Compact, Ultra-Wideband Butler Matrix Beamformers for the Advanced 5G Band FR3—Part I

Abstract

Butler Matrix networks are well established as beamforming networks for phased antenna arrays. The challenge we address in this work is to cover the entire (advanced 5G or 6G) FR3 band (7–24 GHz) with a single network, while retaining low losses and minimal size. The employed multilayer topology is also well established; however, the matching between the utilized hybrid couplers and the phase shifters constitutes a major challenge for such a wideband operation. This is achieved herein by employing meander lines with appropriate curvature and introducing two distinct design methods for the Butler Matrix. The first method focuses on designing individual components separately, followed by their integration into the overall Butler Matrix structure. This approach is demonstrated through the design, prototyping, measurements, and validation of an 8 × 8 Butler Matrix beamformer, which operates across the 6–16 GHz band (FR3 Low). The second method introduces a wideband-matching technique which simplifies the implementation process by designing the Butler Matrix as a single, unified structure. This technique is applied to both 4 × 4 and 8 × 8 Butler Matrices, which are implemented and simulated for the low FR3 band. Both design methods result in wideband operation and compact size and meet the desired performance criteria.

1. Introduction—FR3 Use for eMMB via Analog Beamformers

It is anticipated that the mid-band spectrum FR3 (spectrum between FR1 and FR2) will be the Urban Extreme Capacity for 6G [1,2]. A simple examination of the 5G occupied bands shows that the FR3 band is the only one left for enhanced Mobile Broadband (eMMB) use, and, thus, it has an enormous potential in terms of coverage and capacity. This frequency band facilitates co-deployment, enabling 5G, advanced 5G, and 6G cellular infrastructure to operate at the same cell sites, thereby minimizing deployment costs to practically zero. While the eMBB demands significant increase in capacity, throughput, and spectral efficiency, massive multiple input–multiple output (MIMO) or GIGA MIMO will be the key enabling technology for the realization of advanced 5G and 6G wireless communication [3,4]. Massive MIMO exploits the advantages of beamforming technologies to enhance the efficiency of wireless communication. To fulfill the requirements of modern wireless systems, the deployment of multibeam antenna arrays, employing either passive beamforming with fixed beams or active beamforming with flexible beams, emerges as the optimal solution. Thus, the essential part of multibeam antenna arrays is the beamforming network (BFN), which, in general, is classified into digital BFN [5], analogue BFN [6], and the combination of the two as a hybrid BFN [7]. Analogue BFNs, compared to digital BFNs, are cost effective and more energy-efficient and preferable in large-scale applications due to their simplicity [8].

Popular analogue BFNs or RF-type BFNs are Blass [9], Nolen [10], and Butler Matrix (BM) [11]. BMs are the most widely used BFNs for feeding antenna arrays due to their easy fabrication, low cost, and low profile [12], and they may offer ultra-wideband (UWB) characteristics given certain topologies.

Figure 1 depicts an example of a 3D beamforming topology which could increase the capacity and efficiency of a system, supporting both transmit and receive functions with narrow beams in both azimuth and elevation angles. The analog three-dimensional beamformer is built using a stack of vertical BMs driven by either a single horizontal BM or a stack of horizontal BMs. Ultra-wideband planar-printed Vivaldi antenna elements form the planar array. It is clear from the above discussion that to deliver a fully functional and efficient BFN for the currently anticipated 6G-enhanced mobile broadband use case, we need to develop an UWB system covering the 7–24 GHz band. Moreover, the RF-BFN, the BM in our case, should be of compact size and have UWB characteristics providing a fixed radiation pattern through amplitude and phase accuracy. Thus, the miniaturization of the BM, while preserving or achieving the needed performance, is the most critical point for the realization of a multibeam antenna array similar to that of Figure 1.

Numerous miniaturization efforts for the BM are reported in the literature and reviewed next. The recent advancements in 4 × 4 BM are summarized in

Table 1. Performance summary of state-of-the-art multilayer 4 × 4 BM. Acronyms are denoted as follows: OB, operating band; NS, normalized size; FA, footprint area; IL_Av ± ΔA, average insertion loss with amplitude deviation; PA, phase accuracy.

Reference	Technology	OB (GHz)	f0 (GHz)	FBW (%)	Size (mm2)	NS (λ)	FA (λ2)	ILAv ± ΔA (dB)	PA (°)
[20]	Microstrip to Slot	27–31	29	13.8		3.49 × 2.2	7.67	−9 ± 1.4	±8
[21]	LTCC	26–30	28	14	22.02 × 5.93	2.05 × 0.55	1.12	−9 ± 3.5	±9
[22]	GCPW	23.75–31	27	26.5	19.2 × 29.84	1.8 × 2.8	5.04	−9 ± 4	-
[23]	FWDC	1.9–3.1	2.5	47	81.65 × 71.56	0.68 × 0.59	0.4	−7 ± 3	±6
[24]	Microstrip to Slot	2–3	2.45	40	100 × 120	0.8 × 0.98	0.78	−7 ± 3	±7
This Work	Microstrip to Slot	6–16	9.8	102	39.4 × 17.9	1.28 × 0.58	0.74	−7 ± 2	±9

, and those of 8 × 8 BM are depicted in

Table 2. Performance summary of state-of-the-art multilayer 8 × 8 BM. Acronyms are denoted as follows: OB, operating band; NS, normalized size; FA, footprint area; IL_Av ± ΔA, average insertion loss with amplitude deviation; PA, phase accuracy.

