10.1002/2016RS005999

Figure 3. Physical structure in all-inductive dual-mode waveguide technology implementing the coupling topology of

Figure 2.

Once the equivalent circuit of the whole triplexer is obtained, the second step is the physical implementation of the structure in a given technology. Physically, junctions in Figure 1 will be designed using E- and H-plane waveguide T-junctions, while channel filters, with the topology shown in Figure 2, will be implemented by using the already mentioned all-inductive dual-mode technology [Guglielmi et al., 2001; Martinez Mendoza et al., 2015]. The physical structure implementing the selected coupling topology is presented in Figure 3. In this structure, each mode resonating inside a dual-mode cavity is isolated from each other (no coupling between resonances (2) and (3) or between (4) and (5)). Input and output single-mode cavities are included in the design to provide for the resonances (1) and (6) shown in Figure 2.

For the physical design of the channel filters using all-inductive dual-mode technology the technique described in Martinez Martinez et al. [2014] and Martinez Mendoza et al. [2015] is used. This technique is computer-based, which means that optimization operations are needed to obtain the desired response. However, the employed strategy allows the optimization of individual parts of the structure separately, with a small number of parameters at each step. The optimization of the whole structure is thus avoided, and it is substituted by a sequential optimization of small parts of the filter. This makes extremely simple the optimization steps, which usually converge very fast without the risk of being trapped into local minima.

The design example given in this paper will target a noncontiguous triplexer centered in the X band at 11 GHz. The CPRL is fixed to −20 dB with a rejection level between channels of −35 dB. This high rejection level between channels will be possible thanks to the implementation of two transmission zeros in the insertion loss response of the channel filters on both sides of each passband. The validity and usefulness of the novel manifold multiplexer with all-inductive dual-mode rectangular waveguide filters has been proven by means of full-wave rigorous frequency response simulations, obtained with the commercial software FEST3D [Carceller et al., 2014].

2. Theory

This section is focused on the study of multiplexing networks based on the manifold configuration. Particularly, design strategies are covered to use for the first time all-inductive dual-mode rectangular waveguide filters with a manifold multiplexer configuration (see Figure 1). The complete design involves the synthesis of an equivalent circuit model of the structure. The second step is the physical implementation using all-inductive dual-mode waveguide technology. Both steps are covered in this section.

The synthesis of the equivalent circuit starts from defining a suitable filter topology, as shown in Figure 2. By using the techniques reported in Martinez Martinez et al. [2014] and Martinez Mendoza et al. [2015], the coupling matrix representing the channel filters is obtained as

Figure 4. Equivalent circuit for the selected filter topology. Mij values are directly the J-admittance inverters, and Mii values are constant susceptances (Bi) for each resonator (FIR elements). The values of the shunted LC elements are the same for all resonators.

By applying general coupling matrix theory [Cameron et al., 2007] it is possible to obtain an explicit equivalent circuit model corresponding to the coupling matrix shown in equation (1). The equivalent circuit model chosen in this work represents each resonator with a shunted LC circuit and a Frequency Invariant Reactance (FIR) element. Couplings between each resonator will be represented by J-admittance inverters. The explicit equivalent circuit corresponding to the filter topology shown in Figure 2 is presented in Figure 4.

The values of the inductors (L) and capacitors (C) of the resonators are calculated directly by using the low-pass to bandpass transformation, once the bandwidth (BW) and center frequency of each channel (fc) are specified. In the design example the central channel is placed at 11 GHz with BW = 36 MHz bandwidth per channel. Since it is a noncontiguous triplexer, the center frequencies of adjacent channels will be separated by 75 MHz, leading to the following central frequencies for each channel: 10.925 GHz, 11 GHz, and 11.075 GHz. This represents a narrowband multiplexer having a relative bandwidth of 2.3%.

With this information, the LC values of all resonators in a given filter can be computed as

It is important to note that all resonators in a given filter have the same LC values. The FIR elements of the equivalent circuit are in charge of taking into account for the exact resonant frequency of each resonator in asynchronously tuned topologies. The shift of each resonant frequency with respect to the center frequency (fc) of the filter is given by the diagonal terms of the coupling matrix in equation (1), and therefore, the FIR elements take the form

The last elements of the equivalent circuit that need to be computed are the J-admittance inverters controlling the couplings between resonators. These are directly related to the o -diagonal terms of the coupling matrix (1). Then, one can directly write

Moreover, it is important to mention that the previous expressions apply to a coupling matrix with all low-pass circuit elements normalized to unity, including normalized source and load terminal impedances to 1 Ω. This is the reason why the elements in equations (3) and (4) need not to be scaled to any value. Only the values of the LC resonators are changed according to the standard low-pass to bandpass transformation, as shown in equation (2).

For illustration, the response of the circuit of Figure 4, for the central channel filter, is shown in Figure 5. A six-pole filter response with two transmission zeros to increase the selectivity of the filter out of the passband can be clearly observed. Also, the expected 20 dB return loss is obtained inside the passband, within the specified bandwidth (BW = 36 MHz).

