applied sciences

Article Optimized Design of OTA-Based Gyrator Realizing Fractional-Order Inductance Simulator: A Comprehensive Analysis †

David Kubanek 1,* , Jaroslav Koton 1, Jan Dvorak 1, Norbert Herencsar 1 and Roman Sotner 1,2

1 Department of Telecommunications, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technicka 3082/12, 616 00 Brno, Czech Republic; [email protected] (J.K.); [email protected] (J.D.); [email protected] (N.H.); [email protected] (R.S.) 2 Department of Radio Electronics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technicka 3082/12, 616 00 Brno, Czech Republic * Correspondence: [email protected] † This paper is an extended version of our paper published in 2020 43rd International Conference on Telecommunications and Signal Processing (TSP), Milan, Italy, 7–9 July 2020.

Abstract: A detailed analysis of an operational transconductance amplifier based gyrator implement- ing a fractional-order inductance simulator is presented. The influence of active element non-ideal properties on the gyrator operation is investigated and demonstrated by admittance characteristics and formulas for important values and cut-off frequencies in these characteristics. Recommendations to optimize the performance of the gyrator in terms of operation bandwidth, the range of obtainable admittance magnitude, and signal dynamic range are proposed. The theoretical observations are verified by PSpice simulations of the gyrator with LT1228 integrated circuit.



Keywords: fractional-order capacitor; fractional-order circuit; fractional-order impedance; fractional-
 order inductor; gyrator; operational transconductance amplifier

Citation: Kubanek, D.; Koton, J.; Dvorak, J.; Herencsar, N.; Sotner, R. Optimized Design of OTA-Based Gyrator Realizing Fractional-Order 1. Introduction Inductance Simulator: A Fractional-order (FO) systems, i.e., systems of non-integer order described by non- Comprehensive Analysis . Appl. Sci. integer order differential and integral equations, are able to provide more arbitrary prop- 2021, 11, 291. https://doi.org/ erties and characteristics in comparison with their integer-order counterparts [1,2]. Thus, 10.3390/app11010291 FO systems are employed in many branches, such as signal processing, modeling vari- ous real-world phenomena, or system control [3]. Analog electrical circuits can also be Received: 30 November 2020 designed with fractional order, and for their implementation, elements with FO impedance Accepted: 25 December 2020 are frequently employed. These elements are also often referred to as fractional-order Published: 30 December 2020 elements (FOEs) [4] or fractors [5]. The impedance of FOE can be expressed in the s-domain

Publisher’s Note: MDPI stays neu- by the following non-integer power-law dependence tral with regard to jurisdictional clai- 1 ms in published maps and institutio- Z(s) = , (1) sαF nal affiliations. where the real number α is fractional exponent or order of FOE and the positive constant F represents fractance having the unit Ω−1·secα or F·secα−1; whereas, here “F” is farad. After the substitution s = jω, the relation (1) can be written in the magnitude-phase format in the Copyright: © 2020 by the authors. Li- frequency domain: censee MDPI, Basel, Switzerland. 1 απ This article is an open access article Z(jω) = − . (2) ωαF ∠ 2 distributed under the terms and con- ditions of the Creative Commons At- The roll-off of the impedance magnitude is −20α dB per decade of frequency, and tribution (CC BY) license (https:// the phase remains constant regardless of frequency. If α > 0, the FOE has a capacitive creativecommons.org/licenses/by/ character and is labeled as FO capacitor or capacitive fractor. Its impedance magnitude 4.0/). decreases with increasing frequency, and the phase angle is negative. If α < 0, the FOE

