And

${F}^{\text{T}}=\left[\begin{array}{cc}-& -\\ -& +\end{array}\right]$

For a 4 × 4 matrix arrangement, that is using Equation (3) above, that product of $F{F}^{\text{T}}$ gives [8]

$F{F}^{\text{T}}=F\otimes {F}^{\text{T}}=\left[\begin{array}{cc}+& +\\ +& -\end{array}\right]\otimes \left[\begin{array}{cc}-& -\\ -& +\end{array}\right]=nI$ (4)

Therefore,

$F{F}^{\text{T}}=\left[\begin{array}{cc}\begin{array}{cc}+& +\\ +& -\end{array}& \begin{array}{cc}+& +\\ +& -\end{array}\\ \begin{array}{cc}+& +\\ +& -\end{array}& \begin{array}{cc}-& -\\ -& +\end{array}\end{array}\right]$ (5)

Equation (5) implies that F is nonsingular, and has an inverse ${n}^{-1}{F}^{\text{T}}$ ; consequently

${F}^{\text{T}}F=nI$

But in this paper, we considered a uniform planar array antenna with all elements energised. From Hadamard, let ${F}_{1}\otimes {F}_{2}=A\otimes B$

$A\otimes B\left[\begin{array}{cc}{a}_{11}B& {a}_{12}B\\ {a}_{21}B& {a}_{22}B\end{array}\right]\otimes \left[\begin{array}{cc}{a}_{11}B& {a}_{12}B\\ {a}_{21}B& {a}_{22}B\end{array}\right]=nI$ (6)

Then

$A\otimes B=\left[\begin{array}{cc}\begin{array}{cc}{a}_{11}B& {a}_{12}B\\ {a}_{21}B& {a}_{22}B\end{array}& \begin{array}{cc}{a}_{13}B& {a}_{14}B\\ {a}_{23}B& {a}_{24}B\end{array}\\ \begin{array}{cc}{a}_{31}B& {a}_{32}B\\ {a}_{41}B& {a}_{42}B\end{array}& \begin{array}{cc}{a}_{33}B& {a}_{34}B\\ {a}_{43}B& {a}_{44}B\end{array}\end{array}\right]=nI$ (7)

According to Hadamard matrix, $F={f}_{ij}$ is a Hadamard matrix of order N × M if the entries of F are either +1 or −1. In this paper we considered in Equation (7) where all the elements will to be +1, that is,

${a}_{11}B={a}_{12}B={a}_{13}B={a}_{14}B=+1$ (8)

${a}_{21}B={a}_{22}B={a}_{23}B={a}_{24}B=+1$ (9)

${a}_{31}B={a}_{32}B={a}_{33}B={a}_{34}B=+1$ (10)

${a}_{41}B={a}_{42}B={a}_{43}B={a}_{44}B=+1$ (11)

From the above Equations (8)-(11) it therefore implies that, in a uniform planar array antenna with all the elements in the array matrix energised and represented by “1” we have;

$F=\left[\begin{array}{cc}\begin{array}{cc}1& 1\\ 1& 1\end{array}& \begin{array}{cc}1& 1\\ 1& 1\end{array}\\ \begin{array}{cc}1& 1\\ 1& 1\end{array}& \begin{array}{cc}1& 1\\ 1& 1\end{array}\end{array}\right]$ (12)

For clarity and easy placement of these elements in the planar array form, the nodal elements are numbered from 1 - 16 according the element’s position in the matrix array as shown in Figure 1.

From Figure 1 below, the element of each node $E\left({a}_{x},{a}_{y}\right)$ is the element’s position number in the uniform planar array matrix. The inter-element spacings ${a}_{x}\lambda $ and ${a}_{y}\lambda $ are spacings along x and y-axis respectively.

