Waveguide divider design based on null of electric field

Nowadays, the design of dividers is based on electromagnetic software that optimizes some geometric parameters to obtain the required performance. The choice of the geometry of the discontinuities contained in the divider and of the optimization initial point is quite critical to satisfy the divider requirements. In the last years, it is quite rare to find in the literature a theoretical approach helping the designers in the choice of the divider geometry. Helpful suggestion can derive by the analysis of the electric field in a trial divider that satisfies power division among the output ports in a thin band. In fact, the electric field null can be filled with metallic septa that ensure the same behavior at any frequency. The optimization of the septa position/form with numerical electromagnetic software permits to obtain divider with large bandwidth. A further analysis of the electric field null in the divider permits to add lateral metallic septa that further enlarge the transmission band. Finally, the design of an input matching network increases the transmitted power to the desired value.


Introduction
Waveguide dividers and couplers have been studied since '40s. Typical examples are Riblet's coupler, Bethe coupler, Wilkinson coupler [1][2][3]. These kinds of devices are based on fine and winning theoretical approaches. Nowadays, theoretical approaches to the study of couplers and dividers are quite rare [4][5][6], being replaced by "black box" approaches based on numerical software as CST, HFSS and others [7][8][9][10][11][12][13][14]. These softwares are able to design devices by means of an optimization of the geometrical parameters of a trial device until the desired functional is minimized. Usually, the functional is related to the performance required by the device, as the minimization of reflection and equiamplitude of the output power, in case of dividers. Many examples of dividers based on this approach are discussed in the literature, and they can be divided in two classes. The first analyzes couplers based on top coaxial input that feeds output ports usually distributed along a ring [4][5][6] while the second is based on dividers designed in the E or H plane where an input port feeds the opposite output ports. These dividers can be based on Riblet's theory [9,12], where insulated ports are closed on matched waveguide, or Bagley's theory [14] or other approaches based on the interactions between discontinuities to obtain power division [8,10,13].
In the designs that require use of simulation software, the main problem in finding the solution is the choice of the initial point for the geometrical optimization. In fact, the probability to incur in local minimum is strictly related to the starting point of the optimization. Hence, this choice is quite important to obtain the required performance for the device under study.
This paper focuses on equi-amplitude 1:N waveguide dividers, starting from a theoretical circuit approach that permits to obtain a single-frequency divider satisfying the required performances. Obviously, this divider has a thin bandwidth, but it contains a fundamental information to design a larger bandwidth divider. In fact, this trial divider is used to find the positions where the electric field vanishes in the divider core. These positions are filled with central metallic septa that ensure electric field null for any frequency, enlarging the bandwidth of the trial divider. A further analysis of the electric field in the divider core with central septa permits to understand the placement of lateral septa that further enlarge the bandwidth. The final stage is the design of a matching network that increases the power transmitted to each output port to the theoretical maximum value for a lossless divider, 10 log 10 ( 1 N ) dB, where N is the number of the output ports.

Theory
The goal is the design of H-plane waveguide equi-amplitude dB) with a band of 10% centered at 10 GHz. Many realizations of the divider could be proposed and, for example, a possible realization is shown in Fig. 1, which is formed by the divider core, where the electromagnetic field interacts with divider ports yielding to power division between outputs, and an input matching network that ensures matching and maximum power transfer to the divider core and, indirectly, to the output ports. The divider is represented by a 4 × 4 scattering matrix S where the terms S k1 , k = 2, 3, 4 represent the transmission toward the output ports when the first port is excited. The simplest configuration of the divider shown in Fig. 1, i.e., the empty divider without iris ( w iris = a ), does not satisfy the required power division and an optimization of h t does not permit to reach this goal. Anyway, there is a very fast approach to reach the required power division among the output ports also for this structure. The approach is similar to that used in [15] to design four and six-ports hybrid couplers. In fact, the required power division can be obtained placing a proper iris at port 3 (central output port). In fact, as discussed in [16], a divider can be represented with a four-port equivalent circuit based on a polygon with four normalized susceptances placed at sides ( b 12 , b 23 , b 34 , b 14 ) and two susceptances on the diagonals ( b 13 , b 24 ). Moreover, four input transmission lines of electrical length 1 , 2 , 3 , 4 connect each side to the actual four ports of the device, as shown in Fig. 2. The susceptances are normalized to the modal impedance of the TE 10 mode by the transformers. Once the equivalent circuit has been defined, and setting b 23 = b 14 and b 13 = b 24 by symmetry, we can evaluate the particular value of the susceptance b 34 that satisfies | | S 21 With simple circuit/mathematical manipulations, we can write the first column of the scattering matrix of the divider as:  N 11 is not reported because its knowledge is not important at this stage and Δ is the real denominator of | | S k1 | | . The key to obtain an equi-amplitude divider is to impose N 21 = N 31 , obtaining two possible solutions: These solutions permit to design a trial divider, placing an iris at a proper distance from the third port, to change the global value of b 34 in order to satisfy (5) or (6) [15].
