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Jou nal o Supe conduc i i y: Inco po a ing No el Magne ism, Vol. 16, No. 5, Oc obe 2003 ( C
°2003)
Expe imen s and Model o In e modula ion Dis o ion
in a Ru ile Resona o wi h Y-Ba2-Cu3-O7−δEndpla es
J. Ma eu,1C. Collado,2O. Men´
endez,2and J. M. O’Callaghan2
Recei ed Sep embe 30, 2002
We ha e de i ed gene al equa ions o calcula e in e modula ion dis o ion in esona o s wi h
high empe a u e supe conduc ing (HTS) ilms which a e no es ic ed o a speci ic esona o
shape and may be used whene e he ields in a esona o gene a e cu en s on he su ace
o one o mo e HTS ilms. These equa ions a e applied o u ile-loaded ca i ies wi h one o
wo 10 ×10 mm2Y-Ba2-Cu3-O7−δendpla es and a e used o ex ac pa ame e s cha ac e izing
he nonlinea i ies o hese ilms om in e modula ion measu emen s. E en hough he ilms
ha e simila small-signal pe o mance, we ha e ound la ge a ia ion in he s eng h o hei
nonlinea i ies.
KEY WORDS: nonlinea measu emen s; dielec ic ca i y; in e modula ion measu emen s; supe con-
duc o s.
1. INTRODUCTION
High Tempe a u e Supe conduc ing (HTS) hin
ilms a e being used in na owband mic owa e il e s
due o hei low losses and small olume. The use
o HTS ma e ials in his applica ion migh be limi ed
by hei nonlinea i ies, which gi e ise o in e modu-
la ion dis o ion (IMD) and o he undesi ed e ec s
[1]. Al hough mos o he expe imen al wo k on HTS
in e modula ion has been done on plana pa e ned
de ices [2–4], his ype o cha ac e iza ion migh no
be ep esen a i e o he p ope ies o he whole HTS
ilm because he nonlinea p ope ies o he es de-
ices a e domina ed by he high cu en densi ies a
he edges o he pa e n, whe e he p ope ies o he
ilm may be a ec ed by he e ching p ocess. In his
wo k we desc ibe a echnique ha migh con ibu e
o o e come hese di icul ies, which is based on he
use o a u ile esona o o make a small-a ea cha -
ac e iza ion o he in e modula ion p ope ies o un-
pa e ned HTS ilms.
1CTTC-Cen e Tecnol `ogic de Telecomunicacions de Ca alunya,
Edi ici NEXUS, C/G an Capi `a 2–4, 08034 Ba celona, Ca alunya,
Spain.
2Uni e si a Poli ecnica de Ca alunya, Campus No d UPC-D3,
C/Jo di Gi ona 1, 08034 Ba celona, Spain.
2. A GENERAL FORMULATION FOR HTS
NONLINEARITIES
We assume ha , a high cu en s, he elec ic ield
a he su ace o he supe conduc o is he sum o a
linea con ibu ion due o he su ace impedance o
he ilm, plus an addi ional elec ic ield −→
eNL caused by
he nonlinea i ies [5, 6] which can be exp essed as
−→
eNL(js)=aNL(js)E
js+∂
∂ £bNL(js)E
js¤, (1)
whe e E
jsis he su ace cu en densi y, aNL(js) ep-
esen s he nonlinea esis i e e m, and bNL (js) he
nonlinea eac i e e m. Bo h e ms depend on he
magni ude o he su ace cu en jsand ha e o be
ze o o js=0 o make −→
eNL anish a a bi a ily small
cu en s.
An al e na i e o m o desc ibing he nonlinea -
i ies in he HTS consis s in quan i ying he change in
su ace impedance as a unc ion o he su ace cu -
en densi y (o magne ic ield), i.e., in equency do-
main we would ha e E
ENL =1Rs(Js)E
Js+j1Xs(Js)E
Js
[3], whe e 1Rs(Js)+j1Xs(Js) is he a ia ion in su -
ace impedance wi h espec he small signal alue.
