Calculated (K_{r}a) of annular ring microstrip antennas.
1. Introduction
During the last decades, superconducting antenna was one of the first microwave components to be demonstrated as an application of hightemperature superconducting material [13]. High
The numerical results for the resonant frequency, bandwidth and radiation pattern of microstrip antennas with respect to anisotropy ratio of the substrate, are presented. The Influence of a uniaxial substrate on the radiation of structure has been studied. To include the effect of the superconductivity of the microstrip patch in the full wave spectral analysis, the surface complex impedance has been considered. The effect of the temperature and thickness of HTS thin film on the resonant frequency and bandwidth have been presented. Computations show that, the radiation pattern of the antenna do not vary significantly with the permittivity variation perpendicular to the optical axis. Moreover, it is found to be strongly dependent with the permittivity variation along the optical axis. The computed data are found to be in good agreement with results obtained using other methods. Also, the TM and TE waves are naturally separated in the Green's function. The stationary phase method is used for computing the farzone radiation patterns.
2. Theory
The antenna configurations of proposed structures are shown in Figure 1. The superconducting patches are assumed to be located on grounded dielectric slabs of infinite extent, and the ground planes are assumed to be perfect electric conductors. The substrates of thickness d are considered to be a uniaxial medium with permittivity tensor:
Where ε_{jx}= ε_{jy}≠ ε_{jz} (j=1, 2), and the permeability will be taken as μ_{0}.
Starting from Maxwell’s equations in the Hankel transform domain, we can show that the transverse fields inside the uniaxial anisotropic region can be written as [12]:
We can put these equations in the following form:
That is
The kernel of the vector Hankel transform is given by [12]:
And
Where
In the spectral domain, the relationship between the patch current and the electric field on the microstrip is given by [10, 11]:
Here a_{np} and b_{nq} are unknown coefficients.
For superconducting annular ring patch,
The Hankel transforms of ψ_{np} and φ_{nq} functions are described as [12]:
Where
Using the same procedure, the basis functions for superconducting circular patch, are given by the expressions [11]:
Where
J_{n}(.) and N_{n} (.) are the Bessel functions of the first, and second kind of order n.
Where
In order to incorporate the finite thickness, the dyadic Green’s function is modified by considering a complex boundary condition. The surface impedance of a hightemperature superconductors (HTSs) material for a plane electromagnetic wave incident normally to its surface is defined as the ratio of E to H on the surface of the sample [13]. It is described by the equation:
Where
If the thickness t of the strip of finite conductivity σ is greater than three or four penetration depths, the surface impedance is adequately represented by the real part of the wave impedance [13].
If t is less than three penetration depths, a better boundary condition is given by [13]:
Where the conductivity
For superconductors, a complex conductivity of the form
Now, we have the necessary Green’s function, it is relatively straightforward to formulate the moment method solution for the antenna characteristics. The boundary condition at the surface of the microstrip patch is given by:
Here
And for superconducting circular microstrip antenna, we have:
Where
Galerkin’s method is employed to solve the coupled vector integral equations of (29)(32). Substituting the Hankel transform current expansion of (15) into (29) and (32). Next, multiplying the resulting equation by
Each element of the submatrices
Where C and D represent either ψ or φ, for every value of the integer n.
The integration path for the integrals of (32) is, in general, located in the first quadrant of the complex plane k_{ρ}. This integration path must remain above the pole and the branch point of
In general, the roots of this equation are complex numbers indicating, that the structure has complex resonant frequencies (
Where
The losses in the antenna comprise dielectric loss P_{d}, the conductor loss P_{c}, and the radiation loss P_{r}, are given by [46]:
The efficiency of an antenna can be expressed by:
3. Convergence and comparison of numerical results
Computer programs have been written to evaluate the elements of the impedance, resistance matrices, and then solve the matrix equation (35). To enhance the accuracy of the numerical calculation, the integrals of the matrix elements (33) are evaluated numerically along a straight path above the real axis with a height of about 1k_{0} (Figure 2). In this case, the effects of the surface waves are included in the calculation and knowledge of the pole locations is not required, while the length of the integration path is decided upon by the convergence of the numerical results. The time required to compute the integral depends on the length of the integration path. It is found that length of the integration path required reaching numerical convergence at 100k_{0}, also Muller’s method that involves three initial guesses, is used for root seeking of (35).
To check the correctness of our computer programs, our results are compared with results of other authors. The comparisons are shown in Table 1 for imperfectly conducting microstrip annular ring antennas. The resonant wave number times the inner radius of the ring is




Results of [14] 
Results of [12] 




