Cavity element model

Here, a rotationally symmetric connection of three coaxial lines is considered, in which the cross-sectional areas of the end lines are within the cross-section area of the middle coaxial line section. This type of scattering element is called “cavity” here, although it can be dielectric filled as well. The problem of modal scattering on a generalized cavity connection of three uniaxial waveguides with arbitrary cross sections is solved in [Reference Fabrizio De Paolis, Goulouev, Zheng and Yu5] in rigorous terms of multimoded non-normalized admittance matrix (Y-matrix). Since the cavity connection is rotationally symmetric, no coupling or conversion between modes of different azimuthal indexing (n-index in common indexation TE_nm and TM_nm defined for coaxial waveguides [Reference Marcuvitz6] including TEM-mode associated with n = 0) occurs. Therefore, the entire multimodal Y-matrix can be divided into independent submatrices corresponding to the scattering of modal groups of modes having different n-indexes. Within a modal group, the Y-submatrix can be further normalized and truncated to a 2 × 2 matrix corresponding to the scattering of the dominant (first order) mode accessible from both sides (node 1 and node 2). In this case, for each modal group with index n, the Y-matrix can be expressed in simplified terms as follows:

(1)$$Y = i\left[{\matrix{ {-\displaystyle{1 \over {\upsilon_1}}\mathop \sum \limits_m y_m\displaystyle{{{( {\alpha_m^{( 1 ) } } ) }^2} \over {\tan ( {\beta_mL} ) }}} & {\displaystyle{1 \over {\sqrt {\upsilon_1\upsilon_2} }}\mathop \sum \limits_m y_m\displaystyle{{\alpha_m^{( 1 ) } \alpha_m^{( 2 ) } } \over {\sin ( {\beta_mL} ) }}} \cr {\displaystyle{1 \over {\sqrt {\upsilon_2\upsilon_1} }}\mathop \sum \limits_m y_m\displaystyle{{\alpha_m^{( 2 ) } \alpha_m^{( 1 ) } } \over {\sin ( {\beta_mL} ) }}} & {-\displaystyle{1 \over {\upsilon_2}}\mathop \sum \limits_m y_m\displaystyle{{{( {\alpha_m^{( 2 ) } } ) }^2} \over {\tan ( {\beta_mL} ) }}} \cr } } \right], \;$$

where the α-values are the aperture integrals on both accessible ports (node 1 and node 2) and defined as

(2)$$\alpha _m^{( {1, 2} ) } = \mathop \smallint \limits_{r_{in}^{( {1, 2} ) } }^{r_{ex}^{( {1, 2} ) } } \mathop \smallint \limits_0^{2\pi } \overrightarrow {e^{( {1, 2} ) }( {r, \;\varphi } ) } \cdot \overrightarrow {E_m( {r, \;\varphi } ) } \cdot r\cdot d\varphi \cdot dr$$

and y _m, β _m, and $\overrightarrow {E_m( {r, \;\varphi } ) }$ are the wave admittance, propagation constant, and normalized transverse electric field vector function m-th mode in the middle waveguide section (cavity). $\overrightarrow {e^{( {1, 2} ) }( {r, \;\varphi } ) }$ and υ_1,2 are the transverse normalized electric field vector functions and wave admittances of the dominant waveguide modes in node 1 and node 2, respectively (see Fig. 1). Similar rectangular “corrugation” cavity models in [Reference Fabrizio De Paolis, Goulouev, Zheng and Yu5, Reference Hauth, Keller and Rosenberg12], where the sums are truncated with a limited number of localized modes (<200) are found very fast in simulations and practically accurate and showing a good correlation with full-wave simulation tools and experimental data. This model also has great advantages over the models of traditional filter elements (waveguide or coaxial impedance steps, irises, and stubs) used in distributed networks circuits [Reference Levy3], as it considers the internal resonances caused by high-order waveguide modes.

Fig. 1.

Coaxial cavity element cross section (left) and 3D appearance (right).