While inding he equi alence be ween his o mula-
ion and he one in Eq. (1) is s aigh o wa d (see Fig.
873
0896-1107/03/1000-0873/0 C
°2003 Plenum Publishing Co po a ion
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874 Ma eu, Collado, Men´
endez, and O’Callaghan
Fig. 1. Compa ison be ween aNL( )js( ) in ime domain (solid line
in igh mos g aph) and he de ia ion o he su ace esis ance om
i s small-signal alue (1Rsp opo ional o he ampli ude o he
sinusoid d awn wi h do ed line in he igh mos g aph).
1), we p e e ha in Eq. (1) since i gi es all he de ails
o he ime-domain unc ion, whe eas 1Rs,1Xsonly
e e o i s i s ha monic.
Ano he exis ing o mula ion o HTS nonlinea -
i ies ha can be ela ed o ha in Eq. (1) is he one
desc ibed in [7, 8], acco ding o which he pene a ion
dep h depends on he cu en densi y. This o mula-
ion has been ex ensi ely used o jus i y he gene a-
ion o IMD in HTS ilms, and ou p e ious wo k [9],
shows how i can be ela ed o he nonlinea eac i e
e m bNL(js) in Eq. (1). Equa ion (1) is hus a e y
gene al ma hema ical desc ip ion o HTS nonlinea i-
ies ha can be pa icula ized o a ious models and
should accoun o he gene a ion o bo h ha monics
and in emodula ion p oduc s.
Wi h li le loss in gene ali y, in his wo k we
assume ha aNL,(j
s
), bNL,(j
s
) can be exp essed
as aNL(js)=1Rα|js|,bNL(js)=1Lα|js|α, whe e α,
1Rα,1Lαa e pa ame e s ha cha ac e ize he
s eng h o he nonlinea i y in he HTS. Equa ion (1)
will hen ead
−→
eNL =1Rα|js|αE
js+1Lα
d
d ¡|js|αE
js¢.(2)
3. INTERMODULATION IN HTS
RESONATORS
In his sec ion, we will de i e gene al equa ions
o he powe o he hi d-o de in e modula ion
p oduc s coupled ou o a esona o due o nonlin-
ea i ies in he HTS. The analysis applies o any es-
ona o whe e he ield con igu a ion o he esonan
mode is such ha i gene a es cu en s on he su ace
o an HTS ma e ial. The dielec ic-loaded ca i y es-
ona o s used o Rsmeasu emen o HTS ilms a e an
ob ious example, bu he analysis could also be ap-
plied o a disk esona o o o plana , pa e ned HTS
esona o s.
3.1. Elec ic Field due o HTS Nonlinea i ies
In any esona o he cu en a he su ace o he
HTS ilm can be w i en as
E
js(E , )=A( ) (E )ˆu(E ), (3)
we e ˆu(E ) is a uni ec o , (E )ˆu(E ) desc ibes he spa-
ial dependence o he cu en densi y, and A( ) i s
ime dependence. Fo he case o an in e modula ion
expe imen A( ) will consis o he sum o wo sinu-
soidal signals wi h ampli udes j1and j2:
A( )=j1cos ω1 +j2cos ω2 .(4)
Combining Eqs. (2)–(4) we ind ha he elec ic ield
a he su ace o he supe conduc o can be w i en as
−→
eNL(E , )=½1Rα|A( )|αA( )+1Lα
d
d (|A( )|αA( ))¾
×| (E )|
α (E )ˆu(E ).(5)
No e ha he spa ial dependence o −→
eNL ( he e m
| (E )|α (E )ˆu(E ) in Eq. (5)) di e s om ha o he
cu en E
js. F om Eq. (5) we can calcula e he elec ic
ield a 2ω1−ω2:
E
E2ω1−ω2=[1Rα+j(2ω1−ω2)1Lα]2C2ω1−ω2| (E )|α
× (E )ˆu(E ), (6)
whe e 2C2ω1−ω2is he in e modula ion p oduced by
he unc ion |A( )|αA( )a 2ω
1−ω
2
and has he uni s
o (A/m)α+1, i.e.
