Synthesis and Optimization of Fractional-Order Elements Using a Genetic Algorithm

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 VOLUME 7, 2019 This wo k is licensed unde a C ea i e Commons A ibu ion 3.0 License. Fo mo e in o ma ion, see h p://c ea i ecommons.o g/licenses/by/3.0/ 80233 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 VOLUME 7, 2019 80235 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 VOLUME 7, 2019 80241