pith. sign in

arxiv: 2605.18492 · v1 · pith:TQXVCRRJnew · submitted 2026-05-18 · ⚛️ physics.comp-ph · physics.app-ph· physics.class-ph

Simulation of S-parameters of general multilayer boxed PCBs with the method of moments and the scattering matrix algorithm

A. O. Makarenko , P. Zheglova , R. Gaponenko , R. V. Salimov , R. I. Tikhonov , A. A. Shcherbakov This is my paper

Pith reviewed 2026-05-20 01:44 UTC · model grok-4.3

classification ⚛️ physics.comp-ph physics.app-phphysics.class-ph
keywords printed circuit boardmethod of momentsscattering matrixGreen's functionS-parametersmultilayer structureselectromagnetic modelingshielded media
0
0 comments X

The pith

A scattering matrix approach derives the full dyadic Green's function for 2.5D simulation of multilayer PCBs.

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

The paper presents a method to simulate S-parameters of general multilayer boxed printed circuit boards using the method of moments combined with the scattering matrix algorithm. It derives an analytic expression for the complete dyadic Green's function in a layered waveguide by expressing it through three sets of S-matrices linked to the source and observation layers. This provides a numerically stable way to handle both transverse and longitudinal currents in thin conductive components. The approach includes overlap integrals for rooftop, pulse, and linear basis functions to model planar layers and vertical interconnects. Numerical examples validate the method for shielded PCB geometries.

Core claim

The Green's function in a layered waveguide is calculated in a straightforward manner using the S-matrix formalism, resulting in an expression involving three sets of S-matrices associated with the PCB layers containing the current source and the field observation point. This enables the 2.5D MoM solution for integral equations in general multilayer boxed PCB structures that include both transverse and longitudinal currents.

What carries the argument

Three sets of S-matrices for the layers of the electric current source and the electric field observation point, used to compute the complete dyadic Green's function in a layered waveguide.

If this is right

  • The method supports modeling of planar metallization layers with surface rooftop basis functions and vertical interconnects with volume pulse and linear basis functions.
  • Overlap integrals are provided for these basis functions to facilitate implementation.
  • The approach maintains numerical stability inherent to the S-matrix method while allowing intuitive calculation of the Green's function.
  • It can be extended to other basis functions for simulating objects of various shapes in PCB designs.

Where Pith is reading between the lines

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

  • This could lead to faster design iterations in PCB manufacturing by allowing pre-manufacturing optimization through accurate simulations.
  • Similar S-matrix techniques might apply to other layered electromagnetic problems beyond PCBs, such as integrated circuits or antennas in enclosures.
  • Further development could incorporate more complex material properties or non-uniform layers to broaden applicability.

Load-bearing premise

The PCB geometry can be modeled as thin highly conductive components enclosed in a simply shaped dielectric host that fits the layered waveguide approximation.

What would settle it

If measurements on a physical multilayer PCB with vias show significant deviation from the simulated S-parameters, particularly for structures involving longitudinal currents, the validity of the Green's function derivation would be questioned.

Figures

Figures reproduced from arXiv: 2605.18492 by A. A. Shcherbakov, A. O. Makarenko, P. Zheglova, R. Gaponenko, R. I. Tikhonov, R. V. Salimov.

Figure 1
Figure 1. Figure 1: An example PCB structure. While neither this problem nor our general approach are new, the contribution of our work lies in an alternative solution method characterized by transparency, conciseness, elegance, generality and numerical stability. The paper is organized as follows. In section II, EFIE formulation of the problem is given. In section III, we show the derivation of the multilayer Green’s functio… view at source ↗
Figure 4
Figure 4. Figure 4: S A and S B matrices associated with layers [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 2
Figure 2. Figure 2: S-matrices associated with layers l and l ′ , when zv and zs are in different layers: (a) zv > zs; (b) zv < zs [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: S-matrices associated with layer l, when zv and zs are in the same layer: (a) zv ≥ zs; (b) zv ≤ zs. To compute the three introduced sets of matrices, S A l , S B l , and S P ll′ we assume a waveguide to have (L + 1) layers, L inner interfaces between different media, supported with the top and bottom boundary conditions defined in Eq. (17). As an example, Figs. 4 and 5 show a structure with L = 8 interface… view at source ↗
Figure 7
Figure 7. Figure 7: Filter with three layers and six metal strips: frequency dependence of [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 6
Figure 6. Figure 6: Filter with three layers and six metal strips: model geometry, [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: Filter with three layers and six metal strips [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Filter with three layers, six metal strips and two interconnect sections: [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
read the original abstract

