Brief theory of Butler matrix

According to the SIW Butler matrix shown in Fig. 2, it consists of two hybrid couplers, two crossovers, two −45-degree phase shifters and two 0-degree phase shifters. The Butler matrix will be connected to an antenna array, forming a multi-beam antenna as shown in Fig. 1. The direction of rotation of the beam can be calculated using Formula 1 [Reference Pezhman, Heidari and Ghafoorzadeh-Yazdi13].

(1)\begin{equation}{\theta _{{\text{direction}}}} = 90^\circ - {\cos ^{ - 1}}\left( {\frac{{\lambda \phi }}{{2\pi d}}} \right), - 90^\circ \leqslant {\theta _{{\text{direction}}}} \leqslant 90^\circ \end{equation}

where λ represents the wavelength, d is the distance between two adjacent elements (usually equal to or less than λ/2), and φ is the phase difference between any two adjacent antennas in the array. The beam steering direction is achieved through the phase difference between neighboring antennas, and the phase difference between antennas is controlled by the Butler matrix.

Butler matrix design process

In this paper, the Butler matrix is simulated and designed using a Rogers Duriod 5880 substrate, with a loss tangent of 0.0009 and a relative dielectric constant of 2.2. The phase differences between consecutive antennas are set at −45°, 135°, −135°, and 45°, resulting in theoretical beam angles of 14.4°, −48.6°, 48.6°, and −14.4°, respectively.

According to the structural characteristics of the SIW-based Butler matrix, the cross-coupler is formed by connecting two hybrid couplers together, and the size and structure of the phase shifter are constrained by the hybrid coupler and crossover. As a result, the size of the hybrid coupler determines the overall size of the Butler matrix in this study. Additionally, since the phase shifter needs to change phase, a straight-line structure is not feasible, and a bending structure is required for its implementation. However, in the proposed Butler matrix, to avoid increasing its size, the phase shifter’s structure is positioned toward the overall center, preventing the matrix from expanding its dimensions outward.

Hybrid coupler

To achieve desired phase differences of −45°, 135°, −135°, and 45° between consecutive antennas, a hybrid coupler with a phase difference of −90° is used in this study. The hybrid coupler utilizes metal via holes with a size of 0.4 mm and a spacing of 0.8 mm between adjacent metal via holes. The dimensions and geometry of the designed hybrid coupler are shown in Fig. 3 and Table 1.

Figure 3.

Schematic diagram of the dimensions of the hybrid coupler.

Table 1.

Hybrid coupler dimensions

In this study, the required frequency band, reflection loss, transmission coefficient, and output phase difference are achieved by adjusting the length, width, and position of the metal vias in the hybrid coupler. For instance, shifting the entire row of L5 metal vias horizontally by 0.4 mm toward Ports 1 and 2 increases the frequency of the hybrid coupler, resulting in a phase difference of −270°. Conversely, shifting the vias 0.4 mm toward Ports 3 and 4 decreases the frequency, resulting in a phase difference of +90°. The frequency of the proposed hybrid coupler lies between these two scenarios, maintaining a phase difference of −90°, as shown in Figs. 4 and 5. This demonstrates that adjusting the position of the metal vias is crucial for meeting the design requirements of the hybrid coupler.

Figure 4.

Simulated S-parameter of the hybrid coupler.

Figure 5.

Schematic diagram of the phase difference between S31 and S41 for the hybrid coupler.

Figure 4 shows that the reflection coefficient S11 and isolation S21 of the proposed hybrid coupler are below −10 dB in the frequency range of 25–37.5 GHz. When the frequency of the proposed hybrid coupler is 28 GHz, the through Port S31 is −2.63 dB, and the coupled Port S41 is −4.13 dB, as depicted in Fig. 5. The phase difference between Port 1 and Port 3 is approximately −90.7°.

Crossover

The structure of the cross-coupler in this paper is slightly modified from the typical configuration, where two hybrid couplers are connected in series. In the modified design, the last metal via hole of the first hybrid coupler is shared with the first metal via hole of the second hybrid coupler, as shown in Fig. 6. This approach helps to slightly reduce the size of the cross-coupler.

Figure 6.

The cross-coupler dimensions.

