Analysis of High 𝑇_𝑐 Superconducting Rectangular Microstrip Patches over Ground Planes with Rectangular Apertures in Substrates Containing Anisotropic Materials

Abstract

A rigorous full-wave analysis of high 𝑇_𝑐 superconducting rectangular microstrip patch over ground plane with rectangular aperture in the case where the patch is printed on a uniaxially anisotropic substrate material is presented. The dyadic Green’s functions of the considered structure are efficiently determined in the vector Fourier transform domain. The effect of the superconductivity of the patch is taken into account using the concept of the complex resistive boundary condition. The accuracy of the analysis is tested by comparing the computed results with measurements and previously published data for several anisotropic substrate materials. Numerical results showing variation of the resonant frequency and the quality factor of the superconducting antenna with regard to operating temperature are given. Finally, the effects of uniaxial anisotropy in the substrate on the resonant frequencies of different TM modes of the superconducting microstrip antenna with rectangular aperture in the ground plane are presented.

1. Introduction

Microstrip patch resonators offer many attractive features such as low profile, light weight, low cost, and the ease with which they can be integrated with printed feeding networks and active circuits. They can be used either as antennas or as components of oscillators and filters in microwave-integrated circuits. When a microstrip patch resonator acts as an antenna, the microstrip patch can be fed through an aperture cut into a microstrip line ground plane. Several advantages have been obtained by using this feeding configuration [1, 2]. Such advantages include weak parasitic radiation in the useful direction with respect to conventionally fed antennas and optimal performance for both the feeding network and antenna element. In addition, the presence of aperture on the ground plane adds new design parameters that can be used to tune the antenna impedance and resonance frequency, without modifying the patch itself. Since ground-plane apertures can play a role in the design of microstrip patch antennas, the algorithms developed for the analysis of these antennas should be able to account for the effect of possible apertures existing in the ground planes of the resonators.

Since the discovery of the high 𝑇_𝑐 superconducting materials which have critical temperatures above the boiling point of liquid nitrogen, the development of microwave application of high 𝑇_𝑐 superconductors has been extremely rapid and numbers of highly sophisticated subsystem level modules have been generated. Superconducting passive microwave devices such as antennas, filters, transmission line, and phase shifters have shown significant superiority over corresponding devices fabricated with normal conductors such as gold, silver, or copper due to the low surface resistance of superconductors. The low surface resistance corresponds to a large quality factor and improved performance such as higher gain and lower insertion loss in passive microwave devices [3–7].

Setting aside the topic of superconducting materials, in the last few years, there has been a growing interest in studying how the performance of microstrip antennas is affected when anisotropic dielectrics are used as substrates of those antennas [8–14]. We bear in mind that some dielectric substances used as substrates of microstrip antennas exhibit anisotropy due to their natural crystal structures or as the result of their production processes. Isotropic substances may also exhibit anisotropy at high frequencies. The overall results of the previous studies indicate that operational behavior of such structures may not be as expected especially at higher millimetre wave frequencies when anisotropy of the substrate material is ignored.

This paper presents a rigorous full-wave analysis of high 𝑇_𝑐 superconducting rectangular microstrip patch over ground plane with rectangular aperture. The rectangular patch is printed on a uniaxially anisotropic substrate material. As far as the authors know, this subject has not been reported in the open literature; the only published results on the full-wave analysis of superconducting microstrip antennas with apertures in the ground planes refer to isotropic substrates [15]. This paper is organized as follows. In Section 2, the dyadic Green’s functions of the considered structure are efficiently determined in the vector Fourier transform domain. The effect of the superconductivity of the patch is taken into account using the concept of the complex resistive boundary condition. Various numerical results are given in Section 3. Finally, concluding remarks are summarised in Section 4.

