Waffle-Iron Harmonic (Low-Pass) Filters

Where the spurious comes from

By Rousslan A. Goulouev

Home Page | Site Map | Historic Place ! It was 264x60 Google AdSense banner here from 10/10/09 to 11/01/09. Do you know ? 1. Waffle-iron filters can be often replaced with corrugated waveguide filters, which perform better. 2. Corrugated harmonic filter can be designed online using WR-Connect.

Figure 1: Internal surface of a waffle-iron filter

Summary

The waffle-iron filter was firstly introduced by Cohn [1] in 1962. The conventional structure was represented as uniform two-dimensional periodic structure of rectangular teeth put into a waveguide and coupled with interface by E-plane stepped transformers. A design method based on representing the waffle iron structure as uniform waveguide was presented in [2]. Limited power handling is considered as major disadvantage of the waffle-iron filter of Cohn’s design. In addition to power handling limitation spurious spikes in pass-band, roll-off and stop-band are reported [3]. Those spikes are caused by excitation of waveguide modes of higher order, which are not taken into account in known design methods. Later modifications applied by Levy [4] and Sharp [5] do not considerably overcome those disadvantages.

Principles of Operation

Figure 2: Top and side view of classic waffle-iron structure

The waffle-iron filter is based on two-dimensional slow-wave delay structure of rectangular teeth mounted on wide walls of rectangular waveguide and matched with external waveguide line by quarter-wave stepped transformers. Propagation in “waffle-iron” section can be described by two orthogonal wave numbers k_x (transverse) and k_y (longitudinal) as functions of wave number in space (k). Then cut-off frequency of quasi-TE_nm mode corresponding to “waffle-iron” waveguide (waveguide of cross-section shown on Figure 2 (left)) can be expressed

kc_TEnm = m-th root(k_x(k)-nπ/Ax). (1)

Here k_x is propagation number corresponding to one-dimensional periodic corrugated structure (vane-type, or corrugated plane) forming waffle-iron on x-direction. It is known that transmission through the corrugated plane structure turns to zero and “the first cutoff occurs approximately when the stub becomes resonant, i.e. k(B-b)=π/2” [6]. Simple analysis shows that quasi-TE_no-mode with n < ½ Ax/(B-b) has lower cut-off frequency than prototype TE_no-mode of rectangular waveguide and quasi-TE_no-mode with n > ½ A_x/(B-b) has higher cut-off frequency than prototype TE_no-mode of rectangular waveguide. This is basic idea of operation of waffle-iron filter is based on moving the spurious pass-bands of equivalent corrugated filter (same filter without longitudinal slots) up or down from the design stop-band. If the spurious moves higher than stop-band (condition n > ½ A_x/(B-b)), it is OK. If the spurious moves lower than stop-band (condition n < ½ Ax/(B-b)), it seams to be also OK because the spurious modes cannot be theoretically excited (prototype TEno-mode is evanescent in transformers and interface waveguides [1,2,5]).

Closed Modes

Assuming the spurious modes TE_no (n>1) being shorted at transformer apertures we can apply similar approach for evaluation of spectrum of their resonances and obtain the following expression

k_xynm = (k_xn²+k_ym²) ^½ , (2)

Where k_xynm is plurality of wave numbers corresponding to modes with transverse and longitudinal resonant numbers n and m, and

k_xn = roots(k_x(k)-nπ/Ax), (3)

k_ym= roots(k_y(k)-mπ/Ay).

The resonances can be computed for two-dimensional periodic waffle-iron structure using simple mode-matching procedure presented in [8].

Corrugated Filter

In case of periodic corrugated structure, when k_xn = nπ/Ax, spurious resonances are grouped into separate frequency bands representing TE_no-modes (n=2,3,4,..) (see Figure 3) well known as “spurious responses” [4] (see more in [7]).

Figure 3: Layout of spurious resonances on frequency axis for a Ku-band corrugated filter

Waffle-Iron Filter

In case of two-dimensional periodic waffle-iron structure, spurious resonances are grouped into two frequency bands representing lower quasi-TE_no-modes (n=2,3,4,..,Nc) and higher quasi-TE_no-modes (n=Nc, Nc+1,…) ( Nc=int{½ Ax/(B-b)} ) separated by blank zone with no resonances (see Figure 4). It can be noted that longitudinal slots have not removed the spurious shown on Figure 3 but moved it left or right. It can be also noted that the lower spurious resonances have moved to the pass-band and roll-off zones of the filter.

Figure 4: Layout of spurious resonances on frequency axis for a Ku-band waffle-iron filter (the same corrugated structure longitudinally slotted)

Spikes in Pass-Band

In accordance with basic theory of waffle-iron filters [1,2,6] the resonances should not be excited because of two reasons;

1. A spurious mode TE_no capable to generate any of lower resonances (Figure 4) is evanescent in transformers and input/output rectangular waveguides.
2. The dominant mode TE₁₀ cannot couple with any of lower resonances because its symmetry order is different.

Practically any of the spurious resonances can be easy excited by coupling with the dominant mode (TE₁₀) caused by practical asymmetry (tolerances, offset, shift, etc.). The resonances having odd n-index greater than 1 (3,5,…) can be directly coupled with TE₁₀-mode and excited if even the filter is built ideally. If excited, the spurious resonances can cause spikes in frequency response similar to shown on Figure 5.

