P og ess In Elec omagne ics Resea ch, Vol. 105, 31–48, 2010
DESIGN OF A BAND-PASS FILTER USING STEPPED
IMPEDANCE RESONATORS WITH FLOATING CON-
DUCTORS
M. D. C. Vel´azquez-Ahumada
Depa men o Elec onics and Elec omagne ism
Uni e si y o Se ille
A . Reina Me cedes s/n, Se ille 41012, Spain
J. Ma el
Depa men o Applied Physics 2
Uni e si y o Se ille
A . Reina Me cedes s/n, Se ille 41012, Spain
F. Medina
Depa men o Elec onics and Elec omagne ism
Uni e si y o Se ille
A . Reina Me cedes s/n, Se ille 41012, Spain
F. Mesa
Depa men o Applied Physics 1
Uni e si y o Se ille
A . Reina Me cedes s/n, Se ille 41012, Spain
Abs ac —A new ype o minia u ized s epped impedance esona o
(SIR) o bandpass il e applica ions is p oposed in his pape . The
new esona o inco po a es a g ound plane window wi h a loa ing
conduc o in he backside o he subs a e. The g ound plane window
inc ease he cha ac e is ic impedance o he lines used o implemen
he induc i e egion o he quasi-lumped esona o , hus allowing some
size educ ion. Mo eo e , he p esence o a loa ing conduc ing pa ch
p in ed below he capaci i e egion o he esona o pushes up he
i s spu ious band o he il e . A meaning ul imp o emen o i s
ou -o -band ejec ion le el is hen achie ed. The coupling be ween
adjacen esona o s is also enhanced hus leading o wide achie able
Co esponding au ho : F. Medina ([email p o ec ed]).
32 Vel´azquez-Ahumada e al.
bandwid hs. Some il e designs using he new esona o and o he
s anda d esona o s a e included o compa ison pu poses.
1. INTRODUCTION
Minia u iza ion o p in ed plana mic owa e il e s has been a popula
esea ch opic in ecen yea s owing o he g owing expansion o
wi eless and mobile communica ion sys ems ope a ing a he lowe
end o he mic owa e spec um. Recen examples o his kind o
esea ch can be ound in [1–7]. Con en ional dis ibu ed esona o s
a e oo la ge o be used a low equencies when minia u iza ion is
a key issue. Thus, hese esona o s ha e been p og essi ely eplaced
by smalle s uc u es. Hal -wa eleng h esona o s o iginally used o
implemen pa allel-coupled line il e s [8] we e subs i u ed by hai pin
esona o s such as hose epo ed in [9]. A mo e compac e sion
o he hai pin esona o is he squa e open-loop esona o [10], bu
his esona o s ill wo ks as a dis ibu ed de ice ha yields oo la ge
layou s a low equencies. An addi ional d awback o dis ibu ed
esona o s is he deg ada ion o he ou -o -band ejec ion le el because
o he exis ence o undesi ed spu ious ha monics. This p oblem
can be o e come by using compac quasi-lumped esona o s a he
p ice o inc easing losses. Fo ins ance, he olded hai pin esona o
in [11] is much smalle han he con en ional hal emai pin o open-
loop esona o s owing o inc eased capaci ance associa ed o wo
closely coupled sec ions. A simila e ec can be a ained using he
minia u ized open-loop esona o s p oposed in [12], la e modi ied by
o he au ho s [13] and now called olded s epped impedance esona o s
( olded SIR) [14]. This esona o can be iewed as a high cha ac e is ic
impedance ansmission line loaded wi h a low impedance coupled
line pai . Fo ela i ely high alues o he capaci ance o he wide
coupled lines sec ion and o he lowes esonance equency, he
s uc u e beha es as an LC quasi-lumped esona o . In such case, he
induc ance, L, is p o ided by he sho sec ion o high cha ac e is ic
impedance mic os ip line and he capaci ance, C, is associa ed wi h
he elec ically sho pai o coupled lines. The quasi-lumped na u e
o he esona o au oma ically alle ia es he p oblem associa ed wi h
he p esence o he i s spu ious ansmission band [15]. Mo e
sophis ica ed e sions o SIR’s ha e been applied o il e design looking
o compac ness and wide uppe s opband beha io [16].
