**Full-Wave Spectral Analysis of Resonant Characteristics and Radiation Patterns of High Tc Superconducting Circular and Annular Ring Microstrip Antennas**

Ouarda Barkat

[5] Kin-Lu WongCompact and bradband miccrostrip antennas", (2002). by John Wiley &

[6] Pozar, D. M. Microstrip Antenna Aperture-Coupled to a microstrip line", Electron.

[7] Pozar, D. M. A Reciepprocity Mothod of Analisis for Printed Slot and Slot-Coupled

[8] Wood, C. Improved Bandwidth of Microstrip Antennas using Parasitic Elements",

[9] Pozar David MA Microstrip Antenna Aperture Coupled to a Microstrip line", Elec‐

[10] Borowiec, R, & Slobodzian, P. Multilayer microstrip structure" Phd. dissertation In‐

[11] Bugaj, M, & Wnuk, M. The analysis of microstrip antennas with the utilization the FDTD method" in XII Computational Methods Experimental Measurements, WIT

[12] Kubacki, R, Nowosielski, L, & Przesmycki- Technique, R. for the electric and magnet‐ ic parameter measurement of powdered materials, Computotional methods and ex‐ perimental measurements XIV (2009). Wessex Institute of Technology, Anglia, str.

[13] Kubacki, R, Nowosielski, L, Przesmycki, R, & Frender- The, R. technique of electric and magnetic parameters measurements of powdered materials, Przegląd Elektro‐

[14] Przesmycki, R, Nowosielski, L, & Kubacki-, R. The improved technique of electric and magnetic parameters measurements of powdered materials, Advances in Engi‐

[15] Gupta, K, & Benalla, A. Microstrip Antenna Design", Artech House, London (1980).

techniczny (Electrical Review), 0033-2097R. 85 NR 12/(2009).

neering Software, November (2011). 0965-9978, 42(11)

Antennas," IEEE Trans.Antennas Propaga., Dec. (1986). , AP-34(12)

stitute, of Telecommunications and Acoustics, University of Wroclaw

Sons, Inc., New York, 0-47122-111-2

56 Advancement in Microstrip Antennas with Recent Applications

Lett., (1985). , 21, 49-50.

IEE Proc., Pt.H, 4/(1980). , 127

Press 1746-4064page 611-620

241÷250, 0174-3355X,., 48

tron. Lett.Jan.17 (1985). , 21, 49-50.

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/54665

#### **1. Introduction**

During the last decades, superconducting antenna was one of the first microwave compo‐ nents tobedemonstratedas anapplicationofhigh-temperature superconductingmaterial[1-3]. High *Tc* superconducting microstrip antennas (HTSMA) are becoming popular and getting increased attention in both theoretical research and engineering applications due to their excellent advantages. Various patch configurations implemented on different types of substrates have been tested and investigated. In the design of microstrip antennas, anisotrop‐ ic substances have been increasingly popular. Especially the effects of uniaxial type anisotro‐ py have been investigated due to availability of this type of substances. These structures characterized by their low profile, small size, light weight, low cost and ease of fabrication, which makes them very suitable for microwave and millimeter-wave device applications [4-6]. HTSMA structures have shown significant superiority over corresponding devices fabricated with normal conductors such as gold, silver, or copper. Major property of superconductor is very low surface resistance; this property facilitates the development of microstrip antennas with better performance than conventional antennas. Compared to other patch geometries, the circular and annular ring microstrip patches printed on a uniaxial anisotropic substrate, have been more extensively studied for a long time by a number of investigators [7-11]. Circular microstrip patch can be used either as radiating antennas or as oscillators and filters in microwave integrated circuits (MIC's). The inherent advantage of an annular ring antenna is: first, the size of an annular ring for a given mode of operation is smaller than that of the circular disc resonating at the same frequency. Second, The presence of edges at the inner and outer radii causes more fringing than in the case of a circular microstrip antenna, in which fringing occurs only at the outer edge, and this implies more radiation from these edges, that leads to

© 2013 Barkat; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Barkat; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

high radiation efficiency. These structures are quite a complicated structure to analyze mathematically. Different models are available to model a microstrip antenna as the transmis‐ sion-line model and the cavity model in simple computer aided design formulas. However, the accuracy of these approximate models is limited, and only suitable for analysing simple regularly shaped antenna or thin substrates. The full-wave spectral domain technique is extensivelyusedinmicrostripantennasanalysisanddesign.Inthismethod,Galerkin'smethod, together with Parsval's relation in Hankel transform domain is then applied to compute the resonant frequency and bandwidth. The integral equation is formulated with Hankel trans‐ forms which gives rise to a diagonal form of the Green's function in spectral domain.

0

transverse fields inside the uniaxial anisotropic region can be written as [12]:

ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ

*j n n j nn n jx*

*j n n j nn n jx*

φ ρ ρ ρ ρ ρ ρ ρ

*j n n j nn*

*j nn j nn*

*z e dk k J k J k k z J k J k k k*

(,, ) . . ( ). (,, ) ,

*H z e dk k k <sup>z</sup> i kz*

=-¥

+¥ +¥ =-¥

f

+¥ +¥ =-¥

*n*

*n*

+¥ +¥

f

å ò **Hn**

f

+¥ +¥

=-¥

*<sup>z</sup> e dk k k <sup>z</sup> k z*

0 (,, ) (,) (,, ) . . ( ). (,, ) ( , )/

> <sup>0</sup> (,, ) . . ( ). ( , ) *jn j n*

> <sup>0</sup> (,, ) . . ( ). ( , ) *jn j n*

*z e dk k k h k z*

*H z e dk k k e k z* f

**H E**

*z e dk k J k J k k z J k J k <sup>k</sup>*

*z e dk k i J k J k k z J k J k <sup>k</sup>*

*z e dk k J k J k k z J k J k <sup>k</sup>*

ee

Where εjx= εjy≠ εjz (j=1, 2), and the permeability will be taken as μ0.

0

0

0

¥ ¥

0

  f

 f

> f

 f

> f

> > f

 f

f

*j*

¥ ¥

¥ ¥

=-¥

=-¥

¥ ¥

*n*

=-¥

*n*

=-¥

*j*

 f

 f

**E**

That is

 f

f

 f

 f

> f

ωμ

e e

w e

We can put these equations in the following form:

**ρ jz**

0

0

*jx j jy*

 e

Starting from Maxwell's equations in the Hankel transform domain, we can show that the

e

0 0 00 0 0

Full-Wave Spectral Analysis of Resonant Characteristics and Radiation Patterns of High Tc Superconducting Circular…

*jz*

( ) ( <sup>ρ</sup> ) <sup>φ</sup> <sup>0</sup>

( ) ( <sup>ρ</sup> ) <sup>φ</sup> <sup>0</sup> ρ ρ ρρ ρ ρ ρ ρ ρ


( ( ) ( <sup>ρ</sup> ) <sup>φ</sup> jz

( ) ( <sup>ρ</sup> ) <sup>φ</sup> jz ρ ρ ρ ρ ρ ρ ρ ρ ρ ε i


1 1 1 1

0ρ <sup>0</sup> iω

*j j jn*

 

  *j n j z k z*

é ù é ù <sup>=</sup> ê ú <sup>=</sup> ê ú ê ú ¶ ¶ ë û ë û å ò **Hn**

   e e

**<sup>E</sup> <sup>H</sup>** (6)

  **-H H z** (7)

 

 