Printed circuit board (PCB) modelling is an important part of the PCB production process, in which the designer aims to optimize the desired output characteristics prior to physical PCB manufacturing. Due to the specific shape of PCBs, namely, thin and highly conductive components enclosed within a relatively simply shaped dielectric host, the PCB modelling problem is amenable to solution by the so-called 2.5D Method of Moments (MoM) applied to the integral equation solution of Maxwell's equations. For this purpose, an analytic expression for the Green's function of the host medium needs to be derived. Many studies exist in which expressions are derived for the transverse Green's function components in a waveguide, used for modelling planar metallization layers in shielded layered media. Works containing the full Green's function that allows modelling of both longitudinal and transverse currents are much fewer. In this study, we propose a tool to solve the shielded PCB modelling problem involving both transverse and longitudinal currents, with the Green's function in a layered waveguide derived using the S-matrix formalism. Our approach combines a straightforward, intuitive way of calculating the complete dyadic Green's function in a layered waveguide with the inherent numerical stability of the S-matrix method. The Green's function is expressed in terms of three sets of S-matrices associated with the PCB layers in which the electric current source and the electric field observation point are located. The MoM is implemented using surface rooftop, volume pulse, and linear basis functions, for which we provide the overlap integrals, to model planar metallization layers and wire-like vertical interconnects. The validity of the method is demonstrated on two numerical examples. The method can be extended to other bases to model objects of various shapes.

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.

Referee Report

1 major / 1 minor

Summary. The manuscript develops a 2.5D Method of Moments formulation for computing S-parameters of general multilayer boxed PCBs. The central contribution is an analytic construction of the complete dyadic Green's function for a layered waveguide obtained via the scattering-matrix algorithm; the Green's function is expressed using three sets of S-matrices associated with the layers containing the electric-current source and the field observation point. The formulation accommodates both transverse and longitudinal currents through surface rooftop, volume pulse, and linear basis functions, for which explicit overlap integrals are supplied. Validity is illustrated by two numerical examples.

Significance. If the derivation and demonstrations hold, the work supplies a numerically stable route to the full dyadic Green's function in shielded layered media, directly addressing the modeling of both planar metallization and vertical interconnects inside rectangular enclosures. The S-matrix construction avoids the instabilities that arise in some alternative layered-media formulations, and the provision of overlap integrals for standard basis functions lowers the barrier to implementation. These features constitute a practical advance for the targeted class of thin, highly conductive PCB structures.

major comments (1)
  1. Numerical Examples section: the two demonstrations are described at a high level, yet no quantitative error metrics (e.g., maximum |S_{ij} - S_{ij}^{ref}| or RMS deviation against a reference solver) are reported. Without such measures the claim that validity is demonstrated cannot be assessed quantitatively.
minor comments (1)
  1. Abstract: the statement that 'validity is demonstrated on two numerical examples' would be strengthened by a brief indication of the quantities compared and the observed level of agreement.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation and the recommendation of minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: Numerical Examples section: the two demonstrations are described at a high level, yet no quantitative error metrics (e.g., maximum |S_{ij} - S_{ij}^{ref}| or RMS deviation against a reference solver) are reported. Without such measures the claim that validity is demonstrated cannot be assessed quantitatively.

    Authors: We agree that the current presentation of the numerical examples is primarily qualitative. In the revised manuscript we will expand the Numerical Examples section to include direct quantitative comparisons against reference solutions (commercial full-wave solver or independent MoM implementation). For each example we will report the maximum absolute deviation and RMS error in the S-parameter values over the frequency range of interest, together with the reference data source. These additions will permit a quantitative assessment of accuracy while preserving the existing figures. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper derives the dyadic Green's function explicitly via the S-matrix formalism applied to a layered waveguide, expressing it in terms of three sets of S-matrices tied to source and observation layers. It supplies the MoM implementation details including overlap integrals for rooftop, volume pulse, and linear basis functions, plus two numerical validation examples. No quoted step reduces a claimed prediction or result to a fitted parameter or self-citation by construction; the central construction is presented as an independent analytic derivation rather than a renaming or tautology. The layered-waveguide model is stated as an assumption matching the target PCB class, not derived from the outputs. This is a standard self-contained computational electromagnetics method paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard electromagnetic theory for stratified media and the assumption that thin conductive PCB features inside a simply shaped enclosure can be treated with 2.5D MoM.

axioms (2)
  • standard math Maxwell's equations in integral form govern the fields produced by currents in the layered dielectric host.
    Invoked as the starting point for deriving the Green's function via S-matrix recursion.
  • domain assumption PCB metallization layers are thin and highly conductive relative to the dielectric host, justifying the 2.5D surface/volume current model.
    Stated in the opening paragraph describing why the PCB modelling problem is amenable to 2.5D MoM.

pith-pipeline@v0.9.0 · 5876 in / 1472 out tokens · 48086 ms · 2026-05-20T01:44:28.750943+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

83 extracted references · 83 canonical work pages

  1. [1]

    Ultra-fine feature printed circuits and multi-chip modules,

    N. Chandler and S. Tyler, “Ultra-fine feature printed circuits and multi-chip modules,”Microelectronics Journal, vol. 26, no. 4, pp. 393–404, 1995. [Online]. Available: https://www.sciencedirect.com/ science/article/pii/002626929598940S

    work page arXiv 1995

  2. [2]

