Microwave Journal
www.microwavejournal.com/articles/37286-to-6-ghz-4-%C3%97-4-and-8-%C3%97-8-butler-matrices-based-on-slot-coupled-technology-and-a-flexible-design-method

2 to 6 GHz, 4 × 4 and 8 × 8 Butler Matrices Based on Slot-Coupled Technology and a Flexible Design Method

December 14, 2021

Wideband (2 to 6 GHz) 4 × 4 and 8 × 8 Butler matrices without crossover circuits are implemented on a three metal-layer structure using hybrid couplers and phase shifters realized with elliptically-shaped coupling structures. A flexible design method for phase compensation is used based on the spacing of the hybrid couplers. Measurements agree with simulation, demonstrating the amplitude and phase performance of the ports. Compared to other Butler matrix configurations, the elliptical coupling structure without crossover circuits has an ultra-wide bandwidth of several octaves.

Butler matrices are used in a wide variety of antenna feed applications, such as beamforming networks. Conventional microstrip line Butler matrices1-4 use crossover circuits, which are relatively narrowband and difficult to route. Abbosh5,6 and Bialkowski5 describe an elliptical slot-coupled technology for the required 90-degree hybrid couplers and phase shifters. The slot-coupled technology is appropriate for broadband devices, and the multi-layer structure is helpful for designing Butler matrices without crossover circuits.7-9

In this work, wideband (2 to 6 GHz) 4 × 4 and 8 × 8 Butler matrices were designed based on slot-coupled technology. A flexible design method for phase compensation was used based on the spacing of the hybrid couplers. Accordingly, each component—the 90-degree hybrid couplers, phase shifters and phase-compensation circuits—can be individually simulated and optimized. Measurement results demonstrate a fractional bandwidth (FBW) of 100 percent.

Figure 1

Figure 1 4 × 4 Butler matrix.

Figure 2

Figure 2 8 × 8 Butler matrix.

BUTLER MATRIX ARCHITECTURES

The Butler matrix is widely used in analog beamforming networks and is composed of N inputs and N outputs. In this article, N is 4 or 8. The number of 90-degree hybrid couplers equals (N/2) log2N, and the number of 45-degree phase shifters is (N/2) [log2N-1]. One of the N inputs produces uniform amplitudes at the output ports with a determined phase difference.7 The phase difference between output ports is different for every input port excitation; with four or eight specific values of the phase differences, the antenna array in a switched beamforming system synthesizes four or eight corresponding beams evenly pointing in different directions. The architectures of the 4 × 4 and 8 × 8 Butler matrices are illustrated in Figures 1 and 2, respectively, and their ideal operation summarized in Tables 1 and 2. A 4 × 4 Butler matrix is composed of four 90-degree hybrid couplers and two 45-degree phase shifters, while an 8 × 8 Butler matrix is composed of twelve 90-degree hybrid couplers, four 45-degree, two 22.5-degree and two 67.5-degree phase shifters.

Table1

 

Table 2
Figure 3

Figure 3 Elliptical coupling structures: hybrid coupler (a), phase shifter (b) and dimensions (c).

Elliptical slot-coupled technology is used for both the 90-degree hybrid couplers and phase shifters.5,6 The multi-layer structure enables the design of the Butler matrices without crossover circuits. Figure 3 shows the elliptically-shaped coupling structures of the hybrid coupler and phase shifter and their respective dimensions. The phase shifter of Figure 3b can be viewed as the hybrid coupler of Figure 3a with two ports terminated by open circuits. The last row of Tables 1 and 2 list the phase differences between adjacent antenna ports in Figures 1 and 2 for each beam direction.

Based on the above discussion, prototype 4 × 4 and 8 × 8 Butler matrices and their components were designed and simulated using a Rogers RO4003C substrate 1.016 mm thick, using a dielectric constant of 3.55 and a loss tangent of 0.0027.



90-DEGREE HYBRID COUPLER

The even and odd mode characteristic impedances, respectively denoted by Z0e and Z0d, of the slot-coupled line are5

where K(k) is an elliptical integral of the first kind and.