<sup>=</sup> å ò **Hn <sup>E</sup>** % (9)

jzρ ρ

**jz**

iω <sup>0</sup>

%

<sup>=</sup> å ò **Hn** % (8)

e e

% % **jz jz**

%

**jz**

ρ ρ ρ ρ <sup>2</sup>

**H E <sup>z</sup>** (5)

(,, ) ( / ). ( , ) /

é ù é ù ¶ ¶ = = ê ú ê ú ê ú ë û ë û

**E E z**

*jz jx <sup>j</sup> <sup>j</sup> jn*

*j n j z i k z*

*j in*


1 1 1 1

ρ ρ ρ ρ 2k

**jz**

**H E <sup>z</sup>** (4)

*j in*

*jz j in*

ρ ρ ρ ρ 1 1 1 1

% % % % % % **jz**

e

% % % % % % **jz**

e

e

e

% % % % % % **jz**

% % % % % % **jz**

<sup>ω</sup> ρ ρ ρ ρ 1 1 1 1

**E H <sup>z</sup>** (2)

*jz j in*


, (,, ) [ ( ( ) ( )) , ( ( ) ( )) ] 2 2k

, (,, ) [ ( ( ) ( )) , ( ( ) ( )) ] 2 k <sup>2</sup>

ρ ρ iω 1

, (,, ) [( ( ) ( )) , ( ( ) ( )) ] <sup>2</sup>

, (,, ) [ ( ( ) ( )) , ( ( ) ( )) ] <sup>2</sup>

¶ <sup>=</sup> +× + - ¶ <sup>å</sup> <sup>ò</sup>

¶ <sup>=</sup> - × + +× ¶ <sup>å</sup> <sup>ò</sup>

¶ <sup>=</sup> -× - + ¶ <sup>å</sup> <sup>ò</sup>

**jz**

**E H <sup>z</sup>** (3)

¶ <sup>=</sup> + × +- + ¶ <sup>å</sup> <sup>ò</sup>

**jz**

= (1)

*i k z*

**E**

http://dx.doi.org/10.5772/54665

59

**E**

**H**

**H**

( ) <sup>ρ</sup>

*k z*

*k z*

*k z*

e

The numerical results for the resonant frequency, bandwidth and radiation pattern of micro‐ strip 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 superconduct‐ ing 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:

**Figure 1.** Cross section of a superconducting microstrip patch on uniaxial anisotropic media. (a) circulaire ; (b) annu‐ laire ring

Full-Wave Spectral Analysis of Resonant Characteristics and Radiation Patterns of High Tc Superconducting Circular… http://dx.doi.org/10.5772/54665 59

$$
\overline{\boldsymbol{\varepsilon}}\_{f} = \boldsymbol{\varepsilon}\_{0} \begin{vmatrix}
\boldsymbol{\varepsilon}\_{j\times} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \boldsymbol{\varepsilon}\_{j\times} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \boldsymbol{\varepsilon}\_{j\times}
\end{vmatrix} \tag{1}
$$

Where εjx= εjy≠ εjz (j=1, 2), and the permeability will be taken as μ0.

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]:

$$\mathbf{E}\_{\rho}(\rho,\phi,z\_{j}) = \sum\_{n=-n}^{n} \epsilon^{inq} \int\_{0}^{n} dk\_{\rho} k\_{\rho} \left. \mathbb{I} \frac{\alpha\mu\_{\text{H}}}{2k\_{\rho}} (\mathbf{J}\_{n-1}(k\_{\rho}\rho) + \mathbf{J}\_{n+1}(k\_{\rho}\rho)) \right. \\ \left. \cdot \hat{\mathbf{H}}\_{j\mu} \left(k\_{\rho}z\_{j}\right) + \frac{i\varepsilon\_{jz}}{2\mathbf{k}\_{\rho}\varepsilon\_{jz}} (-\mathbf{J}\_{n+1}(k\_{\rho}\rho) + \mathbf{J}\_{n-1}(k\_{\rho}\rho)) \frac{\partial \cdot \bar{\mathbf{E}}\_{j\mu} \left(k\_{\rho}z\_{j}\right)}{\partial z \partial \mathbf{z}} \right|^{2} \, \tag{2}$$

$$\mathbf{E}\_{\phi}(\rho,\phi,z\_{j}) = \sum\_{n=-\alpha}^{n} \epsilon^{in\eta} \int\_{0}^{n} dk\_{\rho} k\_{\rho} \left[ \mathbf{i} \frac{\epsilon \mu\_{0}}{2} (\tilde{l}\_{n-1}(k\_{\rho}\rho) - \tilde{l}\_{n+1}(k\_{\rho}\rho)) \cdot \mathbf{\hat{R}}\_{\hat{\mu}\nu} \left(k\_{\rho},z\_{j}\right) - \frac{\mathcal{E}\_{\hat{\mu}z}}{2\varepsilon\_{\hat{\mu}}k\_{\rho}} (\tilde{l}\_{n+1}(k\_{\rho}\rho) + \tilde{l}\_{n-1}(k\_{\rho}\rho)) \frac{\partial \,\mathbf{\hat{E}}\_{\hat{\mu}}\left(k\_{\rho},z\_{j}\right)}{\partial z} \right] \tag{3}$$

$$\mathbf{H}\_{\boldsymbol{\uprho}}(\boldsymbol{\rho},\boldsymbol{\phi},\boldsymbol{z}\_{\boldsymbol{\uprho}}) = \sum\_{n=-n}^{n} \epsilon^{\mathrm{imp}} \int\_{0}^{n} dk\_{\boldsymbol{\uprho}} k\_{\boldsymbol{\uprho}} \left[ \mathbf{l} \left( \frac{\boldsymbol{\uprho}\_{\boldsymbol{\uprho}} \mathbf{z}\_{0}}{2\mathbf{k}\_{\boldsymbol{\uprho}}} \right) \tilde{\boldsymbol{\uprho}}\_{n-1}(\mathbf{k}\_{\boldsymbol{\uprho}} \boldsymbol{\uprho}) - \tilde{\boldsymbol{I}}\_{n+1}(\mathbf{k}\_{\boldsymbol{\uprho}} \boldsymbol{\uprho}) \right. \\ \left. + \frac{1}{2\mathbf{k}\_{\boldsymbol{\uprho}}} (\tilde{\boldsymbol{I}}\_{n-1}(\mathbf{k}\_{\boldsymbol{\uprho}} \boldsymbol{\uprho}) + \tilde{\boldsymbol{I}}\_{n+1}(\mathbf{k}\_{\boldsymbol{\uprho}} \boldsymbol{\uprho})) \cdot \frac{\partial \, \mathbf{H}\_{\boldsymbol{\uprho}} \left( \mathbf{k}\_{\boldsymbol{\uprho}},\mathbf{z}\_{\boldsymbol{\uprho}} \right)}{\partial \mathbf{z}} \right] d\mathbf{z}\_{0} \right] d\mathbf{z}\_{0}$$

$$\mathbf{H}\_{p}(\rho,\phi,z\_{j}) = \sum\_{n=-n}^{n} \epsilon^{im} \int\_{0}^{n} dk\_{\rho} k\_{\rho} \left[ \frac{\alpha \varepsilon\_{\mu} \varepsilon\_{0}}{2k\_{\rho}} (\mathbf{J}\_{n-1}(k\_{\rho} \rho) + \mathbf{J}\_{n+1}(k\_{\rho} \rho)) \cdot \mathbf{E}\_{\mathbf{j}\mathbf{z}} \left(k\_{\rho}, z\_{j}\right) \right.\\ \left. + \frac{i}{2k\_{\rho}} (\mathbf{J}\_{n+1}(k\_{\rho} \rho) - \mathbf{J}\_{n-1}(k\_{\rho} \rho)) \frac{\partial \cdot \mathbf{H}\_{\mathbf{j}\mathbf{z}} \left(k\_{\rho}, z\_{j}\right)}{\partial \cdot \mathbf{z}} \right] \tag{5}$$

