Ping Zhao, Ke Wu
IMAGE COURTESY OF SHANGHAI PINTU NETWORK TECHNOLOGY CO., LTD.
Filters are one of the most fundamental and indispensable components in wireless systems for various communications (5G/6G), radar sensing, and electromagnetic (EM) imaging applications. The development of a quality filter usually involves a substantial amount of design, modeling, and optimization procedures commensurate with its electrical performance and physical realization based on selected high-frequency building blocks, such as resonators and coupling sections over a specific frequency range of interest.
A coupling matrix is a powerful tool for the synthesis and design of narrowband coupled-resonator microwave filters. When the concept was first proposed [1], [2], the coupling matrix described a synchronously tuned bandpass prototype circuit, where series LC resonators were intercoupled by transformers. Today, the coupling matrix is usually constructed from a lowpass prototype circuit model related to the bandpass prototype by the bandpass-to-lowpass frequency mapping. With the inclusion of hypothetical frequency-invariant susceptances (FISs) [3], asynchronously tuned resonators can be incorporated in the lowpass prototype circuit, admitting the realization of asymmetric filter responses [4], [5]. This extension makes the coupling matrix model more popular among filter practitioners.
In the classical coupling matrix model, all the couplings are assumed frequency-invariant. Indeed, such a hypothesis makes sense for a narrowband filter and often presents an excellent approximation to practical situations. Cross-coupling is the most commonly used approach to introducing finite-position transmission zeros (TZs) with frequency-invariant couplings. Fruitful research outcomes are available in the synthesis of multiple-coupled resonator filters. For example, the folded and arrow forms were studied in [6]. These two canonical forms are often used as the starting point of subsequent matrix transformations for advanced coupling topologies. In [7], the cul-de-sac and extended-box coupling topologies were synthesized through a sequence of matrix rotations from the folded form. These two coupling configurations are difficult to be synthesized by sequential circuit element extraction. A general coupling matrix transformation strategy was proposed for cascaded configurations starting from the canonical arrow form [8]. Transversal-like forms are obtained through the diagonalization of coupling matrices [9]. Coupling matrix synthesis approaches for many useful coupling topologies are systematically introduced in [10].
With emerging wireless applications and new generations of communication systems, the radio frequency spectrum has become very congested, and the requirements for filter performance are more stringent. Introducing more TZs is a common approach to enhancing filters’ selectivity and stopband rejection. However, the number of cross-couplings is equal to the number of TZs in many practical filter configurations. It means selective filters with many TZs must be realized in complicated coupling topologies. Sometimes, it is impossible to realize all the required cross-couplings due to the limitations of filter layout and physical implementation. Therefore, it is desirable to develop a new synthesis and design theory to relieve such pain.
The frequency-dependent coupling (FDC) is also named frequency-variant coupling or dispersive coupling in literature. These three terminologies emphasize the frequency-varying nature of coupling coefficients omitted by classical constant coupling models. In the lowpass prototype circuit, a constant coupling is modeled by a frequency-independent J-inverter with a ${\pi}$-equivalent network consisting of three FISs, as shown in Figure 1(a). Similarly, the ${\pi}$-equivalent network of a linear FDC is shown in Figure 1(b). The shunt capacitors introduce frequency-dependent nature to the inverter, making the coupling coefficient linearly dependent on the frequency.
Figure 1. (a) ${\pi}$-Equivalent circuit of a J-inverter, (b) ${\pi}$-equivalent circuit of a linear FDC. (c) Bandpass circuit transformed from a linear FDC.
The lowpass prototype is related to the bandpass circuit through the well-known bandpass-to-lowpass frequency mapping: \[{\omega}_{L} = \frac{1}{FBW}{\left(\frac{{\omega}_{B}}{{\omega}_{0}} - \frac{{\omega}_{0}}{{\omega}_{B}}\right)} \tag{1} \] where ${\omega}_{L}$ is the angular frequency in the lowpass frequency domain, ${\omega}_{B}$ is the angular frequency in the bandpass frequency domain, ${\omega}_{0}$ is the angular center frequency, and FBW is the fractional bandwidth.
The frequency mapping bridges the gap between lowpass and bandpass circuit models. If we replace the angular frequency variable ${\omega}_{L}$ in Figure 1(b) with the expression in (1), it is evident that a capacitor in the lowpass prototype becomes a shunt LC resonator in the bandpass prototype. The shunt FIS introduces a resonant frequency offset to the associated resonator. The circuit transformation shows that a linear FDC can be physically realized as a mixed electric-magnetic coupling, which is also called resonant coupling. Many successful filter designs have been proposed based on mixed (resonant) couplings in a variety of technology, such as microstrip [11], [12], coaxial combline [13], waveguide [14], [15], substrate integrated waveguide (SIW) [16], low-temperature cofired ceramic [17], and 3D printing [18].
The study of FDCs begins in [19], which found that a direct-coupled waveguide filter with FDCs can produce TZs. Several years later, the concept of linear FDC was used to explain asymmetric responses produced by in-line filters [20]. Since then, many filters have been designed based on the FDC model, such as SIW [21], [22], waveguide [23], [24], and monoblock dielectric filters [25]. These filter designs show that if linear FDCs are introduced at proper positions, more TZs can be realized with the same coupling routing diagram. Given many successful design examples, researchers began to explore coupling matrix synthesis techniques for filters with FDCs because, for the development of filters, synthesis techniques are vital in predicting the optimal achievable performance and guiding the physical design [26].
