Properties of Building Blocks Comprising Strongly Interacting Posts and Their Consideration in Advanced Coaxial Filter Designs

Mustafa Bakr; Smain Amari; Uwe Rosenberg

Properties of Building Blocks Comprising Strongly Interacting Posts and Their Consideration in Advanced Coaxial Filter Designs

T0 review · 0 major / 8 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper argues that strongly coupled coaxial post pairs and triples must be modeled by the eigen-resonances of the whole block — the ones satisfying the cavity boundary conditions — not by one resonance per post, and that doing so lets…

desk verdict Clear, practical explanation of why strongly coupled post blocks should be modeled with whole-structure eigenmodes; worth careful refereeing despite no new hardware. read the letter →

arxiv 2505.15729 v1 pith:M7VN6MEZ submitted 2025-05-21 physics.class-ph

Smain Amari , Mustafa Bakr , Uwe Rosenberg This is my paper · ORCID

classification physics.class-ph

keywords coaxialfiltercomb-linestronglycoupledpostsdual-postbuildingblocktriple-posttransversalcouplingmatrixsimilaritytransformationtransmissionzeroshifting

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Strongly coupled dual-post and triple-post coaxial resonators cannot be represented by assigning one resonance to each post: the fields of the posts share one volume, and localized 'post resonances' obtained by a 45-degree rotation of the true even and odd modes do not satisfy the boundary conditions. The paper's central claim is that the physically correct basis is the set of eigen-resonances of the complete block — the solutions of Maxwell's equations in that housing — and that the equivalent circuit should be a transversal doublet or triplet built from those modes. In that basis, standard coupling-matrix design methods work: the location of transmission zeros is controlled by simple formulas, and the zero-shifting property, moving a TZ across the passband by changing resonance signs or post heights, is preserved. A localized-resonator coupling matrix obtained by similarity transformations reproduces the overall frequency response by construction, but it obscures the internal physics and can predict wrong behavior, such as a TZ on the wrong side of the passband or a spurious dependence of the TZ on a distant mode. A 2nd-order filter, a triple-post doublet, a 3rd-order triplet, and a 4th-order box-section are presented as evidence that classical design plus a final dimension adjustment is enough.

What carries the argument

The load-bearing object is the set of eigen-resonances of the complete post-plus-housing structure, obtained from a full-wave eigenmode solution: for the transverse dual-post unit the even mode $\phi_e$ and odd mode $\phi_o$, and for the triple-post unit the three orthogonal modes of the footprint. These functions are eigenfunctions of the operator $\mathcal{L}$ with eigenvalues $\omega_e^2$ and $\omega_o^2$. The central identity is the 45-degree rotation $\phi_1=(\phi_e+\phi_o)/\sqrt{2}$, $\phi_2=(\phi_e-\phi_o)/\sqrt{2}$, which produces functions that are eigenfunctions only if the modes are degenerate; the size of the deviation is exactly the coupling coefficient $k=(\omega_e^2-\omega_o^2)/(\omega_e^2+\omega_o^2)$. Port coupling is described by the parameter $p=M_{s1}M_{1L}/(M_{s2}M_{2L})$ in the doublet and by the analogous constrained ratio in the triplet, and these ratios set the transmission-zero position through equations (8)-(10). Physically, the odd mode of the dual-post unit is excited through the evanescent $\mathrm{TE}_{20}$ waveguide mode while the $\mathrm{TE}_{10}$ mode provides an unavoidable bypass coupling; this is the mechanism by which moving the input and output ports relative to the symmetry plane changes $p$ and therefore moves the TZ from one side of the passband to the other. The resulting equivalent circuit is a transversal doublet or triplet, meaning each physical eigenmode connects directly to source and load with its own coupling coefficients rather than through a chain of localized resonators.

What would settle it

Take the triple-post configuration of Fig. 6 and, in a full-wave solver, move its spurious fundamental mode close to the passband (for example by increasing the spacing between the strongly coupled posts or changing their heights), then check whether the transmission-zero location still follows $\omega_z=(\omega_1+p\,\omega_2)/(1+p)$ independently of the spurious frequency. The paper predicts a growing deviation as the spurious mode approaches the band; if the truncated doublet model still predicts the full-wave TZ accurately with a nearby spurious mode, the central claim is wrong. A complementary check on the zero-shifting property: if adjusting only the post heights still moves the TZ to the other side of the passband while the spurious mode sits close, the paper's mechanism would be contradicted.