Recen ly, a numbe o compac p in ed esona o s ha e p o i ed
om he pa e ning o he backside me alliza ion o he subs a e [17–
20] in o de o enhance hei pe o mance. These a e known as de ec ed
P og ess In Elec omagne ics Resea ch, Vol. 105, 2010 33
(a) (b)
A
A’
l4
l3
l2
l1
w0
w
l
s
s0
s
Figu e 1. Folded s epped impedance esona o wi h g ound plane
window and loa ing conduc o . (a) Top iew; (b) bo om iew. To
cla i y he window posi ion in he bo om iew, he con ou o he
uppe side esona o has been d awn using dashed lines.
g ound s uc u es (DGS). Following ha esea ch line, in his pape
we p opose he new esona o shown in Fig. 1. This esona o , a
olded SIR wi h loa ing conduc o , can be conside ed a modi ied
e sion o he esona o s p oposed in [12, 14]. The g ound plane
a ea below he esona o is pa ially emo ed, in such a way ha a
me allic loa ing ec angula pa ch emains inside he g ound window
and below he coupled lines sec ion. The p esence o he window
inc eases he cha ac e is ic impedance o he single s ip sec ion [21–
23] hus p o iding some size educ ion. Bu wha is mo e impo an
in he con ex o his pape , he loa ing conduc o p o ides a way o
imp o e he ou -o -band ejec ion le el o il e s based on his s uc u e.
This is a e y impo an issue o a oid in e modula ion dis o ion [24].
Roughly speaking, he i s esonance equency associa ed wi h he
desi ed pass band is close o he one o he con en ional olded SIR,
while he second esonance equency meaning ully pushed up. This
ac allows a be e ejec ion le el abo e he i s band. As addi ional
bene i s, he g ound plane window opens a new coupling mechanism
imp o ing he coupling ac o be ween neighbo esona o s [25] hus
making possible wide bandwid hs. In b ie , he new esona o allows
us o buil il e s sligh ly smalle han he ones based on con en ional
olded SIRs wi h la ge achie able bandwid hs and be e ejec ion
le el abo e he i s pass band.
34 Vel´azquez-Ahumada e al.
coupled lines
single line
Zse
s
,ε
Zee
e
,ε
Zoe
o
,ε
AA’
Figu e 2. T ansmission line ci cui model o any olded SIR. The
model is alid o con en ional olded SIR and app oxima ely accoun s
o he beha io o he olded SIR wi h loa ing conduc o .
2. CHARACTERIZATION OF SINGLE RESONATORS
2.1. T ansmission Line Ci cui Model
Fig. 2 shows a ansmission line ci cui model ha easonably
accoun s o he elec omagne ic beha io o olded SIR’s. Fo he
olded SIR wi h loa ing conduc o s udied in his pape his is jus
an app oxima e model because he coupled sec ion in ol es h ee
conduc o s plus g ound. Th ee, ins ead o wo, quasi-TEM modes
a e hen in ol ed. Howe e , i has been shown in he pas ha he
e en/odd mode heo y easonably applies o his case [26, 27]. The
cha ac e is ic impedance and he e ec i e dielec ic cons an o he
single ansmission line sec ion, p o iding he induc i e con ibu ion,
a e Zsand εs
e espec i ely. The capaci i e pa o he esona o can be
iewed as a coupled pai o mic os ip lines whose elec ical pa ame e s
a e he e en (e) and odd (o) mode cha ac e is ic impedances, Zeand
Zo, and he modal e ec i e pe mi i i ies, εe
e and εo
e . No e ha his
is he same ci cui p oposed in [11] o a olded hai pin esona o . The
symme y o he s uc u e wi h espec o he AA0plane in Fig.1 allows
us o conside sepa a e e en and odd mode exci a ions [11, 28]. Two
P og ess In Elec omagne ics Resea ch, Vol. 105, 2010 35
independen esonance condi ions a e hen ob ained:
an θs
2 an θo=Rzo (odd esonances) (1)
an θe=−Rze an θs
2(e en esonances) (2)
whe e Rzo =Zo/Zs,RzeCon =Ze/Zs,θsis he elec ical leng h o he
single line sec ion, and θe,o a e he elec ical leng hs o he e en (odd)
modes o he coupled lines sec ion. The i s esonance equency, 0,
co esponds o he i s odd mode esonance. This is he undamen al
equency de e mining he ope a ion band o any il e based on hese
esona o s. The i s spu ious esonance equency, 1, co esponds o
he i s e en mode esonance.