C2ω1−ω2=lim
T→∞
1
TZT
2
−T
2
|A( )|αA( )e−j(2ω1−ω2) d .(7)
Figu e 2 shows he dependence o he elec ic
ield ampli ude a 2ω1−ω2and a 2ω2−ω1as a unc-
ion o he su ace cu en ampli ude j1(i.e., C2ω1−ω2
and C2ω2−ω1 s. j1). When j1=j2bo h C2ω1−ω2and
C2ω2−ω1a e p opo ional o jα+1
1, so a loga i hmic plo
o C2ω1−ω2 e sus j1o C2ω2−ω1 e sus j1has a slope
o α+1. When j1¿j2,C2ω1−ω2∝j2
1j2and C2ω2−ω1∝
j1j2
2so, i j2is kep cons an , he slopes in Fig. 2 a e 2:1
and 1:1, espec i ely, ega dless o he alue o α.In
he opposi e egime, when j1Àj2,C2ω1−ω2∝jα
1jα−1
2
and C2ω2−ω1∝jα−1
1jα
2so, in a loga i hmic scale, he
magni ude o he elec ic ield a he in e modula ion
equency changes wi h slopes o αand α−1 wi h he
cu en ampli ude j1when j2is ixed. The dependence
o C2ω1−ω2and C2ω2−ω1on j1and j2desc ibed he e ac-
coun s o he expe imen al esul s desc ibed in he
ollowing sec ions o his wo k and o hose epo ed
in [10].
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Expe imen s and Model o In e modula ion Dis o ion in a Ru ile Resona o 875
Fig. 2. Dependence o he elec ic ield a he in e modula ion e-
quencies (2C2ω1−ω2and 2C2ω2−ω1) on he ampli ude o he su ace
cu en j1.
When he wo cu en ampli udes j1,j2a e kep
equal o each o he , he ac ion C2ω1−ω2/jα
1j2only
depends on he o de o he nonlinea i y α. Hence,
ins ead o C2ω1−ω2(j1,j2,α), we will use he pa ame e
T
2ω1−ω2(j1,j2,α)≡2C2ω1−ω2(j1,j2,α)
jα
1j2
(8)
in subsequen equa ions. Figu e 3 shows he depen-
dence o T
2ω1−ω2on α o balanced cu en ampli udes.
In summa y, Eqs. (6)–(8) gi e a gene al se o
equa ions ela ing he IMD elec ic ield a he su -
ace o he supe conduc o wi h he ampli ude o he
su ace cu en densi y. These equa ions a e no e-
Fig. 3. Dependence o T
2ω1−ω2on he o de o he nonlinea i y
α o balanced cu en ampli udes. In his case, he ampli ude o
he elec ic ield a he su ace o he HTS a he in e modula ion
equency will be p opo ional o T
2ω1−ω2(α)jα+1
1
s ic ed o a pa icula shape o esona o and should
be alid whene e aNL(js)=1Rα|js|αand bNL(js)=
1Lα|js|α.