Appl. Sci. 2021, 11, 291. https://doi.org/10.3390/app11010291 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, 291 2 of 19

is called as an FO inductor or inductive fractor, its impedance magnitude increases with increasing frequency, and the phase angle is positive. The values of α are considered −1 < α < 1 here. The special cases with α = 0, 1, or −1 denote resistor, classic capacitor, and classic inductor, respectively. Currently, the research of FO capacitors is particularly conducted as surveyed, e.g., in Reference [4], though it is also necessary to study implementations of FO inductors, as their importance is growing in various applications. FO inductors are used, for example, in parallel and series FO LC resonant circuits [6–8] and FO frequency filters [9,10], where they enable achieving high-quality factor. Furthermore, FO inductors are part of electrical equivalent models of various systems, e.g., in medicine, they are used for modeling mechanical impedance of respiratory systems [11,12]. A design example of an FO inductor for this application is presented in this paper. The FO inductors were also found to very accurately describe the real behavior of coils with ferromagnetic cores, eddy currents, and hysteresis losses [13]. The employment of FO inductors in wireless power transmission systems has been investigated recently, leading to higher power efficiency and lower switching frequency than conventional methods [14]. FO inductors can be approximated by passive ladder circuits consisting of resistors and standard inductors [15–17]. However, the implementation of inductors by conventional passive coils is currently avoided, due to their drawbacks, such as bulky dimensions, complicated production, incompatibility with integrated technologies, low-quality factor, and other parasitic electrical properties. A class of active FO inductance simulators studied, e.g., in Reference [18], consists of an FO differentiator or integrator with a voltage/current converter connected in a feedback loop. Here the FO differentiator or integrator is emulated by multiple-feedback structure, such as follow-the-leader feedback (FLF). Their advantage is the use of integer-order elements only; however, the circuit complexity grows when an accurate operation in a wide frequency band is required. Since FO capacitor as a solid-state discrete device is expected to be obtainable soon, another type of active FO inductance simulator is becoming attractive. It is implemented by connecting a FO capacitor to a circuit that provides immittance inversion, as is the case of the gyrator, immittance inverter, or generalized immittance converter (GIC) [19]. This approach using a gyrator with two operational transconductance amplifiers (OTAs) is also discussed in this article. As it is not possible to manufacture perfectly ideal active elements, it is appropriate to analyze the influence of parasitic properties of these elements on the function of the gyrator. Thus, limitations on the performance of a gyrator transforming FO capacitor to FO inductor, due to non-ideal OTA properties are presented here in detail. Deviations in magnitude and phase admittance characteristics are investigated, and guidelines for minimizing these errors are provided. Moreover, design steps for optimal utilization of OTA signal dynamic range are given. A similar approach of obtaining FO inductor by means of immittance inversion has also been presented in several recent works [20–22]. In Reference [20], an opamp-based GIC is employed to obtain grounded FOEs with impedance phasor in any of the four quadrants, therefore, FO inductor is also included. The guidelines for obtaining passive element parameters for the required fractance value are given; however, there is no thorough analysis of the effect of opamp non-idealities. The paper [21] is devoted to the optimal design of a grounded FO inductor using GIC, and a detailed study of the influence of opamp non-idealities is given there. This work aims to provide similar analysis and design guidelines; however, here we deal with a gyrator, which is a more suitable solution compared to GIC regarding the number of elements. It is based on only one loading passive element and two OTAs; whereas, the FO inductor in Reference [21] contains five passive elements and two opamps. In contrast to Reference [21], the circuit in this work provides a FO inductor with electronically adjustable fractance, and it can be easily converted to a floating variant by using one of the OTAs with differential outputs. Among the works dealing with FO inductors, which contain an analysis of active element parasitic properties, Appl. Sci. 2021, 11, x FOR PEER REVIEW 3 of 21