Reference	Technology	OB (GHz)	f0 (GHz)	FBW (%)	Size (mm2)	NS (λ)	FA (λ2)	ILAv ± ΔA (dB)	PA (°)
[25]	Slotline Transition—Magit T	7–13	10	85	140 × 97	4.7 × 3.2	15.04	−12 ± 2.5	±20
[26]	Microstrip	3.6–4.7	4.2	26	160 × 160	2.1 × 2.1	4.4	−12 ± 3	±15
[27]	Stripline	2.3–2.5	2.4	5.5	250 × 160	2 × 1.28	2.56	−12 ± 1.9	±12
[28]	Stripline		3	33	170 × 145	1.7 × 1.45	2.46	±0.5	±10
[29]	Stripline	2.5–3.5	3	33	130 × 100	1.3 × 1	1.3	±0.45	±7.5
This Work	Microstrip to Slot	6–16	9.8	92	58.46 × 90 (39.9 × 53.52)	1.91 × 2.94 (1.3 × 1.74)	5.6 (2.26)	−12 ± 3	±21.5
This Work	Microstrip to Slot	6–16	9.8	92	40.1 × 84.26 (29.55 × 53.6)	1.3 × 2.75 (0.96 × 1.75)	3.57 (1.68)	−12 ± 3	±21

. These approaches are grouped into single-layer (SL) [13,14,15,16,17,18,19] and multilayer (ML) [20,21,22,23,24,25,26,27,28,29] configurations. In [13,14], an artificial transmission-line branch-line coupler (BLC) and a swapped ports-coupled line coupler, respectively, were utilized to achieve a reduction in the size of the BM from 21% to 10% (0.11 λ²–0.05 λ²) when compared to the conventional BM, while maintaining a fractional bandwidth (FBW) of 30%. In [15,16,17], elimination of the phase shifter and crossovers approach is enforced, using 45° and 90° couplers only, to build the BM. Therein, an improvement in the FBW is reported (45–63%) with miniaturized sizes ranging from 1.41 ${\mathsf{\lambda }}_0^2$ to 1.22 ${\mathsf{\lambda }}_0^2$. While References [13,14,15,16,17] refer to 4 × 4 BM, in [18], a conventional 8 × 8 BLC BM printed in glass substrate was presented, with a narrow FWB of 2.6% and a large size of 33.5 ${\mathsf{\lambda }}_0^2$. Nevertheless, in [19], two asymmetrical 4 × 4 BMs consisting of BLCs were employed to form an 8 × 8 BM with a reduced size of 4.14 ${\mathsf{\lambda }}_0^2$ and an FBW equal to 24%.

Regarding multilayer configurations, a folded 4 × 4 BM with eliminated crossover couplers was newly proposed in [20]. Stacking half components of the BM on different layers, while combining them with a multilayer transition scheme, achieved a Footprint Area (FA) of 7.67 ${\mathsf{\lambda }}_0^2$ and an FBW of 14%. A similar FBW was achieved in [21], where the components of a conventional 4 × 4 BM were stacked up in four different layers using the low-temperature co-fired ceramics (LTCCs) process, thus achieving a significantly reduced size, with an FA of 1.12 ${\mathsf{\lambda }}_0^2$. An increase in the FBW to 26.5% was reported in [22], adopting the grounded coplanar waveguide (GCPW) technology. Using bent hybrid couplers, a curved-line phase shifter, and multilayer transitioned crossovers, a size with an FA of 5.04 ${\mathsf{\lambda }}_0^2$ is obtained. A good compromise between size and BW is reported in [23,24]. In [23], a BM composed of forward-wave directional couplers (FWDCs) combined with microstrip-line phase shifters and via crossovers is proposed, resulting in a compact size of 1.19 ${\mathsf{\lambda }}_0^2$ and an FBW of 47%. In [24], dual transmission-line couplers and circular patched phase shifters combined with elliptic and circular slots, respectively, resulted in a compact BM with an FA of 0.78 ${\mathsf{\lambda }}_0^2$ and FBW of 40%. Concerning 8 × 8 BM designs, slot-line transition and magic tees were utilized in [25], where an 8 × 8 BM with 3 dB 180° couplers and several phase shifters (PSs) was proposed. The achieved FBW is 60%, and the FA is equal to 15.04 ${\mathsf{\lambda }}_0^2$. A conventional 8 × 8 BM comprising BLCs and Schiffman phase shifters was proposed in [26]. The multilayered configuration was achieved using vias to connect the components at different layers, achieving an FA size of 4.4 ${\mathsf{\lambda }}_0^2$ and an FBW of 32%. Stripline technology was employed in [27,28,29] for the implementation of compact 8 × 8 BMs. Specifically, in [27], two 4 × 4 BMs using both 90° and 180° couplers combined with an output network of the same couplers and several different PSs were proposed. Despite the reduced number of crossovers, the inclusion of an output network and multiple PSs increases the complexity of the proposed design and the size (FA = 2.56 ${\mathsf{\lambda }}_0^2$), while the FBW is only 5.5%. In [28], an 8 × 8 BM composed of coupled-line quadrature directional couplers and coupled-line Schiffman C-sections PS was proposed. The designs featured a reduced number of crossovers, and thus the obtained size was 2.46 ${\mathsf{\lambda }}_0^2$, and a significantly enhanced FBW of 73% was achieved. An optimal trade-off between BW and size was reported in [29]. Therein, an 8 × 8 BM with single Section 3 dB/90° directional couplers and PSs realized in tandem connection of two 3 dB/90° couplers was proposed. The achieved size was 1.3 ${\mathsf{\lambda }}_0^2$, with an FBW of 66%.