The basic equivalent circuits of the three channel filters can now be connected to the manifold structure, obtaining the overall network shown in Figure 1 [Cameron and Yu, 2007]. In the proposed implementation,

Figure 5. Ideal isolated filter response calculated from the equivalent circuit in Figure 4.

using waveguide technology, the manifold is composed of several waveguide T-junctions connected through di erent waveguide sections. For compactness, a first design was performed using E-plane T-junctions. To model the elements of the manifold at the circuit level, we have used for the waveguide sections the standard transmission line equivalent circuit for the dominant TE10 mode in a rectangular waveguide [Pozar, 2005]. On the contrary, the waveguide T-junctions are modeled by extracting their single-mode scattering parameters using the full-wave electromagnetic simulator FEST3D [Carceller et al., 2014].

The characterization of the waveguide T-junctions through the full-wave scattering matrix is convenient, since then the influence of this component on the performance of the manifold is more accurately taken into account, as compared to using approximate equivalent circuits [Tao et al., 2009]. Also, the waveguide T-junctions need not to be modified along the whole design process of the manifold. Consequently, the full-wave characterization of these components is performed only once at the beginning of the design process, and it is not repeated subsequently.

For the design of the equivalent circuit of the complete triplexer the procedure described in Cameron and Yu [2007] has been used. This is an iterative procedure based on optimizing di erent parts of the structure sequentially. As described in Cameron and Yu [2007], the first step is to optimize the line lengths between T-junctions and the length of the line terminated in short circuit (lengths l1, l2, and l3 in Figure 1). In a second step the line lengths connecting the filters to the T-junctions are optimized (lengths ls1, ls2, and ls3 in Figure 1). In the third step the same lengths are optimized again, but this time including the first three coupling elements of each filter. This step is performed sequentially, to improve the response of the triplexer in each channel at a time. It means that by applying once the above procedure, the response of the triplexer is improved in all channels, but in general, the required specifications will not be met. However, the same steps are now repeated in an iterative fashion to improve the response of the structure. At each iteration the response is improved, until eventually the initial specifications are met in all considered channels.

After applying this iterative procedure, the obtained electrical lengths for the waveguide sections of the manifold (Eln, Elsn, n = 1, 2, 3) are collected in Table 1. Here we use the notation (Eln, Elsn) to refer to the electrical lengths corresponding to the physical lengths shown in Figure 1.

It is also important to note that during the optimization process, some of the couplings of the filters are also modified to compensate for the strong interactions occurring through the manifold. Consequently,

Table 1. Manifold Electrical Lengths Corresponding to the Optimized Equivalent Circuit

Model (in Degrees)

PONS ABENZA ET AL. MULTIPLEXER WITH ALL-INDUCTIVE FILTERS 1069

Figure 6. Response of the optimized equivalent circuit of the triplexer.

the original coupling matrix in (1) is now di erent for each one of the three channel filters. For completeness, the final coupling matrices for the three filters when they are connected to the manifold are given as

To show the e ectiveness of the design process, we present in Figure 6 the response of the optimized equivalent circuit of the triplexer. In general, a good response is obtained, with all three channels fulfilling the initial specifications in terms of center frequency, bandwidth, and return losses. Also, very high isolation better than 35 dB is obtained between adjacent channels.

Once the equivalent circuit of the triplexer is obtained, it can be used to perform the physical design of the structure in waveguide technology. As already said, the channel filters will be implemented using all-inductive

Figure 7. Responses of the three channel filters isolated from the manifold. Solid lines indicate ideal responses given by the coupling matrices.

dual-mode filters, as first introduced in Guglielmi et al. [2001]. With this technology, the coupling topology shown in Figure 2 is implemented with the waveguide structure sketched in Figure 3.

It can be observed that the structure is composed of single-mode cavities at the input and output, implementing resonators (1) and (6) of Figure 2. These single-mode cavities operate on the fundamental resonant mode (TE101). Inside the structure, two dual-mode cavities of extended widths are cascaded to implement resonators (2,3) and (4,5) of Figure 2, respectively. The couplings to both modes of the dual-mode cavities are controlled with the o sets (xi), widths (wi), and thicknesses (ti) of the inductive rectangular windows acting as coupling elements. At this point, it is important to remark that in order to keep orthogonal, the two modes of the dual-mode cavities, and avoid unwanting couplings between both resonant modes, it is necessary to fix the widths of the coupling windows (wi) to a suitable value and to adjust the final coupling values by acting on the thicknesses of the windows (ti, i = 2, 3, 4). Moreover, the resonant frequencies of both resonators can independently be adjusted by acting on the length (di) and width (ai) of the corresponding cavity. Note that each dual-mode cavity is also responsible for one transmission zero, due to the interference between the two orthogonal resonating modes inside the cavity (TE201 and TE102) [Guglielmi et al., 2001;

Martinez Martinez et al., 2014].