this work provides a FO inductor with electronically adjustable fractance, and it can be Appl. Sci. 2021, 11, x FOR PEER REVIEWeasily converted to a floating variant by using one of the OTAs with differential outputs.3 of 21 Among the works dealing with FO inductors, which contain an analysis of active element Appl. Sci. 2021, 11, 291 3 of 19 parasitic properties, the paper [22] can also be mentioned. However, here a lossy FO in- ductor is implemented by a current feedback operational amplifier (CFOA) and FO ca- this work provides a FO inductor with electronically adjustable fractance, and it can be pacitor. The analysis of parasitic properties is performed only at the level of symbolic easily converted to a floating variant by using one of the OTAs with differential outputs. therelations paper, [and22] canthe alsoinfluence be mentioned. of individual However, CFOA here parasitics a lossy FOon inductorthe properties is implemented of the FO Among the works dealing with FO inductors, which contain an analysis of active element byinductor a current is not feedback explained. operational amplifier (CFOA) and FO capacitor. The analysis of parasitic properties, the paper [22] can also be mentioned. However, here a lossy FO in- parasiticThis properties paper is organized is performed as onlyfollows: at the In levelSection of symbolic 2, the theory relations, of OTA and-based the influence gyrator ductor is implemented by a current feedback operational amplifier (CFOA) and FO ca- ofimplementing individual CFOA FO inductance parasitics on simulator the properties is described. of the FO The inductor influence is not of explained. OTA parasitic pacitor. The analysis of parasitic properties is performed only at the level of symbolic propertiesThis paper on the is organized gyrator function as follows: is analyzed In Section, and2, thedynamic theory properties of OTA-based are studied gyrator to relations, and the influence of individual CFOA parasitics on the properties of the FO implementingprevent the amplifiers FO inductance from overloading. simulator isDesign described. guidelines The influenceand compensation of OTA parasiticpossibili- inductor is not explained. propertiesties are given on theto minimize gyrator function the effect is of analyzed, the parasitic and properties. dynamic properties Section 3 arecontains studied verifi- to This paper is organized as follows: In Section 2, the theory of OTA-based gyrator preventcation of the the amplifiers theoretical from findings overloading. by PSpice Design simulations. guidelines andThe compensationpractical design possibilities of FO in- implementing FO inductance simulator is described. The influence of OTA parasitic areductor given for to minimizean electrical the model effect of of the the parasitic human properties. respiratory Section system3 contains is demonstrated verification as ofan properties on the gyrator function is analyzed, and dynamic properties are studied to theexample. theoretical Finally, findings our conclusions by PSpice simulations. are given in TheSection practical 4. design of FO inductor for an electricalprevent the model amplifiers of the humanfrom overloading. respiratory systemDesign isguidelines demonstrated and compensation as an example. possibili- Finally, ourties2. Methods conclusionsare given to areminimize given in the Section effect4 .of the parasitic properties. Section 3 contains verifi- cation of the theoretical findings by PSpice simulations. The practical design of FO in- 2.1. Ideal OTA-Based Gyrator 2.ductor Methods for an electrical model of the human respiratory system is demonstrated as an 2.1.example. IdealIdeal OTA-Based Finally, OTA [23]our Gyrator conclusions(Figure 1) is are a differentialgiven in Section voltage 4. -controlled current source with output current given by the following relation Ideal OTA [23] (Figure1) is a differential voltage-controlled current source with output current2. Methods given by the following relation i g v v  , (3) 2.1. Ideal OTA-Based Gyrator OUT m i = g (v − v ), (3) Ideal OTA [23] (Figure 1) is OUTa differentialm + voltage− -controlled current source with output current given by the following relation where gm is transconductanceISET/VSET gain, which is usually electronically adjustable by an external DC currentOTIASET or DC voltage VSET. Input and output terminals of an ideal OTA have infinite v+ iOUT g m  v v  , (3) internal impedance. The gyrator circuit employing two OTAs is depicted in Figure2[23]. + iOUT gm v‒ _ ISET/VSET OTA v+ + iOUT Figure 1. OTA schematic symbol. gm v‒ _ where gm is transconductance gain, which is usually electronically adjustable by an ex- ternal DC current ISET or DC voltage VSET. Input and output terminals of an ideal OTA

have infinite internal impedance. The gyrator circuit employing two OTAs is depicted in FigureFigureFigure 1.1. 2 OTA [23] .schematicschematic symbol.symbol.

