Cylindrical microstrip array: normalized current excitations
1. Introduction
Owing to their electrical and mechanical attractive characteristics, conformal microstrip antennas and their arrays are suitable for installation in a wide variety of structures such as aircrafts, missiles, satellites, ships, vehicles, base stations, etc. Specifically, these radiators can become integrated with the structures where they are mounted on and, consequently, do not cause extra drag and are less visible to the human eye; moreover they are lowweight, easy to fabricate and can be integrated with microwave and millimetrewave circuits [1,2]. Nonetheless, there are few algorithms available in the literature to assist their design. The purpose of this chapter is to present accurate design techniques for conformal microstrip antennas and arrays composed of these radiators that can bring, among other things, significant reductions in design time.
The development of efficient design techniques for conformal microstrip radiators, assisted by stateoftheart computational electromagnetic tools, is desirable in order to establish clear procedures that bring about reductions in computational time, along with high accuracy results. Nowadays, the commercial availability of high performance threedimensional electromagnetic tools allows computeraided analysis and optimization that replace the design process based on iterative experimental modification of the initial prototype. Software such as CST^{®}, which uses the Finite Integration Technique (FIT), and HFSS^{®}, based on the Finite Element Method (FEM), are two examples of analysis tools available in the market [3]. But, since they are only capable of performing the analysis of the structures, the synthesis of an antenna needs to be guided by an algorithm whereby iterative process of simulations, result analysis and model’s parameters modification are conducted until a set of goals is satisfied [4].
Generally, the design of a probefed microstrip antenna starts from an initial geometry determined by means of an approximate method such as the Transmissionline Model [57] or the Cavity Model [8]. Despite their numerical efficiency, i.e., they are not timeconsuming and do not require a powerful computer to run on, these methods are not accurate enough for the design of probefed conformal microstrip antennas, leading to the need of antenna model optimization through the use of fullwave electromagnetic solvers in an iterative process. However, the fullwave simulations demand high computational efforts. Therefore, it is advantageous to have a design technique that employs fullwave electromagnetic solvers for accuracy purposes, but requires a small number of simulations to accomplish the design. Unfortunately, the approximated methods mentioned before provide no means for using the fullwave solution data in a feedback scheme, what precludes their integration in an iterative design process, hence restricting them just to the initial design step. In this chapter, in order to overcome this drawback and to reduce the number of fullwave simulations required to synthesize a probefed conformal microstrip antenna with quasirectangular patch, a circuital model able to predict the antenna impedance locus calculated in the fullwave electromagnetic solver is developed with the aim of replacing the fullwave simulations for the probe positioning. This is accomplished by the use of a transmissionline model with a set of parameters derived to fit its impedance locus to the one obtained in the fullwave simulation [4]. Since this transmission line model adapts its input impedance to fit the one from the fullwave simulation, at each algorithm iteration, it is an adaptive model per nature, so it was named ATLM – Adaptive Transmission Line Model. In Section 2, the ATLM is described in detail and some design examples are given to demonstrate its applicability.
Similar to what occurs with conformal microstrip antennas, the literature does not provide a great number of techniques to guide the design of conformal microstrip arrays. Among these design techniques, there are, for example, the DolphChebyshev design and the Genetic Algorithms [9]. However, the results provided by the DolphChebyshev design are not accurate for beam steering [10], once it does not take the radiation patterns of the array elements into account in its calculations, i.e., for this pattern synthesis technique, the array is composed of only isotropic radiators; hence it implies errors in the main beam position and sidelobes levels when the real patterns of the array elements are considered. On the other hand, the Genetic Algorithms can handle well the radiation patterns of the array elements and guarantee that the sidelobes assume a level better than a given specification
Another concern for designing conformal microstrip arrays is how to implement a feed network that can impose appropriate excitations (amplitude and phase) on the array elements to synthesize a desired radiation pattern. Some microstrip arrays used in tracking systems, for example, employ the Butler Matrix [11] as a feed network. Nevertheless, this solution can just accomplish a limited set of look directions and cannot control the sidelobes levels. Hence, in this work, in order not to limit the number of radiation patterns that can be synthesized, an active circuit, composed of phase shifters and variable gain amplifiers, is adopted to feed the array elements. Expressions for calculating the phase shifts and the gains of these components are addressed in Section 4, as well as some design examples are provided to demonstrate their applicability.
