The design of waveguide filters based on cross‐coupled resonators

This article addresses the physical realization of crosscoupled waveguide filters based on electromagnetic (EM) simulations. For this design procedure, the filter structure is simulated by successively adding one resonator at a time. The desired filter response is achieved without the need of a global optimisation on all the mechanical dimensions within an EM simulator. This reduces the design time required for a crosscoupled waveguide filter and allows the possibility of building high‐order waveguide filters with complex crosscouplings. A sixth order X‐band dual‐band filter with a center frequency of 10 GHz and a fractional bandwidth of 1% is designed using this procedure and presented here as an example. Excellent agreement between simulation results and theoretical results from coupling matrix verifies the proposed approach. © 2014 Wiley Periodicals, Inc. Microwave Opt Technol Lett 56:3–8, 2014


INTRODUCTION
A microwave filter is a two-port network used to transmit and attenuate signals in specified frequency bands. Microwave filters have found wide applications in modern communication systems, radar systems, and laboratory measurement equipments [1]. Filters based on crosscoupled resonators, with real or complex transmission zeros (TZs), have been extensively used to (i) improve the close-to-band selectivity; (ii) achieve in-band group delay linearity; (iii) divide the single-band into multiple passbands. However, compared with the conventional in-line resonator coupled filter, the crosscoupled filter is more difficult to be physically implemented, due to the interactions introduced by the crosscouplings.
Traditionally, the design methods for direct coupled filters have been applied to extract the dimensions for crosscoupled filters. This design process usually involves the following four main steps: (i) identify the filter order and filter functions according to specification requirements; (ii) synthesis or optimise the coupling coefficients (M i,j ) and external quality factors (Q e ) that can realize the desired filter function; (iii) choose the filter type (waveguide, microstrip, etc), and obtain dimensions which can achieve desired specified Q e and M i,j from electromagnetic (EM) simulations on one resonator and two weakly coupled resonators; (iv) construct the filter in the simulator to get its initial responses [1,2]. For crosscoupled filters, this approach ignores the influences from crosscouplings, and therefore normally requires a global optimization on all the physical dimensions. This global optimization is time-consuming, and in some special scenario, where the filter consists of a large number of resonators and/or complex crosscouplings, the final optimization may fail to converge to an acceptable solution, due to the large amount of control parameters. In [3], a design procedure, which eliminates the need of global optimization, has been presented for a crosscoupled folded waveguide filter. A fourth order and a sixth order crosscoupled single-band waveguide filters have been successfully demonstrated using this design approach. However, this design approach is limited to waveguide filters with folded topologies.
In this article, we present an EM-based design approach for determining the physical dimensions of a crosscoupled waveguide filter with any type of topology. This design procedure enables us to account for the attributions of crosscouplings, and provides precise desired dimensions without the need of a final global optimization. This approach may find useful application in the design of resonator based crosscoupled waveguide filters or multiplexers [4].

DESIGN STEPS
The design approach is demonstrated by a sixth order X-band dual-band waveguide filter. Figure 1 illustrates the topology and the structure of this filter. This filter is designed to have the following specifications: the center frequency is 9.965 GHz for the first passband and 10.035 GHz for the second passband, both passbands have a desired return loss of 20 dB and the same bandwidth of 30 MHz, the attenuation level for the middle stopband is better than 26 dB. The N 3 N coupling matrix of this dual-band filter, as depicted below, is generated by a synthesis technique described in [5]. Their corresponding S-parameter responses can be found in Figure 2. A pair of symmetrical TZs positioned at 9.995 and 10.005 GHz occur at the in-band to split it into two symmetrical passbands. These two TZs are attributed to the crosscoupling between resonators 2 and 5. As all the coupling coefficients are positive, thereby all inductive irises have be utilized by this filter.
The design can be divided into six substeps, as shown in Figure 3. At work step, rather than optimizing dimensions for the entire structure, only the dimensions of one cavity and its connecting irises are significantly tuned toward the desired responses. This reduces the number of dimensions to be adjusted during the design, which in return yields faster and more reliable convergence. Especially for large scale filter structure, in which case it is virtually impossible to optimize all the mechanical dimensions at the same time. The calculation of the physical dimensions for the sixth order dual-band filter shown in Figure  1 comprises the following steps.
1. Calculate the approximate initial dimensions for all the resonators and irises using the equivalent circuit models based on the coupling matrix as described in [1,6,7].   For this design approach, the middle stage S-parameter responses are calculated from their corresponding coupling coefficients, and act as the objective responses for the tuning. To plot the desired responses at each stage, the inner coupling coefficient needs to be converted into external quality factor. For instance, at Step 1, Q e2 should be calculated from M 12 . After expressing both the external quality factor (Q e ) and internal coupling coefficients (M ij ) using inverter value K [1], the relationship between M ij and Q e can be found as: where k g is guided wavelength and k is the free-space wavelength. For an X-band waveguide filter operating at a center frequency of 10 GHz, M ij 2 3 Q e is calculated to be 0.3625. Consequently, M ij can be converted to its corresponding Q e . It is interesting to note that, the value of M ij 2 3 Q e does not depend on the fractional bandwidth (FBW) of the filter.
In additional, it may be observed from Figure 3 that, for work steps 2 to 5, there are three external ports. However, both equations of N 3 N [2] and (N 1 2) 3 (N 1 2) [1] matrix are derived for a two-port network circuit. In the following, the equations for (N 1 3) 3 (N 1 3) matrix, which can be applied to calculate S-parameter responses of a three-port filter network, will be derived and given.
Ref. 4 reports equations for computing three-port filter network S-parameter responses. However, these equations are derived following the similar approach to a N 3 N coupling matrix [2], and therefore has a restriction that the resonators number should be larger than the number of external ports. In the work presented here, there exists a case that the resonator number is less than the number of ports [see Fig. 3(b)], thereby a similar approach to the (N 1 2) 3 (N 1 2) matrix synthesis is applied in this work to derive the equations. Here, the relationship between the S-parameters and the coupling matrix is extracted by analyzing the node voltage and current of the three-port network's equivalent circuits, as described in detail in [1,2,4]. The matrix m for a general three-port network consists of N coupled resonators, one input port (S) and two output ports (L1, L2) can be written in the following form: The above matrix is symmetrical about the principal diagonal and it includes the couplings between external ports and the internal resonators. Additionally, it is also possible to accommodate the direct couplings between external ports, such as m S,L1 , m S,L2 , and m L1,L2 . The dual-band filter presented here does not include any direct coupling between ports, as shown in Figure 3, therefore m S,L1 , m S,L2 . and m L1,L2 are assigned to zero here. The highlighted part (using grey colour) represents the core N 3 N matrix, whose entries are normalized coupling coefficients (m ij 5 M ij /FBW). The coupling coefficients between external ports and inner resonators can be calculated by where q ei is the normalized external quality factors of the external port i (q ei 5 Q ei 3 FBW), N 1 and N 2 refer to the resonator number connecting to the output ports (L1 and L2). For instance, at Step 3 as shown in Figure 3(c), N 1 5 2, N 2 5 3. The general matrix A can be expressed as below where U is similar to a (N 1 3) 3 (N 1 3) unit matrix, except that U(1,1) 5 U (N 1 2, N 1 2) 5 U (N 1 3, N 1 3) 5 0, R is a (N 1 3) 3 (N 1 3) matrix whose only nonzero entries are R(1,1) 5 R (N 1 2, N 1 2) 5 R (N 1 3, N 1 3) 5 1, p is the lowpass frequency variable, which can be written in terms of FBW and the filter center frequency (x 0 ) as Then the S-parameter responses of a three-port filter network can be expressed as: For the case where there are more than three external ports, similar equations can be derived accordingly by adding extra rows at the bottom and extra columns at the right, to the coupling matrix shown in Eq. (2).