Re (k_{r}a) 
Im (k_{r}a) 
Re (k_{r}a) 
Im (k_{r}a) 
Re (k_{r}a) 
Im (k_{r}a)  Re (k_{r}a) 
Im (k_{r}a) 
Re (k_{r}a) 
Im (k_{r}a) 

0.005  3.26  0.002  3.24  0.002  3.257  0.002  0.67  1,6.10^{4}  0.676  1,6. 10^{4} 
0.01  3.24  0.003  3.23  0.002  3.248  0.003  0.68  1,7. 10^{4}  0.682  1,8. 10^{4} 
0.05  3.13  0.008  3.10  0.006  3.085  0.006  0.70  5,4. 10^{4}  0.695  5,5. 10^{4} 
0.1  3.01  0.014  2.96  0.0103  2.968  0.016  0.71  0.0012  0.705  0.0012 
In table 2, we have calculated the resonant frequencies for the modes (TM_{11}, TM_{21}, TM_{31}, and TM_{12}) for perfect conducting circular patch with a radius 7.9375mm, is printed on a substrate of thickness 1.5875mm. These values are compared with theoretical and experimental data, which have been suggested in [10]. Note that the agreement between our computed results, and the theoretical results of [10], is very good.
In our results, we need to consider only the P functions of ψ_{np}, and the Q functions of φ_{nq}. The required basis functions for reaching convergent solutions of complex resonant frequencies, using cavity model basis functions are obtained with (P=5, Q=0).




Resonant Frequency (GHz) 
Quality factors (Q) 
Resonant Frequency (GHz) 
Quality factors (Q) 

TM_{11}  6.1703  19.105  6.2101  19.001 
TM_{12}  17.056  10.324  17.120  10.303 
TM_{21}  10.401  19.504  10.438  19.366 
TM_{01}  12.275  8.9864  12.296  8.993 
4. Resonant frequency of superconducting patch antenna
Shown in Figures 34, is the dependence of resonant frequency on the thickness t of superconducting patch of the antennas. It is observed that, when the film thickness (t) increases, the resonant frequency increases quickly until, the thickness t reaches the value penetration depth (λ_{0}). After this value, the increase in the frequency of resonance becomes less significant.
Figures 56 demonstrated relations between the real part of frequency resonance, and the normalized temperature (T/Tc), where the critical temperature used here for our data (89K). The variations of the real part of frequency due to the uniaxial anisotropy decrease gradually with the increase in the temperature. This reduction becomes more significant for the values of temperature close to the critical temperature. These behaviours agree very well with those reported by Mr. A. Richard for the case of rectangular microstrip antennas [2].
5. Radiations patterns and efficiency of superconducting patch antenna
The calculated radiations patterns (electric field components, Eθ; E_{φ}), of the microstrip antenna on a finite ground plane in the E plane, and in the H plane are plotted in Figs. 7 10, printed on an uniaxial anisotropy substrate thickness (d=254μm). The mode excited for superconducting annular ring patch antenna is the TM_{12} and for superconducting circular patch antenna is the TM_{11}. It is seen that the permittivity ε_{z} has a stronger effect on the radiation than the permittivity ε_{x}. The radiation pattern of an antenna becomes more directional as its ε_{z} increases. Another useful parameter describing the performance of an antenna is the gain. Although the gain of the antenna is related to the directivity, the gain of an antenna becomes high as its ε_{z} increases.
In calculation of losses, we have found that, the values of dielectric loss (P_{d}), the conductor loss (P_{c}), and the radiation loss (P_{r}) will depend on frequency. We use results precedents to calculate the variation of radiation efficiency as a function of resonant frequency, for various isotropic dielectric substrates. Our results are shown in Figs. 11 and 12. It is seen that the efficiency increases with decreasing frequencies. The same behaviour is found by R. C. Hansen [1].
6. Conclusion
This work presents a fullwave analysis for the superconducting microstrip antenna on uniaxial anisotropic media. The complex resonant frequency problem of structure is formulated in terms of an integral equation. Galerkin procedure is used in the resolution of the electric field integral equation, also the TM, TE waves are naturally separated in the Green’s function. In order to introduce the effect of a superconductor microstrip patch, the surface complex impedance has been considered. Results show that the superconductor patch thickness and the temperature have significant effect on the resonant frequency of the antenna. The effects of a uniaxial substrate on the resonant frequency and radiation pattern of structures are considered in detail. It was found that the use of such substrates significantly affects the characterization of the microstrip antennas, and the permittivity (ε_{z}) along the optical axis has a stronger effect on the radiation of antenna. Thus, microstrip superconducting could give high efficiency with high gain in millimeter wavelengths. A comparative study between our results and those available in the literature shows a very good agreement.
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