On the Design of Conical Antennas for Broadband Impedance Matching Performance

Abstract

In the scope of broadband radiators, the biconical antenna, or its monopole conical counterpart, is long known to be a proper choice. One common form of such radiator, the spherically capped conical antenna (SCCA), has closed-form solution to its input impedance, from which the broadband performance potential is easily verified. Nonetheless, from the design perspective, apart from a few clues inferred from existing solutions, little is found to accurately guide the choice of the main geometrical parameters of the antenna that will enable it to comply with a set of imposed bandwidth requirements. This paper proposes a simple 10-step sequence to derive conical or biconical antenna design charts. These charts provide straightforward information on the geometrical limits within which the required antenna impedance matching broadband performance is achieved. The method is assessed for the SCCA and the open conical antenna (OCA) using theoretical and simulated estimates of the input impedance. A discussion on the impact of the cap and the feed gap is included.

[ProQuest: [...] denotes non US-ASCII text; see PDF]

Francisco Estêvão Simão Pereira 1 and Maurício Henrique Costa Dias 1

Academic Editor:Ikmo Park

Military Institute of Engineering, 22290-270 Rio de Janeiro, RJ, Brazil

Received 29 December 2016; Accepted 7 February 2017; 27 February 2017

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Biconical and conical antennas are among the most widely known radiators. They are natural choices for RF communication, broadcasting, or EMI testing, whenever omnidirectional radiation pattern and broadband performance are needed. The basic conical geometry (or biconical in its dipole equivalent) is typically assembled either by a wire grid or by a continuous metal surface [1, 2].

The ideal biconical geometry is actually a frequency-independent antenna, though it is not a feasible one, since it extends to infinity in the axial direction. Its input impedance does not change with frequency. Realistic biconical antennas must be truncated, leading to a broadband rather than frequency-independent response [1, 2].

Truncated versions of the biconical and conical antenna have been addressed as early as the 1940s by Schelkunoff [3], Smith [4] and Papas and King [5, 6], providing analytic approaches to calculate the antenna impedance and field pattern dependence on frequency and on the main geometric parameters (length and flare angle). Those antennas have still drawn attention through the following decades up to the present days. Recent analytical works on the subject may be found, for instance, in [7-9]. Nevertheless, information regarding the design point of view of such antennas, as in [10], for instance, is still scarce in the literature, especially when specific broadband performance criteria must be fulfilled.

A typical question that may arise for the designer of such antenna is how its geometric parameters should be chosen from start. Yet, how can one combine compactness and broadband performance up to the required levels considering such a structure? Though a few clues may be inferred from the mere observation of commercially available antennas, reliable open-source information to accurately aid the antenna engineer in this sense is not easily found.

In the present scope, this work discusses a simple method to derive bandwidth compliance charts for the design of conical and biconical antennas. These design charts set the limits that the main geometric parameters must fall within so that the antenna should be able to comply with a given imposed bandwidth constraint. The method is easily applied to any variant of the conical or biconical antenna, provided that a series of impedance estimates spanning a frequency band large enough and some different flare angle values is obtained. Such estimates may be derived analytically, for instance, when closed-form equations are available, as in the case of the spherically capped conical antenna (SCCA) addressed in [5]. Otherwise, they can be collected by measurements or computed with the aid of antenna analysis tools, such as CST MW Studio, FEKO, HFSS, and NEC implementations. In this work, focus is given to the SCCA and to the open conical antenna (OCA), first analytically for the former, then with data generated by CST MW Studio for both of them. The influence of the cap and the feed gap is also discussed.

Section 2 summarizes the theoretical approach of Papas and King [5] on the SCCA, providing further insight on its bandwidth performance, from the antenna designer perspective. The method to derive design charts for broadband performance is described in the following section, taking as reference the design of an SCCA to be broadband matched to a 50 Ω load. Section 4 applies the proposed method to derive charts for the design of an OCA matched to 50 Ω, using impedance estimates calculated by simulation on CST MW Studio. Both SCCA and OCA bandwidth performances are analyzed, taking into account also the cap and the feed gap impact. Section 5 concludes this paper.