Figure 7 presents the simulation results, indicating that the reflection coefficient S11 and isolation levels S21 and the crossover Port S31 are all below −10 dB in the frequency range of 26–33 GHz. At 28 GHz, through Port S41 is −0.47 dB. These simulation results demonstrate that the cross-coupler meets the design requirements.

Figure 7.

Simulated S-parameter of the crossover.

Phase shifter

The phase shifter is a critical component in the Butler matrix, used in conjunction with the phase difference of the hybrid coupler to achieve fixed output phase shifts. In this design, the hybrid coupler has a phase difference of −90°, so the required corresponding phase shifters are −45° and 0°. The size of the phase shifter is constrained by the dimensions of the hybrid and crossover and cannot be adjusted by altering the length or width of the substrate.

Additionally, the inner side of the phase shifter shares one side of the metal via holes with the cross-coupler, making it impossible to change the position of the inner metal via holes.

Several studies have suggested different approaches to design the phase shifter, such as bending the outer row of metal via holes outward or inserting multiple metal via holes internally. However, bending the metal via holes outward would increase the overall size of the structure and adding additional metal via holes would introduce additional fabrication costs and time. In this paper, without increasing the overall size or using additional metal via holes, the approach of moving the positions of the outer metal via holes of the phase shifter inward is adopted to achieve the design of the −45-degree and 0-degree phase shifters.

−45-degree phase shifter

The structure of the −45-degrees phase shifter has the same length as the cross-coupler and a width equal to the size of the hybrid couplers it connects to. The overall size of the −45-degrees phase shifter is depicted in Fig. 8.

Figure 8.

Dimensions diagram of the −45-degree phase shifter.

As shown in Fig. 9, the simulation results indicate that the reflection coefficient (S11) of the −45-degrees phase shifter is below −10 dB in the frequency range of 25–35 GHz. At 28 GHz, the transmission coefficient (S21) is −0.24 dB, as depicted in Fig. 9. Furthermore, at 28 GHz in Fig. 10, the phase shift is simulated to be −44.28°, confirming that the −45-degree phase shifter meets the design requirements.

Figure 9.

Simulated S-parameter of −45-degree phase shifter.

Figure 10.

The phase of a −45-degree phase shifter.

0-degree phase shifter

The structure of the 0-degree phase shifter is designed under the same constraints as the 45-degree phase shifter. Therefore, the same design approach as used for the 45-degree phase shifter to implement the 0-degree phase shifter. The overall size of the 0-degree phase shifter is shown in Fig. 11.

Figure 11.

Dimensions of a 0-degree phase shifter.

As depicted in Fig. 12, the simulation results demonstrate that the reflection coefficient (S11) of the 0-degree phase shifter is below −10 dB in the frequency range of 25–35 GHz. At 28 GHz, the transmission coefficient (S21) is −0.24 dB, as shown in Fig. 12. Moreover, at 28 GHz in Fig. 13, the phase shift is simulated to be −0.23°, which is very close to 0°. These simulation results confirm that the 0-degree phase shifter meets the design requirements.

Figure 12.

Simulated S-parameter of 0-degree phase shifter.

Figure 13.

The phase of a 0-degree phase shifter.

Slot antenna

Slot antennas have been widely used in SIW-form Butler matrices, and their design typically involves a size of about half a wavelength. This allows for smaller antenna dimensions and extends the SIW’s structure, increasing the isolation between adjacent antenna elements. Moreover, when the adjacent antenna elements are spaced half a wavelength apart, the radiated electromagnetic waves can be aggregated, leading to an increase in antenna gain.

In this paper, the design of the Butler matrix took the requirements of the slot antennas into consideration before designing the hybrid coupler. In other words, the slot antenna was the first element to be considered and designed in this study. The structure and dimensions of the antenna slot used in this article are depicted in Fig. 14 and Table 2.

Figure 14.

The dimensions of the SIW slot antenna.

Table 2.

SIW slot antenna dimensions

As shown in Fig. 15, the simulation results of this slot antenna indicate that the reflection coefficient is below −10 dB in the frequency ranges of 27.55–28.75 GHz, 30.78–31.32 GHz, and 34.65–35.28 GHz.

Figure 15.

The reflection coefficient of the slot antenna.

