# Circuit Simulation of Dual-mode Waveguide Cavity Filters

Dual-mode circular resonator (DMCR)-based waveguide filters, introduced in the beginning of the 1970s,^{1-6} have well known advantages, compared to traditional waveguide filters. DMCR-based filters allow for two orthogonal TE_{111} modes to be employed by each circular cavity, thereby reducing by a factor of two the number of actual resonant cavities while maintaining a necessary number of filter sections. This results in a significant filter size reduction. The Q-factor of circular resonators employing TE_{111} modes is approximately twice that of the rectangular resonators employing TE_{101} modes,^{7 }which has a direct effect on the filter insertion loss. Finally, the major advantage of the DMCR-based waveguide filters is that such a structure allows for the cross-coupling between electrically non-adjacent elements.

The subject of this article is a transmission line (TL) circuit model representation of the dual-mode resonator-based waveguide filters in a variety of configurations. The TL circuit models presented allow for the evaluation of the expected filter parameters, based on an accurate solution, which includes a passband amplitude response, skirt selectivity, filter asymmetry, true real frequency transmission zeroes (reject notches) location, group delay flatness, phase slope, etc. The cross-coupling nature (capacitive or inductive) and filter configuration/topology related phasing issues are also reflected in the modeling, which makes it a useful tool during the design path. A separate analysis of the filter constituent elements provides the design engineer with coupling coefficients and loaded input/output Q-factors, the information necessary for the actual filter development. The experimental filter responses compared against the circuit simulation are also presented.

#### General Considerations and Basic Filter Configurations

**Figure 1** Four-element dual-mode filter.

* Figure 1* illustrates a four-element dual-mode circular resonator-based waveguide elliptic filter structure and the mode formation in the resonators as described by Williams.

^{1}The incident wave applied to the input magnetic slot excites a vertical TE

_{111v}mode, whose frequency is controlled by the tuning screw #1. The coup-ling screw #2, located at +45° to the vertical plane, produces an angular E-field component and, consequently, contributes to the orthogonal horizontal TE

_{111h}resonant mode controlled by tuning screw #3. The horizontal TE

_{111h}mode (second filter resonance element) of the first circular resonator is coupled to the third resonance element, which belongs to the second circular resonator via the vertical slot as shown, and the further mode formation in the second cavity occurs. The cross-coupling between vertical TE

_{111v}modes of the first and second cavities (or between first and forth filter elements of the filter) is implemented via the horizontal magnetic slot, which is much smaller than a vertical slot of the main filter path.

**Figure 2** Four-element dual-mode filter circuit.

The phase and response analysis of the dual-path signal propagation in the filter structure is illustrated by the circuit represented in * Figure 2*. Shown here are four half wavelength resonators each formed by two quarter-wave pieces of physical transmission lines (slightly shorter for tuning purposes) and coupled to each other and to the input/output transmission lines via inductive or capacitive elements. The frequency tuning of each filter element/resonance mode is provided by capacitors (C

_{1}, C

_{2}, C

_{3}and C

_{4}) as shown. The first and second filter elements TE

_{111}vertical and horizontal modes of the first resonator are coupled through the capacitive (electric coupling) tuning screw #2 (C

_{12}of the circuit) as shown in Figures 1 and 2, so they maintain a 270° electrical phase shift at the center frequency.

^{8,9}The second and third filter elements, TE

_{111}horizontal modes of the first and second resonators, are coupled through the vertical magnetic slot of the main path in the partition separating two cavities and represented by L

_{23}inductance, so the phase between them is 90°. Coupling between horizontal and vertical modes of the second resonator is provided by the capacitive coupling screw #5 located at -45° or +135° angle to the vertical plane. The angular location of this coupling screw (C

_{34}) assures the out-of-phase excitation of the vertical TE

_{111v}mode

^{1}and can be introduced by a 180° electrical phase inverter in the circuit.

