Electromagnetic Radiation from Arbitrarily Shaped Microstrip Antenna Using the Equivalent Dipole-Moment Method

Abstract

The equivalent dipole-moment method (EDM) is extended and applied in the analysis of electromagnetic (EM) radiation by arbitrarily shaped microstrip antenna in this paper. The method of moments (MoM) is used to solve the volume-surface integral equation (VSIE). A strip model is applied in the treatment of the feeding probe of the microstrip antenna, in which the discretized triangular elements of the excitation source are equivalent as dipole models. The proposed approach is sufficiently versatile in handling arbitrarily shaped microstrip antenna and is easily constructed through a simple procedure. Numerical results are given to demonstrate the accuracy and efficiency of this method.

1. Introduction

Conformal microstrip structures have received increasing attention in the last decades, due to the need of antennas and devices that can be mounted on curved surfaces. Analysis and design of conformal microstrip antenna are more complicated than planar microstrip antenna [1]. The method of moments (MoM) [2] is commonly used in the full-wave methods owing to its accuracy. A number of the method of moments (MoM)-based full-wave solutions have been proposed to analyze the microstrip antennas with various shapes [3–6]. Among them, the volume-surface integral equation (VSIE) method [5, 6] is one of the most attractive approaches due to its flexibility and easy conjunction. However, the conventional VSIE approache to computing the MoM impedance matrix elements consumes a considerable portion of the total solution time, and this, in turn, can place an inordinately heavy burden on the CPU in terms of memory and time as the problem dimensions become large in terms of the wavelength. Fortunately, several efficient VSIE-based methods have been studied in recent years such as the fast multipole method (FMM) and the multilevel FMM [7], the precorrected-FFT method [5], the integral equation fast Fourier transform algorithm [8], and the characteristic basis function method [9]. These methods play a great role in saving CPU time or reducing the memory requirement to some extent.

More recently, the equivalent dipole-moment method [10, 11] has been proposed to efficiently analyze the electromagnetic scattering by the composite metallic and dielectric objects. In this paper, the equivalent dipole-moment method is extended to analyze the EM radiation problems of arbitrary shaped microstrip antenna. From the scheme and formulation, we can find that though the problem becomes much more complicated for the arbitrary shaped microstrip antenna, each impedance matrix element can still be expressed in a unified simple form. The article is organized as follows. Section 2 introduces the theory of the EDM and its voltage model in detail, and Section 3 shows the numerical results. The final section presents the conclusions.

2. Theory

In this section, we present the theoretical foundation of the procedure for efficient generation of MoM matrices associated with the coupled VSIE. The detailed derivation of the VSIE and the definition of RWG (f_sm) and SWG (f_vm) basis functions can be found in [12–14]. The VSIE [14] can be expressed as^𝑁𝑠_𝑛=1𝐼_𝑠𝑛𝑍^𝑠𝑠_𝑚𝑛+^𝑁𝑣_𝑛=1𝐼_𝑣𝑛𝑍^𝑠𝑣_𝑚𝑛=𝐄^𝑖,𝐟_𝑠𝑚,_{𝑁𝑠𝑛=1}𝐼_𝑠𝑛𝑍^𝑣𝑠_𝑚𝑛+^𝑁𝑣_𝑛=1𝐼_𝑣𝑛𝑍^𝑣𝑣_𝑚𝑛=𝐄^𝑖,𝐟_𝑣𝑚𝑚=1,2,…,𝑁, (1) where𝑍^𝑠𝑠_𝑚𝑛=−𝐄^𝑠_𝑠𝑛,𝐟_s𝑚(𝐫),𝑍^𝑠𝑣_𝑚𝑛=−𝐄^𝑠_𝑣𝑛,𝐟_𝑠𝑚(𝐫),𝑍^𝑣𝑠_𝑚𝑛=−𝐄^𝑠_𝑠𝑛,𝐟_𝑣𝑚(𝐫),𝑍^𝑣𝑣_𝑚𝑛=𝐟_𝑣𝑛(𝐫)/𝑗𝜔𝜀(𝐫),𝐟_𝑣𝑚(𝐫)−𝐄^𝑠_𝑣𝑛,𝐟_𝑣𝑚(𝐫), (2)(3)(4)(5) where the “⟨⋯⟩” is the mathematical symbol of interproduct, 𝑍^𝑡𝑢_𝑚𝑛(𝑡=𝑠,𝑣,𝑢=𝑠,𝑣)stands for the impedance matrix element in row 𝑚 and column n. 𝐄^𝑠_𝑢𝑛(𝑢=s,𝑣) stands for the radiated electric field from the equivalent current of the 𝑛th RWG or SWG element.