The proposed slot antenna was simulated with input phase settings of −45, −90, −135, and −180° for Ant. 1, Ant. 2, Ant. 3, and Ant. 4, respectively, to form a beam angle of +14° and a gain of 12.94 dB at the frequency of 28 GHz.

Similarly, for Port 2, the input phase settings were set to −135°, −0°, −225°, and −90° for Ant. 1, Ant. 2, Ant. 3, and Ant. 4, respectively, resulting in a beam angle of −48° and a gain of 13.76 dB at a frequency of 28 GHz.

For Port 3, the input phase settings were −90°, −225°, −0°, and −135° for Ant. 1, Ant. 2, Ant. 3, and Ant. 4, respectively, to achieve a beam angle of 47° and a gain of 13.24 dB at the frequency of 28 GHz.

Finally, for Port 4, the input phase settings were −180°, −135°, −90°, and −45° for Ant. 1, Ant. 2, Ant. 3, and Ant. 4, respectively, resulting in a beam angle of −15° and a gain of 13.26 dB at the frequency of 28 GHz. The simulation results are illustrated in Fig. 16.

Figure 16.

The beam angle and gain of the proposed slot antenna for analogue Ports 1–4 at 28 GHz.

Although this slotted antenna design has a narrower bandwidth and does not meet the 5G n257 band, the gain of this slotted antenna is significantly higher than that of some other slotted antenna designs reported in the reference [Reference Desai, Tsao and Hsu27]. Additionally, the final antenna bandwidth can be adjusted to cover the 5G n257 band by tuning the SIW transmission channel, as discussed later.

Simulation of the Butler matrix

In this section, the previously designed hybrid couplers, crossovers, 45-degree phase shifter, and 0-degree phase shifter, each with their respective dimensions and configurations as described earlier, have been successfully combined following the structure shown in Fig. 1. The same metal via is used between adjacent elements, with the first metal via of the preceding element connected to the last metal via of the subsequent element. The size and spacing of the metal vias are maintained as in the original design to ensure phase consistency and proper transmission. The general dimensions of the combined Butler matrix are shown in Fig. 17.

Figure 17.

The dimensions of the Butler matrix.

In Fig. 17, the Butler matrix shows a symmetric structure between Port 1 and Port 4, as well as between Port 2 and Port 3. As indicated in the simulation results, the trend of the reflection coefficients S11 and S44 for Port 1 and Port 4 is highly consistent. Similarly, the trend of reflection coefficients S22 and S33 for Port 2 and Port 3 is also in close agreement, reflecting the mutual correspondence between Port 1 and Port 4, and Port 2 and Port 3 due to their symmetric configuration. Moreover, the reflection coefficient S11 for Port 1 remains below −10 dB in the frequency range of 26.67–34.73 GHz, while the reflection coefficient S44 for Port 4 remains below −10 dB in the frequency range of 26.61–33.63 GHz, as shown in Fig. 18.

Figure 18.

The reflection coefficient of the Butler matrix.

As shown in Fig. 19, the final design resulted in reflection losses of −10 dB or below at a frequency of 28 GHz when Ports 1–4 were excited, using the original Butler matrix combined slot antenna and 1.85 mm coaxial connectors, along with the adjusted positions of the metal via holes in the SIW transmission channel.

Figure 19.

The Butler matrix combined the slot antenna with the adjusted SIW transmission channel and 1.85 mm coaxial connectors.

Figure 20 illustrates the frequency ranges for Ports 1– 4, where Port 1 operates in the frequency range of 26.4 –32.52 GHz, Port 2 in the range of 27.1–30.56 GHz, Port 3 in the range of 27.1–30.56 GHz, and Port 4 in the range of 27.12–30.56 GHz.

Figure 20.

The schematic diagram of the reflection coefficient for Ports 1–4 after excitation is shown in Figure 19.

Figure 21 presents the performance results when each Port (Port 1–4) is individually excited. At a frequency of 28 GHz, Port 1 achieved a maximum gain of 10.57 dB with a beam angle of 44°. Port 2 demonstrated a maximum gain of 10.56 dB and a beam angle of −19°. Port 3 exhibited a maximum gain of 10.34 dB with a beam angle of 19°. Finally, Port 4 showed a maximum gain of 11.35 dB with a beam angle of −43°. These results closely align with the theoretical beam angles for Ports 1–4.

