Research Article  Open Access
Zusheng Jin, Giorgio Montisci, Giovanni Andrea Casula, Hu Yang, Junqi Lu, "Efficient Evaluation of the External Mutual Coupling in DielectricCovered Waveguide Slot Arrays", International Journal of Antennas and Propagation, vol. 2012, Article ID 491242, 7 pages, 2012. https://doi.org/10.1155/2012/491242
Efficient Evaluation of the External Mutual Coupling in DielectricCovered Waveguide Slot Arrays
Abstract
An accurate and efficient procedure is devised to evaluate the mutual coupling in dielectriccovered planar arrays of longitudinal slots. This approach takes full advantage of the discrete complex image method to cast the spatialdomain Green's functions into closed forms, and hence a direct calculation of mutual coupling in the spatialdomain is available. The computation time reduces significantly compared to the previous spectraldomain procedure, without any loss in the accuracy, rendering this approach very attractive for the design of large dielectriccovered planar arrays.
1. Introduction
Broadwall waveguide slots are widely employed as radiating elements in arrays for remote sensing, communication, and radar systems up to the millimeter wave frequency band. The advantages of these antennas are the high efficiency, the high gain, a good polarization purity [1], and a considerable mechanical strength.
The analysis and synthesis procedures available in the open literature [2–12] allow to achieve a very good accuracy in the design of waveguide slot arrays. Moreover, the recent advances in manufacturing technology assure high precision in the realization of these antennas even in the millimeter wave range [13]. Different kinds of broadwall radiating slots have been proposed in the past literature: longitudinal slots [1], inclined slots for linear [8], or circular polarization [14], Ushaped slots for circular polarization [15]. Among them, longitudinal slots are the most popular solution, since they allow the higher polarization purity than other waveguide slots. Therefore, their use, started in the 40s, is still continuing nowadays.
The design of planar arrays of longitudinal slots reached a state of maturity with the work by Elliott and his team, and the proposed iterative synthesis procedure has become a standard for the design of these antennas [1, 2].
In many practical applications, waveguide slot arrays need to be covered by a dielectric material for protection, aerodynamic purposes, or pressurization of the radiating waveguides [5, 10, 16]. The extension of the Elliott design procedure for such case has been recently proposed by Casula and Montisci [5] for a small planar array.
As it is well known, the external mutual coupling has a significant effect on the performance of a slot array, causing the slots to be detuned at the design frequency. Therefore, an accurate evaluation of the external mutual coupling is required in order to comply with the design specifications on the farfield pattern and on the impedance matching. Moreover, since the iterative synthesis procedure for large arrays is computationally expensive, efficient computation of the external mutual coupling is also required.
In [5], the external mutual coupling is evaluated resorting to the spectraldomain by using the Parseval identity and through numerical integration. A specially tailored numerical technique, the socalled Weighted Average Algorithm (WAA) [17], has been devised, and the integration of the highly oscillatory integrand has been accelerated by using the Shank's transform [18]. However, this approach cannot be considered entirely satisfactory, in particular for the designing of large arrays. Actually, the convergence of the integral in the mutual coupling expression deteriorates rapidly as the distance between the slots increases. As a result, its evaluation becomes timeconsuming, and it is hard to ensure the accuracy for large distances. Moreover, the WAA requires to extensively investigate the behavior of the integrand, rendering the spectraldomain approach inefficient and not robust. In fact, the oscillation period depends in a complicated manner on the spacing between the slots, and the WAA must be tailored to this behavior.
These drawbacks can be avoided if the calculation is performed directly in the spatialdomain, which is the aim of this paper. An alternative spatialdomain approach for the computation of the external mutual coupling has been proposed with the help of the discrete complex image method (DCIM). Numerical examples show that the computational time required for an accurate evaluation of the mutual coupling is only a small fraction of that required by the previous spectraldomain procedure. Moreover, the computational time and the accuracy are independent of the distance between the slots, and no special treatments are needed for large distances, as required by the procedure presented in [5].
The spatialdomain approach is described in detail in Section 2. Section 3 provides several numerical examples to validate the efficiency and accuracy of this procedure, as compared to the spectraldomain approach of [5].
2. Evaluation of the External Mutual Coupling
The external mutual coupling between two arbitrary slots in a dielectriccovered planar array, as depicted in Figure 1, is proportional to the reaction integral between the magnetic field of one slot (slot ) and the equivalent current distribution of another (slot ) [2] which can be represented as where is a suitable normalization coefficient; and are the normalized equivalent magnetic currents on the slot and , respectively; is the magnetic field in the dielectric cover due to the equivalent magnetic current and can be expressed through the vector potential wherein is the angular frequency; and denote the permittivity and permeability of the dielectric cover, respectively.
