima.The lobe structure for small is not unexpected because the number of oscillations and the magnitude near of the Associated Legendre Function increase as increases.Consequently, as increases, must increase to get accurate results.One major feature that distinguishes the pattern of a wide-angle bicone for high frequencies from the patterns at other frequencies is the migration of the direction of peak radiation away from broadside.In Fig. 3(e), the peak occurs well off broadside at .As increases, one expects the peak to approach the cone's surface at since by ( 50) and ( 52), as for .The limiting value in Fig. 4 is the triangular region bounded by (the vertical line segment ), the line , and the line .Consequently, as , the pattern calculated with (52) will approach the pattern of Fig. 4. Fig. 3(e) is consistent with Fig. 4 in that most of the energy is radiated in and the five local maxima are very close to the vertical line .In case 2), the summations in (52) are truncated at terms and the resulting fraction is denoted .Although not shown, the pattern looks like that of the short dipole for .As increases from unity through transitional values, the effect of distinct bicone angles is manifested in asymmetrical patterns relative to the axis.Specifically, in the limit as , .Moreover, the pattern approaches the triangular region bounded by the lines , , and , which are indicated by the dashed line segments in Fig. 5.To illustrate the behavior for large , Fig. 5 also displays the pattern for and (solid curve).This pattern has many small amplitude lobes for and .For angles between 60 and 108 , several relative maxima and minima occur near the line , with the two largest maxima at and .As and increase, the number of these local extrema increases and the associated approaches the line with the largest and second largest values of approaching the edges of the upper and lower cones, respectively.The two obvious differences from case 1) are: a) the loss of symmetry in the pattern about the axis of case 2) and b) the peak radiation occurs near the cone with the smaller cone angle in case 2).
These examples provide analytical justification of Sandler and King's numerically based observations [8] that the wideangle bicone behaves like a short dipole at low frequencies (38b) and has its direction of maximum radiation moving away from broadside toward the bicone's surface for high frequencies (50).Consequently, the tacit assumption in [9, Appendixes B and C] that the peak radiation occurs at for electrically large bicones is not justified.Since an ultrawide-band input signal's passband [13] may contain both low-and high-frequency spectral components relative to the bicone's passband, the explicit analytical characterizations of the frequency-dependent behavior contained herein are essential in analyzing radiation of ultrawide-band signals.

III. TIME HISTORY OF RADIATED AND RECEIVED FIELDS ASSOCIATED WITH ELECTRICALLY SMALL AND ELECTRICALLY LARGE WIDE-ANGLE BICONES
Thus far, the discussion of electrically small and large wide-angle conical antennas has concentrated on the impact to quantities in the frequency domain.Of interest, especially when the input is wide-band or ultrawide-band, is the temporal behavior of the radiated field and received voltage.This section expresses the time-domain radiated field of wide-angle conical antennas in terms of the input voltage and expresses the time-dependent load voltage of a receiving conical antenna in terms of the incident field.In particular, four special cases are considered: 1) transmission for ; 2) transmission for ; 3) reception for ; and 4) reception for .

Fig. 1 .
Fig.1.Cross-sectional view of spherically capped biconical antenna with equal slant heights denoted by a and with unequal cone angles 1 and 2 in rectangular (x; y; z) and spherical (r; ; ) coordinates.The x axis points out of the paper and the angle about the z axis is measured from the x axis toward the y axis.The dotted curve represents a sphere of radius a and the lengths h 1 and h 2 are the altitudes of the top cone and bottom cone, respectively.
Equation (2b) satisfies the boundary condition that vanishes on the surface of the bicone for and .In general, the index runs over a countable number of noninteger values, which are determined by solving the transcendental equation for each .Moreover, taking the limit as approaches in ( that vanishes on the spherical caps of the metallic bicone yield(9) where (see (10a) at the bottom of the page) and (10b)

Fig. 3 (
b)-(d) agrees well with [8, Fig. 5(c)].Sandler and King don't indicate how many terms of the series they use to generate their figures; however in the Mathematica calculations used to generate Fig. 3, one obtains good graphical depictions for when , respectively.As Table I indicates, Fig. 3(a)-(d) is generated for 30 terms of each series

Fig. 4 .
Fig.4.E-plane radiation pattern in the high-frequency limit (ka !+1) for a spherically capped biconical antenna with equal cone angles in free space.The cone angles 1 and 2 are equal to 53.1 and the slant height (height) is 20 in (12 in).

Fig. 5 .
Fig.5.Approximate pattern with ka = 106:3950 and N = 120 (solid curve) and high-frequency limit pattern (dashed curves) for a spherically capped biconical antenna with unequal cone angles in free space.The cone angles 1 and 2 are 53.1 and 70 , respectively, and the slant height (height) is 20 in (12 in).
If the first term on the right side of (23) is called the zeroorder solution, then the second expression may be called the correction term.

TABLE I CORRESPONDENCE
AMONG UPPER LIMIT M OF APPROXIMATE RADIATION DISTRIBUTION FUNCTION R M , ka, AND FREQUENCY f IN FIG. 3.