    Bond testing enters mainstream PCB assembly,

    P. Walter, “Bond testing enters mainstream PCB assembly,”Micro- electronics Journal, vol. 27, no. 1, pp. i–v, 1996. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S002626929690018X

    work page 1996

  3. [3]

    Dispensing answers the call: Speed, accuracy, and flexibility meet PCB assembly needs,

    T. Prentice, “Dispensing answers the call: Speed, accuracy, and flexibility meet PCB assembly needs,”Microelectronics Journal, vol. 28, no. 1, pp. vii–xi, 1997. [Online]. Available: https: //www.sciencedirect.com/science/article/pii/S002626929787869X

    work page 1997

  4. [4]

    Some issues for microsystem packaging in plastic and 3D,

    A. Morrissey, G. Kelly, J. Alderman, J. Barrett, C. Lyden, and L. O’Rourke, “Some issues for microsystem packaging in plastic and 3D,”Microelectronics Journal, vol. 29, no. 9, pp. 645–650, 1998, low Dimensional Structures and Devices: Micromachined Devices. [Online]. Available: https://www.sciencedirect.com/science/article/pii/ S0026269298000299

    work page 1998

  5. [5]

    A 8-10 GHz compact low noise amplifier MMIC with high linearity based on GaAs technology,

    X. Bai, Y . Wu, S. Zhen, Z. Gao, and W. Wang, “A 8-10 GHz compact low noise amplifier MMIC with high linearity based on GaAs technology,”Microelectronics Journal, vol. 162, p. 106742, 2025. [Online]. Available: https://www.sciencedirect.com/science/article/pii/ S1879239125001912

    work page 2025

  6. [6]

    A novel minimized millimeter-wave on-chip spoof surface plasmon polariton and its applications based on IPD technology,

    Y . Zeng, Y . Wu, Y . Yang, L. Pan, and W. Wang, “A novel minimized millimeter-wave on-chip spoof surface plasmon polariton and its applications based on IPD technology,”Microelectronics Journal, vol. 162, p. 106755, 2025. [Online]. Available: https: //www.sciencedirect.com/science/article/pii/S1879239125002048

    work page 2025

  7. [7]

    Applications of advanced shielded planar EM analysis techniques to antenna analysis,

    B. J. Rautio, “Applications of advanced shielded planar EM analysis techniques to antenna analysis,”2nd European Conference on Antennas and Propagation (EuCAP 2007) Proceedings, 2007

    work page 2007

  8. [8]

    W. C. Gibson,The Method of Moments in electromagnetics. Boca Raton, FL: Chapman & Hall/CRC, 2008

    work page 2008

  9. [9]

    D. G. Swanson Jr and W. J. R. Hoefer,Microwave circuit modeling using electromagnetic field simulation, ser. Microwave Library. Norwood, MA: Artech House, May 2003

    work page 2003

  10. [10]

    V olume 2: Field representations and the Method of Moments,

    V . Lancellotti, “V olume 2: Field representations and the Method of Moments,” inAdvanced Theoretical and Numerical Electromagnetics. London, United Kingdom: SciTech Publishing, 2022

    work page 2022

  11. [11]

    Okhmatovski and S

    V . Okhmatovski and S. Zheng,Theory and Computation of Electromag- netic Fields in Layered Media. Wiley-IEEE Press, 2024

    work page 2024

  12. [12]

    Efficient Method-of- Moments formulation for the modeling of planar conductive layers in a shielded guided-wave structure,

    A. I. Khalil, A. B. Yakovlev, and M. B. Steer, “Efficient Method-of- Moments formulation for the modeling of planar conductive layers in a shielded guided-wave structure,”IEEE Transactions on Microwave Theory and Techniques, vol. 47, no. 9, pp. 1730–1736, 1999

    work page 1999

  13. [13]

    D. B. Davidson,Computational Electromagnetics for RF and Microwave Engineering. Cambridge University Press, 2005

    work page 2005

  14. [14]

    Analysis of multilayer boxed printed circuits,

    P. Crespo-Valero, M. Mattes, I. Stevanovi ´c, and J. R. Mosig, “Analysis of multilayer boxed printed circuits,”Proceedings of the Mediterranean Electrotechnical Conference - MELECON, vol. 2006, no. 4, pp. 206– 209, 2006

    work page 2006

  15. [15]

    Simple and efficient full wave analysis of electromagnetic coupling in multilayer printed circuit board layouts,

    M. R. Abdul-Gaffoor, “Simple and efficient full wave analysis of electromagnetic coupling in multilayer printed circuit board layouts,” Ph.D. dissertation, University of Mississippi, 2000

    work page 2000

  16. [16]

    Simple and efficient full-wave modeling of electromagnetic coupling in realistic RF multilayer PCB layouts,