Reflection and transmission zeros

Based on the s-parameters analysis of (A.1) corresponding to 0-index modal group in (1), a symmetric coaxial cavity shows reflection zero at DC and capacitive response in the vicinity of DC when the transmission coefficient magnitude is reducing while the frequency is increasing (see Figs 2 and 3) until the first standing wave resonances corresponding to TEM and TM_0m excite. In a common coaxial filter concept [Reference Matthaei, Young and Jones1–Reference Levy3], those resonances are generally not utilized, create unwanted spurious responses limiting the stopband bandwidth and quality. Based on this EM model (A.1), it is analytically confirmed (see the Appendix), that the first two resonant modes TM₀₁₀ and TEM₁ can be coupled and utilized to achieve two transmission zeros in a remote frequency zone. Those transmission zeros can be utilized to gain the attenuation in three ways such as closely coupled (Fig. 2a), regenerated (Fig. 2b), and slightly attenuated (Fig. 2c).

Fig. 2.

Reflection and transmission magnitudes (dB) versus frequency (GHz) characteristics for the first-order cavity elements having two coupled transmission zeros (a), regenerated transmission zeros (b), and no transmission zeros (c).

Fig. 3.

Reflection and transmission magnitudes (dB) versus frequency (GHz) characteristics for the second-order cavity elements having two coupled transmission zeros (a), regenerated transmission zeros (b), and no transmission zeros (c).

These conditions occur if TM₀₁-mode is excited on TEM-mode scattering on the end junctions of the cavity. The TM₀₁ mode excitation can be eliminated if the corresponding terms in Y-matrix sums (A.1–A.3) turn into zero (see the Appendix). In this case, similar transmission responses with coupled transmission zeros (see Fig. 3) can be achieved by utilization of TM₀₂₀ and TEM₁ resonances instead. Because these resonances are higher on the frequency scale, these types of cavity elements can be used in LPFs where a higher cut-off is required.

Periodic LPF configuration

A periodic structure consisting of two ports, as known [Reference Collin7], can be represented as a transmission line with a corresponding propagation constant and impedance. Based on formulation in the Appendix, in case of N symmetric ( y ₁₁ = y ₂₂) two ports with connection nodes with length d, the appropriate expressions for transmission coefficient and effective impedance can be derived as follows:

(3)$$T_N = {\rm exp}( {-j\cdot N\cdot \theta } ) , \;$$

(4)$${\rm cos}( \theta ) = \displaystyle{{\,j\cdot ( {y_{12}^2 -y_{11}^2 -1} ) \cdot {\rm sin}( \varphi ) -2y_{11}\cdot {\rm cos}( \varphi ) } \over {2y_{12}}}, \;$$

(5)$$\eqalign{z_p = & z_n \cr & \cdot \displaystyle{{\,j\cdot ( {y_{11}{\rm sin}( \varphi ) -y_{12}{\rm sin}( \theta ) } ) + 1-( {y_{11}^2 -y_{12}^2 + 1} ) {\rm si}{\rm n}^2( {\varphi /2} ) } \over {\,j\cdot ( {y_{11}{\rm sin}( \varphi ) -y_{12}{\rm sin}( \theta ) } ) -1 + ( {y_{11}^2 -y_{12}^2 + 1} ) {\rm co}{\rm s}^2( {\varphi /2} ) }}, \;}$$

where

$$\varphi = \sqrt {\varepsilon _r^{\left( {1,2} \right)} \mu _r^{\left( {1,2} \right)} } \cdot k_0\cdot d\;{\rm and}\;z_n = \displaystyle{{Z_0} \over {2\pi }}\sqrt {\displaystyle{{\mu _r} \over {\varepsilon _r}}\cdot } {\rm ln}\left( {\displaystyle{{r_{ex}} \over {r_{in}}}} \right).$$

A simple analysis of expression (3) shows that a cavity transmission zero (when y ₁₂ = y ₂₁ = 0) remains in the periodic structure composed of such cavities except for an uncertainty case when the numerator in (4) turns into zero at the same frequency point. The actual stopband, where θ is imaginary, is expected to be significantly wider and roughly cover the entire cavity 3 dB cutoff bandwidth, for example from f ₀ to f ₁ as shown in Fig. 5a. Nevertheless, some spurious responses may be present in frequency spots where θ becomes real. Using electrically short nodes (φ ≪ 1) allows us to push those spurious responses up out or to the vicinity of cavity transmission zeros where they become very narrow or vanish. Based on the above considerations, a simple filter design procedure can be applied.

