MODELING OF MICROSTRIP ANTENNA ARRAYS

E. Schneider, R. Staraj, E. Cambiaggio, O. Benevello

Laboratoire d'Electronique - Université de Nice - Sophia Antipolis
Unité de Recherche Associée au CNRS n° 1400 - 250, rue A. Einstein, 06560 Valbonne, France
Tel.: 92.94.28.30 - Fax: 92.94.28.12

**Abstract** - A new model is developed from the "element by element" approach. It consists in a decomposition in sub-arrays in order to reduce the time of calculation in the modeling of finite arrays. In this modeling we consider that the surface currents are identical on patches which have the same neighbourhood configuration. This approach simplifies the solving of the integral equation by moment method in reducing the number of unknown coefficients. An application is presented in order to characterize the bandwidth of an array in terms of its geometry.

**Résumé** - Une nouvelle méthode est développée à partir de l'approche "élément par élément". Elle consiste à décomposer les réseaux finis d'antennes imprimées en un ensemble de sous-réseaux afin de réduire le temps de calcul. Pour cela, on considère que les courants de surface sont identiques sur les éléments qui ont la même configuration de voisinage. Cette approche simplifie la résolution des équations intégrales grâce à une diminution du nombre d'inconnues. Cette méthode est ensuite appliquée à l'étude de la bande passante d'un réseau en fonction de sa disposition géométrique.

I. INTRODUCTION

Using the "element by element" approach [1], the analysis of a N patches finite array needs the calculation of the mutual coupling between each pair of elements, and the number of unknown coefficients in the Moment Method, which is at least equal to N in the simplest cases (only one expansion mode per element and only one patch fed) may become very large when both the number of fed elements and expansion modes increase. Thus, this "element by element" approach is very time-consuming in the analysis of large arrays. In more classical studies the behaviour of all the elements of a finite array of microstrip patches is often assumed to be the same than those of the single isolated element. This approximation is not too bad for the array radiation pattern, but can be very far from the experiment for the input impedance of each element. In a second hand, this classical method can not be used with parasitic elements. For all these reasons, we have developed a new method, using the "element by element" model, but with a decomposition in sub-arrays to reduce the calculation time. Surface currents are assumed to be equal on all the patches which have the same number and configuration of neighbours. Thus, the number of unknown coefficients can be easily reduced by using a single basis function to describe the surface currents on these patches. Indeed, the integral equation can be simplified by factorizing the equal unknown coefficients.

Section II presents quickly the theory of a decomposition of a linear array in sub-arrays (fig. 1). The theoretical study can be done for multilayer structures (fig. 2) located in the coordinate system of figure 3.

Section III presents the experimental validation of our model. Theoretical results are compared with the experiment. A good agreement between theory and experiment is obtained. Then, an application is presented in order to characterize bandwidth enhancement for an array with parasitic elements.

<figure><img src="image_1.png" /><figcaption>Figure 1 : Decomposition of a five-element linear array in three sub-arrays</figcaption></figure>
<figure><img src="image_2.png" /><figcaption>Figure 2: Antenna structure</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Figure 3: Spherical coordinates system</figcaption></figure>

## II. THEORY

### A. Calculation of the Input Impedance

The Moment Method solution is a Galerkin solution of the electric field integral equation with the dielectric slab Green's functions [2]. The unknown basis function coefficients are found as the solution of the matrix equation:

$$ [Z] [I] = [V] \qquad (1) $$

$$ \text{where} \quad Z_{li,mj} = - \iint F_i(\alpha, \beta) Q_{ij}(\alpha, \beta) F_j(-\alpha, -\beta) e^{jk_0(\alpha(x_{ol}-x_{om}) + \beta(y_{ol}-y_{om}))} d\alpha d\beta \qquad (2) $$

$$ \text{and} \quad V_{li} = I_a e^{-j\phi} \iint F_i(\alpha, \beta) Q_{iz}(\alpha, \beta) e^{jk_0(\alpha(x_{ol}-x_p) + \beta(y_{ol}-y_p))} d\alpha d\beta \qquad (3) $$

$F_i(\alpha, \beta)$ and $F_j(\alpha, \beta)$ are the Fourier Transform of an ith basis function or a jth test function on the element l or m whose the positions are $(x_{ol}, y_{ol})$ and $(x_{om}, y_{om})$. $(x_p, y_p)$ the feed point on the centre patch. The feed current is 1A. By gathering together some elements in symmetrical sub-arrays (figure 1), we can simplify the integral equations system of the Moment Method. Indeed, the Z terms represent the mutual coupling between sub-arrays and the V terms represent the excitation on each sub-array. Then, we obtain the new matrix equation :

$$ [Z] [I]^k = [V]^k \qquad (4) $$

where

$$ Z_{ig,jk} = - \iint F_i^g(\alpha, \beta) Q_{ij}(\alpha, \beta) F_j^k(-\alpha, -\beta) d\alpha d\beta \qquad (5) $$

and

$$ V_{ig} = \iint F_i^g(\alpha, \beta) Q_{iz}(\alpha, \beta) e^{-jk_0(\alpha x_p + \beta y_p)} d\alpha d\beta \qquad (6) $$

$Z_{ig,jk}$ is now the term corresponding to the mutual coupling between the ith expansion mode on the sub-array g and the jth expansion mode on the k one. $V_{ig}$ is the excitation term for the ith mode on the sub-array g. The Fourier transform of an ith basis function is easily obtained with the Fourier transform of its elements :

$$ F_i^g(\alpha, \beta) = F_i(\alpha, \beta) \sum_{a=i}^{N_a} e^{-jk_0(\alpha x_a + \beta y_a)} \qquad (7) $$

