# Design of Power Dividers for Finite Dipole Subarrays with Complex Impedances

## Article information

## Abstract

In this study, we propose a method for designing a feeding network for all port-matched and high gain dipole subarray antennas excited by uniform current mode (UCM) by considering the mutual coupling effect. By analyzing the array impedance matrix, we first calculated the active input impedance and the required excitation power. Following this, we transformed and combined the complex impedances using the proposed power divider, thus preserving the required excitation power. The design process and results of this study are demonstrated stepwise through circuit simulation. Subsequently, we fabricated and evaluated the proposed subarrays. The impedance matching performance of the UCM was found to be better than that of the conventional subarray, with their array gains being 11.5 dBi and 9.8 dBi, respectively.

**Keywords:**Dipole Antenna; Feeding Network; Finite Array; Gain Maximization; Impedance Matching; Power Divider; Subarray

## I. 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.

## II. 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* × (*h*_{1} + *h*_{2} + *h*_{3}) in an infinitely periodic boundary, as shown in Fig. 1(a). This antenna comprised three layers (M1, M2, and M3) with *v** _{d}* = 0.48,

*v*

*= 0.2,*

_{r}*d*

*= 1.03,*

_{l}*d*

*= 0.8,*

_{w}*d*

*= 0.47,*

_{a}*d*

*= 0.1,*

_{g}*w*= 3.2,

*r*= 0.5,

*f*

*= 0.34,*

_{l}*f*

*= 0.2,*

_{r}*h*

_{1}= 1.57,

*h*

_{2}= 0.06,

*h*

_{3}= 0.25, and

*A*

*= 32 mm. Taconic TLY-5 dielectric boards of thickness*

_{w}*h*

_{1}and

*h*

_{3}(

*ɛ*

*= 2.2, tanδ = 0.0009) were utilized, with an adhesive of thickness*

_{r}*h*

_{2}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

*i*th port can be defined as [11]:

where *S*_{i}_{,}* _{n}* denotes the scattering parameter between the

*i*th and

*n*th 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 {*Z** _{a}*} and ARC {Γ

*} can be expressed as:*

_{a}where “./” denotes the element-wise division between two vectors, {*Z*_{0}} is a vector comprising the referenced port impedances, and {*Z** _{a}*} generally represents a complex vector, since [

*Z*] and {

*V*} are complex. Furthermore, the required incident power wave {

*a*} was determined from {

*Z*

_{0}}, 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), {*Z** _{a}*} was determined to be complex. In addition, {

*Z*

_{0}} in Eq. (5) was set to {

*Z*

*}*

_{a}*, 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:** {*Z** _{a}*} and {

*a*} were determined through UCM excitation. In Fig. 3, the complex {

*Z*

*} was transformed into real impedance R*

_{a}_{n}without perturbation of {

*a*}, while parallelly combining it with (R

*//R*

_{n}

_{n+}_{1}). As a result, R

*should be determined in accordance with the following rule:*

_{n}where *k*_{12} and *k*_{34} can be chosen as arbitrary real numbers, making R* _{n}* 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 (*Z** _{a}*)

*=(R*

_{n}*)*

_{a}*+*

_{n}*j*(X

*)*

_{a}*and 0 ≤*

_{n}*θ*

*< 180°. Generally, since*

_{n}*θ*

*was different at each port, the phase of the traveling signal {*

_{n}*a*} via the transformer should be different. However, due to the phase discontinuity that occurred at the boundary between (

*Z*

*)*

_{a}*and (*

_{n}*Z*

*)*

_{t}*, Δ*

_{n}*θ*

*partially canceled this difference, as follows:*

_{n}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 R_{1}//R_{2} and R_{3}//R_{4}. Similarly, the following transformers were designed using the following equations:

where R_{p} 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.

**Step 3:** Similarly, owing to the design rule in Eq. (10), {*a*} did not change. As mentioned previously, R_{p} is the final input impedance. Therefore, we set R_{p} at 50 Ω to match the final port impedance. The ARCs of all the transformed ports are shown in Fig. 3 (bottom right).

## III. 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.

## IV. 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.

## Acknowledgments

This work was supported by the Institute of Information & Communications Technology Planning & Evaluation (IITP) grant funded by the Korean government (MSIT) (No. 2020-0-00858, Millimeter-wave Metasurface-based Dual-band Beamforming Antenna-on-Package Technology for 5G Smartphone).

## References

## Biography

Seongjung Kim, https://orcid.org/0000-0001-5189-8023 received his B.S. degree in electronic and electrical engineering from Hongik University, Seoul, South Korea, in 2017. He received his M.S. and Ph.D. degrees in electrical engineering and computer science from Seoul National University, Seoul, South Korea, in 2023. Since 2023, he has been a staff engineer in the Department of System Large Scale Integration, Samsung Electronics, Hwaseong, South Korea. His main research interests are phased array antenna theory and design.

Sangwook Nam, https://orcid.org/0000-0003-3598-1497 received his B.S. degree in electrical engineering from Seoul National University, Seoul, Korea, in 1981. In 1983, he received his M.S. degree in electrical engineering from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea. He completed his Ph.D. in electrical engineering from the University of Texas at Austin, Austin, TX, USA, in 1989. From 1983 to 1986, he worked as a researcher at the Gold Star Central Research Laboratory in Seoul, South Korea. Since 1990, he has been a professor at the School of Electrical Engineering and Computer Science, Seoul National University. His research interests include the analysis and design of electromagnetic structures, antennas, and microwave active/passive circuits.