In [27], a zero-pole optimization technique was proposed to synthesize filters with FDCs. One year later, filters with direct source-load resonant couplings were synthesized by a similar coupling matrix optimization framework [28]. Unlike optimization methods, analytical coupling matrix techniques do not rely on the choice of initial values and quickly give deterministic answers to synthesis problems. Moreover, once a coupling matrix corresponding to one possible filter topology is synthesized, matrix operations can be applied to reconfigure the filter to many other forms. For these reasons, a lot of research effort has been devoted to developing analytical coupling matrix synthesis techniques for generalized Chebyshev filters with linear FDCs in various topologies, such as the lattice form [29], mainline FDCs [30], [31], and cascaded blocks with FDCs [32], [33], [34], [35]. The recent advancement of microwave filter theory further extends the linear FDC model to arbitrary FDCs, where a nonlinear FDC can generate more TZs but is more complicated to synthesize. Currently, filters with arbitrary FDCs are synthesized by solving an inverse nonlinear eigenvalue problem [36]. Analytical synthesis techniques for nonlinear FDCs need further research in the future.
This feature article reviews some practical coupling configurations with linear FDCs that emerged in recent years. In particular, coupling matrix transformation strategies based on elementary matrix operations for synthesizing filters with linear FDCs are introduced. Node scaling, matrix rotation, and row/column addition are used as three elementary matrix operations, as each of them has only one free parameter apart from pivot indices. Two waveguide filters realizing the same filtering function but with different coupling topologies are presented to demonstrate the great design flexibility granted by FDCs.
Matrix rotation is intensively used as an elementary transformation to reconfigure conventional coupling matrices with constant couplings [10]. However, the coupling matrix synthesis techniques presented in this article are based on three kinds of matrix operations: i.e., node scaling, row/column addition, and matrix rotation. These three operations are elementary because each has only one free parameter apart from pivot indices. They can modify coupling topologies and coefficients while preserving network characteristics [37]. To facilitate the discussion of coupling matrix synthesis procedures for filters with FDCs, we will first define the notations for these three matrix operations. In the discussions hereafter, we assume the Nth degree filter is described by an ${N} + {2}$ coupling matrix, where the first and last nodes are the source and load terminations, respectively.
The first kind of elementary matrix operation is node scaling: \[{[}{\bf{M'}},\,{\bf{C'}}{]} = {\text{NodeScale}}{(}{\bf{M}},{\bf{C}},{i},{\alpha}{)}\] where i is the pivot index ${(}{i}\,{≠}\,{1}$ or ${N} + {2}$) and ${\alpha}$ is the scaling factor. In this operation, all the entries in the ith row and column of M and C are multiplied by ${\alpha}$ (the ith diagonal entry is multiplied by ${\alpha}$ twice). ${\bf{M'}}$ and ${\bf{C'}}$ are the matrices we obtain after the operation. This operation has been widely used in the synthesis of both narrowband and wideband filters and for circuit models in the lowpass or bandpass frequency domain.
The second kind of elementary matrix operation is matrix rotation: \[{[}{\bf{M'}},\,{\bf{C'}}{]} = {\text{Rotate}}{(}{\bf{M}},{\bf{C}},{i},{j},{\theta}{)}. \]
Both M and C are pre- and postmultiplied by a rotation matrix R and ${\bf{R}}^{\text{T}}$, respectively. The entries of R are the same as an identity matrix except that ${R}_{ii} = {\cos}\,{\theta}$, ${R}_{ij} = {-}{\sin}\,{\theta}$, ${R}_{ji} = {\sin}\,{\theta}$, and ${R}_{jj} = {\cos}\,{\theta}$, where (i, j) is the pivot index (i, ${j}\,{≠}\,{1}$ or ${N} + {2}$), and ${\theta}$ is the rotation angle. Matrix rotation is widely used in the coupling matrix synthesis of filters whose coupling coefficients are constant. In those cases, C takes the special form ${\bf{C}} = {\text{diag}}{\left\{{\left[{0},\,{1},\,{1},\,{\ldots},\,{1},\,{0}\right]}\right\}}$ so it is invariant under matrix rotation. We only need to compute matrix multiplications for M in those cases.
The third kind of elementary matrix operation is row/column addition. In this operation, ${\beta}$ times the ith row is added to the jth row, and then ${\beta}$ times the ith column is added to the jth column. (i, j) is the pivot index (i, ${j}\,{\ne}\,{1}$ or ${N} + {2}$) and ${\beta}$ is the multiplier. We introduce the following notation for this row/column addition: \[{[}{\bf{M'}},\,{\bf{C'}}{]} = {\text{NodeAdd}}{(}{\bf{M}},{C},{i},{j},{\beta}{)}. \]
Levy used the row/column addition operation in his study of cascaded trisection and cascaded quartet filters [38], [39], [40], [41]. Row/column addition operation is particularly useful in coupling matrix synthesis for filters with FDCs, as will be discussed later.
All of these three elementary transformations can be cast into a general form: \[{\bf{M'}} = {\bf{PMP}}^{\text{T}},\,{\bf{C'}} = {\bf{PCP}}^{\text{T}} \tag{2} \] where P is the transformation matrix. The transformation matrix P of node scaling is almost identical to an identity matrix, except the ith diagonal entry is ${\alpha}$ instead of 1. The transformation matrix P of matrix rotation is the rotation matrix R. For row/column addition, the multiplier ${\beta}$ goes into ${P}_{ji}$ of the transformation matrix P, and all the other entries of P are the same as an identity matrix.