2. Algorithm for conformal microstrip antennas design
The main property of the proposed ATLM is to allow the prediction of the impedance locus determined in the antenna fullwave analysis when one of its geometric parameters is modified, for instance, the probe position, thereby replacing fullwave simulations in probe position optimization. It results in a dramatic computational time saving, since a circuital simulation is usually at least 1000 times faster than a fullwave one. In this section, the ATLM is described in detail and some design examples are provided to highlight its advantages.
2.1. Algorithm description
In order to describe the algorithm for the design of conformal microstrip antennas, for the sake of simplicity, let us first consider a probefed planar microstrip antenna with a gular patch of length
It is convenient to write both the probe position
and the patch width as follows
Therefore, the standard set of control variables is composed of
In this work, it is considered that the resonant frequency
in which Γ
Since the antennas design will be conducted in an iterative manner, the optimization process of the model needs to be evaluated against optimization goals in order to set a stop criterion. Therefore, let the frequency error be defined as
and its maximum value specified as
The second optimization goal is expressed by means of
where Γ
Now that the main parameters of the design algorithm have been derived, let us focus on the Adaptive Transmission Line Model, depicted in Figure 2. As can be seen, this circuital model is composed of two microstrip lines, μ
where
in which Γ
Once the fullwave simulation Γ
Consider the generalized load reflection coefficient [16] that is written as
in which
after the optimization process.
As we want to ensure that Γ
which is the generator impedance utilized during the ATLM synthesis. On the other hand, for the circuital simulation afterwards,
Besides, to find a meaningful solution from a physical standpoint, the following two constraints are ensured during the ATLM synthesis
The complete probefed microstrip antenna design algorithm is depicted through the flowchart in Figure 3, which can be summarized as follows: perform a fullwave antenna simulation for a given patch length and probe position at a certain frequency range (simulation domain), which results in accurate impedance locus data; synthesize the ATLM based on the most updated fullwave simulation data available; optimize the probe position in order to match the antenna to its feed network through circuital simulation and evaluate the resonant frequency; perform patch length scaling; update the fullwave model with the new values of patch length and probe position; and repeat the whole process in an iterative manner until the goals are satisfied.
Generally, it is difficult to get the input impedance of the circuital model perfectly matched to the one obtained from fullwave simulation over the entire simulation domain [
The previous goal contributes to reduce the number of iterations required by the Gradient optimization tool to determine the set of parameters. It was found that, in general, the required time for the synthesis of the ATLM is at most 5% of the time spent for one fullwave simulation.
Regarding the probe position optimization, algorithm step 3b, it can be performed manually by means of a tuning process, a usual feature found in circuit simulators. Thus,
2.2. Applications
To illustrate the use of the technique proposed before, let us first consider the design of a cylindrical microstrip antenna (Figure 1(b)) with a quasirectangular metallic patch mounted on a cylindrical dielectric substrate with a thickness
Following the algorithm (Figure 3), a model was built (step 4a) in the CST^{®} software with
Hence, at the first decision point of the algorithm, the reflection coefficient magnitude at resonance is not lower than Γ
With the circuital model available, the probe position was optimized through manual tuning of the variable
Now let us design a probefed spherical microstrip antenna, such as the one illustrated in Figure 1(c). A copper sphere (ground layer) of 120.0mm radius is covered with a dielectric substrate of constant thickness
As a last example, let us consider the design of a conical microstrip antenna with a quasirectangular metallic patch, as shown in Figure 1(d). It is composed of a conical dielectric substrate of constant thickness
3. Radiation pattern synthesis of conformal microstrip arrays
The previous section addressed a computationally efficient algorithm for assisting the design of probefed conformal microstrip antennas with quasirectangular patches. In order to demonstrate its applicability, three conformal microstrip antennas were synthesized: a cylindrical, a spherical and a conical one. According to what was observed, the algorithm converges very fast, what expedites the antennas’ design time.