In this paper, we propose two different design approaches for the BM beamformers, targeting the FR-3 band and utilizing the microstrip to slot technology. The BMs are realized using multilayer 3 dB/90° hybrid couplers (HCs) and PSs with elliptical geometry. The first design approach focuses on designing the components comprising the BM (HC, PS, etc.) separately and then integrating them to form the overall BM structure. Thus, an 8 × 8 BM operating in the 6–16 GHz band was successfully designed and prototyped. Except for the UWB characteristics where an FBW of 92% was achieved, a compact size was delivered, replacing the reference microstrip lines associated with the PSs and required for the appropriate differential phase, with meander lines (MLs). The second method introduces a novel design approach utilizing the wideband-matching technique. This technique simplifies the implementation process by designing the BM as a single, unified structure and contributes to a further reduction in the size of the BM, while maintaining the same and even a higher FBW. For this case, 4 × 4 and 8 × 8 BMs were successfully implemented and simulated for the FR3 band, achieving compact size and UWB operation, while meeting the desired performance criteria. Table 1 and Table 2 provide a comprehensive summary of the performance metrics for the mentioned multilayer designs concerning the 4 × 4 and 8 × 8 BMs, respectively. Therein, except for the size and FBW, various characteristics, including technology, amplitude imbalance, and phase accuracy, are listed in comparison with the herein contributed designs.

As shown in Table 1 and Table 2, our proposed designs exhibit the best performance in terms of BW and offer a comparable size to those of the reported works. Concerning the phase accuracy, the deviation listed in Table 2 is expected given the large BW of operation. Notably, our proposed BM topologies integrate an equiphased transmission line (TL) network, leading to a significant increase in the overall size of the BMs. The size of the BMs alone (excluding the TL network) is provided in parenthesis in Table 2 for clarity.

The rest of the paper is organized as follows: Section 2 demonstrates the BM design methods, while Section 3 presents the results of the implemented and prototyped designs. Finally, Section 4 draws conclusions regarding the proposed work.

2. Butler Matrix Design

Figure 2 depicts the topology of the proposed BM, which consists of N = 2ⁿ = 8 (n = 3) input ports or excitation ports and N = 2ⁿ = 8 output ports or antenna ports. Since each input port is electrically connected to the output ports, exciting one of its N inputs produces uniform amplitudes at the outputs with phase difference, Δφ. Thus, for every input port excitation, there is a different Δφ between the output ports, and, hence, a different beam is produced in the desired direction. After exciting the eight input ports separately, eight orthogonal beams are produced at the output, where their radiation field is in the form of sinx/x. The phase difference, Δφ, at the output ports is calculated as follows:

$$\mathsf{\Delta }\mathsf{\phi }=\pm \frac{2\mathrm{p}-1}{\mathrm{N}}180°$$

(1)

where N = 8, p = 1, 2, ….., (n + 1) n = 7 for 8 × 8 BM. The beam angular direction, θ_p, is given as follows:

$$\sin {\mathsf{\theta }}_{\mathrm{p}}=\frac{\mathsf{\lambda }}{\mathrm{d}}\frac{\mathsf{\Delta }\mathsf{\phi }}{360°}$$

(2)

where λ is the free-space wavelength of the operating frequency, and d is the distance of antenna elements at the output ports. Using Equation (2), and with respect to Figure 2, the phase relations between the input and output ports and the produced beam’s angle are easily derived, and they are given in

Table 3. Excitation phase distribution in an ideal 8 × 8 BM.

Antenna Ports	Beam Ports
	1L	4R	3L	2R	2L	3R	4L	1R
1	90	−180	157.5	−112.5	135	−135	157.5	−112.5
2	112.5	22.5	−90	−180	−157.5	112.5	−45	−135
3	135	−135	22.5	112.5	90	0	112.5	−157.5
4	157.5	67.5	135	45	−22.5	−112.5	−90	−180
5	−180	−90	−112.5	−22.5	45	135	67.5	157.5
6	−157.5	112.5	0	−90	112.5	22.5	−135	135
7	−135	−45	112.5	−157.5	−180	−90	22.5	112.5
8	−112.5	157.5	−135	135	−112.5	157.5	−180	90
Δφ	22.5	−157.5	112.5	−67.5	67.5	−112.5	157.5	−22.5
Beam angle	−6°	46°	−31°	18°	−18°	31°	−46°	6°

.