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Extended reading notes

Core claim

The paper establishes that, for building blocks of two or three closely spaced posts in a metallic enclosure, the only resonances that faithfully represent the structure are the eigen-resonances of the whole block that satisfy all boundary conditions; these are the even and odd modes of the dual-post unit (and the three orthogonal modes of the triple-post unit), which are uncoupled by orthogonality. A similarity transformation to a basis of localized 'resonances' associated with individual posts produces a coupling matrix that yields the same overall frequency response by construction, but the new functions are not eigenfunctions unless the original modes are degenerate. When the coupling is strong, that error is large, and the localized matrix misrepresents sections where resonators share the same volume: it predicts, for example, a transmission zero below the passband for three identical posts with predominantly magnetic coupling, contrary to full-wave simulation and experiment. The correct transversal equivalent circuit — each physical eigenmode coupled directly to the input and output — yields explicit design formulas, namely the doublet TZ location $\omega_z=(\omega_{od}+p\,\omega_{sp})/(1+p)$ with $p=M_{s1}M_{1L}/(M_{s2}M_{2L})$, and the triple-post TZ location $\omega_z=(\omega_1+p\,\omega_2)/(1+p)+\mathcal{O}(1/\omega_{sp})$, and it preserves the zero-shifting property by which changing the signs of the resonance frequencies moves the TZ across the passband. The paper concludes that filters containing these blocks can be designed by well-established methods as long as the equivalent circuit contains only the physical resonances that contribute to the passband; the far-away spurious resonance should be pushed away and its small effect compensated by final dimension adjustments, not treated as a controllable extra resonator.

Load-bearing premise

The argument assumes that the two (or three) selected eigen-resonances of the block are the only modes relevant in the frequency range of interest; if any other mode of the structure moves close to the passband, the truncated doublet or triplet equivalent circuit and the derived transmission-zero formulas lose validity.

Editorial extensions

If this is right

Dual-post and triple-post blocks can be inserted into higher-order filters and designed with conventional coupling-matrix synthesis, provided the matrix is written in the whole-block eigenmode basis; the paper demonstrates this on 2nd-, 3rd- and 4th-order examples.
For a transverse dual-post unit, putting the input and output ports on the same side of the symmetry plane places the transmission zero below the passband, opposite sides places it above, and moving the ports toward the symmetry plane brings the TZ closer to the band regardless of the spurious even mode's frequency.
An in-line dual-post unit has $p=-1$, which puts the TZ at infinity (an all-pole response), so it cannot create a finite TZ unless higher-order evanescent modes carry enough energy around the odd-mode resonance.
For a triple-post unit, the single TZ's location is set by the two in-band resonances and is insensitive to the far-away spurious fundamental mode, so that mode can be left out of the design model and its effect absorbed by final dimension adjustments.
The zero-shifting property — moving a TZ across the passband by changing the signs of the resonance frequencies, realized physically by adjusting post heights — is a real feature of these blocks and is captured by the transversal equivalent circuit, not by the similarity-transformed localized circuit.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the whole-block eigenmode principle is right, the same reasoning should apply to other multi-resonator assemblies whose fields share one volume, such as dielectric-loaded cavities or strongly coupled waveguide resonators: any localized-resonance model used in that regime should be checked against boundary-condition-satisfying modes before being trusted for design.
The paper's formulas connecting port offset to $p$ and $p$ to TZ location could be turned into a direct pre-design mapping from geometry to transmission-zero frequency, avoiding optimization loops whenever the spurious-mode separation assumption holds.
A testable criterion follows from the paper's argument: a coupling matrix is physically trustworthy for a strongly coupled section only if its entries can be varied independently by geometry; the paper's account predicts that localized dual-post and triple-post matrices will violate this independence because boundary conditions tie several elements together.
The zero-shifting property, realized by post-height tuning, points toward tunable transmission zeros in reconfigurable coaxial filters, since it changes only resonance frequencies and leaves the coupling topology intact; the paper does not itself address tunable devices.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The short version: this is a solid conceptual paper that should be published, but it is more of a clear explanation of how to think about strongly coupled post blocks than a genuinely new physics result. The main takeaway—that for strongly coupled dual-post and triple-post units the physically meaningful equivalent circuit uses the eigenmodes of the whole structure, not localized per-post resonances—is well argued and practically important for filter designers. The derivation of the coupling coefficient expression k = D/S from the eigenproblem is clean and more general than the usual textbook approach. The explicit TZ formulas (7)-(10) and the zero-shifting analysis are useful and not something I've seen put this concisely for these structures.