The filter design process begins from designing a single-cavity element having a 3-dB stopband bandwidth roughly covering the design stopband. Next, the length of cavity extension nodes d is determined to match the period impedance (5) with the external interface impedance (commonly 50 Ω) at a middle frequency point f _c of the design passband. Finally, the cascade is connected to the I/O coaxial lines directly or indirectly using additional matching elements. The resulted filter is expected to demonstrate a broadband and high attenuation over the design stopband and reasonable return loss at the vicinity of f _c. Some additional fine adjustments or optimization, however, might be further required in order to achieve a “perfectionistic” near-band performance. Illustrational simulations are performed for cascades (N = 7) of the first (Fig. 2a) and the second (Fig. 3a) order cavities connected by short nodes with lengths d = 2 mm and d = 1.3 mm, respectively. Both cascades are directly connected with coaxial lines having same filling (polytetrafluoroethylene or Teflon (PTFE)) and dimensions matching exterior and corresponding to 50-Ω impedance (r _in = 3.5 mm, r _ex = 12 mm and r _in = 2 mm, r _ex = 6.437 mm, respectively). The simulation results are displayed in Figs 4 and 5.

Fig. 4.

Typical reflection (a) and transmission (a, b) magnitudes (dB) versus frequency (GHz) characteristics for the periodic cascade of seven cavities first order (Fig. 2a) corresponding to TEM mode (a) and first three TE modes (b).

Fig. 5.

Reflection (a) and transmission (a, b) magnitudes (dB) versus frequency (GHz) characteristics for the periodic cascade of seven second-order cavities (Fig. 3a) elements corresponding to TEM mode (a) and first three TE modes (b).

Higher order mode suppression

Most of coaxial low-frequency LPF applications are intended to operate within the frequency range when single TEM-mode propagates. Since the common LPF build utilizes radially narrow coaxial nodes (with low impedance), the TE_{n 1}-modal (n = 1, 2…) family can be considered as a prime spurious modal transit. A method of suppression of the TE_{n 1} modes used here is very similar to the one used for suppression of the TE_{n 0} modes in a corrugated waveguide filter [Reference Fabrizio De Paolis, Goulouev, Zheng and Yu5, Reference Levy8]. Then, under assumption of periodicity of cavity elements, a TE_{n 1}-mode propagation starts at the cut-off frequency corresponding to cavity connection nodes and then can be approximated as

(6)$$f_{cn}\approx \displaystyle{1 \over {\sqrt {\varepsilon \cdot \mu } }}\cdot \displaystyle{{n\cdot c} \over {2\pi \cdot r_{mid}}}, \;$$

where r _mid is a median radius defined as r _mid = (r _ex + r _in)/2. Furthermore, we can apply the frequency transformation function defined in [Reference Levy9] for corrugated waveguide filters as

(7)$$f_t( {n, \;f} ) = \sqrt {\,f^2 + f_{cn}^2 } .$$

If we have an original transmission response (TEM-mode) of a structure composed of such cavities as a function of frequency, for example T(f), the transmission response of a spurious TE_n1-mode, ideally, would be T _n(f) = T(f _t(n, f)).