The parameters k1 and k2 are calculated from

where h is thickness of substrate, ωp is the width of the top and bottom equivalent microstrip patches and ωs is the width of the equivalent rectangular slot. Their relationships with the actual dimensions (see Figure 3c) are

Using equations 1 through 7, the initial dimensions Dm, DS and DL were determined and then optimized using ANSYS software. For the 90-degree hybrid coupler, the optimized dimensions were Dm=6, DS=8 and DL=12.8 mm. The simulated performance of the hybrid coupler is shown in Figure 4.

Figure 4

Figure 4 Simulated transmission (a), phase (b), isolation and reflection (c) characteristics of the 90-degree hybrid coupler.

Figure 5

Figure 5 Phase shifter structure.

 

PHASE SHIFTERS

f6.jpg

Figure 6 Simulated transmission (a), phase (b) and reflection (c) characteristics of the 45-degree phase shifter.

The phase shifter shown in Figure 2b can be viewed as the hybrid coupler in Figure 2a with two ports terminated by open circuit impedances. Its phase shift is6

where βef = β0 √(εr), βm is the corresponding microstrip propagation constant, l=λm/4, λm is the effective microstrip wavelength and lm is the microstrip line reference length. Δφ is the phase difference between two transmission paths (see Figure 5).

Equation 8 shows there are two degrees of freedom to achieve the desired phase shift: the coupling coefficient C and lm. In actual application, lm must be calibrated with a phase-compensated microstrip line. Thus, the phase shift Δφ is determined only by C, and its value is inversely proportional to C. C is defined by Z0e and Z0o as

Equations 1 through 9 define the relationships between the phase shift Δφ and the dimensions Dm, DS and DL. Three new variables Dpmi, DpSi and DpLi (see Figure 5) are used for the corresponding dimensions of the phase shifters, distinguishing them from those of the hybrid coupler. Similarly, the initial values were calculated and then optimized using ANSYS. The optimized dimensions in mm were:

  • 45-degree phase shifter: Dpm45=4.35, DpS45=7.4, DpL45=12.1
  • 22-degree phase shifter: Dpm22=11.2, DpS22=13.2, DpL22=12.8
  • 67.5-degree phase shifter: Dpm67=2.7, DpS67=6.2, DpL67=12.2.

The simulated performance is shown in Figures 6, 7 and 8, respectively.



4 × 4 BUTLER MATRIX SYNTHESIS

After determining the dimensions of the 90-degree hybrid couplers and the 45-degree phase shifters, the 4 × 4 Butler matrix (see Figure 9) was synthesized using the topology in Figure 1. A flexible layout method places components based on the spacing of adjacent couplers. With this approach, each component, including 90-degree hybrid couplers, 45-degree phase shifters and phase-compensation circuits, can be individually simulated and optimized. Also, the input and output ports of the Butler matrix can be flexibly placed.

Figure 7

Figure 7 Simulated transmission (a), phase (b) and reflection (c) characteristics of the 22.5-degree phase shifter.

Figure 8

Figure 8 Simulated transmission (a), phase (b) and reflection (c) characteristics of the 67.5-degree phase shifter.

Figure 9

Figure 9 Layout and dimensions of the 4 × 4 Butler matrix.

The phase shifters have phase differences with respect to a microstrip line of fixed length. Unfortunately, the lengths of the through lines in the non-phase-shifting paths between the corresponding connected two couplers are lengthy and dependent (see Line 0 in Figure 9). Therefore, to provide the needed phase shift, the extra microstrip lines, whose lengths are determined by the spacing of the connected couplers, must be compensated in the corresponding phase-shifting paths. Ideally, the phase-compensation circuits should be designed so as to not affect the overall layout. The 45-degree phase-shifting paths all have the same trace format of line 1 with the extended lines of length LC. Thus, when the spacing of adjacent couplers is determined, denoted by Sh and Sv in Figure 9, the length of line 0 is determined, and the value of LC is easily adjusted to guarantee a phase difference of 45 degrees between lines 1 and 0. Line 0 can be flexibly routed to maintain a certain line spacing within the same metal layer.

