A 4 × 4 Butler matrix with switching/steering beams based on new tunable phase difference couplers

Abstract

Basically, a 4 × 4 Butler matrix (BM) connected to an antenna array allows to have four beams, each oriented in a specific direction depending on the excitation port. In this paper, an almost continuously steerable beam system based on a conventional 4 × 4 BM with adjustable phase shift is presented and demonstrated. Here, varicap diodes are used instead of an additional phase shifter. Under different bias levels applied to the couplers throughout these varicap diodes, an output variable phase difference was obtained. A prototype of the proposed tunable BM integrated with an antenna array operating at 3.5 GHz was fabricated and tested. The experimental results show a good agreement with those simulated. A reflection and isolation coefficient better than −15 dB over the entire desired frequency band and an amplitude imbalance lower than ±1.5 dB were achieved. The measured radiating beam under different DC biasing can be oriented from ±6° to ±18° when port 1 or 4 is excited and oriented from ±32° to ±43° for ports 2 and 3.

Introduction

Beamforming is a signal processing technique to send or receive directional signals. The idea is to combine radiation elements in an array in such a way as to create constructive interference for signals at desired angles and destructive interference for the rest of the signals. This technique has become the core technology in modern mobile communication systems, which are developing rapidly.

The Blass [1], Nolen [2], and Butler matrices [3] are famous analog solutions that provide multibeams through the alternative selection of the input excitation. Between these, the Butler matrix (BM) has been implemented in most beam-switching array systems due to its straightforward use, low-loss nature, and easy implementation. The BM can be used in many applications, such as satellite applications [4], massive multi-input multi-output (MIMO) systems [5], and for sending and receiving data between several users [6].

A classical BM is a 2N × 2N network (N is an integer greater than or equal to 1). In a 2N × 2N BM, only 2N spatially orthogonal independent beams can be generated. For example, when N = 2, we will have the best-known 4 × 4 BM, consisting of four couplers, two crossovers, and two phase shifters, along with four input and output ports. By exciting one of the input ports, four signals with equal amplitudes and a phase difference of ±45° or ±135° will be generated at the output ports [4]. Thus, we can produce four different radiating beams when the 4 × 4 BM is associated with an antenna array. However, with the emergence of the Internet of Things and 5G technology, more channel capacity is required. Increasing the number of beams by adopting higher-order Butler arrays (N = 8, 16,…), could be a solution. Nevertheless, the number of couplers, crossovers, and phase shifters increases considerably when the number of beams increases, which results in the circuit size, higher transmission loss, and greater complexity of the circuit design. To reduce design complexity, several efforts have been reported [7–15]. These efforts can be divided into two categories. The first category consists in reducing or avoiding the use of crossovers. For example, a BM based on double-layer structure [7, 8], or the use of multilayer Complementary Metal Oxide Semiconductor (CMOS) technology [9]. The second one consists in eliminating, in terms of appearance, phase shifters by integrating them with couplers. For example, by using couplers with quasi-arbitrary phase differences [10, 11], one can have a BM without phase shifters [8]. The abovementioned studies eliminate phase shifters or crossovers, but the number of couplers remains the same as those in the conventional BM. Other studies have focused on improving the capacity while keeping the order of the matrix [12–15]. These studies are based on the application of additional phase shifting devices. In [12], phase reconfigurable synthesized transmission lines connected to the outputs of a 4 × 4 BM were reported. In a few other studies [13–15], additional phase shifters are integrated with the BM structure instead of placing them at the output ports. These extra phase shifters could be beneficial in increasing the beam’s pointing angle range, usually at the expense of design complexity and additional power loss. However, it would not improve the intrinsic properties of the matrix. Furthermore, another study focused on reconfigurable directional couplers that are used in a BM system is reported in [16, 17], here the author relied on double section couplers with six positive intrinsic negative (PIN) diodes for each coupler, which causes design complexity with high cost. In this paper, a 4 × 4 BM based on couplers with a continuously variable output phase difference is presented. The characteristics of the proposed BM allow a better spatial coverage compared to the conventional 4 × 4 BM while avoiding to increase circuit size and design complexity. The theoretical design analysis of the proposed 4 × 4 BM is presented and discussed. To verify the proposed design, a 4 × 4 BM for 3.5 GHz applications is fabricated and measured. To validate the performance of the beam steering over a wide spatial coverage, the integration of the proposed BM along with a planar antenna array is presented.