In fact, the equivalent circuit of the empty core divider shown in Fig. 1, without iris and matching network ( w iris = a, a m1 = a m2 = a ), can be easily identified solving four polynomial equations that permit to obtain the values of the four transmission lines lengths with simple circuit manipulations [16]. Hence, the scattering matrix of the empty core divider at the reference planes, S empty , can be numerically evaluated by CST at 10 GHz for s = 3, a = 22.86 (dimensions are in mm throughout the paper), and it can be used to identify the elements of the equivalent circuit shown i n Fi g . = 4.7 • . As previously discussed, equi-amplitude division is not obtained with the empty divider, but the presence of an iris at a proper distance from port 3, as shown in Fig. 1, could permit to reach the required goal.
In fact, if we choose an iris with susceptance b iris = −b , satisfying (5). Now, we must choose the iris dimensions to synthesize the obtained value for b iris . The first choice should be to select a capacitive iris, with vertical diaphragms, but this would require a global divider in the H-plane with an iris in the E-plane. Our goals is to obtain a divider in the H-plane, and the desired susceptance b iris can be realized with an inductive iris with value −b iris , surrounded by two lines of lengths rev = − arctan 2∕b iris = 50.4 • and 180 • − rev [15]. Hence, the distance of the inductive iris from the third input port is: iris = 180 • − empty 3 − rev = 124.9 • = 2.18 rad = L iris = 2 g L iris which corresponds to L iris = 13.78 mm. Finally, the width of the iris is obtained by a simple evaluation [2], obtaining w iris = 11.61 mm with thickness t = 0.2 mm.
increase to the desired value −4.77 dB at 10 GHz. The divider performance can be increased if the iris at port 3 is optimized in order to obtain a centered band ( L iris = 12.5 , w iris = 11.46 ) with a new matching network ( L m0 = 9.5 , L m1 = 18 , L m2 = 11.5 , a m1 = 16.66 , a m2 = 17.66 ), as shown in Fig. 3b. In this case, the transmission band is 600 MHz again, but it is centered at 10 GHz, as required.
The previous example, even if not able to satisfy requirements, is important to understand how to enhance the performance of the divider. In fact, if we compare the electric field distribution in the empty divider, Fig. 3c, and the field in the divider with the iris, Fig. 3d, we can observe the presence of a null (dark blue) in the core of the divider near the third port and symmetrically placed. This null is the main effect of the iris placed at the third port which ensures that power is equally distributed to the output ports. It is evident that this electric field null can change its position and intensity by varying frequency. Hence, if a metallic septum is placed in the null zone, it ensures null of electric field at any frequency. At the same time, the iris can be removed, being useless for the presence of the septum. This placement permits to enhance divider performance by means of an optimization of the metallic septum position and dimension.
Following this approach, we try to optimize the core divider, i.e., the divider without the input matching network, to enlarge the transmission band, acting on the position and dimension of a metallic septum in symmetric position, in the absence of the iris at port 3, as shown in Fig. 4a. Hence, the goal is to obtain a quasi-flat transmission for  Fig. 1 with iris at port 3. Scattering parameters amplitude | | S k1 | | for excitation at port 1 (dimensions expressed in mm): a iris by (5); b iris optimized to center transmission band. c Electric field in the empty divider at f = 10 GHz; d electric field in the divider with iris at f = 10 GHz (Color figure online) reflection |S11 core | |S11 div | |S21 core |=|S41 core | |S21 div |=|S41 div | |S31 core | |S31 div | |S21 core |dB-|S31 core |dB |S21 div |dB-|S31 div |dB desired value of −4.77 ± 0.1 dB. This design is omitted, at this moment, to try to further increase transmission band. To do this, the amplitude of the electric field in the divider core are shown in Fig. 5a-c at 9.5, 10 and 10.5 GHz, for an excitation at port 1.