We can put these equations in the following form:

$$\mathbf{E}(\rho,\phi,z\_{j}) = \begin{bmatrix} \mathbf{E}\_{\rho}(\rho,\phi,z\_{j}) \\ \mathbf{E}\_{\phi}(\rho,\phi,z\_{j}) \end{bmatrix} = \sum\_{\mu=-\infty}^{+\infty} e^{\mathbf{j}\imath\phi} \int\_{0}^{+\infty} dk\_{\rho} \cdot \mathbf{k}\_{\rho} \cdot \mathbf{H}\_{\mathbf{n}}(k\_{\rho}\rho) \cdot \begin{bmatrix} (\mathbf{i}z\_{j\varepsilon}/z\_{j\varepsilon})\boldsymbol{\mathcal{J}}\cdot\mathbf{\hat{E}}\_{\mathbf{j}\mathbf{k}}(k\_{\rho},z\_{j})/\boldsymbol{\mathcal{J}}\cdot\mathbf{z} \\ \mathbf{i}z\_{0}\mathbf{j}\mathbf{\hat{H}}\_{\mathbf{j}\mathbf{k}}\left(k\_{\rho},z\_{j}\right) \end{bmatrix} \tag{6}$$

$$H(\rho,\phi,z\_{j}) = \begin{bmatrix} \mathbf{H}\_{\rho}(\rho,\phi,z\_{j}) \\ \mathbf{H}\_{\rho}(\rho,\phi,z\_{j}) \end{bmatrix} \\ = \sum\_{n=-\alpha}^{+\alpha} e^{in\theta} \int\_{0}^{+\alpha} dk\_{\rho} \, k\_{\rho} \, \mathbf{\tilde{H}}\_{\mathbf{n}}(k\_{\rho}\rho) \cdot \begin{bmatrix} \mathbf{i} & \mathbf{e}\_{j\mathbf{e}} \ \mathbf{\tilde{e}}\_{\mathbf{j}\mathbf{e}}(k\ \mathbf{z},z\_{j}) \\ \mathbf{i}\boldsymbol{\mathcal{O}} \ \mathbf{\tilde{H}}\_{\mathbf{j}\mathbf{e}}(k\_{\rho}z\_{j}) / \boldsymbol{\mathcal{O}} \mathbf{z} \end{bmatrix} \tag{7}$$

That is

high radiation efficiency. These structures are quite a complicated structure to analyze mathematically. Different models are available to model a microstrip antenna as the transmis‐ sion-line model and the cavity model in simple computer aided design formulas. However, the accuracy of these approximate models is limited, and only suitable for analysing simple regularly shaped antenna or thin substrates. The full-wave spectral domain technique is extensivelyusedinmicrostripantennasanalysisanddesign.Inthismethod,Galerkin'smethod, together with Parsval's relation in Hankel transform domain is then applied to compute the resonant frequency and bandwidth. The integral equation is formulated with Hankel trans‐

forms which gives rise to a diagonal form of the Green's function in spectral domain.

zone radiation patterns.

58 Advancement in Microstrip Antennas with Recent Applications

**2. Theory**

laire ring

The numerical results for the resonant frequency, bandwidth and radiation pattern of micro‐ strip 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 far-

The antenna configurations of proposed structures are shown in Figure 1. The superconduct‐ ing 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

Patch

<sup>φ</sup> b a

t

d,,ε <sup>01</sup>

Plan de masse

(a) (b)

**Figure 1.** Cross section of a superconducting microstrip patch on uniaxial anisotropic media. (a) circulaire ; (b) annu‐

considered to be a uniaxial medium with permittivity tensor:

a <sup>φ</sup>

d,,ε <sup>01</sup>

t

$$H(\rho,\phi,z\_{\nearrow}) = \sum\_{n=-\alpha}^{+\alpha} e^{jn\phi} \int\_0^{+\alpha} dk\_{\rho} \, k\_{\rho} \, \mathbf{H}\_{\mathbf{n}}(k\_{\rho}\rho) \, \mathbb{J}\_{n}(k\_{\rho},z) \tag{8}$$

$$\mathbf{E}(\rho,\phi,z\_{\boldsymbol{\gamma}}) = \sum\_{n=-\alpha}^{+\alpha} e^{in\phi} \int\_{0}^{+\alpha} dk\_{\rho} \, k\_{\rho} \, \mathbf{H}\_{\mathbf{n}}(k\_{\rho}\rho) \, \tilde{h}\_{n}(k\_{\rho},z) \tag{9}$$

The kernel of the vector Hankel transform is given by [12]:

$$\overline{H}\_{n}(k\_{\rho}\rho) = \begin{vmatrix} \dot{f}\_{n}(k\_{\rho}\rho) & -\text{in } f\_{n}(k\_{\rho}\rho)/k\_{\rho} \\ \text{in } f\_{n}(k\_{\rho}\rho)/k\_{\rho}\rho & \dot{f}\_{n}(k\_{\rho}\rho) \end{vmatrix} \tag{10}$$

Here anp and bnq are unknown coefficients.

'

( )

î

( ) <sup>ρ</sup> ρ ρ

*nq nq*

ï

k α

ï

*np*

**Ψ**%

(*βnpb* / *a*) =0.

The Hankel transforms of ψnp and φnq functions are described as [12]:

*np np*

na <sup>k</sup> Y (k ) (β

*np*

2 2

*k a*

Ψ ββ

ρ

*nq*

j

*k*

0

( ) ( ( / ))

*φn*(*αnqb* / *a*) =0 and *ψ<sup>n</sup>*

Where

Where

*J* ˙

by the expressions [11]:

*<sup>n</sup>*(*βnpa*)=0 and *Jn*(*αnqa*)=0 .

For superconducting annular ring patch, (*αnq*, *βnp*) are the roots of dual equations

Full-Wave Spectral Analysis of Resonant Characteristics and Radiation Patterns of High Tc Superconducting Circular…

np ρ np ρ

β a <sup>ρ</sup>

ìé ù ïê ú ¢ <sup>ï</sup> - <sup>ï</sup> < < <sup>ï</sup> <sup>=</sup> <sup>í</sup>

*a k <sup>b</sup>*

0 ,

<sup>ï</sup> > < <sup>î</sup>

Using the same procedure, the basis functions for superconducting circular patch, are given

ρ ρ ρ

= ê ú

ρ ρ

*n nq n*

é ù = ê ú - ë û

2 2 ( )( ) () 0

*nq*

ρ α α

*k aJ a J k a <sup>k</sup> k*

&

*n np np n np n*

*J ka in k aJ a J ka*

ρ ρ

*kka* é ù


*T*

*np np*

β β 22 2 ( ) () ( ) ( )

<sup>ì</sup> é ù <sup>ï</sup> ê ú <sup>ï</sup> < < <sup>ï</sup> <sup>=</sup> <sup>í</sup> - ë û <sup>ï</sup>

ïë û <sup>ï</sup> > < <sup>ï</sup>

k a<sup>a</sup> <sup>ρ</sup> <sup>Z</sup>

*<sup>b</sup> <sup>k</sup>*

npρ ( ) <sup>ρ</sup> <sup>ρ</sup> ψ β( / ) ( ) ( ) ( ) ψ β *Y k b a J kb J ka n np n n np n* = - (18)

nqρ ( ) <sup>ρ</sup> <sup>ρ</sup> Zφ α ( / ) ( / ) ( ) ( ) ( ) φ α *n nq n n nq n k b a b a J kb J ka* = ¢ ¢ - (19)

*T*

% (21)

& % (20)

**<sup>Φ</sup>**% (17)

,

*b a*

ρ ρ

*b a*

(16)

61

http://dx.doi.org/10.5772/54665

(β Y (k )

2 2 / ) (( / ) )

*a*

)

0 ρ ρ

*nq*

And

$$\tilde{e}\_n(k\_\rho, z) = \overline{\mathbf{A}}\_{n\circ} \left(k\_\rho \right) e^{-i\int\_{\overline{\mathbf{A}}} \overline{\mathbf{E}}\_{\overline{\mu}} z} + \overline{\mathbf{B}}\_{n\circ} \left(k\_\rho \right) e^{i\int\_{\overline{\mu}} \overline{\mathbf{E}}\_{\overline{\mu}} z} \tag{11}$$

$$\tilde{h}\_n(k\_\rho, z) = \overline{\mathbf{g}}\_j \text{ (k}\_\rho\text{)} \text{ (}\overline{\mathbf{A}}\_{nj} \text{ (}k\_\rho\text{)} \text{ }e^{-i\sqrt{k}\_\rho z} - \overline{\mathbf{B}}\_{nj} \text{ (}k\_\rho\text{)} \text{ }e^{i\sqrt{k}\_\rho z} \text{ }\tag{12}$$

*<sup>A</sup>***¯** and *B***¯** are two unknown vectors and *<sup>g</sup>* **¯**(*kρ*) is determined by:

$$\overline{\mathbf{g}}\_{f}(k\_{\rho}) = \begin{vmatrix} \alpha \mathbf{e}\_{\neq \mathbf{e}} \mathbf{e}\_{\neq 0} \ / k\_{\neq \mathbf{e}}^{h} & \mathbf{0} \\ \mathbf{0} & k\_{\neq \mathbf{e}}^{e} \ / \alpha \mathbf{u}\_{\mathbf{0}} \end{vmatrix} \tag{13}$$

Where

*k*0 : is the free space wavenumber,

*k jz <sup>e</sup>* =(*<sup>ε</sup> jxk*<sup>0</sup> <sup>2</sup> <sup>−</sup>(*<sup>ε</sup> jxk<sup>ρ</sup>* <sup>2</sup> / *<sup>ε</sup> jz*)) 1/2 : is TM propagation constants in the uniaxial substrate.

