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

90-DEGREE HYBRID COUPLER

The even and odd mode characteristic impedances, respectively denoted by Z_0e and Z_0d, of the slot-coupled line are⁵

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

The parameters k₁and k₂ are calculated from

where h is thickness of substrate, ω_p is the width of the top and bottom equivalent microstrip patches and ω_s 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 D_m, D_S and D_L were determined and then optimized using ANSYS software. For the 90-degree hybrid coupler, the optimized dimensions were D_m=6, D_S=8 and D_L=12.8 mm. The simulated performance of the hybrid coupler is shown in Figure 4.

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

Figure 5 Phase shifter structure.

PHASE SHIFTERS

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 β_ef= β₀ √(ε_r), β_m is the corresponding microstrip propagation constant, l=λ_m/4, λ_m is the effective microstrip wavelength and l_m 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 l_m. In actual application, l_m 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 Z_0e and Z_0o as

Equations 1 through 9 define the relationships between the phase shift Δφ and the dimensions D_m, D_S and D_L. Three new variables D_pmi, D_pSi and D_pLi (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: D_pm45=4.35, D_pS45=7.4, D_pL45=12.1
22-degree phase shifter: D_pm22=11.2, D_pS22=13.2, D_pL22=12.8
67.5-degree phase shifter: D_pm67=2.7, D_pS67=6.2, D_pL67=12.2.