**Table 1** Phase Shift for two Paths

The resulting phase shifts for the main and cross-coupled paths are presented in * Table 1*. Incorporation of an out-of phase cross-coupling in reality delivers an electrically in-phase “skip two” (quadruple) cross-coupling that results in two real frequency transmission zeroes (jω-axis) on the filter skirts,

^{8}as shown in

*. Analogously, removal of the 180° phase inverter is equivalent to a +45° location of the coupling screw #5, which leads to the out-of-phase condition for both paths at the center frequency and consequently,*

**Figure 3**^{8}a flattened group delay and sluggish skirt selectivity responses.

**Figure 3** Four-element dual-mode filter circuit amplitude response.

For an accurate representation of the waveguide resonators, pieces of physical transmission lines used for simulation were modeled with characteristic impedances, electrical lengths and insertion losses (optional) dependent on the cut-off frequencies and Q-factors of the actual resonators. Shown below is a typical Genesis simulation software equation block containing all necessary variables:

f=FREQ/1000

l1= 14.2

l2=17.2

F0=13.7425

Fcoin=7.87

Fcoout=7.87

fco1=9.261

fco2=10.943

Zw1=377/SQR(1-(fco1/f)^2)

Zw2=377/SQR(1-(fco2/f)^2)

Zin = 377/SQR(1-(fcoin/f)^2)

Zout=377/SQR(1-(fcoout/f)^2)

WGWL1=300/SQR(f^2-fco1^2)

Tet1=0.5*l1*360/WGWL1

WGWL2=300/SQR(f^2-fco2^2)

Tet2=0.5*l2*360/WGWL2

where

- FREQ=Genesys frequency definition in MHz.
- l1, l2=an actual physical length of the first and second resonators (mm).
- fcoin, fcoout=cut-off frequency of the input/output waveguides (WR-75).
- fco1, fco2=cut-off frequencies of both resonators, directly related to the diameters of the actual resonators. The varying resonator diameters are used for a practical consideration in order to suppress an undesired high-order resonant mode, especially TE
_{010}of the circular resonator whose frequency is determined by the cavity diameter only. - Zw1, Zw2=characteristic impedances of the circular waveguides the resonators are based on.
- Zin, Zout=characteristic impedances of the input / output waveguides.
- WGWL1, WGWL2=waveguide wavelengths.
- Tet1, Tet2=cut-off frequency dependent electrical lengths of the resonators.

**Figure 4** Four-element dual-mode filter prototype response.

The input/output port impedances for this particular example are assigned as a WR-75 waveguide impedance at the center frequency 13.74 GHz. * Figure 4* shows an actual prototype response for the four-element dual-mode filter circuit.

**Figure 5** Five-element dual-mode filter circuit.

**Figure 6** Five-element dual-mode filter prototype.

It should be mentioned that the capacitive coupling element in cascade with the 180° phase inverter representing the out-of-phase excitation of the vertical TE_{111v} mode in the circuit can be replaced with an inductive element with a minimal coupled resonator length (frequency) adjustment. An example of this statement, shown in * Figure 5*, is a five-element dual-mode filter circuit resulting from the addition of a single resonance element to the four-element filter similar to one discussed above. The actual prototype is shown in

*. Here the capacitive coupling screw located at +135° (–45°) and delivering the out-of-phase excitation of the vertical TE*

**Figure 6**_{111v}mode is replaced by an inductive element (L

_{34}). Both theoretical and experimental responses (location of the transmission zeros, asymmetry and rejection overshoot) shown in

*and*

**Figures 7***demonstrate a great deal of similarity.*

**8****Figure 7** Five-element dual-mode filter circuit response.

**Figure 8** Five-element dual-mode filter prototype response.

#### Six-element Dual-mode Circular Waveguide Filter

**Figure 9** Six-element dual-mode filter.

* Figure 9* illustrates a six-element DMCR-based waveguide elliptic filter structure and the mode formation inside the resonators. The practical importance of this filter topology, mentioned in a number of papers,

^{8,10,11}is based on the fact that a single negative (out-of-phase) cross-coupling delivers an improved skirt selectivity along with a flattened group delay response.