2. Theory

The problem to be solved is illustrated in Figure 1. We have a high 𝑇_𝑐 superconducting rectangular microstrip patch of thickness 𝑒 over a ground plane with a rectangular aperture. The substrate material is uniaxially anisotropic with the optical axis being normal to the patch. The uniaxial substrate is characterized by the free-space permeability 𝜇₀ and a permittivity tensor of the form 𝜺=𝜀₀𝜀_𝑥000𝜀_𝑥000𝜀_𝑧, (1) where 𝜀₀ is the free-space permittivity. Equation (1) can be specialized to the isotropic substrate by allowing 𝜀_𝑥=𝜀_𝑧=𝜀_𝑟. All fields and currents are time harmonic with the 𝑒^𝑖𝜔𝑡 time dependence being suppressed. The transverse fields inside the anisotropic region (0<𝑧<𝑑) can be obtained via the inverse vector Fourier transforms as shown in [16] 𝐄𝐫_𝑠=𝐸,𝑧^𝑥𝐫_𝑠𝐸,𝑧_𝑦𝐫_𝑠=1,𝑧4𝜋^2+∞_−∞𝐅𝐤_𝑠,𝐫_𝑠⋅𝐞𝐤_𝑠,𝑧𝑑𝑘_𝑥𝑑𝑘_𝑦,𝐇𝐫_𝑠=𝐻,𝑧^𝑦𝐫_𝑠,𝑧−𝐻_𝑥𝐫_𝑠=1,𝑧4𝜋^2+∞_−∞𝐅𝐤_𝑠,𝐫_𝑠⋅𝐡𝐤_𝑠,𝑧𝑑𝑘_𝑥𝑑𝑘_𝑦, (2) where 𝐅(𝐤_𝑠,𝐫_𝑠) is the kernel of the vector Fourier transform [16], and 𝐞𝐤_𝑠=𝑒,𝑧^𝑒𝐤_𝑠𝑒,𝑧_ℎ𝐤_𝑠=𝑖,𝑧𝑘_𝑠𝜀_𝑧𝜀_𝑥𝜕𝐸_𝑧𝐤_𝑠,𝑧𝜕𝑧𝜔𝜇₀𝑘_𝑠𝐻_𝑧𝐤_𝑠,𝐡𝐤,𝑧_𝑠=ℎ,𝑧^𝑒𝐤_𝑠ℎ,𝑧_ℎ𝐤_𝑠=,𝑧𝜔𝜀₀𝜀_𝑧𝑘_𝑠𝐸_𝑧𝐤_𝑠𝑖,𝑧𝑘_𝑠𝜕𝐻_𝑧𝐤_𝑠,𝑧.𝜕𝑧 (3) The superscripts 𝑒 and ℎ denote the TM and TE waves, respectively, and 𝐸_𝑧 is the scalar Fourier transform of 𝐸_𝑧. The general form of 𝐸_𝑧 and 𝐻_𝑧 is [17] 𝐸_𝑧𝐤_𝑠,𝑧=𝐴^𝑒𝑒^{−𝑖𝑘𝑒𝑧𝑧}+𝐵^𝑒𝑒^{𝑖𝑘𝑒𝑧𝑧},𝐻_𝑧𝐤_𝑠,𝑧=𝐴^ℎ𝑒^{−𝑖𝑘ℎ𝑧𝑧}+𝐵^ℎ𝑒^{𝑖𝑘ℎ𝑧𝑧}, (4) where 𝐴^𝑒, 𝐵^𝑒, 𝐴^ℎ, and 𝐵^ℎ are the field spectral amplitudes and 𝑘^𝑒_𝑧=𝜀_𝑥𝑘²₀−𝜀_𝑥𝜀_𝑧𝑘²_𝑠^1/2,𝑘^ℎ_𝑧=𝜀_𝑥𝑘²₀−𝑘²_𝑠^1/2,𝑘²₀=𝜔²𝜀₀𝜇₀. (5)𝑘^𝑒_𝑧 and 𝑘^ℎ_𝑧 are, respectively, propagation constants for TM and TE waves in the uniaxially anisotropic substrate [2]. After substitution of the expressions of 𝐸_𝑧 and 𝐻_𝑧 given by (4) into (3), we get [17] 𝐞𝐤_𝑠=𝑒,𝑧^𝑒𝐤_𝑠𝑒,𝑧_ℎ𝐤_𝑠,𝑧=𝑒^{−𝑖𝐤𝑧𝑧}⋅𝐀𝐤_𝑠+𝑒^{𝑖𝐤𝑧𝑧}⋅𝐁𝐤_𝑠,𝐡𝐤_𝑠=ℎ,𝑧^𝑒𝐤_𝑠ℎ,𝑧_ℎ𝐤_𝑠=,𝑧𝐠𝐤_𝑠⋅𝑒^{−𝑖𝐤𝑧𝑧}⋅𝐀𝐤_𝑠−𝑒^{𝑖𝐤𝑧𝑧}⋅𝐁𝐤_𝑠. (6) In (6), 𝐀, and 𝐁 are two-component unknown vectors and 𝐤_𝑧=𝑘^𝑒_𝑧00𝑘^ℎ_𝑧,𝐠𝐤_𝑠=𝜔𝜀₀𝜀_𝑥𝑘^𝑒_𝑧00𝑘^ℎ_𝑧𝜔𝜇₀. (7) Writing (6) in the planes 𝑧=0 and 𝑧=𝑑, and by eliminating the unknowns 𝐀 and 𝐁, we obtain the matrix form 𝐞𝐤_𝑠,𝑑⁻𝐡𝐤_𝑠,𝑑⁻=𝐞𝐤𝐓⋅^𝑠,0⁺𝐡𝐤_𝑠,0⁺ (8) with 𝐓=𝐓¹¹𝐓¹²𝐓²¹𝐓²²=cos𝐤_𝑧𝑑−𝑖𝐠⁻¹⋅sin𝐤_𝑧𝑑−𝑖𝐠⋅sin𝐤_𝑧𝑑cos𝐤_𝑧𝑑, (9) which combines 𝐞 and 𝐡 on both sides of the anisotropic region as input and output