Recei ed Ap il 9, 2019, accep ed May 21, 2019, da e o publica ion June 14, 2019, da e o cu en e sion July 2, 2019.
Digi al Objec Iden i ie 10.1109/ACCESS.2019.2923166
Syn hesis and Op imiza ion o F ac ional-O de
Elemen s Using a Gene ic Algo i hm
ASLIHAN KARTCI 1,2, (S uden Membe , IEEE), AGAMYRAT AGAMBAYEV 3,
MOHAMED FARHAT3, NORBERT HERENCSAR 2, (Senio Membe , IEEE),
LUBOMIR BRANCIK1, (Senio Membe , IEEE), HAKAN BAGCI 3, (Senio Membe , IEEE),
AND KHALED N. SALAMA3, (Senio Membe , IEEE)
1Depa men o Radio Elec onics, B no Uni e si y o Technology, 616 00 B no, Czech Republic
2Depa men o Telecommunica ions, B no Uni e si y o Technology, 616 00 B no, Czech Republic
3Compu e , Elec ical and Ma hema ical Sciences & Enginee ing Di ision, King Abdullah Uni e si y o Science and Technology, Thuwal 23955, Saudi A abia
Co esponding au ho : Aslihan Ka ci ([email p o ec ed].cz)
Resea ch desc ibed in his pape was suppo ed by he King Abdullah Uni e si y o Science and Technology, Saudi A abia, he Na ional
Sus ainabili y P og am unde g an LO1401, he Minis y o Educa ion, and he Czech Science Founda ion unde g an no. 19-24585S.
Fo he esea ch, in as uc u e o he SIX Cen e was used.
ABSTRACT This s udy p oposes a new app oach o he op imiza ion o phase and magni ude esponses o
ac ional-o de capaci i e and induc i e elemen s based on he mixed in ege -o de gene ic algo i hm (GA),
o e a bandwid h o ou -decade, and ope a ing up o 1 GHz wi h a low phase e o o app oxima ely ±1◦.
I p o ides a phase op imiza ion in he desi ed bandwid h wi h minimal b anch numbe and a oids he use o
nega i e componen alues, and any complex ma hema ical analysis. S anda dized, IEC 60063 complian
comme cially a ailable passi e componen alues a e used; hence, no co ec ion on passi e elemen s
is equi ed. To he bes knowledge o he au ho s, his app oach is p oposed o he i s ime in he
li e a u e. As alida ion, we p esen nume ical simula ions using MATLAB R
and expe imen al measu emen
esul s, in pa icula , he Fos e -II and Valsa s uc u es wi h i e b anches o p ecise and/o high- equency
applica ions. Indeed, he esul s demons a e excellen pe o mance and signi ican imp o emen s o e he
Ous aloup app oxima ion, he Valsa ecu si e algo i hm, and he con inued ac ion expansion and he
adap abili y o he GA-based design wi h i e di e en ypes o dis ibu ed RC/RL ne wo k.
INDEX TERMS Caue ne wo k, cons an phase elemen , con inued ac ion expansion, dis ibu ed RC
ne wo k, dis ibu ed RL ne wo k, Fos e ne wo k, ac ional-o de capaci o , ac ional-o de elemen ,
ac ional-o de induc o , gene ic algo i hm, impedance op imiza ion, phase op imiza ion, RC ne wo k, RL
ne wo k, ecu si e algo i hm, Valsa ne wo k.
I. INTRODUCTION
Recen ly, emendous e o s ha e been made o design
ac ional-o de elemen s (FOEs). Indeed hey o e addi-
ional deg ees o eedom and e sa ili y in elec ical ci -
cui s [1]–[4], such as il e s [5]–[7], oscilla o s [8]–[11],
con olle s [12]–[15], bioimpedance modeling [16], li hium-
ion ba e y modeling [17], ansmission line design [18],
eluc ance induc i e ansduce ealiza ion [19], dc–dc boos
con e e s [20] and e e ences ci ed he ein. The e sa ili y
o ac ional-o de ci cui s leads esea che s o belie e ha
he u u e o disc e e elemen ci cui design will unde go a
The associa e edi o coo dina ing he e iew o his manusc ip and
app o ing i o publica ion was Hisao Ishibuchi.