10 elements with (+) sign before them, that is, elements 1, 2, 3, 4, 5, 9, 11, 13 are “on” while 6 elements with (−) sign before them, that is, elements 7, 8, 10, 12, 14, and 15 are “off” of the 4 × 4 array. The total elements involved in this planar array evaluation were 16 elements.

3. Derivation of the Formula for Planar Array Factor AF_{planar} from Linear Array Antenna

Linear antenna array provides a good background in array theory because of the insights that leads into formation of beam and it relates to the array excitation functions and the radiation pattern obtained [6] .

Consider an isotropic radiator located at the origin of the linear array, the E-field may be written as (assuming θ-polarization)

${E}_{\theta}={I}_{0}\frac{{\text{e}}^{-jkd}}{4\pi d}$ (13)

where,

${I}_{0}$ is the complex excitation of the isotropic radiator, k is the free space wave number and d is distance of the observation point from the origin.

Assume that the N elements of the array are uniformly spaced with a separation

Figure 1. Placement of element in planar array form [1] .

distance d. The uniformly spaced linear array antenna is presented in Figure 2 below [6] .

This arrangement is necessary showing that the configuration of a planar array antenna is made up of two linear antennas placed at orthogonally to each other. The nodes, 1 - 4 represent the four elements of the linear array and a, is the inter-element spacing. While d_{1} - d_{4} is the far field approximation.

From Figure 2 below, the far field approximations of the array is

${d}_{1}=d$ (14)

${d}_{2}\approx d-a\mathrm{cos}\theta $ (15)

${d}_{3}\approx d-2a\mathrm{cos}\theta $ (16)

${r}_{N}\approx d-\left(N-1\right)a\mathrm{cos}\theta $ (17)

The magnitudes of current of the array elements was assumed to be equal and the current on the array element located at the origin was used as the phase reference (zero phase) [3] see Figure 1.

Therefore, the currents ${I}_{1}={I}_{0}{I}_{2}={I}_{0}{e}^{j{\varphi}_{2}}\cdots {I}_{N}{e}^{j{\varphi}_{N}}$ .

The far E-field of the individual elements is:

${E}_{{\theta}_{1}}={I}_{0}\frac{{\text{e}}^{-jkd}}{4\pi d}={E}_{0}$ (18)

${E}_{{\theta}_{2}}={I}_{0}{\text{e}}^{j{\varphi}_{2}}\frac{{\text{e}}^{-jk\left(r-a\mathrm{cos}\theta \right)}}{4\pi d}={E}_{0}{\text{e}}^{-jk\left(r+kac\mathrm{cos}\theta \right)}$

${E}_{{\theta}_{3}}={I}_{0}{\text{e}}^{j{\varphi}_{3}}\frac{{\text{e}}^{-jk\left(d-2ac\mathrm{cos}\theta \right)}}{4\pi d}={E}_{0}{\text{e}}^{-jk\left(d+2kac\mathrm{cos}\theta \right)}$ (19)

${E}_{{\theta}_{N}}={I}_{0}{\text{e}}^{j{\varphi}_{N}}\frac{{\text{e}}^{-jk\left(d-\left(N-1\right)a\mathrm{cos}\theta \right)}}{4\pi d}={E}_{0}{\text{e}}^{j\left[{\varphi}_{N}+\left(N-1\right)ka\mathrm{cos}\theta \right]}$

The total array far field could be found by using superposition:

${E}_{0}={E}_{{\theta}_{1}}+{E}_{{\theta}_{2}}+{E}_{{\theta}_{3}}+\cdots +{E}_{{\theta}_{N}}$ (20)

$={E}_{0}\left[1+{\text{e}}^{j\left({\varphi}_{2}+ka\mathrm{cos}\theta \right)}+\cdots +{\text{e}}^{j\left[{\varphi}_{N}+\left(N-1\right)ka\mathrm{cos}\theta \right]}\right]={E}_{0}\left[AF\right]$ (21)

Figure 2. Uniform linear array antenna [1] .

