Introduction
Phased array antennas have garnered significant attention for various applications, such as wireless communication systems and modern radars, that require high bandwidth, high gain, and wide coverage [1‒4]. One of the most interesting observations regarding phased arrays is impedance matching in finite arrays, especially when the mutual coupling between its elements is strong. In this regard, the characteristic mode (CM), which accounts for mutual coupling effects, is considered a suitable impedance matching technique [5]. It solves the eigenvalue problem of the mutual-impedance matrix of finite arrays, wherein each element requires a different port impedance and incident power. However, the CM analysis approach focuses only on impedance matching, not on designing a directive beam shape.
Subarray techniques are often used to decrease the array antenna implementation cost of phased array antennas by reducing the number of active ports and TR modules [6, 7]. However, it may have strong mutual coupling between elements to reduce the area of the subarray. Usually, the power divider used in conventional subarrays [8, 9] is designed to excite each element uniformly, assuming that each element has the same active input impedance. In this study, conventional subarrays using power dividers were called to be excited by the uniform power divider mode (UPDM). Notably, mutual coupling between the elements of a finite array led to differences in the active input impedance of each element. As a result, UPDM failed to produce uniform incident power for each element, resulting in bad impedance matching and aperture efficiency.
Therefore, in this work, we propose a methodology to excite the uniform current mode (UCM) on the array aperture using a power divider to yield maximum array gain. Furthermore, we verify whether this method simultaneously matches the impedances of all antenna elements.
Design of UCM-Excited Subarrays
1. Basic Theory of the Array
Using Computer Simulation Technology (CST)’s full-wave simulator, we designed a dipole antenna unit cell (UC) structure of size w × w × (h1 + h2 + h3) in an infinitely periodic boundary, as shown in Fig. 1(a). This antenna comprised three layers (M1, M2, and M3) with vd = 0.48, vr = 0.2, dl = 1.03, dw = 0.8, da = 0.47, dg = 0.1, w = 3.2, r = 0.5, fl = 0.34, fr = 0.2, h1 = 1.57, h2 = 0.06, h3 = 0.25, and Aw = 32 mm. Taconic TLY-5 dielectric boards of thickness h1 and h3 (ɛr = 2.2, tanδ = 0.0009) were utilized, with an adhesive of thickness h2 applied between the boards. To prevent the generation of the common mode, two shorting vias were drawn from M1 to M2 (ground plane) [10]. The left feeding via went through M1 to M3 and was terminated by a discrete port with 50 Ω. As shown in Fig. 1(d), the active reflection coefficient (ARC) of the UC at the discrete port was under −10 dB across 20–25 GHz. When uniformly exciting the array for broadside radiation, the ARC at ith port can be defined as [11]:
where Si,n denotes the scattering parameter between the ith and nth ports and N is the total number of elements.
For the UC, N is considered an infinite number. Fig. 1(b) and 1(c) show a 4 × 4 finite dipole array comprising the UCs. Notably, the ARC at the discrete ports (blue circles) was significantly different from that of the infinite UC structure. Additionally, since the ARCs differed based on the element position, the current distribution on the finite array aperture was not uniform (rather irregular).
For the N-port network of an array antenna, an N × N impedance matrix [Z], induced port voltage {V}, and current {I} can be related as follows:
where [Z] = [R] + j[X], with [R] and [X] being the real matrices. Note that these parameters indicate the functions of the operating frequency, with the vector quantities and matrices denoted by { } and [ ], respectively. Notably, since this equation denotes a quantity that already accounts for the mutual coupling effect between antenna elements, the active input impedance {Za} and ARC {Γa} can be expressed as:
where “./” denotes the element-wise division between two vectors, {Z0} is a vector comprising the referenced port impedances, and {Za} generally represents a complex vector, since [Z] and {V} are complex. Furthermore, the required incident power wave {a} was determined from {Z0}, as follows [12]:
where “×” denotes the element-wise multiplication between two vectors.
2. Design of a Power Divider for UCM Excitation
For UCM excitation, {I} was set to {1}, which is a vector comprising only one element. In other words, the current distribution on the array aperture was set to be uniform to achieve maximum array gain. Subsequently, using Eqs. (2) and (3), {Za} was determined to be complex. In addition, {Z0} in Eq. (5) was set to {Za}*, where “*” denotes conjugation. Thus, the antennas and reference ports were conjugately matched to ensure maximum power transfer [12]. When the finite array was excited by the UCM, it ensured that all ports (1–16) matched simultaneously at the center frequency (22.5 GHz), as shown in Fig. 2.
Step 1: {Za} and {a} were determined through UCM excitation. In Fig. 3, the complex {Za} was transformed into real impedance Rn without perturbation of {a}, while parallelly combining it with (Rn//Rn+1). As a result, Rn should be determined in accordance with the following rule:
where k12 and k34 can be chosen as arbitrary real numbers, making Rn a reasonable value. Furthermore, the electrical length and characteristic impedance of the transformer were uniquely determined after transforming the complex impedance into real impedance [13] using the following equations:
where (Za)n=(Ra)n+j(Xa)n and 0 ≤ θn < 180°. Generally, since θn was different at each port, the phase of the traveling signal {a} via the transformer should be different. However, due to the phase discontinuity that occurred at the boundary between (Za)n and (Zt)n, Δθn partially canceled this difference, as follows:
The remaining phase difference was approximately neglected. Fig. 3 (bottom left) depicts the ARCs of all the transformed ports, obtained using an Advanced Design System (ADS) circuit simulator with the extracted impedance matrix [Z].