    M. R. Abdul-Gaffoor, H. K. Smith, A. A. Kishk, and A. W. Glisson, “Simple and efficient full-wave modeling of electromagnetic coupling in realistic RF multilayer PCB layouts,”IEEE Transactions on Microwave Theory and Techniques, vol. 50, no. 6, pp. 1445–1457, 2002

    work page 2002

  17. [17]

    Multilayered media Green’s functions in integral equation formulations,

    K. A. Michalski and J. R. Mosig, “Multilayered media Green’s functions in integral equation formulations,”IEEE Transactions on Antennas and Propagation, vol. 45, no. 3, pp. 508–519, 1997

    work page 1997

  18. [18]

    Kinayman and M

    N. Kinayman and M. I. Aksun,Modern Microwave Circuits. Artech House, 2005

    work page 2005

  19. [19]

    A three- dimensional precorrected FFT algorithm for fast method of moments solutions of the mixed-potential integral equation in layered media,

    V . Okhmatovski, M. Yuan, I. Jeffrey, and R. Phelps, “A three- dimensional precorrected FFT algorithm for fast method of moments solutions of the mixed-potential integral equation in layered media,” IEEE Transactions on Microwave Theory and Techniques, vol. 57, no. 12, pp. 3505–3517, 2009

    work page 2009

  20. [20]

    W. Chew, E. Michielssen, J. Song, and J. Jin,Fast and Efficient Algorithms in Computational Electromagnetics. Artech House, 2001

    work page 2001

  21. [21]

    Calculation of CFIE impedance matrix elements with RWG and n/spl times/RWG functions,

    P. Yla-Oijala and M. Taskinen, “Calculation of CFIE impedance matrix elements with RWG and n/spl times/RWG functions,”IEEE Transac- tions on Antennas and Propagation, vol. 51, no. 8, pp. 1837–1846, 2003

    work page 2003

  22. [22]

    Comparative study of acceleration tech- niques for integrals and series in electromagnetic problems,

    N. Kinayman and M. I. A., “Comparative study of acceleration tech- niques for integrals and series in electromagnetic problems,”Radio Science, vol. 30, no. 6, pp. 1713–1722, 1995

    work page 1995

  23. [23]

    On the summation of double infinite series field computations inside rectangular cavities,

    S. Hashemi-Yeganeh, “On the summation of double infinite series field computations inside rectangular cavities,”IEEE Transactions on Microwave Theory and Techniques, vol. 43, no. 3, pp. 641–646, 1995

    work page 1995

  24. [24]

    A fast integral equation technique for shielded planar circuits defined on nonuniform meshes,

    G. Eleftheriades, J. Mosig, and M. Guglielmi, “A fast integral equation technique for shielded planar circuits defined on nonuniform meshes,” 15 IEEE Transactions on Microwave Theory and Techniques, vol. 44, no. 12, pp. 2293–2296, 1996

    work page 1996

  25. [25]

    Generalized Poisson summation formula for tempered distributions,

    H. Q. Nguyen and M. Unser, “Generalized Poisson summation formula for tempered distributions,” in2015 International Conference on Sam- pling Theory and Applications (SampTA), 2015, pp. 1–5

    work page 2015

  26. [26]

    Analytical technique to evaluate the asymp- totic part of the impedance matrix of Sommerfeld-type integrals,

    S.-O. Park and C. Balanis, “Analytical technique to evaluate the asymp- totic part of the impedance matrix of Sommerfeld-type integrals,”IEEE Transactions on Antennas and Propagation, vol. 45, no. 5, pp. 798–805, 1997

    work page 1997

  27. [27]

    A novel efficient technique for the calculation of the Green’s functions in rectangular waveguides based on accelerated series decomposition,

    F. J. P. Soler, F. D. Q. Pereira, D. C. Rebenaque, A. A. Melcon, and J. R. Mosig, “A novel efficient technique for the calculation of the Green’s functions in rectangular waveguides based on accelerated series decomposition,”IEEE Transactions on Antennas and Propagation, vol. 56, no. 10, pp. 3260–3270, 2008

    work page 2008

  28. [28]

    Efficient calculation of the 3-D rectangular waveguide Green’s functions derivatives by the Ewald method,

    A. M. H. de la Cruz, C. G. Molina, F. D. Quesada Pereira, A. A. Melcon, and V . E. B. Esbert, “Efficient calculation of the 3-D rectangular waveguide Green’s functions derivatives by the Ewald method,”IEEE Transactions on Microwave Theory and Techniques, vol. 71, no. 12, pp. 5171–5181, 2023

    work page 2023

  29. [29]

    A time-harmonic electromagnetic analysis of shielded microstrip circuits,

    J. C. Rautio, “A time-harmonic electromagnetic analysis of shielded microstrip circuits,” Ph.D. dissertation, Syracuse University, 1986

    work page 1986

  30. [30]

    An efficient algorithm for the three-dimensional analysis of passive microstrip components and discontinuities for mi- crowave and millimeter-wave integrated circuits,