where gm is transconductance gain, which is usually electronically adjustable by an ex- OTA2 ternal DC current ISET or DC voltage VSET. Input and output terminals of an ideal OTA + have infinite internal impedance. The gyrator circuit employing two OTAs is depicted in gm2 Figure 2 [23]. _

OOTTAA2 1 + + YIN ggmm21 _ _ Y OTA1 + Figure 2. Gyrator with two OTAs. FigureYIN 2. Gyratorgm1 with two OTAs. _ This gyrator providesY inversion of the loading admittance Y as apparent from the following relation for its input admittance. Figure 2. Gyrator with two OTAs. g g Y (s) = m1 m2 . (4) IN Y(s) Appl. Sci. 2021, 11, x FOR PEER REVIEW 4 of 21

This gyrator provides inversion of the loading admittance Y as apparent from the following relation for its input admittance.

ggm1 m2 YsIN    . (4) Appl. Sci. 2021, 11, 291 Ys  4 of 19

A capacitor is most often connected as the loading element, which yields the induc- tive character of the input admittance. When connecting a FO capacitor with admittance Y(s) =A s capacitorαF and 0 < is α most < 1 as often the gyrator’s connected load, as the the loading resulting element, input which admittance yields is the inductive character of the input admittance. When connecting a FO capacitor with admittance α F gg Y(s) = s F and 0 < α < 1 as the gyrator’sYs load,IN them1 resulting m2 input admittance is IN    (5) s s F , F g g Y (s) = IN = m1 m2 , (5) which represents FO inductor admittanceIN swithα thes magnitudeαF and phase which represents FO inductor admittanceF  withgg the magnitude  and phase Yj IN   m1 m2   IN    (6) F 22απ gF g . απ Y (jω) = IN − = m1 m2 − . (6) IN ωα ∠ 2 ωαF ∠ 2

2.2.2.2. PracticalPractical OTA-BasedOTA-Based GyratorGyrator ParasiticParasitic propertiesproperties ofof aa realreal OTAOTA areare commonlycommonly modeledmodeled byby shuntshunt resistancesresistances andand capacitancescapacitances of of the the input input and and output output terminals. terminals. Input resistance is usually very large, large, especiallyespecially inin CMOSCMOS technology;technology; whereas,whereas, inputinput capacitancecapacitance isis typicallytypically inin thethe rangerange ofof tenstens to to the the low low hundreds hundreds of of femtofarads femtofarads when when integrated integrated on on aa substratesubstrate of of standard standard CMOSCMOS processes.processes. The outputoutput resistance of OTAOTA generallygenerally reachesreaches severalseveral hundredshundreds ofof kiloohmskiloohms andand outputoutput capacitancecapacitance tenstens toto aa fewfew hundredshundreds ofof femtofaradsfemtofarads [[24,25]24,25].. ForFor somesome OTAOTA implementations,implementations, thethe parasiticparasitic propertiesproperties maymay bebe dependentdependent onon thethe OTAOTA operatingoperating conditions,conditions, e.g., e.g., the the terminal terminal conductance conductance may may change change depending depending on onthe the set transconduc-set transcon- tance,ductance, which which is also is also the casethe case of the of the amplifier amplifier used used in the in the computer computer simulation simulation example example in thisin this article. article.

2.2.1.2.2.1. InfluenceInfluence ofof OTAOTA TerminalTerminalImpedances Impedances ConsideringConsidering the the aforementioned aforementioned OTA OTA parasitic parasitic properties, properties, the gyrator the gyrator schematic schematic from Figurefrom Figure2 can be 2 can modified be modified to the formto the in form Figure in 3Figure. 3.

OTA2 + gm2 _

GP2 CP2

OTA1 + YIN' gm1 _

GP1 CP1 Y

FigureFigure 3.3. GyratorGyrator withwith OTAOTA parasiticparasitic properties.properties.

