Wideband impedance matrix representation of passive waveguide components based on cascaded planar junctions

Authors


Abstract

[1] A very efficient technique for the full-wave analysis of passive waveguide components, composed of the cascade connection of planar junctions, is presented. This novel technique provides the wideband generalized impedance matrix representation of the whole structure in the form of pole expansions, thus extracting the most expensive computations from the frequency loop. For this purpose, the structure is segmented into planar junctions and uniform waveguide sections, which are characterized in terms of wideband impedance matrices. Then, an efficient iterative algorithm for combining such matrices, and finally providing the wideband generalized impedance matrix of the complete structure, is followed. Two waveguide filters have been used to prove the accuracy and efficiency of this novel technique, which is specially suited for modeling complex geometries.

1. Introduction

[2] Modern microwave and millimeter-wave equipment, present in mobile, wireless and space communication systems, employ a wide variety of waveguide components [Uher et al., 1993]. Most of these components are based on the cascade connection of waveguides with different cross-section, as recognized by Conciauro et al. [2000]. Therefore, the full-wave modal analysis of such structures has received a considerable attention from the microwave community [Sorrentino, 1989; Itoh, 1989; Boria and Gimeno, 2007]. The numerical efficiency of these methods has been substantially improved by Mansour and MacPhie [1986] and Alessandri et al. [1988, 1992] by means of the segmentation technique, which consists of decomposing the analysis of a complete waveguide structure into the characterization of its elementary key building blocks, i.e. planar junctions and uniform waveguides.

[3] The modeling of planar junctions between waveguides of different cross-section has been widely studied in the past through modal analysis methods, where higher-order mode interactions were already considered by Wexler [1967]. For instance, in order to represent such junctions, the well-known mode-matching technique has been typically formulated in terms of the generalized scattering matrix [Safavi-Naini and MacPhie, 1981, 1982; Eleftheriades et al., 1994]. Alternatively, the planar waveguide junction can be characterized using a generalized admittance matrix or a generalized impedance matrix, obtained either by applying the general network theory of Alvarez-Melcón et al. [1996] or by solving integral equations of Gerini et al. [1998] A common drawback to all the previous techniques is that any related generalized matrix must be recomputed at each frequency point.

[4] In the last two decades, several works have been focused on avoiding the repeated computations of the cited generalized matrices within the frequency loop. For instance, frequency-independent integral equations have been set up when dealing, respectively, with inductive (or H plane) and capacitive (or E plane) discontinuities in the works of Guglielmi and Newport [1990] and Guglielmi and Alvarez-Melcón [1993], steps in the works of Guglielmi et al. [1994] and Guglielmi and Gheri [1994], and posts in the work of Guglielmi and Gheri [1995]. On the other hand, following the Boundary Integral-Resonant Mode Expansion (BI-RME) technique developed at University of Pavia (Italy), a generalized admittance matrix in the form of pole expansions has been derived for arbitrarily shaped H plane [Conciauro et al., 1996] and E plane components [Arcioni et al., 1996], as well as for 3-D resonant waveguide cavities in the work of Arcioni et al. [2002].

[5] More recently, the BI-RME method has been combined by Mira et al. [2006] with an integral equation technique in order to provide a generalized impedance matrix, in the form of quasi-static terms and a pole expansion, for planar waveguide junctions. However, in order to get convergent results, such method requires a bigger number of accessible modes than the original formulation of the integral equation technique [Gerini et al., 1998]. Furthermore, the cascade connection of planar waveguide junctions was solved by Mira et al. [2006] within the frequency loop. Therefore, the implementation of the wideband solution proposed by Mira et al. [2006] was not very computationally efficient.

[6] In this paper, we present two novel contributions in order to increase the numerical efficiency related to the pole expansion technique proposed by Mira et al. [2006]. First of all, by considering additional static (frequency independent) terms in the integral equation formulation, we are able to reduce the number of accessible modes required in the convergent characterization of planar waveguide junctions. Secondly, a very efficient iterative algorithm is proposed for combining the generalized impedance matrices of planar junctions and interconnecting waveguide sections, thus providing a wideband representation of any passive device based on the cascaded connection of such basic elements. For validation purposes, this wideband modeling technique has been successfully applied to the full-wave analysis of two waveguide filter examples.

