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A numerical approach for the evaluation of the nonlinear effects on the attenuation constant in high-temperature superconducting transmission lines

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UNIVERSITY OF TRENTO
DIPARTIMENTO DI INGEGNERIA E SCIENZA DELL’INFORMAZIONE
38123 Povo – Trento (Italy), Via Sommarive 14 http://www.disi.unitn.it
A NUMERICAL APPROACH FOR THE EVALUATION OF THE NONLINEAR EFFECTS ON THE ATTENUATION CONSTANT IN HIGH TEMPERATURE SUPERCONDUCTION TRANSMISSION LINES S. Caorsi, M. Donelli, A. Massa, and M. Pastorino December 2001 Technical Report # DISI-11-097
.
1
A numerical approach for the evaluation of the nonlinear effectson the attenuation constant in high temperaturesuperconducting transmission lines.
S. Caorsi
1
Department of Electronics, University of Pavia, Pavia, Italy
M. Donelli, A. Massa, and M. Pastorino
Department of Biophysical and Electronic Engineering, University of Genoa, Genoa, Italy
Abstract.
Superconducting materials exhibit an experimentally veriﬁed nonlinear dependencewith respect to the magnetic ﬁeld. In this paper, this nonlinearity is taken into account in theevaluation of the attenuation constant in propagating structures of pratical usage. Quadratic andcubic nonlinearities are considered and an iterative numerical procedure is applied to calculate theattenuation constant. The nonlinearity in the penetration depth is also considered. In the resultssection, some typicalstructuresare investigated. Inparticular,parallel-planetransmissionlines ﬁlledby dielectric materials, microstrip lines, and striplines are considered. Comparisons with exsistingresults show that this nonlinear behavior cause signiﬁcant changes in the attenuation parameters.
1. Introduction
In the past years there has been a growing interest insuperconducting materials and their applications. In par-ticular, the discoveryof high-
T
c
superconductorsstronglychanged the possibility of using these materials in the de-sign of advanced devices for microwave electronics andother engineering applications [
Van Duzer and Turner,
1981]. Since superconducting materials are character-ized by a very small resistance, the design and realiza-tion of typical propagating structures (e.g, stripline, mi-crostripline, etc.) by using these materials result in agreat reduction of loss; in particular, the attenuation fac-tors turn out to be some degrees lower comparing withthose of normal conductors. For further developmentsin this area, it is necessary to accurately model the elec-tromagneticbehaviourofhightemperaturesuperconduct-ing (
HTS
) materials and, in particular, to devise method-ologies for studing propagating structures which must beable to take into account this behaviour. In this respect, inthe ﬁeld of classical guided propagation, many efﬁcientnumerical approches have been recently proposed. Forexample, a method to calculate the resistance and induc-tance of trasmission lines with rectangular cross sections[
Weeks et al.,
1991] is based on the network theory andpermits to calculate the frequency-dependent resistance
1
AlsoatDepartment of Biophysical andElectronicEngineer-ing, University of Genoa, Genoa Italy
2and inductance per unit length matrices for transmission-line systems consisting of conductors with rectangularcross sections. The above method was extended in or-der to calculate resistance and inductance for a systemof coupled superconducting transmission lines [
Sheen et al.,
1991]. This goal has been accomplished by usingthe constitutive relation between the current density inthe superconducting lines and the electric ﬁeld, modeledby using the two ﬂuid model. Nonlinear propagatingstructure have also widely numerically and experimen-tally studied (the reader can be referred to [
Lee and Itoh
1989;
El-Ghazaly et al.
1992;
Oates et al.,
1991;
Choud-hury et al.