Another concern in the design of conformal radiators is how to determine the current excitations of a conformal microstrip array to synthesize a desired radiation pattern, in which both the main beam position and the sidelobes levels can be controlled. This section is dedicated to the presentation of a technique employed for the design of conformal microstrip arrays. It is based on the iterative solution of linearly constrained least squares problems [12], so it has closedform solutions and exhibits fast convergence, and, more important, it takes the radiation pattern of each array element into account in its code, what improves its accuracy. These radiation patterns are determined from the output data obtained through the conformal microstrip array analysis in a fullwave electromagnetic simulator, such as CST^{®} and HFSS^{®}. Once those data are available, polynomial interpolation is utilized to write simple closedform expressions that represent adequately the far electric field radiated by each array element, which makes the technique numerically efficient.
The developed design technique was implemented in the Mathematica^{®} platform giving rise to a computer program – called CMAD (Conformal Microstrip Array Design) – capable of performing the design of conformal microstrip arrays. The Mathematica^{®} package, an integrated scientific computer software, was chosen mainly due to its vast collection of builtin functions that permit implementing the respective algorithm in a short number of lines, in addition to its many graphical resources. At the end of the section, to illustrate the CMAD ability to synthesize the radiation pattern of conformal microstrip arrays, the synthesis of the radiation pattern of three conformal microstrip array topologies is considered. First, a microstrip antenna array conformed onto a cylindrical surface is analysed. Afterwards, a spherical microstrip array is studied. Finally, the synthesis of the radiation pattern of a conical microstrip array is presented.
3.1. Algorithm description
The far electric field radiated by a conformal microstrip array composed of
where the constant
with
in which
Based on (14), the radiation pattern of a conformal microstrip array can be promptly calculated using the relation
where the complex weight
Once the array elements are chosen and their positions are predefined, to determine the vector v (θ_{,} ϕ) tor
From the array fullwave simulation data, polynomial interpolation is applied to generate simple closedform expressions that represent adequately the far electric field (amplitude and phase) radiated by each array element. In this work, the degree of the interpolation polynomials is established from the analysis of the
Considering the previous scenario, to synthesize a radiation pattern in a given plane, it just requires the determination of the current excitations
Based on (17) and following [12], a constrained least squares problem is established in order to locate the main beam at the α direction,
subject to the constraints
in which
with the angles θ
In order to find a closedform solution to the problem defined by (18) to (20), we determine its real counterpart [21], that is,
subject to the following linear constraints
where
with
The closedform solution to the problem (22) and (23) is
from which the complex weight
After solving the problem (18)(20) the main beam is located at the αdirection. Nevertheless, it cannot be assured that the sidelobes levels are below the threshold
A constrained least squares problem, similar to (18)(20), that ensures the sidelobes levels, is set up for the purpose of calculating the residual complex weights Δ
subject to the constraints
in which
where
It is important to point out that the constraints (34) and (35) retain the main beam located at the αdirection, and the ones in (36) are responsible for conducting the sidelobes levels to the threshold
The radiation pattern synthesis technique described before was implemented in the Mathematica^{®} platform with the aim of developing a CAD – called CMAD – capable of performing the design of conformal microstrip arrays. The inputs required to start the design procedure in the CMAD program are the Text Files (.txt extension) containing the points that describe the complex patterns of each array element – obtained from the conformal microstrip array simulation in CST^{®} package –, the look direction α, the maximum sidelobes level
3.2. Cylindrical microstrip array
To illustrate the described pattern synthesis technique, let us first consider the design of a fiveelement cylindrical microstrip array, such as the one shown in Figure 7(a). For this array, the cylindrical ground layer is made out of copper cylinder with a 60.0mm radius and a 300.0mm height. The employed dielectric substrate has a relative permittivity ε_{r} = 2.5, a loss tangent tan δ = 0.0022 and its thickness is
It is important to point out that the elements close to the ends of the ground cylinder have significantly different radiation patterns than those close to the centre of this cylinder; however, the technique developed in this chapter can handle well this aspect, different from the common practice that assumes the elements’ radiation patterns are identical [22]. To clarify this difference among the patterns, Figure 8 shows the radiation patterns of the elements number 1 and 5. In Figure 8(a) they were evaluated in CST^{®} and in Figure 8(b) they were determined from the interpolation polynomials. As observed, there is an excellent agreement between the radiation patterns described by the interpolation polynomials and the ones provided by CST^{®}, even in the back region, where the radiation pattern exhibits low level and oscillatory behaviour. It validates the use of polynomial interpolation functions to represent the far electric field radiated by the conformal array elements.