Design and analysis

Couplers with a tunable phase difference present more challenges than those with a tunable frequency or power division ratio, as reflected by the limited studies in research articles. For the first time, a 3-dB coupler offering a continuously tunable phase difference was achieved using a tunable phase shifting unit [18]. Enhancements in bandwidth were attained by inserting a phase-tunable transmission line between the branch line segments, as described in [19]. To achieve a phase adjustment span of 180°, in [20] a design that integrates two tunable units, comprising open/short stubs and varactors, between the coupled lines was reported. The main advantage of our proposed tunable coupler based on varicap diodes is its simplicity, ease of manufacture, and low cost.

Coupler design

Figure 1 shows a conventional coupler and its equivalent model. Each branch of the coupler is represented by its Pi-network circuit equivalent [21]. To calculate the impedance Z ₂ and electrical length θ ₂ of the coupler from its equivalent circuit, the equations already obtained in [21] are required:

(1)\begin{equation}\frac{{{C_0}}}{2} = \frac{{\tan \left( {\frac{{{\theta _2}}}{2}} \right)}}{{{Z_2}\omega }}\end{equation}

(2)\begin{equation}{L_2} = \frac{{{Z_2}{\text{sin}}\left( {{\theta _2}} \right)}}{\omega }\end{equation}

Figure 1. (a) Conventional coupler and (b) its equivalent in a Pi-network circuit.

The proposed coupler consists in adding two variable capacitors to each of the two vertical branches, as shown in Fig. 2. This will provide a new electrical length and impedance. Note that the couple (θ, Z) and (θ′, Z) are the new electrical lengths and impedance, respectively, of the left and right vertical branches of the proposed coupler after adding the variable capacitors, C ₁ and C ₂. θ represents the branch with C ₁, θ′ is the branch with C ₂, and Z is the same for both.

Figure 2. (a) The proposed coupler and (b) its equivalent in a Pi-network circuit.

To calculate θ and Z, it can be deducted from equations (1) and (2):

(3)\begin{equation}\frac{{{C_x}}}{2} = \frac{{\tan \left( {\frac{\theta }{2}} \right)}}{{Z\omega }}\end{equation}

(4)\begin{equation}{L_x} = \frac{{Z{\text{sin}}\left( \theta \right)}}{\omega }\end{equation}

where C_x and L_x are, respectively, the equivalent capacitor and the inductance of the left branch after adding C ₁.

Note that ${C_x} = {C_1} + \frac{{{C_0}}}{2}$ and ${L_x} = {L_2}$. This allows writing equations (3) and (4) as follows:

(5)\begin{equation}\frac{{{C_1}}}{2} + \frac{{{C_0}}}{4} = \frac{{\tan \left( {\frac{\theta }{2}} \right)}}{{Z\omega }}\end{equation}

(6)\begin{equation}{L_2} = \frac{{Z{\text{sin}}\left( \theta \right)}}{\omega }\end{equation}

Replacing C ₀ and L ₂ by their values in equations (1) and (2):

(7)\begin{equation}\frac{{{C_1}}}{2} + \frac{{\tan \left( {\frac{{{\theta _2}}}{2}} \right)}}{{2{z_2}\omega }} = \frac{{\tan \left( {\frac{\theta }{2}} \right)}}{{Z\omega }}\end{equation}

(8)\begin{equation}\frac{{{Z_2}{\text{sin}}\left( {{\theta _2}} \right)}}{\omega } = \frac{{Z{\text{sin}}\left( \theta \right)}}{\omega }\end{equation}

from equations (7) and (8), it can be deducted as follows:

(9)\begin{equation}\theta = 2{\text{ta}}{{\text{n}}^{ - 1}}\left[ {\frac{{Z\omega {C_1}}}{2} + \frac{Z}{{2{Z_2}}}{\text{tan}}\left( {\frac{{{\theta _2}}}{2}} \right)} \right]\end{equation}

(10)\begin{equation}Z = \frac{{{Z_2}{\text{sin}}\left( {{\theta _2}} \right)}}{{{\text{sin}}\left( \theta \right)}}\end{equation}

where Z ₂ is the impedance before adding the variable capacitors, Z ₂ = 35,35 Ω.

To calculate the electrical length of the right vertical branch θ′, simply replace C₁ with C ₂ in equation (9).

Once the electrical lengths θ and θ′ have been determined according to the added variable capacitors, it will be necessary to proceed with the analysis in even–odd mode to obtain the parameters of the proposed coupler. For this purpose, we can reuse all the results obtained after the analysis of the odd–even mode in [8, 9], precisely the closed-form equations:

(11)\begin{equation}P = \left| {\frac{{{S_{21}}}}{{{S_{41}}}}} \right|\end{equation}

(12)\begin{equation}\psi = \angle {S_{41}} - \angle {S_{21}}\end{equation}

P and $\psi $ are, respectively, the power division ratio and the phase difference between the output ports. Port 1 is chosen as the input, 2 and 4 are the output ports, and port 3 is the isolated one.