Although there are three elementary operations, we can accomplish the coupling matrix transformation using node scaling and one of the other two elementary operations. To see this, let us examine the ${2}\,{\times}\,{2}$ submatrices comprising the pivot rows and columns of the transformation matrix P in the following discussion. The other rows and columns are omitted because they are unaffected by the transformations.
A matrix rotation can be decomposed into a sequence of node scaling and row/column addition as: \[{\left[\begin{array}{cc}{c}&{-s}\\{s}&{c}\end{array}\right]} = {\left[\begin{array}{cc}{1}&{0}\\{\frac{s}{c}}&{1}\end{array}\right]}{\left[\begin{array}{cc}{c}&{0}\\{0}&{\frac{1}{c}}\end{array}\right]}{\left[\begin{array}{cc}{1}&{{-}\frac{s}{c}}\\{0}&{1}\end{array}\right]} \tag{3} \] where ${c} = {\cos}\,{\theta}$ and ${s} = {\sin}\,{\theta}$. Equation (3) shows that a matrix rotation is equivalent to a row/column addition, followed by two scaling operations at the pivot nodes and one more row/column addition. For the trivial case of ${c} = {0}$, the rotation becomes a permutation and a scaling with the factor −1. The permutation reorders the nodes but does not alter the coupling topology or coefficients.
A row/column addition can also be decomposed into a sequence of node scaling and matrix rotation operations as: \[{\left[\begin{array}{cc}{1}&{0}\\{\beta}&{1}\end{array}\right]} = {\left[\begin{array}{cc}{{\alpha}_{2}}&{0}\\{0}&{{\alpha}_{3}}\end{array}\right]}{\left[\begin{array}{cc}{{c}_{2}}&{{-}{s}_{2}}\\{{s}_{2}}&{{c}_{2}}\end{array}\right]}{\left[\begin{array}{cc}{{\alpha}_{1}}&{0}\\{0}&{1}\end{array}\right]}{\left[\begin{array}{cc}{{c}_{1}}&{{-}{s}_{1}}\\{{s}_{1}}&{{c}_{1}}\end{array}\right]} \tag{4} \] where ${c}_{i} = {\cos}\,{\theta}_{i}$ and ${s}_{i} = {\sin}\,{\theta}_{i}$, ${i} = {1},\,{2}$. Given any real-valued ${\beta}$, there are many solutions to (4). One of the solutions is constructed as follows: \begin{align*}{s}_{1} & = \sqrt{\frac{1}{2}{(}{1} + \frac{\beta}{\sqrt{{1} + {\beta}^{2}}}{)}},\,\,{c}_{1} = \sqrt{\frac{1}{2}{(}{1}{-}\frac{\beta}{\sqrt{{1} + {\beta}^{2}}}{)}},\,\,{\theta}_{2} = \frac{\pi}{4}, \\ {\alpha}_{1} & = {-}\sqrt{{1} + {\beta}^{2}} + {\beta},\,\,{\alpha}_{2} = \frac{{-}{1}}{{(}\sqrt{{1} + {\beta}^{2}}{-}{\beta}{)}^{\frac{1}{2}}{(}{1} + {\beta}^{2}{)}^{\frac{1}{4}}}, \\ {\alpha}_{3} & = \frac{{(}{1} + {\beta}^{2}{)}^{\frac{1}{4}}}{{(}\sqrt{{1} + {\beta}^{2}}{-}{\beta}{)}^{\frac{1}{2}}}{.} \tag{5} \end{align*}
The above result shows that a single row/column addition can be equivalently realized by a first matrix rotation with angle ${\theta}_{1}$, followed by scaling the first pivot node with the factor ${\alpha}_{1}$, a second matrix rotation with angle ${\theta}_{2}$, and two more scaling operations at the two pivot nodes with scaling factors ${\alpha}_{2}$ and ${\alpha}_{3}$.
However, node scaling cannot be replaced by a sequence of matrix rotation and row/column addition operations. The reason is that, in general, the determinant of the scaling matrix is not 1. However, the determinants of rotation matrices and row/column addition matrices are 1. According to the property of determinants, it is impossible to write a scaling matrix as the product of transformation matrices of rotation and row/column addition.
We may insist on using node scaling and matrix rotation exclusively to synthesize filters, as we have done to conventional coupling matrices. However, if we embrace all three elementary operations, the transformation procedure can be greatly simplified, especially for synthesizing filters with FDCs.
An elementary transformation is simple as it involves only one transformation parameter besides pivot indices. Usually, multiple elementary transformation steps are taken to synthesize the target coupling topology. Each step achieves a short-term goal of annihilating an unwanted coupling without introducing any new one. The coupling matrix synthesis procedure is finished after all unwanted couplings are annihilated.
As explained by (2), the kth step of elementary operations amounts to pre- and postmultiplying M and C by a transformation matrix ${\bf{P}}_{k}$ and its transpose ${\bf{P}}_{k}^{T}$. It is possible to accomplish the desired coupling matrix reconfiguration by a single step with a transformation matrix ${\bf{P}}_{\text{total}}$, which is equal to the product of ${\bf{P}}_{k}$’s in the reverse order. Thus, the step-by-step procedure can be compressed into a single transformation where both M and C are pre- and postmultiplied by ${\bf{P}}_{\text{total}}$ and ${\bf{P}}_{\text{total}}^{T}$. This operation executes all the elimination steps at once. Therefore, an alternative approach is to solve ${\bf{P}}_{\text{total}}$ directly. Compared with a sequence of elementary matrix operations, this approach is straightforward, as it takes only one transformation step. Besides, the attempt to directly solve ${\bf{P}}_{\text{total}}$ is useful for searching the complete solution set of a particular coupling matrix synthesis problem [42], [43].