The paper does well on validation within its scope: full-wave simulations back the qualitative claims, and the design examples (2nd-order filter, 4th-order box-section, triplet) show that the classic synthesis method works when the correct basis is chosen. The authors are honest about the lack of new hardware, citing prior measurements from [5-9]. That's a real limitation but not a fatal one, since the point is explanatory.

Where are the soft spots? The central claim is conditional on the assumption that all non-selected resonances are far from the passband; the authors state this but don't really explore what happens when that assumption fails. The TZ formulas (8) and (10.a) are derived under that truncation, and the paper doesn't give quantitative bounds on the error. Also, the phrase "violation of boundary conditions" in Section II is imprecise—the rotated basis functions still satisfy the metallic boundary conditions; they just aren't single-frequency eigenmodes. That wording could confuse a careful reader even though the practical conclusions don't depend on it. A referee should ask the authors to tighten that language. Minor presentational issues: some figures are unnumbered or referenced as 'Fig. xx', and the coupling matrix notation in the 2-order example is hard to parse.

Overall, this is a useful paper for microwave filter engineers, especially those who design with strongly coupled resonator pairs/triplets. It deserves serious review and, with modest revision, publication. I'd send it to a competent referee and let the author address the truncation caveat and the wording.

Referee Report

0 major / 8 minor

Summary. This paper analyzes building blocks composed of strongly coupled coaxial posts, arguing that equivalent circuits based on individual post resonances are unreliable when the posts are strongly coupled because the local 'resonances' are not eigenmodes of the full structure. The authors derive a coupling coefficient k = D/S from the eigenproblem (Eq. 6) and provide formulas for transmission-zero locations in dual-post (Eqs. 7-9) and triple-post (Eq. 10) units. They then demonstrate a systematic design workflow using these physical resonances, with full-wave verified examples including a 2-order filter, a 4th-order box-section, and a triplet filter. The central claim is that using eigen-resonances of the complete structure allows standard filter design methods to succeed, whereas localized-resonance models obscure the physics and can fail to predict local behavior.

Significance. If the claims hold, this is a valuable conceptual contribution to filter design: it provides a first-principles justification for using transversal equivalent circuits in strongly coupled resonator configurations and demonstrates a practical design path that avoids overdetermined models. The derivation of k from the eigenproblem (Eq. 6) is elegant and general for linear two-state systems, and the design examples substantiate the qualitative claims. The paper also builds on prior experimental validation in [5]-[9], which is appropriate given its scope. The main limitation, acknowledged by the authors, is that the truncated equivalent circuit is valid only when non-selected resonances are sufficiently far from the passband; the paper states this assumption explicitly.

minor comments (8)

[Section II, paragraph 2] The phrase 'violation of the boundary conditions' is imprecise: the rotated functions φ1 and φ2 are linear combinations of eigenfunctions and therefore satisfy the same metallic boundary conditions; they are not, however, single-frequency solutions of the eigenproblem (1). Please rephrase this sentence and the related conclusion (b).
[Section VI.B, Fig. 13 caption] The caption refers to 'configuration in Fig. xx'; please insert the correct figure number.
[Section VII.B, Figs. 23 and 24 captions] Both captions refer to 'Fig. 14' but should reference the triple-post configuration in Fig. 22.
[Section VII.C, Fig. 27 caption] The caption refers to 'inset Fig. 18'; this should be 'inset Fig. 26'.
[Section VI, 2-order filter example] The normalized coupling matrix displayed after the specification of the 2-order filter example is typeset in a single line and is difficult to read; please present it as a standard 4x4 matrix.
[Fig. 18 caption] Please correct the typo 'Retrun loss' to 'Return loss'.
[Section VII.C] In the sentence 'The single posts are conductively couplet with the coaxial interfaces', 'couplet' should be 'coupled'.
[Section VI.B, last paragraph] The sentence 'There is no advantage to a more elaborate higher order equivalent circuit model' is too categorical; the following sentences show only that the authors found no advantage for this configuration, so consider softening it.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central derivations are self-contained, and the validation is external.