On the conducting surface S, a RWG [12] element 𝑛 with two triangles 𝑇^±_𝑛 adjacent to the edge of length 𝑙_𝑛is shown in Figure 1(a). The 𝑛th dipole moment m_s is obtained by the integration of the surface current, corresponding to edge element n, over the element surface𝐦_𝑠=_{𝑇+𝑛+𝑇−𝑛}𝐉_𝑠𝑛𝐫^′𝑑𝑠^′=𝐼_𝑠𝑛𝐦_𝑠𝑛, (6) where 𝐦_𝑠𝑛≈𝑙_𝑛𝐫_𝑛^𝑐−−𝐫_𝑛^𝑐+, (7)

(a)

(b)

(c)

Figure 1 

(a) Pair of triangle patches with the 𝑛th common edge   𝑙𝑛. (b) Pair of tetrahedrons associated with the 𝑛th common face. (c) The 𝑛th boundary face associated with one tetrahedron.

In the dielectric body V, described by a SWG [13] element 𝑛 with two inner tetrahedrons 𝑇^±_𝑛 adjacent to the common face of area 𝑎_𝑛 is shown in Figure 1(b). If the 𝑛th face is a boundary face, which is associated with only one tetrahedron, this corresponding tetrahedron is always assumed as 𝑇⁺_𝑛, as shown in Figure 1(c). The 𝑛th dipole moment m_v is obtained by the integration of the volume current, corresponding to face element n, over the element volume𝐦_𝑣=_{𝑇+𝑛+𝑇−𝑛}𝐉_𝑣𝑛𝐫^′𝑑𝑣^′=𝐼_𝑣𝑛𝐦_𝑣𝑛, (8) where𝐦_𝑣𝑛≈𝑎_𝑛𝜅(𝐫)𝐫_𝑛^𝑐−−𝐫_𝑛^𝑐+𝑇^±_𝑛𝑎∈𝑉,_𝑛𝜅(𝐫)𝐫^𝑐_𝑛𝑠−𝐫_𝑛^𝑐+𝑇⁺_𝑛∈𝑉,𝑇⁻_𝑛∉𝑉, (9) where the dielectric material of each dipole model is assumed uniform. The𝐫_𝑛^𝑐−, 𝐫_𝑛^𝑐+and 𝐫^𝑐_𝑛𝑠 are the centroid radius vectors of  𝑇⁺_𝑛,𝑇⁻_𝑛  and the 𝑛th boundary face, respectively, as shown in Figure 1.

Referring to [10], the radiated electric fields of the 𝑛th infinitesimal dipole at the field point 𝐫can be expressed as𝐄^𝑠_𝑢𝑛(𝐫)=𝜂𝐌4𝜋^𝑢𝑛−𝐦_𝑢𝑛𝑗𝑘𝑅+𝐶+2𝐌_𝑢𝑛𝐶𝑒^{−𝑗𝑘𝑅},𝑢=𝑠,𝑣 (10) in which1𝐶=𝑅²11+,𝐌𝑗𝑘𝑅_𝑢=𝐑⋅𝐦_𝑢𝐑𝑅², (11) where 𝐑=𝐫−𝐫′ and𝑅=|𝐑|. Equation (10) is the exact expressions and valid at arbitrary distances from the dipole, not only in the far field. However, if the observation point is close to the dipole for a separation distance of less than a threshold distance between the source and the testing function locations, the SWG or RWG element can not be regarded as an infinitesimal dipole model, so that (10) will not hold. In this case, the conventional MoM is still used to calculate the impedance matrix element, otherwise the MoM matrix elements are computed by the EDM method directly.