For this cylindrical array, let us consider that the radiation pattern in the


1  1.0_{}∠_{}0.0° 
2  0.800_{}∠_{}82.394° 
3  0.360_{}∠_{}6.211° 
4  0.781_{}∠_{}90.315° 
5  0.617_{}∠_{}172.593° 
3.3. Spherical microstrip array
Another conformal microstrip array topology used to demonstrate the CMAD’s ability to synthesize radiation patterns is the fiveelement spherical microstrip array, which operates at 3.5 GHz, illustrated in Figure 9(a). For this array, the selected ground layer is a copper sphere with a radius of 120.0 mm. A typical microwave substrate (ε_{r} = 2.5, tan δ = 0.0022 and
In this case, the synthesized radiation pattern in the


1  0.680_{}∠_{}264.460° 
2  0.252_{}∠_{}6.059° 
3  0.160_{}∠_{}156.639° 
4  1.0_{}∠_{}0.0° 
5  0.728_{}∠_{}36.758° 
3.4. Conical microstrip array
Finally, let us consider the radiation pattern synthesis of the fourelement conical microstrip array presented in Figure 10(a). For this array, the ground layer is a 280.0mmhigh cone made of copper with a 40.0° aperture. This cone is covered with a dielectric substrate of constant thickness
The radiation pattern specifications for this synthesis are: main beam direction α = 70° and maximum sidelobe level
4. Active feed circuit design
As can be seen, the radiation pattern synthesis technique presented in the previous section is suitable for applications that require electronic radiation pattern control, for example. However, it only provides the array current excitations, i.e., to complete the array design it is still necessary to synthesize its feed network. A simple active circuit topology dedicated to feed those arrays can be composed of branches having a variable gain amplifier cascaded to a phase shifter, both controlled by a microcontroller, and a 1 :
At the end of this section, to illustrate the synthesis of the proposed active feed network (Figure 11), the design of the active beamformers of the three conformal microstrip arrays (cylindrical, spherical and conical) that appear along the chapter is described. Furthermore, to validate the phase shifts and gains calculated, the designed feed networks are analysed in the ADS^{®} package.


1  0.574_{}∠_{}7.835° 
2  0.875_{}∠_{}0.149° 
3  1.0_{}∠_{}0,0° 
4  0.625_{}∠_{}6.561° 
4.1. Design equations
For the analysis conducted here the phase shifters are considered perfectly matched to the input and output lines and produce zero attenuation. Based on these assumptions the scattering matrix (
with ϕ
The variable gain amplifiers are also considered perfectly matched to the input and output lines and they are unilateral devices, i.e.,
in which
Let us examine the operation of the
where
in which
Alternatively, the input power at the terminals of the
with
Combining (41) and (43) results in an expression to evaluate the incident power at the terminals of the
which is equal to the
Based on (45), an equation to determine the gain of the
Notice that to evaluate (46) it is necessary to choose one of the circuit branches as a reference, i.e., the gain of the
It is important to highlight that this formulation has relevant importance for arrays whose mutual coupling among elements is strong [23], since it takes this effect into account. For arrays whose mutual coupling among elements is weak and the array elements selfimpedances are close to
Now, to determine the phase shifts ϕ
in which
Once the currents
with
Also for the determination of the phase shift ϕ
For arrays whose mutual coupling among elements is weak and the array elements selfimpedances are close to
The expressions for evaluating the gains
4.2. Examples
The normalized current excitations found in Tables 1 to 3 and the scattering parameters of the three conformal microstrip arrays synthesized in this chapter (evaluated in CST^{®}) were provided to the CMAD. As results, it returned the gains and phase shifts of the active feed networks that implement the radiation patterns shown in Figures 7(b), 9(b) and 10(b). These values are listed in Table 4.