pa adigm shi in a o o FOEs [21]. Fig. 1shows he unda-
men al componen s and possible FOEs in ou quad an s [22],
[23]. Thei impedance is desc ibed as Z(s)=Ksα, whe e
ωis he angula equency in s=jω, and ob iously has a
eal pa dependen on he non-ze o equency. In pa icula ,
he impedance o Type IV FOEs, i.e. ac ional-o de capac-
i o s (FOCs), is p o ided wi h an o de o −1< α < 0 and
pseudocapaci ance o Cα=1/K, whe eas ac ional-o de
induc o s (FOIs) in quad an I (Type I) ha e an o de o 0 <
α < 1 and pseudoinduc ance o Lα=K. These wo FOEs
a e he key componen s in ac ional-o de ci cui design and
ou main objec o in es iga ion in his wo k. Thei cha ac e -
is ics such as pseudocapaci ance, pseudoinduc ance, cons an
phase zone (CPZ), cons an phase angle (CPA – de ined phase
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A. Ka ci e al.: Syn hesis and Op imiza ion o F ac ional-O de Elemen s Using a Gene ic Algo i hm
FIGURE 1. Desc ip ion o ac ional-o de elemen s in ou quad an s
[22], [23].
angle in CPZ), and phase angle de ia ion (PAD – maximum
di e ence be ween a designed/measu ed phase and a a ge
phase) p o oundly impac he ans e unc ion o hose sys-
ems [1], [2], [4], [22].
Di e en app oaches o he ealiza ion o an FOC using
dielec ic ma e ials [24]–[27], a liquid elec ochemical capac-
i o [28], [29], and o he ac i e emula o s [30] exis in
he open li e a u e. Howe e , hey a e in he concep ual
s age and s ill equi e much wo k be o e hey mee he
expec a ions o ci cui designe s. One me hod o imple-
men ing an FOC is o app oxima e he a ional unc-
ion o sin a desi ed bandwid h using he me hods o
Ca lson [31], Ma suda [32], El-Khazali [33], Maione [34],
Ous aloup [35], Con inued F ac ion Expansion (CFE) [36],
and Recu si e Algo i hms (RAs) [37], [38] among o h-
e s [39]–[43]. Once a a ional unc ion is ob ained, i can
be syn hesized by he use o a weigh ed sum o i s -
o de il e sec ions [44], ac i e building blocks [45], con-
olle s [46], bilinea sec ions [47], ee s uc u es [48],
RC ladde ne wo ks [36], o a Valsa s uc u e [37]. Sub-
sequen ly, by de ining he Cα, hei o e all ans e unc-
ion mimics he impedance o FOCs a a speci ic equency
ange. On he o he hand, he e a e ew app oaches ha
mimic he FOI esponse such as RL-ne wo ks [20], equi ip-
ple and El-Khazali app oxima ions [33], using gene alized
impedance con e e s (GICs) [19], [23], [23], [49]–[52],
o o he ac i e emula o s [45]–[47]. Howe e , he ade-o
be ween an ope a ing equency ange and he numbe o
ci cui elemen s equi ed in he FOC o FOI design limi s
he use o passi e o ac i e emula o s in many applica ions.
In addi ion, all abo e men ioned app oxima ion me hods a e
analy ical and equi e no s anda d IEC 60063 complian
alues o esis o s, capaci o s, and induc o s o ha e be e
esul s. I he used alues a e eplaced by he closes s an-
da dized alues, he accu acy o he app oxima ion dec eases,
which leads o an inc ease o PAD and o e all deg ada ion o
he pe o mance o he FOE ( o ins ance [15], [20], [23]).