Based on this assumption, we can define a discrete “bandwidth function” (see Fig. 6) corresponding to TEM (n = 0) or TE_{n 1} (n ≠ 0) transmission, which is equal to unity over the passband and zero everywhere else as follows:

(8)$$BW( {n, \;f} ) = \left\{{\matrix{ 1 & {{\rm if}} & {\,f_{cn} \le f \le \sqrt {\,f_0^2 + f_{cn}^2 \;} \;\mathop \cup \nolimits^ \;f \ge \sqrt {\,f_1^2 + f_{cn}^2 \;} } \cr 0 & {{\rm if}} & {{\rm else}} \cr } } \right..$$

Fig. 6.

Diagram representation for LPF transmission response corresponding TEM mode and a TE_{n 1} mode (a). Block diagram of a composite filter with extended stopband (b).

As an illustration, both periodic LPFs considered in the previous section are simulated for transmission responses (Figs 4b and 5b) corresponding to TE₁₁, TE₂₁, and TE₃₁ modes.

Quasi-periodic LPF configuration

A quasi-periodic and tapered arrangement of cavity elements is applied to the corrugated waveguide LPF [Reference Fabrizio De Paolis, Goulouev, Zheng and Yu5] in order to extend the stopband and eliminate stopband spurious responses (corresponding to the dominant mode) by scattering the cavity transmission zeros. The inhomogeneous structure profile, as is found in [Reference Levy8], also narrows the spurious responses corresponding the spurious modes. A similar approach is also applied here to the coaxial low-pass structures considered above. The main reason of tapering remains to scatter the cavity transmission zeros over targeted spots of design stopband and thus enwiden the stopband bandwidth (reducing f ₀ and increasing f ₁) for the dominant TEM mode and narrow the spurious passbands (8) corresponding to higher order modes of TE_{n 1} family in the same manner as done in [Reference Fabrizio De Paolis, Goulouev, Zheng and Yu5, Reference Collin7]. The matching method used here, however, differs, and still based on keeping the local characteristic impedance defined in (5) matching the interface value.

Composite LPF configuration

Let us consider a composite LPF [Reference Fabrizio De Paolis, Goulouev, Zheng and Yu5] composed of N sub-filters and connected with external cables (can be represented as “all-pass” sub-filters with f ₀ = ∞ and f ₁ = ∞). Then the resulting bandwidth function can be then expressed as:

(9)$$BW( {n, \;f} ) = \mathop \prod \limits_{i = 0}^{N + 1} BW_i( {n, \;f} ) , \;$$

where BW _i(n, f) corresponds to the i-th sub-filter in the cascade including I/O interface (i = 0, i = N + 1). It should be also note that the whole cascade keeps the rotational symmetry preventing cross modal coupling. The considered composite LPF configuration is, however, limited to two sub-filters, which are uniaxially cascaded and called “lower LPF” and “upper LPF” here as shown in Fig. 6b. The lower LPF is based on cavity elements of first type that utilize TEM₁–TM₀₁₀ resonances and covers the first frequency octave of TEM stopband (n = 0). The upper LPF is composed of the cavities of second type, which operate on TEM₁–TM₀₂₀ resonances and roughly covers the second stopband octave if same cavity radii are implemented. Regarding the TE_{n 1}-modes propagation bands (n ≠ 0), the prime spurious passbands (9) corresponding to both sub-filters are also expected to be about octave separated and therefore do not have intersect bands. In both cases, the spurious responses corresponding to TEM and TE_{n 1} responses can be further reduced or completely eliminated, if the filter channel is configured in a quasi-periodic tapered way as discussed above.

Multipactor breakdown

Two analysis methods are used for multipactor breakdown power handling evaluation. The first method is based on empirical parallel planes’ multipaction breakdown charts with fringing effects factor (see Fig. 7b, [Reference Mader, Dillenbourg, Labourdette, Lepeltier, Sinigaglia, Smits, Puech, Gizard, Lopez, Anderson, Lisak, Rakova and Semenov10]) added. In order to estimate the multipactor breakdown power handling for a single cavity, four possible breakdown directions are considered (see Fig. 7b).

Fig. 7.