2. SCCA Input Impedance

2.1. Theoretical Model

The reference antenna in the present work is the SCCA fed by a coaxial line depicted in Figure 1. The ground plane is supposed to be ideal, that is, infinitely extended. The cone and the ground plane are also assumed to be perfect electrical conductors (PEC). This conical antenna configuration was assessed by Papas and King in [5] to derive an equation of the input impedance _Zin of an antenna of length a and flare angle _θ0 , leading to [figure omitted; refer to PDF] where [figure omitted; refer to PDF] ^hn(2) is the spherical Hankel function of the second kind, and _Pn (cos[...]_θ0 ) is the Legendre polynomial of order n. The summation in (3) must be over odd integral values. Yet, _Z0 is defined as the characteristic impedance of the antenna and ka is the wave number k=2π/λ=2πf/c multiplied by the sphere radius or cone side length a (λ is the free-space wavelength, f the frequency, and c the speed of light in vacuum).

Figure 1: Vertical plane cut of an SCCA fed by a coaxial feed line. The antenna is a body of revolution over the z-axis.

[figure omitted; refer to PDF]

It must be remarked that (1) holds for any reference plane along the coaxial line, provided that its characteristic impedance is equal to the antenna's _Z0 . Otherwise, (1) represents the impedance at the junction reference plane in Figure 1, and the proper correction factor must be applied at other reference planes along the cable, as addressed in [7].

If the biconical equivalent of the SCCA is to be analyzed instead, the set of equations (1) to (4) still holds, replacing (2) by [figure omitted; refer to PDF]

Figure 2 illustrates _Zin =_Rin +i_Xin as a function of ka for three different flare angles: 15°, 45°, and 75°. The summation in (3) was truncated with a relative error of 10^-4 . The broadband potential is clear as the antenna is large (high a) or the frequency is high or both, since _Rin converges to _Z0 while _Xin converges to 0. It is also worth remarking how _Z0 decreases with _θ0 and how the damped oscillations are relatively high at the first two cycles. Such features are paramount from the design perspective, especially if antenna compactness is a major requirement.

Figure 2: SCCA input impedance versus ka for _θ0 = 15°, 45°, and 75°: (a) resistance (_Rin ) and (b) reactance (_Xin ). _Z0 for each _θ0 is represented by a solid horizontal line in (a).

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

2.2. Bandwidth Performance

A proper assessment of how broadband the antenna may be depends not only on the antenna impedance itself but also on the reference impedance _Zr to which it should be matched. Yet, it depends on a reflection coefficient (_S11 ) threshold, below which antenna matching is regarded as achieved for practical purposes. In this work, the reference impedance is _Zr = 50 Ω and two _S11 thresholds are considered: -6 and -10 dB. These values are typical samples of what is seen in commercial antenna brochures and in antenna literature in general for different target applications.

Figure 3 illustrates _S11 for the same impedance responses plotted in Figure 2. The SCCA with 45° flare angle reaches the -6 dB threshold at ka = 0.94 and the -10 dB one at ka = 1.2. From those points on, the antenna is matched. It is expected that the best broadband performances be achieved for the flare angles providing _Z0 close to the 50 Ω reference impedance. Such will take place roughly for angles between 30° and 60° for the SCCA. Nonetheless, the 15° angle is still able to provide broadband behavior from ka = 2.88 on, regarding the -6 dB limit, but it is not compliant with the more rigorous -10 dB level. On its turn, with _θ0 = 75°, the matching threshold would have to be as high as -3.5 dB for the antenna to present acceptable wideband matching from the second resonance on.

Figure 3: _{S 11} of the SCCA versus ka for _θ0 = 15°, 45°, and 75° for _Zr = 50 Ω.

