II. ARRAY STRUCTURE

A circular antenna-array of three rectangular patch antennas, equiangular to each other (Fig. 1), is the primary structure of the proposed design. The patch antennas possess directional radiation pattern in azimuthal plane [Reference Waterhouse12] thus these are chosen for illustration. The radius of circular antenna-array (say r) does not contribute to any significant difference (if frequency compensates for it), thus the three patches can be mounted on a pyramidal body as long as they are placed in the configuration mentioned in Fig. 1. It increases the ease of installation and volumetric efficiency [Reference Waterhouse12]. For cylindrical body such a configuration is used in [Reference Ioannides and Balanis13] but it has different objectives and working principles.

During simulation the three rectangular antenna-array elements are standing perpendicular to XY-plane (in Cartesian coordinate system) at a distance r from the origin, such that lower- and upper halves of each of the element should lye below and above the XY-plane, respectively. All three perpendiculars from three patch elements, each passing through intersect-points of respective two-diagonals of each of the elements, lye in XY-plane, and makes 2π/3 radian angle with each other. One of the elements is parallel to YZ-plane and X-axis pass through intersect-point of two-diagonals of this element.

III. BEAM ROTATION MECHANISM

Consider three phasors oriented in same plane, making 2π/3 angle from each other. The observation is, when their amplitudes is made to vary sinusoidally, with a 2π/3 phase difference, the resultant vector rotates with the angular frequency of the sinusoid. For the purpose, one may take the rotation vector as:

(1)$$\eqalign{& A=\underbrace {\cos \lpar 2\pi \, f\, t\rpar }_{{\rm Cosine}\, \, {\rm at}\, \, 0^\circ \, \, {\rm phase}}\underbrace {e^0 }_{{\rm At}\, \, 0^\circ \, \, {\rm angle}}+\underbrace {\cos \lpar 2\pi \, f\, t+2\pi /3\rpar }_{{\rm Cosine}\, \, {\rm at}\, \, 120^\circ \, \, {\rm phase}}\underbrace {e^{j2\pi /3} }_{{\rm At}\, \, 120^\circ \, \, {\rm angle}} \cr & \quad +\underbrace {\cos \lpar 2\pi \, f\, t+4\pi /3\rpar }_{{\rm Cosine}\, \, {\rm at}\, \, 240^\circ \, \, {\rm phase}}\underbrace {e^{j4\pi /3} }_{{\rm At}\, \, 240^\circ \, \, {\rm angle}}\comma \; }$$

where f is frequency of transmission, and t is time variable. The same observation is applied to the considered structure (Fig. 1), where the three patches oriented in 2π/3 angle from each other are excited with three sinusoids of 2π/3 phase shifted with respect to each other. The patch antenna may be used as the array-element because of its ease of fabrication and high volumetric efficiency [Reference Waterhouse12]. As, the radiation pattern from (ideal) the patch cannot be behind the ground plane, the cosine terms in (1) may be replaced with modulus of cosine, to get:

(2)$$\eqalign{& A = \left\vert {\cos \lpar 2\pi \, f\, t\rpar } \right\vert {e^0} + \left\vert {\cos \lpar 2\pi \, f\, t + 2\pi /3\rpar } \right\vert {e^{j2\pi /3}} \cr & \quad + \left\vert {\cos \lpar 2\pi \, f\, t + 4\pi /3\rpar } \right\vert {e^{j4\pi /3}}.}$$

A better analysis of the structure can be done by studying the AF of the array. The AF of circular array is given as [Reference Balanis6]:

(3)$$AF\lpar \theta \comma \; \, \varphi \rpar = \sum\limits_{n = 1}^N {{a_n}{e^{jkr\sin \theta \cos \lpar \varphi - {\varphi _n}\rpar }}}\comma$$

where θ and φ are spherical coordinates; φ_n is the angular position of nth element on XY plane; a _n is the excitation coefficient of nth array-element; k is the wave-number; r is the radius of circular array pattern; and the phase difference between two consecutive array-elements φ_n = 2πn/N, N being number of array-elements.