Up un il now, e olu iona y compu ing algo i hms ha e
been used o educe he d awbacks in adi ional op imiza ion
me hods and o sol e complex p oblems whe e con en ional
echniques ail in many a eas o he ac ional-o de domain
such as chaos [53], con ol [54], o ex ac ing he design
pa ame e s o il e s [55]. In his ega d, lowe pollina ion
algo i hms [56], pa icle swa m op imiza ion [57], o gene ic
algo i hms (GA) [58] a e used. In his wo k, a mixed in ege -
o de GA in MATLAB R
is used. Ins ead o app oxima ing
sαusing he abo e men ioned app oxima ions a a ce ain
equency (o bandwid h), we op imize he phase and/o mag-
ni ude esponses o RC/RL ne wo ks in he whole desi ed
equency ange. In b ie , he GA is a powe ul compu a ional
echnique, which mimics he p ocess o na u al selec ion he-
o y. I consis s o a popula ion o ep esen a ions o candida e
solu ions o an op imiza ion p oblem, which e ol e owa d
enhanced solu ions. I is impo an o men ion ha he GA
uses he objec i e unc ion i sel , no de i a i es o o he aux-
ilia y knowledge based on p obabilis ic/de e minis ic cha -
ac e iza ion. These ea u es make his op imiza ion me hod
he mos sui able echnique o op imize he CPA in dis-
ibu ed RC/RL ne wo ks. Fu he mo e, he equi ed alues
a e ob ained wi h GA, e en i he passi e componen alues
a e es ic ed o comme cially a ailable ki alues de ined by
s anda d IEC 60063, and s ill main ain supe b esul s. Hence,
he pape aims o in oduce an FOE op imiza ion me hod ha
achie es a b oad ope a ing equency ange wi h CPA de i-
a ion o app oxima ely ±1◦using comme cially a ailable
passi e componen alues in RC and RL s uc u es wi h i e
b anches o Fos e -I, Fos e -II, Caue -I, Caue -II, and Valsa
ne wo ks. Mos c ucially, he p esen ed app oach a oids he
use o nega i e componen alues, GICs, o andom passi e
elemen alues. Thus, his a icle deals wi h he op imal
emula ion o an FOE cu en ly a ailable in he li e a u e.
In pa icula , Fos e -II and Valsa ne wo ks a e selec ed as
ou main objec i e, because he o me o e s a minimum
o al capaci ance alue and he la e p o ides a minimum
CPA de ia ion. He e i is also wo h no ing ha , o he bes
knowledge o au ho s, an FOI design using he lis ed i e RL
ne wo ks is s udied o he i s ime in he li e a u e.
The es o his wo k is o ganized as ollows. Sec ion II
desc ibes he p oposed op imiza ion app oach using GA as
well as design conside a ions. In Sec ion III, i e ypes o
RC ne wo ks o FOC design wi h op imized passi e com-
ponen alues a e analyzed nume ically and he pa icula
cases expe imen ally. Fu he mo e, he FOC esul s o com-
pa isons wi h o he algo i hms (CFE, Ous aloup, and RA)
a e p esen ed. A nume ical s udy o FOI wi h i e ypes o
RL ne wo ks using GA is p esen ed o he i s ime in he
li e a u e in he Sec ion IV. As an example, he beha io o he
op imized Valsa s uc u e is measu ed. Sec ion Vdiscusses
he esul s and pe o mance cha ac e is ics o all examples,
whe eas he Sec ion VI concludes he s udy.
II. DESCRIPTION OF THE GA APPROACH USED IN
OPTIMIZATION OF FOE
Table 1summa izes he FOC and FOI app oxima ion me h-
ods used in his wo k wi h hei syn hesized RC and RL ne -
80234 VOLUME 7, 2019
A. Ka ci e al.: Syn hesis and Op imiza ion o F ac ional-O de Elemen s Using a Gene ic Algo i hm
TABLE 1. FOC and FOI app oxima ion me hods used in his s udy (no e: all below ne wo ks a e op imized using GA).
Algo i hm 1 Gene ic Algo i hm Pseudocode
1: o i=1 o NumO Gene a ions (o un il an
accep able solu ion is ound) do
2: i i s gene a ion hen
3: Gene a e he ini ial popula ion wi h p imi i es
(CPZ, CPA, pseudocapaci ance, pseudoinduc ance,
numbe o b anches, esis o , capaci o , and
induc o se )
4: else
5: Wi h cu en popula ion, gene a e a new one using
c osso e and mu a ion ope a o s
6: end i
7: Calcula e i ness o popula ion membe s
8: i i ness 6= 0 hen
9: Re u n o gene a e new popula ion
10: else
11: B eak he loop
12: end i
13: end o
Re u n bes indi idual in las popula ion
wo ks and equi alen admi ances o impedances. The admi -
ances o some o he RC ne wo ks can be ound in [4], [36].