*k jz <sup>h</sup>* =(*<sup>ε</sup> jxk*<sup>0</sup> <sup>2</sup> <sup>−</sup>*k<sup>ρ</sup>* 2 ) 1 <sup>2</sup> : is TE propagation constants in the uniaxial substrate.

In the spectral domain, the relationship between the patch current and the electric field on the microstrip is given by [10, 11]:

$$\vec{\mathbf{E}}\left(\mathbf{k}\_{\rho}\right) = \vec{\mathbf{G}}\left(\mathbf{k}\_{\rho}\right)\vec{\mathcal{K}}\left(\mathbf{k}\_{\rho}\right) \tag{14}$$

*<sup>K</sup>*˜(*<sup>k</sup>ρ*)is the current on the microstrip which related to the vector Hankel transform of K(ρ). The unknown currents are expanded, in terms of a complete orthogonal set of basis functions, issued from the magnetic wall cavity model. It is possible to find a complete set of vector basis functions to approximate the current distribution, by noting that the superposition of the currents due to TM and TE modes of a magnetic-wall cavity form a complete set. The current distribution of the nth mode of microstrip patch can be written as [11, 12]:

$$\mathcal{K}\_n\left(\boldsymbol{\rho}\right) = \sum\_{p=1}^P a\_{np} \boldsymbol{\Psi}\_{np}\left(\boldsymbol{\rho}\right) + \sum\_{q=1}^Q b\_{nq} \boldsymbol{\Theta}\_{nq}\left(\boldsymbol{\rho}\right) \tag{15}$$

Here anp and bnq are unknown coefficients.

For superconducting annular ring patch, (*αnq*, *βnp*) are the roots of dual equations *φn*(*αnqb* / *a*) =0 and *ψ<sup>n</sup>* ' (*βnpb* / *a*) =0.

The Hankel transforms of ψnp and φnq functions are described as [12]:

$$
\hat{\Psi}\_{np}(\mathbf{k}\_{\rho}) = \begin{bmatrix}
\left[\frac{\mathfrak{P}\_{np}/a}{\left(\mathfrak{P}\_{np}/a\right)^2 - \boldsymbol{k}\_{\rho}^2}\right] \mathbf{Y}\_{np}'(\mathbf{k}\_{\rho}) \\
\left[\frac{\mathbf{na}}{\mathfrak{P}\_{np}k\_{\rho}}\right] \mathbf{Y}\_{np}(\mathbf{k}\_{\rho}) \\
\mathbf{0} \\
\mathbf{0} \\
\end{bmatrix} \rho \qquad \qquad \text{a} < \text{ } < \text{b}
$$

$$\left| \Phi\_{nq} \left( k\_{\rho} \right) = \left| \begin{array}{c} 0 \\ \frac{k\_{\rho} \mathbf{a}}{\left(k\_{\rho}^{2} - \left(a\_{nq}/a\right)^{2}\right)} \mathbf{Z}\_{nq}(k\_{\rho}) \\\\ 0 \\\\ 0 \end{array} \right| \tag{17}$$

Where

The kernel of the vector Hankel transform is given by [12]:

60 Advancement in Microstrip Antennas with Recent Applications

*<sup>A</sup>***¯** and *B***¯** are two unknown vectors and *<sup>g</sup>*

*k*0 : is the free space wavenumber,

<sup>2</sup> / *<sup>ε</sup> jz*)) 1/2

<sup>2</sup> <sup>−</sup>(*<sup>ε</sup> jxk<sup>ρ</sup>*

microstrip is given by [10, 11]:

<sup>2</sup> <sup>−</sup>*k<sup>ρ</sup>* 2 ) 1

And

Where

*k jz <sup>e</sup>* =(*<sup>ε</sup> jxk*<sup>0</sup>

*k jz <sup>h</sup>* =(*<sup>ε</sup> jxk*<sup>0</sup> ρ

% = + **A B** (11)

*g k <sup>k</sup>* <sup>=</sup> (13)

) <sup>=</sup> ( ).*K*( ) % % (14)

 = **gA B**- % (12)

& (10)

   

 

 

**¯**(*kρ*) is determined by:

0

: is TM propagation constants in the uniaxial substrate.

= + å å **Ψ Φ** (15)

ωμ

 

ρ ( ) ( )/ ( ) ( )/ ( ) *n n <sup>n</sup>*

 

ρ

( )


*n n J k in J k k H k in J k k J k* 

( ) ( ) - ( ,) *jz jz ik z ik z <sup>n</sup> nj nj ek z k e k e*

( ) ( ) - ( , ) (k ) .( *jz jz ik z ik z <sup>n</sup> <sup>j</sup> nj nj hk z ke ke*

0

<sup>2</sup> : is TE propagation constants in the uniaxial substrate.

**Ek Gk k** ( 

distribution of the nth mode of microstrip patch can be written as [11, 12]:

(ρ ρρ ) ( ) ( ) 1 1

*P Q n np np nq nq p q Ka b* = =

 

*<sup>K</sup>*˜(*<sup>k</sup>ρ*)is the current on the microstrip which related to the vector Hankel transform of K(ρ). The unknown currents are expanded, in terms of a complete orthogonal set of basis functions, issued from the magnetic wall cavity model. It is possible to find a complete set of vector basis functions to approximate the current distribution, by noting that the superposition of the currents due to TM and TE modes of a magnetic-wall cavity form a complete set. The current

In the spectral domain, the relationship between the patch current and the electric field on the

/ 0

*jz*

0 / *h jx jz j e*

*k*

ωε ε

$$\Psi\_{\mathfrak{pp}}\begin{pmatrix} k \ \end{pmatrix} = \Psi\_n(\mathfrak{P}\_{np}b \ / \ a) \int\_{\mathbb{R}^n} (k \ \ b) - \Psi\_n(\mathfrak{P}\_{np}) \int\_{\mathbb{R}^n} (k \ \ a) \tag{18}$$

$$\text{id}\_{\mathfrak{p}\mathfrak{q}}(\not k \ ) = (b \not/ a) \ \ \_n (\! \begin{array}{c} \ \text{\$ }\_n (\! \begin{array}{c} \ \text{\$ }\_n \text{\$ }\_n (k \ ) \end{array} \text{\$ }\_n (k \ ) - \ \text{\$ }\_n (\! \begin{array}{c} \ \text{\$ }\_n (k \ ) \end{array} \text{\$ }\_n (k \ ) \end{array} \tag{19}$$

Using the same procedure, the basis functions for superconducting circular patch, are given by the expressions [11]:

$$
\tilde{\Psi}\_{np}(k\_{\rho}) = \mathfrak{P}\_{np} a I\_n(\mathfrak{P}\_{np} a) \left[ \frac{\not{p}\_n(k\_{\rho} a)}{\not{p}\_{np}^2 - k\_{\rho}^2} \quad \frac{in}{k\_{\rho} \not{p}\_{np}^2 a} J\_n(k\_{\rho} a) \right]^T \tag{20}
$$

$$\tilde{\phi}\_{nq}(k\_{\rho}) = \left[ 0 \quad \frac{k\_{\rho} a \dot{l}\_{n}(a\_{nq} a) I\_{n}(k\_{\rho} a)}{k\_{\rho}^{2} - a\_{nq}^{2}} \right]^{T} \tag{21}$$

Where

$$\dot{J}\_n(\beta\_{np}a) = 0 \text{ and } J\_n(\alpha\_{nq}a) = 0 \text{ .}$$

Jn(.) and Nn (.) are the Bessel functions of the first, and second kind of order n.