**Figure 10** Six-element dual-mode filter circuit.

**Table 2** Six-Element Filter Phase Shifts

The vertically polarized signal of the main path (black arrows) propagates through the resonators (exciting a TE_{111v} mode) and the horizontal coupling irises in the direction indicated by black dashed arrows. Coupled by the angular coupling screw (+45°) of the last (3rd) resonance cavity into the horizontal mode TE_{111h}, the main path signal travels back (red arrows) through the same cavities and vertical coupling slots. The out-of-phase cross-coupled signal delivered by the angular coupling screw (–45°) of the 1st resonance cavity is superimposed with the main path TE_{111h} mode. The transmission line circuit for this configuration is presented in * Figure 10*. Applying the same argument as before and assuming that the center frequency phase shifts between resonators are related to the coupling means (inductive L

_{16}, L

_{12}to L

_{56}, –90° and capacitive C

_{34}, –270°) between resonance elements, the overall phase shift can be found (see

*).*

**Table 2****Figure 11** Six-element dual-mode filter circuit amplitude and GD responses.

Two signal paths, main and cross-coupled, maintain out-of-phase conditions provided by the angular coupling screw at the center frequency, which produces a pair of real axis (equalization) and real frequency (jw-axis) transmission zeroes. Due to a significant phase slope difference between the two paths, the in-phase signal summation occurs near the passband edges, but does not have any noticeable effect on the overall response.^{8} Typical amplitude and group delay responses of this filter topology are shown in * Figure 11*.

#### Eight-element Dual-mode Circular Waveguide Filter

**Figure 12** Eight-element dual-mode filter.

Shown in * Figure 12* are the configuration and the signal propagation for a four-cavity/eight-element DMCR-based filter with an elliptic function response.

^{12}Such a topology allows three cross-couplings between non-adjacent resonators that place three pairs of real frequency transmission zeroes on both sides of the passband, thus significantly improving skirt selectivity and out-of-band rejection. The cross-coupling topology

^{12}is shown in

*, the transmission line-based circuit in*

**Figure 13***and its simulated response in*

**Figure 14***.*

**Figure 15****Figure 13** Eight-element dual-mode circular waveguide filter cross-coupling.

**Figure 14** Eight-element dual-mode filter circuit configuration for elliptic function response.

**Figure 15** Eight-element dual-mode filter response.

With such a complicated nested topology (one loop inside another), when a specific transmission zero is not trivially conditioned by a particular cross-coupling, it is very important to verify the filter response of the circuit model as it relates to the cross-coupled loop phase shifts. The analysis of the transmission line circuit was performed based on the phase shift between resonance elements and conditioned by inductive or capacitive couplings. This analysis proved that all three cross-coupling loops deliver in-phase dual-path signal propagation which, in its turn, results in the desired response. It should be mentioned that formal selection of coupling elements for the transmission line circuit (Lij, Cij or Cij + 180° inverter) makes no difference as long as the loop phase requirement is met. For example, in order to provide in-phase cross-coupling between elements 1 and 8, the capacitive elements C_{18} and C_{12} of the circuit can be replaced with inductances, which after some tuning and optimization would result in the same response. As demonstrated in the section below, the required cross-coupling phasing may be determined at the initial design stage based on the simplified L-C circuit. However, it is a good practice to set the circuit coupling (inductive or capacitive) between adjacent elements (main path) according to their waveguide implementation, that is inductance for the slots and capacitance for the screws.

The actual dual-mode filter design is based on the transmission line circuit representation and topology, with cross-coupling loops maintaining a required phase shift. It is assumed that angular coupling screws located at +45° produce a -270° electrical phase shift at the center frequency and screws located at -45° (135°) produce -90° (-270°+180° phase inversion =–90°). The horizontally polarized input signal (indicated by red arrows) is launched to the first resonator where it excites the TE_{111h} resonance mode.