It can be observed that the proposed multiplexer has a total fractional bandwidth of 2.3%, involving narrowband channel filters. However, wideband multiplexers can also be designed with the proposed structure up to fractional bandwidths of 12%. In principle, dual-mode cavities in all-inductive technology have some limitations in the bandwidth that can be achieved. However, this is the case when dual-mode cavities are placed at the input and output of the filter [Martinez Mendoza et al., 2015]. In the filter structure proposed in this work, single-mode cavities are placed at the input and output sections of the filter, as it is shown in Figure 2. This removes the bandwidth limitations of this technology, which can achieve bandwidths comparable to the standard all-inductive waveguide technology.

The first task in the procedure is to design the three channel filters, isolated from the manifold, and represented with the modified coupling matrices shown in equations (5)–(7), using the physical implementation presented in Figure 3. This task could be quite complex, since the responses of the filters are electrically asymmetric, and they do not exhibit the typical equiripple response within the passband. However, by using a segmentation technique described in Martinez Martinez et al. [2014] and Martinez Mendoza et al. [2015], each cavity and coupling element of the filter is systematically optimized until the whole structure exhibits the right response given by the corresponding coupling matrix. For illustration purposes, we include in Figure 7 the responses of the three isolated filters obtained after applying this design process. In solid lines we also include the ideal responses calculated from the three coupling matrices given in equations (5)–(7). It can be observed that good agreement has been obtained, indicating that the design process was successful.

Figure 10. Phase adjustment of the isolated waveguide filters.

3.1. E-Plane Manifold Multiplexer

With the design strategy explained, the obtained geometrical dimensions for the manifold structure are collected in Table 2.

Table 3 collects the dimensions of the three designed channel filters.

The response of the triplexer, directly after the assembling process is shown in Figure 11. It can be observed that the obtained response is already quite close to the final target specifications, indicating the e ectiveness of the design strategy. In particular, most of the filter poles in each channel can be di erentiated, with a level of return loss better than 10 dB. Also, center frequency and bandwidth are quite close to the specified values in all three channels.

Since the performance of the structure is quite close to the target response, a final optimization procedure can be applied quite e ectively. This final optimization procedure is carried out with the full-wave simulator, following the iterative strategy already introduced at the circuit level [Cameron and Yu, 2007]. This strategy will assure fast convergence of the procedure, limiting at each step the number of variables to be optimized.

In general, the transmission line sections of the manifold are first optimized alone. This simple optimization already introduces a big improvement in the response of the multiplexer. This is because the interactions between the channel filters through the manifold are very strong, and small imprecisions during the design procedure produce rather large deviations in the expected behavior of the structure. At a later stage, the first few elements of each channel filter (coupling windows and cavities) are also introduced in the optimization process. For the channel filters proposed in this work, this step results to be also simple. This is because the filter structure is terminated by single-mode first and last cavities. In this way, the optimization of the more complex dual-mode cavities, placed internally in the filter structure, requires very small adjustments.

If necessary, these steps are repeated in an iterative fashion until all channels are correctly recovered. In Figure 12 we include the full-wave response of the triplexer after this final optimization process. It can be observed very good electrical performance of the structure. By comparing the response of the final triplexer with the behavior of the ideal circuit model in Figure 6, it can be verified that all initial specifications are correctly met in the three considered channels.

Table 2. Geometrical Parameters in (mm) of the Manifold Line Sections Obtained Directly

After the Design Process

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The final relevant lengths of the mainfold are collected in Table 4.

Finally, for the sake of completeness, Tables 5–7 collect the final dimensions of the three channel filters including the relative errors with respect to the initial design.

In general, it can be noticed that the first and last filters contain the largest errors. This is probably due to stronger interactions of these filters with the input port and the manifold short-circuited termination. The second filter contains relative errors consistently below 0.6%. A 3-D sketch of the designed triplexer is shown in Figure 13.

An additional important consideration that should be taken into account during the analysis of this structure is that the manifold, being in E-plane, spoils the H-plane symmetry of the filters. Consequently, the analysis of the whole device should be accomplished using the full set of TE and TM modes. Still, each channel filter can be optimized separately from the manifold using the described procedure. Following this strategy, the filters can be analyzed and optimized using 2-D models when they are separated from the manifold, which considerably increases numerical e ciency.

The structure was designed using the commercial software FEST3D [Carceller et al., 2014]. This software tool implements an integral equation formulation based on a multimode network representation of the waveguide discontinuities. Basically, accuracy is controlled with the number of basis functions used in the expansion on the unknown magnetic currents at the discontinuities and with the number of modes used to sum the Green’s functions of the integral equation. An additional important parameter is the number of accessible modes, which controls how many modes below cuto interact with nearby discontinuities. For the analysis of the whole triplexer, convergence was reached with 30 accessible modes, 250 basis functions, and 1500 Green’s functions modes.

Figure 11. Electrical response of the triplexer directly after connecting the isolated filters through the manifold structure.