2. Theoretical Description of Basic Blocks

[7] The structure under study is composed of the cascade connection of planar junctions between two different waveguides of lengths l1 and l2 (see Figure 1), whose equivalent circuit representation can be found in Figure 2. Our aim is to represent each basic building block of this equivalent circuit in terms of a wideband generalized impedance matrix (or Z matrix) in the form of pole expansions

equation image

where k = ωequation image, η = equation image, and A, B, C, Δ and U are frequency independent matrices (the meaning and structure of these matrices are detailed by Arcioni and Conciauro [1999] for the dual case of the wideband admittance matrix formulation). In particular, A and B are square symmetric matrices of size N (being N the total number of accessible modes considered in each building block), C is a matrix of size N × Q, with Q the number of terms included in the pole expansion, Δ is a diagonal matrix with the values of the poles, and U is the identity matrix of size Q.

Figure 1.

Planar junction between two waveguides of lengths l1 and l2.

Figure 2.

Equivalent circuit representation of the structure shown in Figure 1.

[8] According to Figure 2, we will compute the wideband impedance matrix in the form of (1) for the uniform waveguide sections (Zw1 and Zw2), for each planar junction (Zst), and also for the two sets of asymptotic modal admittances generated by the integral equation technique [Gerini et al., 1998], which are denoted as Za1 and Za2, respectively. Next, we concentrate on the novel aspects related to the efficient computation of all such matrices.

2.1. Generalized Z Matrix of Planar Waveguide Steps

[9] First, we consider the planar junction between two arbitrary waveguides shown in Figure 1 for l1 = l2 = 0. Following the integral equation technique described by Gerini et al. [1998], such junction can be represented in terms of a generalized Z matrix (Zst), and two sets of asymptotic modal admittances (see Figure 2), which are determined as follows

equation image

where Ym(δ) and κm(δ) represent, respectively, the modal admittance and the cutoff wavenumber of the m-th mode at waveguide port δ (δ = 1, 2), whose definitions for TE and TM modes are given by Conciauro et al. [2000].

[10] In order to derive the expressions for the elements of the generalized Z matrix of the planar junction (Zst in Figure 2), the next integral equation set up for the magnetic field at the junction plane must be solved (see more details about its derivation in the work of Gerini et al. [1998])

equation image

where hn(γ) is the normalized magnetic field related to the n-th mode at waveguide γ [Conciauro et al., 2000], and Mn(γ) is the unknown tangent magnetic field at the junction plane.

[11] If we want to find an expression for Zst in the form of (1), we must express the kernel of the previous integral equation as a sum of terms depending on k and 1/k. Taking into account (2), the first summation of (3) fulfills such condition directly. Regarding the second summation in (3), since Ym(ζ) → Ŷm(ζ) when m → ∞, we can approximate the term within parenthesis by its Taylor series

equation image

where the values of the first coefficients cr for the TE and TM modes are shown in Table 1. Then, if we consider a k2 frequency dependency for TE modes and all contributions from TM modes are set to be frequency independent (due to the definitions of the asymptotic modal admittances given in (2) and the expression for the second summation in (3)), we can rewrite the previous Taylor series as follows

equation image

where k0 corresponds to the value of k at the center point of the frequency range. Proceeding in this way, we manage to express the second series of (3) as the required combination of terms with k and 1/k dependence.

Table 1. Values of the First Coefficients cr for TE and TM Modes
rTETM
11/2−1/2
21/8−3/8
31/16−5/16

[12] It is important to highlight that the method proposed by Mira et al. [2006] only considers the first term in (5) for TE modes, and no contribution from TM modes. As it will be proved in section 4 with several practical examples, the more rigorous approach proposed in this paper involves a reduction in the number of accessible modes, i.e. N(γ) in (3), required to obtain an accurate representation of the planar junction in the whole frequency range. Therefore, the computational effort related to this new characterization technique will be decreased.