Our intent is to observe where the electric field goes to zero (dark blue): this occurs symmetrically in the right/left part of the divider near port 2 and 4. These zones change their form with frequency as highlighted by the red arrows. As in the case of the electric field null due to the iris at port 3, we can try to add a metallic septum to enforce null of the electric field in these zones at any frequency. To do this, two lateral metallic septa of length w p with thickness t = 0.2 are placed near ports 2 and 4, as shown in Fig. 6a, and a global optimization on h t , w s , L s , w p is performed with CST, obtaining: h t = 13.14, w s = 3.94, L s = 8.94, w p = 3.94 . The amplitude of the scattering parameters for excitation at port 1 are shown in Fig. 6b for the core divider ( a = a m1 = a m2 ). | | | S core 11 | | | < −16 dB and the transmission band is greater than 2 GHz. These results show that the introduction of the lateral metallic septa where the electric vanishes is a good strategy to enforce vanishing of the electric field at any frequency, increasing transmission band. This is confirmed by the behavior of the electric field in Fig. 7a-c at 9.5 GHz, 10 GHz and 10.5 GHz: the distribution of the electric field is quite similar for the three analyzed frequencies.
The last step is the design of a matching network based on two optimized g ∕4 transformers [17]. This design is not performed with CST to reduce computational time. In fact, the optimization with CST could require too many steps to converge to the best result in terms of matching network. Moreover, if the first matching stage is placed at a Frequency(GHz) 9 9.5 10 10.5 11 (b) Fig. 6 a Divider with central and lateral metallic septa. b Scattering parameters amplitude | | S k1 | | for excitation at port 1 for the core divider ( h t = 13.14, w s = 3.94, L s = 8.94, w p = 3.94, a = a m1 = a m2 ) proper distance L m0 from the input port of the divider core to neglect effect of higher order modes, as shown in Fig. 6a, the matching network does not alter the field distribution in the divider core or the transmission band, but lowers | | S 11 | | and increases | | S 21 | | , | | S 31 | | to the desired values. Hence, the matching network is designed evaluating at first the scattering matrix S core at the input port of the unmatched divider core in the band of interest with CST.
Then, the transition between two waveguides with different width is evaluated with the MEN approach [18,19], considering three accessible modes at the input and output ports, obtaining the 6x6 scattering matrix S wr 1 →wr 2 . This analysis is needed to simulate the transition between the stages of the g ∕4 transformers. Then, a database of the 6x6 scattering matrix S wr 1 →wr 2 is evaluated for different width in the range 18.86 ÷ 26.86 mm (steps 0.1 mm). The database requires short computational time with a Fortran code (about 15 s for the full band for each transition). The database building is performed only one time, and it is used in the design of any matching network in the following of the paper. Finally, the possible combinations of the previous transitions separated by lengths L m0 , L m1 , L m2 are evaluated by cascading the relative scattering matrices S wr 1 →wr 2 , shifted by the lines j 10 L , 30 L , 50 L , where 10 , 30 , 50 correspond to the three propagating, or not, accessible modes and L to L m0 , L m1 , L m2 . The scattering matrix cascade of all combinations yields to a database of the scattering matrix of the matching network S mn shown in Fig. 6a. The evaluation of the database of S mn requires about ten minutes.
The last step is the choice of the best matching network S mn for the scattering matrix of the divider core S core . The reflection coefficient of the global divider (matching network and divider core) is obtained as combination of S mn and S core : The choice of the best matching network is done starting from (7), using the database corresponding to all cases of matching network. The computation time is very fast (some minutes to obtain the best configuration for the matched divider). This numerical matching technique is applied to the divider core of Fig. 6a, obtaining: L m0 = 13.5 , L m1 = 8 , L m2 = 13.5 , a m1 = 18.26 , a m2 = 19.66 . The scattering coefficients of the matched divider are shown in Fig. 8 Fig. 8. Hence, we can assume a global bandwidth of 1.8 GHz (9.1÷10.9) that is more than required. Following the previous example, we can define an optimization procedure in three steps: first, an iris is placed at port 3 to equally share power among outputs and the electric field in the divider core is analyzed; second, the transmission to the output ports is optimized in the required band by placing proper metallic discontinuities in the position of the divider core where the electric field vanishes; third, a proper matching network is placed at the input port of the divider core to ensure low reflection in the required band. These three tasks are performed separately enhancing the optimization time. In fact, the first is based on a theoretical approach, while the other two tasks can be analyzed in two different phases being independent the two goals, as long as the transmission to the output ports is analyzed and insulation between ports is not taken into account.