The impedance and phase op imiza ion o all s uc u es using
TABLE 2. Gene ic algo i hm pa ame e s.
he GA is ob ained wi h p ede ined Rand C alues. The
desi ed cons an phase and/o pseudocapaci ance, pseudoin-
duc ance, numbe o b anches, and equency ange (i.e. CPZ)
a e de ined as design pa ame e s.
To p o ide mo e de ail ega ding he exac s eps ha we e
pe o med by he GA app oach, we p esen i s pseudocode in
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A. Ka ci e al.: Syn hesis and Op imiza ion o F ac ional-O de Elemen s Using a Gene ic Algo i hm
Algo i hm 1 based on [58]. Table 2p esen s he pa ame e s
employed du ing he aining phase o he GA app oach. Fi -
ness, also known as he cos unc ion o he solu ion se ,
is de e mined using he ollowing equa ion:
F=
ϕsim −ϕ a g
,(1)
whe e ϕsim and ϕ a g exp ess he simula ed and a ge phase,
espec i ely.
III. OPTIMIZATION AND VERIFICATION OF FOC
As an exempla y s udy, p ima ily he Fos e -II [36] and
Valsa [37] ne wo ks a e op imized wi hin his sec ion. Fo
he eason ha ; Fos e -II ne wo k o e s a minimum o al
capaci ance alue and Valsa ne wo k p o ides a minimum
PAD.
A. OPTIMIZATION OF FOSTER-II STRUCTURE FOR FOC
DESIGN
Fo a Fos e -II ealiza ion, he componen alues a e gi en by
he pa ial ac ion expansion and i s admi ance is exp essed
in Table 1. He e, nis he numbe o b anches, R0is he
ini ial esis o , and Riand Cia e he esis ances and capaci-
ances o i- h b anch. Fi s ly, he pe o mance o he ne wo k
ob ained using he GA wi h he Ous aloup and CFE me hods
is compa ed o show he ad an age o he GA. The desi ed
bandwid h, numbe o b anches which is equi alen o a i h-
o de admi ance unc ion (n=5), and CPA a e espec i ely
se as 100 Hz−1 MHz, 5, and −45◦wi h a pseudocapaci ance
o Cα=100 nF·s−0.5. As a popula ion, he andom and
comme cially a ailable passi e elemen s de ined in Table 2
a e used. The cen al equency in case o CFE is se o
10 kHz. I can be obse ed om Fig. 2(a) ha all h ee
app oxima ions p o ide a cons an phase esponse wi h a ge
CPA nea a cen al equency, speci ically be ween 1 kHz
and 100 kHz. Howe e , e o s in phase o he app oxi-
ma ion models inc ease signi ican ly when he equency is
2 decades abo e and below he cen al equency, whe eas
he phase esponse ob ained using he GA is sa is ied in he
whole equency ange o in e es . Fu he mo e, Fig. 2(b)
shows ela i e phase e o s and co esponding no malized
his og ams (%) o phase angle de ia ion om CPA as an
inse . I can be seen ha he maximum de ia ion in he GA
is limi ed o only ±2◦, whe eas in bo h CFE and Ous aloup,
±25◦e o s occu . Because no di ec con ol exis s o e he
Rand C alues ob ained om he las wo app oxima ions,
a co ec ion o use he comme cially a ailable RC ki alues
is obliga o y o build he FOCs. Howe e , his co ec ion is
no needed o he esul s ob ained by he GA since i di ec ly
p o ides he s anda d IEC 60063 complian RC alues as he
esul s. Indeed, i is possible o include in he popula ion,
i.e. a ailable Rand C alues o MATLAB R
and he GA
pe o ms he op imiza ion using only gi en alues. Fig. 2(c)
shows he simula ed phase esponse o co ec ed RC ne wo k
alues using he Ous aloup, CFE, and op imized ne wo k
using he GA, while he comme cially a ailable 0603 size
Rand Cki alues de ined in Table 2a e used. The es o
FIGURE 2. (a) Nume ical phase esponse plo s o he Fos e -II RC ne wo k
using he Ous aloup, CFE, and GA me hods wi h andom alues,
(b) ela i e phase e o s and co esponding no malized his og ams (%) o
phase angle de ia ion om CPA as inse , (c) phase angle esponse o he
RC ne wo k using he Ous aloup and CFE me hods a e RC alue
co ec ion, and he GA op imized o comme cially a ailable RC ki alues,
(d) ela i e phase e o s and co esponding no malized his og ams (%) o
phase angle de ia ion om CPA as inse . Phase esponses a e op imized
in he equency ange o 100 Hz–1 MHz.