**Table 3** Eight-Element Filter Phase Shifts

The -45° angular coupling screw (C12) transforms the horizontal into TE_{111v} vertical mode of the same circular cavity that propagates through the resonators and horizontal coupling irises (L_{23}, L_{34} and L_{45}) in the direction indicated by black dashed arrows. The angular coupling screw located at +45° (C56) of the last cavity transforms the TE_{111v} into the TE_{111h} horizontal mode, which propagates back through the vertical coupling slots (L_{67}, and L_{78}) to the output as indicated by the red dashed arrows. That is how the main path of the signal is formed. Two coupling screws, -45° in cavity #3 and +45° in cavity #2, along with a small size vertical slot in the partition separating first and second cavities, deliver a required in-phase cross-coupling between 4 to 7, 3 to 8 and 1 to 8 elements of the filter, respectively. The phase shifts for all three loops are presented in * Table 3*.

Graphically, the 180° cross-coupling phase inversion provided by the angular -45° (135°) coupling screws can be seen as a main path electrical vectors superimposed with an oppositely directed (electrically 180° inversion may deliver both in and out-of-phase dual-path signal propagation) cross-coupled vector.

#### Design Steps

**Preliminary Circuit Analysis**

A simplified circuit analysis should be performed at the initial stage of the design in order to generate a response similar to the required one and verify phase shifting between cross-coupled and main paths. Shown in * Figure 16* is the eight-element L-C filter circuit with three nested cross-coupling loops, which produces a similar (but not accurate) response to that shown previously. This configuration is a starting point in the design of the eight-element dual-mode circular waveguide filter discussed above. Such a circuit can be easily generated using, for example, Genesis filter synthesis and optimization tools.

^{8}Simplicity of the implementation and the analysis of the circuit presented allows one to determine crucial factors for the end filter design, such as the total number of elements, number and type of the cross-couplings (positive for in-phase, negative for out-of-phase, etc.), as well as preliminary coupling coefficients between elements. It should be noted that the dual-mode waveguide filter topology assumes only an even number of elements skipped for structural reasons.

^{13}

**Transmission Line Circuit Synthesis**

The transmission line circuit synthesis and simulation is tied to the circular waveguide pieces selected for filter resonators. Diameters of the actual waveguides are introduced to the circuit by the cut-off frequencies and characteristic impedances of the transmission lines, and set in the equation block as illustrated in the general considerations and basic filter configurations covered previously. The Q-factors of the circular resonators utilized in the design are translated into the transmission line loss in dB. In the beginning of the simulation process, inductive and capacitive coupling elements of both main and cross-coupled paths as well as input/output Q-factors can be set approximately, based on the J-invertors or coupling coefficients^{7} obtained from the L-C circuit as follows:

where

L_{ij} and C_{ij} = inductive or capacitive couplings,

are the slope parameter of the L-C shunt resonators,

k_{ij} = a coupling coefficient between L-C shunt resonators,

G_{0} = 1/Z_{0} is an input transmission line (port) admittance and

Q_{e1} = input (or output) external Q-factor.

The process of the TL circuit synthesis may take a few steps, including:

**Figure 17** Example of thee input/output element external Q-factor (red) and two adjacent element coupling coefficient verification.

a) Adjustment of the each main path adjacent pair of TL resonator coupling responses. This adjustment is performed in order to meet k_{ij} value of the preliminary L-C circuit. A typical coupling response is shown in * Figure 17*.

b) Adjustment of the input/output element S_{11} responses of the input/output inductive element for the determination of the external Q-factor.

c) Adjustment of the cross-coupling path pairs of TL elements coupling responses, similar to step a.

d) Full circuit response simulation and optimization.

**Practical Aspects of the TL Circuit Analysis**

**Figure 18** Example of the circuit fragments adjusted for input external Q (a) and element coupling coefficient (b and c) verification.

After the desired response of the transmission line-based circuit is achieved (return loss, bandwidth, rejection, etc.), actual couplings and external Qs for further 3D or filter prototype development can be de-embedded. Shown in * Figure 18* are resonance elements extracted from the eight-element dual-mode filter circuit for coupling and external Q determination. Since the element resonance frequencies extracted from the circuit will slightly shift, they need to be re-adjusted to the center frequency of the filter using tuning capacitors. In case of paired element coupling value verification, the input shunt inductance should be minimized for better accuracy. All unterminated ends of the transmission lines should be shortened to the ground as shown. The typical S

_{11}responses for a singly loaded resonator having a finite unloaded Q-factor and paired transmission line resonators are demonstrated in the figure. The methodology described by Matthaei, et al.