Referring again to the BM topology in Figure 2, it comprises a BFN and an equiphased TL network. The BFN is composed of 12 HCs 3 dB/90°, 8 PSs (values shown in Figure 2), and their corresponding reference lines. The TL network is used to align and order, in sequence, the BM outputs ports to match to the linear array ports, thus enabling their integration.

The BM design is initially carried out by estimating the theoretical dimensions of the HC and PSs, following the methodology outlined in our previous works [30,31]. Subsequently, we introduce two distinct design approaches for the BM. The first approach involves the design of each component (HC, PSs, and TL network) of the 8 × 8 BM separately and then integrating them to form the overall BM structure. The second approach incorporates the wideband-matching technique, as adopted from our preceding work [32]. With this technique, the separate design of each component is avoided since the BM is implemented as a single structure using the theoretical values of the core components and the estimated geometrical characteristic of the interconnecting TLs between them. Moreover, this technique not only facilitates the further miniaturization of the BM by using only the appropriate interconnecting TLs but also demonstrates adaptability to various passive devices, including six-port networks and beyond.

2.1. Butler-Matrix Components’ Design

2.1.1. The 3 dB/90° Hybrid Coupler

Figure 3a depicts the multilayer structure of the adopted HC, which consists of three conductive layers interleaved with two Rogers 4003C dielectric layers. Elliptically shaped patches are printed on the top and bottom layers, which are coupled through an elliptical slot etched on the middle metallic (ground) layer. Thus, as shown in Figure 3a, a signal applied to input port P1 is equally divided between output P2 and coupled port P3 with a phase shift of −90°. Ideally, there is no signal at the isolated port P4, which is not illustrated in Figure 3b, as it is below −20 dB, with an average value of −20.12 dB.

The coupling C between the P1 input and P3 output, as well as the UBW operation of the HC, is totally controlled by its geometrical characteristics (Figure 3a), which are the widths, C_w and C_s, of the elliptical patch and slot, respectively, and the coupler length, C_l. In the design process of the HC, the desired coupling coefficient, C = S₃₁, is initially chosen, which defines the output power ratio, K = P₂/P₃. For an ideal coupler with perfect isolation (S₄₁ = 0), and ignoring the presence of small loses, the conservation of power yields (K + 1)C² = 1, and therefore it is |S₂₁|² = K/(K + 1). From the coupling coefficient, the even and odd impedances are derived, since the HC of Figure 3a exhibits a symmetry between ports, thus supporting the even and odd modes. Subsequently, the parameters C_w, C_s, and C_l are calculated using formulas outlined in our previous work [30], where a detailed design methodology of the HC and PSs and the calculation of their geometrical characteristics are given. The theoretically estimated and optimized values of the HC parameters are depicted in

Table 4. Theoretical and simulated values of the HC parameters of Figure 3a.

HC	Cl	Cw	Cs
Theoretical value	4.57	2.29	3.44
Simulated value	4.97	2.35	3.6

. The performance of the HC in terms of S-parameters and phase accuracy is illustrated in Figure 3b. As shown, the transmission parameters S21 and S31 are −3.5 dB for the entire frequency range (6–16 GHz), with a negligible deviation of ±0.5 dB, while the reflection coefficient at the input ports is below −15 dB at the same frequency range. Concerning the phase difference between P2 and P3, it is stable at 90°, with an acceptable deviation of ±3°.

2.1.2. Phase Shifter and Meander Lines

The same multilayer technology used for the HC is applied to the PSs, with the exception of open-ended ports and the removal of the TLs from ports P2 and P4 of the HC shown in Figure 3a. The obtained two-port PS is shown in Figure 4a. It consists of top and bottom elliptical patches coupled through an elliptical slot located in the middle layer.

The phase shift φ_p inserted by the PS is referenced to the phase shift φ_m of a corresponding microstrip line of length l_m = 2l₁ + 3l_h + 4l_v + 8(w + r) in order to achieve the desired differential phase, Δφ.

$$\mathsf{\Delta }\mathsf{\phi }=\angle {\mathsf{\phi }}_{\mathrm{p}}-\angle {\mathsf{\phi }}_{\mathrm{m}}$$

(3)

The differential phase is influenced by both the coupling coefficient, C, of the PS and the length of the microstrip line, which is also shown in Figure 4a. The coupling coefficient, in turn, is selected based on the required length dictated by the layout. Consequently, the lengths of the patches and slots are set equal to a quarter of the effective wavelength (λ_eff/4), while the widths (Cw and Cs) of the patches and slots, respectively, are determined using, once again, the formulas provided in [30]. The lengths of the remaining TLs at ports P1 and P2 are adjusted to align with the length of the reference transmission line. In this work, the reference TLs are intentionally meandered to shrink the size of the BM without sacrificing its performance. As outlined in our previous work [32], when a microstrip line is meandered, it achieves wideband-matching characteristics through conjugate matching and admittance compensation, just treating the ML as a shunt component. Figure 5a illustrates a TL with a length (l) alongside its meandered counterpart, which maintains the same length while reducing its footprint to nearly half. In Figure 5b, the simulated Voltage Standing Wave Ratio (VSWR) of both lines is depicted. Notably, the ML demonstrates an equivalent or even superior response within the 6-to-15 GHz range compared to the straight-line configuration, while it remains well below the limit of VSWR = 2 over the entire band of 6–16 GHz.