3.2. Resonan IMD Magne ic Field
on he HTS Su ace
F om he discussion abo e, we ha e de e mined
ha he IMD elec ic ield on he HTS su ace will be
o he o m
E
E2ω1−ω2=E2ω1−ω2| (E )|α (E )ˆu(E ), (9)
wi h
E2ω1−ω2=[1Rα+j(2ω1−ω2)1Lα]
×T
2ω1−ω2(j1,j2,α)jα
1j2.(10)
I ω1≈ω2≈2ω1−ω2 his ield will couple o he
same mode a which ω1and ω2 esona e and will gen-
e a e a magne ic ield a 2ω1−ω2whose spa ial dis-
ibu ion has o be he same han ha o E
js(E , )in
Eq. (3), i.e.,
E
H2ω1−ω2=H2ω1−ω2 (E )ˆ (E ), (11)
whe e ˆ (E ) is a uni ec o on he HTS su ace pe -
pendicula o ˆµ(E ). The magne ic ield on he HTS
a 2ω1−ω2is hus ully known i we can de e -
mine i s ampli ude H2ω1−ω2. To do ha we conside
he powe gene a ed a he su ace o he HTS a
2ω1−ω2:
P=1
2ZS
E
E2ω1−ω2×E
H∗
2ω1−ω2dE
S(12)
and ela e i o he esona o Q. This powe will be
ei he dissipa ed inside he esona o o coupled ou -
wa ds, so i has o sa is y
P=(2ω1−ω2)W
QL
, (13)
whe e Wis he ene gy s o ed in he esona o a
2ω1−ω2and QLis he loaded quali y ac o . I we
no malize his ene gy wi h espec o he ield ampli-
ude |H2ω1−ω2|2we ge W0=W/|E
H2ω1−ω2|2and
P=1
2ZS
E
E2ω1−ω2×E
H∗
2ω1−ω2dE
S
=(2ω1−ω2)W0
QL
H∗
2ω1−ω2H2ω1−ω2.(14)
Using Eqs. (9)–(11) in Eq. (14) we ind H2ω1−ω2:
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876 Ma eu, Collado, Men´
endez, and O’Callaghan
E
H2ω1−ω2(E )=1
2[1Rα+j(2ω1−ω2)1Lα]
T
2ω1−ω2(j1,j2,α)jα
1j2
QL
(2ω1−ω2)W0
ZS
{| (E )|α (E ) (E )dS} (E )E .(15)
No e ha he e m 1
2RS| (E )|α (E ) (E )dS, is a so o
coupling coe icien be ween he spa ial dis ibu ion
o he elec ic ield on HTS (| (E )|α (E )) and ha o
he magne ic ield o he esonan mode in he s uc-
u e ( (E )). I we exp ess his e m as 0α, Eq. (15) can
be w i en as
E
H2ω1−ω2(E )=[1Rα+j(2ω1−ω2)1Lα]
T
2ω1−ω2(j1,j2,α)jα
1j2
0α
QL
(2ω1−ω2)W0
(E )E .(16)
No e ha we a e assuming ha he in e modu-
la ion p oduc s a e only gene a ed by he mixing o
he undamen al signals. This means ha we a e no
conside ing highe o de e ec s ha could con ibu e
o E
H2ω1−ω2like o example he mixing o highe o de
spu ious signals.
3.3. IMD Powe Coupled Ou o he Resona o
In his sec ion we calcula e he powe o he in-
e modula ion p oduc coupled ou o he s uc u e
as a unc ion o he a ailable powe o he sou ces a
undamen al equencies ω1and ω2(P0,ω1and P0,ω2,
espec i ely) and he ex e nal coupling ac o s.
We i s analyze he case o a one-po esona o .
In his esona o he coupling ac o κis he a io o
he powe coupled o he load (PL) o he powe dis-
sipa ed in he esona o (Pd) [11], so
PL=κPd=κ(2ω1−ω2)W
Q0
=κ(2ω1−ω2)W0
Q0
|H2ω1−ω2|2, (17)
whe e |H2ω1−ω2|is gi en by Eq. (16).
We now ha e o ela e j1and j2in Eq. (16) wi h
he powe a ailable om he sou ces a ω1and ω2
(P0,ω1and P0,ω2). To do his we use he equa ion ha
ela es dissipa ed powe (Pd) wi h a ailable powe
(P0) in a one-po esona o :
Pd=4P0κ
(1 +κ)2(18)
so, o ω1
4P0,ω1κ
(1 +κ)2=ω1W0
Q0
|j1|2(19)
and simila ly o ω2.
Using Eqs. (16), (18), and (19) o ω1and ω2, and
assuming ω1≈ω2≈2ω1−ω2≈ω0,wege
P
L=(4P0,ω1)α(4P0,ω2)µQ0
ω0W0¶α+2·κ
(1 +κ)2¸(α+2)
|[1Rα+j(2ω1−ω2)1Lα]T
2ω1−ω2(j1,j2,α)0α|2.