Note that the array factor (AF) is independent of the antenna type, assuming all of the elements are identical. The array factor for a uniformly-spaced N-element linear array is [3] :

$AF=\left[1+{\text{e}}^{j\left({\varphi}_{2}+ka\mathrm{cos}\theta \right)}+\cdots +{\text{e}}^{j\left[{\varphi}_{N}+\left(N-1\right)ka\mathrm{cos}\theta \right]}\right]$ (22)

We consider a uniform array defined by uniformly-spaced identical elements of equal magnitude with a linearly progressive phase from element to element:

${\varphi}_{1}=0,{\varphi}_{2}=\rho ,{\varphi}_{3}=2\rho ,\cdots ,{\varphi}_{N}=\left(N-1\right)\rho $ (23)

Inserting this linear phase progression into the formula for the general N-element array gives

$AF=\left[1+{\text{e}}^{j\left({\varphi}_{2}+ka\mathrm{cos}\theta \right)}+\cdots +{\text{e}}^{j\left[{\varphi}_{N}+\left(N-1\right)ka\mathrm{cos}\theta \right]}\right]$

$AF=\left[1+{\text{e}}^{j\rho}+{\text{e}}^{j2\rho}+{\text{e}}^{j3\rho}+{\text{e}}^{j4\rho}+{\text{e}}^{j5\rho}+\cdots +{\text{e}}^{j\rho \left(N-1\right)}\right]$ (24)

where,

$\rho ={\varphi}_{N}+ka\mathrm{cos}\theta $

And thus,

$AF={\displaystyle \underset{n=1}{\overset{N}{\sum}}{\text{e}}^{j\rho \left(N-1\right)}}$ (25)

where $\rho $ is a function which defined as the array phase function and is a function of [6] the element spacing, phase shift, frequency and elevation angle. If the array factor of Equation (25) is multiplied by ${\text{e}}^{j\rho}$ the result is

$AF{\text{e}}^{j\rho}=\left[{\text{e}}^{j\rho}{\text{e}}^{j\rho}+{\text{e}}^{j2\rho}+{\text{e}}^{j3\rho}+{\text{e}}^{j4\rho}+{\text{e}}^{j5\rho}+\cdots +{\text{e}}^{jN\rho}\right]$ (26)

The subtraction of the array factor from Equation (26) above gives

$AF\left({\text{e}}^{j\rho}-1\right)=\left({\text{e}}^{jN\rho}-1\right)$ (27)

Therefore, according to [2] and [6]

$AF=\frac{{\text{e}}^{jN\phi}-1}{{\text{e}}^{j\phi}-1}=\frac{{\text{e}}^{jN\frac{\rho}{2}}}{{\text{e}}^{j\frac{\rho}{2}}}\frac{{\text{e}}^{jN\frac{\rho}{2}}-{\text{e}}^{-jN\frac{\rho}{2}}}{{\text{e}}^{j\frac{\rho}{2}}-{\text{e}}^{-j\frac{\rho}{2}}}={\text{e}}^{j\left(N-1\right)\frac{\rho}{2}}\frac{\mathrm{sin}\left(N\frac{\rho}{2}\right)}{\mathrm{sin}\left(\frac{\rho}{2}\right)}$ (28)

The complex exponential term ${\text{e}}^{j\left(N-1\right)\frac{\rho}{2}}$ in Equation (28) is the phase shift (array phase factor) of the array phase center relative to the origin. If the position of the array is shifted so that the center of the array is located at the origin, this phase term goes to zero. That is if $\left(N-1\right)\frac{\rho}{2}=0$ ,

then Equation (23) becomes;

$AF=\frac{\mathrm{sin}\left(\frac{N\rho}{2}\right)}{\mathrm{sin}\left(\frac{\rho}{2}\right)}$ (29)

The array factor is normalized so that the maximum value for any value of N is unity [6] . The normalized array factor therefore is given as

$AF=\frac{1}{N}\frac{\mathrm{sin}\left(\frac{N\rho}{2}\right)}{\mathrm{sin}\left(\frac{\rho}{2}\right)}$ (30)

The antenna is rectangular array antenna as shown in Figure 3. The interest is the arrangement of a linear array which is of N × N matrix with M number of element in x equals N number elements in y [4] .