*<sup>G</sup>***¯**(*kρ*) is the spectral dyadic Green's function, and after some simple algebraic manipulation, we determine the closed form of the spectral Green dyadic at z=d for a grounded uniaxial substrate.

$$\mathbf{G} = \begin{bmatrix} \mathbf{G}^{\mathrm{TM}} & \mathbf{0} \\ \mathbf{0} & \mathbf{G}^{\mathrm{TE}} \end{bmatrix} \tag{22}$$

For superconductors, a complex conductivity of the form

Full-Wave Spectral Analysis of Resonant Characteristics and Radiation Patterns of High Tc Superconducting Circular…

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

field is enforced to satisfy the impedance boundary condition on the microstrip patch, and the current vanishes off the microstrip patch, to give the following set of vector dual integral

(ρ ρ ) ρρ ρ ρ ( ) ( ) ρ ρ

(ρ ρ ) ρρ ρ ρ ( ) ( ) ρ

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, multi‐

Parseval's theorem for VHT, we obtain a system of Q+ P linear algebraic equations for each mode n which may be written in matrix form. Following well known procedures, we obtain

<sup>+</sup> (*ρ*) (p=1, 2,....... P) and *ρφnq*

*K dk k H k K k n nn a*

, *K dk k H k K k n nn b a*

equations [3]. For superconducting annular ring microstrip antenna, we have:

(ρ ρ ) ρρ ρ ρ ( ) ( ) ( <sup>ρ</sup> ) ρ

(ρ ρ ) ρρ ρ ρ ( ) ( ) ( <sup>ρ</sup> ) ρ

() *<sup>n</sup> <sup>n</sup> S n e dk k H k G k Z K k a*

( ) a *<sup>n</sup> <sup>n</sup> S n e dk k H k G k Z K k b*

, where σ<sup>n</sup> is often associated with the normal state

http://dx.doi.org/10.5772/54665

63

0 *scat inc n* + -× = *Z K* **<sup>s</sup> E E** (28)

*scat* are tangential components of incident and scattered electrics fields. Electric

<sup>=</sup> ×- = á < ò **<sup>0</sup>** (29)

= ×- = < ò **<sup>0</sup>** (31)

= ×= > ò **<sup>0</sup>** (32)

<sup>+</sup> (*ρ*) (q=1, 2,....... Q) and using

= ×= > á ò **<sup>0</sup>** (30)

2

conductivity at Tc and λ0 is the effective field penetration depth.

*σ* =*σn*(*T* / *Tc*)<sup>4</sup> −*i*(1−(*T* / *Tc*)4)/ *ωμ*0*λ*<sup>0</sup>

of the microstrip patch is given by:

0

0

plying the resulting equation by *ρψnp*

the following system of linear algebraic equations:

¥

0

And for superconducting circular microstrip antenna, we have:

0

¥

¥

¥

Here *E***¯**

Where

*Zs* 0 0 *Zs*

*Z*¯ *s* = *inc* and *E***¯**

Where

$$\mathbf{G}^{\text{TM}} = \frac{k\_0}{\dot{\mathbf{u}}} \frac{k\_\varepsilon^\varepsilon \sin(k\_\varepsilon^\varepsilon d)}{\varepsilon\_\text{u} \cdot \varepsilon\_\text{x} \cdot k\_0 \cdot \cos(k\_\varepsilon^\varepsilon d) + ik\_\varepsilon^\varepsilon \cdot \sin(k\_\varepsilon^\varepsilon d)}\tag{23}$$

$$\mathbf{G}^{\text{TE}} = \frac{k\_0^2}{\dot{\mathbf{u}} \cdot \varepsilon\_0} \frac{\sin(k\_{\frac{\hbar}{2}}^\hbar d)}{k\_{\frac{\hbar}{2}}^\hbar \cos(k\_{\frac{\hbar}{2}}^\hbar d) + ik\_0 \sin(k\_{\frac{\hbar}{2}}^\hbar d)}\tag{24}$$

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 high-temperature 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:

$$Z\_s = \mathcal{R}\_s + i \; X\_s \tag{25}$$

Where *Rs* and *Xs* are the surface resistance and the surface reactance.

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 impe‐ dance [13].

$$Z\_s = \sqrt{\alpha \mu\_0 / (\Im \sigma)}\tag{26}$$

If t is less than three penetration depths, a better boundary condition is given by [13]:

$$Z\_s = 1/\,\text{t}\sigma\tag{27}$$

Where the conductivity *σ* =*σ<sup>c</sup>* is real for conventional conductors. These approximations have been verified for practical metallization thicknesses by comparison with rigorous mode matching result.

For superconductors, a complex conductivity of the form *σ* =*σn*(*T* / *Tc*)<sup>4</sup> −*i*(1−(*T* / *Tc*)4)/ *ωμ*0*λ*<sup>0</sup> 2 , where σ<sup>n</sup> is often associated with the normal state conductivity at Tc and λ0 is the effective field penetration depth.

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:

$$
\overline{\mathbf{E}}\_{scat} + \overline{\mathbf{E}}\_{inc} - \overline{Z}\_{\mathbf{a}} \cdot \overline{K}\_{\mathbf{n}} = \mathbf{0} \tag{28}
$$

Here *E***¯** *inc* and *E***¯** *scat* are tangential components of incident and scattered electrics fields. Electric field is enforced to satisfy the impedance boundary condition on the microstrip patch, and the current vanishes off the microstrip patch, to give the following set of vector dual integral equations [3]. For superconducting annular ring microstrip antenna, we have:

$$\mathcal{L}\_n\left(\boldsymbol{\rho}\right) = \bigcap\_{\boldsymbol{0}}^n d k\_{\boldsymbol{\rho}} k\_{\boldsymbol{\rho}} \overline{H}\_n\left(k\_{\boldsymbol{\rho}} \boldsymbol{\rho}\right) \cdot \left(\overline{\mathbf{G}}\left(k\_{\boldsymbol{\rho}}\right) - \overline{Z}\_S\right) \overline{K}\_n\left(k\_{\boldsymbol{\rho}}\right) = \mathbf{0} \tag{29} \tag{29}$$

$$\mathcal{K}\_n\left(\rho\right) = \bigcap\_{0}^{\alpha} k\_{\rho} k\_{\rho} \,\overline{H}\_n\left(k\_{\rho} \rho\right) \cdot \overline{K}\_n\left(k\_{\rho}\right) = \mathbf{0} \tag{30} \tag{31} \\ \qquad \qquad \qquad \rho > b, \quad \rho \langle a \tag{30} \rangle$$

And for superconducting circular microstrip antenna, we have:

$$\rho\_n(\rho) = \bigcap\_{\rho}^{\alpha} dk\_{\rho} k\_{\rho} \overline{H}\_n \left( k\_{\rho} \rho \right) \cdot \left( \overline{G} \left( k\_{\rho} \right) - \overline{Z}\_{\overline{\pi}} \right) \overline{K}\_n \left( k\_{\rho} \right) = \mathbf{0} \tag{31} \tag{31}$$

$$\mathcal{K}\_n\left(\boldsymbol{\rho}\right) = \bigcap\_{\boldsymbol{0}}^{\boldsymbol{\alpha}} d k\_{\boldsymbol{\rho}} k\_{\boldsymbol{\rho}} \overline{H}\_n\left(k\_{\boldsymbol{\rho}} \boldsymbol{\rho}\right) \cdot \overline{\mathcal{K}}\_n\left(k\_{\boldsymbol{\rho}}\right) = \mathbf{0} \tag{32}$$

Where

Jn(.) and Nn (.) are the Bessel functions of the first, and second kind of order n.

substrate.

Where

by the equation:

dance [13].

matching result.