(20)
Fo a wo-po esona o we use he inpu and ou pu
coe icien κin and κou [11]. In his case he powe
coupled o he load is
PL=κou Pd=κou
(2ω1−ω2)W0
Q0
|H2ω1−ω2|2(21)
and, simila ly o Eq. (19), j1,j2can be ela ed o he
powe a ailable om he sou ces h ough
4P0,ω1κin
(1 +κin +κou )2=ω1W0
Q0
|j1|2(22)
and he esul ing equa ion o PLis
PL=(4P0,ω1)α(4P0,ω2)µQ0
ω0W0¶α+2
κou κα+1
in
(1 +κou +κin)2(α+2)
|[1Rα+j(2ω1−ω2)1Lα]T
2ω1−ω2(j1,j2,α)0α|2.
(23)
Equa ions (20) and (23) a e he gene al equa ions ha
ela e he measu ed in e modula ion powe wi h he
pa ame e s o he ilm 1Rα,1Lα, and α. In hese
equa ions 1Rαand 1Lαonly a ec he le el o he
in e modula ion p oduc s, bu no he slope o hei
a ia ion wi h he sou ce powe . On he o he hand, α
a ec s bo h he IMD le el and he slope. This can also
be seen om he equa ions o he in e modula ion
elec ic ield (Eqs. (4), (6), and (7)), and om hose
o he in e modula ion magne ic ield (Eqs. (8) and
(16)).
No e ha we a e assuming ha he su ace cu -
en ampli udes j1,j2in Eq. (16) (o , equi alen ly,
he angen ial componen o he magne ic ields on
he HTS su ace Hω1,Hω2) a e iden ical o hose ha
would be on he HTS i i was comple ely linea (see
Eqs. (19) and (22)). In o he wo ds, we do no ake
in oaccoun hecomp ession e ec ha occu sa high
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Expe imen s and Model o In e modula ion Dis o ion in a Ru ile Resona o 877
powe le els, when a signi ican ac ion o he powe
ha is injec ed o he esona o a ω1and ω2is ans-
e ed o o he equencies (2ω1−ω2,2ω
2−ω
1
, and
o he s) by he e ec o he nonlinea i ies, he eby e-
ducing he ampli ude o he ields and cu en s a ω1
and ω2.
4. ANALYSIS OF A RUTILE-LOADED CAVITY
In his sec ion we will apply he gene al analy-
sis desc ibed in he p eceding one o he dielec ic-
loaded esona o ca i y o Fig. 4. We will analyze a
TE011 u ile esona o simila o he one desc ibed in
[12], bu he analysis may also be applied o simila
esona o s wi h o he dielec ic ma e ials (like sap-
phi e) which a e commonly used in mic owa e HTS
cha ac e iza ion. We ha e chosen o use a u ile di-
elec ic because i a oids high inging ields in he
no mal-me al housing o he esona o , hus limi -
ing he e ec o housing loss in he o e all Qo he
esona o despi e he small dimensions o he ca i y.