In order to implement the present case of a circular array of three array-elements, one may take N = 3, φ_n = 2πn/3 (where n = 1,2,3), and the excitation coefficients: a ₁ = e ^jωt, a ₂ = e ^j(ωt+2π/3), and a ₃ = e ^{j(ωt+4π/3−α)} (ω and α being angular frequency of the beam-rotation and correction parameter for getting circular rotation, respectively), therefore, in the azimuthal plane (θ = π/2), one may write (3) as:

(4)$$\eqalign{& AF\left({\displaystyle{\pi \over 2}\comma \; {\mkern 1mu} \varphi } \right)= {e^{j\omega t}}\left[{{e^{jkr\cos \lpar \varphi - 2\pi /3\rpar }}} \right. + {e^{j\lpar kr\cos \lpar \varphi - 4\pi /3\rpar + 2\pi /3\rpar }} \cr & \quad \left. {+ {e^{j\lpar kr\cos \lpar \varphi - 2\pi \rpar + 4\pi /3 - \alpha \rpar }}} \right].}$$

Under the initial boundary condition φ = 0 and α = 3kr/2, which makes the contribution of each array-element 2π/3 phase shifted with respect to other two, one may write (4) as:

(5)$$AF\left({\displaystyle{\pi \over 2}\comma \; \, 0 } \right)= {e^{j\lpar \omega t - kr/2\rpar }}\left[{1 + {e^{j2\pi /3}} + {e^{j4\pi /3}}} \right].$$

The equation (5) shows that the AF has three constituent phasors of equal magnitude but having a phase difference of 2π/3 with each other. Thus, the proposed assembly should produce a circularly rotating instantaneous electric field having a rotational frequency equal to the frequency of feed signal. That would make the radiation pattern of the described array omni-directional over the horizontal plane. The radius of the array governs the phase difference between the observed radiation pattern and the feed signal. Although discussing array structure, it is assumed that the radius of the circular antenna-array does not contribute to any significant difference; very specifically, whatever difference it creates is reflected equally on all the array elements. Therefore, it only changes the amplitude of the pattern, speed of instantaneous E-field rotation (3) and the overall phase (4).

The parameter α marks the difference between a perfect circular pattern and the pattern obtained by the array, and does not have any major role as its effect become negligible after incorporation of the frequency reduction mechanism (Section IV), and compel the entire pattern to rotate. Therefore, it may be ignored for further discussion. In order to present a qualitative view of the shape and rotation of the pattern, one may plot AF using (4). Since the elements chosen are directional, the radiation pattern will be more directional than the obtained plot (Fig. 2).

Fig. 2.

AF patterns obtained at various instant: (i) ωt = 2nπ (solid curve), (ii) ωt = 2nπ + 2π/3 (dotted curve), and ωt = 2nπ + 4π/3 (dashed curve), using (4) with α = 0.

The wireless devices are, usually, designed to work at GHz frequencies and, hence, the speed of rotation is several billion rotations per second. Since the frequency of beam-rotation achieved using (4) and (5) is same as the frequency of operation of antenna (usually, GHz), i.e. ~109 rotations per second. Although observing for the radiation pattern of the array structure, the power density is calculated by taking the time average of Poynting vector over the complete cycle, as [Reference Balanis6, Reference Waterhouse12–Reference Pozar14]: W _av(x, y, z) = [W _av(x, y, z, t)]_av = (1/2)Re[E × H*], where E and H are the electric and magnetic field intensities. Now, the aim is to reduce the frequency of rotation. A technique for the same is discussed in Section IV.

For typically chosen design frequency 2.4 GHz, the typical size of rectangular antenna-element is 49.4 mm × 41.4 mm. The thickness and the dielectric constant of the substrate are taken as 1.5 mm and 2.2, respectively. The basic parameters are tabulated [Reference Ali, Kamarudin, Md Tan and Rahman15] (Table 1). The radius of bends is taken as 0.65 mm. The obtained result shows a return loss as low as −54.6 dB and other S-parameters (in dB) are tabulated (Table 2). The phase shift obtained for S ₁₁ is −22°, and for S ₂₁, S ₃₁, and S ₄₁ is −130°. The design is implemented with 50 Ω characteristic impedance, so that it complies with the standards. The other parameters are calculated using [Reference Pozar14] and the radiation pattern in horizontal plane is plotted using simulator (Fig. 3).

Fig. 3.

Radiation pattern (in dB) in horizontal plane, obtained using FEKO EM simulator.