[13] After inserting (2) and (5) into (3), we solve the resulting integral equation by means of the Method of Moments (Galerkin approach). Following the procedure detailed by Mira et al. [2006], we obtain the linear system of equations

equation image

where α(γ) contains the expansion coefficients αq,n(γ) related to the unknown Mn(γ), and Q(γ) and P matrix elements are computed as indicated next

equation image
equation image
equation image
equation image

where p, q = 1,…, Q (Q being the number of vector basis functions used to expand Mn(γ)), δp,q stands for the Kronecker's delta (δp,q = 1 if p = q and δp,q = 0 if pq), and

equation image

[14] As indicated by Mira et al. [2006], the matrix P can be inverted by solving an eigenvalue problem. Then, the elements of the generalized Z matrix of the planar step can be finally derived through the evaluation of

equation image

thus obtaining the following final expressions for all possible combinations of TE and TM modes

equation image
equation image
equation image
equation image

where we have that yi(δ) = {Q11(δ)}Txi and yi(δ) = {E12(δ)}Txi, E12(δ) = Q12(δ)S12S22−1Q22(δ), ki and xi are, respectively, the eigenvalues and eigenvectors related to the inversion of the matrix P, the subscripts 1 and 2 refer, respectively, to TE and TM modes, and Q1 is the number of the Q vector basis functions corresponding to TE modes.

[15] If we recall the expression (1) for the generalized impedance matrix in the form of a pole expansion, we find the following frequency independent blocks for the planar waveguide junction under study

equation image
equation image
equation image

where Y(δ) = [y1(δ)yQ1(δ)] and Y(δ) = [y1(δ)yQ1(δ)] for δ = 1, 2.

2.2. Generalized Z Matrix of Asymptotic Admittances

[16] Each set of asymptotic modal admittances in Figure 2 can be seen as a two-port network, which can be easily characterized by a generalized Z matrix (Za) whose elements are defined as follows

equation image

[17] The previous expression is suitable for the representation of the generalized impedance matrix as indicated by (1). In this case, the pole expansion is not present and therefore

equation image
equation image

2.3. Generalized Z Matrix of Uniform Waveguides

[18] Now, we derive the expressions for the generalized impedance matrix of a uniform waveguide section of length l. Since the modes of this element are uncoupled, the only nonzero entries of such Z matrix (Zw) are those relating voltages and currents of the same mode, which are computed as

equation image
equation image

where γm(1) = [(κm(1))2k2]1/2, and κm(1) and Zm(1) are, respectively, the cutoff wavenumber and the characteristic impedance of the m-th mode considered in the waveguide section [Conciauro et al., 2000].

[19] Considering that the modes of the waveguide section can be of type TE or TM, we obtain the following expressions

equation image
equation image

[20] In order to express the Z matrix elements of the waveguide section in the form of the pole expansion collected in (1), we will make use of the theorem of Mittag-Leffler [Spiegel, 1991]

equation image

where Res(f, zp) are the residues of function f related to their poles zp, which in our case are defined as

equation image

with s = 0, 1, 2,… for TE modes and s = 1, 2, 3,… for TM modes. It can be seen that the values for the poles are directly related to the resonant wavenumbers of the open-circuited waveguide.

[21] Then, applying the theorem of Mittag-Leffler to the functions f in equations (25) and (26), we can obtain

equation image
equation image
equation image
equation image

where εs means the Neumann's factor (i.e. εs = 1 if s = 0 and εs = 2 if s ≠ 0).

[22] For the case of the frTM and ftTM functions we need further treatment of the previous expressions. In particular, we must extract the low frequency contribution from the series in (30) and (32). Then, after solving analytically the infinite summations when k → 0 [Gradstheyn and Ryzhik, 1980], we can obtain that

equation image
equation image

[23] Finally, if we introduce the previous expansions (29), (31), (33) and (34) into (25) and (26), we obtain the Z matrix representation in the form of (1), where the entries of the frequency independent matrices are

equation image
equation image
equation image
equation image
equation image

and the elements of the matrices C(1) and C(2) are defined as follow

equation image

3. Efficient Cascade Connection of Z Matrices

[24] Once the expressions for the generalized Z matrices of all basic blocks of the structure shown in Figure 1 have been presented, we proceed to combine them in order to determine the wideband Z matrix representation of the complete structure. For such purpose, we first outline a procedure to solve the combination of two cascaded Z matrices in the form of pole expansions, which follows a dual formulation to the one derived by Arcioni and Conciauro [1999] for the admittance matrix case. Then, a novel efficient algorithm, which allows to reduce the effective number of poles to be considered after connecting two wideband Z matrices, is fully described.