The TE011 mode used has he ad an age o a oiding
“w ap a ound” e ec s which—as desc ibed in [13]—
enhance he nonlinea i ies in he de ice and make
he IMD e y s ongly dependen on he p ope ies
o na ow a eas o he HTS ilm. In hese condi ions,
he cu en densi y on he HTS endpla e in an in e -
modula ion expe imen can be w i en as
E
js(ρ, )=(j1cos ω1 +j2cos ω2 ) (ρ)ˆ
φ, (24)
whe e ˆ
φis he uni ec o in he azimu h di ec ion
and (ρ) desc ibes he adial dependence o he TE011
mode which, acco ding o [14], is
(ρ)=
β
ξ1
J1(ξ1ρ)ρ≤a
β
ξ2
J0(ξ1ρ)
F0(ξ2ρ)F1(ξ2ρ)b>ρ>a
, (25)
whe e βis he z-di ec ion p opaga ion cons an ,
ξ1and ξ2a e he ρ-di ec ion wa e numbe s
(inside and ou side he dielec ic, espec i
ely), F0(ξ2ρ)=I0(ξ2ρ)+K0(ξ2ρ)I1(ξ2b)/K1(ξ2b),
F1(ξ2ρ)=−I
1
(ξ
2
ρ)+K
1
(ξ
2
ρ)I1(ξ2b)/K1(ξ2ρ) being
J0,I1,I2,K0,K1 he co esponding Bessel and Han-
kel unc ions. F om his cu en dis ibu ion and
analyzing he ca i y as desc ibed in he p e ious
sec ion, we ob ain
E
H2ω1−ω2(ρ)=[1Rα+j(2ω1−ω2)1Lα]T
2ω1−ω2
(j1,j2,α)jα
1j20α
QL
(2ω1−ω2)W0
(ρ)ˆρ,
(26)
whe e
0α=π"¯¯¯¯
β
ξ1¯¯¯¯
α+2
IJ2+α+¯¯¯¯
β
ξ2
J0(aξ1)
F0(aξ2)¯¯¯¯
α+2
IF2+α#
(27)
and IJ
2+α=Rα
0|J
1(ξ
1ρ)|
2+αρdρ,IF
2+α=Rb
a|F
1
(ξ
2ρ)|2+αρdρ. The esul ing powe coupled ou he
ca i y a 2ω1−ω2is ob ained om Eq. (20).
Once we know he E
H2ω1−ω2gene a ed by one
HTS endpla e, we can easily ex end ou s udy o a
ca i y ha ing wo nonlinea endpla es. In his case,
he in eg al in Eq. (14) has o be done o e he wo
HTS endpla es. As a esul he alue o H2ω1−ω2is
he sum o he con ibu ion o each endpla e, and
can be calcula ed by applying Eq. (26) o each o
hem.
5. MEASUREMENTS AND DATA FITTING
Measu emen s we e made wi h he esona o
shown in Fig. 4, which has a 3-mm high u ile od
(4-mm diame e ) o ε ∼110 a 77 k, sandwiched be-
ween wo 10 ×10 mm2endpla es. The esonan e-
quency o he TE011 mode is close o 8 GHz.
To make he IMD measu emen s, he inpu po
is ed wi h wo signals a equencies 1and 2, bo h
wi hin he passband o he esona o . The powe o
hese sou ces is ei he a ied simul aneously (bal-
anced measu emen s), o is changed in only one o
he sou ces, lea ing he second one powe cons an
(unbalanced measu emen s). Coupling is made by an
adjus able coupling loop. The in e modula ion p od-
uc s a e measu ed wi h a one-po se up (see Fig. 5)
in which a spec um analyze (SA) is used o mea-
su e he signals going ou o he esona o . The SA
is kep isola ed om he gene a o s p oducing he
signals a 1and 2so ha , i he esona o is c i i-
cally coupled, he powe o he signals a 1and 2
Fig.4. Dielec ic esona o s uc u eused o IMDcha ac e iza ion
o HTS ilms.
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878 Ma eu, Collado, Men´
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Fig. 5. Block diag am o he one po measu emen se up.
eaching he SA is e y small, and he IMD o he SA
does no al e he measu emen s o he powe o he
in e modula ion p oduc s a 2 1− 2and 2 2− 1.
Fu he mo e, in his a angemen he spec al pu i y
o he sou ces is no as c i ical as i is in ansmission
measu emen s, whe e he in e modula ion p oduc s
a e easily masked by he phase noise o he sou ces i
| 1− 2|is no su icien ly la ge.
All he measu emen s made o cha ac e iza ion
we e done wi h he esona o imme sed in liquid ni-
ogen. Howe e , compa a i e measu emen s we e
also made in a closed-cycle c yos a o ensu e ha
he mal e ec s we e no dominan in he gene a ion
o in e modula ion p oduc s. In hese measu emen s,
i he powe applied o he esona o was su icien ly
la ge, we obse ed a ia ions o he esonan e-
quency which we a ibu ed o hea ing e ec s on he
u ile [12]. This a ia ions we e small i he sou ce
powe was su icien ly low. In hese condi ions he
IMD p oduc s measu ed compa ed well (wi hin 0.5
dB) wi h hose made in liquid ni ogen imme sion.