full rationale

The paper's main derivations do not reduce to their inputs. The coupling coefficient k = D/S is derived from the eigenproblem in Eqs. (3)-(6), not fitted to the target response. The transmission-zero formulas in Eqs. (7)-(10) follow from direct circuit analysis of the doublet and the transversal three-resonator model, with the stated large-spurious-resonance approximation made explicit. The paper's physical conclusions are tested against an external full-wave solver (µWaveWizard) and against measured results in the cited literature [5-9], which are not authored by the present authors. The self-citations [13,14] are used for known properties such as the TZ-shifting property of a doublet, but this property is also demonstrated by full-wave examples in Section VII, so the argument does not rest solely on self-citation. The statement that similarity transformations 'yield the correct frequency response (by construction)' is a mathematical fact about similarity transformations, not a circular definition of the paper's conclusions. The only notable assumption is that all non-selected resonances are far enough from the passband; the paper states this condition explicitly in Section II, and the conclusions are conditional on it. No step was found where a prediction is equivalent to a fitted parameter or where a load-bearing premise is imported only from the authors' prior work.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted to data in the derivations. The axioms are standard domain simplifications explicitly stated in the paper. No new physical entities are introduced; the transversal circuit is a modeling choice, not a new entity.

assumptions (5)

domain assumption The structure is lossless and homogeneous.
Stated in Section II: 'We assume that the structure is lossless and homogenous.' This simplifies the eigenproblem to real eigenfrequencies.
domain assumption Input and output couplings are weak enough not to significantly affect the eigenmode field distributions.
Stated in Section III.A: 'We assume that the coupling between the unit and the input and output is weak enough for the modal field distributions not to be significantly affected.' This justifies the fixed eigenmode basis.
standard math Orthogonality of Maxwell eigenmodes in a given volume means the two (or three) resonances of the block are not coupled to each other.
Used in Section II to assert that the even and odd modes of the dual-post structure are uncoupled, forming the basis of the doublet equivalent circuit.
domain assumption All other resonances (spurious modes) are far enough from the frequency range of interest that they can be neglected.
Stated in Section II: 'assuming that all other resonances are far enough from the frequency range of interest.' This truncation is load-bearing for the two- or three-resonance circuit models.
domain assumption The narrowband coupling-matrix representation is valid near the passband.
Stated in the Introduction (footnote 3): the transversal coupling matrix is a narrow-band limit, valid for the realizable characteristics close to the passband.

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Cite this review

Pith. "Pith review of Properties of Building Blocks Comprising Strongly Interacting Posts and Their Consideration in Advanced Coaxial Filter Designs." pith.science (2026). https://pith.science/paper/M7VN6MEZ

@misc{pith2026250515729,
  author       = {Pith},
  title        = {Pith review of: Properties of Building Blocks Comprising Strongly Interacting Posts and Their Consideration in Advanced Coaxial Filter Designs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M7VN6MEZ}},
  note         = {Machine review of arXiv:2505.15729}
}

read the original abstract

Building blocks containing strongly coupled posts offer new possibilities for advanced coaxial (comb-line) filter designs. Equivalent circuits based on the individual resonances of the posts cannot be used to reliably describe the behavior of these structures because of the strong coupling between the posts. Instead, sets of electromagnetic (EM) resonances that satisfy the boundary conditions are used. The resulting equivalent circuit is either a fully transversal circuit or contains locally transversal sub-circuits depending on the strength of the coupling between the cascaded blocks. The validity of similarity transformations that result in topologies with unusual strong coupling coefficients is questionable despite the fact that they yield the correct frequency response. Such coupling matrices obscure the physics of the problem and fail to predict the correct behavior of filtering structures. However, topologies that match the layout of the posts can be used to optimize the filter in connection with a full-wave solver or measurement. Examples of dual-post and triple-post units are used to illustrate the key findings. The basic knowledge of the real functionality of these special resonator configurations allows their consideration in advanced filter implementations by well-established classic design methods, without limitation by the design approach. This is demonstrated by an example of a 2-order in-line filter implementation providing one transmission zero by using the combination of single and transverse dual-post resonators. This fundamental understanding of the special properties provides the pre-requisite for a variety of novel filter solutions.

Figures

Figures reproduced from arXiv: 2505.15729 by the authors.