It is also possible to develop transformation strategies combining a few elementary operations into compound transformations. In general, fewer steps of high-level transformations are required to achieve the same transformation than elementary transformations. However, the compound transformation matrix has more than one free variable. In the worst case, all the ${N}^{2}$ parameters in the core ${N}\,{\times}\,{N}$ matrix in the transformation matrix P need to be solved. Multivariate nonlinear equations or vector operations are required to find a suitable transformation matrix, and sophisticated mathematical skills are involved in these approaches.
On the contrary, an elementary matrix operation only has one free variable apart from pivot indices, and the value of the free variable can be determined from the current coupling or capacitance matrix according to the short-term goal. Besides, node scaling and row/column addition can be directly executed by manipulating the rows and columns without computing matrix multiplications. Therefore, the coupling matrix synthesis procedures are amenable for programming after the elementary matrix operations are realized as callable functions.
Since coupling matrix synthesis for various filter topologies with constant couplings has been well-established [10], [44] for a long time, the following sections will focus on transformation procedures for synthesizing FDCs. In developing these transformation strategies, our attention is paid to the nonzero entry patterns of the coupling and capacitance matrices. In the complete transformation procedures, we first apply matrix rotations to reconfigure the transversal coupling matrix to a suitable form to prepare for the synthesis of FDCs. Then we execute node scaling and row/column addition by directly manipulating the rows and columns instead of working on the transformation matrix.
An N-tuplet consists of N cross-coupled resonators. It can be used alone as an N-pole filter or cascaded with other blocks to construct higher-order filters. If all the interresonator couplings are constant, an N-tuplet can realize ${N}{-}{2}$ TZs according to the minimum-path rule [45]. However, it is shown that if the outermost cross-coupling becomes linearly frequency-dependent, an N-tuplet can realize ${N}{-}{1}$ TZs. The N-tuplet with an FDC can be transformed from an ${(}{N} + {1}{)}$-tuplet with constant couplings.
Figure 2 shows the ${(}{N} + {1}{)}$-tuplet in the general even-order and odd-order cases. The ${N} + {1}$ resonators are coupled in a topology like the conventional folded form, except that the orientations of some diagonal cross-couplings are different. The synthesis of the ${(}{N} + {1}{)}$-tuplet in Figure 2 follows a similar procedure to the folded form [10, pp. 264–265] with matrix rotations. The difference is that we alternately annihilate elements in two adjacent columns from top to bottom and two adjacent rows from right to left.
Figure 2. ${(}{N} + {1}{)}$-tuplet. (a) Odd-order block. (b) Even-order block. Nodes represent resonators and line segments are constant couplings.
Let us take a septet ${(}{N} = {6}{)}$ as an example to demonstrate the synthesis procedure for a lattice coupling topology with FDCs. The coupling matrix M(0) and the capacitance matrix C(0) corresponding to the topology in Figure 2(a) are shown in Figure 3(a). At the beginning, all the diagonal entries of C(0) are one. The nonzero entry pattern of the matrices suggests using the following row/column addition to annihilate ${M}_{17}$: \[{[}{\bf{M}}^{(1)},{\bf{C}}^{(1)}{]} = {\text{NodeAdd}}{\left({\bf{M}}^{(0)},{\bf{C}}^{(0)},{6},{1},{-}{M}_{17}^{(0)} / {M}_{67}^{(0)}\right)}{.}\]
Figure 3. Coupling topology, coupling and capacitance matrices during the transformation from an ${(}{N} + {1}{)}$-tuplet with constant couplings to an N-tuplet with FDCs. (a) Septet with constant couplings. (b) M17 is annihilated and an FDC is generated between nodes 1 and 6. (c) M15 is annihilated and an FDC is generated between nodes 2 and 5. (d) Final lattice form with three FDCs. Black nodes are resonators. Solid line segments are constant couplings. Line segments with crossed arrows are FDCs. Crosses denote nonzero entries of M. Circles denote nonzero entries in C. Blank cells are zeros.
The resultant matrices M(1) and C(1) are shown in Figure 3(b). This operation makes the coupling between nodes 1 and 6 frequency-dependent without introducing any new coupling. This single row/column addition accomplishes the transformation from a conventional ${(}{N} + {1}{)}$-tuplet into an N-tuplet with an FDC. Then one more node scaling operation can be applied to normalize the shunt capacitor at node 1.
This procedure can be recursively applied to the embedded lower-order subnetworks to further simplify the coupling topology. For example, nodes 1 to 5 in Figure 3(b) constitute a quintet. Therefore, the following operation can be applied to annihilate ${M}_{15}$: \[{[}{\bf{M}}^{(2)},{\bf{C}}^{(2)}{]} = {\text{NodeAdd}}{\left({\bf{M}}^{(1)},{\bf{C}}^{(1)},{2},{5},{-}{M}_{15}^{(1)} / {M}_{12}^{(1)}\right)}{.}\]
The resultant matrices M(2) and C(2) are shown in Figure 3(c). Then, nodes 3 to 5 form a trisection, which can be transformed into an FDC by the following operation: \[{[}{\bf{M}}^{(3)},{\bf{C}}^{(3)}{]} = {\text{NodeAdd}}{\left({\bf{M}}^{(2)},{\bf{C}}^{(2)},{4},{3},{-}{M}_{35}^{(2)} / {M}_{45}^{(2)}\right)}{.}\]
The final matrices M(3) and C(3) are shown in Figure 3(d). Compared with the initial septet in Figure 3(a), the number of cross-couplings is reduced by three. Meanwhile, three of the interresonator couplings become frequency-dependent.