he simula ion se up is iden ical o he simula ion se up o
Fig. 2(a). Fig. 2(d) plo s he ela i e phase e o s and co e-
sponding no malized his og ams (%) o phase angle de ia ion
om CPA as an inse . As i can be obse ed, he maximum
de ia ion in he GA is limi ed o ±2.8◦, whe eas ±30◦e o
is ob ained in bo h Ous aloup and CFE app oxima ion esul s.
No ably, he maximum e o in he phase ob ained om bo h
app oxima ion me hods a e u he inc eased compa ed wi h
he esul s in Fig. 2(b) wi h no RC alue co ec ion. Howe e ,
no signi ican change is obse ed in he phase o he ci cui
ob ained using he GA.
Figs. 3(a) and (c) show he a ge , simula ed, and mea-
su ed phase angle and pseudocapaci ance esponses o he RC
ne wo k op imized using he GA. The same passi e elemen
alues a e used om he comme cially a ailable RC ki s
as depic ed in Fig. 2(c) (see ‘‘This wo k’’) wi h he se up
lis ed in Appendix A. The expe imen al e i ica ion uses he
Agilen 4294A P ecision Impedance Analyze . S anda d cali-
b a ion es s (open and sho ci cui s) o he Keysigh 16048G
Tes Leads a e pe o med o calib a e he ins umen . F om
he esul s in Figs. 3(b) and (d), i can be seen ha he
maximum CPA de ia ion be ween a ge (ideal) and simu-
la ed as well as measu ed alues is only ±2.8◦and ±3.2◦,
espec i ely, whe eas he pseudocapaci ance is ±6.6 nF·s−0.5
and ±7.3 nF·s−0.5.
80236 VOLUME 7, 2019
A. Ka ci e al.: Syn hesis and Op imiza ion o F ac ional-O de Elemen s Using a Gene ic Algo i hm
FIGURE 3. Ta ge (ideal), simula ed, and measu ed (a) phase esponses,
(b) ela i e phase e o s and co esponding no malized his og ams (%) o
phase angle de ia ion om CPA as an inse , (c) pseudocapaci ance
esponses, and (d) ela i e pseudocapaci ance e o s and co esponding
no malized his og ams (%) o pseudocapaci ance de ia ion om CPA as
an inse , espec i ely, o he Fos e -II ne wo k op imized using GA.
Impedance and phase esponses a e op imized in he equency ange
o 100 Hz–1 MHz.
S a is ical analysis o Mon e Ca lo (MC) was pe o med
in O CAD PSpice R
simula ion so wa e wi h passi e ele-
men ole ances based on hei da ashee s [59], [61] and
200 uns o obse e e ec s due o manu ac u ing p ocesses.
The his og am shown in Fig. 4demons a es he a ia ion
o he phase a 100 kHz o he Fos e -II ne wo k op imized
using GA. The mean alue wi h s anda d de ia ion 0.555 is
−44.8109◦, which is e y close o he heo e ical alue −45◦
con i ming ha he p oposed ne wo k has low sensi i i y
cha ac e is ic on passi e componen s. The analysis esul s o
MC o all s udied ne wo ks a hei middle equency a e
lis ed in Appendix A.
B. OPTIMIZATION OF VALSA STRUCTURE FOR FOC
DESIGN
The Valsa ne wo k in Table 1[37] is p oposed o emula e FOC
beha io and ealized using RA. The possibili y o designing
his ne wo k using comme cially a ailable Rand C alues
was claimed by he au ho s [37]. Howe e , he RA allows
us o se only ini ial alues and he emaining b anch alues
mus be adjus ed acco ding o comme cially a ailable passi e
elemen alues. Wi h his in mind, simila o wi h he Fos e -II
s uc u e, he GA is applied o he Valsa ne wo k in his sub-
sec ion. The admi ance unc ion is gi en in Table 1, whe e
compa ed o Fos e -II ne wo k he addi ional C0deno es an
ini ial capaci o . In ou s udy, o p o ide a ai compa ison
wi h [37], he phase esponses o RA and he GA o an
FIGURE 4. Mon e Ca lo analysis: Phase a ia ion a 100 kHz o he
Fos e -II ne wo k op imized using GA ( alues used in Fig. 3(a)).
