Basic Theory of Composite Filters
By RLC Electronics, Inc.

The model of direct coupled cavities has been initiated as base model describing reflection and transmission properties of a filter as non-uniform waveguide. The model is based on EM approach taken modes of higher order into account. Thus the filter is considered as sequence of cavities or sub-filters selected by such order, which provides elimination of spurious pass-bands of some elements by stop-bands of other elements in respect to different waveguide modes. The model of quasi-periodical cavities has been considered as base model describing reflection and transmission properties of a filter as quasi-uniform transmission line in multi-moded condition of propagation.

Filter Cavity Model
The cavity element is represented as three uniform rectangular waveguide sections uniaxially connected to each other and figure of cross section of the middle waveguide includes cross-section figures of the end waveguides. In order to compute s-parameters of the cavity, the multimodal variational method [10] is used.

According to Harrington’s “mode functions”, “mode voltage” and “mode current” concepts [11], the network expressions for transverse E and H – fields in node waveguides are given by

Here vn(0,1) (z) and in(0,1) (z) correspond respectively to the sum and difference of incident and reflected wave amplitudes of the n-the eigenmode at waveguides 0 and 1. The modal surface current density vector is related to
Hn by in(0,1) = hn(0,1) × z

z being the direction of propagation. E and H-fields in the cavity waveguide are represented as superposition of standing waves

Here An(0,1) corresponds to modal amplitude of standing modes of order n shorted at z=0 and z=L respectively, ßn corresponds to propagation number of the mode and yn is wave admittance of the mode related with space wave number by
yn = k/ ßn for TM-modes
yn = ßn /k for TE-modes.

Projecting boundary conditions for electric field at planes of junctions

on egenfunction basis {En} and projecting boundary conditions related to continuity of magnetic field at apertures of junctions

Multimodal equivalent network of the cavity element can be represented as -circuit of three matrix reactances [12] as shown on Figure 2.

S-parameters matrix can be expressed as

and U is unity matrix of rank same with Y matrix.

Multi-moded Scattering
Expanding expression (11) by sub-matrices we obtain expression

for multi-moded scattering matrix as stacked matrix of transmission and reflection matrices.

Transmission Zeros
Simple analysis of expression (12) shows that at points of singularity of matrix function Y0,1(k) appropriate element of transmission matrix S0,1(k) turns to zero. On Figure 3, it is shown a typical transmission/conversion versus frequency response for a symmetric cavity where transmission zeros and critical conversion points are marked. It is also shown below that position of transmission zero on frequency axis can be approximately expressed by waveguide stub formula (13).

Reflection Zeros
In case of symmetric cavities, reflection zeros of particular waveguide modes coincide with cut-off frequencies of those in input/output nodes. It can be also noted on Figure 3 that reflection zeros coincide with critical points of mode-to-mode conversion.

Theoretical Foundation of Spurious-less Filters
It was stated above that conventional design methods do not take into account information about critical points corresponding non-dominant waveguide modes, which form pass-bands or stop-bands in addition to the synthesized one-mode frequency response. However, the only method to design a spurious-less filter must be based on right distribution of those critical points over the bandwidth such a way providing full cover of reflection zeros by transmission zeros. The basic criteria of such approach are presented below.

E-plane Approximation
If relative change of width of input/output of cavity difference is much smaller than change of height, a0,1/A˜1 and b0,1/B<<1, all modes of TEnm and TMnm types of same n-index can be separated into smaller sub-matrices, because the other elements of Y-matrix are smaller. In this case, irregularities corresponding to first transmission zeros can be extracted from Y0,1 sub-matrix as a shorted waveguide stub as shown on Figure 3. T he model of E-plane T-junction [12] loaded with shorted stub can be used.

This T-junction network is similar with E-plane T-junction network in [12] generalized for multi-moded condition. Assuming height of input/output ports much smaller than their width, we can limit our consideration to only TEn0 - modes. Then transmission zero corresponding to TEn0 - mode is determined by effective length L of the stub and cut-off frequency kn0 corresponding to prototype mode in the waveguide forming the stub, so

The reflection zeros kn0 of first TEn0 - modes are generally related to cut-off frequencies of input/output waveguides kn0.