TheThe parasiticparasitic propertiesproperties ofof OTAOTA11 outputoutput andand OTAOTA22 inputinput respectiverespective toto thethe groundground areare togethertogether modeledmodeled withwith GGP1P1 andand CCP1P1 elements;elements; whereas,whereas, thethe parasiticsparasitics ofof OTAOTA11 inputinput andand OTAOTA22 outputoutput areare modeledmodeled withwith GGP2P2 andand CCP2P2. The The relation forfor inputinput admittanceadmittance ofof thethe gyratorgyrator loadedloaded withwith FOFO capacitorcapacitor takingtaking intointo accountaccount thethe OTAOTA parasiticparasiticproperties propertiesis is g g 0( ) ≈ m1 m2 + + YIN s α GP2 sCP2. (7) s F + GP1 + sCP1 This relation needs to be analyzed to determine the effect of the parasitic properties. To illustrate the meaning of (7), the passive circuit with the identical input admittance YIN’(s) can be depicted in Figure4. Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 21 Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 21

gg  m1 m2   YIN s  G P2gg sC P2 . (7) s F G sC m1 m2   P1YIN P1s  G P2 sC P2 . (7) s F GP1 sC P1 This relation needs to be analyzed to determine the effect of the parasitic properties. Appl. Sci. 2021, 11,To 291 illustrate the meaningThis relationof (7), the needs passive to be circuitanalyzed with to thedetermine identical the input effect admittance of the parasitic properties.5 of 19 YIN’(s) can be depictedTo illustrate in Figure the 4.meaning of (7), the passive circuit with the identical input admittance YIN’(s) can be depicted in Figure 4.

ggm1 m2  gg sF m1 m2 sF ggm1 m2 YIN' GP2 sCP2 ggm1 m2 YIN' sCP1 GP2 sCP2 sCP1 ggm1 m2 ggm1 m2 GP1 GP1

Figure 4. Substitute circuit providing the input admittance (7); the relations in the schematic ex- press admittancesFigureFigure of the 4. particular4.Substitute Substitute elements. circuit circuit providing providing the the input input admittance admittance (7); (7); the the relations relations in in the the schematic schematic express ex- admittancespress admittances of the particularof the particular elements. elements. Analyzing (7) and Figure 4, the gyrator transforms the FO capacitor to FO inductor with admittance gm1AnalyzinggAnalyzingm2/(sαF), which (7)(7) andand is FigureFigure in agreement4 ,4, the the gyrator gyrator with thetransforms transforms relation the (5)the FO validFO capacitor capacitor for the to to FO FO inductor inductor αα ideal case. The withparasiticwith admittance admittance propertiesgm1 gm1 gCm2gP1m2/( and/(ss F ),)G, whichP1which are also is is in intransformed agreement agreement with by with thethe the relationgyrator relatio (5) andn valid (5) valid for the for ideal the appear in the case.ideal network The case. parasitic in The Figure parasitic properties 4 as properties an C P1integerand C GP1-orderP1 andare G also inductorP1 are transformed also with transformed admittance by the gyratorby the andgyrator appear and in the network in Figure4 as an integer-order inductor with admittance g g /(sC ) gm1gm2/(sCP1) andappear resistor inwith the conductance network in g m1 Figuregm2/GP1 ,4 respectively as an integer. The-order admittance inductor of the with m1 admittancem2 P1 and resistor with conductance g g /G , respectively. The admittance of the parasitic parasitic elementsgm1 Ggm2P2 /(andsCP1 C) P2and is resistoraccording with to conductance(7)m1 additionalm2 P1 g m1partgm2 of/G P1the, respectively input admittance. The admittance of the elements G and C is according to (7) additional part of the input admittance Y ’ (s) YIN’ (s) and thereforeparasitic, these elementsP2 elements GP2 P2appear and C atP2 theis according overall input to (7) of additional the structure part in of Figure the input admittanceIN 4. andYIN’ therefore,(s) and therefore these elements, these elements appear atappear the overall at the inputoverall of input the structure of the structure in Figure in4 .Figure Considering4. the Considering above-mentioned the above-mentioned typical values of typical the OTA values parasitic of the properties, OTA parasitic the properties, the format of the admittanceformatConsidering of themagnitude admittance the aboveand magnitude phase-mentioned frequency and typical phase characteristics frequencyvalues of the characteristics resulting OTA parasitic from resulting properties, from (7)the can be displayed as the asymptotic Bode plots in Figure5. (7) can be displayedformat as theof theasymptotic admittance Bode magnitude plots in Figure and phase 5. frequency characteristics resulting from (7) can be displayed as the asymptotic Bode plots in Figure 5.