Critical multipactor breakdown voltages in vacuum filled coaxial cavity (a). Iris fringing effect factor versus gap width per iris thickness ratio relation based on NASA data.

For each of the directions, the breakdown power level P _br is approximated as:

(10)$$P_{br}\approx \;\displaystyle{{{( {\sigma \cdot \gamma \cdot f\cdot g} ) }^2} \over {2\cdot Z_{eff}}}, \;$$

where f is the analysis frequency, g is the considered gap (g =  r _ex − r _in for the y-direction and g = L for the z-direction), σ is the iris fringing factor (σ = 1 for the z-direction, but for the r-direction it is defined from the appropriate gap to iris thickness ratio chart in [Reference Mader, Dillenbourg, Labourdette, Lepeltier, Sinigaglia, Smits, Puech, Gizard, Lopez, Anderson, Lisak, Rakova and Semenov10] or Fig. 7b, γ is multipactor breakdown factor (γ = 63 V/(GHz ⋅ mm) for the z-direction and γ = 46 V/(GHz ⋅ mm) for the y-direction where the gap is PTFE filled [Reference Vague, Melgarejo, Guglielmi, Boria, Anza, Vicente, Moreno, Taroncher, Martínez and Raboso13]). The gap effective impedance Z _eff is found from the EM simulation based on how the considered gap voltage V _g is related to the incident wave power P _in delivered to the filter, i.e.

(11)$$Z_{eff} = \displaystyle{{V_g^2 } \over {2\cdot P_{in}}}.$$

Then, the worst case multipactor breakdown power level is taken as minimal value computed over all cavities in the filter channel. The second method is based on Spark 3D software with a high-fidelity electron population evolution analysis. Nevertheless, it could not be accurately realized due to PTFE unknown secondary electron yield (SEY). Therefore, the analysis is performed with either aluminum (conservative) and silver (optimistic) SEY boundary conditions and interpreted as the conservative (lower bound) and optimistic (upper bound) estimations. According to multipactor discharge testing performed on similar structures, the measured breakdown power is close to the geometrical mean between both estimates.

Prototypes

Two coaxial LPF prototypes (UNITS 1 and 2) have been designed, fabricated, and tested. The UNIT 1 is designed as a cascade of two LPF units (see Fig. 6b) as discussed above. Both, lower and upper, sub-filters are designed using domestic Honeywell software based on implementation of conventional design concept and synthesis method [Reference Levy2, Reference Levy3, Reference Cameron, Kudsia and Mansour14] in a computer algorithm and further verified with ANSYS HFSS. The lower LPF is designed based on keeping constant the outer diameter with the inner diameter and dielectric spacers synthesized accordingly. The upper LPF is designed based on keeping the inner diameter constant with the outer diameter and dielectric spacers synthesized in the same manner. UNIT 2 is also designed as a cascade of two sub-filters (see Figs 6b and 8), each of which is based on quasi-periodic coaxial structure of appropriate cavity types utilizing coupled resonances (TEM₁–TM₀₁₀ with Δ ≪ 0 for lower LPF and TEM₁–TM₀₂₀ with Δ ≈ 0 for upper LPF) as discussed above in Section “Theory”. The spurious modes removal technique proposed in Section “Quasi-periodic LPF configuration,” however, had not been implemented by that time.

Key requirements

Both filters are designed to cover two channel OMUX and meet the same requirements and size envelope. The key requirements are to pass the OMUX channel carriers (151 + 158 W) within the 1.14–1.31 GHz frequency range with minimal insertion loss and reject the unwanted harmonics and interference from 2 to 32 GHz.

Fabrication, assembly, and tuning

Both units are integrated in a single assembly realized with two external aluminum body halves, internal wire structure and dielectric (PTFE) rings. Fabrication, assembly, and tuning of UNIT 1, however, faced some complications due to some tiny gaps (0.2 mm) between the internal wire structure and exterior body. As suggested, the undersized PTFE gaps correspond to extremely high impedance ratios defined by the corresponding distributed networks and synthesis method. In contrary, the proposed above design method (with exceptions noted above) leads to an alternative structure utilizing significantly larger PTFE gaps (>1.5 mm), which resolved fabrication and assembly problems with no tuning required (Figs 9 and 10).