The designed principles for planar arrays are similar to those elements placed in two dimensions (Figure 3), the array factor of a planar array can be expressed as the multiplication of the array factors of two linear arrays of Equation (30) above, one along the x-axis and the other one along the y-axis. Therefore, planar array factor $A{F}_{planar}$ can be simply expressed as:

$A{F}_{planar}=\left\{\frac{1}{M}\frac{\mathrm{sin}\left(\frac{M}{2}{\rho}_{x}\right)}{\mathrm{sin}\left(\frac{{\rho}_{x}}{2}\right)}\right\}\left\{\frac{1}{N}\frac{\mathrm{sin}\left(\frac{N}{2}{\rho}_{y}\right)}{\mathrm{sin}\left(\frac{{\rho}_{y}}{2}\right)}\right\}$ (31)

where,

${\rho}_{x}=k{a}_{x}\mathrm{sin}\theta \mathrm{cos}\varphi +{\omega}_{x}$ (32)

${\rho}_{y}=k{a}_{y}\mathrm{sin}\theta \mathrm{cos}\varphi +{\omega}_{y}$ (33)

where, ${\rho}_{x}$ and ${\rho}_{y}$ is the array phase function along x-and-y axis while ${\omega}_{x}$ and ${\omega}_{y}$ is the progressive phase shift along x-and-y axis respectively. ${a}_{x}$ and ${a}_{y}$ , is the inter-element spacing x-and-y axis respectively when ${\rho}_{x}$ and ${\rho}_{y}=0$ can be obtained as

${\omega}_{x}=-k{a}_{x}\mathrm{sin}\theta \mathrm{cos}\varphi $ (34)

${\omega}_{y}=-k{a}_{y}\mathrm{sin}\theta \mathrm{cos}\varphi $ (35)

And

Figure 3. Geometry of the N × M planar array antenna [1] .

$k=2\frac{\pi}{\lambda}$ (36)

The nulls of the array function are found by determining the zeros of the numerator term where the denominator is not simultaneously zero [2] . That is:

$\mathrm{sin}\left(\frac{N\rho}{2}\right)=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\Rightarrow \text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{N\rho}{2}=\pm n\pi \text{\hspace{0.17em}}\text{\hspace{0.17em}}\Rightarrow \text{\hspace{0.17em}}\text{\hspace{0.17em}}\varphi +ka\mathrm{cos}{\theta}_{n}=\pm \frac{2n\pi}{N}$ (37)

${\theta}_{n}={\mathrm{cos}}^{-1}\left[\frac{\lambda}{2\pi a}\left(-\alpha \pm \frac{2n\pi}{N}\right)\right]$ (38)

$n=1,2,3,\cdots $

$n\ne 0,N,2N,3N,\cdots $

The peaks of the array function can be found by determining the zeros of the numerator term where the denominator is simultaneously zero.

${\theta}_{m}={\mathrm{cos}}^{-1}\left[\frac{\lambda}{2\pi a}\left(-\alpha \pm 2m\pi \right)\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}m=1,2,3,\cdots $ (39)

When $m=0$ term,

${\theta}_{m}={\mathrm{cos}}^{-1}\left(\frac{\lambda}{2\pi a}\right)$ (40)

4. The Wavelength (Lambda, 𝜆)

It has been established that the speed of electromagnetic wave on air is 3 × 10^{8} m/s, and can be obtained from the formula,

$V=f\lambda $ (41)

where,

f = the frequency of the electromagnetic wave.

$\lambda $ = Lambda, the wavelength.