Phase{YIN'} |YIN'| (rad) Phase{YIN'} |YIN'| (rad) ggm1 m2 π2 gggg π2 GP1 m1 m2m1 m2  gg GP1  F ωCPm12 m2   F ωCP2 0 GP2 0 GP2 π2 π2 ωGP1 ωGP2 ωCP2 ω  π2 ωGP1ωCP2' ωGP2 ωCP2 ωωGP1  π2 ωGP2 ωCP2 ω ωCP2' ωGP1ωCP2' ωGP2 ωCP2 ω (a) (b) ωCP2' (a) (b) Figure 5. BodeFigure plots 5. Bode of (a )plots magnitude of (a) magnitude and (b) phase and ( admittanceb) phase admittance characteristics characteristics according according to (7). Blue to (7). parts denote ideal Blue parts denote Figureideal function 5. Bode withoutplots of the(a) magnitudeinfluence of and parasitics. (b) phase admittance characteristics according to (7). function without the influenceBlue of parasitics.parts denote ideal function without the influence of parasitics. The lower cut-offThe frequency lower cut-off of the frequency correct gyrator of the correct operation gyrator is approximately operation is approximately equal equal to to The lower cut-off frequency of the correct gyrator operation is approximately equal to   1 1 G α ≈ P1 G  ωGP1 1 . (8)   P1 F (8) GP1 . G  F   P1 (8) GP1 . α At this frequency, the admittance magnitudeF of the gyrator load |ω F| equals to GP1. Below this frequency, the gyrator exhibits parasitic behavior with limited input admittance magnitude gm1gm2/GP1 and zero phase. This value represents the highest achievable admittance magnitude at the gyrator input. The value of GP2 has no effect below the frequency ωGP1, as we assume GP2  gm1gm2/GP1. Above the frequency ωGP1, the operation of the gyrator can be considered correct, i.e., providing the admittance of FO inductor given by (5), which is marked by the blue color in Figure5. The magnitude of the gyrator input admittance decreases with the roll-off rate −20α dB/dec, and the constant phase is −απ/2. Note that the admittance characteristics of Appl. Sci. 2021, 11, 291 6 of 19

an ideal FO inductor can be obtained by linearly extending the blue lines in Figure5 over the entire frequency range. Based on the relationships between the values of the elements in Figure4, the upper cut-off frequency of the correct behavior of the gyrator may be equal to

  1 gm1gm2 α ωGP2 ≈ , (9) GP2F

where the input admittance magnitude |YIN’| decreases to the parasitic conductance GP2. Above this frequency, the input admittance has the magnitude value GP2 and is purely real. Further, above the frequency GP2 ωCP2 ≈ , (10) CP2

the effect of the parasitic capacitance CP2 prevails, and the input admittance YIN’ has purely capacitive character, which even more differs from the required FO inductor. As demonstrated in Figure5 by the dashed part of the characteristic, when the α conductance GP2 is lower than the higher value of the two admittances gm1gm2/(ω F) and ωCP2, the upper cut-off frequency of the correct gyrator behavior is

