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 [Reference Matthaei, Young and Jones21]. To calculate the impedance Z ₂ and electrical length θ ₂ of the coupler from its equivalent circuit, the equations already obtained in [Reference Matthaei, Young and Jones21] 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 [Reference Babale, Abdul Rahim, Barro, Himdi and Khalily8, Reference Nedil, Denidni and Talbi9], 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°.

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 [Reference Ren, Li, Gu and Arigong14], 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