Thus, all the required PSs are designed in conjunction with the MLs while the required coupling and differential phase are achieved by optimizing the PS elliptic geometry and the vertical length of the MLs. At the same time, the footprint of the PS is effectively reduced, resulting further in the compactness of the overall BM, as is shown later, in Section 3. Three PSs were designed and simulated herein. A 22.5° PS is shown in exploded view (Figure 4a), followed by a 45° and a 67.5° PS shown in a layout view (Figure 4b), both sharing the same layer stack as the 22.5° PS. All parameters depicted in Figure 4 and their corresponding values for the three PSs are listed in

Table 5. Final values of the parameters of the three PSs. T denotes theoretical, and S denotes simulated.

	Cw		Cs		Cl		W	l1	l2	l3	lh	lv	r	r1
	T	S	T	S	T	S
22.5°	4.21	3.72	5.21	5.32	5.01	4.5	0.54	1	3	0.807	1.372	1.705	0.06	0.5
45°	3.4	2.7	3.9	3.7	4.7	4.2	0.54	1	7.024	0.643	1.953	4.63	0.06	0.5
67.5°	2.5	1.47	2.75	2.76	4.59	4.5	0.54	1	3	0.807	1.372	1.801	0.06	0.5

, while Figure 6 illustrates the performance of the PSs in terms of S-parameters accuracy and differential phase between the PS path and the corresponding microstrip line or ML. As shown in Figure 6a, the transmission coefficient (S₂₁) for the PSs, as expected, exhibits an insertion loss of −1 dB or less, with a maximum deviation of 0.6 dB at 16 GHz, while the reflection coefficient (S₁₁) is below −10 dB for almost the entire frequency range. The maximum deviation for the transmission coefficient (S₄₃) of MLs is 0.5 dB at 13 GHz from the expected value of −0.3 dB. Excellent agreement between the ideal–theoretical and the simulated differential phase is achieved for the three PSs, as shown in Figure 6b, where a maximum deviation of 3° for the 22.5° PS is observed.

2.1.3. Equiphased Transmission Line Network

Figure 7a illustrates the equiphased TL network which was employed to rearrange the outputs ports of the BM to achieve the desired phase shift, Δφ (Table 3), between them and to facilitate the integration of the linear Vivaldi antenna array. Only four out of the eight lines are depicted in Figure 7a, since our BM exhibits symmetry, as shown in Figure 2. Those lines are structured in a reversed S-shape configuration and positioned at distance d, where d = 0.6λ_min = 0.6c/f_max. This distance allows for a direct connection to the corresponding antenna ports. It is selected to ensure the avoidance of grating lobes in the radiation pattern even for the two far-end steered beams expected at angles θ_max = ±45° from broadside. According to Balanis [33], this restriction reads as follows:

$$\mathrm{d}\le \frac{{\mathsf{\lambda }}_{\min }}{1+\left|{\sin \mathsf{\theta }}_{\max }\right|}{|}_{{\mathsf{\theta }}_{\max }={45}^{\mathrm{o}}}\cong \frac{{\mathsf{\lambda }}_{\min }}{1.7}$$

(4)

Additionally, each of the four lines integrates a unit cell ML which serves to equalize their physical lengths, thereby compensating–zeroing the phase difference between them.

The parameters for the four TLs shown in Figure 7a and their values are listed in

Table 6. Final values of the parameters of the four TLs (dimensions are in mm).

	L1	L2	L3	L4	w	r
Line1	2	0.92	14.34	8	0.54	0.5
Line2	4	0.6	15	6	0.54	0.5
Line3	6	0.469	15.2	4	0.54	0.5
Line4	8	0.06	16	2	0.54	0.5

, while Figure 7b depicts the transmission S-parameters and phase difference of the TLs. A maximum loss is observed for the first line, where the S₅₁ transmission parameter is −0.9 dB at 16 GHz. A maximum phase deviation of 1.5° from zero is observed between lines 5 and 6.

2.2. BM Design Employing Wideband-Matching Technique

The second approach elaborating on the design of BM as a single, unified structure is now presented. Figure 8 illustrates the concept of the wideband-matching technique utilized to design a BM in the 6–16 GHz band. For illustrative purposes, the design of a 4 × 4 BM is presented here; however, the method is identical for the design of an 8 × 8 BM. The 4 × 4 BM is the classical BM consisting of four HCs and two 45° PSs. As shown in Figure 8, the TLs connecting the HC ports to the input and output ports are removed (Sections A and C). Additionally, the interconnecting TLs and those connecting the PS ports to the HC ports (Section B) are also eliminated, leaving only the HCs and PSs as the core components. The theoretical parameters related to the geometry of the core components are given in Table 4 (Section 3.1) for the HC and in Table 5 (Section 3.2) for the PS. Thus, the input impedance at the reference plane, depicted in Figure 8, is initially estimated through a simulation for each core component. The simulated input impedance for the HC and for the PS is depicted in Figure 9, where the real and the imaginary parts are plotted. For the input impedances, the low–min and high–max impedances for each component are easily obtained from the plots as Z_low = R_low + jX_low and Z_high = R_high + jX_high. Subsequently, the hyperbolic mean impedance, Z_m = R_m + jX_m, is computed for the HC and PS from the relation given in our previous work [32]. Due to the symmetry and reciprocity inherent in the core components, it is evident that the hyperbolic mean impedance will be identical for all ports of both the HC and PS. Therefore, we have Z^HC_m1 = Z^HC_m2 = Z^HC_m3 = Z^HC_m4 and Z^PS_m1 = Z^PS_m2. The next step is to determine the appropriate characteristic impedance, Z_o, of the interconnecting TLs between the core components and between the input and output ports using Equation (5) taken from [32] and repeated herein in order to clarify the procedure.