(a)
(b)
Substituting (2) into (1), after an integration by parts of the second term, we obtain with
Using the wellknown assumption of a monomodal electric field distribution on a longitudinal slot [2], we get wherein and are the slot voltages; and are the slot lengths (Figure 1); is the Green’s function of , where denotes the distance between the source point and the field point. Once the spatialdomain Green’s function has been obtained, the double integrals in (6) can easily be solved via a Gaussian quadrature.
The spatialdomain Green’s functions can be obtained from their spectraldomain counterparts by the following Sommerfeld integral where is the zeroordersecondkind Hankel function, and is the spectraldomain Green’s function, which can be obtained analytically both in the free space and in planar media [19]. For the freespace case, with , and therefore with . For the dielectriccovered case, is given by with , ,, where is the relative permittivity of the dielectric cover, and is the dielectric thickness. The constants and are the wave numbers of the free space and of the dielectric medium, respectively. In this case, integrals (6) cannot be solved in closed form, and the direct numerical integration of the Sommerfeld integral is very timeconsuming due to the highly oscillating and slow converging nature of its integrand.
In order to avoid the timeconsuming integration typical of the spectraldomain approach, in this work, the computation of the mutual coupling is performed in the spatialdomain by means of a threelevel DCIM [20].
The DCIM was first proposed by Fang et al. in 1988 [21] and is under development till now [22–24]. The basic idea of DCIM is to approximate the spectraldomain Green's functions in terms of complex exponentials and then transform them into the spatialdomain analytically with the help of the Sommerfeld identity.
Before the implementation of the approximation, the surface wave poles (SWPs) must be extracted from and their contribution will be analytically added after the approximation. This extraction is necessary in order to ensure the accuracy when the field point is far from the source point. Since the spectral function is an even function of , the poles occur in complex conjugate pairs and the surface wave term in the spectraldomain can be represented mathematically as where and are the th SWP and the corresponding residue. Substituting (9) into (7), the SWP contributions are obtained as
The remaining part , denoted as , can be approximated by complex exponentials via a threelevel generalized pencil of function (GPOF) method [20, 25]. The socalled threelevel GPOF, an extension of the previous twolevel version, is capable of approximating the lateral waves by introducing an additional path around the branch point. There may be many different parameterizations of the paths, and the one applied here, first proposed and demonstrated by Alparslan et al. [20, Equation 13], provides very accurate closedform Green's functions over all ranges of distances.
By performing the threelevel GPOF, can be approximated in terms of complex exponentials. Then, using the wellknown Sommerfeld identity, the spatialdomain counterpart of , denoted as , can be written as where is a complex function of . The complex coefficients and the complex exponents are obtained through the GPOF procedure; denotes the number of exponentials used in the approximation in the th level.
Finally, combining all the approximating exponentials, together with the SWP contributions, the spatialdomain Green’s function can be expressed in closed form .
Once the spatialdomain Green’s function is available, the computation of the mutual coupling (6) is straightforward. A good accuracy is expected for all slots spacings, since the spatialdomain Green’s functions can be calculated accurately in all ranges via the threelevel DCIM [20]. Moreover, the coefficients and exponents in the closedform expression of the Green’s function need to be generated for only once, leading to a very efficient calculation.
3. Results
In all the simulations presented in this section, we consider two longitudinal slots with lengths mm on a standard WR90 waveguide (22.86 mm × 10.16 mm). Let be the guided wavelength of the radiating guide fundamental mode and the angle between the waveguide axis and the line connecting the centers of the slots (Figure 1).
In order to assess the procedure of this paper, first of all, we test its accuracy against the expression of the mutual coupling obtained by Elliott in of [2, Equation 30] for an array radiating in free space, that is, for (see Figure 1). The operating frequency is 9 GHz and °. The results are reported in Figures 2 and 3, where we show the mutual coupling coefficient as a function of the normalized distance between the slots . An excellent agreement can be observed.
Then, the spatialdomain procedure has been compared against the spectraldomain approach, which has been validated for a small array in [5]. A number of simulations have been performed (on an Intel Core I7 CPU at 2.80 GHz with 8 GB RAM) at the operating frequency of 9.4 GHz. The dielectric cover thickness is mm, and its permittivity is .