quantities. Now that we have the matrix representation of the anisotropic substrate, it is easy to derive the dyadic Green’s functions of the problem. Let 𝐉₀(𝑥,𝑦) be the surface current density on the ground plane with rectangular aperture, and let 𝐉(𝑥,𝑦) be the surface current density on the superconducting rectangular patch. Also, let 𝐄(𝑥,𝑦,0) and 𝐄(𝑥,𝑦,𝑑) be the values of the transverse electric field at the plane of the aperture and at the plane of the superconducting patch, respectively. Following a mathematical reasoning similar to that shown in [16], we can obtain a relation among 𝐉₀(𝑥,𝑦), 𝐉(𝑥,𝑦), 𝐄(𝑥,𝑦,0), and 𝐄(𝑥,𝑦,𝑑) in the vector Fourier transform domain given by 𝐞𝐤_𝑠,𝑑=𝐆𝐤_𝑠⋅𝐣𝐤_𝑠+Γ𝐤_𝑠⋅𝐞𝐤_𝑠,0,𝐣₀𝐤_𝑠=−Γ𝐤_𝑠⋅𝐣𝐤_𝑠+𝐘𝐤_𝑠⋅𝐞𝐤_𝑠,0, (10)(11) where the 2 × 2 diagonal matrices 𝐆(𝐤_𝑠),Γ(𝐤_𝑠), and 𝐘(𝐤_𝑠) stand for a set of dyadic Green’s functions in the vector Fourier transform domain. It is to be noted that 𝐆(𝐤_𝑠) is related to the patch current and 𝐘(𝐤_𝑠) is related to the aperture field. The matrix Γ(𝐤_𝑠) represents the mutual coupling between the patch current and aperture field. Considering the superconducting effect, we need simply to modify (10) by replacing 𝐆(𝐤_𝑠) by 𝐆_𝑠(𝐤_𝑠)=𝐆(𝐤_𝑠)−𝑍_𝑠⋅𝐈, where 𝐈 stands for the 2×2 unit matrix and 𝑍_𝑠 is the surface impedance of the superconducting patch. When the thickness of the superconducting patch is less than three times the zero-temperature penetration depth (𝜆₀), 𝑍_𝑠 can be expressed as follows [7, 15]: 𝑍_𝑠=1,𝑒𝜎 (12) where 𝜎 is the complex conductivity of the superconducting film. It is determined by using London’s equation and the Gorter-Casimir two-fluid model as [7, 15] 𝜎=𝜎_𝑛𝑇𝑇_𝑐⁴−𝑖1−𝑇/𝑇_𝑐⁴𝜔𝜇₀𝜆²₀, (13) where 𝑇 is the temperature, 𝑇_𝑐 is the transition temperature, 𝜎_𝑛 is the normal state conductivity at 𝑇=𝑇_𝑐, and 𝜔 is the angular frequency. Now that we have included the effect of the superconductivity of the rectangular patch in the Green’s functions formulation, the well-known Galerkin procedure of the moment method can be easily applied to obtain the resonant frequencies and quality factors of the resonant modes of the high 𝑇_𝑐 superconducting rectangular microstrip patch shown in Figure 1.