3.1. Combination of Two Generalized Z Matrices

[25] Let us consider two cascaded building blocks of the structure shown in Figure 1, whose generalized Z matrices (named as ZI and ZII in Figure 3) are given in the previous form of (1). As it can be inferred from Figure 3, the voltages and the currents at the external ports are grouped into the vectors v(1), i(1), v(2) and i(2), and the currents at the connected ports are collected into the vector i(c). If we consider that the currents i(1) and i(2) are incoming, respectively, to the blocks I and II, and the currents i(c) are incoming to the block II, we can write

equation image
equation image

where

equation image

and the matrices Zee, Zec and Zcc are given by

equation image
Figure 3.

Two elementary building blocks connected in cascade.

[26] Our final goal is to obtain the overall matrix Ztot, relating the vectors v and i, in the form of (1). For such purpose, we first express the matrices Zee, Zec and Zcc in the form of pole expansions. Taking into account the definitions of the matrices included in Table 2, we can easily write that

equation image
equation image
equation image
Table 2. Definition of Matrices
A MatricesB MatricesC and Δ Matrices
Aee = equation imageBee = equation imageequation image
Acc = [AI(c,c) + AII(c,c)]Bcc = [BI(c,c) + BII(c,c)]Cc = [−CI(c)CII(c)]
Aec = equation imageBec = equation imageΔ =equation image

[27] Then, the currents at the connected ports can be arranged in the following way

equation image

where i1c and i2c contain, respectively, the currents corresponding to TE (subscript 1) and TM (subscript 2) modes. According to such arrangement, the related matrices of Table 2 can be partitioned as indicated in Table 3, where zero matrices appear in the partitioning of Acc and Aec when TE modes are involved (remember the expressions collected in (13)–(16), (21), (35) and (36)). At this point, the problem to be solved is completely dual to the one considered by Arcioni and Conciauro [1999] for the admittance matrix formulation. Therefore, following a dual procedure, we can easily deduce the Z matrix for the cascaded connection of the building blocks in the required form of pole expansions

equation image

where

equation image
equation image
equation image

and

equation image
equation image
equation image
Table 3. Partitioning of Matrices
A MatricesB MatricesC Matrices
Acc = equation imageBcc = equation imageCc = equation image
Aec = equation imageBec = equation image 

[28] Furthermore, the K = diag{ki} is a diagonal matrix with the eigenvalues, and Xi and Xw are matrices with the eigenvectors, corresponding to the solution of the problem

equation image

where

equation image
equation image
equation image

and the auxiliary vector wc is defined as follows

equation image

3.2. Efficient Characterization of Passive Structures

[29] When two building blocks of a structure are cascaded, the number of terms in the resulting pole expansion is equal to the number of poles for each block plus the number of TM accessible modes at the common port (see the generalized eigenvalue problem raised in (56), as well as the definition and partitioning of the involved matrices in Tables 2 and 3). If the structure is composed of many blocks, the total number of poles will become very high, thus reducing the efficiency of the algorithm due to the size of the successive eigenvalue problems.

[30] For avoiding such drawback, we propose to limit the number of eigenvalues considered after each connection. When two different blocks are connected, we obtain the entries of the generalized Z matrix in the following form

equation image

[31] In this equation, the higher terms of the sum have a lower contribution to the final result. Due to this fact, we can only consider Q′ eigenvalues in the sum, and approximate the contribution of the remaining QQ′ eigenvalues by

equation image

where k0 corresponds to the value of k at the center point of the frequency band. Proceeding in such a way, the eigenvalues with lower weight are included within the linear term

equation image

thus obtaining a reduced size for the eigenvalue problem to be solved during the next connection, whereas very good accuracy is still preserved. This technique can also be applied to reduce the number of poles involved in the Z matrix characterization of each single building block, i.e. waveguide steps and uniform waveguide sections.

4. Validation Results

[32] First of all, we proceed to validate the novel contribution of this paper related to the use of more terms (k0 ≠ 0) in the series expansions collected in (5). For this purpose, we have considered a simple planar junction between two rectangular waveguides (see geometry in Figure 4). In this figure, we show the results obtained for 1, 8 and 15 accessible modes (N), whereas the total number of basis functions (Q) is equal to 100 (Q1 = 58), R = 3 in (5), and the infinite series in (9) and (10) are summed up with 600 terms. With regard to the convergent evaluation of the frequency-dependent series in (13)–(16), only the first 25 terms (poles) have been required. For comparative reasons, we also include the results for k0 = 0 in (5), which correspond to the ones provided by the method proposed by Mira et al. [2006], and the convergent results obtained with the original integral equation technique using only one accessible mode [Gerini et al., 1998]. It can be concluded that the use of k0 clearly improves the convergence rate of our method, thus involving a reduction in the required number of accessible modes (N) when compared to the previous approach of Mira et al. [2006].