$${\mathrm{Z}}_{\mathrm{o}}=\sqrt{\frac{\mathrm{real}\left\{\left({\mathrm{Z}}_{\mathrm{A}}^{*}{\mathrm{Z}}_{\mathrm{B}}^{*}\right)\left({\mathrm{Z}}_{\mathrm{A}}-{\mathrm{Z}}_{\mathrm{B}}\right)\right\}}{\mathrm{real}\left({\mathrm{Z}}_{\mathrm{A}}-{\mathrm{Z}}_{\mathrm{B}}\right)}}$$

(5)

In Equation (5), Z_A and Z_B represent the hyperbolic mean impedances of the components to be matched. Thus, to compute, for example, the characteristic impedance of the TL connecting the HC’s port P1 to connector port 1L (Figure 8), a complex-to-real matching is applied, since it is connected to a 50 Ω connector and, hence, in Equation (5), Z_A = 50 Ω and Z_B = Z^HC_m1. A complex-to-real matching is applied also for the estimation of TLs connecting the HC ports to the output connector ports (ANT1–ANT4). A complex-to-complex matching is applied for the interstage components. Specifically, to match, for example, PS port P1 to HC port P2 (Figure 8), it should be Z_A = Z^HC_m2 and Z_B = Z^PS_m1, while to estimate the matching reference line connecting the HC’s port P3 to the next HC’s port P1, it should be Z_A = Z^HC_m3 and Z_B = Z^HC_m1. Except for the characteristic impedance of the TLs, the electrical length of them is also estimated using Equation (17) given in [32]. However, for the BM in Figure 8, only the length of the TLs connecting the HC’s ports to the input connectors (1L, 2R, 2L, and 1R) is determined. This is because the lengths of the interstage TLs, as well as those at the output ports, are dictated by the required differential phase and the geometry of the equiphased TL network, respectively. Therefore, their estimation is omitted herein. The last step is to compute the appropriate width (W) and length (L) of the interconnecting TLs using well-known equations found in [34].

Table 7. Estimated hyperbolic mean impedance for the HC and PS and their corresponding TL’s geometrical characteristics.

	Section	Zlow (Ω)	Zhigh (Ω)	Zm (Ω)	Matching TL
					Zo (Ω)	θ (deg)	W (mm)	L (mm)
HC	A and C	54.63 + j1.56	62.52 − j6.46	58.57 − j2.18	54.3	103	0.49	5.48
HC	B	54.63 + j1.56	62.52 − j6.46	58.57 − j2.18	58.51	-	0.42	-
PS	B	26.13 − j16.5	49.6 + j4.1	37.3 − j9.4	44.16	-	0.68	-

depicts the Z_low, Z_high, and estimated Z_m of the HC and PS ports, as well as the geometrical characteristics of their corresponding interconnecting TLs.

Thus, using the theoretical values for the core components and the estimated geometrical characteristics (width and length) of the interconnecting TLs, a 4 × 4 BM was designed and is shown in Section 3.2. The same design approach was employed to design an 8 × 8 BM operating, again, in the 6–16 GHz band.

3. Butler Matrix Results

3.1. First Design Approach’s 8 × 8 BM Results

Figure 10 shows a schematic view of our proposed BM. The multilayer structure comprises three conductive layers interleaved with two dielectric layers. Notably, the dielectric layers are not shown here in order to provide a clear view of the top and bottom patches, as well as the ground layer. It is worth mentioning that each key component was individually designed, as demonstrated in Section 2.1, with a 50 Ω characteristic impedance for both input and output lines—connectors. Consequently, when these components are assembled to create the overall structure, there is a degree of mismatch at the connection points due to slight deviations from the exact 50 Ω characteristic impedance at the input and output connectors of each component. This mismatch results in a slight variation in the transmission coefficient for each path, altering the phase as well. Hence, the parameters denoted in Figure 10 which are related to PSs and MLs geometry are optimized to obtain their final values, which are listed in

Table 8. Final optimized parameters for the 8 × 8 Butler Matrix (dimensions are in mm). PD denotes percent difference.