*<sup>G</sup>***¯**(*kρ*) is the spectral dyadic Green's function, and after some simple algebraic manipulation, we determine the closed form of the spectral Green dyadic at z=d for a grounded uniaxial

0

*G* é ù = ê ú ê ú ë û

*TE*

.sin( ) . .cos( ) .sin( ) *e e*

*hh h zz z*

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 high-temperature 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

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 impe‐

> s

Where the conductivity *σ* =*σ<sup>c</sup>* is real for conventional conductors. These approximations have been verified for practical metallization thicknesses by comparison with rigorous mode

<sup>0</sup> / (2. ) *Zs* = w

If t is less than three penetration depths, a better boundary condition is given by [13]:

1 / *Z t <sup>s</sup>* = s

*eee x zzz*

*k k d ik k d* <sup>=</sup> <sup>+</sup> (23)

*k k d ik k d* <sup>=</sup> <sup>+</sup> (24)

*Z R iX ss s* = + (25)

(26)

(27)

(22)

0 *TM*

*<sup>G</sup> <sup>G</sup>*

<sup>0</sup> iω ε 0

e

iω 2 0

Where *Rs* and *Xs* are the surface resistance and the surface reactance.

e

*G*

62 Advancement in Microstrip Antennas with Recent Applications

*G*

0

*k k kd*

0 0 sin( ) . cos( ) sin( ) *h*

*k k d*

*TM z z*

*TE z*

$$
\overline{Z}\_s = \begin{bmatrix} Z\_s & 0 \\ 0 & Z\_s \end{bmatrix}
$$

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, multi‐ plying the resulting equation by *ρψnp* <sup>+</sup> (*ρ*) (p=1, 2,....... P) and *ρφnq* <sup>+</sup> (*ρ*) (q=1, 2,....... Q) and using Parseval's theorem for VHT, we obtain a system of Q+ P linear algebraic equations for each mode n which may be written in matrix form. Following well known procedures, we obtain the following system of linear algebraic equations:

$$
\begin{bmatrix}
\left(\mathbf{Z}^{\cdot\cdot\mathbf{q}\cdot\mathbf{q}}\right)\_{P\times P} & \left(\mathbf{Z}^{\cdot\cdot\mathbf{q}\cdot\mathbf{q}}\right)\_{P\times Q}
\\
\left(\mathbf{Z}^{\cdot\cdot\mathbf{q}\cdot\mathbf{q}}\right)\_{Q\times P} & \left(\mathbf{Z}^{\cdot\cdot\mathbf{q}\cdot\mathbf{q}}\right)\_{Q\times Q}
\end{bmatrix}
\begin{bmatrix}
\left(\mathbf{A}^{\cdot}\right)\_{P\times 1} \\
\left(\mathbf{B}^{\cdot}\right)\_{Q\times 1}
\end{bmatrix} = \mathbf{0}
\tag{33}
$$

2 2 *c s P Z H H dS* j 

Full-Wave Spectral Analysis of Resonant Characteristics and Radiation Patterns of High Tc Superconducting Circular…

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 pathabove the real axis withaheight of about 1k0 (Figure 2).Inthis case,the effects ofthe surface waves are included in the calculation and knowledge ofthe pole locations is notrequired, 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 foundthatlengthofthe integrationpathrequiredreachingnumerical convergence at 100k0, also

100 k0

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 *kra* (*kra* =2*π f <sup>r</sup>a εxε*0*μ*0), as functions of different sizes of the ratio of the substrate thickness d normalized by the inner radius a is fixed of (0.71cm), and an outer radius of b=2a, the relative permittivity was ε1z=ε1x=2.65. Annular ring microstrip antenna is excited in the TM11 and TM12 modes. We found that, for the TM11 mode, the real part of (kra) increases as d/a increases. At the same time,forthismode,the imaginarypartof(kra),whichincludes the lossesbyradiationofthe

Muller's method that involves three initial guesses, is used for root seeking of (35).

( / 2) *d z P tg E dV* = we d

The efficiency of an antenna can be expressed by:

1 k0

Im(k )

**Figure 2.** Integration path used for computing the integrals in the complex kρ plane

**3. Convergence and comparison of numerical results**

2

= + òò (38)

/( ) *r rcd efficiency P P P P* = ++ (40)

òò (39)

http://dx.doi.org/10.5772/54665

65

Re(k )

Each element of the submatrices *Z***¯***CD* is given by:

$$\overline{Z}\_{\stackrel{\scriptstyle \rm d}{\dot{\rm d}}}^{\rm C,CD} = \bigcap\_{0}^{\circ} d\boldsymbol{k}\_{\rho} \boldsymbol{k}\_{\rho} \mathbf{C}\_{\boldsymbol{m}}^{+} \left(\boldsymbol{k}\_{\rho}\right) \cdot \left(\overline{\mathbf{G}}\left(\boldsymbol{k}\_{\rho}\right) - \overline{Z}\_{\rm S}\right) \cdot \mathbf{D}\_{\boldsymbol{n}\dot{\boldsymbol{\eta}}}\left(\boldsymbol{k}\_{\rho}\right) \tag{34}$$

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 kρ. This integration path must remain above the pole and the branch point of *<sup>G</sup>*¯. Although other choices of branch cut are possible, the choice made in this paper is very convenient, this treated in section 3. Once the impedance and the resistance matrices have been calculated, the resulting system of equations is then solved, for the unknown current modes on the microstrip patch. Nontrivial solutions can exist, if the determinant of Eq. (35) vanishes, that is:

$$\det[\overline{\mathbf{Z}}^\*(f)] = 0 \tag{35}$$

In general, the roots of this equation are complex numbers indicating, that the structure has complex resonant frequencies ( *f* = *f <sup>r</sup>* + *i f <sup>i</sup>* ). The bandwidth of a structure operating around its resonant frequency, can be approximately related to its resonant frequency, though the wellknown formula (*BW* =2 *f <sup>i</sup>* / *f <sup>r</sup>*). Once the problem is solved for the resonant frequency, far field radiation, co-polar and cross- polar fields in spherical coordinates are given from.

$$
\begin{bmatrix} E\_{\boldsymbol{\theta}}(\overline{\boldsymbol{\tau}}) \\ E\_{\boldsymbol{\phi}}(\overline{\boldsymbol{\tau}}) \end{bmatrix} = \sum\_{n=-\alpha}^{+\alpha} e^{in\boldsymbol{\rho}} \cdot (-i)^{n} \cdot e^{\delta n} \cdot \overline{\mathcal{T}}(\boldsymbol{\theta}) \cdot \overline{\mathcal{V}}(k\_{\boldsymbol{\rho}}) \cdot (\overline{\mathcal{G}}\left(k\_{\boldsymbol{\rho}}\right) - \overline{Z}\_{\mathbb{S}}) \cdot \overline{K}\_{n}(k\_{\boldsymbol{\rho}}) \tag{36}
$$

Where

$$
\overline{V}(k\_{\rho}) = \begin{bmatrix} 1 & 0 & -\frac{\mathbf{k}\_{\rho}}{\mathbf{k}\_{z}} \\ 0 & 1 & 0 \end{bmatrix} \text{and} \begin{pmatrix} \theta \end{pmatrix} = \begin{bmatrix} \cos \theta & 0 & -\sin \theta \\ 0 & 1 & 0 \end{bmatrix}.
$$

The losses in the antenna comprise dielectric loss Pd, the conductor loss Pc, and the radiation loss Pr, are given by [4-6]:

$$P\_r = (1/4\,\eta) \left[ \left| E\_{\theta} E\_{\theta} \right. \right. + E\_{\phi} E\_{\phi} \left. \frac{\ast}{\ast} \right| r^2 \sin\theta \, d\theta \, d\phi \, \tag{37}$$

Full-Wave Spectral Analysis of Resonant Characteristics and Radiation Patterns of High Tc Superconducting Circular… http://dx.doi.org/10.5772/54665 65

$$P\_c = Z\_s \iint \left| H\_\phi^2 + H\_\rho^2 \right| dS \tag{38}$$

$$P\_d = (\alpha \varepsilon t \lg \delta / 2) \left[ \left\| \begin{bmatrix} E\_z \\ \end{bmatrix} \right\|^2 dV \right] \tag{39}$$

The efficiency of an antenna can be expressed by:

( ) ( ) ( ) ( )

Each element of the submatrices *Z***¯***CD* is given by:

64 Advancement in Microstrip Antennas with Recent Applications

complex resonant frequencies ( *f* = *f <sup>r</sup>* + *i f <sup>i</sup>*

φ

k*ρ* kz

0 1 0

loss Pr, are given by [4-6]:

*E r*

é ù

*n*

+¥ =-¥

that is:

Where

*<sup>V</sup>*¯(*<sup>k</sup>ρ*)= <sup>1</sup> 0 -

0

¥

Where C and D represent either ψ or φ, for every value of the integer n.