In Figures 4 and 5, we show the comparison between the spatialdomain procedure proposed in this paper and the procedure used in [5] for . The curves have been divided in two figures with different scales in order to better show the behavior of the mutual coupling for large values of . The mutual coupling coefficient computed either with the DCIMbased procedure or with the spectraldomain approach of [5] virtually overlap until . For larger values, the results of the procedure used in [5] are not satisfactory, according to what we have pointed out in the Introduction.
(a)
(b)
(a)
(b)
For small values of (near zero), the results of the comparison are similar to those of Figures 4 and 5, though the convergence of [5, Equation 6] is achieved for a slightly larger value of the maximum distance between the slots. In Figures 6 and 7, we show the comparison for °. In this case, a good agreement can be observed until . For around 90°, the procedure of [5] is more stable. In Figures 8 and 9, the comparison for ° is shown: in this case, the results of the procedure in [5] are accurate enough until around .
As apparent in Figures 4–9, the deterioration of the mutual coupling coefficient computed with the spectraldomain approach of [5] is mainly due to the contribution of its real part.
In [5], the mutual coupling coefficient has been expressed in the spectraldomain by means of the Parseval identity and the resulting integral has been evaluated by dividing the integration domain into three intervals (see [5, Section 2]). We have found that the integral in the unbounded region, wherein the integrand is highly oscillatory and convergence is very slow, is always real, and this is the reason for the strong instability of the real part of the mutual coupling coefficient.
The proposed spatialdomain procedure has been tested and compared with the spectraldomain approach of [5] also for different dielectric cover permittivity and thickness. The same behavior as for the case investigated above has been observed.
We have found that the DCIMbased spatialdomain approach is not only stable but also efficient. Actually, the convergence of the results is assured regardless both of the distance between the slots and of the angle . Here are the parameters of the DCIM for the related simulations presented in this section: , , , , , , and , , . denotes the number of exponentials used in the approximation in the th level and is defined in (11). According to [20], , and are the truncation points for the parametrization of the paths , , and ; , , and are the number of samples of the GPOF approximation in the paths , , and , respectively. With this choice, the spatialdomain procedure takes about 0.05 seconds to compute the mutual coupling coefficient, and, as pointed out before, this computational time does not depend on the relative position between the slots.
On the other hand, if the mutual coupling is calculated in the spectraldomain by using [5, Equation 6], the integration parameters strongly depend on and . According to [5], we have selected here the parameters of the WAA in order to obtain the convergence of the integrations for each and a maximum value of equal to . In particular, 15 iterations have been used in the WAA. In this case, the computational time required for the evaluation of the mutual coupling is always about tens of times slower than the spatialdomain procedure. If a larger than is required, in order to achieve a good accuracy, the integration parameters in the spectraldomain approach of [5] must be suitably recomputed. However, since the convergence of the integral in the mutual coupling expression deteriorates rapidly as the distance between the slots increases, in this case, its evaluation becomes very timeconsuming.
To rate the improvements in the computational time of the approach proposed in this paper, we take a general case (Figures 4 and 5) as an example. The results are shown in Table 1. The larger the maximum distance between two arbitrary slots, the larger the computational time required for the convergence of the integration technique of [5]. On the contrary, the approach presented in this paper, based on the DCIM, takes about 0.05 seconds to compute the mutual coupling with the same accuracy of [5], and this time is independent of the distance between the slots. Therefore, the technique proposed in this paper is significantly faster than that of [5] and the improvement increases for larger arrays.

4. Conclusion
A spatialdomain approach has been proposed to evaluate the external mutual coupling in dielectriccovered waveguide slot planar arrays. The presented procedure is based on the use of the DCIM for the approximation of the spatialdomain Green’s function. The computational time and the accuracy are independent of the distance between the slots, leading to an accurate and efficient evaluation for all ranges of distances. Numerical examples show that the computational time reduces significantly with respect to the spectraldomain approach without any loss in the accuracy.
Acknowledgments
The authors would like to express their appreciation to Ye Liangfeng of NUDT, Antenna Research Center, and to Professor G. Mazzarella for helpful discussion and suggestions throughout the study. The authors also wish to acknowledge Professor Z. H. Firouzeh of Amirkabir University of Technology for his instructive discussion pertaining to GPOF, and Xi He of Hangzhou Normal University for her instructive suggestions.
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Copyright
Copyright © 2012 Zusheng Jin et al. 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.