Figure 1 

Geometrical structure of a high 𝑇𝑐 superconducting rectangular microstrip patch over a ground plane with a rectangular aperture in the case where the superconducting patch is printed on an anisotropic dielectric substrate.

Using the moment method, with weighting modes chosen to be identical to the expansion modes, (10) and (11) are reduced to a system of linear equations which can be written compactly in matrix form as [17] 𝐙⋅𝐂=𝟎, (14) where 𝐙 is the impedance matrix and the elements of the vector 𝐂 are the modes expansion coefficient to be sought [17]. The system of linear equations given in (14) has nontrivial solutions when det𝐙(𝜔)=0. (15) Equation (15) is an eigenequation for 𝜔, from which the resonant frequency and quality factor of the structure of Figure 1 can be obtained. In fact, let 𝜔=2𝜋(𝑓_𝑟+𝑖𝑓_𝑖) be the complex root of (15). In that case, the quantity 𝑓_𝑟 stands for the resonant frequency and the quantity 𝑄=𝑓_𝑟/(2𝑓_𝑖) stands for the quality factor.

3. Numerical Results and Discussion

3.1. Comparison of Numerical Results

In this study, the considered mode is the TM₀₁ mode with the dominant current in the 𝑦 direction (higher order TM modes are considered only in Section 3.4). The basis functions considered here for approximating the unknown current on the superconducting rectangular patch are formed by the set of TM modes of a rectangular cavity with magnetic side walls and electric top and bottom walls. Also, the same basis functions are used for approximating the magnetic current density on the aperture in accordance with the concept of complementary electromagnetic structures [1, 2].

To check the correctness of our computer program, our numerical results are compared with those obtained from the magnetic-wall cavity model of Richard et al. [18], when there is no aperture in the ground plane. Numerical evaluations have been done with a patch of dimension 1.5 cm × 1 cm printed on anisotropic substrates. The patch is fabricated with a YBCO superconducting thin film with parameters 𝜎_𝑛=10⁶ S/m, 𝜆₀=140 nm, 𝑇_c=89 K, and 𝑒=350 nm. The operating temperature is 𝑇=60 K. Table 1 summarizes our computed resonant frequencies and those obtained via the magnetic-wall cavity model [18] for four different nonmagnetic anisotropic substrate materials. These anisotropic materials are Sapphire, Epsilam-10, Pyrolytic boron nitride, and PTFE. It is clear from Table 1 that the agreement between our results and those obtained via the magnetic-wall cavity model [18] is very good since the discrepancies between the two sets of results are below 1%. Note that in order to make the magnetic-wall cavity model of Richard et al. [18] able to account for uniaxial anisotropy in the substrate, we have associated with this model the electromagnetic knowledge [13]. The idea is to determine effective parameters for the uniaxially anisotropic substrate using [13, Equations (3) and (4)], and then we use these parameters in the magnetic-wall cavity model of Richard et al. [18].

Table 1 

Comparison of our calculated resonant frequencies with those obtained via the cavity model [18] combined with electromagnetic knowledge [13] for various anisotropic substrate materials; 𝑎×𝑏=1.5 cm × 1 cm, 𝑎_𝑎=0, 𝑑=0.2 mm, 𝜎_𝑛= 10⁶ S/m, 𝜆₀=140 nm, 𝑇_𝑐=89 K, 𝑒=350 nm, and 𝑇=60 K.