YY YF

' ' '

**Z Z A**

´ ´ ´ FY FF ´ ´ ´ é ù é ù ê ú× = ê ú

 ' ' *P P P Q P <sup>Q</sup> Q P Q Q*

' () *CD Z dk k k G k Z k ij*

 

ë û ë û

( ) ( ) <sup>1</sup> 1

( ) ( <sup>ρ</sup> ) ( )

*ni S nj*

The integration path for the integrals of (32) is, in general, located in the first quadrant of the complex plane kρ. This integration path must remain above the pole and the branch point of *<sup>G</sup>*¯. Although other choices of branch cut are possible, the choice made in this paper is very convenient, this treated in section 3. Once the impedance and the resistance matrices have been calculated, the resulting system of equations is then solved, for the unknown current modes on the microstrip patch. Nontrivial solutions can exist, if the determinant of Eq. (35) vanishes,

In general, the roots of this equation are complex numbers indicating, that the structure has

resonant frequency, can be approximately related to its resonant frequency, though the wellknown formula (*BW* =2 *f <sup>i</sup>* / *f <sup>r</sup>*). Once the problem is solved for the resonant frequency, far field

ρρ ρ

ê ú = - × × -× ê ú ë û <sup>å</sup> (36)

qqj\* \* <sup>=</sup> <sup>+</sup> òò (37)

*S n*

radiation, co-polar and cross- polar fields in spherical coordinates are given from.

<sup>θ</sup> ( ) .( ) . . ( ) ( ) ( ) ( ) ( )

*e i e T Vk Gk Z K k E r*

01 0

<sup>2</sup> (1 / 4 ) sin *<sup>r</sup> P EE EE r d d* qq

 jj

The losses in the antenna comprise dielectric loss Pd, the conductor loss Pc, and the radiation

( ) <sup>θ</sup> <sup>φ</sup>

and (*θ*)= cos*<sup>θ</sup>* 0 -sin*<sup>θ</sup>*

h

*in n ikr*

'

ê ú

**0 Z Z <sup>B</sup>** (33)

<sup>+</sup> <sup>=</sup> × -× ò **C D** (34)

det[ ' ] 0 **Z** ( *f* ) = (35)

). The bandwidth of a structure operating around its

$$\text{efficiency} = P\_r \left( (P\_r + P\_c + P\_d) \right) \tag{40}$$

#### **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 pathabove the real axis withaheight of about 1k0 (Figure 2).Inthis case,the effects ofthe surface waves are included in the calculation and knowledge ofthe pole locations is notrequired, 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 foundthatlengthofthe integrationpathrequiredreachingnumerical convergence at 100k0, also Muller's method that involves three initial guesses, is used for root seeking of (35).

**Figure 2.** Integration path used for computing the integrals in the complex kρ plane

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 *kra*

(*kra* =2*π f <sup>r</sup>a εxε*0*μ*0), as functions of different sizes of the ratio of the substrate thickness d normalized by the inner radius a is fixed of (0.71cm), and an outer radius of b=2a, the relative permittivity was ε1z=ε1x=2.65. Annular ring microstrip antenna is excited in the TM11 and TM12 modes. We found that, for the TM11 mode, the real part of (kra) increases as d/a increases. At the same time,forthismode,the imaginarypartof(kra),whichincludes the lossesbyradiationofthe structure, is approximately zero. This means that, the TM11 mode has narrow bandwidth and weak-radiation. In addition, we observe that, for the TM12 mode, the real part of (kra) decreases as (d/a) increases, the TM12 has relatively wide bandwidth and high-radiation. Therefore, the TM11 mode is good for resonator applications and the TM12 mode for antennas. It is for that one does the applications of the annular ring microstrip antenna in the TM12 mode, better than in the TM11 mode. Thus, for this reason, the considered mode in this work is the TM12 mode. It is clear from Table 1 that our results agree very well with results obtained by other authors [14, 12].

**4. Resonant frequency of superconducting patch antenna**

Shown in Figures 3-4, is the dependence of resonant frequency on the thickness t of super‐ conducting 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 (λ0). After this value, the increase in the frequency of resonance becomes less significant.

Full-Wave Spectral Analysis of Resonant Characteristics and Radiation Patterns of High Tc Superconducting Circular…

http://dx.doi.org/10.5772/54665

67

**Figure 3.** Real part of resonant Frequency against thickness of superconducting annular ring patch antenna. (d=254µm, a=815µm, b=2a, T/ TC=0.5, λ0=1500Å, σn=210S/mm). (–––– ) εx= 9.4, εz= 11.6;(– – – ) εx= 11.6, εz= 11.6;

**Figure 4.** Real part of resonant Frequency against thickness of superconducting circular patch antenna. (d=254µm, a=815µm, TC =89K, T/ TC=0.5, λ0=1500Å, σn=210S/mm).(–––– ) εx= 9.4, εz= 11.6;(– – – ) εx= 11.6, εz= 11.6; (―∙―∙) εx=

(―∙―∙) εx= 13, εz=10.3; (……) εx=10.3, εz= 10.3.

13, εz=10.3; (……) εx=10.3, εz= 10.3.


**Table 1.** Calculated (Kra) of annular ring microstrip antennas.

In table 2, we have calculated the resonant frequencies for the modes (TM11, TM21, TM31, and TM12) 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 ψnp, and the Q functions of φnq. The required basis functions for reaching convergent solutions of complex resonant frequencies, using cavity model basis functions are obtained with (P=5, Q=0).


**Table 2.** Comparison of resonant frequencies of the first four modes of a perfect conducting circular patch printed on a dielectric substrate (a=7.9375mm, ε*<sup>x</sup>* =ε*<sup>z</sup>* =2.65, d=1.5875 mm).

#### **4. Resonant frequency of superconducting patch antenna**

structure, is approximately zero. This means that, the TM11 mode has narrow bandwidth and weak-radiation. In addition, we observe that, for the TM12 mode, the real part of (kra) decreases as (d/a) increases, the TM12 has relatively wide bandwidth and high-radiation. Therefore, the TM11 mode is good for resonator applications and the TM12 mode for antennas. It is for that one does the applications of the annular ring microstrip antenna in the TM12 mode, better than in the TM11 mode. Thus, for this reason, the considered mode in this work is the TM12 mode. It is clear from Table 1 that our results agree very well with results obtained by other authors [14, 12].

**Mode TM12 Mode TM11**

**Our results Results of [14] Our results**

Im (kra)

Re (kra)

Im (kra)

Quality factors (Q)

Re (kra)

**d/a**

**Mode**

Results of [14]

> Im (kra)

66 Advancement in Microstrip Antennas with Recent Applications

Re (kra) Results of [12]

> Im (kra)

Re

(kra) Im (kra)

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 (TM11, TM21, TM31, and TM12) 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,

In our results, we need to consider only the P functions of ψnp, and the Q functions of φnq. The required basis functions for reaching convergent solutions of complex resonant frequencies,

> Quality factors (Q)

TM11 6.1703 19.105 6.2101 19.001 TM12 17.056 10.324 17.120 10.303 TM21 10.401 19.504 10.438 19.366 TM01 12.275 8.9864 12.296 8.993

**Table 2.** Comparison of resonant frequencies of the first four modes of a perfect conducting circular patch printed on

**Results of [10 ] Our results**

Resonant Frequency (GHz)

Re (kra)

**Table 1.** Calculated (Kra) of annular ring microstrip antennas.

and the theoretical results of [10], is very good.

Resonant Frequency (GHz)

a dielectric substrate (a=7.9375mm, ε*<sup>x</sup>* =ε*<sup>z</sup>* =2.65, d=1.5875 mm).

using cavity model basis functions are obtained with (P=5, Q=0).

Shown in Figures 3-4, is the dependence of resonant frequency on the thickness t of super‐ conducting 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 (λ0). After this value, the increase in the frequency of resonance becomes less significant.

**Figure 3.** Real part of resonant Frequency against thickness of superconducting annular ring patch antenna. (d=254µm, a=815µm, b=2a, T/ TC=0.5, λ0=1500Å, σn=210S/mm). (–––– ) εx= 9.4, εz= 11.6;(– – – ) εx= 11.6, εz= 11.6; (―∙―∙) εx= 13, εz=10.3; (……) εx=10.3, εz= 10.3.