Figure 5: Spike of spurious quasi-TE₄₀-mode in pass-band of a Ku-band waffle-iron filter caused by tolerances

Spikes in Stop-Band

Spurious responses can appear in design stop-band, if cut-off frequency of at least one spurious mode (1) resides there. The amplitude and bandwidth of such spurious will depend on asymmetry of structure or existence of the prototype mode in the interface waveguides. The nature of the spurious is similar to spurious pass-bands of a corrugated filter, which are images of the pass-band of the dominant mode (see [7] for more information). Therefore it is theoretically impossible to design a spurious-less waffle-iron filter. However it might be possible to move the spurious out of spec bands.

Figure 6: Spike of spurious mode (likely quasi-TE₂₀) in pass-band of a C-band waffle-iron filter [3]

Peak Power Handling

As the stop-bandwidth (see Figure 4) directly depends on “slowness” of waffle-iron structure in two directions, the gap (b for asymmetric filters and 2b for vertically symmetric filters) must be very small in order to move the spectrum of spurious resonances (see Figure 4) below the design stop-band and/or reduce margin between pass-band and stop-band. For example for the Ku-band waffle-iron filter (see Figure 4) designed for pass-band 13.5 – 14.5 GHz and stop-band 17 – 33 GHz, the gap dimension cannot be chosen greater than 0.03’’ (b<0.018λ or 2b<0.036λ). Besides to small “voltage gap”, smallness and sharpness of teeth also increase the maximum value of strength of electrical field in several times relatively to equivalent plane waveguide having the same gap dimension. Therefore the waffle-iron filter cannot be used in multi-carrier space applications, because of very low peak power handling capacity.

Production Sensitivity

As the gap dimension is small, it can be practically very difficult to keep its uniformity over the waffle-iron structure. Even small tolerance +/-0.001’’ randomly applied to teeth can significantly worsen VSWR (return loss) for Ku-band and Ka-band filters, as it is large relatively to tiny dimensions. In addition to degrading pass-band so usual for all type of filters, the random tolerances can cause spurious spikes in pass-band (see Figure 5) by exciting “high Q” resonances of closed odd modes (see Figure 4). The resonances can be also caused by other nature of asymmetry (for example twisting, offsetting, deformation, temperature expansion, mechanical tension, etc.)

Design Methods

The known design methods based on synthesis of equivalent periodic waveguide [2] or distributed circuit [4] taking into account only the dominant mode. Nevertheless, analysis of spurious modes is much more sophisticated than in case of corrugated filters. The analysis must include:

1. Solving problem of eigen modes of waffle-iron waveguide in order to make sure no spurious mode has cut-off frequency in spec bands.
2. Computing and analyzing frequency responses corresponding to other TE_no-modes.
3. Performing detailed sensitivity analysis simulating effect of random and worst case (+/-) tolerances applied to heights and positions of teeth of waffle-iron structure.
4. Testing for spurious modes [ 9 ].

Fixing Spurious Problems

It is theoretically impossible to fix spurious spikes and responses (Figure 6) in stop-band of waffle-iron filter without significant modifications of basic structure of the filter. However, the spurious can be reduced or even eliminated by changing the system outside the filter and reducing presence of the spurious modes. For example, removing “asymmetric” components (H-plane bends, twists, etc.) and making the system symmetric eliminates excitation of odd waveguide modes causing the spurious spikes. However, the spikes of pass-band (Figure 5) might be reduced by using more accurate manufacturing methods. Sometimes, the spurious caused by quasi-TE20-mode can be fixed by two offset tuning screws. Although the fixing methods listed above do not seem to be much useful - there is no a good way to fix a bad thing, except creating a good thing.

Possible Design Replacement

A bi-corrugated [10,12] or inhomogeneous stepped-Impedance corrugated filter [11] can be a better design solution because of not having the in-band spurious problems and being more predictable for the out-of-band spurs. The design process is also straight forward and simple, if using right software (see WR-Connect page)

References

[1] S.B. Cohn, US Patent 3,046,503, July 1962

[2] C.G. Matthaei, L. Young, and E. M. T. Jones, “Microwave Filters, Impedance Matching Networks, and Coupling Structures”, New York, McGraw-Hill, 1964.

[3] 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

[4] Ralph Levy, “Aperiodic Tapered Corrugated Waveguide Filter”, US Patent 3,597,710, Nov. 28, 1969.

[5] E.D. Sharp “A High-Power Wide-Band Waffle-Iron Filter”, IEEE, Trans. Microwave Theory and Tech., March, 1963, pp.  [2] R. Levy “Tapered Corrugated Waveguide Low-Pass Filter”, IEEE Trans. Microwave Theory Tech., MTT-21, August 1973, pp. 526-532.

[6] R.E. Collin “Foundation for microwave engineering”, McGRAW-Hill, 1966

[7] R. A. Goulouev “Corrugated Low-Pass Filter Design Workshop”, Online Resources, www.goulouev.com, 2002.

[8] R. A. Goulouev “Waffle-iron Filter. Analysis of Spurious.”, Online Resources, www.goulouev.com, 2004.

[9] R. A. Goulouev “Harmonic Rejection Test Procedure”, Online Resources, www.goulouev.com, 2001

[10] W. Hauth, R. Keller, U. Rosenberg, "The Corrugated-Waveguide Band-Pass Filter - A New Type of Waveguide Filter", 18th EUMC Proc., Stockholm, 1988, pp. 945-949.

[11] R. Levy, "Inhomogeneous Stepped-Impedance Corrugated Waveguide Low-Pass Filters", Microwave Symposium Digest, 2005 IEEE MTT-S International Volume, Issue , 12-17 June 2005 Page(s): 4 pp.

[12] R. A. Goulouev “Broad Wall Stepped Corrugated Filters. Typical Performance.”, Online Resources, www.goulouev.com, 2006.