arxiv: 2604.18353 · v1 · submitted 2026-04-20 · 💻 cs.ET · cs.AR

Recognition: unknown

Scattering-Matrix-Based Parametric Characterization of a Two-Port Bridged-T Network for Microstrip Filter Applications

Naser Khatti Dizabadi , Douglas Jussaume

Authors on Pith no claims yet

Pith reviewed 2026-05-10 02:58 UTC · model grok-4.3

classification 💻 cs.ET cs.AR

keywords bridged-T networkscattering parametersmicrostrip filterhigh-pass filtertransmission matrixS11 polynomialparametric characterization

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The pith

The bridged-T network acts as a high-pass filter when its two inductors are set equal, because this cancels all even coefficients in the numerator of the S11 transfer function.

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

The paper derives the full set of scattering parameters for a two-port bridged-T network by first writing its transmission matrix and then converting to the scattering matrix. With the frequency normalized, the authors obtain closed-form expressions for the magnitude and phase of S11 and S21. When the two series inductors are identical, every even-powered term disappears from the numerator polynomial of S11, leaving only odd powers; the resulting response is that of a high-pass filter whose corner frequency is set by the common inductor value and the shunt capacitor. The circuit is then simulated at a 1 GHz corner, producing roll-off slopes near -30 dB/GHz for S11 and -32 dB/GHz for S21.

Core claim

The central claim is that a bridged-T network whose inductors L1 and L2 are identical yields an S11 transfer function whose numerator contains only odd powers of normalized frequency. This symmetry property converts the network into a high-pass filter whose scattering parameters can be calculated directly from the T-to-S conversion and whose simulated response at 1 GHz shows the expected sharp roll-off.

What carries the argument

The numerator polynomial of the S11 transfer function obtained after T-to-S matrix conversion; the equality L1 = L2 removes every even-powered coefficient from that polynomial.

If this is right

The bridged-T topology can be used directly for microstrip high-pass filters once L1 is set equal to L2.
The magnitude and phase of both S11 and S21 can be computed parametrically after frequency normalization.
Roll-off rates of approximately -30 dB/GHz for S11 and -32 dB/GHz for S21 are obtained at a 1 GHz corner frequency.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same symmetry cancellation may appear in other bridged or lattice networks if corresponding element values are made equal.
At frequencies well above 1 GHz the ideal lumped-element assumption will fail and full-wave simulation will be required to retain the odd-only property.
The closed-form S-parameter expressions allow rapid tuning of the corner frequency by scaling the common inductor value while keeping the ratio of other elements fixed.

Load-bearing premise

The derivations and simulations treat all circuit elements as ideal lumped components with no parasitic reactance or distributed-line effects at 1 GHz.

What would settle it

Fabricate the bridged-T circuit with equal inductors, measure the actual S11 response versus frequency, and check whether even-powered terms appear in the extracted numerator polynomial.

Figures

Figures reproduced from arXiv: 2604.18353 by Douglas Jussaume, Naser Khatti Dizabadi.

**Figure 1.** Figure 1: A two-port capacitive bridged-T network simulation results, confirming the bridged-T network’s ability to act as a high-pass filter. Detailed explanations and simulation results follow, providing a comprehensive overview of the bridged-T network’s performance and its significance for high-pass filtering. 2. The Transmission and Scattering Matrices [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Figure 2: Laplace equivalent impedance of the bridged-T network [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: Laplace equivalent impedance of the bridged-T network [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: High pass filter performance using a bridged-T network [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

**Figure 5.** Figure 5: S11 and S21 versus normalized frequency After establishing the corner frequency at about 1 GHz, [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗

**Figure 6.** Figure 6: Magnitude and phase of S11 versus the normalized frequency 12 [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗

**Figure 7.** Figure 7: Magnitude and phase of S21 versus normalized frequency 13 [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗

read the original abstract

The purpose of this study is to characterize a two-port Bridged-T network using transmission (T) and scattering (S) matrices. Using mathematical derivations, scattering parameters including S11, S12, S21, and S22 have been derived from the T and S matrices to permit a detailed investigation of the network's performance. As two of the most relevant parameters in the design of microstrip filters, both the magnitude and phase of S11 and S21 have been parametrically calculated after normalizing the frequency. Furthermore, when the inductors L1 and L2 are identical, all even coefficients of the numerator polynomial in the S11 transfer function are eliminated, leaving only the odd coefficients behind. Based on this feature, the bridged-T circuit is designed to operate as a high-pass filter. Therefore, the magnitude and phase of both S11 and S21 have been simulated for the designed filter with a corner frequency of 1 GHz. Simulation results performed by Keysight ADS show that S11 and S21 for the high-pass filter built upon the bridged-T network have sharp roll-off ratios of -30dB/GHz and -32dB/GHz respectively.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Standard lumped T-to-S derivation on a bridged-T network with an algebraic high-pass observation when L1 equals L2, but the microstrip application at 1 GHz rests on an ideal-element assumption that likely does not hold.

read the letter

The core observation is straightforward: setting the two inductors equal cancels the even-powered terms in the numerator of the derived S11 rational function, leaving an odd-only polynomial that produces high-pass behavior. They then pick a 1 GHz corner and run ADS simulations on the ideal circuit to show roll-offs around 30 dB/GHz. That algebraic step is correct on its own terms and the parametric plots of magnitude and phase are what you would expect from the topology once the T-matrix is converted to S-parameters in the usual way. The paper therefore gives a clean, if unsurprising, closed-form characterization for anyone who needs the explicit expressions for this exact bridged-T arrangement. Beyond that, there is little new; bridged-T networks and their filter uses are textbook material, and the equal-inductor trick follows directly from the admittance matrix without requiring new machinery. The main soft spot is the gap between the model and the stated microstrip application. The derivations treat the inductors and capacitors as perfect lumped elements whose T-matrix converts directly to S-parameters. At 1 GHz, any microstrip realization replaces those inductors with finite-length lines whose electrical length is not negligible, introducing distributed effects, frequency-dependent impedance, and parasitics that the ideal T-to-S conversion omits. The simulated sharp roll-off is therefore guaranteed only inside the lumped abstraction and may soften or shift once the layout is drawn. There are also no measured data or even a full microstrip layout shown, only ADS runs of the schematic. This leaves the central performance claim moderately supported at best. The work is mainly useful to RF designers who already work with bridged-T topologies and want the normalized S-parameter curves or the equal-L high-pass shortcut written out. It does not open new design spaces or resolve open questions in filter theory. I would not bring it to a reading group and would not cite it. It does not look strong enough to justify referee time in its current form.

Referee Report

2 major / 2 minor

Summary. The manuscript derives the scattering parameters (S11, S12, S21, S22) of a two-port bridged-T network from its transmission (T) matrix using standard T-to-S conversion formulas. It parametrically examines the magnitude and phase of S11 and S21 after frequency normalization and shows that equating the two series inductors L1 = L2 eliminates all even-powered coefficients in the numerator polynomial of the S11 transfer function. This algebraic property is exploited to realize a high-pass filter with a 1 GHz corner frequency, whose simulated S-parameters in Keysight ADS exhibit roll-off rates of -30 dB/GHz for S11 and -32 dB/GHz for S21.

Significance. If the lumped-element assumptions hold for the intended microstrip realization, the work supplies closed-form S-parameter expressions and a parameter-free design rule (inductor equality) that directly yields a high-pass response with sharp roll-off, offering a compact algebraic route to filter synthesis that could be useful for microstrip applications.

major comments (2)

[Derivation of S-parameters and high-pass filter design] The derivation of the S-parameters and the subsequent claim of a sharp high-pass roll-off rest on the assumption that the bridged-T network is exactly representable by ideal lumped inductors and capacitors whose T-matrix converts directly to S-parameters. At the stated design frequency of 1 GHz, any microstrip implementation replaces the inductors with finite-length transmission-line sections whose electrical length is non-negligible; the resulting distributed parasitics, frequency-dependent characteristic impedance, and higher-order modes are absent from the T-to-S conversion and are not addressed in the simulation.
[Simulation results] The ADS simulation results that report the -30 dB/GHz and -32 dB/GHz roll-offs are presented without indicating whether the circuit was modeled with ideal lumped elements or with distributed microstrip transmission-line models; if the former, the central claim for microstrip filter applications lacks direct numerical support.

minor comments (2)