A second echnique o check o he mal e ec s
is o change he equency o he en elope o he sig-
nal coupled in o he esona o (| 1− 2|/2). We ha e
pe o med such measu emen s by changing | 1− 2|
be ween 1 and 40 KHz, and ound consis en esul s
(i.e., a ia ions wi h | 1− 2|, wi hin a 0.5 dB).
Fou di e en endpla es ha e been used in
hese measu emen s. Th ee o hem (samples HTS1,
HTS2, and HTS3) a e 700-nm hick Y-Ba2-Cu3-O7−δ
(YBCO) ilms g own on 0.5-mm MgO subs a es by a
comme cial supplie (The a), and a ou h one made
o coppe . The HTS ilms we e used as deli e ed by
he supplie , wi h no u he p ocessing on ou pa .
We made a ound- obin se ies o small signal Qmea-
su emen s using all he HTS endpla es o ule ou a
la ge sp ead in Rsamong he ilms. The a e age Rsa
77 K was 0.3 mÄand we can gua an ee ha he sp ead
is less han 25%, bu ou capabili y o assu e his is lim-
i ed by he loss o he u ile, so he eal sp ead in Rs
migh belowe .Nonlinea measu emen swhe emade
wi h he combina ion o endpla es shown in Table I.
Table I. Endpla es Used in he Nonlinea
Resona o Measu emen s (T=77 K)
Se Films QLQ0
1 HTS1–Cu 8000 16000
2 HTS2–Cu 6500 16000
3 HTS1–HTS217500 80000
4 HTS3–Cu 8000 16000
5 HTS2–HTS316000 77000
Figu e 6 shows he esul s o he IMD measu e-
men s o se s 1 and 2, which e eal e y s ong di e -
ences in he in e modula ion pe o mance o samples
HTS1and HTS2. Fo example, he in e modula ion
p oduc s ob ained om balanced measu emen s a 0
dBm in se 1 a e 20 dB highe han hose o se 2,
which canno be jus i ied by he di e ence in QL.
By applying Eq. (20) o se 1 and se 2 and
assuming ha esis i e nonlinea i ies a e negligi-
ble [2,4] we ha e ex ac ed he nonlinea pa am-
e e s o HTS1and HTS2. The esul ing alues
a e α1=0.2, 1Lα1=5×10−16 Hm0.2/A
0.2and α2=
1, 1Lα2=3.5×10−19 Hm/A, espec i ely. Figu e 7
shows he esul ing bNL(js) o each o he wo se s in
he ange o cu en densi ies used in ou expe imen s.
Fig. 6. Resul s o measu emen s and i ings (lines) o se 1 (a) and
se 2 (b). Ci cles: measu ed powe o in e modula ion p oduc s a
2ω1−ω2as a unc ion o he sou ce powe when he powe o
bo h sou ces was inc eased simul aneously. Squa es and diamonds
ep esen he measu ed powe o he in e modula ion p oduc s a
2ω1−ω2and 2ω2−ω1, espec i ely, as a unc ion o he sou ce
powe a ω1when he powe o he sou ce a ω2was kep cons an
a −2 dBm in se 1 and 4 dBm in se 2. The di e ence be ween he
wo sou ce equencies is 20 KHz.
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Expe imen s and Model o In e modula ion Dis o ion in a Ru ile Resona o 879
Fig. 7. Plo o bNL(js) o se 1 and se 2 o he ange o sou ce
powe used in he measu emen s.
Once HTS1and HTS2a e cha ac e ized we ha e
ex ended ou expe imen s o a esona o ha ing bo h
HTS1and HTS2as endpla es (se 3 in Table I).