**Figure 4.** Real part of resonant Frequency against thickness of superconducting circular patch antenna. (d=254µm, a=815µm, TC =89K, T/ TC=0.5, λ0=1500Å, σn=210S/mm).(–––– ) εx= 9.4, εz= 11.6;(– – – ) εx= 11.6, εz= 11.6; (―∙―∙) εx= 13, εz=10.3; (……) εx=10.3, εz= 10.3.

Figures 5-6 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; Eφ), of the microstrip anten‐ na 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 TM12 and for superconducting circular patch antenna is the TM11. It is seen that the permittivity εz has a stronger effect on the radiation than the permittivi‐ ty εx. The radiation pattern of an antenna becomes more directional as its εz 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 εz increases.

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**Figure 7.** Radiation pattern versus angle θ of superconducting annular ring patch antenna at φ=0° plane (a=815µm, b=2a, t=0.02µm,d=254µm, λ0=1500Å, σn=210S/mm). (–––– )εx= 9.4, εz= 11.6; ( – – – )εx= 11.6, εz= 11.6;(― ∙ ― ∙) εx= 13,

**Figure 8.** Radiation pattern versus angle θ of superconducting annular ring patch antenna at φ=90° plane (a=815µm, b=2a, t=0.02µm, d=254µm, λ0=1500Å, σn=210S/mm).(–––– )εx= 9.4, εz= 11.6;( – – – )εx= 11.6, εz= 11.6; (― ∙ ― ∙) εx= 13,

In calculation of losses, we have found that, the values of dielectric loss (Pd), the conductor loss (Pc), and the radiation loss (Pr) 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].

εz=10.3 ;(……) εx=10.3, εz= 10.3.

εz=10.3 ; (……) εx=10.3, εz= 10.3.

**Figure 5.** Real part of resonant frequency of superconducting annular ring patch antenna against T/TC (d=254µm, a=815µm, b=2a, t=0.02µm, λ0=1500Å, σn=210S/mm). (–––– ) εx= 9.4, εz= 11.6;(– – – ) εx= 11.6, εz= 11.6; (―∙―∙) εx= 13, εz=10.3; (……) εx=10.3, εz= 10.3.

**Figure 6.** Real part of resonant frequency of superconducting circular patch antenna against T/TC (d=254µm, a=815µm, TC =89K, h= 0.02µm, λ0=1500Å, σn=210S/mm ). (–––– ) εx= 9.4, εz= 11.6;(– – – ) εx= 11.6, εz= 11.6; (―∙―∙) εx= 13, εz=10.3; (……) εx=10.3, εz= 10.3.

#### **5. Radiations patterns and efficiency of superconducting patch antenna**

Figures 5-6 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

**Figure 5.** Real part of resonant frequency of superconducting annular ring patch antenna against T/TC (d=254µm, a=815µm, b=2a, t=0.02µm, λ0=1500Å, σn=210S/mm). (–––– ) εx= 9.4, εz= 11.6;(– – – ) εx= 11.6, εz= 11.6; (―∙―∙) εx= 13,

**Figure 6.** Real part of resonant frequency of superconducting circular patch antenna against T/TC (d=254µm, a=815µm, TC =89K, h= 0.02µm, λ0=1500Å, σn=210S/mm ). (–––– ) εx= 9.4, εz= 11.6;(– – – ) εx= 11.6, εz= 11.6; (―∙―∙) εx=

εz=10.3; (……) εx=10.3, εz= 10.3.

13, εz=10.3; (……) εx=10.3, εz= 10.3.

reported by Mr. A. Richard for the case of rectangular microstrip antennas [2].

68 Advancement in Microstrip Antennas with Recent Applications

The calculated radiations patterns (electric field components, Eθ; Eφ), of the microstrip anten‐ na 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 TM12 and for superconducting circular patch antenna is the TM11. It is seen that the permittivity εz has a stronger effect on the radiation than the permittivi‐ ty εx. The radiation pattern of an antenna becomes more directional as its εz 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 εz increases.

**Figure 7.** Radiation pattern versus angle θ of superconducting annular ring patch antenna at φ=0° plane (a=815µm, b=2a, t=0.02µm,d=254µm, λ0=1500Å, σn=210S/mm). (–––– )εx= 9.4, εz= 11.6; ( – – – )εx= 11.6, εz= 11.6;(― ∙ ― ∙) εx= 13, εz=10.3 ;(……) εx=10.3, εz= 10.3.

**Figure 8.** Radiation pattern versus angle θ of superconducting annular ring patch antenna at φ=90° plane (a=815µm, b=2a, t=0.02µm, d=254µm, λ0=1500Å, σn=210S/mm).(–––– )εx= 9.4, εz= 11.6;( – – – )εx= 11.6, εz= 11.6; (― ∙ ― ∙) εx= 13, εz=10.3 ; (……) εx=10.3, εz= 10.3.

In calculation of losses, we have found that, the values of dielectric loss (Pd), the conductor loss (Pc), and the radiation loss (Pr) 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].

**Figure 9.** Radiation pattern versus angle θ of superconducting circular patch antenna at φ=0° plane (a=815µm, t=0.02µm, T/ TC=0.5, d=254µm, λ0=1500Å, n=210S/mm). (–––– ) εx= 9.4, εz= 11.6; ( – – – ) εx= 11.6, εz= 11.6; (― ∙ ― ∙) εx= 13, εz=10.3 ; (……) εx=10.3, εz= 10.3.

**Figure 12.** Superconducting circular patch antenna efficiency for the mode TM11 ( (d=254µm, a=815µm,h=0.02µm, T/

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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 (εz) 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.

Electronics Department, University of Constantine, Constantine, Algeria

TC=0.5, λ0=1500Å, σn=210S/mm, εx= εz=11.6, δ=0.0024).

**6. Conclusion**

**Author details**

Ouarda Barkat\*

**Figure 10.** Radiation pattern versus angle θ of superconducting circular patch antenna at φ=π/2 plane (a=815µm, h=0.02µm, T/ TC=0.5, d=254µm, λ0=1500Å, n=210S/mm). (–––– ) εx= 9.4, εz= 11.6; ( – – – ) εx= 11.6, εz= 11.6; (― ∙ ― ∙) εx= 13, εz=10.3 ; (……) εx=10.3, εz= 10.3.

**Figure 11.** Superconducting annular ring patch antenna efficiency for the mode TM12 (d=254µm, a=815µm, b=2a, t=0.02µm, λ0=1500Å, σn=210S/mm, εx= εz=11.6, δ=0.0004).

**Figure 12.** Superconducting circular patch antenna efficiency for the mode TM11 ( (d=254µm, a=815µm,h=0.02µm, T/ TC=0.5, λ0=1500Å, σn=210S/mm, εx= εz=11.6, δ=0.0024).

#### **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 (εz) 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.

#### **Author details**

Ouarda Barkat\*

**Figure 11.** Superconducting annular ring patch antenna efficiency for the mode TM12 (d=254µm, a=815µm, b=2a,

**Figure 9.** Radiation pattern versus angle θ of superconducting circular patch antenna at φ=0° plane (a=815µm, t=0.02µm, T/ TC=0.5, d=254µm, λ0=1500Å, n=210S/mm). (–––– ) εx= 9.4, εz= 11.6; ( – – – ) εx= 11.6, εz= 11.6; (― ∙ ― ∙)

**Figure 10.** Radiation pattern versus angle θ of superconducting circular patch antenna at φ=π/2 plane (a=815µm, h=0.02µm, T/ TC=0.5, d=254µm, λ0=1500Å, n=210S/mm). (–––– ) εx= 9.4, εz= 11.6; ( – – – ) εx= 11.6, εz= 11.6; (― ∙ ― ∙)

t=0.02µm, λ0=1500Å, σn=210S/mm, εx= εz=11.6, δ=0.0004).

εx= 13, εz=10.3 ; (……) εx=10.3, εz= 10.3.

70 Advancement in Microstrip Antennas with Recent Applications

εx= 13, εz=10.3 ; (……) εx=10.3, εz= 10.3.

Electronics Department, University of Constantine, Constantine, Algeria

#### **References**

[1] Hansen, C R. Electrically Small, Superdirective, and Superconducting Antennas. John Wiley& Sons, Inc, Hoboken, New Jersey, (2006).

**Section 2**

**Multiband Planar Antennas**