The prototype 4 × 4 Butler matrix was designed and simulated on a Rogers 4003C substrate. Each component, including the 90-degree hybrid coupler, 45-degree phase shifter and the phase-compensation circuit were individually simulated and optimized. Full-wave simulations and optimizations were performed using ANSYS software. Most of the physical parameters have been noted; the remaining physical parameters in Figure 9 are: LC=6.95, W50=1.15, Sh=45.6 and Sv=34.1 mm.

Figure 10

Figure 10 Layout and dimensions of the 8 × 8 Butler matrix

8 × 8 BUTLER MATRIX SYNTHESIS

In a similar fashion, the prototype 8 × 8 Butler matrix was designed and simulated, using Rogers 4003C substrate and each component individually simulated and optimized. The full-wave simulations and optimizations were performed, and the realizable physical parameters are shown in Figure 10. As with the 4 x 4 matrix, most of the physical parameters have been listed previously. The remaining are: L1=9.53, L2=6.35, L3=6.35, W50=1.15, Sh8=46.2 and Sv8=50.85 mm.

Figure 11

Figure 11 Fabricated 4 × 4 Butler matrix.

Figure 12

Figure 12 Fabricated 8 × 8 Butler matrix.

 

MEASURED RESULTS

The fabricated 4 × 4 and 8 × 8 matrices are shown in Figures 11 and 12, respectively. The sizes are approximately 89 × 54 mm for 4 × 4 Butler matrix and 212 × 118 mm for the 8 × 8 matrix.

Figure 13

Figure 13 Simulated (solid black) vs. measured (dotted red) characteristics of the 4 × 4 Butler matrix: transmission (a), reflection (b) and port 1 coupling (c).

Measurements with a Keysight vector network analyzer are in agreement with the simulations, as shown in Figures 13 and 14. As the 4 × 4 Butler matrix has a plane of reflection symmetry, shown in Figure 9, the simulated and measured results are only presented when port 1 is excited. The differential phases are shown with ports 1 and 2 excited. In Figure 13a, S51, S61, S71 and S81 are shown, indicated about 7 dB for the insertion loss with approximately 1 dB of amplitude imbalance. Figure 13b shows greater than 15 dB return loss at all input ports over the entire band from 2 to 6 GHz. Figure 13c shows the isolation characteristics of port 1 from the other three input ports: 15 dB for port 2, 20 dB for port 3 and 30 dB for port 4. The differential phases of beam 1L (with an ideal differential phase of 135 degrees) and 2L (with an ideal differential phase of 45 degrees) are plotted in Figures 14a and b, respectively, showing 5-degree phase imbalance for the 135-degree phase shift of beam 1L and 3-degree phase imbalance for the 45-degree phase shift of beam 2L. The 8 × 8 Butler matrix has similar characteristics (see Figure 15).

Figure 14

Figure 14 Simulated (solid black) vs. measured (dotted red) differential phase between adjacent antenna ports of the 4 × 4 Butler matrix: beam 1L with 135-degree differential phase (a) and beam 2L with 45-degree differential phase (b).

CONCLUSION

Slot-coupled technology was used to design 2 to 6 GHz 4 × 4 and 8 × 8 Butler matrices, and a three metal-layer structure avoided crossover circuits. Phase-compensation circuits were added based on the spacing of adjacent couplers, a helpful design approach. Measurement results agree with the simulations. A 100 percent fractional bandwidth was achieved, which is attractive for wideband beamforming systems.

Figure 15

Figure 15 Measured performance of the 8 × 8 Butler matrix: transmission (a), reflection (b) and differential phase (c) for each beam.

ACKNOWLEDGMENT

This work was supported in part by the National Natural Science Foundation of China under Grant 61671149, Grant 61861136002 and Grant 61701110.

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