FIGURE 5. (a) Measu ed phase esponses o he Valsa s uc u e using he
RA and GA me hods (comme cially a ailable ki s a e used), and
(b) ela i e phase e o s and co esponding no malized his og ams (%) o
phase angle de ia ion om CPA as an inse . Impedances a e measu ed in
he equency ange o 100 Hz–10 MHz.
o de o α= −0.67 using comme cially a ailable 0603 size
RC ki alues a e expe imen ally e alua ed. The used pas-
si e elemen alues a e lis ed in Appendix B (see ‘‘Fig. 5’’
columns). Du ing he expe imen al e i ica ion, he same
ins umen s lis ed in Sec ion III(A) a e used. Wi h he phase
e o equal o ±2.1◦, he app oxima ion wi h he GA expe -
imen ally eaches a wide bandwid h o 100 Hz −5 MHz,
as shown in Fig. 5(a). Conside ing he ull equency band up
o 10 MHz, he e o is s ill only ±3.2◦(see Fig. 5(b)).
Fu he mo e, he measu emen esul s o α=−0.5 o de
FOCs using an ENA Se ies Ne wo k Analyze E5071C
(300 kHz−20 GHz) in h ee di e en equency anges [case
s udy (a) in 1 MHz −100 MHz, (b) 5 MHz −500 MHz,
and (c) 50 MHz −1 GHz] a e shown in Fig. 6. Two a ian s
o he FOE de ice wi h dimensions o 20 mm ×20 mm
we e designed ( o 0402 and 0603 size passi e compo-
nen s) employing a subminia u e e sion A (SMA) coax-
ial RF connec o . The ab ica ed p in ed ci cui boa d o
0402 size ki alues is shown in Fig. 6(c) as an inse . Con-
side ing an inpu impedance 50 o he connec o , he phase
is measu ed by de ining he equa ion o impedance as
Z=50·[(1 +S11)/(1 −S11)]. As passi e elemen s, RF- ype
esis o s om Vishay [60] and capaci o s om Keme [62]
a e used. Because o he p oduce s ab ica ion bounda ies,
used passi e componen s ha ing CPA in limi ed equency
VOLUME 7, 2019 80237

A. Ka ci e al.: Syn hesis and Op imiza ion o F ac ional-O de Elemen s Using a Gene ic Algo i hm
FIGURE 6. Measu emen esul s o an α= −0.5 o de FOC implemen ed using he Valsa ne wo k op imized using GA o wo decades in di e en
equency anges: (a) 1 MHz–100 MHz, (b) 5 MHz–500 MHz, and (c) 50 MHz–1 GHz.
FIGURE 7. Mon e Ca lo analysis: Phase a ia ion a 30 MHz o he Valsa
ne wo k op imized using GA ( alues used in Fig. 6(a)).
ange, ope a e up o a maximum o 5 GHz. In addi ion, his
equency ange is in e sely p opo ional o he esis ance
alues. Fo ins ance, a 100  esis o wo ks un il 8 GHz,
whe eas a 1 k esis o has a cons an ze o-deg ee phase
esponse up o 800 MHz and so o h. A high equencies,
he ansmission line e ec becomes dominan ; he e o e,
we main ain he dis ance be ween passi e elemen s he leas .
Despi e he abo e men ioned limi a ions, we ob ained he
esul s un il 1 GHz wi h low phase angle de ia ions as shown
in Figs. 6(a)–(c).
MC analysis was pe o med in O CAD PSpice R
simula-
ion so wa e wi h 0402 ki esis o s [60] and capaci o s [62]
wi h ole ance acco ding o hei da ashee s, and 200 uns.
The his og am shown in Fig. 7demons a es he a ia ion o
he phase a 30 MHz wi h alues used in Fig. 6(a). The mean
alue wi h s anda d de ia ion 0.793 is −44.8853◦, which is
again e y close o he heo e ical alue −45◦.