Figure 21.

The beam angle schematic diagram for Ports 1– 4 after excitation in Figure 19.

Furthermore, as shown in Fig. 22, this study also explored the original Butler matrix combined slot antenna with end launch coaxial connectors and adjusted the positions of the metal via holes in the SIW transmission channel to reduce transmission losses. The goal was to maintain the correct beam angles when Ports 1–4 were excited and achieve reflection losses of −10 dB or lower at a frequency of 28 GHz, and even extend the reflection losses to cover the n257 frequency band. Furthermore, as shown in Fig. 22, this study also explored the original Butler matrix combined slot antenna with end launch coaxial connectors and adjusted the positions of the metal via holes in the SIW transmission channel to reduce transmission losses. The goal was to maintain the correct beam angles when Ports 1– 4 were excited and achieve reflection losses of −10 dB or lower at a frequency of 28 GHz, and even extend the reflection losses to cover the n257 frequency band.

Figure 22.

The original Butler matrix combined the slot antenna with the adjusted SIW transmission channel and the end-launch coaxial connectors.

Figure 26 illustrates the frequency ranges for Ports 1– 4, where Port 1 operates in the frequency range of 27.3–32.5 GHz, Port 2 in the range of 26.65– 30.3 GHz, Port 3 in the range of 26.6–30.3 GHz, and Port 4 in the range of 27.3–32.5 GHz.

Figure 26.

Measured and simulated reflection loss of the Butler matrix antenna.

Figure 28 presents the performance results when each Port (Port 1–4) is individually excited. At a frequency of 28 GHz, Port 1 achieved a maximum gain of 10.8 dB with a beam angle of 48°. Port 2 demonstrated a maximum gain of 9.85 dB and a beam angle of −19°. Port 3 exhibited a maximum gain of 10.4 dB with a beam angle of 19°. Finally, Port 4 displayed a maximum gain of 10.27 dB with a beam angle of −39°. Although the beam angle for Port 4 differed by 6° from the theoretical value, the original Butler matrix combined slot antenna with end launch coaxial connectors offered better bandwidth. Consequently, this design was selected for subsequent measurements.

Figure 28.

The measured and simulated beam angle of the Butler matrix antenna.

Measurement and comparison of the Butler matrix antenna

The proposed Butler matrix array antenna was manufactured and measured as shown in Fig. 23, with overall dimensions identical to those depicted in Fig. 22. The measurements were made using a Keysight N5293A network analyzer, and Fig. 24 provides a photograph of the measured reflection losses.

Figure 23.

The fabricated sample of the Butler matrix antenna with the end-launch coaxial connectors connected.

Figure 24.

Measurement of the reflection coefficients of the Butler matrix antenna sample.

For measuring antenna gain and radiation patterns, an Atenlab R3 anechoic chamber was used. Figure 25 presents a photograph of the measurement setup for the radiation patterns and gain. During the measurement process, all ports (Port 1–Port 4) were connected using end launch coaxial connectors.

Figure 25.

Measurement of radiation patterns and gain of the Butler matrix antenna sample.

As shown in Fig. 26, the results of the reflection coefficient measurement for the sample of the Butler matrix array antenna indicate that they are below −10 dB. The frequency ranges for each port are as follows:

Port 1 operates in the range of 26.8–31.8 GHz.

Port 2 operates in the range of 25.8–32.8 GHz.

Port 3 operates in the range of 25.8–31.4 GHz. The reflection loss of Port 3 is slightly higher than −10 dB in the frequency range of 29.8–30.5 GHz.

Port 4 operates in the frequency range of 26.8–31.8 GHz.

Figure 27 presents the measured gain results for Ports 1– 4 of the Butler matrix array antenna sample at a frequency of 28 GHz. The gains are as follows: Port 1-11.57 dB, Port 2-14.28 dB, Port 3-11.94 dB, and Port 4-12.84 dB. Additionally, the maximum gains at the frequency point of 28 GHz are: Port 1-15.71 dB, Port 2-17.848 dB, Port 3-14.02 dB, and Port 4-14.96 dB.

Figure 27.

The measured gain of a sample of the Butler matrix antenna.