    A. Hill and V . Tripathi, “An efficient algorithm for the three-dimensional analysis of passive microstrip components and discontinuities for mi- crowave and millimeter-wave integrated circuits,”IEEE Transactions on Microwave Theory and Techniques, vol. 39, no. 1, pp. 83–91, 1991

    work page 1991

  31. [31]

    The Unified-FFT algorithm for fast electromagnetic analysis of planar integrated circuits printed on layered media inside a rectangular enclosure,

    B. J. Rautio, V . I. Okhmatovski, A. C. Cangellaris, J. C. Rautio, and J. K. Lee, “The Unified-FFT algorithm for fast electromagnetic analysis of planar integrated circuits printed on layered media inside a rectangular enclosure,”IEEE Transactions on Microwave Theory and Techniques, vol. 62, no. 5, pp. 1112–1121, 2014

    work page 2014

  32. [32]

    The unified-FFT grid totalizing algorithm for fast O(N logN) method of moments electro- magnetic analysis with accuracy to machine precision,

    B. Rautio, V . I. Okhmatovski, and J. K. Lee, “The unified-FFT grid totalizing algorithm for fast O(N logN) method of moments electro- magnetic analysis with accuracy to machine precision,”Progress in Electromagnetics Research, vol. 154, no. November, pp. 101–114, 2015

    work page 2015

  33. [33]

    A Proposal for Toeplitz matrix calculations,

    G. Strang, “A Proposal for Toeplitz matrix calculations,”Studies in Applied Mathematics, vol. 74, no. 2, pp. 171–176, 1986

    work page 1986

  34. [34]

    Toeplitz equations by conjugate gradients with circulant precon- ditioner,

    ——, “Toeplitz equations by conjugate gradients with circulant precon- ditioner,”SIAM Journal on Scientific and Statistical Computing, vol. 10, no. 1, pp. 104–119, 1989

    work page 1989

  35. [35]

    Thin-stratified medium fast-multipole algorithm for solving microstrip structures,

    J.-S. Zhao, W. C. Chew, C.-C. Lu, E. Michielssen, and J. Song, “Thin-stratified medium fast-multipole algorithm for solving microstrip structures,”IEEE Transactions on Microwave Theory and Techniques, vol. 46, no. 4, pp. 395–403, 1998

    work page 1998

  36. [36]

    On the application of symmetry conditions and the convergence rate of modal series in the MOM-based integral equation analysis of laterally shielded multilayered media,

    L. Golestanirad, M. Mattes, and J. R. Mosig, “On the application of symmetry conditions and the convergence rate of modal series in the MOM-based integral equation analysis of laterally shielded multilayered media,”Microwave and Optical Technology Letters, vol. 52, no. 1, pp. 221–226, 2010. [Online]. Available: https: //onlinelibrary.wiley.com/doi/abs/10....

    work page doi:10.1002/mop.24868 2010

  37. [37]

    An efficient technique to accelerate the simulation of passive shielded microwave circuits,

    ——, “An efficient technique to accelerate the simulation of passive shielded microwave circuits,”Proceedings of the Fourth European Conference on Antennas and Propagation, pp. 1–4, 2010

    work page 2010

  38. [38]

    An electromagnetic time-harmonic analysis of shielded microstrip circuits,

    J. C. Rautio and R. F. Harrington, “An electromagnetic time-harmonic analysis of shielded microstrip circuits,” pp. 726–730, 1987

    work page 1987

  39. [39]

    J. C. Rautio,Applied Numerical Electromagnetic Analysis for Planar High Frequency Circuits. John Wiley and Sons, Ltd, 2005. [Online]. Available: https://onlinelibrary.wiley.com/doi/abs/10.1002/0471654507. eme040

    work page doi:10.1002/0471654507 2005

  40. [40]

    Efficient CAD of boxed microwave circuits based on arbitrary rectangular elements,

    A. Melcon, J. Mosig, and M. Guglielmi, “Efficient CAD of boxed microwave circuits based on arbitrary rectangular elements,”IEEE Transactions on Microwave Theory and Techniques, vol. 47, no. 7, pp. 1045–1058, 1999

    work page 1999

  41. [41]

    Integral equation modeling of waveguide- fed planar antennas,

    I. Stevanovic, F. Merli, P. Crespo-Valero, W. Simon, S. Holzwarth, M. Mattes, and J. R. Mosig, “Integral equation modeling of waveguide- fed planar antennas,”IEEE Antennas and Propagation Magazine, vol. 51, no. 6, pp. 82–92, 2009

    work page 2009

  42. [42]

    Modal transmission line theory of plane wave excited layered media with multiple conductive anisotropic sheets at the interfaces,

    K. A. Michalski, “Modal transmission line theory of plane wave excited layered media with multiple conductive anisotropic sheets at the interfaces,”Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 226, pp. 19–28, 2019. [Online]. Available: https: //www.sciencedirect.com/science/article/pii/S0022407318308987

    work page 2019

  43. [43]