**Figure 1.** Basic building blocks and pre-requisites (coupling signs) for the realization of TZs. A MA2 M2B B MA1 M22 M11 M1B MA1 M1B M11 MAB A B A B M11 M22 MA1 M2B M12 MAB Note: A and B are resonating (R), non-resonating or interface nodes (NRN) Singlet ➔ A, B = NRN Triplet ➔ A, B = R Doublet ➔ A, B = NRN Box-section ➔ A, B = R Quadruplet ➔ A, B = NRN or R 1 TZ below passband: 1 TZ above passband: 1 TZ above or below passband… view at source ↗

**Figure 2.** Dimensions of dual-post configurations for analyses of basic behavior; left: inline; right: transverse (housing height 22.86mm, height of posts 22.00mm) 8.7 3.87 22.86 3.87 3.87 22.868.7 3.87 [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** EM field patterns of dual post fundamental mode (resonance [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (19 more)

**Figure 5.** Figure 5: Coupling scheme of dual-post with ∅1 and ∅2 in equations (2) used as basis [PITH_FULL_IMAGE:figures/full_fig_p003_5.png]

**Figure 6.** Figure 6: Dimensions of triangular post configurations for eigenmode [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]

**Figure 9.** Figure 9: Triple-post represented by localized ‘resonances’. M1 B2 M2 M3 [PITH_FULL_IMAGE:figures/full_fig_p006_9.png]

**Figure 10.** Figure 10: Alternative coupling scheme of triple-post unit (after [9]). . Ms1 M1L Ms2 M2L M23 M11 M22 M33 [PITH_FULL_IMAGE:figures/full_fig_p007_10.png]

**Figure 12.** Figure 12: Configuration for input coupling investigation [PITH_FULL_IMAGE:figures/full_fig_p007_12.png]

**Figure 14.** Figure 14: Dual-post singlet; analyzed response with coupling stub offset in same direction (cf. inset) [PITH_FULL_IMAGE:figures/full_fig_p008_14.png]

**Figure 15.** Figure 15: Dual-post singlet; analyzed response with coupling stub offset in opposite direction (cf. inset) -50 -25 0 S11 & S21 Parameter in dB 2.0 3.0 Frequency in GHz 2.2 2.4 2.6 2.8 -50 -25 0 S11 & S21 Parameter in dB 2.0 3.0 Frequency in GHz 2.2 2.4 2.6 2.8 [PITH_FULL_IMAGE…

**Figure 16.** Figure 16: Configuration for investigation of inter [PITH_FULL_IMAGE:figures/full_fig_p009_16.png]

**Figure 17.** Figure 17: Coupling values of inter-resonator couplings; dual-post configuration, basic mode red; 2nd mode green; magenta: single post configuration with same envelop configuration and center offset, post height (20.85mm) adjusted (fc=2.72GHz); blue: centered single posts in hal…

**Figure 18.** Figure 18: Dual-post doublet; analyzed response with opposite offset spacing of coupling stubs (cf. inset) Frequency in GHz Retrun loss, Selectivity in dB -80 -60 -40 -20 0 2 2.2 2.4 2.6 2.8 3 [PITH_FULL_IMAGE:figures/full_fig_p010_18.png]

**Figure 19.** Figure 19: Initial filter configuration after assessment of resonator, [PITH_FULL_IMAGE:figures/full_fig_p010_19.png]

**Figure 22.** Figure 22: Triple-post (doublet) filter configuration using coupling stubs at both sides [PITH_FULL_IMAGE:figures/full_fig_p011_22.png]

**Figure 23.** Figure 23: Response of configuration in Fig. 14 [PITH_FULL_IMAGE:figures/full_fig_p011_23.png]

**Figure 24.** Figure 24: Response of configuration in Fig. 14 [PITH_FULL_IMAGE:figures/full_fig_p011_24.png]

**Figure 25.** Figure 25: Narrow-band responses of initial doublet filter ( [PITH_FULL_IMAGE:figures/full_fig_p011_25.png]

**Figure 26.** Figure 26: ; they coincide accurately with the ideal response of the coupling matrix. As expected, the spurious fundamental mode resonance of the triple-post unit appears far below the passband resulting in an impairment of the ideal (‘box-section’) filter response with increasi…

**Figure 27.** Figure 27: 4-order filter design according inset [PITH_FULL_IMAGE:figures/full_fig_p012_27.png]

**Figure 30.** Figure 30: Analyzed wideband response of triplet filter design [PITH_FULL_IMAGE:figures/full_fig_p013_30.png]

**Figure 31.** Figure 31: Analyzed wide response of triplet filter design in Fig.27 but [PITH_FULL_IMAGE:figures/full_fig_p013_31.png]

Discussion (0). Continue with ORCID to comment.

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Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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