Another commonly used configuration is the mainline FDC, which can independently control one TZ. If there is an isolated mainline FDC, it can be synthesized from a trisection following the procedure introduced in the previous section. An N-pole filter can have up to ${N}{-}{1}$ consecutive mainline FDCs generating ${N}{-}{1}$ TZs. When consecutive mainline FDCs are desired, the synthesis starts from the arrow canonical coupling matrix.
Let us take a 5-4 filter for example. Its coupling matrix M(0) in the arrow canonical form and capacitance matrix C(0) are shown in Figure 4(a). Nodes 1 and 7 are the two terminations and are nonresonant, whereas nodes 2 to 5 are resonators. Synthesis of the arrow canonical form follows a sequence of matrix rotations with a regular pattern of pivots and rotation angles [10, pp. 350–351]. Then a trisection is formed among resonators 4 to 6 to realize a designated TZ at ${\omega} = {\omega}_{z1}$ (the TZ must be pure real in ${\omega}$-domain) with the following matrix rotation [8]: \[{[}{\bf{M}}^{(1)},{\bf{C}}^{(1)}{]} = {\text{Rotate}}{(}{\bf{M}}^{(0)},{\bf{C}}^{(0)},{5},{6},{\theta}_{1}{)}\]
Figure 4. Coupling topology, coupling and capacitance matrices in the transformation from the arrow canonical form to consecutive mainline FDCs. (a) Arrow canonical form where all the coupling coefficients are constant. (b) The first TZ at ${\omega} = {\omega}_{z1}$ is realized by the trisection 2–4. (c) After annihilation of M24 and M27, the first mainline FDC is created between nodes 2 and 3. (d) Final M and C where the four TZs are realized by four consecutive mainline FDCs. Black nodes stand for resonators. Solid line segments are constant couplings. Line segments with crossed arrows are FDCs. Crosses denote nonzero entries of M. Circles denote nonzero entries in C. Blank cells are zeros.
where the rotation angle is given by \[{\theta}_{1} = {\tan}^{{-}{1}}{\left[\frac{{M}_{56}^{(0)}}{{\omega}_{z1} + {M}_{66}^{(0)}}\right]}{.} \tag{6} \]
Two more matrix rotations are applied to M(1) and C(1) to pull up the trisection along the main diagonal of the coupling matrix, and then the resultant coupling matrix M(2) and capacitance matrix C(2) are shown in Figure 4(b). Up to now, the capacitance matrix remains to be ${\bf{C}} = {\text{diag}}{\left\{{\left[{0,\,1,\,1,\,1,\,1,\,1,\,0}\right]}\right\}}$ because only matrix rotations are applied. When the embedded trisection is about to be pulled out of the arrow structure, as shown in Figure 4(b), the entries of the coupling matrix satisfy ${M}_{24} / {M}_{34} = {M}_{27} / {M}_{37}$. Therefore, the following row/column addition can annihilate ${M}_{24}$ and ${M}_{27}$ simultaneously, and it will create an FDC between nodes 2 and 3: \[{[}{\bf{M}}^{(3)},{\bf{C}}^{(3)}{]} = {\text{NodeAdd}}{\left({\bf{M}}^{(2)},{\bf{C}}^{(2)},{3},{2},{-}{M}_{24}^{(2)} / {M}_{34}^{(2)}\right)}{.}\]
The resultant coupling and capacitance matrices are shown in Figure 4(c).
This transformation strategy can be recursively applied to the remaining arrow subnetwork, i.e., the subnetwork comprising nodes 3 to 7. Each time a TZ is pulled out from the arrow structure and assigned to a mainline FDC, the number of cross-couplings in the remaining arrow subnetwork is reduced by one. Finally, all the TZs are realized by mainline FDCs, and there is no cross-coupling left. The final coupling and capacitance matrices are shown in Figure 4(d).
A box section is similar to a quartet in that both of them consist of four resonators positioned at the corners of a square. However, the input and output of a box section exist at two diagonal nodes, whereas those of a quartet are connected to two adjacent nodes. According to the minimum-path rule, the box section can realize only one TZ if all its couplings are constant. It is found that if an FDC is introduced to a box section, two TZs will be created.
A box section with an FDC can be synthesized from a parallel coupling topology, whose coupling and capacitance matrices M(0) and C(0) are shown in Figure 5(a). Nodes 3 and 4 form two first-degree resonant paths between nodes 2 and 5, and they are bypassed by the cross-coupling ${M}_{25}$. An observation on the nonzero entry pattern of M(0) and C(0) suggests the following row/column addition to annihilate ${M}_{25}$: \[{[}{\bf{M}}^{(1)},{\bf{C}}^{(1)}{]} = {\text{NodeAdd}}{\left({\bf{M}}^{(0)},{\bf{C}}^{(0)},{4},{2},{-}{M}_{25}^{(0)} / {M}_{45}^{(0)}\right)}{.}\]
Figure 5. Coupling topology, coupling and capacitance matrices in the transformation from the parallel form to a box section with an FDC. (a) Parallel topology with constant couplings. (b) Box section with an FDC. Black nodes are resonators. Solid line segments are constant couplings. Line segments with crossed arrows are FDCs. Crosses denote nonzero entries of M. Circles denote nonzero entries in C. Blank cells are zeros.