(13)\begin{equation}{Z_1} = {Z_0}P\left| {{\text{sin}}\psi } \right|\end{equation}

(14)\begin{equation}Z = \frac{{{Z_0}P\left| {{\text{sin}}\psi } \right|}}{{\sqrt {1 + {P^2}{\text{si}}{{\text{n}}^2}\psi } }}\end{equation}

where Z ₀ and Z ₁ are, respectively, the input impedance and the impedance of the two horizontal branches, with Z ₀ = Z ₁ = 50 Ω.

(15)\begin{equation}{\theta _1} = \frac{\pi }{2}\end{equation}

(16)\begin{equation}\theta + \theta ' = \pi \end{equation}

(17)\begin{equation}\theta = {\text{ta}}{{\text{n}}^{ - 1}}\left( {\frac{{{Z_0}{\text{tan}}\psi }}{{{Z_1}}}} \right)\end{equation}

(18)\begin{equation}\theta ' = \pi - {\text{ta}}{{\text{n}}^{ - 1}}\left( {\frac{{{Z_0}{\text{tan}}\psi }}{{{Z_1}}}} \right)\end{equation}

Since Z ₀ = Z ₁, equation (17) can be simplified as:

(19)\begin{equation}\psi = \theta \end{equation}

Since ψ = θ, it is clear from equations (10) and (14) that Z will depend on the coupler’s output phase difference ψ. For example, when ψ = 90° (classical case), Z will be equal to 35,36 Ω. Since the proposed coupler consists in having a variable output phase difference, Z will be fixed at 46 Ω after optimization, independently of the ψ value. More explanations are given later.

Equation (13) shows that to maintain a tolerable coupling coefficient P of −3 dB ± 0.5 dB where Z ₀ = Z ₁ = 50 Ω, the output phase difference ψ has to be limited to 90° ± 40°.

By replacing θ by its value in equation (9), equation (19) became:

(20)\begin{equation}\psi = 2{\text{ta}}{{\text{n}}^{ - 1}}\left[ {\frac{{Z\omega {c_1}}}{2} + \frac{Z}{{2{Z_2}}}{\text{tan}}\left( {\frac{{{\theta _2}}}{2}} \right)} \right]\end{equation}

Equation (20) shows that the output phase difference of the coupler depends mainly on C ₁, with the other parameters remaining constant. Also, the values of capacitors C ₂ depend on C ₁ and are used to obtain the θ′ required to satisfy the condition (18).

To control the two variable capacitors independently, two biasing voltages are required. At 3.5 GHz, the optimized values of θ ₂ and Z ₂, constituting the θ and Z of the proposed hybrid coupler, are, respectively, 79° and 46 Ω. These values have been optimized to keep having good performances of a classical 3-dB/90° hybrid coupler in terms of good matching, good isolation, and a good power ratio, while the phase shift is now ψ (see equation 20) instead of 90°. This optimization was validated by CST simulation software, and the values are shown in Table 1, while Fig. 2 illustrates the definitions of each parameter.

Table 1. Optimized parameters of the proposed coupler

BM design

As a recall, a classic 4 × 4 BM has four input ports and four output ports (Fig. 3(a)). It consists of four 90° couplers, two 45° phase shifters, and two crossovers. When the BM is connected to an antenna array, it can provide four beams, each directed in a specific direction, as shown in Fig. 3(a). Comparing this conventional matrix with the 4 × 4 BM proposed in this work, as shown in Fig. 3(b), both matrices have the same structure, except that the proposed matrix allows having four beams, each of which can be oriented continuously in different directions. These directions are controlled by the voltages applied to the couplers.

Figure 3. Schematic of (a) a conventional 4 × 4 Butler matrix and (b) the proposed 4 × 4 Butler matrix.

Here, the proposed BM consists of two couplers with a phase difference of ${\beta _1}$, two couplers with a phase difference of β ₂, two crossovers, and two 45° phase shifters. The input ports are denoted (1–4) and the output ports are (5–8), as presented in Fig. 3(b). Table 2 summarizes the phase responses obtained at the output ports and the corresponding excited input ports.