Corrugated Waveguide
Similar approach can be applied to infinite wave guiding line consisting from considered elements connected to each other periodically. In this case, we can obtain

where j is phase shift of waveguide mode per element and U is unity matrix. Thus field structure of n-th eigen mode can be expressed by n-th normalized eigenvector Vn of matrix

Y0,1 Y0,0 . Then wave admittance for the n-th eigen mode can be defined as scalar

It can be noted that propagation of particular waveguide mode is impossible in vicinity of frequency points of singularity of matrix Y0,0 corresponding to the mode. It can also be shown that in case of E-plane uniformity the eigen vectors corresponding to modes of different index n are not coupled. This means that in case of E-plane structures the scattering parameters of particular mode of index n are function of only bn,0. This property has been reported by Levy in his article [3].

Corrugated Structures with Distributed Transmission Zeros
Here quasi-periodic corrugated waveguides are considered as part of harmonic filters. In this case principle of partial reflections applied to tapered transmission lines [13]

where Lw is length of the tapered quasi-periodic waveguide loaded with semi-infinitive periodic waveguides, Gni is input reflection at input corresponding to n-th mode, fn(z) and Ync(z) are phase (14) and wave admittance (15) interpolated at z-section. Transmission of the mode through quasi-periodic waveguide can be also approximately expressed by

If transmission factor of at least one cavity element turns to zero at frequency point k, the transmission factor of whole tapered corrugated structure also turns to zero. If each cavity element eigen value of matrix

-Y0,1 Y0,0

corresponding to particular eigen mode is greater than 1 in given frequency range [k0,k1], the mode is attenuated through the corrugated structure in the frequency range. In the vicinity of transmission zero ktz we define loaded Q-factor of a single cavity as

If transmission zeros are distributed over frequency range [k0,k1] as close to each other as

the absolute value of transmission coefficient is less than specified Tsp.

Junctions
Tapered corrugated waveguides can be connected to each other directly matching wave admittances (15) or via step transformers. Those methods are widely known and, therefore not described here. However conversion of waveguide modes of order higher than dominant can be computed using expressions (10,11) if cross-section of one of port waveguides coincide with cross-section of the cavity waveguide.

E-plane Corrugated Filter
E-plane corrugated waveguide filter can be synthesized minimizing reflection function (16) over given pass-band [kp0,kp1] and transmission function (17) over specified stop-band [ks0,ks1] using appropriate functional and variational procedure. Typical TE10-mode response for a E-plane corrugated filter is shown on Figure 4. The frequency response can be characterized by the following features:

• Pass-band from mode excitation to vicinity of the first transmission zero
• Stop-band from the first to the end of vicinity of the last transmission zero
• Zone of dense spectrum of transmission zeros with high attenuation of the mode.
  Here Q-factors (11) of cavities are high and transmission zeros are close to each
  other Zone of sparse spectrum of transmission zeros with low attenuation of the mode.
  Here Q-factors (11) of cavities are lower and transmission zeros not so close.
• TE30-mode excitation caused by not-Eplane elements (transformer junctions)

Theoretically this kind of filter can be synthesized matching any not intersecting stop-band and pass-band for particular waveguide mode. However, the same frequency response features shown by the dominant mode are related to any other waveguide mode having different n-index (n-modes). In other words, frequency response of any n-mode is an image of frequency response of the dominant mode transferred using frequency transformation function [4] as

Thus particular n-mode shows pass-band in frequency range [kn(kp0),kn(kp1)], and stop-band in frequency range [kn(ks0),kn(ks1)], which are predictable but cannot be removed. Typical frequency response of E-plane corrugated filter for different waveguide modes is shown on Figure 5.

It can be noted the filter designed to match particular transmission requirements cannot match them for the other n-modes. As pass-band of an n-mode [kn(kp0),kn(kp1)] is determined by transfer function (19), it might be impossible to design a filter providing rejection for in wide stop-band for all n-modes simultaneously. However, the filter can be used as a partial element of a composite filter presented here.