[figure omitted; refer to PDF]

As expected from the damped oscillatory behavior of the SCCA impedance, the most critical region for achieving wideband matching is at the lowest values of ka. In this sense, the lowest impedance matching ka (k_aL ) is a relevant parameter from the design point of view. In cases such as the 45° angle, though the bandwidth is theoretically unlimited, it has a starting point that defines the lowest operation frequency if the antenna length is fixed, or, alternatively, how small can the antenna length be, for a given desired lowest operation frequency.

To sum up, the antenna designer cannot rely exclusively on _Z0 to choose the geometric parameters. A closer look to the first cycles of the impedance response is crucial to define the lowest operation limit. Furthermore, other practical aspects will result in changes both on the lowest and on the highest limits of the passband, requiring more realistic impedance estimates than the ones derived by Papas and King's model.

3. Design Chart for Broadband Performance

As addressed in the previous section, the broadband antenna design point of view goes beyond what is inferred from the impedance response alone. The impedance dependence on the antenna main geometrical parameters must be previously mapped and translated to bandwidth performance that must comply with imposed project requirements, expressed in terms of a reference impedance and reflection coefficient thresholds.

In this sense, a general method to derive bandwidth compliance charts to aid the conical or biconical antenna design is proposed step by step, as follows. It is explained taking the SCCA theoretical model briefly reproduced in Section 2 as reference but may easily be extended to other potentially broadband antennas, even if no analytical model is available to calculate its impedance. The method is exemplified with specific reference values while the sequence is chained.

(i) Calculate a reasonable number N of impedance responses within practical limits of the flare angle, say N=71 responses from 10° to 80°, as considered in the present example.

(ii) From the design requirements, define the reference impedance (_Zr = 50 Ω, e.g., as considered in the present example) and calculate the corresponding _S11 for the N impedance responses.

(iii) Yet, from the design requirements, define a reference passband threshold, say -6 or -10 dB, and a maximum ka of interest (k_amax ). Alternatively, for a given fixed a, the maximum frequency of interest _fmax must be chosen. In the present example, both -6 and -10 dB are chosen.

(iv) Choose a criterion to define the lowest operation ka or frequency (k_aL or _fL , resp.). It can be the lowest ka or frequency that cuts the threshold (1st criterion) or the lowest ka or frequency of the largest observable subband (2nd criterion), as depicted in Figure 4. The former is chosen for the remainder of this work.

Figure 4: Lowest operation k_aL definition criteria.

[figure omitted; refer to PDF]

(v) Choose the bandwidth metric, which could be either the relative bandwidth _BWr or the k_aH /k_aL ratio (or _fH /_fL for a given fixed a). The latter is chosen for the present example. _BWr is computed as [figure omitted; refer to PDF] or for a given fixed a [figure omitted; refer to PDF] It is worth remarking that a common broadband criterion is that _BWr > 2/3 or, equivalently, _fH /_fL >2 [1, 2].

(vi) For each of N _S11 responses available, find k_aH and k_aL (or _fH and _fL for a given fixed a) that cross the reference _S11 threshold.

(vii) Plot k_aL (or its normalized version k_aL /2π=_aL /_λL ) × _θ0 , as illustrated in Figure 5 for this example. If a fixed length a is given, plot _fL ×_θ0 instead. The apparently odd behavior of k_aL at 27-33° in Figure 5 is a mere consequence of _S11 response of the present example, as seen in Figure 6.

Figure 5: k _{a L} × _{θ 0} for the theoretical SCCA example.

[figure omitted; refer to PDF]

Figure 6: _{S 11} of the theoretical SCCA example versus ka for _θ0 = 28° and 32° for _Zr = 50 Ω.