Step 2: Owing to the design rule noted in Eq. (6), {a} did not change despite the parallel combination of R1//R2 and R3//R4. Similarly, the following transformers were designed using the following equations:
where Rp is an arbitrary constant denoting the final input impedance of 50 Ω. Subsequently, the characteristic impedance of the transformer was given by:
The ARCs of all transformed ports are presented in Fig. 3 (bottom middle), showing that they were well preserved at the center frequency.
Simulated and Experimental Verification of Performance of the Two Types of 2 × 2 Subarrays
Fig. 4 depicts the simulated structure of the power dividers and arrays for the two cases. The UPDM structure comprised a general power divider for exciting the conventional subarrays [8, 9]. This power divider contained only quarter-wave transformers (70.7 Ω), which transformed 50 Ω to 100 Ω. In other words, all elements were assumed to have an active input impedance of 50 Ω, and each transformed 100 Ω was combined parallel to 50 Ω. In the second case pertaining to the UCM, the power divider comprised transformers featuring individual lengths and characteristic impedances (line widths) to match complex active input impedances, with only the second-stage transformers exhibiting quarter wavelengths.
Since the unit cell size was doubled by a subarray, mutual coupling between the E-plane elements, which exhibited the strongest coupling, declined from −9 dB to −13 dB, while those for the others (H- and D-planes) were below −20 dB. Furthermore, the power dividers of ports 1, 2, 3, and 4 were asymmetric to those of ports 5, 6, 7, and 8. However, this asymmetricity did not affect the design process of the power dividers since it was based on the entire structure in itself, [Z]16×16, which did not assume any symmetricity. Finally, the results for {I} = {1} and simultaneous impedance matching for gain improvement were obtained.
In this context, it should be noted that the UPDM could not provide uniform incident power to each element, meaning that its current distribution was irregular, as shown in Fig. 4 (bottom). This is also evident in Table 1 (left), in which the current values correspond to the antenna position in Fig. 4 (bottom). The values noted in the table were calculated by conducting contour integration of the surface current on the antenna feeding via. In contrast, the UCM generated a nearly uniform current distribution ({I} = {1}) on the array aperture. Therefore, it is evident that UCM can potentially provide higher array directivity than UPDM.
The three fabricated case prototypes are shown in Fig. 5(a). Although their front views appear to be similar, their back views are different. Notably, 2.92-mm vertical launch connectors operating at up to 40 GHz were used. The connectors and the feeding lines connecting to the power divider were calibrated using the method (using direct and extended thru lines) [14]. The radiation patterns were measured in an anechoic chamber, which comprised a reflector to secure the far-field distance between the horn antenna and the antenna under test (AUT), as shown in Fig. 5(b).
In Fig. 6, the simulated (Fig. 4) and measured (Fig. 5) results are shown and compared. It is observed that the measured UCM shifted by 1 GHz from the full-wave simulated result, possibly due to the fabricated dimension (length and size) error. Barring this shift, the measured result is found to be in good agreement with the simulated result. As expected, the ARC of the UCM improved across the wideband compared to that of the UPDM. In other words, the simulated matching efficiency increased by 0.53 dB, while the radiation efficiency remained almost similar. Moreover, due to the uniformity of the current distribution, as shown in Fig. 4 and Table 1, the simulated directivity of the UCM improved by 1.9 dB to reach 12.3 dBi. As a result, the simulated broadside gain of the UCM, which refers to directivity + matching efficiency + radiation efficiency in the dB scale, also improved significantly by 2.43 dB, attaining 11.5 dBi at 22.5 GHz. Furthermore, the measured gain increased from 9.8 dBi to 11.5 dBi at resonant frequencies of 22.5 GHz and 23.5 GHz, respectively. The simulated and measured radiation patterns are shown in Fig. 7. Notably, the measured backlobes differ from the simulated ones because of disturbances in the backside radiation caused by the positioner, as shown in Fig. 5(b). Apart from this, the measured beam patterns seem to be in good agreement with the simulated patterns.
Conclusion
In this study, we propose a novel methodology for designing a feeding network to excite all port-matched UCM by considering the mutual coupling effect. Port matching was confirmed in a stepwise manner. Furthermore, subarrays with a feeding network were realized through full-wave simulation, which demonstrated uniform current distribution for the UCM, leading to a significantly higher broadside gain achieved by the UCM than the UPDM (it improved by 1.7 dBi). The proposed method with full impedance matching can potentially be applied and extended for implementation in various synthesized arrays. In addition, due to its high gain and low-profile characteristics, it can be expected to significantly benefit high-resolution radars, satellite communications, and point-to-point wireless data transfer applications.
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