The resultant coupling topology and matrices are shown in Figures 5(b). We can see that the coupling between nodes 2 and 4 becomes frequency-dependent. Meanwhile, the coupling topology is simplified with the cross-coupling ${M}_{25}$ annihilated.
Alternatively, in the beginning, we can take any one of the following row/column addition operations to annihilate ${M}_{25}$: \begin{align*}{[}{\bf{M}}^{(2)},{\bf{C}}^{(2)}{]} & = {\text{NodeAdd}}{\left({\bf{M}}^{(0)},{\bf{C}}^{(0)},{3},{2},{-}\frac{{M}_{25}^{(0)}}{{M}_{35}^{(0)}}\right)} \\ {[}{\bf{M}}^{(3)},{\bf{C}}^{(3)}{]} & = {\text{NodeAdd}}{\left({\bf{M}}^{(0)},{\bf{C}}^{(0)},{3},{5},{-}\frac{{M}_{52}^{(0)}}{{M}_{32}^{(0)}}\right)} \\ {[}{\bf{M}}^{(4)},{\bf{C}}^{(4)}{]} & = {\text{NodeAdd}}{\left({\bf{M}}^{(0)},{\bf{C}}^{(0)},{4},{5},{-}\frac{{M}_{52}^{(0)}}{{M}_{42}^{(0)}}\right)}{.} \end{align*}
By one of the above operations, an FDC will be created at one of the four couplings in the box section.
Once the coupling and capacitance matrices have been synthesized, the filter designer is ready for the physical realization of the filter. The coupling matrix model is valid for various types of resonators and coupling structures. Therefore, this section discusses the general procedure of filter dimensioning according to synthesized matrices. The designer should choose the most appropriate technology to realize the filter based on system requirements, such as mass, volume, cost, insertion loss, power handling capability, and ease of integration.
Synthesis results of filters with linear FDCs include the normalized coupling and capacitance matrix. Therefore, conventional filter design techniques [10], [46], [47] based on coupling matrices, as briefly described below, can still be applied for preliminary dimensioning filters with FDCs. A diagonal entry of the coupling matrix determines the resonant frequency shift from the designated center frequency. An off-diagonal entry corresponding to a constant coupling is denormalized to the physical coupling coefficient realized by a pure electric or magnetic coupling structure. The constant coupling design is typically carried out by building a pair of symmetric resonators, and the coupling coefficient is extracted from the resonant frequencies of the even and odd eigenmodes. An input/output coupling is translated into the external Q, which can be estimated from the group delay of the reflection coefficient.
The most critical step for realizing the desired filter response is dimensioning the FDCs. If transmission line-type resonators are used, such as uniform waveguides and microstrip lines, the inverter parameter J/K can be calculated from simulated S-parameters [48]. The simulation is conducted with the discontinuity that realizes the FDC in the middle of a uniform transmission line segment. By studying the variation of the inverter parameter J/K over frequency, we can estimate the slope and value of the realized FDC [14], [24]. When nontransmission line-type resonators are used, we should build a coupled resonator pair with the FDC in-between. The value of the FDC can be diagnosed from the Y-parameters seen from auxiliary ports [29]. Alternatively, the linear FDC can be characterized by two quantities. The first is the coupling value at the center frequency, and the second is the frequency point where the coupling coefficient vanishes [30]. We can adjust the dimensions of the coupling structure until the two quantities dictated by the synthesized coupling matrix are well realized.
The above procedure provides a set of initial dimensions for the filter design. However, when all of the resonators and coupling structures are assembled to construct the whole filter, the resonant frequencies shift, and coupling coefficients vary due to the presence of other parts and the change in the EM environment. As a result, further fine-tuning or optimization of the dimensions is needed. Our recommended approach is to fit the simulated filter responses by rational functions and extract a coupling and capacitance matrix in the correct form [49], [50]. By comparing the extracted matrices to the synthesized ones, we can quickly identify which dimensions need to be adjusted according to the one-to-one relationship between the coupling matrix entries and physical structures. The filter response can gradually approach desired ones by repeated extraction and tuning to minimize the difference between extracted and synthesized matrices. This procedure also applies to the tuning of filter hardware.
The following two sections will present two rectangular waveguide filter design examples. Both filters contain FDCs and realize the same filtering functions but with different coupling topologies.
The first filter design example realizes a generalized Chebyshev characteristic with a 22-dB return loss level and three TZs at ${s} = {j}{2}$ and ${\pm}{1}{-}{j}{0.14}$ in the normalized frequency domain. The first TZ improves the near-skirt selectivity of the upper stopband, and the paraconjugate TZ pair gives a group delay equalization over 70% of the passband.
S-parameter polynomials in the form of \[{\left[{S}\right]} = \frac{1}{E}{\left[\begin{array}{cc}{\frac{F}{{\varepsilon}_{R}}}&{\frac{P}{\varepsilon}}\\{\frac{P}{\varepsilon}}&{\frac{{F}_{22}}{{\varepsilon}_{R}}}\end{array}\right]} \tag{7} \] can be constructed from the generalized Chebyshev function [10]. E, P, F, and ${F}_{22}$ are polynomials on the complex frequency variable ${s} = {j}{\omega}_{L}$. $\varepsilon$, and ${\varepsilon}_{R}$ are real-valued normalization constants. The coefficients multiplied to different powers of s in the polynomials are listed in Table 1. Then, the S-parameter polynomials are converted to Y-parameters for the synthesis of a transversal coupling matrix [5], and the transversal coupling matrix is given in Table 2.
