The Invariance Properties of a Multivalue n-Port in a Linear Embedding

Heuven, J. van; Rozzi, T.

doi:10.1109/tct.1972.1083435

Cited by 18 publications

(6 citation statements)

References 9 publications

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“…At this point, it is important to highlight that both T and L are similar matrices [28], which means that their traces and determinants coincide, and their value is given by…”

Section: A Bianco and Parodi Two Lines Methodsmentioning

confidence: 99%

Parameters Characterization of Dielectric Materials Samples in Microwave and Millimeter-Wave Bands

Pérez-Escribano

Márquez-Segura

2021

IEEE Trans. Microwave Theory Techn.

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In this work, an optimized broadband method using multilayer transmission lines to characterize dielectric permittivity and loss tangent of material samples is presented. For this purpose, a microstrip line loaded with a piece of the selected dielectric to be characterized is used. From two-port measurements, and using different length lines, the propagation constant can be obtained. To minimize random errors and to improve the accuracy, an over determination of the method increasing the number of lines measured and a criteria to choose the optimal line lengths is considered. Firstly, the measurement method itself is applied to uncovered microstrip lines and an accurate model of the substrate is obtained. Secondly, the lines are covered with several materials, made by FDM additive manufacturing technique, such as Acrylonitrile Butadiene Styrene (ABS), Polylactic Acid (PLA), High Impact Polystyrene (HIPS), Thermoplastic Polyurethane (TPU), Copolyester (CPE), FLEX, Polyethylene Terephthalate Glycol (PETG) and Nylon. A model of the transmission line considering the cover is developed and an electromagnetic simulator is used to indirectly determine the cover material electrical parameters. Results show excellent agreement with electromagnetic simulations in the 0.1-to 67-GHz frequency band, so they assess the suitability of the proposed method.

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“…At this point, it is important to highlight that both T and L are similar matrices [28], which means that their traces and determinants coincide, and their value is given by…”

Section: A Bianco and Parodi Two Lines Methodsmentioning

confidence: 99%

Parameters Characterization of Dielectric Materials Samples in Microwave and Millimeter-Wave Bands

Pérez-Escribano

Márquez-Segura

2021

IEEE Trans. Microwave Theory Techn.

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“…Actually, we can just measure the following N

\times

N admittance matrix

Y_{meas}

Y_{meas} = Y_{EE} - Y_{EI} {false[Y_{DUT} + Y_{II} false]}^{- 1} Y_{EI}^{T} .

…”

Section: Identification Methodology Based On De‐embedding Theorymentioning

confidence: 99%

A Methodology for RF modeling of packages with external pin measurements

Ballicchia

Farina

Morini

et al. 2014

Int J RF and Microwave Comp Aid Eng

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This article presents a methodology that allows the determination of the matrix representation of a package. The methodology is based on a set of calibration measurements, picked up at the external pins of the package, of a set of known integrated circuits. The representation of the package can be used both for a correct measurement of the embedded devices and to improve the design of integrated circuits. The effectiveness of the technique is demonstrated by the de-embedding of a packaged passive integrated inductor and the design of a low noise amplifier. The measurement sets were obtained through computer simulations. The effects of added noise on data, like measurement errors, are also investigated, and an approximate methodology, able to reduce these effects, is suggested

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“…Comparing T1 and T2 to (41) and (42) respectively, one might argue that there is an intrinsic relation to another group that contains Z and P as operators acting on 2 x 2 block matrices. In fact, { 1, T1, T2, T3 ) is isomorphic to the group [I, I, P, ZP) , where I is the identity and ZP denotes composition of I and P.…”

Section: L ( Z ) = Jxr'(z)gmentioning

confidence: 99%

“…(2) A reversal of the 2n-port amounts to (41) for scattering matrices, whereas the corresponding relation (40) for transfer matrices is obtained by applying the operator IP.…”

Section: L ( Z ) = Jxr'(z)gmentioning

confidence: 99%

Darlington's theorem and complex normalization

Pauli

1989

Circuit Theory & Apps

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SUMMARYTwo novel proofs are presented to establish the equivalence between complex normalization and Darlington representation of the loads. Schwarz reflection of analytic functions and Hermitian metrics serve in these proofs as natural concepts for the analysis of lossless multiports. By use of these tools, the algebraic structure that underlies the physical notion of losslessness is analysed, and a class of matrix representations of lossless 2n-ports is derived. The fundamental transformations between these matrices are established to facilitate easy conversion of results related to different choices of co-ordinates (voltages/currents, waves, impedances, reflectances). By virtue of this theory, complex normalization is tacitly extended to include not only impedances but also reflectances as references. Explicit expressions for a 2n-port normalized to complex loads are derived. Invariance of the complex normalized scattering matrix in a lossless cascade circuit is shown to be the most comprehensive generalization of the well known invariance of scalar reflection magnitudes. The relation between this viewpoint and the general theory of invariants of linear fractional maps in terms of cross-ratios is pointed out explicitly. The factorization problem, common to complex normalization as well as Darlington representations, is also reviewed, and it is shown that the concept of minimal factorization (well known in linear prediction theory) leads to more satisfactory results than those of established factorization recipes in complex normalization.

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The Invariance Properties of a Multivalue n-Port in a Linear Embedding

Abstract: A set of four n-port networks placed in an identical linear time-invariant m+n-port

Cited by 18 publications

References 9 publications

Parameters Characterization of Dielectric Materials Samples in Microwave and Millimeter-Wave Bands

Parameters Characterization of Dielectric Materials Samples in Microwave and Millimeter-Wave Bands

A Methodology for RF modeling of packages with external pin measurements

Darlington's theorem and complex normalization

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