For the purpose of this investigation, the frequency is 2.5 GHz, therefore, the wavelength, $\lambda $ can be obtained as follows;

$\lambda =\frac{v}{f}=\frac{3\times {10}^{8}}{2.5\times {10}^{9}}=\frac{3\times {10}^{-1}}{2.5}=0.12\text{\hspace{0.17em}}\text{m}$

The (AF) in this research is calculated using the normalized (AF) obtained in Equation (28)

$A{F}_{n}\left(\theta ,\varphi \right)=\left\{\frac{1}{M}\frac{\mathrm{sin}\left(\frac{M}{2}{\rho}_{x}\right)}{\mathrm{sin}\left(\frac{{\rho}_{x}}{2}\right)}\right\}\left\{\frac{1}{N}\frac{\mathrm{sin}\left(\frac{N}{2}{\rho}_{y}\right)}{\mathrm{sin}\left(\frac{{\rho}_{y}}{2}\right)}\right\}$

where,

${\rho}_{x}=k{a}_{x}\mathrm{sin}\theta \mathrm{cos}\varphi +{\omega}_{x}$

${\rho}_{y}=k{a}_{y}\mathrm{sin}\theta \mathrm{cos}\varphi +{\omega}_{y}$

For a beam pattern broadside

${d}_{x}={d}_{y}=\frac{\lambda}{2}$ , ${\omega}_{x}={\omega}_{y}=0$ and $M=N=4$

Let ${\rho}_{x}=kd\mathrm{cos}\theta +\omega ={\rho}_{y}$ for a uniform array and $M=N=4$ . ${\omega}_{x}={\omega}_{y}=0$ , $\theta ={45}^{\circ}$ .

5. Determination of the Directivity D_{x} and D_{y} of a Linear Array

The directivity of linear array antenna D, is determined by the formula,

$D=\frac{2{R}_{0}^{2}}{1+\left({R}_{0}^{2}-1\right)f\frac{\lambda}{L+a}}$ (42)

where, ${R}_{0}$ is the voltage ratio, L is the length of the linear array, f is the broadening factor of broad array and a is the distance between elements in the linear array [2] . Let ${R}_{0}=20$ voltage ratio, $L=4,a=0.47,f$ .

6. Determination of the Directivity D_{0} of Planar Array

Since the configuration of a planar array antenna is consists of two linear antennas placed orthogonally to each other, the directivity of a planar array antenna, ${D}_{0}$ can be obtained as [2] :

${D}_{0}={D}_{x}{D}_{y}\mathrm{cos}\theta $ (43)

where, ${D}_{x}$ = The directivity of a linear array along x-axis.

${D}_{y}$ = The directivity of a linear array along y-axis.

Table 1 is the result of the computation of varying the Inter-Element spacing

Table 1. Result of varying the inter-element spacing.

of the Planar Array of Monopole Antenna in both x and y-directions keeping progressive phase shift, theta/phi = 45, number of elements N = M = 4 constant throughout the variation. The table contain the values of number elements along both x and y-directions, progressive phase-shift and directivity of fields.

It also shows a decrement on the array factor, $A{F}_{n}{\left(\theta ,\varphi \right)}_{xy}$ as the inter-element spacing was increased, for instance, when the element spacing of 0.47λ was mathematically evaluated, the correspondent value of array factor is 0.9984 as the value of the list spacing considered in this evaluation. While 0.9924 is obtained as the array factor, $A{F}_{n}{\left(\theta ,\varphi \right)}_{xy}$ when 1.0λ spacing is considered as the widest distance between the elements. On the other hand, the numerical value of directivity increases as the inter-element spacing increases.

The below Figure 4(a) and Figure 4(b) are obtained from plotting Array

Figure 4. Graph of array factor against (a) directivity (b).

Factor against Directivity and inter-element spacing of the Table 1 above. The two graphs obtained show that as the inter-element spacing of the matrix array is increased, the Array Factor decreases and the Directivity of the radiated pattern of the field increases.