Table 1. Coefficients of S-parameter polynomials.
Table 2. Transversal coupling matrix of design A.
The transversal coupling matrix is first transformed into a cascaded form [8], as shown in Figure 6, whose coupling matrix ${\bf{M}}^{(0)}$ is given in Table 3. The associated capacitance matrix is ${\bf{C}}^{(0)} = {\text{diag}}{\left\{{\left[{0},{1},{1},{1},{1},{1},{1},{0}\right]}\right\}}$. In the coupling topology in Figure 6, nodes 2 to 4 constitute a trisection realizing the TZ on the imaginary axis, and nodes 4 to 7 form a quartet realizing the self-equalized TZ pair. Node 4 is heavily loaded as it is coupled to five resonators. A physical realization based on this coupling topology is impractical.
Figure 6. A conventional cascaded topology of the 6-1-2 filter.
Table 3. Cascaded coupling matrix M(0), corresponding to Figure 6.
With the matrix synthesis technique for N-tuplet with FDCs, we can execute the following two operations to transform the trisection into a mainline FDC: \begin{align*}{[}{\bf{M}}^{(1)},{\bf{C}}^{(1)}{]} & = {\text{NodeAdd}}{\left({\bf{M}}^{(0)},{\bf{C}}^{(0)},{3},{2},{-}\frac{{M}_{24}^{(0)}}{{M}_{34}^{(0)}}\right)} \\ {[}{\bf{M}}^{(2)},{\bf{C}}^{(2)}{]} & = {\text{NodeScale}}{\left({\bf{M}}^{(1)},{\bf{C}}^{(1)},{2},\frac{1}{\sqrt{{C}_{22}^{(1)}}}\right)}{.} \end{align*}
Then, two more matrix operations transform the quartet into a trisection with a frequency-dependent cross-coupling: \begin{align*}{[}{\bf{M}}^{(3)},{\bf{C}}^{(3)}{]} & = {\text{NodeAdd}}{\left({\bf{M}}^{(2)},{\bf{C}}^{(2)},{6},{4},{-}\frac{{M}_{47}^{(2)}}{{M}_{67}^{(2)}}\right)} \\ {[}{\bf{M}}^{(4)},{\bf{C}}^{(4)}{]} & = {\text{NodeScale}}{\left({\bf{M}}^{(3)},{\bf{C}}^{(3)},{2},\frac{1}{\sqrt{{C}_{44}^{(3)}}}\right)}{.} \end{align*}
The resultant coupling and capacitance matrices are shown in Tables 4 and 5, respectively, and the corresponding coupling topology is depicted in Figure 7. After two FDCs are introduced, node 4 is coupled to three nodes, making the filter layout easier to be realized.
Figure 7. A cascaded topology of the 6-1-2 filter with FDCs.
Table 4. Coupling matrix M(4), corresponding to Figure 7.
Table 5. Capacitance matrix C(4), corresponding to Figure 7.
After the coupling and capacitance matrices are synthesized, we can proceed with the physical realization. This section presents a waveguide filter design whose center frequency and bandwidth are ${f}_{0} = {10}{\text{ GHz}}$ and ${BW} = {0.2}{\text{ GHz}}$. Figure 8 shows the EM model and geometrical dimensions of the filter. The filter is designed with the standard WR90 waveguide. The FDCs are implemented by partial-height metal posts, and the constant couplings are designed as inductive irises. Both metal posts and H-plane irises are simple structures suitable for low-cost milling. Node 5 is implemented by a ${\text{TE}}_{102}$ mode cavity, whereas all the other resonant nodes operate in the fundamental ${\text{TE}}_{101}$ mode. The small cylinders in the EM model are tuning screws of a diameter of 2.29 mm.
Figure 8. EM model of filter design A. The small cylinders on the top are tuning screws with a diameter of 2.29 mm. (a) Perspective view. (b) Top view (units: millimeters).
Figure 9 shows the photograph of a prototyped filter according to the EM model in Figure 8. After the filter is fine-tuned with the aid of the model-based vector fitting technique [49], [50], we obtain the measurement results shown in Figure 10. The ideal coupling matrix responses are superimposed on measurement results for direct comparison. A decent agreement between measured and theoretical responses is achieved. We can see that the desired stopband TZ and the group delay equalization are well realized.
Figure 9. Photograph of the prototyped filter A (from [32]).
Figure 10. Measurement results of prototyped filter A. (a) Magnitude. (b) Group delay. Solid lines: measurement results. Dashed lines: responses of ideal coupling matrices.
The previous section presents the design of a 6-1-2 generalized Chebyshev filter in a cascaded topology with FDCs. This section will demonstrate the realization of the same filtering function in a lattice topology. Due to the coupling matrix approach, we do not need to restart the synthesis from the very beginning. Instead, we can start from the transversal coupling matrix and apply a different sequence of elementary matrix transformations.