2.2. Minimum Size Condi ions
The deg ee o compac ness o olded SIR’s can be desc ibed ollowing
he me hod in [28], whe e he physical leng h o he esona o is
compa ed wi h he leng h o an open-loop esona o ha ing he same
undamen al equency. This analysis yields he ollowing op imal
alues o he elec ical leng hs o he single and coupled line sec ions
Figu e 3. No malized elec ical leng h a he i s spu ious esonance
equency ( 1) o SIR’s e sus θ1
susing he a io Rze as a pa ame e .

36 Vel´azquez-Ahumada e al.
(Rzo <1 is assumed):
θ0
s= 2 an−1sRzo(RzoPs−Po)
RzoPo−Ps
(3)
θ0
o= an−1sRzo(RzoPo−Ps)
RzoPs−Po
(4)
whe e Ps=qε e
e /εs
e and Po=qε e
e /εo
e ,ε e
e being he e ec i e
pe mi i i y o he ansmission line used o implemen he open-loop
esona o (no e ha i Ps=1 we eco e he esul s in [29] and i
Ps=Po=1 we a e in he case ea ed in [30]). In b ie , we conclude
ha , o minia u iza ion pu poses, he SIR mus be designed so as o
ge alues o Rzo,Psand Poas small as possible whe eas θ0
sand θ0
o
ul ill he minimum condi ions in (3) and (4).
2.3. Pushing up he Fi s Spu ious Resonance
I is ou goal o maximize he a io 1/ 0. In o he wo ds, he elec ical
leng h o he SIR a 1should be as la ge as possible when compa ed
wi h he elec ical leng h o ou e e ence hal -wa e esona o a 0
(i.e., π). Le us de ine he no malized elec ical leng h a he i s
spu ious esonance o a SIR as θN1 = (θ1
s+ 2θ1
e)/π. The elec ical
leng hs θ1
sand θ1
ea e, espec i ely, he leng hs o he single line sec ion
and he e en mode elec ical leng h o he coupled sec ion o he SIR
a 1. In Fig. 3, we plo he alues o θN1 as a unc ion o θ1
susing
he a io Rze as a pa ame e . The case Rze = 1 co esponds o he
hal -wa e open-loop esona o and, ob iously, θN1 = 2 in such case. I
Rze <1θN1 eaches a maximum when an(θ1
s/2) = 1/√Rze (in ha
case θ1
e=θ1
s/2 + π/2) and a minimum when an(θ1
s/2) = −1/√Rze (in
ha case θ1
e=θ1
s/2−π/2). On he con a y, i Rze >1, θN1 eaches a
minimum when an(θ1
s/2) = 1/√Rze (in ha case θ1
e=θ1
s/2+π/2) and
a maximum when an(θ1
s/2) = −1/√Rzo (in ha case θ1
e=θ1
s/2−π/2).