Composite E-plane Corrugated Filter
The filter presented here is based on two E-plane uniform quasi-periodic tapered waveguides (partial filters) connected to each other directly and coupled with external waveguide line using conventional transforming methods (see Figure 6). The basic principles of design method are based on the conditions:

• The partial filters are connected to each other matching input/output wave admittance
  (15) criterion
• Corrugations of each partial filter are tapered from ends to center such a way providing
  distribution transmission zeros over design stop-band
• Widths of partial filters are selected to coincide pass-band zone corresponding to
  particular spurious n-mode of one partial filter with stop-band zone corresponding to
  the n-mode of the other partial filter
• Any of n-mode excited by asymmetry of junction of partial filters is rejected by one of
  the partial filters

Spectrum Superposition
The multi-moded transmission responses (see Figure 5) can be schematically drawn as block graphs for the two filters on the same chart as shown on Figure 7 and Figure 8. Selecting partial filters with different a-dimension the pass-bands corresponding to spurious n-modes could be positioned relatively to each other such a way when they could not have mutual frequency points as shown on Figure 7. In this case any of n-modes carrying spurious frequency spectrum in one of the partial filters will be rejected by the other partial filter. This condition is called “strong” here, because it guarantees rejection of spurious frequency spectrum carried by all modes except the dominant one.

If positions of pass-bands of spurious modes corresponding to both partial filters intersect each other, but the intersecting modes have different symmetry n-indexes (see Figure 8), this condition is called “weak” here. Weak condition also guarantees no spurious mode can propagate through both partial filters if they keep ideal symmetry. Both conditions of superposition of spectrum can be used for design. However, it has to be mentioned that the weak condition provides practically less rejection because some asymmetry caused by production inaccuracy can potentially cause reduction of rejection due to conversion between symmetric and asymmetric modes. From other hand the “weak” filters are expected to demonstrate less insertion loss and more power.

Example Filters and Comparison Criteria Designed Filters
Three harmonic filters have been originated, designed and proposed in order to demonstrate advantages of the composite E-plane filters relatively to traditional harmonic filters used and known in microwave engineering applications. All three filters have been designed using the principles, theory, design methods and computer algorithms developed during this work.

References
[1] S.B. Cohn, “A theoretical and experimental study of a waveguide filter structure,” Office Naval Res., Cruft Lab., Harvard Univ., Cambridge, Mass., Rep. 39, Apr. 25, 1948
[2] C.G. Matthaei, L. Young, and E.M.T. Jones, “Microwave Filters, Impedance Matching Networks, and Coupling Structures,” McGraw-Hill, New York, 1964.
[3] R. Levy, “Tapered Corrugated Waveguide Low-Pass Filter,” IEEE Trans. Microwave Theory Tech., MTT-21, August 1973, pp. 526-532.
[4] Ralph Levy, “A Periodic Tapered Corrugated Waveguide Filter,” US Patent 3,597,710, Nov. 28, 1969.
[5] S.B. Cohn, US Patent 3,046,503. July 1962.
[6] E.D. Sharp, “A High-Power Wide-Band Waffle-Iron Filter,” IEEE, Trans. Microwave Theory and Tech., March 1963, pp. 111-119.
[7] H. Chapell, “Waveguide Low-Pass Filter,” US Patent 3,949,327. 1976.
[8] J. Rodgers, Y. Carmel, P. O’Shea, “Electromagnetic Effects on Integrated Circuits and Systems at Microwave Frequencies,” Institute for Research in Electronics and Applied Physics, University of Maryland, 2001.
[9] Saad, Low Pass Filters with Finite Transmission Zeros in Evanescent Modes,” US Patent 4,646,039. 1987.
[10] J.W. Tao, H. Baudrand, “Multimodal Variational Analysis of Uniaxial Waveguide Discontinuity,” IEEE, Trans. MTT, v. 39, No. 3, March 1991.
[11] R. Harrington, “Time Harmonic Electromagnetic Fields,” McGraw-Hill, New York, 1961.
[12] N. Marcuvitz, “Waveguide Handbook,” Peter Peregrinus Ltd., 1986.
[13] R.E. Collin, “Foundation for Microwave Engineering,” McGraw-Hill, 1966.

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