One o he ad an age o he p oposed GA o design FOC
is i s sui abili y o any RC ladde opology, such as Caue -I,
Caue -II, o Fos e -I. In gene al, by eplacing he admi ance
unc ion o he desi ed opology, i is possible o de e mine
he equi ed esis ance and capaci ance alues o build an
FOC wi h desi able elec ical p ope ies. No ably, he lis
o admi ances o lis ed ne wo ks can be ound in Table 1.
Fig. 8shows he phase and pseudocapaci ance esponses
FIGURE 8. (a) Simula ed phase and (b) ela i e phase e o s and
co esponding no malized his og ams (%) o phase angle de ia ion om
CPA as an inse , (c) pseudocapaci ance esponses, (d) ela i e
pseudocapaci ance e o s and co esponding no malized his og ams (%)
o pseudocapaci ance de ia ion om CPA as an inse , espec i ely, o ou
RC ne wo ks op imized using GA. Responses a e op imized in he
equency ange o 100 Hz–1 MHz.
wi h co esponding ela i e e o s and no malized his og ams
(%) o de ia ions o ou RC opologies while he a ge
phase, pseudocapaci ance, and equency bandwid h a e se
o −45◦, 10 nF·s−0.5, and 4 decades in he equency ange o
100 Hz−1 MHz, espec i ely. The la ges de ia ion be ween
he desi ed and simula ed phase alues in all opologies is up
o ±2.5◦wi h low pseudocapaci ance de ia ion.
IV. OPTIMIZATION AND VERIFICATION OF FOI
The mos popula echnique o mimic an induc o is using
a GIC employing Op-Amps, esis o s, and capaci o s [19],
80238 VOLUME 7, 2019
A. Ka ci e al.: Syn hesis and Op imiza ion o F ac ional-O de Elemen s Using a Gene ic Algo i hm
TABLE 3. Compa ison o simula ion and measu emen esul s o used me hods o FOC design.
FIGURE 9. Nume ical simula ion esul s o i e-b anches Valsa RL
ne wo k using 0603 ki R and L alues o FOI design: (a) phase,
pseudoinduc ance, and magni ude esponses, (b) ela i e phase e o s
and co esponding no malized his og ams (%) o h ee di e en o de s in
he equency ange o 10 kHz–10 MHz.
[20], [33], [45]–[47], [49]– [52]. Howe e , he pe o mances
o hese GIC-based ac i e induc ance simula o s o en su e
om he non-ideali ies o Op-Amps. The e o e, his sec ion
deals wi h he op imal emula ion o an FOI o he i s ime
in he li e a u e. The FOI design using he GA is s udied
nume ically and expe imen ally e i ied.
A. OPTIMIZATION OF VALSA STRUCTURE FOR FOI DESIGN
The Valsa RC ne wo k in Table 1is modi ied o an RL- ype
s uc u e by eplacing all capaci o s wi h induc o s as shown
in co esponding igu e. I s equi alen impedance unc ion is
gi en in Table 1, whe e nis he numbe o b anches, R0is
he ini ial esis o , L0is he ini ial induc o , Riand Lia e he
esis ances and induc ances o he i- h b anch, espec i ely,
while he i ness unc ion is desc ibed as (1). The equency
esponse o i e-b anch FOIs wi h h ee di e en angles using
FIGURE 10. Measu emen esul s o an α=0.5 o de FOI om Fig. 9and
he ab ica ed de ice wi h dimensions o 15 mm ×17 mm as in inse
(blue line - impedance esponse; ed line - phase esponse).
0603 ki R[59] and L[63] alues is s udied nume ically
and shown in Fig. 9. The pseudoinduc ances o o de s α=
{0.25,0.5,0.75}a e 8.52 mH·s−0.75, 834.62 µH·s−0.5, and
89.62 µH·s−0.25, which a e cons an wi h small de ia ions
in he whole equency ange. Fu he mo e, he slope o
magni ude in he inse o Fig. 9(a) shows ha he induc i e
eac ance (impedance) o he FOI inc eases as he supply e-
quency ac oss i inc eases. To es ima e he equi alen o de α,
he simula ed magni ude esponses a e i ed o he unc ion
log10|Z| = αlog +log10(2π)αLαusing he linea leas
squa es me hod. The equi alen equa ions om i ing he
magni ude a e p o ided inside Fig. 9(a). The maximum PAD
and ela i e phase e o s o he ela ed o de s a e {±1.84◦,
±1.66◦,±1.55◦} and {±8.16%, ±3.68%, ±2.29%}, espec-
i ely, as depic ed in Fig. 9(b). The ope a ing equency
ange is chosen be ween 10 kHz and 10 MHz because o
he wo king equency ange o he 0603 ki ce amic chip
induc o s [63]. Conside ing ha he maximum PAD is a ound
±2◦, o de o 0.25 is limi ed om 12 kHz. The used passi e
elemen alues a e lis ed in Appendix C.