Fig. 9.

3D EM model of UNIT 2 cut on symmetry plane and drawn in ANSYS HFSS.

Fig. 10.

UNIT 2 assembly photo image.

Key RF parameters’ test results

Both prototypes have passed the RF requirements for rejection with minor non-conformances. Nevertheless, when performing multipactor testing, UNIT 1 has failed, but UNIT 2 has passed with adequate margins. Therefore, UNIT 2 has been further selected as a baseline design concept. Here, in Figs 11 and 12, the appropriate measured and simulated s-parameter magnitudes are plotted over different frequency ranges of interest. As observed the LPF measured bandwidth (Fig. 11a) and roll-off (Fig. 12a) are slightly deviated from the originally simulated state due (as suggested) to fabrication factors (tolerances on dimensions and PTFE dielectric constant). It can be also noted that the measured transmission response is spiky in comparison with the simulation. The origin of the spike responses is due to the resonances occurring between setup components and device under test (DUT) caused by high-order modes during calibration and measurement. Therefore, such transmission response measurement cannot be accurate [Reference Morini, Farina and Guglielmi11] if realized without special multi-modal calibration kits. Such calibration kits, however, are not widely available, difficult, expensive, or impossible to practically realize. Nevertheless, it is still very common, when the measurement of transmission in overmoded waveguides is performed based on just though response calibration. Therefore, such measurement procedure cannot be accurate, but still can be often used as for verification and correlation purposes [Reference Fabrizio De Paolis, Goulouev, Zheng and Yu5]. Taking those factors into consideration, UNIT 2 test and simulation results well correlate with each other within the simulation, fabrication, and measurement errors.

Fig. 11.

Measured (black) and simulated (gray) reflection (a) and transmission (b) magnitudes (dB) versus frequency (GHz) plots over the near band.

Fig. 12.

Measured (black) and simulated (gray) transmission magnitudes (dB) over near (a) and far (b) frequencies (GHz) plots.

Multipactor test results

UNIT 2 has been high-power tested for multipactor breakdown. The tests were conducted with pressure <10⁻⁵ mbar, radioactive sources three ⁹⁰Sr, vacuum prior testing >14 h, baking temperature +75°C × 10 h, test temperature +22 and +55°C using output/reverse power nulling, harmonic detection, electron avalanche monitoring, and optical detection systems. In most cases, the discharge happened in OMUX channel filters and not detected in the output LPF. It should be noted that the multipactor breakdown power level was previously limited by the LPF when conventional design was used. According to test results, UNIT 2 has passed simple pulse (2% duty cycle) of 1254 MHz single carrier with power up to 1955 W with no discharge detected. This power level was limited to the setup capability. UNIT 2 was also twice with two simple pulsed carriers of 1146 and 1254 MHz with the corresponding maximum power levels of 676 and 892 W (3.12 kW peak), which caused a discharge after 3 min. The repeated test, however, did not show any discharge during 130 min. The high level of power handling of UNIT 2 is also confirmed by both analysis methods described in Section “Multipactor breakdown” when applied to the EM model at 1.30 MHz. In that case, the analytical formulation (10) resulted a 2.26 kW power handling capacity and the second method based on Spark 3D simulations resulted a 3.99 kW with the corresponding 1.92 kW lower (aluminum) and 8.31 kW upper (silver) bounds. Both, test, and simulation results, confirm a significant gain (3–8 dB) of multipactor breakdown power handling of UNIT 2 relatively to UNIT 1 under the same applied conditions. As already claimed, the higher power handling of UNIT 2 in comparison with UNIT 1 is due to larger vacuum filled gaps and volumes achieved between internal and external conductors in the coaxial LPF structure, while keeping similar passband and stopband quality when applying the new design concept and method proposed here.