Substituting (10) into (2)–(5) and referring to [12, 13], the expressions of the impedance matrix element Z_𝑚,𝑛 are calculated byZ_𝑚,𝑛≈−𝑙_𝑚𝑒^{−𝑗𝑘𝑅}𝐄^𝑠_𝑢𝑛(𝑅)_{𝑅=|𝐫′𝑚−𝐫′𝑛|}⋅𝐫_𝑚^𝑐−−𝐫_𝑚^𝑐+𝑇^±_𝑚∈𝑆,−𝑎_𝑚𝑒^{−𝑗𝑘𝑅}𝐄^𝑠_𝑢𝑛(𝑅)_{𝑅=|𝐫′𝑚−𝐫′𝑛|}⋅𝐫_𝑚^𝑐−−𝐫_𝑚^𝑐+𝑇^±_𝑚∈𝑉,−𝑎_𝑚𝑒^{−𝑗𝑘𝑅}𝐄^𝑠_𝑢𝑛(𝑅)_{𝑅=|𝐫′𝑚−𝐫′𝑛|}⋅𝐫^𝑐_𝑚𝑠−𝐫_𝑚^𝑐+𝑇⁺_𝑚∈𝑉,𝑇⁻_𝑚∉𝑉, (12) where the𝐫_𝑚^𝑐−, 𝐫_𝑚^𝑐+,and 𝐫^𝑐_𝑚s are the centroid radius vectors of 𝑇⁺_𝑚,𝑇⁻_𝑚,  and the mth face, respectively. 𝐫^′_𝑛 and 𝐫^′_𝑚are the centre radius vector of the 𝑛th and the mth equivalent dipole model respectively,𝐫′_𝑢=𝐫_𝑢^𝑐++𝐫_𝑢^𝑐−2𝑇^±_𝑢𝐫∈𝑉,𝑆,_𝑢𝑐++𝐫^𝑐_𝑢𝑠2𝑇⁺_𝑢∈𝑉,𝑇⁻_𝑢∉𝑉.𝑢=𝑚,𝑛 (13)

A simple equivalent strip model [6] is applied in the treatment of the feeding probe, where the wire-surface junction is changed to a surface-surface junction. The feeding probe of diameter 𝑑 is replaced with its equivalent strip structure with the width 𝑤=2𝑑. There are 3 triangles, marked as 1, 2, 3, sharing the surface-surface junction, of   𝑙_𝑛, which are viewed as 2 RWG basis functions, that is, (1,3), (2,3). In EDM, the two triangle pairs, (1,3) and (2,3), are viewed as the equivalent electric dipole model, respectively, as shown in Figure 2. The excitation source locates at the junction and the 𝐄^𝑖 in (1) can be established by using the delta-gap voltage model [10]𝐄^𝑖=𝑉𝛿(𝑙)𝐧_𝑙, (14) where 𝑉 is the voltage across the gap, and 𝐧_𝑙 is normal to the junction 𝑙_𝑛.

Figure 2 

Dipole model in the excitation source of antenna. (a) Wire-surface junction. (b) Surface-surface junction and its equivalent electric dipole model.

3. Numerical Results

In this section, some numerical results are presented to validate the algorithm, and the efficiency of the method is demonstrated through comparison.