Table 1.

Parameters for Wilkinson power divider, used in FEKO EM simulator.

Table 2.

S-parameters (S _pq) (in dB) as obtained using FEKO EM simulator.

IV. REDUCTION IN ANGULAR FREQUENCY

The signal to be transmitted is split into three equal parts as described (Section III) and attenuated to the ratio of a₁, a₂, and a₃ and fed to each of the patches (Fig. 4). The parameters a₁, a₂, and a₃ are varied sinusoidally. (In a practical case, for this purpose, a PIN diode may be used as voltage controlled variable attenuator, for that the voltage may be generated from a microprocessor to be fed to a PIN diode to vary the attenuation sinusoidally. However, the insertion loss needs to be tightly controlled otherwise it may drastically affect the system performance.) The frequency of the variation is kept same as the desired frequency of rotation. The sinusoids are 2π/3 phase separated with respect to themselves. This amplitude modulates the signal, envelop being the sinusoid of desired frequency of rotation (Fig. 5). This makes the average amplitude of excitation to antenna vary as desired with low frequency and, hence, the pattern rotates with the desired frequency.

Fig. 4.

Functional block diagram for the final design for reducing the frequency of rotation.

Fig. 5.

Selection and variation of the excitation coefficients a₁, a₂, and a₃, the magnitudes of excitation of each array.

Under the considered condition the AF may be written as:

(6)$$\eqalign{& AF\left({\displaystyle{\pi \over 2}\comma \; \, \varphi } \right)= \cos \lpar \omega \, t\rpar \, {e^{j\lpar \cos \varphi + 2\pi f{\kern 1pt} t\rpar }} + \cos \left({\omega \, t + 2\pi /3} \right)\, \cr & \quad {e^{j\left[{\cos \left({\varphi - 2\pi /3} \right)+ 2\pi f{\kern 1pt} t} \right]}} + \cos \left({\omega \, t + 4\pi /3} \right)\, {e^{j\left[{\cos \left({\varphi - 4\pi /3} \right)+ 2\pi f{\kern 1pt} t} \right]}}.}$$

As kr(=2πr/λ, λ being wavelength) is governed by the ratio r/λ, and introduces an additional phase shift. Here, for the simple qualitative analysis, kr is taken as 1 in (6), which will not introduce any additional phase shift.The time averaged AF over time-period of 1/f is:

(7)$$\eqalign{\left\langle {AF\left({\displaystyle{\pi \over 2}\comma \; \varphi } \right)} \right\rangle & = f\left[{\vint_0^{1/f} {\cos \lpar \omega \, t\rpar \, {e^{j\lpar \cos \varphi + 2\pi f{\kern 1pt} t\rpar }}} dt} \right. + \cr & \quad \vint_0^{1/f} {\cos \left({\omega \, t + 2\pi /3} \right)\, {e^{j\left[{\cos \left({\varphi - 2\pi /3} \right)+ 2\pi f{\kern 1pt} t} \right]}}} dt \cr & \left. { + \vint_0^{1/f} {\cos \left({\omega \, t + 4\pi /3} \right)\, {e^{j\left[{\cos \left({\varphi - 4\pi /3} \right)+ 2\pi f{\kern 1pt} t} \right]}}} dt} \right]}$$

Since, ω ≪ 2πf, cos(ωt) is nearly constant in the period of e ^j2πft. Applying this to (7) and a little mathematical manipulation will result in:

(8)$$\eqalign{& \big\langle {AF\bigg({\displaystyle{\pi \over 2}\comma \; \, \varphi } \bigg)} \big\rangle \, = K\big[{\cos \lpar \omega \, t\rpar \, {e^{j\cos \varphi }}} \cr & \quad + \cos \big({\omega \, t + 2\pi /3} \big)\, {e^{j\cos \big({\varphi - 2\pi /3} \big)}} \cr & \quad{ + \cos \big({\omega \, t + 4\pi /3} \big)\, {e^{j\cos \big({\varphi - 4\pi /3} \big)}}} \big]\comma \; }$$

where K is a complex number, which will add to phase difference. The obtained (8) is similar to (1), which corresponds to a rotating field with angular frequency of ω instead of 2πf, which may be used to plot the AF pattern (Fig. 6(a)).