    A generalized scattering matrix method using the Method of Moments for electromagnetic analysis of multilay- ered structures in waveguide,

    A. I. Khalil and M. B. Steer, “A generalized scattering matrix method using the Method of Moments for electromagnetic analysis of multilay- ered structures in waveguide,”IEEE Transactions on Microwave Theory and Techniques, vol. 47, no. 9, pp. 2151–2157, 1999

    work page 1999

  44. [44]

    Electromagnetic mod- eling of a waveguide-based strip-to-slot transition module for application to spatial power combining systems,

    A. Yakovlev, A. Khalil, C. Hicks, and M. Steer, “Electromagnetic mod- eling of a waveguide-based strip-to-slot transition module for application to spatial power combining systems,” inIEEE Antennas and Propagation Society International Symposium. 1999 Digest. Held in conjunction with: USNC/URSI National Radio Science Meeting (Cat. No.99CH37010), vol. 1, 1...

    work page 1999

  45. [45]

    The generalized scattering matrix of closely spaced strip and slot layers in waveguide,

    A. Yakovlev, A. Khalil, C. Hicks, A. Mortazawi, and M. Steer, “The generalized scattering matrix of closely spaced strip and slot layers in waveguide,”IEEE Transactions on Microwave Theory and Techniques, vol. 48, no. 1, pp. 126–137, 2000

    work page 2000

  46. [46]

    Electric dyadic Green’s functions for modeling resonance and coupling effects in waveguide-based aperture-coupled patch arrays,

    A. Yakovlev, S. Ortiz, M. Ozkar, A. Mortazawi, and M. Steer, “Electric dyadic Green’s functions for modeling resonance and coupling effects in waveguide-based aperture-coupled patch arrays,”Aces Journal, vol. 17, no. 2, pp. 123–133, 2002

    work page 2002

  47. [47]

    Input impedance of a probe-excited semi-infinite rectangular waveguide with arbitrary multilayered loads: Part II - A full-wave analysis,

    L.-W. Li, P.-S. Kooi, M.-S. Leong, T.-S. Yeo, and S.-L. Ho, “Input impedance of a probe-excited semi-infinite rectangular waveguide with arbitrary multilayered loads: Part II - A full-wave analysis,”IEEE Transactions on Microwave Theory and Techniques, vol. 45, no. 3, pp. 321–329, 1997

    work page 1997

  48. [48]

    Multimode equivalent network for boxed multilayer arbitrary planar circuits,

    C. Gomez Molina, F. Q. Pereira, A. A. Melcon, S. Marini, M. A. Sanchez-Soriano, V . E. Boria, and M. Guglielmi, “Multimode equivalent network for boxed multilayer arbitrary planar circuits,”IEEE Transac- tions on Microwave Theory and Techniques, vol. 68, no. 7, pp. 2501– 2514, 2020

    work page 2020

  49. [49]

    Novel waveguide filters with multiple attenuation poles using dual-behavior resonance of frequency-selective surfaces,

    M. Ohira, H. Deguchi, M. Tsuji, and H. Shigesawa, “Novel waveguide filters with multiple attenuation poles using dual-behavior resonance of frequency-selective surfaces,”IEEE Transactions on Microwave Theory and Techniques, vol. 53, no. 11, pp. 3320–3326, 2005

    work page 2005

  50. [50]

    Efficient electromagnetic analysis of line- fed aperture antennas in thick conducting screens,

    I. Stevanovic and J. Mosig, “Efficient electromagnetic analysis of line- fed aperture antennas in thick conducting screens,”IEEE Transactions on Antennas and Propagation, vol. 52, no. 11, pp. 2896–2903, 2004

    work page 2004

  51. [51]

    An integral-equation technique for solving thick irises in rectangular waveguides,

    I. Stevanovic, P. Crespo-Valero, and J. Mosig, “An integral-equation technique for solving thick irises in rectangular waveguides,”IEEE Transactions on Microwave Theory and Techniques, vol. 54, no. 1, pp. 189–197, 2006

    work page 2006

  52. [52]

    The unified-FFT method for fast solution of integral equations as applied to shielded-domain electromagnetics,

    B. J. Rautio, “The unified-FFT method for fast solution of integral equations as applied to shielded-domain electromagnetics,” Ph.D. dis- sertation, Syracuse University, 1986

    work page 1986

  53. [53]

    Design of EMI and suppression structure based on bar-via,

    N. Li and M. Miao, “Design of EMI and suppression structure based on bar-via,”Microelectronics Journal, vol. 112, p. 105049, 2021. [Online]. Available: https://www.sciencedirect.com/science/article/pii/ S0026269221000604

    work page 2021

  54. [54]

    V olume rooftop basis functions in shielded layered media,

    J. C. Rautio and M. Thelen, “V olume rooftop basis functions in shielded layered media,”2019 IEEE MTT-S International Conference on Nu- merical Electromagnetic and Multiphysics Modeling and Optimization, NEMO 2019, vol. c, pp. 1–4, 2019

    work page 2019

  55. [55]