	C22.5w			C22.5s			l22.5v
	C45w			C45s			l45v
	C67.5w			C67.5s			l67.5v
	Initial	Final	PD (%)	Initial	Final	PD (%)	Initial	Final	PD (%)
22.5°	3.72	3.4	8.6	5.32	4.68	12	1.705	1.816	6.5
45°	2.7	2.52	6.6	3.7	3.4	8	4.63	4.47	3.4
67.5°	1.47	1.23	16.3	2.76	2.77	0.36	1.801	1.48	17.8

. Specifically, the vertical lengths l_22.5v, l_45v, and l_67.5v are optimized to keep the differential phase between output ports around the target value, while the widths C_w and C_s for each PS are optimized to reduce the ripples of each Δφ. As shown in Table 8, the maximum percentage change occurs for the vertical length l_67.5v, which is 17%. For adequate grounding and to minimize parasitic effects, additional 4 mm long lines with 0.4 mm diameter through-plated via holes were placed at each port, as shown in Figure 10. These modifications facilitate measurements using SMA or SMP connectors. The total achieved size of the BM is 58.46 mm × 90 mm, or 5.6 ${\mathsf{\lambda }}_0^2$ FA. Notably, since the length of the vertical side of the BM is defined by the length of the TL network, miniaturization is achieved (through the integration of the MLs) on the horizontal side of the BM, effectively reducing its FA.

The fabricated BM prototype is depicted in Figure 11, printed into separate boards, which are aligned appropriately and bonded together in a back-to-back configuration. Due to the limited amount of space at the input side of the BM, the SMA connectors that correspond to the input ports (bottom side of Figure 11b) of the bottom layer were attached vertically. Rogers 4350B with ε_r = 3.34 and a thickness h of 0.25 mm was utilized for the two dielectric layers.

Figure 12a,b illustrate the simulated and measured reflection coefficients, as well as the simulated and measured coupling of the input ports for the 8 × 8 BM shown in Figure 10 and Figure 11, respectively. Given the symmetry of the BM’s topology shown in Figure 2, the response from four out of eight ports suffices for illustration purposes. Consequently, the responses of the remaining ports are identical. Hence, S₁₁ = S₈₈, S₂₂ = S₇₇, S₃₃ = S₆₆, and S₄₄ = S₅₅; and S₂₁ = S₈₇, S₃₂ = S₇₆, S₄₃ = S₆₅, and S₅₄ = S₄₅. As shown in Figure 12a, both the simulated and measured reflection coefficient remains below −10 dB for the entire frequency range from 6 GHz to 16 GHz. Concerning the coupling between input ports, it is maintained below −15 dB for almost the entire range of operation.

Figure 13, Figure 14, Figure 15 and Figure 16 illustrate the simulated and measured transmission responses and phase differences obtained at the output ports when input ports 1, 2, 3, and 4 are excited. Due to the symmetrical design, the responses for ports 5, 6, 7, and 8 are identical. The expected transmission coefficients to the output ports are around −12 dB. This is because a signal applied to the input ports will traverse three HCs and two PSs, or a PS with an ML, or two MLs to reach the output ports. Notably, an ideal 8 × 8 BM divides the input power equally to its eight output ports; thus, an S₂₁ of −9 dB would result. Hence, the measured −12 dB observed includes 3 dB losses across the entire path. Observing Figure 13a, Figure 14a, Figure 15a and Figure 16a, we see that there is a good agreement between simulated and measured transmission coefficients from input to output ports when input ports 1 to 4 are excited, respectively. Their magnitude is around −12 dB, with a ±3 dB deviation across the frequency ranging from 6 to 15 GHz. Considering the 6–15 GHz range as the band of operation, the achieved FBW of the proposed BM is 92%, which is the highest when compared against the related works in Table 2.

Regarding the phase difference at output ports for each input port’s excitation, as depicted in Figure 13b, Figure 14b, Figure 15b and Figure 16b, the maximum deviation, Δφ, from the ideal values occurs for the measured phase differences. Specifically, for port 1, its phase deviation is Δφ¹ = 15°; for port 2, Δφ² = 16°; for port 3, Δφ³ = 20°; and for port 4, Δφ⁴ = 21.5°. The phase variations of the proposed BM are relatively high, which was anticipated given the UWB operation. Phase accuracy tends to vary linearly with BW; hence, the wider the BW, the more significant the phase variations.

Moreover, despite the phase variations, the generated beam pattern of the BM experiences minimal impact, as illustrated in Figure 17. This figure illustrates the gain of each beam when the output ports of the BM feed into a uniform linear array. For this purpose, the simulated and measured S-parameters were imported in an antenna array simulation tool representing the excitation amplitude and phase. As shown in Figure 17, a very good agreement is obtained between the theoretical (Table 3), the simulated, and the measured beam angles.

3.2. BM Results of the 2nd Design Approach

Using the estimated geometrical characteristics of the interconnecting TLs from Table 7, a 4 × 4 BM was implemented, as shown in Figure 18. Each section (input, output, and interstage) is designed given the widths tabulated in Table 7. Additionally, the reference TLs connecting the HCs are meandered once again, resulting in a BM size of 39.4 mm × 17.1 mm, or 0.74 ${\mathsf{\lambda }}^2$. The theoretical parameters of the HC and 45° PS (shown in the parenthesis of

Table 9. Final parameters of the HC and PS. Dimensions are in mm.