1   + 0 gm1gm2 α 1 ωCP2 ≈ . (11) CP2F

In this case, the FO inductance appearing at the input port of the gyrator and the parasitic capacitance CP2 create a parallel resonant circuit, which can be a potential source of instability. Analyzing the relation (7), the real part of YIN’ cannot be lower than GP2, which means that the parasitic conductance GP2 ensures damping of the resonance. The value of α also has an impact on the gyrator stability. The more it is less than one, the more the resonance is damped. The standard inductor in Figure4 resulting from immittance inversion of the parasitic capacitance CP1 affects the input admittance of the gyrator negligibly. Its effect would be in decreasing the input admittance above a cut-off frequency that is higher than ωCP2; however, in this frequency band, the input admittance grows, due to the admittance of CP2, which outweighs. The following recommendations can be drawn from the previous analysis to im- prove the performance of the gyrator mainly as far as operation bandwidth, and range of obtainable admittance magnitude are concerned:

• If there is a possibility to modify the fractance F, increase the values of both (gm1gm2) and F such that the ratio (gm1gm2)/F remains constant. This decreases the lower cut-off frequency ωGP1, and thus, expands the band of correct operation of the gyrator to lower frequencies without affecting the FO inductance. Moreover, the maximum obtainable magnitude of input admittance gm1gm2/GP1 increases. • From the OTA parasitic properties, the main effort should be to decrease GP1, which also brings a decrease of ωGP1 and an increase of maximum input admittance mag- nitude. Adherence to the recommendation on decreasing GP1 will also efficiently increase ωGP2, as due to the gyrator topology, we may assume GP1 ≈ GP2. This can be achieved by choosing an OTA structure that offers low parasitic conductance at input and output terminals. An alternative way to reduce a parasitic conductance in the gyrator is connecting a negative conductance in parallel as described in Section 2.2.2. To further optimize the gyrator operation at high frequencies in terms of increasing ωCP2, it is recommended to choose an OTA structure with low CP2. It can also be observed that for lower values of α, the bandwidth of the correct gyrator operation is increased, which is related to the lower slope of the admittance magnitude characteristic. Thus, a gyrator implementing a FO inductor provides higher bandwidth than a gyrator implementing a classic inductor. Appl. Sci. 2021, 11, x FOR PEER REVIEW 7 of 21

nitude. Adherence to the recommendation on decreasing GP1 will also efficiently in- crease ωGP2, as due to the gyrator topology, we may assume GP1 ≈ GP2. This can be achieved by choosing an OTA structure that offers low parasitic conductance at in- put and output terminals. An alternative way to reduce a parasitic conductance in the gyrator is connecting a negative conductance in parallel as described in Section 2.2.2. To further optimize the gyrator operation at high frequencies in terms of in- creasing ωCP2, it is recommended to choose an OTA structure with low CP2. It can also be observed that for lower values of α, the bandwidth of the correct gy- Appl. Sci. 2021, 11, 291 rator operation is increased, which is related to the lower slope of the admittance 7mag- of 19 nitude characteristic. Thus, a gyrator implementing a FO inductor provides higher bandwidth than a gyrator implementing a classic inductor.

2.2.2.2.2.2. CompensationCompensation ofof ParasiticParasitic ConductanceConductance WhenWhen aa parasiticparasitic conductance conductance of of a gyratora gyrator node node is tois beto decreasedbe decreased as described as described in the in previousthe previous section, section, a simple a simple compensation compensation circuit with circuit negative with negativeinput conductance, input conductance shown in, Figureshown6 ,in can Figure be used. 6, can This be circuit used. isThis connected circuit is with connected its COMP with terminal its COMP to the terminal node where to the the parasitic conductance is to be compensated. node where the parasitic conductance is to be compensated.

COMP ISETC/VSETC

OTAC + gmC _

FigureFigure 6.6. CircuitCircuit withwith groundedgrounded negativenegative conductanceconductance atat terminalterminal COMP.COMP.