    A volume current based Method of Moments analysis of shielded planar 3-D circuits in layered media,

    ——, “A volume current based Method of Moments analysis of shielded planar 3-D circuits in layered media,” in2020 IEEE/MTT-S International Microwave Symposium (IMS), 2020, pp. 146–149

    work page 2020

  56. [56]

    Method of Moments analysis of arbitrary structures in shielded layered media,

    J. C. Rautio and M. A. Thelen, “Method of Moments analysis of arbitrary structures in shielded layered media,”IEEE Transactions on Microwave Theory and Techniques, vol. 69, no. 1, pp. 509–517, 2021

    work page 2021

  57. [57]

    Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media. I. Theory,

    K. Michalski and D. Zheng, “Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media. I. Theory,”IEEE Transactions on Antennas and Propagation, vol. 38, no. 3, pp. 335– 344, 1990

    work page 1990

  58. [58]

    Surface-volume-surface electric field integral equation for solution of scattering problems on 3-D dielectric objects in multilayered media,

    S. Zheng, R. Gholami, and V . I. Okhmatovski, “Surface-volume-surface electric field integral equation for solution of scattering problems on 3-D dielectric objects in multilayered media,”IEEE Transactions on Microwave Theory and Techniques, vol. 66, no. 12, pp. 5399–5414, 2018

    work page 2018

  59. [59]

    Three-dimensional planar radiating structures in stratified media,

    P. Gay-Balmaz and J. R. Mosig, “Three-dimensional planar radiating structures in stratified media,”International Journal of Microwave and Millimeter-Wave Computer-Aided Engineering, vol. 7, no. 5, pp. 330–343, 1997. [Online]. Available: https://onlinelibrary.wiley.com/doi/ abs/10.1002/%28SICI%291522-6301%28199709%297%3A5%3C330% 3A%3AAID-MMCE3%3E3.0.CO%3B2-L

    work page 1997

  60. [60]

    Full-wave analysis of antennas containing horizontal and vertical metallizations embedded in planar multilayered media,

    T. Grzegorczyk and J. Mosig, “Full-wave analysis of antennas containing horizontal and vertical metallizations embedded in planar multilayered media,”IEEE Transactions on Antennas and Propagation, vol. 51, no. 11, pp. 3047–3054, 2003

    work page 2003

  61. [61]

    A rigorous and efficient analysis of 3-D printed circuits: Vertical conductors arbitrarily dis- tributed in multilayer environment,

    T. Onal, M. I. Aksun, and N. Kinayman, “A rigorous and efficient analysis of 3-D printed circuits: Vertical conductors arbitrarily dis- tributed in multilayer environment,”IEEE Transactions on Antennas and Propagation, vol. 55, no. 12, pp. 3726–3729, 2007

    work page 2007

  62. [62]

    Analysis of 3-D metallization structures by a full-wave spectral-domain technique,

    T. Becks and I. Wolff, “Analysis of 3-D metallization structures by a full-wave spectral-domain technique,”IEEE Transactions on Microwave Theory and Techniques, vol. 40, no. 12, pp. 2219–2227, 1992. 16

    work page 1992

  63. [63]

    Full-wave analysis of three-dimensional microstrip discontinuities by a new spectral domain approach,

    H. Aroudaki and V . Hansen, “Full-wave analysis of three-dimensional microstrip discontinuities by a new spectral domain approach,”IEEE AP-S International Symposium, Proceedings, vol. 3, p. 1694–1697, 1994

    work page 1994

  64. [64]

    An efficient application of the discrete complex image method for quasi-3-D microwave circuits in layered media,

    W.-H. Tang and S. D. Gedney, “An efficient application of the discrete complex image method for quasi-3-D microwave circuits in layered media,”IEEE Transactions on Microwave Theory and Techniques, vol. 55, no. 8, pp. 1723–1729, 2007

    work page 2007

  65. [65]

    Hybrid dyadic-mixed-potential and combined spectral-space domain integral-equation analysis of quasi-3-D structures in stratified media,

    M. Vrancken and G. Vandenbosch, “Hybrid dyadic-mixed-potential and combined spectral-space domain integral-equation analysis of quasi-3-D structures in stratified media,”IEEE Transactions on Microwave Theory and Techniques, vol. 51, no. 1, pp. 216–225, 2003

    work page 2003

  66. [66]

    A pre-corrected FFT algorithm for fast electromagnetic modeling of three-dimensional integrated passives in multilayered substrates,

    V . Okhmatovski, M. Yuan, I. Jeffrey, and R. Phelps, “A pre-corrected FFT algorithm for fast electromagnetic modeling of three-dimensional integrated passives in multilayered substrates,” in2009 IEEE MTT-S International Microwave Symposium Digest, 2009, pp. 1581–1584

    work page 2009

  67. [67]