^{7}for a singly loaded resonator can be utilized for the input/output external Q-factor definition. The coupling value between two resonance elements is found as followed:

k_{ij} = (f_{j} - f_{i}) / f_{0} (2)

The coupling values found are a key factor for the actual prototype development. The non-tunable coupling slots between either pair of actual coupled resonators of the filter should be adjusted prior to the full assembly so as to produce responses approaching the de-embedded ones from the circuit. When all actual coupling and Q values are obtained experimentally from pre-assembled parts and all slot dimensions are finalized, the fully assembled unit prototype may only need a minor tuning provided by capacitive screws.

#### References

1. A.E. Williams, “A Four-cavity Elliptic Waveguide Filter,” *IEEE Transactions on Microwave Theory and Techniques,* Vol. 18, No. 12, December 1970, pp. 1109-1114.

2. A.E. Atia and A.E. Williams, “New Types of Waveguide Bandpass Filters for Satellite Transponders,” *COMSAT Technical Review*, Vol. 1, No. 1, Fall 1971.

3. A.E. Atia and A.E. Williams, “Narrow Bandpass Waveguide Filters,” *IEEE Transactions on Microwave Theory and Techniques*, Vol. 20, No. 4, April 1972, pp. 258-265.

4. A.E. Atia and A.E. Williams, “Nonminimum-phase Optimum-amplitude Bandpass Waveguide Filters,” *IEEE Transactions on Microwave Theory and Techniques*, Vol. 22, No. 4, April 1974, pp. 425-431.

5. R.D. Wanselow, “Prototype Characteristics for a Class of Dual-mode Filters” *IEEE Transactions on Microwave Theory and Techniques*, Vol. 23, No. 8, August 1975, pp. 708-711.

6. A.E. Williams and A.E. Atia, “Dual-mode Canonical Waveguide Filters,” *IEEE Transactions on Microwave Theory and Techniques*, Vol. 25, No. 12, December 1977, pp. 1021-1026.

7. G. Matthaei, L. Young and E.M.T. Jones, *Microwave Filters, Impedance-matching Networks and Coupling Structures*, Artech House Inc. Norwood, MA, 1985.

8. A.D. Lapidus and C. Rossiter, “Cross-coupling in Microwave Bandpass Filters,” *Microwave Journal*, Vol. 47, No. 11, November 2004, pp. 22-46.

9. J.B. Thomas, “Cross-coupling in Coaxial Cavity Filters: A Tutorial Overview,” *IEEE Transactions on Microwave Theory and Techniques*, Vol. 51, No. 4, April 2003, pp. 1368-1376.

10. R. Levy, “Filters with Single Transmission Zeros at Real or Imaginary Frequencies,” *IEEE Transactions on Microwave Theory and Techniques*, Vol. 24, No. 4, April 1973, pp. 172-181.

11. K.T. Jokela, “Narrow-band Stripline or Microstrip Filters with Transmission Zeroes at Real and Imaginary Frequencies,” *IEEE Transactions on Microwave Theory and Techniques*, Vol. 28, No. 6, June 1980, pp. 542-547.

12. J. Uher, J. Bornemann and U. Rosenberg, *Waveguide Components for Antenna Feed Systems: Theory and CAD*, Artech House Inc, Norwood, MA, 1993.

13. S.T. Kahng, M.S. Uhm and S.P. Lee “A Dual-mode Narrow-band Channel Filter and Group-delay Equalizer for a Ka-band Satellite Transponder,” *ETRI Journal*, Vol. 25, No. 5, October 2003.

**Alex D. Lapidus** *received his BS and MS degrees in electrical engineering from the University of Technology, Nizhny Novgorod, Russia. He is currently involved in the design and development of various filtering systems and passive components with L-3 Communications, Narda Microwave-West, where he holds the position of engineering manager.*