[figure omitted; refer to PDF]

(viii) Plot k_aH /k_aL (or _fH /_fL for a given fixed a) × _θ0 . Figure 7 illustrates such a graph for the present example. It is worth remarking that k_aH is actually equal to k_amax in _θ0 range for which k_aH /k_aL >10. This is a consequence of the convergence of the SCCA impedance to _Z0 , as seen in Figures 2 and 3. In this example, k_amax = 52.36, or equivalently _fmax = 1 GHz for a = 2.5 m. However, such behavior does not necessarily hold for more realistic impedance estimates, as discussed ahead in Section 4.

Figure 7: k _{a H} / k _{a L} × _{θ 0} for the theoretical SCCA example.

[figure omitted; refer to PDF]

(ix) From the design requirements, define the minimum desired target for k_aH /k_aL ratio (or _fH /_fL for a given fixed a), _DFmin , say _DFmin = 20 for the present example.

(x) From the intermediary design charts k_aL ×_θ0 and k_aH /k_aL ×_θ0 , generate a M×N truth table, M being a chosen number of samples of k_aL within expected values from the design requirements, where each cell is true whenever k_aH /k_aL (k_aL ,_θ0 )≥_DFmin . This resulting matrix provides a chart like the one illustrated in Figure 8(a). The vertical axis is represented in terms of the normalized version of k_aL (k_aL /2π=a/_λL , the relative length) in the example. Alternatively, instead of sampling k_aL , the length a may be sampled over _fL ×_θ0 and _fH /_fL ×_θ0 plots, resulting in a chart like the one in Figure 8(b). In this example, _fL = 30 MHz and _fmax = 1 GHz. This absolute version of the chart is actually the most helpful for the antenna design, providing straightforwardly the length and angle values for which the desired bandwidth performance is achieved, for a given set of requirements.

Figure 8: Design chart for the theoretical SCCA example with _DFmin = 20: (a) relative and (b) absolute length versus _θ0 .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

The specific results of this example show that the SCCA is able to provide 50 Ω broadband impedance matching even with lengths smaller than _λL /4. For the -6 dB threshold, the length can be as small as 0.12_λL and _θ0 as narrow as 19°, while for -10 dB the antenna must have at least 0.19_λL and _θ0 = 33° to comply. A note is due regarding the biconical counterpart. Since its _Z0 is twice the value of the conical antenna's, the design chart for 50 Ω direct matching is not expected to be as good as the one in Figure 8. In fact, commercial biconical antennas typically have a matching circuit to compensate that.

The availability of a closed-form solution for the impedance of the SCCA (or its biconical counterpart) surely eases the generation of design charts such as the proposed. Nonetheless, the rationale still applies to other variants of the conical antenna, such as the skeleton biconical antenna [11, 12] or the OCA, provided the impedance estimates needed in step (i) are obtained otherwise. They could be derived by measurements or from antenna analysis software simulations. Actually, more realistic design charts are expected to be generated this way, since theoretical models usually disregard a few practical aspects. In this sense, the next section presents design charts for the OCA based on simulations from CST MW Studio, including the influence of another important geometrical parameter, the feed gap length.

4. Open Conical Antenna Design

4.1. Outline

The OCA configuration, depicted in Figure 9, is actually easier and cheaper to assemble than the SCCA. Since the opening is in line with the axial direction and recalling that the radiation intensity towards the z-axis is inherently low in relation to the horizontal plane, as assessed in [3, 6], the OCA tends to perform very similarly to the SCCA, either impedance-wise or radiation-wise.

Figure 9: (a) Vertical plane cut of an OCA fed by a coaxial feed line and (b) a close-up of its junction. The antenna is a body of revolution over the z-axis.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

In order to apply the design chart procedure of Section 3 to a more realistic configuration, impedance estimates of the OCA were derived with the aid of CST MW Studio. In addition, the simulations incorporated the feed gap influence on the impedance. In the theoretical model of the SCCA in [5, 6], the gap length g was assumed to be infinitesimally small, and for that its influence was not assessed.