We have also compared our results with experimental data available in the literature [18]. Table 2 shows the comparison between our calculations and the theoretical and experimental results reported in [18]. It is clear from Table 2 that our results are better than the theoretical values in [18]. The previous comparisons show a very good agreement between our results and those of the literature. This validates the theory proposed in this paper.

Table 2 

Comparison of our calculated resonant frequencies with the theoretical and experimental data reported in [18]; 𝑎×𝑏=1630 μm × 935 μm, 𝜀_𝑟=23.81, 𝑑=254 μm, 𝑎_𝑎=0, 𝜎_𝑛=10⁶ S/m, 𝜆₀=140 nm, 𝑇_𝑐=89 K, and 𝑒=350 nm.

3.2. Influence of the Temperature on the Resonant Frequency and Quality Factor

Now, we investigate the influence of the operating temperature on the resonant frequency and quality factor of the superconducting rectangular microstrip antenna with a rectangular aperture in the ground plane. The patch of size 1630 𝜇m × 1870 𝜇m is made of 350 nm thick YBCO thin film with a normal state conductivity at the transition temperature 𝜎_𝑛=10⁶ S/m, a zero-temperature penetration depth 𝜆₀=140 nm, and a transition temperature 𝑇_c=89 K. The rectangular aperture has a size of 53 𝜇m × 187 𝜇m. The resonant frequency and the quality factor of the superconducting antenna against operating temperature for two different anisotropic substrate materials are shown in Figures 2(a) and 2(b), respectively. The first uniaxially anisotropic material has the pair of relative permittivities (𝜀_𝑥,𝜀_𝑧)=(6.04,7.35). The second uniaxially anisotropic material is characterized by an electric anisotropy of negative type [14] ((𝜀_𝑥,𝜀_𝑧)=(8.66,7.35)). Each dielectric substrate has a thickness of 56.1 𝜇m. From the results of Figure 2(a) (Figure 2(b)), it is found that the resonant frequencies (quality factors) obtained when the superconducting patch is printed on the first anisotropic material are higher (lower) than those obtained when the superconducting patch is printed on the second anisotropic material because the effective relative permittivity of the first uniaxial medium is lower than the one of second uniaxial medium. Concerning the influence of the operating temperature on the resonant frequency and quality factor of the superconducting microstrip patch shown in Figure 1, it can be seen that the effect of varying the temperature on the resonant frequency and quality factor is significant only for temperatures near the transition temperature. Note that the steep change in the resonant frequency and quality factor at temperatures near 𝑇_c can be attributed to a change in the magnetic penetration depth of the YBCO [18].

(a)

(b)

Figure 2 

Resonant frequency and quality factor of the high 𝑇𝑐 superconducting rectangular microstrip patch over ground plane with rectangular aperture against operating temperature. The superconducting patch is printed on two different anisotropic substrate materials; 𝑎×𝑏=1630 𝜇m × 1870 μm, 𝑎𝑎×𝑏𝑎=53 𝜇m × 187 𝜇m, 𝑑=56.1 𝜇m, 𝜎𝑛=106 S/m, 𝜆0=140 nm, 𝑇c=89 K, and 𝑒=350 nm.

3.3. Influence of Uniaxial Anisotropy in the Substrate on the Resonant Frequency

In Table 3, results are presented for the resonant frequencies of rectangular microstrip patch over ground planes with and without rectangular apertures in the case where the high 𝑇_𝑐 superconducting patch is printed on an anisotropic dielectric substrate, that is, Pyrolytic boron nitride, which exhibits a negative uniaxial anisotropy. The patch of size 1.5 cm × 1 cm is fabricated with a YBCO superconducting thin film with parameters 𝜎_𝑛=10⁶ S/m, 𝜆₀=140 nm, 𝑇_c=89 K, and 𝑒=350 nm. The substrate has a thickness of 1 mm. The operating temperature is 𝑇=50 K. In Table 3, the results obtained for the high 𝑇_𝑐 superconducting patch printed on anisotropic Pyrolytic boron nitride are compared with the results that would be obtained if the anisotropy of Pyrolytic boron nitride were neglected. In the case where 𝑎_𝑎×𝑏_𝑎=𝑎×𝑏, the differences between the results obtained considering anisotropy and neglecting anisotropy are 5.45%. However, in the other considered cases, these differences are much smaller with the maximum change being 2.70% when 𝑎_𝑎×𝑏_𝑎=0.75𝑎×0.75𝑏. Therefore, dielectric anisotropy effect is especially significant when the size of the aperture is similar to that of the high 𝑇_𝑐 superconducting patch. This result agrees with that discovered theoretically for perfectly conducting rectangular microstrip patches over ground planes with rectangular apertures [2].