As shown in Fig. 28, at a frequency of 28 GHz, the measured beam angles for the sample antenna were as follows:

Port 1 had a beam angle of 44°.

Port 2 had a beam angle of −14°.

Port 3 had a beam angle of 15°.

Port 4 had a beam angle of −50°.

In Fig. 28, the discrepancy between the measured and simulated gains, where the measured gain is significantly higher, can be attributed to two main factors: differences in the testing environment and manufacturing tolerances.

Firstly, testing environment differences can lead to variations in measured results, as real-world testing conditions may introduce reflections or interference that are not accounted for in the simulation model. These external influences can enhance the measured gain, creating a discrepancy when compared to idealized simulation results.

Secondly, manufacturing tolerances can also impact the antenna’s performance. Minor variations in the fabrication process, such as slight deviations in component dimensions or material properties, can affect the antenna’s gain characteristics. These tolerances may lead to improved gain in the measured prototype compared to the theoretical model used in simulations.

The azimuth plane beam coverage at a frequency of 28 GHz is depicted in Fig. 29. After normalizing the antenna gains of Ports 1–4, the beam coverage range is more prominently shown to be between −50° and 44°, as illustrated in Fig. 29.

Figure 29.

Normalized beam coverage of the Butler matrix antenna in the azimuth plane at a frequency of 28 GHz.

The comparison between simulation and measurement results reveals that the beam angles of Ports 1– 3 in the sample antenna differ by approximately 4°, while Port 4’s beam angle differs by 11°. Additionally, the measured frequency ranges for Ports 1–4 show a slight increase, which may be attributed to manufacturing discrepancies.

Discussion and comparison

In this paper, similar studies are compared with the proposed design in Table 3. The comparison highlights several advantages of the design presented in this study. First, the sample from this work covers a frequency range of approximately 26–32 GHz, offering a competitive bandwidth that is comparable to or wider than other single-layer and multilayer designs listed in Table 3. For instance, the bandwidth of the proposed design is wider than those reported in references [Reference Xu, Shen, Xue and Hu7–Reference Qin, Chen and Xiang19, Reference Han, Wei and Wang21–Reference Wang, Dai and Sun24] [Reference Lee, Lee and Shin12], and comparable to multilayer designs such as [Reference Liu, Chen and Qin20].

Table 3.

Antenna comparison table for samples and other studies

Second, the size of the proposed antenna (56.8 × 21.2 mm) is relatively compact, particularly when compared to multilayer designs like [Reference Van Messem, Moerman, Caytan, de Paula, Hoflack, Stroobandt, Lemey, Moeneclaey and Rogier18–Reference Han, Wei and Wang21, Reference Yang, Ban, Lian, Yu and Wu23, Reference Wang, Dai and Sun24], which have a significantly larger footprint, which also has larger dimensions. This compact size, combined with the single-layer structure, makes the proposed design well-suited for practical applications requiring miniaturization.

In terms of peak gain, while the gain of the sample (14.28 dB) is slightly lower than those reported in references [Reference Qin, Chen and Xiang19] (14.37 dB) and [Reference Liu, Chen and Qin20] (15.81 dB), it still performs competitively. It surpasses the gain achieved by several other single-layer designs, such as references [Reference Lee, Lee and Shin12] (6.7 dB) and [Reference Pezhman, Heidari and Ghafoorzadeh-Yazdi11] (10.2 dB), demonstrating the effectiveness of the inward-positioned metal vias in enhancing performance without increasing the complexity.

Moreover, the single-layer design presented in this work offers advantages in terms of fabrication simplicity and cost-efficiency, as it avoids the increased complexity and potential dielectric losses associated with multilayer and CPW-based designs, such as those in references [Reference Wang, Polat, Tesmer, Maune and Jakoby16, Reference Van Messem, Moerman, Caytan, de Paula, Hoflack, Stroobandt, Lemey, Moeneclaey and Rogier18], and [Reference Kong, Hu, Li, Zhang and Hong22]. This design choice enhances its suitability for large-scale production and commercial applications.

Overall, the results indicate that the proposed design exhibits favorable characteristics, including broader bandwidth, compact size, and competitive gain, when compared to other research designs. The single-layer structure further enhances the feasibility of this design for mass production and potential integration into commercial 5G systems.