To verify the validity of the results found in Table 4, the designed active feed networks were analysed in the ADS^{®} package. As an example, Figure 12 shows the simulated feed network for the conical microstrip array. In this circuit, the array is represented through a 4port microwave network, whose scattering parameters are the same as the ones used by the CMAD, it is fed by a 30dBm power source with a 50ohm impedance, and there are four current probes to measure the currents at the terminals of the 4port microwave network, which correspond to the array current excitations. Table 5 summarizes the current probes readings for the three analysed feed networks. The comparison between the currents given in Table 5 and the ones presented in Tables 1 to 3 shows that these currents are in agreement, thereby validating the design equations derived before.












1  7.4  351.1  11.1  114.6  0.0  0.0 
2  5.7  267.6  3.8  205.0  4.2  7.0 
3  0.0  0.0  0.0  0.0  5.4  4.1 
4  4.9  260.6  14.0  210.3  1.5  356.6 
5  2.4  164.0  10.9  173.0  –  – 




1  0.160_{}∠_{}10.37°  0.249_{}∠_{}113.2°  0.106_{}∠_{}9.417° 
2  0.128_{}∠_{}92.76°  0.092_{}∠_{}157.3°  0.161_{}∠_{}1.432° 
3  0.058_{}∠_{}4.156°  0.059_{}∠_{}5.375°  0.184_{}∠_{}1.581° 
4  0.125_{}∠_{}100.7°  0.366_{}∠_{}151.3°  0.115_{}∠_{}8.141° 
5  0.099_{}∠_{}162.2°  0.267_{}∠_{}172.0°  – 
5. Conclusion
In summary, a computationally efficient algorithm capable of assisting the design of probefed conformal microstrip antennas with quasirectangular patches was discussed. Some examples were provided to illustrate its use and advantages. As seen, it can result in significant reductions in design time, since the required number of fullwave electromagnetic simulations, which are computationally intensive – especially for conformal radiators –, is diminished. For instance, the proposed designs could be performed with only three fullwave simulations. Also in this chapter, an accurate design technique to synthesize radiation patterns of conformal microstrip arrays was introduced. The adopted technique takes the radiation pattern of each array element into account in its code through the use of interpolation polynomials, different from the common practice that assumes the elements’ radiation patterns are identical. Hence, the developed technique can provide more accurate results. Besides, it is able to control the sidelobes levels, so that optimized array directivity can be achieved. This design technique was coded in the Mathematica^{®} platform giving rise to a computer program, called CMAD, that evaluates the array current excitations responsible for synthesizing a given radiation pattern. To show the potential of the CMAD program, the design of cylindrical, spherical and conical microstrip arrays were exemplified. Finally, an active feed network suitable for applications that require electronic radiation pattern control, like tracking systems, was addressed. The expressions derived for the synthesis of this circuit take into account the mutual coupling among the array elements; therefore they are also suited for array configurations in which the mutual coupling among the elements is strong. These design equations were incorporated into the CMAD code adding to it one more project tool. In order to validate this new CMAD feature, the feed networks of the three conformal microstrip arrays described along the chapter were designed. The obtained results were validated through the feed networks’ simulations in the ADS^{®} software.
Acknowledgments
The authors would like to acknowledge the support given to this work, developed under the project "Adaptive Antennas and RF Modules for Wireless Broadband Networks Applied to Public Safety", with the support of the Ministry of Communications' FUNTTEL (Brazilian Fund for the Technological Development of Telecommunications), under Grant No. 01.09.0634.00 with the Financier of Studies and Projects  FINEP / MCTI.References
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