$N_a$ is the number of patches of this sub-array and $(x_a, y_a)$ is the position of its elements. $F_j^k(\alpha, \beta)$ is obtained by the same way. In the case of a linear array divided in K sub-arrays, the input impedance is obtained as

$$ Z_{in} = - \sum_{k=1}^{K} \sum_{n=1}^{N_a} I_{in}^{k} V_{in}^{k} \qquad (8) $$

An active reactance term can be added to $Z_{in}$ to better modeling the coaxial probe feed effect [2], [3].
## B Calculation of the Far Field

Using the aperture model, we calculate the radiation field of a single element. Indeed, we obtain the far field from the Fourier Transform of the aperture field. In the spherical coordinates system (fig. 3), the radiation field of the ith element is calculated from $E_{tx}(\alpha, \beta, h)$ and $E_{ty}(\alpha, \beta, h)$, the Fourier Transform of the tangential component of the electric field in the plane $z=h$. h is the total substrate thickness [4]. We write $E_{tx}$ and $E_{ty}$ as a function of the current coefficients we obtain on each patch and the Green's functions [4]:

$$E_{tx}(\alpha, \beta, h) = \frac{-jZ_0}{k_0} \sum_{j=1}^{n_i} I_j \left[ Q_{x3x1}(\alpha, \beta, h)F_{jx}(\alpha, \beta) + Q_{x3y1}(\alpha, \beta, h)F_{jy}(\alpha, \beta) \right] \quad (9)$$

$$E_{ty}(\alpha, \beta, h) = \frac{-jZ_0}{k_0} \sum_{j=1}^{n_i} I_j \left[ Q_{y3x1}(\alpha, \beta, h)F_{jx}(\alpha, \beta) + Q_{y3y1}(\alpha, \beta, h)F_{jy}(\alpha, \beta) \right] \quad (10)$$

$Q_{x3x1}$, $Q_{x3y1}$, ... are the Green's Functions which link the surface currents on the patches on the layer 1 with the tangential component of the electric field on the layer 3. $n_i$ is the number of expansion mode on the ith patch. $I_j$ are the unknown coefficients of these modes. It gives $E_i(r, \theta, \phi)$ the radiation field of the ith element [4].

The total radiation field of the array is obtained as a sum of the radiation field of each element including a phase term due to its relative position $(x_{oi}, y_{oi})$ in the array (of N elements). It follows:

$$E^{\text{tot}}(r, \theta, \phi) = \sum_{i=1}^{N} E_i(r, \theta, \phi) \exp(jk_0(x_{0i}+y_{0i})) \quad (11)$$

## III. RESULTS

This method gives very good results [4]. An example of accuracy in the input impedance and radiation pattern is presented in the case of a strongly coupled three-element array (fig. 4 and 5) ($W_{x1}=W_{x2}=W_{y1}=W_{y2}=18.9$ mm, $Y_p=1.8$ mm, $d1=5$ mm). A good agreement between theory and experiment is obtained with a calculation time 2.1 time less than the "element by element" approach (considering two expansion modes) [4]. This time difference increases rapidly with the number of expansion modes. Good results could be obtain too with more than a single excited element [4]. An example in the case of a four-element linear array with all the patches excited, is given figures 6 and 7. Figure 6 shows the theoretical and experimental input impedances for the two central elements when all the patches are fed. Figure 7 presents the corresponding radiation pattern for all the array. A very good agreement is obtained again between theory and experiment.

Figure 8 shows the effects of the coupling when two elements are added to the first three-element array described at the beginning of this paragraph ($d2=5$ mm). We can see that we have a larger bandwidth.

As we obtain a good accuracy with our model, we can easily modify the geometrical parameters in order to increase the bandwidth of an antenna. Thus, we can obtain for example (fig. 9), for a three-element array ($W_{x1}=12$mm, $W_{x2}=8$mm, $W_{y1}=W_{y2}=50$mm, $Y_p=5$mm, $d1=13$ mm), a bandwidth of 1.45% (SWR<2) instead of 0.4% for a single simple element at these frequencies.

## VI. CONCLUSION

The comparison between input impedance computed with our model and the experiment shows a good accuracy. Also, the decomposition in sub-arrays used to reduce the time of calculation in the modeling of finite arrays, can give good results with a calculation time less than the rigorous "element by element" approach. This method can be so applied in the case of the study of microstrip arrays to determine the effects of the mutual coupling or as presented here, in the case of parasitic sub-arrays used to optimise the bandwidth of the structure.

## REFERENCES

[1] D. M. Pozar "Finite phased arrays of rectangular microstrip patches" IEEE Transactions on Antennas and Propagation, vol AP-34, n° 5, pp 658-665, May 1986.

[2] D. M. Pozar "Input impedance and mutual coupling of rectangular microstrip antennas" IEEE Transactions on Antennas and Propagation, vol AP-30, n° 6, pp 1191-1196, November 1982.

[3] M. D. Deshpande and M. C. Bailey "Input impedance of microstrip antennas" IEEE Transactions on Antennas and Propagation, vol AP-30, n° 4, pp 645-650, July 1982.

[4] E. Schneider, R. Staraj and E. Cambiaggio "Reduced Time Modeling of Microstrip Antenna Arrays using Equal-Current Sub-array Decomposition" IEEE Sixth Biennial Conference on Electromagnetic Field Computation CEFC'94, Aix-les-Bains, France, July 94.
<figure><img src="image_4.png" /><figcaption>Fig 4 : Input impedance of a 3 element array</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig 5 : Radiation pattern of a 3 element array</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 6 : Input impedance for 4 elements</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 7 : Theoretical radiation pattern for 4 elements</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 8 : Input impedance of a 5 element array</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 9 : Theoretical input impedance for 3 elements</figcaption></figure>