Figu e 8 shows he esul s o he balanced and unbal-
anced IMD measu emen s made wi h his se e sus
he heo e ical esul s p edic ed using he alues o α
and 1Lα ound abo e, om se s 1 and 2.
The IMD o se 4 (HTS3and he Cu endpla e)
was below he sensi i i y o ou measu emen se up
(-95 dBm o an a ailable powe o 10 dBm a he es-
ona o po ). Coupling in his se was adjus ed o ob-
ain he same ci cula ing powe han ha in se 1 and
se 2. This indica es ha he HTS nonlinea i ies o he
h ee samples a e e y di e en , and ha he e is no
Fig. 8. Resul s o measu emen s and calcula ions o se 3. Ci -
cles: measu ed powe o in e modula ion p oduc s a 2ω1−ω2as
a unc ion o he sou ce powe when he powe o bo h sou ces
was inc eased simul aneously. Squa es and diamonds ep esen he
measu ed powe o he in e modula ion p oduc s a 2ω1−ω2and
2ω2−ω1, espec i ely, as a unc ion o he sou ce powe a ω1when
he powe o he sou ce a ω2was kep cons an a 3 dBm. The lines
in he igu e a e esul s o calcula ions pe o med wi h pa ame-
e s ob ained om se 1 and se 2. The di e ence be ween he wo
sou ce equencies is 20 KHz.
con ibu ion o he measu emen se up o he IMD
measu emen s o se s 1 and 2. Measu able le els o
IMD we e ob ained by combining HTS3and HTS2in
se 5, and we ob ained esul s ha a e consis en wi h
he o he se s (la ge IMD han se 2). Cha ac e i-
za ion o he nonlinea i ies in sample HTS3 equi es
imp o emen s in ou measu emen se up ha we a e
cu en ly add essing.
6. CONCLUSIONS
We ha e de eloped a gene al echnique o cal-
cula e in e modula ion dis o ion in esona o s wi h
HTS ma e ials. We ha e applied his echnique o a u-
ile esona o o use i o cha ac e iza ion o he non-
linea p ope ies o unpa e ned 10 ×10 mm2HTS
ilms. We ha e ound ha h ee s a e-o - he-a HTS
samples can ha e widely di e en IMD pe o mance,
while ha ing a mode a e sp ead in Rs. We encou age
o he g oups o y his echnique so ha da a can
be accumula ed o a la ge numbe o HTS samples,
and o e ou collabo a ion o supply u he de ails
on he esona o and measu emen se up.
Possible imp o emen s in his echnique a e e-
la ed o he possibili y o ex ending he alidi y o he
ma hema ical o mula ion used o he HTS nonlin-
ea i ies o i he wide ange o expe imen al esul s
ound in he li e a u e. The o mula ion in his wo k
is based on he assump ion ha he same powe law
can be applied o he esis i e and eac i e nonlin-
ea i ies. While his is a se e e es ic ion, i allows o
use ou echnique in he nume ous wo ks ha claim
ha , a low powe s, he in e modula ion p oduc s in
HTS de ices a e due o a cu en dependence o he
pene a ion dep h, since his makes he eac i e non-
linea i ies dominan [7,8].
We also no e ha , while i may be possible o ex-
end he ma hema ical o mula ion in his wo k o
conside nonlinea unc ions o he han he powe
law (aNL(js)=1Rα|Js|α,bNL(js)=1Lα|js|α), and al-
low di e en ypes o nonlinea unc ions in he esis-
i e and eac i e nonlinea i ies (aNL(js) and bNL(js)),
hese ex ended abili ies can al eady be achie ed wi h
nume ical calcula ions as desc ibed in [15].
ACKNOWLEDGMENTS
This wo k has been unded by Spanish Min-
is y o Educa ion and Cul u e h ough P ojec
No. MAT2002–04551–C03–03 and Schola ship AP99
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880 Ma eu, Collado, Men´
endez, and O’Callaghan
78085980, and by he Gene ali a de Ca alunya
h ough Schola ship 2002 FI 00622.
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