    Electromagnetic scattering by surfaces of arbitrary shape,

    S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,”IEEE Transactions on Antennas and Propagation, vol. 30, no. 3, pp. 409–418, 1982

    work page 1982

  68. [68]

    A surface integral equation formulation for efficient simulation of finite-sized multilayered parallel-plate structure,

    Y . Ren, Y . Chen, M. Meng, Y . Liu, K.-D. Xu, and J. Li, “A surface integral equation formulation for efficient simulation of finite-sized multilayered parallel-plate structure,”IEEE Transactions on Microwave Theory and Techniques, vol. 68, no. 7, pp. 2475–2484, 2020

    work page 2020

  69. [69]

    A fast full-wave modeling methodology for stripline structures with ver- tical interconnects in multi-layer dielectrics,

    K. Nikellis, Y . Koutsoyannopoulos, S. Bantas, and N. Uzunoglu, “A fast full-wave modeling methodology for stripline structures with ver- tical interconnects in multi-layer dielectrics,” in2004 Proceedings. 54th Electronic Components and Technology Conference (IEEE Cat. No.04CH37546), vol. 1, 2004, pp. 225–230 V ol.1

    work page 2004

  70. [70]

    Full- wave modeling of stripline structures in multilayer dielectrics,

    K. Nikellis, N. Uzunoglu, Y . Koutsoyannopoulos, and S. Bantas, “Full- wave modeling of stripline structures in multilayer dielectrics,”Progress in Electromagnetic Research, vol. 57, pp. 253–264, 2006

    work page 2006

  71. [71]

    Sonnet suites

    Sonnet Software Inc. Sonnet suites. [Online]. Available: https: //www.sonnetsoftware.com/

    work page

  72. [72]

    Efficient integral equation analysis of arbitrarily shaped rectangular waveguide discontinuities including conducting objects,

    A. M. H. de la Cruz, A. O. Aparicio, F. D. Q. Pereira, A. A. Melcon, and V . E. B. Esbert, “Efficient integral equation analysis of arbitrarily shaped rectangular waveguide discontinuities including conducting objects,” IEEE Journal of Microwaves, vol. 5, no. 3, pp. 739–749, 2025

    work page 2025

  73. [73]

    Scattering matrix method for propagation of radiation in stratified media: attenuated total reflection studies of liquid crystals,

    D. Y . K. Ko and J. R. Sambles, “Scattering matrix method for propagation of radiation in stratified media: attenuated total reflection studies of liquid crystals,”J. Opt. Soc. Am. A, vol. 5, no. 11, pp. 1863–1866, Nov 1988. [Online]. Available: https: //opg.optica.org/josaa/abstract.cfm?URI=josaa-5-11-1863

    work page 1988

  74. [74]

    Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,

    L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,”J. Opt. Soc. Am. A, vol. 13, no. 5, pp. 1024–1035, May 1996. [Online]. Available: https://opg.optica.org/josaa/abstract.cfm?URI=josaa-13-5-1024

    work page 1996

  75. [75]

    Memory sparing, fast scattering formalism for rigorous diffraction modeling,

    W. Iff, T. K ¨ampfe, Y . Jourlin, and A. V . Tishchenko, “Memory sparing, fast scattering formalism for rigorous diffraction modeling,”Journal of Optics, vol. 19, no. 7, p. 075602, jun 2017. [Online]. Available: https://doi.org/10.1088/2040-8986/aa7012

    work page doi:10.1088/2040-8986/aa7012 2017

  76. [76]

    New fast and memory- sparing method for rigorous electromagnetic analysis of 2D periodic dielectric structures,

    A. A. Shcherbakov and A. V . Tishchenko, “New fast and memory- sparing method for rigorous electromagnetic analysis of 2D periodic dielectric structures,”Journal of Quantitative Spectroscopy and Radia- tive Transfer, vol. 113, no. 2, pp. 158–171, 2012. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0022407311003542

    work page 2012

  77. [77]

    First-order rooftop functions for the current discretisation in the method of moments solution of planar structures,

    J. Sercu, N. Fache, and P. Lagasse, “First-order rooftop functions for the current discretisation in the method of moments solution of planar structures,” inProceedings of IEEE Antennas and Propagation Society International Symposium and URSI National Radio Science Meeting, vol. 3, 1994, pp. 2170–2173 vol.3

    work page 1994

  78. [78]

    J. L. V olakis and K. Sertel,Integral Equation Methods for Electromagnetics. Institution of Engineering and Technology, Jan

    work page

  79. [79]

    Available: http://dx.doi.org/10.1049/SBEW045E

    [Online]. Available: http://dx.doi.org/10.1049/SBEW045E

    work page doi:10.1049/sbew045e

  80. [80]

    Marcuvitz,Waveguide Handbook

    N. Marcuvitz,Waveguide Handbook. The Institution of Engineering and Technology, 1986. [Online]. Available: https://digital-library.theiet. org/doi/abs/10.1049/PBEW021E

    work page doi:10.1049/pbew021e 1986

Showing first 80 references.