Table 3 

Resonant frequencies of high 𝑇_𝑐 superconducting rectangular microstrip patch printed on anisotropic Pyrolytic boron nitride over ground planes with and without rectangular apertures; 𝑎×𝑏=1.5 cm × 1 cm, 𝑑 = 1 mm, 𝜎_𝑛=10⁶ S/m, 𝜆₀=140 nm, 𝑇_𝑐=89 K, 𝑒=350 nm, and 𝑇=50 K.

3.4. Effect of the Anisotropy on Different Modes of the Superconducting Antenna

In Table 4, the effect of uniaxial anisotropy in the substrate on the resonant frequencies of superconducting rectangular microstrip patch over ground plane with rectangular aperture is also investigated. In this table, the considered dielectric is Sapphire, which exhibits a positive uniaxial anisotropy. Unlike Section 3.3, both fundamental mode and higher order TM modes are considered. The patch of size 1.5 cm × 1 cm is fabricated with a YBCO superconducting thin film with parameters 𝜎_𝑛=10⁶ S/m, 𝜆₀=140 nm, 𝑇_c=89 K, and 𝑒=350 nm. The substrate has a thickness of 2 mm, and the aperture size is 2.1 mm × 1.4 mm. The operating temperature is 𝑇=40 K. For the modes having the dominant current in the 𝑥 direction (TM₁₀ and TM₂₀), the differences between the results obtained considering anisotropy and neglecting anisotropy are smaller compared to those of the other considered modes. It is also seen from Table 4 that for the mode TM₀₂, the difference between the results obtained considering anisotropy and neglecting anisotropy is 2.27%. However, in the other considered modes, these differences are much smaller with the maximum change being 1.72% for the mode TM₁₁. Therefore, dielectric anisotropy effect is especially significant for the mode TM₀₂.

Table 4 

Resonant frequencies of different TM modes of high 𝑇_𝑐 superconducting rectangular microstrip patch printed on anisotropic Sapphire over ground plane with rectangular aperture; 𝑎×𝑏=1.5 cm × 1 cm, 𝑎_𝑎×𝑏_𝑎=2.1 mm × 1.4 mm, 𝑑 = 2 mm, 𝜎_𝑛=10⁶ S/m, 𝜆₀=140 nm, 𝑇_𝑐=89 K, 𝑒=350 nm, and 𝑇=40 K.

4. Conclusion

We have described an accurate analysis of high 𝑇_𝑐 superconducting rectangular microstrip patch over ground plane with rectangular aperture in the case where the superconducting patch is printed on an anisotropic dielectric substrate. The dyadic Green’s functions of the considered anisotropic structure have been efficiently determined in the vector Fourier transform domain. The effect of the superconductivity of the patch has been taken into account using the concept of the complex resistive boundary condition. Galerkin’s method has been used to solve for the surface current density on the superconducting patch and the transverse electric field at the aperture. The accuracy of the method was checked by performing a set of results in terms of resonant frequencies for various anisotropic substrate materials. In all cases, very good agreements compared with the literature were obtained. Numerical results show that the influence of the operating temperature on the resonant frequency and quality factor of superconducting rectangular microstrip patches over ground planes with rectangular apertures in substrates containing anisotropic materials is especially significant for temperatures near the transition temperature. Other results also have indicated that dielectric anisotropy effect is especially significant when the size of the aperture is similar to that of the high 𝑇_𝑐 superconducting patch. This result agrees with that discovered theoretically for perfectly conducting rectangular microstrip patches over ground planes with rectangular apertures [2]. Concerning the influence of uniaxial anisotropy in the substrate on different TM modes of the superconducting antenna, we have found that dielectric anisotropy effect is especially significant for the mode TM₀₂.