RESULTS
The six substep responses for the structures shown in Figure 3 are depicted in Figure 3. In Figure 4, the dashed lines refer to the theoretical responses plotted using the equations described in section 2, and these responses are served as goals of the dimensional tuning. The lwave wizard [8] based on mode-matching technique is used in the simulations. The simulated responses using the optimized dimensions are denoted as solid lines in Figure 4.
At each substep, initially only one resonator's dimensions (three or less parameters) are tuned in the simulations. Therefore, a desired set of dimensions, whose corresponding responses match the objective ones, can be obtained within a short time. The dimensions from the previous stages may also be slightly altered to tune the responses toward the desired ones, as shown in Tables 1 and 2. It can be observed that, only very small adjustments are required on the dimensions achieved in the foregoing stages, to account for the influence from the subsequent resonators.
The final dimensions of the dual-band filter are shown in Tables 1 and 2 (Step 6), and their corresponding simulation results can be found in Figure 2. It can be observed that, without any global optimization, the final acquired dimensions have extremely close responses with the theory ones from coupling matrix. It should be pointed out that, both lwave wizard and CST microwave studio [9] have been used to simulate the dualband filter. These two EM simulators produce very close results, as shown in Figure 2.

CONCLUSION
A mechanical dimensions calculation method for crosscoupled waveguide filters has been described. During this design procedure, the filter structure is constructed step by step by adding one resonator to the simulated structure at a time. Dimensions of this resonator are tuned toward the desired target middle stage responses. Equations have been derived and provided in this article to plot the middle stage responses from coupling matrix. A sixth order dual-band X-band filter with a pair of symmetrical TZs has been successfully demonstrated using this approach. This approach eliminates the need of a global EM-based dimensional optimization, and therefore leads to a reduction in the time required. Moreover, it also opens the possibility of building highorder waveguide filters with complex crosscouplings.

INTRODUCTION
The most key band pass filters (BPFs), which are microwave devices that are used in wireless communication systems, have a small size, low insertion loss, superior stopband rejection, and high roll-off rate. Therefore, much of the relevant literature is concerned with designs in which high roll-off rate and wide stopband rejection with excellence. Most of the conventional microstrip filters developed to date are based mainly on transmission-line structures [1,2]. These filters require too much area because each resonator needs quarter-or halfwavelength to get resonance. Some of the novel structures proposed for reducing filter size include stepped impedance resonator [3] and defected ground structure [4]. However, the proposed designs are still not size-competitive with LTCC filters. To maximize device miniaturization, many filters are designed on an LTCC substrate.
LTCC BPFs play an important role because they can provide satisfactory electrical performance with a compact size. However, the unstoppable trend toward smaller mobile terminals continuously demands for smaller LTCC BPFs. The size evolution of LTCC BPFs for 2.4-GHz instruments, scientific, and measurements band applications in the electronic market of mass production is from the early 3225 (3.2 3 2.5 mm 2 ), then to 2520 and 2012, and now toward 1608. Every new generation of product reduces the area almost in half when compared to the prior product. Therefore, miniaturization for LTCC BPFs becomes a more and more challenging work.
The previous researches on the miniaturization of LTCC BPFs for wireless communication applications mainly include the second-