A final remark must be done about the efficiency of the proposed approach. The bandwidth can be enlarged Frequency (GHz) 9 9.5 10 10.5 11 Fig. 8 Scattering parameters amplitude | | S k1 | | for excitation at port 1 for the divider shown in Fig. 6a by inserting some septa in the coupler, using an iterative approach, but it is not possible to evaluate a priori the maximum bandwidth that can be obtained. In fact, while the first two-three septa drastically alter the electromagnetic field of the empty device, the septa added at the successive steps can vary only some details of the global electromagnetic field, with a small increase of the bandwidth. Hence, the band enlargement tends to diminish with the number of septa/ iterations.
If the isolation between ports must be taken into account in the design process, matching, power division and insulation must be taken into account at the same time. This approach is very complex as confirmed by the literature, showing that insulation is rarely included in the optimization.
The proposed approach is applied to the four-port star divider shown in Fig. 9a. The goal is to choose proper metallic septa to obtain 1 GHz transmission band centered at 10 GHz.

| | |dB
, the divider core with the iris shows a transmission band of about 600 MHz (9.7÷ 10.3) and its electric field is shown in Fig. 10a. The field behavior suggests to place a central metallic septum ( w s , L s ) as done for the previous divider and shown in Fig. 9a, with w p = 0. The optimization of this septum ( w s = 3.08, L s = 3.78 ) gives the result shown in Fig. 9b with black and red squared lines ("central"): the behavior of | | | S core 21 | | | is more flat than the iris case and the transmission band ("tr. band. central" in the figure) is about 1.1 GHz, centered at 10 GHz. This divider core satisfies the requirements, but it is interesting to discuss the behavior of the electric field shown in Fig. 10b-d (only half divider core by symmetry). The field vanishes between ports 1 and 2 in an area that increases with frequency as highlighted by the arrows. In the previous divider, this occurred near port 3. This is due to the absence of the curved arms that, probably, forced the electric field to "curve" toward them. in the first cavity, between ports 1 and 2, the electric field null changes its extension and form with frequency, and this fact could suggest to add two other lateral septa in that position to further enlarge transmission band. On the other hand, the required goals in terms of reflection and transmission band have been obtained without these new septa; hence, this further design is not performed. These 1:5 dividers can be used to design a 1:25 BFN, as shown in Fig. 17a. The expected transmission band should be 1.2 GHz ( 9.4 ÷ 10.6 ), by the transmission band of the divider results shown in Figs. 15a and 16b. Unfortunately, the coupling between ports of the stage 1 divider reduces this band. To lower this effect, optimal choice of the length of the lines that connect stage 2 to stage 1 has been evaluated ( L ext = 35, L inner = 11, L center = 22 ), and a new input matching network must be designed ( L m0 = 8, L m1 = 9, L m2 = 12 , a m1 = 19.06, a m2 = 19.66 ). The global scattering coefficients of the BFN when port 1 is excited are shown in Fig. 17b

Conclusion
A technique to design divider has been proposed, and it is based on the analysis of the electric field in the inner of the divider core. In fact, the initial design based on the insertion of irises at the output ports permits to identify where trial metallic septa can be placed in order to ensure that the electric field vanishes at any frequency. After an optimization of the septa, another analysis of the electric field gives the trial position of lateral metallic septa that could ensure an enlargement of the transmission band. A final optimization of the septa geometric parameters gives final transmission band that can satisfy the requirement. Finally, a matching network at the input port is designed to ensure that the transmission be at the desired value 10 log 10 ( 1 N ) , where N is the number of output ports.
Funding Open Access funding provided by Università Politecnica delle Marche.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creat iveco mmons .org/licen ses/by/4.0/.