, 1997] and references therein). Although mostof the proposed approaches consider linear propagationonly, nonlinear effects should be taken into account if the propagating structure have to operate under certainconditions [
Van Duzer and Turner,
1981], in particular,when the magnetic ﬁeld is high. The nonlinear effectswere experimentally veriﬁed and further studied in sev-eral works [
Oates et al.,
1991;
Hein et al.,
1997
; Ma and Wolff,
1996
; Talanov et al.,
1999], in particular there hasbeen a debate about the effective degree of nonlinearity,expecially in the light of the recent improved ﬁlm qualityand depending of the patterned and unpatterned charac-ter of the ﬁlm. In several papers, the nonlinear modelwas represented by a quadratic nonlinearity, whereas inother studies, different nonlinear behaviors were foundto be suitable [
Ma and Wolff,
1996
(a
)
; Ma and Wolff,
1996
(b)
], especially in the presence of high magneticﬁelds. A quadratic nonlinearity was also assumed in acomputational approach [
Caorsi et al.,
2001] devised forstudying the interaction between an incident wave and asuperconducting cylinder modeled by a negative permit-tivity. In the same paper, a preliminary result concerningthe guided propagation has been reported with referenceto the same degree of nonlinearity. In the present work,a numerical iterative procedure able to take into accountthe nonlinear effects on systems of multisuperconduct-ing transmission line is presented. Since the main effectrelated to the nonlinearity with respect to the magneticﬁeldisanincreasinginthesurfaceresistanceofthesuper-conductor, the present paper is focused on the evaluationof the attenuation constants of several guided structuresof practical interest, for which the nonlinear behaviorof the superconductors is rigorously taken into accountand modeled by using some results derived from exper-imental data. The mathematical formulation starts fromthe two ﬂuids model and is developed in the framework of the classic electrodynamics [
Mei and Liang,
1991],which allows one to consider a superconducting mate-rial as a material with a complex conductivity
σ
c
. Dif-
3ferent nonlinear behaviors (resulting from experimentalanalyses) can be easily incorporated into the model. Inthe following second- and third-order relations describ-ing the nonlinearity versus the magnetic ﬁeld are usedand the results are compared. Moreover, the effects as-sociated with changes in the penetration depth
λ
, due tothe magnetic ﬁeld, are taken into account, too. Althoughthe rigorous treatment of the nonlinearity would involvethe use of constitutive relationships written in terms of Volterra series [Censor, 1985; Censor, 1987], by using anapproximationsimilar to the one involvedin the so calleddistorted-waveBornapproximation,aniterativeapproachforthe computationoftheattenuationfactoris developed.Finally, someresults areshownconcerningguidingstruc-tures as parallel-plane transmission lines ﬁlled by dielec-tric materials, microstrip lines and striplines.
2. Mathematical Formulation
2.1. Parameters of a multiconductor transmissionline system
Let us considerthepropagatingstructureshownin Fig-ure1, which is constitutedby
M
+
1superconductors,one
Figure 1.
ofthemis usedas referenceandtheothersare usedas sig-nal lines. The cross section of each line is subdividedintosegments and the current ﬂowing into each segment isconsidered to be uniformly distributed over the cross sec-tion of the segment. The generic segment is indicated by
n
, whereas a segment of the reference conductor is cho-sen as reference and indexed as 0. In this segment ﬂowsthe return current, which is the sum of all the currents inall the other
N
segments:
i
tot
=
N
∑
j
=
1
i
j
(1)By takingintoaccountthenonlinearitywithrespecttothemagnetic ﬁeld, a nonlinear complex electric conductivitycan be deﬁned as follows:
σ
nlc
(
H
) =
σ
lin
+
L
(
H
)
(2)where
L
(
H
)
is a nonlinear operator depending on the as-sumed model for the nonlinearity,
σ
lin
is the linear partof
σ
nlc
, which, according to the two ﬂuid model [Mei andLiang, 1991], can be expressed as
σ
lin
=
σ
1
(
T
)
−
j
1
ω
µ
0
λ
2
(
T
)
(3)where
ω
is the angular frequency,
µ
0
is the magnetic per-meability, T is the temperature
(
K
)
, and
λ
is the penetra-

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