Figure 4.

Convergence study of the new method for a single waveguide step, whose dimensions are a1 = 19.05 mm, b1 = 9.525 mm, a2 = 13.0 mm, b2 = 5.5 mm, xs = 4.0 mm and ys = 3.0 mm.

[33] In terms of numerical efficiency, the CPU time required by our method to solve the considered waveguide step in the whole frequency range (201 frequency points) has been equal to 0.10 s for the worst case (N = 15). (All reported CPU times have been obtained with a Pentium4 at 3.2 GHz.) However, the original integral equation technique from Gerini et al. [1998] needed 1.01 s for solving the same planar junction. Therefore, the analysis method of planar waveguide junctions proposed in this paper involves a substantial reduction in the related computational effort, without degrading the accuracy of results.

[34] Next, we have applied the complete technique proposed in this paper to the full-wave characterization of two waveguide filters. In order to get an accurate modelling of all waveguide steps involved in both examples, we have always considered the same values for the set of parameters affecting the convergence of the analysis method, i.e. 20 accessible modes, 150 basis functions (40 terms in the pole expansion), 3 terms in the Taylor series of (5), and 800 terms for summing up the infinite series in (9) and (10). Then, the efficient technique described in section 3 has been followed to determine the wideband impedance matrix representation of each filter.

[35] The practical application of the iterative algorithm of section 3 has been implemented in the same way for both filter examples. First, the cascade connection of the generalized matrices representing each waveguide step, and their related two sets of asymptotic admittances (see Figure 2), has been solved. For this purpose, only four poles (Q′ = 4) have been preserved in (63) after solving the cited connections. Next, the generalized matrices representing the uniform waveguide sections cascaded to the steps have been added. In this case, we have chosen Q′ = 20 in (63) for characterizing each single waveguide section, and the total number of resulting poles after cascading each pair of Z matrices has been always kept under control. In fact, the final number of poles related to the last cascade connection was equal to 100.

[36] Even though the precise wideband characterization of any passive waveguide component requires a detailed convergence study, in order to finely tune the optimal values of all parameters mentioned above, most waveguide filters based on cascaded planar junctions can be accurately modelled as described in the two previous paragraphs. Therefore, such procedure has been first applied to the full-wave analysis of a four-pole filter in WR-75 waveguide (a = 19.05 mm, b = 9.525 mm), whose geometry is shown in Figure 5. The dimensions for the centered coupling windows of this filter are w1 = 9.55 mm, w2 = 6.49 mm and w3 = 5.89 mm, h1 = h2 = h3 = 6.0 mm and d = 2.0 mm, and the lengths of the WR-75 waveguide cavities are l1 = 11.95 mm and l2 = 13.37 mm.

Figure 5.

Geometry of a four-pole filter in rectangular waveguide technology.

[37] In Figure 6, we successfully compare the set of S parameters obtained with our pole expansion technique (solid lines) with those provided by the well-known commercial software Ansoft HFSS (v.10.0) based on the Finite Elements Method (FEM). In order to evaluate the numerical efficiency related to the analysis technique proposed in this paper, we have also made use of a traditional approach based on frequency-by-frequency (point-to-point) calculations for solving the cascade connection of Z matrices [see, e.g., Boria et al., 1997]. As it can be concluded from Figure 6 (see results with stars), the same accurate response can be obtained with the point-to-point cascade connection technique. However, the CPU effort required by such traditional approach to compute the electrical response for 301 frequency values was equal to 0.54 s, whereas following the method proposed in this paper was reduced to only 0.23 s. If the original integral equation technique from Gerini et al. [1998] is used to analyze the planar waveguide steps, together with the point-to-point connection technique, the same accurate results from Figure 6 can be recovered in 2.70 s. Therefore, it can be concluded that our novel analysis technique is the most efficient one in terms of computational effort. Furthermore, we have verified that the CPU time required by our method remains rather stable with the required number of frequency points, thus making it very appropriate for dense simulations in the frequency domain. Finally, we have also included in Figure 6 the results obtained when k0 = 0 in (5) and (6). As it could be expected, such results converge more slowly than those provided by our analysis technique.