This discussion is no comple ely equi alen o he simila analysis
ca ied ou in [28] o olded hai pin esona o s because in [28] only
he ange π < θ1
s<2π(o θ1
s/2> θ1
e) was conside ed. Thus he
au ho s o [28] conclude ha , o inc ease he i s spu ious equency, i
is necessa y ha Rze >1. Howe e , he e e se condi ion, i.e., Rze <1
migh also lead o inc ease he i s spu ious equency in he ange
0< θ1
s< π (o θ1
s/2< θ1
e). An impo an e e ence conce ning his
discussion, bu ocused on he un olded e sion o he con en ional
SIR, can be ound in [31].
P og ess In Elec omagne ics Resea ch, Vol. 105, 2010 37
l3(mm)
0.51.0 1.52.0 2.53.0 3.54.0 4.55.0 5.56.0
1/ 0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
l =5 mm
l =4 mm
l =3 mm
Con en ional
Figu e 4. Va ia ion o 1/ 0as a unc ion o he leng h o he coupled
lines sec ion (l3). The dashed line co esponds o a con en ional olded
SIR. Solid lines co espond o olded SIR wi h loa ing conduc o s
ha ing di e en alues o l and s= 0.1 mm.
In conclusion, he double goal o op imum minia u iza ion and
maximum a io 1/ 0can be eached p o ided he condi ions discussed
in his sec ion and (3) and (4) a e simul aneously ul illed. Bu his is
no a i ial ask because usually he in ol ed elec ical pa ame e s a e
no independen . Fo a gi en subs a e, e ec i e dielec ic cons an s
and cha ac e is ic impedances a e s ongly in e lea ed. Howe e ,
addi ional lexibili y would be ob ained i he alues o Rzo and Rze
could be sepa a ely con olled. In such case, he designe could
independen ly une, wi hin a ce ain ange o alues, he posi ions
o he undamen al and he i s spu ious esonance equencies. This
is he ad an age p o ided by he s uc u e p oposed in his pape .
In he con en ional olded SIR he elec ic ield in bo h he odd and
e en modes is s ongly con ined inside he dielec ic subs a e and Ze
ends o be simila o Zo. The e o e he minia u iza ion leads o a
lack o con ol o he i s spu ious equency because he alue o
Zecan no be uned independen ly o Zo. This is no he case o he
modi ied olded SIR in Fig. 1. Fo he i s (odd) esonance, he loa ing
conduc ing pa ch beha es, oughly speaking, as a g ounded conduc o .
The pa ame e s o he coupled lines Zoand εo
e a e, app oxima ely,
he same as hose he con en ional olded SIR. The e o e, we would
no expec an impo an modi ica ion o he undamen al esonance
equency wi h espec o ha o he con en ional olded SIR. Howe e ,
o he i s e en esonance, he beha io o he loa ing conduc o is
a away o being a i ual g ound plane. In such case, he elec ic ield
38 Vel´azquez-Ahumada e al.
is no con ined in he subs a e and, e en i Zois chosen o be e y low,
a wide ange o alues o Zeis achie able uning he dimensions o he
loa ing conduc o . Thus, he designe has a highe le el o con ol on
he sepa a ion o he wo i s esonance equencies. This ea u e o
he he s uc u e in Fig. 1 is exploi ed in his pape . In Fig. 4, we plo
he a io 1/ 0as a unc ion o l3 o se e al alues o he pa ame e
l (SIR wi h loa ing conduc o ) and o he con en ional olded SIR.
The new s uc u e allows us o ob ain alues o his a io la ge han
he a io p o ided by con en ional olded SIR.
2.4. Some P ac ical Examples
In o de o check he heo e ical conside a ions abo e, we ha e
nume ically s udied he beha io o se e al elec ical pa ame e s o
con en ional olded SIR’s and he modi ied SIR in his pape . The
esona o s a e supposed o be p in ed on a comme cial subs a e o
ela i e dielec ic pe mi i i y ε = 10.2 and hickness h= 0.254 mm.