Mo eo e , he beha io o an α=0.5 o de FOI,
nume ically simula ed in Fig. 9, was e i ied using he
Agilen 4294A p ecision Impedance Analyze . S anda d cal-
ib a ion es s (open and sho ci cui s) o he 16047E Tes
Fix u e we e pe o med o calib a e he ins umen . Du ing
he expe imen al alida ion in he equency ange 400 kHz−
40 MHz (801 loga i hmically spaced poin s in wo decades),
VOLUME 7, 2019 80239
A. Ka ci e al.: Syn hesis and Op imiza ion o F ac ional-O de Elemen s Using a Gene ic Algo i hm
TABLE 4. Passi e elemen alues o FOC using he Fos e -II ne wo k and hei pe o mance cha ac e is ics.
TABLE 5. Passi e elemen alues o FOC using he Valsa ne wo k and hei pe o mance cha ac e is ics.
80240 VOLUME 7, 2019
A. Ka ci e al.: Syn hesis and Op imiza ion o F ac ional-O de Elemen s Using a Gene ic Algo i hm
TABLE 6. Passi e elemen alues o FOI using he RL ne wo k and hei pe o mance cha ac e is ics.
a sinusoidal inpu signal wi h a de aul AC ol age o 500 mV
and a equency o 1 MHz was applied, while one o e minals
was g ounded. The measu emen esul s and a pho og aph o
he ab ica ed de ice wi h dimensions o 15 mm ×17 mm
a e depic ed in Fig. 10. The measu ed PAD in wo decades o
he equency ange o ou in e es is ±5.82◦.
In addi ion o an α=0.5 o de FOI, a MC s a is ical
analysis was also pe o med in he O CAD PSpice R
simu-
la ion so wa e. The passi e elemen ole ances acco ding o
0603 ki da ashee s [59], [63] and 200 uns we e se o obse e
a ec s due o manu ac u ing p ocesses. The his og am shown
in Fig. 11 demons a es he a ia ion o he phase a 3 MHz.
The mean alue wi h s anda d de ia ion 0.49 is 45.1964◦,
which is e y close o he heo e ical alue 45◦con i ming
ha he p oposed ne wo k has low sensi i i y cha ac e is ic
on passi e componen s.
In addi ion, o he i s ime in he li e a u e, he Fos e -I,
Fos e -II, Caue -I, and Caue -II ype o RL ne wo ks a e also
s udied. The impedance unc ion o all ne wo ks op imized
using GA a e gi en in Table 1. Fig. 12 shows he phase
and pseudoinduc ance esponses wi h co esponding ela i e
e o s and no malized his og ams (%) o de ia ions o ou
RL opologies. The a ge phase and equency bandwid h
a e se o 45◦and 3 decades in he equency ange o
10 kHz −10 MHz, espec i ely, wi h no pseudoinduc ance
FIGURE 11. Mon e Ca lo analysis: Phase a ia ion a 3 MHz o he Valsa
RL ne wo k op imized using GA (α=0.5 o de FOI wi h alues used
in Figs. 9and 10).
speci ica ion o ob ain he bes esul . In summa y, he min-
imal e o is ob ained wi h Fos e -I s uc u e while he leas
sp ead o passi e elemen alues a e obse ed wi h he Fos e -
II. The de ailed analysis including MC esul s o all s udied
ne wo ks is p esen ed in Appendix C.
V. BRIEF DISCUSSION OF RESULTS
Table 3compa es he pe o mance o RC ne wo ks buil
using Ous aloup, CFE, RA, and he GA. Fo ins ance, o
he Fos e -II ne wo k composed o he same numbe o
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