Figure 6.

S parameters of a four-pole filter in rectangular waveguide technology.

[38] The second example deals with the full-wave analysis of a triple-mode filter proposed by Lastoria et al. [1998], whose topology is shown in Figure 7. The input and output sections of this filter are WR-75 waveguides (a = 19.05 mm, b = 9.525 mm), and the dimensions of the five inner waveguides are a1 = 12.2 mm, a2 = 19.6 mm, a3 = 19.6 mm, a4 = 15.6 mm and a5 = 5.0 mm for the widths, b1 = 5.0 mm, b2 = 15.6 mm, b3 = 19.6 mm, b4 = 19.6 mm and b5 = 12.2 mm for the heights, l1 = 5.3 mm, l2 = 4.0 mm, l3 = 8.6 mm, l4 = 4.0 mm and l5 = 5.3 mm for the lengths. The input and output coupling windows are centered with regard to the central cavity.

Figure 7.

Geometry of a triple-mode filter in rectangular waveguide technology.

[39] In Figure 8, our results (solid lines) are well compared with those provided by the commercial software HFSS (circles). The same accurate results (see stars in Figure 8) can be obtained if the frequency-by-frequency (point-to-point) technique for cascading Z matrices proposed by Boria et al. [1997] is used. However, our novel analysis method provides such results in only 1.13 s for 301 frequency points, whereas the CPU effort is raised to 2.93 s when using the point-to-point connection technique. Finally, the combination of the original integral equation technique from Gerini et al. [1998] with the cited point-to-point connection technique would provide the same accurate results in 29 s.

Figure 8.

S parameters of the triple-mode cavity filter.

[40] To sum up, we have collected in Table 4 all the CPU times previously reported for the two considered filter examples. For comparative reasons, the CPU time required to obtain convergent results using the commercial software Ansoft HFSS (v.10.0) is also displayed. As it can be seen, the new analysis technique proposed in this work is always more computationally efficient.

Table 4. Comparative Study in Terms of CPU Time
ParameterExample 1Example 2
Four-Pole Direct-Coupled FilterTriple-Mode Filter
Accessible modes2020
Basis functions150150
Series terms800800
Global poles100100
Frequency points301301
CPU time (s)0.231.13
new technique  
CPU time (s)0.542.93
point-to-point cascade  
CPU time (s)2.7029
original integral equation  
CPU time (s) HFSS1490467

[41] Finally, we have studied the accuracy of the proposed method when used to predict the out-of-band response of the triple-mode filter just considered before. In Figure 9, we compare the S parameters of such structure in a very wide frequency band (1000 points comprised between 8 and 18 GHz) when k0 in (63) is chosen to be 0 and equal to the value of the in-band center frequency. As it can be observed, both results are less accurate at very high frequencies (far from the center frequency of the filter), and more accuracy is preserved when additional terms are considered in (63). when k0 ≠ 0. In order to recover more accurate results, even at very high frequencies, it has been needed to increase the number of accessible modes to 25 and the total number of poles to 250, thus involving a CPU effort (1000 points) of 4.9 s.

Figure 9.

Out-of-band response of the triple-mode cavity filter.

5. Conclusion

[42] In this paper, we have presented a very efficient procedure to compute the wideband generalized impedance matrix representation of cascaded planar waveguide junctions, which allows to model a wide variety of real passive components. The proposed method provides the generalized Z matrices of waveguide steps and uniform waveguide sections in the form of pole expansions. Then, such matrices are combined following an iterative algorithm, which finally provides a wideband matrix representation of the complete structure. Proceeding in this way, the most expensive computations are performed outside the frequency loop, thus widely reducing the computational effort required for the analysis of complex geometries with a high frequency resolution. The accuracy and numerical efficiency of this new technique have been successfully validated through the full-wave analysis of two practical examples of rectangular waveguide filters.

Acknowledgments

[43] This work has been supported by the Ministerio de Educación y Ciencia, Spanish Government, under the coordinated Research Project TEC 2007/67630-C03-01.

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