Table 2. The phase response of the proposed Butler matrix

The progressive phase differences Δθ between the output ports depending on the excitation case are obtained as follows:

Case 1: P1 excited

(21)\begin{equation}\Delta {\theta _1} = \angle {S_{61}} - \angle {S_{51}} = \angle {S_{71}} - \angle {S_{61}} = \angle {S_{81}} - \angle {S_{71}}\end{equation}

(22)\begin{equation}\Delta {\theta _1} = - {\beta _1} + 45^\circ = - 45^\circ - {\beta _2} + {\beta _1} = - {\beta _1} + 45^\circ \end{equation}

Case 2: P2 excited

(23)\begin{equation}\Delta {\theta _2} = \angle {S_{62}} - \angle {S_{52}} = \angle {S_{72}} - \angle {S_{62}} = \angle {S_{82}} - \angle {S_{72}}\end{equation}

(24)\begin{equation}\Delta {\theta _2} = {\beta _1} + 45^\circ = - {\beta _1} - 45^\circ - {\beta _2} = {\beta _1} + 45^\circ \end{equation}

Case 3: P3 excited

(25)\begin{equation}\Delta {\theta _3} = \angle {S_{63}} - \angle {S_{53}} = \angle {S_{73}} - \angle {S_{63}} = \angle {S_{83}} - \angle {S_{73}}\end{equation}

(26)\begin{equation}\Delta {\theta _3} = - {\beta _1} - 45^\circ = {\beta _1} + 45^\circ + {\beta _2} = - {\beta _1} - 45^\circ \end{equation}

Case 4: P4 excited

(27)\begin{equation}\Delta {\theta _4} = \angle {S_{64}} - \angle {S_{54}} = \angle {S_{74}} - \angle {S_{64}} = \angle {S_{84}} - \angle {S_{74}}\end{equation}

(28)\begin{equation}\Delta {\theta _4} = - 45^\circ + {\beta _1} = - {\beta _1} + 45^\circ + {\beta _2} = - \,45^\circ + {\beta _1}\end{equation}

From equations (21–28), it and can be deduced that $\Delta \theta $ depends on the value of ${\beta _1}$, and ${\beta _1}$ depends on ${\beta _2}$:

(29)\begin{equation}\Delta {\theta _1} = - {\beta _1} + 45^\circ \end{equation}

(30)\begin{equation}\Delta {\theta _2} = {\beta _1} + 45^\circ \end{equation}

(31)\begin{equation}\Delta {\theta _3} = - {\beta _1} - 45^\circ \end{equation}

(32)\begin{equation}\Delta {\theta _4} = - 45^\circ + {\beta _1}\end{equation}

(33)\begin{equation}{\beta _1} = 45^\circ + \frac{{{\beta _2}}}{2}\end{equation}

From equations (29–33), the case of the classical 4 × 4 matrix can be obtained when β ₁ = β ₂ = 90°. The progressive phase differences between the output ports will therefore be ±45° and ±135°.

Simulation and measurement results

To verify the design concept, a BM connected to an antenna array (half-wavelength distance between adjacent elements) by transmission lines operating at 3.5 GHz is designed and fabricated. Figure 4 shows a photograph of the prototype fabricated on a printed circuit board: Rogers RT5880 substrate with a thickness of 0.508 mm, a dielectric constant of 2.2, and a loss tangent of 0.0009.

Figure 4. (a) Photograph of the fabricated BM integrated with a planar antenna array; (b) schematic diagram of the reconfigurable coupler; and (c) a zoom-in of the reconfigurable coupler.

A commercially available varactor MA46H120, from MACOM Technical Solutions, is used to provide tunability as it provides a capacitance tuning range from 0.165 pF to 1.23 pF at 3.5 GHz by adjusting the DC voltage from 0 to 19 V [22]. This allowed us to have a range of output phase differences from 52° to 128° while keeping a coupling factor of −3 dB (±0.5 dB).

Therefore, to clearly discuss the EM simulations as well as the measurement results, only three output phase shift difference values will be treated and analyzed. Here, the two extreme values (i.e., 52° and 128°) and one 90° allowing to have the conventional structure are chosen to conduct this analysis. Here, the following three configurations will be studied:

1. 1. Config_1: when the phase difference is at its minimum value, ${\beta _2}\,$ = 52$^\circ $.

2. 2. Config_2: when ${\beta _2}\,$ = 90$^\circ $, as already explained here, the conventional case is verified.

3. 3. Config_3: when the phase difference is at its maximum value, ${\beta _2}\,$ = 128$^\circ $.

We can deduce from equation (33) that when ${\beta _2}\,$ = 52$^\circ $ (Config_1), ${\beta _1}$ = 71$^\circ $. When ${\beta _2}\,$ = 90$^\circ $ (Config_2), ${\beta _1}$ = 90$^\circ .\,$When ${\beta _2}\,$ = 128$^\circ $ (Config _3), ${\beta _1}$ = 109$^\circ $.