[Abstract] The abstract states that S-parameters were 'derived mathematically' yet provides no explicit expressions, polynomial coefficients, or error bounds; including at least the final S11 numerator polynomial (with the even terms shown to vanish) would strengthen the central algebraic claim.
[Simulation results] No comparison against measured data or against a full-wave electromagnetic simulation of the microstrip layout is provided; only commercial circuit simulation is reported.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the lumped-element assumptions in our derivations and the need for greater clarity regarding the ADS simulations. These observations help us better delineate the scope of the work. We address each major comment below.

read point-by-point responses

Referee: [Derivation of S-parameters and high-pass filter design] The derivation of the S-parameters and the subsequent claim of a sharp high-pass roll-off rest on the assumption that the bridged-T network is exactly representable by ideal lumped inductors and capacitors whose T-matrix converts directly to S-parameters. At the stated design frequency of 1 GHz, any microstrip implementation replaces the inductors with finite-length transmission-line sections whose electrical length is non-negligible; the resulting distributed parasitics, frequency-dependent characteristic impedance, and higher-order modes are absent from the T-to-S conversion and are not addressed in the simulation.

Authors: The core contribution of the manuscript is the closed-form derivation of the scattering parameters from the T-matrix for the bridged-T network under the standard lumped-element model, together with the algebraic observation that L1 = L2 eliminates even-powered terms in the S11 numerator and produces the reported high-pass response. We agree that this ideal model omits distributed effects that become relevant in a microstrip realization at 1 GHz. The derived expressions and the parameter-free design rule nevertheless supply an exact algebraic route to the observed roll-off within the lumped approximation. In the revised manuscript we will add an explicit discussion of the model limitations, noting that the closed-form results can serve as an initial design that is subsequently refined by electromagnetic simulation to incorporate parasitics and higher-order modes. revision: yes
Referee: [Simulation results] The ADS simulation results that report the -30 dB/GHz and -32 dB/GHz roll-offs are presented without indicating whether the circuit was modeled with ideal lumped elements or with distributed microstrip transmission-line models; if the former, the central claim for microstrip filter applications lacks direct numerical support.

Authors: The Keysight ADS simulations were performed with ideal lumped-element components in order to verify the analytically derived S-parameters and the high-pass behavior obtained when L1 = L2. We acknowledge that this does not constitute a distributed microstrip simulation and therefore does not provide direct numerical support for a fabricated microstrip filter. In the revision we will state explicitly that the ADS results employ ideal lumped elements, clarify that the work presents a parametric characterization and design rule applicable as a foundation for microstrip filter synthesis, and note that full-wave modeling would be required for precise practical implementation. revision: yes

Circularity Check

0 steps flagged

Standard T-to-S conversion and algebraic simplification; derivation is self-contained

full rationale

The paper begins with the conventional T-matrix for the bridged-T topology using ideal lumped L and C elements, applies the standard T-to-S conversion formulas, and then algebraically expands the resulting S11 rational function. When L1 equals L2 the even-powered numerator coefficients cancel by direct polynomial arithmetic, leaving only odd terms; this is a mathematical identity within the model and is not obtained by fitting or redefinition. No parameters are tuned to data and then relabeled as predictions, no self-citations are invoked to justify uniqueness or ansatz choices, and the ADS simulations are presented only as numerical verification of the closed-form expressions. The derivation chain therefore stands independently of its conclusions and contains no load-bearing circular steps.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard two-port network theory and the assumption that the bridged-T topology can be treated with lumped elements at the frequencies of interest.

free parameters (2)

Corner frequency
Chosen as 1 GHz for the example high-pass filter design and simulation.
Inductor equality condition
L1 set equal to L2 to eliminate even coefficients in the S11 numerator.

axioms (2)

standard math Standard conversion formulas between transmission (T) matrix and scattering (S) matrix for two-port networks
Invoked to obtain S11, S12, S21, S22 from the T-matrix of the bridged-T circuit.
domain assumption Lumped-element model remains valid for the microstrip implementation at 1 GHz
Required for the derivation and ADS simulation to match physical behavior.

pith-pipeline@v0.9.0 · 5513 in / 1506 out tokens · 49255 ms · 2026-05-10T02:58:59.633898+00:00 · methodology

discussion (0)

Reference graph

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