7. Radiation Patterns of Varying Element Spacing

Figure 5 below are the simulation results obtained both in 2D plots and polar plots of Array Factor against the elevation (azimuthal). It is the various radiation patterns obtained by varying the inter-element spacing along ${a}_{x}$ and ${a}_{y}$ axis of the matrix array as ${a}_{x}$ and ${a}_{y}$ are varied, while the number of elements, N and progressive phase shift, theta/phi = 45, number of elements N = M = 4 are kept constant.

a_{x}=a_{y}=0.47λ a_{x}=a_{y}=0.50λ a_{x}=a_{y}=0.55λ a_{x}=a_{y}=0.60λ a_{x}=a_{y}=0.65λ a_{x}=a_{y}=0.70λ a_{x}=a_{y}=0.75λ a_{x}=a_{y}=0.80λ a_{x}=a_{y}=0.85λ a_{x}=a_{y}=0.90λ a_{x}=a_{y}=0.95λ a_{x}=a_{y}=1.0λ

Figure 5. Radiation patterns of varying element spacing.

8. Discussion of Results

Figure 2 above shows the various simulated radiation patterns obtained by varying the inter-element spacing of the 4 × 4 matrix array. While the values of progressive phase shift and the number of elements N, is kept constant throughout the variation.

The results have a great effect on the array factor as it is clearly shown by radiation patterns of Figure 2 above. Element spacing of 0.47λ to 0.55λ has broad radiation pattern and occupied large bandwidth, which decrease with increase in the spacing. But from element spacing of 0.60λ, the radiation patterns shows an emergence of grating lobe whose size and level increases with the increase in ${a}_{x}$ and ${a}_{y}$ respectively. When element spacing equals 0.70λ, the size of the grating lobe equals that of the main lobe in opposite direction. Further increase to 0.75λ shows a beginning in the split of the grating lobe. Complete splitting in the grating lobe is eminence when ${a}_{x}$ is increased to 0.85λ, 0.90λ, 0.95λ and 1.0λ. Increasing the element spacing towards l results in an increased directivity and grating lobe effect with a maximum grating lobe amplitude equal to the main lobe magnitude at an element spacing of 1.0λ, see Figure 2.

9. Findings

After a thorough research and simulation of results as can be seen in Table 1 and in Figure 2 above, the following are my findings:

1) As inter-element spacing increases, the grating lobes increase in size, number and levels.

2) Increase in inter-element spacing also increases the directivity of the main lobe.

3) Increasing the spacing to 1.0λ, is a bad element spacing as more grating lobes are developed equal in size and level with the main lobe.

4) Therefore, the best directivity (radiation pattern) can only be obtained when the element spacing is within 0.1 - 0.5λ.

10. Conclusion

In this paper, a low profile planar array of monopole antenna has been presented. The purpose was to design a low level grating lobe with high directivity using planar array of monopole antenna by varying the inter-element spacing within the array matrix. A 4 × 4 matrix array of monopole antennas has been used in order to enhance the directivity and directivity as high as 39.67 dB has been achieved. The results of Table 1 and Figure 1 above show that directivity of the radiated field decreases as the inter-elements spacing is increased.

Acknowledgements

The idea of carrying out the present investigation originated from Prof. S. A. Adekole, FAS, who was the author’s M. Eng. Supervisor and Engr. S. Ojomu who is always ready with his mentorship.

Cite this paper

Omini, O. , Baasey, D. and Adekola, S. (2019) Impact of Element Spacing on the Radiation Pattern of Planar Array of Monopole Antenna.*Journal of Computer and Communications*, **7**, 36-51. doi: 10.4236/jcc.2019.710004.

Omini, O. , Baasey, D. and Adekola, S. (2019) Impact of Element Spacing on the Radiation Pattern of Planar Array of Monopole Antenna.

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