We first transform the transversal coupling matrix in Table 2 into the folded-like form as Figure 2. The resultant coupling matrix ${M}^{(0)}$ is given in Table 6, and the corresponding coupling topology is depicted in Figure 11. Since only matrix rotations are applied to obtain the folded-like form, the capacitive coupling matrix remains to be ${\bf{C}}^{(0)} = {\text{diag}}{\left\{{\left[{0},{1},{1},{1},{1},{1},{1},{0}\right]}\right\}}$.
Figure 11. A folded-like topology of the 6-1-2 filter.
Table 6. Folded-like coupling matrix M(0), corresponding to Figure 11.
The following two matrix operations annihilate ${M}_{37}$ and make the coupling between nodes 3 and 6 frequency-dependent: \begin{align*}{[}{\bf{M}}^{(1)},{\bf{C}}^{(1)}{]} & = {\text{NodeAdd}}{\left({\bf{M}}^{(0)},{\bf{C}}^{(0)},{6},{3},{-}\frac{{M}_{37}^{(0)}}{{M}_{67}^{(0)}}\right)} \\ {[}{\bf{M}}^{(2)},{\bf{C}}^{(2)}{]} & = {\text{NodeScale}}{\left({\bf{M}}^{(1)},{\bf{C}}^{(1)},{3},\frac{1}{\sqrt{{C}_{33}^{(1)}}}\right)}{.} \end{align*}
Then, we apply two more elementary matrix operations to annihilate ${M}_{35}$ and introduce an FDC between nodes 4 and 5: \begin{align*}{[}{\bf{M}}^{(3)},{\bf{C}}^{(3)}{]} & = {\text{NodeAdd}}{\left({\bf{M}}^{(2)},{\bf{C}}^{(2)},{4},{5},{-}\frac{{M}_{35}^{(2)}}{{M}_{34}^{(2)}}\right)} \\ {[}{\bf{M}}^{(4)},{\bf{C}}^{(4)}{]} & = {\text{NodeScale}}{\left({\bf{M}}^{(3)},{\bf{C}}^{(3)},{5},\frac{1}{\sqrt{{C}_{55}^{(3)}}}\right)}{.} \end{align*}
The resultant coupling and capacitance matrices ${\bf{M}}^{(4)}$ and ${\bf{C}}^{(4)}$ are shown in Tables 7 and 8, respectively, and the corresponding coupling topology is depicted in Figure 12.
Figure 12. A lattice topology of the 6-1-2 filter with FDCs.
Table 7. Lattice coupling matrix M(4), corresponding to Figure 12.
Table 8. Lattice capacitance matrix C(4), corresponding to Figure 12.
${\bf{M}}^{(4)}$ and ${\bf{C}}^{(4)}$ are symmetric about the minor diagonal. Therefore, the filter can be designed with a symmetric structure. Note that if the 6-1-2 filter is synthesized in a symmetrical lattice structure with constant coupling coefficients exclusively, there will be diagonal cross-coupling pairs [6], causing difficulty in its physical implementation.
The center frequency and bandwidth of design B are ${f}_{0} = {12}{\text{ GHz}}$ and ${BW} = {0.2}{\text{ GHz}}$, respectively. Although filter B operates in a different frequency band from filter A, these two filters realize the same filtering functions so that their selectivity and phase linearity are identical in theory. The EM model and the geometrical dimensions of the filter are shown in Figure 13. This filter takes a symmetric structure, so the number of design variables is reduced by almost half.
Figure 13. EM model of filter design B. The small cylinders on the top model are M2.5 tuning screws. The filter structure is symmetric about the central plane. (a) Perspective view. (b) Top view (units: millimeters).
We prototype another rectangular waveguide filter according to the EM model in Figure 13 to verify the design. Figure 14 shows a photograph of the fabricated filter. The filter is made of aluminum alloy 6061-t6. The measured magnitude and group delay responses of the fabricated filter are shown in Figure 15. Due to the finite conductivity of the alloy and degradation caused by tuning screws (made of stainless steel), the resonant cavities have an average unloaded quality factor of about 1,000. The measured insertion loss is 2.4 dB at the passband center and 3.4 dB at the upper band edge. The coupling matrix responses with an even ${Q} = {1,000}$ are superimposed in Figure 15 for comparison. It can be seen that measurement results agree with the S-parameters computed from the coupling matrix and capacitance matrix within a reasonable margin of error.
Figure 14. Photograph of the prototyped filter B.
Figure 15. Comparison between measured responses and S-parameters calculated by the coupling matrix model for filter B. (a) Magnitude. (b) Group delay. Solid lines are measurement results. Dashed lines are responses calculated from the synthesized coupling and capacitance matrices with a uniform ${Q} = {1,000}$ for all resonators.
The extension of conventional coupling matrix models to include linear FDCs is straightforward, as we simply allow off-diagonal entries of the capacitance matrix to take nonzero values. This extension grants extra flexibility in the design of microwave filters. If the FDCs are introduced in suitable locations, more TZs can be created than the counterpart topologies without FDCs. Moreover, with FDCs, high-order filters incorporating many TZs can be realized in simpler coupling topologies with fewer cross-couplings.
The coupling matrix is the most convenient tool for the synthesis and design of narrowband coupled-resonator filters with linear FDCs. The synthesis of FDCs needs a few more steps based on conventional coupling matrix transformation strategies. In particular, after the coupling matrix is reduced to a suitable form by matrix rotation, we can apply row/column addition and node scaling operations to introduce FDCs. After the coupling matrix is synthesized, initial filter dimensioning can be carried out in similar ways to conventional filters, and fine-tuning can be performed with the aid of coupling matrix extraction based on the model-based vector fitting technique.
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