Both SCCA and OCA configurations were simulated in CST for N=15 flare angles: _θ0 =^10° ,^15° ,^20° ,...,^80° . Figure 10 reproduces the SCCA and OCA views from the graphical user interface (GUI) of CST. The design requirements were the same from the example in Section 3: _Zr = 50 Ω, _S11 thresholds at -6 and -10 dB, a = 2.5 m, _fmax = 1 GHz, and _fH /_fL > 20. PEC antennas were considered.

Figure 10: CST GUI views of the (a) OCA and (b) SCCA.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

The CST setup for simulation was similar to the ones adopted in [13, 14]. The ground plane was set as PEC and a 50 Ω discrete port was inserted between the cone and the plane. Also, the ground plane was assumed to be perfect, extending to infinity, which is achieved setting xy plane boundary condition to be a PEC. The minimum gap length from which the time domain solver was able to provide valid results was g = 1 mm. The cone shell width w of the OCA was set to 0.5 mm. Mesh size varied from 20 to 26 cells per wavelength (of the maximum frequency, 1 GHz). The solver precision was equal to or better than -60 dB. The frequency span was from 1 MHz to 1 GHz, with 1000 samples.

4.2. SCCA versus OCA

Figures 11-13 present the SCCA and OCA impedances calculated from CST at the same three different flare angles of Section 2: 15°, 45°, and 75°. They also show the corresponding theoretical estimate for the SCCA for comparison purposes. As expected, the OCA impedance does not differ much from the SCCA, indicating that the cap effect is not that significant. What seems to be more relevant in the present comparison is the feed gap effect, present only in the simulated curves. It imposes an almost linear tilt to _Z0 convergence line in _Rin curves, as well as to X = 0 line, which represents the convergence reactance value of the theoretical _Xin . Such effect surely deserves a detailed analysis regarding the angle dependence, but it is out of the scope of the present work. It is worth mentioning that an unsuccessful attempt to fit such effects to a lumped RLC circuit model was made, but the results clearly indicated that a more complex approach is needed.

Figure 11: _{Z in} of the theoretical SCCA and the CST simulated SCCA and OCA for _θ0 = 15°: (a) _Rin and (b) _Xin .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Figure 12: _{Z in} of the theoretical SCCA and the CST simulated SCCA and OCA for _θ0 = 45°: (a) _Rin and (b) _Xin .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Figure 13: _{Z in} of the theoretical SCCA and the CST simulated SCCA and OCA for _θ0 = 75°: (a) _Rin and (b) _Xin .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

The corresponding reflection coefficients for the present example are shown in Figures 14-16. Though the tilt due to the feed gap is more clearly seen at the highest frequencies, the most relevant impact in this sense is on the passband lower limit _fL definition. Such behavior is better evidenced in _fL ×_θ0 chart in Figure 17. As a consequence, the broadband potential of the more realistic conical antennas is lower than the one predicted by the theoretical model, as seen in _fH /_fL ×_θ0 chart in Figure 18. It is also worth noting that the cap effect is also more relevant at the lower part of the spectrum, as seen in the charts of Figures 17 and 18, with the better performance of the SCCA over the OCA, as should be expected from wideband antennas fundamentals [1-3]. Finally, design charts for the simulated SCCA and OCA configurations are shown in Figures 19 and 20, reflecting the effects discussed so far. The same D_Fmin = 20 reference level of the previous section example was taken here.

Figure 14: _{S 11} of the theoretical SCCA and the CST simulated SCCA and OCA for _θ0 = 15° and _Zr = 50 Ω.

[figure omitted; refer to PDF]

Figure 15: _{S 11} of the theoretical SCCA and the CST simulated SCCA and OCA for _θ0 = 45° and _Zr = 50 Ω.

[figure omitted; refer to PDF]

Figure 16: _{S 11} of the theoretical SCCA and the CST simulated SCCA and OCA for _θ0 = 75° and _Zr = 50 Ω.

[figure omitted; refer to PDF]

Figure 17: _{f L} × _{θ 0} chart for the theoretical SCCA and the CST simulated SCCA and OCA for _Zr = 50 Ω.