To verify the correctness and applicability of our theory as well as the program code, we first consider a plane microstrip antenna on a finite substrate. Figure 3 shows the structure of the antenna. The size of the substrate is W_𝑔×𝐿_𝑔=50mm×60mm, while the size of the patch isW×𝐿=30mm×40mm, respectively. The radius of the feed probe is 0.75 mm. The feeding point is on the centerline of the L-side and is𝑑=6.5mm from the center of the patch. The substrate’s thickness is ℎ=1.5mm with a dielectric constant of 𝜀_𝑟=2.55.

Figure 3 

Geometry of a probe feed microstrip antenna.

The mesh of the equivalent strip model should be refined, and the maximum edge length of triangles on strip model and metal surface is chosen as less than 𝜆₀/20 (𝜆₀ is the wavelength in free space) on the excitation source of antenna, while λ_g/15 (λ_g is the wavelength in dielectric) is chosen as the maximum edge length for the tetrahedral cells, respectively. Meanwhile, 0.15 λ₀ is chosen in EDM as the threshold distance for both the metal surface and dielectric volume. The conventional MoM and the EDM-accelerated MoM are applied to compute the input impedance and compared with the results given in [15]. Figure 4 shows the input resistance and the input reactance from 2.7 to 3.1 GHz, respectively, and good agreement is observed.

(a)

(b)

Figure 4 

Input impedance of the planar microstrip antenna. (a) Input resistance. (b) Input reactance.

In the computation, the microstrip antenna is discretized into 502 triangular patches and the 1031 tetrahedrons, yielding a total number of 3163 unknowns. The number of the calculated frequency points is chosen 20, and the total CPU time is 5223 seconds in EDM method, and 7142 seconds in conventional method respectively, which yields a reduction of 26.9%.

In second example, we consider a cylindrically conformal microstrip patch antennas; as shown in Figure 5 the length of the patch is 𝐿×𝑊=30mm×40mm, while the sizes of the substrate are 𝐿_𝑔×𝑊_𝑔=50mm×60mm, the inner radius of the curvature of the substrate is 50 mm, and the feeding point is on the centerline of the curved side and is 10 mm from the center of the patch. The thickness of the substrate is ℎ=0.795mm. The permittivity of the substrate is𝜀_𝑟=2.32.

Figure 5 

Geometry of a conformal microstrip antenna.

The maximum size of mesh discretization and the threshold distance in EDM are the same as the first numerical example. The EDM is applied to compute the return loss and compared with the results from the conventional MoM. Figure 6 shows the return loss from 2.9 to 3.5 GHz. It is observed that the results of the EDM are in better agreement with the conventional MoM.

Figure 6 

Ruturn loss as a function of frequency.

In the computation, the conformal microstrip antenna is discretized into 400 triangular patches and the 1340 tetrahedrons, yielding a total number of 4035 unknowns. The number of the calculated frequency points is chosen 30, and the total CPU time is 11967 seconds in EDM method and 17364 seconds in conventional method, respectively, which yields a reduction of 31.1%.

4. Conclusion

The EDM method has been applied to analyze the EM radiation characteristics of arbitrarily shaped microstrip antenna on finite-sized substrate. The antennas are discretized into tetrahedral volume elements in the material region and into triangle patches on the conducting surface, where the volume-surface integral equation (VSIE) is setup. Then the method of moments (MoM) is employed to solve the VSIE. A simple equivalent strip model has been established and the EDM has been applied in the treatment of the excitation source of the microstrip antenna, where the surface-surface junction associated with three triangles is equivalent as two dipole models. Numerical results validate the accuracy of the equivalent treatment method. The application of the EDM method significantly reduces the complexity of the computation to the impedance matrix as well as the solution time. Even when the problems become more complicated for the introduction of the probe-fed model in the conventional MoM, the expressions of impedance matrix element in the EDM method still keep a simple form and are very easy constructed by using a simple procedure.

Acknowledgments

This work was supported by Natural Science Foundation of Fujian Province of China (2011J01348) and the Foundation of Fuzhou University (XRC-1057).