Table 3 shows in detail the value of ${\beta _1}$ regarding ${\beta _2}\,$ and the required voltage values for each phase difference.

Table 3. Required voltage for each configuration

Note: BM = Butler matrix.

BM

Unfortunately, it will not be possible to have the BM measurements without the antenna array since, in our prototype, they are connected by transmission lines, as shown in Fig. 4. We think it would not be very practical to fabricate another prototype since we can validate the principle of our work by measuring the radiation pattern at different control voltages.

The simulated results of S-parameter and phase difference are depicted in Figs. 5–7, respectively, for Config_1, Config_2, and Config_3. As can be observed, from 3.4 GHz to 3.6 GHz, the worst case of the insertion loss is about −7.7 dB and occurs for Config_1, while the reflection and the isolation are better than −15.5 dB at all ports for the three configurations.

Figure 5. Simulated S-parameters and phase difference of the proposed Butler matrix for Config_1 when Δθ = ±26$^\circ $, ±116$^\circ $.

Figure 6. Simulated S-parameters and phase difference of the proposed Butler matrix for Config_2 (Conventional case) when Δθ = ±45$^\circ $ and ±135$^\circ $.

Figure 7. Simulated S-parameters and phase difference of the proposed Butler matrix for Config_3 when Δθ = ±64$^\circ $, ±154$^\circ $.

The same figures illustrate the simulated progressive phase shift, and it is clearly noted that, as was expected, those values achieved are in great agreement with slight variations to what was theoretically expected. These variations are more seen for Config_1 from 3.40 GHz to 3.425 GHz, a little bit far for the working frequency of 3.5 GHz.

BM with antennas

To verify the radiation performance, a four-element patch linear antenna array is connected to the proposed BM. Each patch has a size of 29 mm × 29 mm, with a half-wavelength between adjacent patches. The comparison of the simulated and measured results of the radiation pattern of the three configurations is presented in Fig. 8(a–c). A good agreement between both simulation and measurement results can be noted, in particular for the beams obtained from the excitation of ports 1 and 4. A slight disagreement between simulations and measurements is observed and is varied between 0° and 4°, while it can reach 7° when the RF signal is from port 2 or 3. This could be due to the wires used for DC voltage and/or the misconnection (i.e., weld the connector) of the SMA connector at the different input ports.

Figure 8. Simulated and measured radiation pattern: (a) Config_1, (b) Config_2, and (c) Config_3.

As shown in Fig. 8(a), the beam directions in Config_1 are oriented at ±6° for ports 1 and 4, and at ±26° (simulated) and ±32° (measured) for ports 2 and 3. In the second configuration, Config_2, as can be seen in Fig. 8(b), the beams are oriented at ±10° when ports 1 and 4 are excited, and at ±30° (simulated) and ±36° (measured) when the signal is from ports 2 and 3.

Figure 8(c) shows the results of the radiation pattern of the third configuration, Config_3. It can be observed that the orientation of the beams obtained from ports 1 and 4 is at ±14° (simulated) and ±18° (measured), and at ±36° (simulated) and ±43° (measured) for ports 2 and 3.

According to Fig. 9, through all sets of measured curves of the three configurations, the three sequences of the beams directions related to each configuration can be easily distinguished. When ports 1 and 4 are the input ports, the measured main beam is steered from ±6° to ±18° and from ±32° to ±43° when ports 2 and 3 are considered as input ports. A comparison with other relative Butler matrices is presented in Table 4. It is clear that the proposed BM has a continuously tunable phase difference when compared with the other proposals. Although in [14], the output phase difference is also continuously tunable, except that external phase shifters had to be added, resulting in increasing design area, which is directly related to the design cost and significant insertion losses compared to our proposed design.

Figure 9. Measured radiation pattern of the three configurations.

Table 4. Comparison between the proposed Butler matrix with the state-of-the-art

Conclusion

A new concept of BM based on the conventional 4 × 4 structure is presented. This new concept demonstrates the feasibility of getting a continuous tunability of beams where it is not possible with the classical structure. The proposed Butler matrix is based on couplers with a tunable phase difference. To control the phase difference, each coupler requires four varactor diodes controlled by two bias voltages. Each of the four diodes of the couplers in the same row is fed by the same DC voltage. The proposed BM has been connected to an antenna array and then fabricated and tested. The prototype has a simple structure and good radiation performance, capable of radiating four beams, where each beam can be steered in a range of 12°. With these results, the proposed system could become an attractive solution in beamforming and steering applications.