	Cw	Cs	Cl
	C45w	C45s	C45l
HC	2.42 (2.29)	3.63 (3.44)	4.5 (4.56)
PS 45°	3.39 (3.4)	3.7 (3.9)	4.2 (4.7)

) are optimized, with their final values provided in Table 9. At the same time, to achieve the desired differential phase at the output ports, the length of the vertical sections of the ML (denoted as l45v in Figure 18) is optimized. The final optimized value is provided in the caption of Figure 18. Figure 19 depicts the performance of the 4 × 4 BM in terms of S-parameters and phase accuracy. Since the network is symmetric, the results for ports 1 and 2 only are illustrated. As shown in Figure 19a, the insertion loss of the BM exciting ports P1 and P2 is as expected, around −7 dB, with a ±2 dB deviation in the entire 6–16 GHz band. The reflection coefficient of the BM is below −10 dB for the entire band of operation. Regarding the phase difference at the output ports, Δφ = 45° is expected between them when port P1 is excited, while Δφ = 135° is expected when port P2 is excited. Indeed, as shown in Figure 19b, the obtained phase difference is around 45° and 135°, exciting P1 and P2, respectively, with a maximum deviation of 9° for the P1 excitation. Considering the S-parameter and phase response of the 4 × 4 BM, the achieved FBW is 102%, which is the highest compared to the works reported in Table 1. Additionally, the design maintains a compact size, comparable to or even smaller than those reported in Table 1.

Following the same procedure used for the 4 × 4 BM, an 8 × 8 BM was designed by utilizing the wideband-matching technique described in Section 2.2. Herein, in the estimation of the geometrical characteristics of interconnecting TLs, an extra section (Section D) is added, as shown in Figure 20, which contains the 22.5° and 67.5° PSs and their related reference TLs. The estimated widths are given in the caption of Figure 20. The theoretical parameters (shown in the parenthesis of

Table 10. Final parameters of the 22.5° and 67.5° PSs. Dimensions are in mm.

	C22w	C22s	C22l
	C67w	C67s	C67l
PS 22.5°	5.19 (4.21)	5.82 (5.21)	4.5 (4.5)
PS 67.5°	1.87 (2.5)	2.76 (2.75)	4.5 (4.5)

) of the 22.5° and 67.5° PSs are optimized, with their final dimensions provided in Table 10. The lengths of the vertical sections of the MLs (denoted as l22v, l45v, and l67v in Figure 20) are optimized in order to obtain the desired differential phase at the output ports. Their final optimized values are provided in the caption of Figure 20. Since BM is designed as a single, unified structure using only the appropriate TLs to interconnect the core components, the total achieved size is 84.2 mm × 40.1 mm or 3.57 ${\mathsf{\lambda }}^2$. This design represents an approximate 20% miniaturization compared to the previous 8 × 8 BM design outlined in Section 2.1.

Figure 21 depicts the simulated reflection coefficient at the input ports of the 8 × 8 BM presented in Figure 20. As shown, the reflection coefficient is maintained below −10 dB across the entire operational bandwidth. Figure 22 and Figure 23 illustrate the simulated transmission responses and phase differences obtained at the output ports when input ports 1, 2, 3, and 4 are excited. Due to the symmetrical design, the responses for ports 5, 6, 7, and 8 are identical. The simulated transmission coefficients from input to output ports for each port excitation are approximately −12 dB, with a deviation of a ±3 dB within the frequency range from 6 to 15 GHz, as demonstrated in Figure 22a and Figure 23a. Considering its ideal value of −9 dB resulting from the 1-to-8 power division, the additional −3 dB corresponds to power losses. Concerning the phase difference, Δφ, at output ports for each input port’s excitation, as illustrated in Figure 22b and Figure 23b, a maximum deviation of 21° from the ideal values occurs between output ports P11 and P12 when port P3 is excited. The same FBW is achieved as the BM designed in Section 2.1. Notably, a reduction in footprint area (FA) by 36% is attained in this design, resulting in the best compromise between FBW and size compared to the reported works in Table 2.

4. Conclusions

This paper introduced two distinct design approaches for Butler Matrix beamformers operating in the low FR3 (6–16 GHz) band. The first approach is based on designing individual components and utilizing the advantages of MLs. This method led to the successful design and construction of an 8 × 8 BM, operating in the 6-to-16 GHz frequency range. Both simulated and measured results demonstrate UWB operation where an FBW of 92% was achieved, which is the highest among the reported works. Furthermore, this approach achieved a remarkably compact size, with an FA of 5.6 λ², surpassing previous works. The amplitude and phase accuracy were also well within the expected values. The second proposed approach simplifies the implementation process by using wideband-matching techniques to design the BM as a single, unified structure. To validate this design approach, 4 × 4 and 8 × 8 BMs were successfully implemented and simulated for the FR3 band. The 4 × 4 BM achieved a 102% FBW and a compact size, with an FA of 0.74 λ², offering the best compromise between bandwidth and size compared to related works. Additionally, the amplitude and phase accuracy were maintained at acceptable levels, with deviations of ±3 dB and ±9°, respectively. Regarding the 8 × 8 BM, a further size reduction of 20% (FA = 3.57 λ²) was attained compared to the 8 × 8 BM implemented with the first design approach, while the same FBW, amplitude accuracy, and phase accuracy were achieved, thereby demonstrating the validity of the second design approach.