This subs a e has been used la e o ob ain expe imen al esul s.
The p in ed ci cui is enclosed in a me allic box whose dimensions
ha e been aken in o accoun in he calcula ions. The physical size
o he induc i e egion has been chosen o be he same o all he
esona o s. The e o e, he dimensions w0= 0.37 mm, l1= 3.075 mm,
l2= 1.68 mm a e he same o all o hem. We also keep cons an
he alue l4= 3.025 mm. Using he as sol e desc ibed in [32] we
can compu e he equi ed elec ical pa ame e s. Fo he con en ional
SIR we ha e Zs= 39.72 Ω, εs
e = 7.06, Zo= 7.69 Ω, εo
e = 8.09,
Ze= 8.93 Ω and εe
e = 9.33. No e ha , as p edic ed, Zeand Zo
a e low, as equi ed o minia u iza ion, bu close o each o he , wha
is no sui able o ge high alues o 1/ 0. Fo he olded SIR wi h
loa ing conduc o Zs= 61.85 Ω, εs
e = 4.97. Fo his s uc u e, he
alues o Zo,εo
e ,Zeand εe
e depend on he dimensions o he loa ing
conduc o , sand l in Fig. 1. Ou compu a ions show ha odd mode
pa ame e s o he coupled sec ion sligh ly depend on sbu a e in a ian
wi h espec o l . Fo alues o s anging om 0.1 mm o 1.0 mm, Zo
a ies be ween 7.7 and 9.2 Ω. These alues a e simila o he alue o Zo
o he con en ional olded SIR. Al hough εs
e is smalle o he newly
p oposed esona o , he ela i ely high alue o Zscompensa es his
e ec and makes 0sligh ly smalle in he case o he SIR wi h loa ing
conduc o . Howe e , i is he e ec on 1wha is mo e impo an . The
alues o Ze o ou esona o quickly inc eases wi h he alue o l and
is no di icul o ob ain alues o his pa ame e 5 o 6 imes highe
han he alue o 8.93 Ω co esponding o he con en ional olded SIR.
P og ess In Elec omagne ics Resea ch, Vol. 105, 2010 39
d(mm)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
d
d
Coupling ac o ,K
Figu e 5. Coupling ac o e sus he dis ance be ween adjacen
esona o s o he di e en esona o geome ies used o design il e
A. Black solid line: olded SIR wi h loa ing conduc o ; g ey solid line:
con en ional olded SIR; black dashed line: open-loop esona o .
3. EXAMPLES OF BANDPASS FILTER DESIGNS
Se e al il e s wi h simila speci ica ions ha e been designed and
ab ica ed using con en ional open-loop, olded SIR esona o s, and
he newly p oposed olded SIR wi h loa ing conduc o .
3.1. Example 1: Th ee Poles Fil e
Le us conside he ollowing specs o il e A: o de N= 3, cen al
equency 0= 900 MHz, ipple = 0.1 dB and ac ional bandwid h
= 4%. The same subs a e conside ed in he p e ious sec ion has
been used. The il e s we e placed inside a me allic enclosu e whose
dimensions we e 40 ×20 mm2. The hicknesses o he ai laye s below
and abo e he subs a e we e chosen o be 2.0 mm. I is wo h
men ioning ha he e ec o he me allic enclosu e on he pass band
and i s spu ious band is ma ginal because he elec omagne ic ield
is highly con ined a ound me allic s ips and slo s. The esponse o
he il e a hose equencies is go e ned by he p in ed me alliza ion
and is no meaning ully in luenced by he enclosu e. No e ha his is
no he case o o he suspended subs a e s uc u es [14], whe e he
enclosu e dimensions a e c i ical o he design. This is a p ac ical
ad an age o ou p oposal since he il e can be in eg a ed as a
pa o a mo e complex sys em wi hou modi ica ions associa ed o
housing. Ne e heless, in en ionally designed small housing can be
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