Since the ratio of excitation coefficients, a₁:a₂:a₃ governs the direction of beam, fixing the values of the excitation coefficients a₁, a₂, and a₃ can make the beam stop at a particular angle. In the case of stopping the beam at a particular angle, by fixing the values of the excitation coefficients, the corresponding phase shift may be given and there is no more sinusoidal variation of these coefficients. The variation in the values of excitation coefficients can be achieved with the use of voltage controlled variable RF attenuator circuit, which can be easily realized using PIN diodes [Reference Zhuo, Yan and Yu16]. This low frequency signal can be generated by a digital device making the device programed for different frequencies and different directions. This makes the device highly versatile to be used in large number of applications. Many protocols can be developed to use the devices installed with this mechanism to make it power efficient and to maximize throughput of closely spaced networks. The same mechanism is tested on FEKO EM simulator. Since, the radiation pattern of dipole in horizontal plane is omni-directional, one may model the element of the array as a dipole antenna to study AF of the array, and the radiation patterns are plotted for five different combinations of the excitation coefficients a1, a2, and a3 by sampling the sinusoids five times in a complete half cycle (Fig. 6(b)).

It would be of interest to see a comparison between the results obtained using analytical equation and that using FEKO EM simulator. For a case study, one may consider an element of the array to be a half wavelength dipole antenna, the radiation pattern of which may be written as:

$${E_\theta } = j\displaystyle{{{I_0}{e^{ - jk\, r}}} \over {2\pi \, r\sin \theta }}\left[{\cos \left({\displaystyle{{k\lambda } \over 4}\cos \theta } \right)- \cos \left({\displaystyle{{k\lambda } \over 4}} \right)} \right].$$

One can note that the radiation pattern is independent of φ, which makes it omni-directional. Therefore, the radiation pattern of the array in the azimuthal plane (θ = π/2) becomes:

$$\eqalign{& {E_\theta }\, \lpar \theta = \pi /2\rpar = jK\displaystyle{{{I_0}{e^{ - jk\, r}}} \over {2\pi \, r}}\left[{\cos \lpar \omega \, t\rpar \, {e^{j\cos \varphi }}} \right. + \cr & \quad \cos \lpar \omega \, t + 2\pi /3\rpar \, {e^{j\lpar \cos \varphi - 2\pi /3\rpar }}\left. { + \cos \lpar \omega \, t + 4\pi /3\rpar \, {e^{j\lpar \cos \varphi - 4\pi /3\rpar }}} \right].}$$

The additional terms get absorbed in K, and contribute uniformly to amplification and phase shift. This transformation only sets an initial offset to the rotation given by arg(K′), where K′ = jKI ₀e ^−jkr/2πr. Normalizing the radiation pattern for qualitative overview and plotting generate Fig. 6. One may pick a curve for AF pattern obtained at the instant 2π/3ω from Fig. 6(a) and the corresponding result obtained using FEKO EM simulator from Fig. 6(b) to present the comparison of the two methods (Fig. 7). The deviation of presented analytical result with that using FEKO EM simulator (Fig. 7) may be understood due to the mutual impedance effect ignored in analytical equation, which was discussed for uniform circular arrays in [Reference Ioannides and Balanis13].

Fig. 6.

AF patterns obtained (a) using (8) at various instant: (i) 2π/3ω (dotted curve), (ii) 4π/3ω (solid curve), and 2π/ω (dashed curve); and (b) using FEKO EM simulator for different combinations of the excitation coefficients a₁, a₂, and a₃.

Fig. 7.

A comparative AF pattern obtained at the instant 2π/3ω using analytical equation (dotted curve) and the corresponding result obtained using FEKO EM simulator (solid curve).

One may think for a circular antenna-array of rectangular microstrip antennas with number of elements more than three (N > 3), each fed with phase difference of 2π/N between two consecutive array elements, however, increasing number of elements will need more space as the size of each element is assumed to be same which might be impracticable [Reference Ahmed and Ali17]. The proposed design can be easily extended for very directive antennas simply by adopting a suitable geometrical array structure. It will still be better than the presently used smart antennas that employ tens of elements and switch each of them for different phase in order to achieve relatively small directional flexibility but accommodating large structured antenna on mobile devices may not be practicable. The paper uses the array factor based formulation for circular antenna-array.