[figure omitted; refer to PDF]

Figure 18: _{f H} / _{f L} × _{θ 0} chart for the theoretical SCCA and the CST simulated SCCA and OCA for _Zr = 50 Ω.

[figure omitted; refer to PDF]

Figure 19: Design chart for the CST simulated SCCA example with _DFmin = 20.

[figure omitted; refer to PDF]

Figure 20: Design chart for the CST simulated OCA example with _DFmin = 20.

[figure omitted; refer to PDF]

4.3. Feed Gap Influence

The previous analysis pointed out the gap feed length as a relatively more impacting parameter than the spherical cap. Though the relevance of the feed gap to the conical antenna impedance and radiation performances should be expected, as addressed in [14], its dominance over the cap was not foreseen beforehand.

In this sense, an extension to the feed gap analysis was carried out for the OCA configuration, calculating the impedance for two further gap lengths: 2 and 4 mm. Figures 21-23 show the impedance for the same three flare angles of the previous example. Though the most visible trend difference is the tilt at higher frequencies, the lower frequency band is also affected increasingly with longer gaps. The corresponding _S11 curves in Figures 24-26 give the same impression.

Figure 21: _{Z in} of the OCA for 3 different gap lengths and _θ0 = 15° (_Rin : curves above, _Xin : curves below).

[figure omitted; refer to PDF]

Figure 22: _{Z in} of the OCA for 3 different gap lengths and _θ0 = 45° (_Rin : curves above, _Xin : curves below).

[figure omitted; refer to PDF]

Figure 23: _{Z in} of the OCA for 3 different gap lengths and _θ0 = 75° (_Rin : curves above, _Xin : curves below).

[figure omitted; refer to PDF]

Figure 24: _{S 11} of the OCA for 3 different gap lengths, _θ0 = 15° and _Zr = 50 Ω.

[figure omitted; refer to PDF]

Figure 25: _{S 11} of the OCA for 3 different gap lengths, _θ0 = 45° and _Zr = 50 Ω.

[figure omitted; refer to PDF]

Figure 26: _{S 11} of the OCA for 3 different gap lengths, _θ0 = 75° and _Zr = 50 Ω.

[figure omitted; refer to PDF]

The actual impact on the broadband performance is seen in the intermediary design charts in Figures 27 and 28. At least for the design requirements considered, the higher frequency tilt is still not enough to make _fH < _fmax , as seen in Figure 27. On the other hand, _fL is pushed above for increasing gap lengths. Anyway, the overall impact on the relative bandwidth chart is low, as seen in Figure 28, not enough to inhibit the broadband potential of the OCA.

Figure 27: _{f L} × _{θ 0} chart for the OCA for 3 different gap lengths and _Zr = 50 Ω.

[figure omitted; refer to PDF]

Figure 28: _{f H} / _{f L} × _{θ 0} chart for the OCA for 3 different gap lengths and _Zr = 50 Ω.

[figure omitted; refer to PDF]

5. Conclusion

In this paper, a 10-step sequence was proposed to derive charts that directly relate impedance matching bandwidth compliance to the main geometrical parameters of conical or biconical antennas. The motivation came from the challenge faced by the antenna engineer when specific broadband impedance requirements are imposed. From the design perspective, the choice for the conical or biconical configuration leads to the subsequent questions on how long and how opened must the antenna be to meet the specifications. The proposed charts try to provide straightforward answers to those queries.

The method was assessed taking the SCCA and the OCA configurations, using theoretical and simulated estimates of the input impedance. A hypothetical set of requirements was imposed, with a reference impedance of 50 Ω, -6 and -10 dB _S11 thresholds, and a minimum relative bandwidth such that _fH /_fL was greater than 20. The impact of the spherical cap and the feed gap was also addressed, concluding that the broadband performance of those antennas is still achieved for a relatively large set of angles and lengths, as seen in the design charts.

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