TheThe conductanceconductance ofof thethe COMPCOMP terminalterminal relativerelative toto groundground equalsequals toto

G  g  G (12) GCOMPCOMP= −gmC mC+ GP (12)

and can be set to an appropriate negative value by the electronic setting of gmC. Here GP is and can be set to an appropriate negative value by the electronic setting of gmC. Here GP is the total parasitic conductance of OTAC input and output terminals. If the implementa- the total parasitic conductance of OTAC input and output terminals. If the implementation tion of OTAC is identical with the OTAs of the gyrator, all the parasitic conductances are of OTAC is identical with the OTAs of the gyrator, all the parasitic conductances are equal equal (GP = GP1 = GP2). From (8), it is apparent that when the admittance GP1 is compen- (GP = GP1 = GP2). From (8), it is apparent that when the admittance GP1 is compensated sated in this way to almost zero value, the cut-off frequency ωGP1 can be theoretically in this way to almost zero value, the cut-off frequency ωGP1 can be theoretically shifted toshifted very lowto very values, low andvalues the, and maximum the maximum magnitude magnitude of the gyrator of the gyrator input admittance input admittance can be increasedcan be increased significantly. significantly. However, However, the demands the demands on the accuracy on the ofaccuracy the compensation of the compensa- circuit increase,tion circuit and increase it is important, and it is to important prevent reaching to prevent a negativereaching conductance a negative conductance of the node. of The the node. The conductance GP2 is worth to be compensated when it affects the gyrator input conductance GP2 is worth to be compensated when it affects the gyrator input admittance GP2 CP2 atadmittance high frequencies at high betweenfrequenciesωGP2 betweenand ωCP2 ω , i.e., and when ω , the i.e., solid when line the (and solid not line the (and dashed not line)the dashed in Figure line)5 holds. in Figure Remember 5 holds. that Remember when α tendsthat when to unity, α tends the conductanceto unity, theG conduct-P2 is an importantance GP2 is dampingan important factor damping of the above-mentioned factor of the above LC-mentioned resonance, LC and resonance thus, it cannot, and thus be, excessivelyit cannot be reduced. excessively reduced. ItIt should should also also be be noted noted that that when when connecting connecting the thecircuit circuit in Figure in Figure6 to a 6 gyrator to a gyrator node, thenode, parasitic the parasitic capacitance capacitance of this of node this node increases increases by the by own the own parasitic parasitic capacitance capacitance of the of compensationthe compensation circuit, circuit which, which is equal is equal to CP1 toand CP1C andP2, assumingCP2, assuming that all that OTAs all OTAs are the are same. the Thus,same. itThus, is necessary it is necessary to take to this take property this property into accountinto account in the in relations the relations containing containing the parasiticthe parasitic capacitance capacitance of the of the node node being being compensated. compensated. Note Note that that the the circuit circuit in in Figure Figure6 6 cannotcannot bebe usedused forfor compensatingcompensating thethe gyratorgyrator parasiticparasitic capacitance.capacitance.

2.2.3.2.2.3. DynamicDynamic PropertiesProperties NextNext toto the the non-ideal non-ideal frequency frequency analysis, analysis, it is it also is also important important to investigate to investigate the dynamic the dy- propertiesnamic properties of the gyrator of the suchgyrator that such the amplifiersthat the amplifiers operate in operate linear mode in linear and themode maximum and the voltage and current values of their terminals are not exceeded. As far as common OTAs are concerned, the most limiting is the non-linearity of their input transistor stage, which takes effect already at differential input voltages of several tens of millivolts. Let us denote the maximum allowed amplitude of OTA input voltage for linear operation VOTA,MAX. The output drive current of OTA is also limited to a maximum value IOTA,MAX; whereas, it holds IOTA,MAX = gmVOTA,MAX. The value of VOTA,MAX, and also maximum allowable transconductance gm can be determined from the respective OTA documentation. The following two conditions must be met to avoid the non-linear operation of the gyrator:

• The gyrator input voltage VIN must not be higher than the allowed input voltage range of OTA1 VOTA1,MAX VIN ≤ VOTA1,MAX. (13) Appl. Sci. 2021, 11, 291 8 of 19

• The second condition for VIN provides that the voltage swing across the gyrator loading admittance Y does not exceed the maximum possible OTA2 input voltage VOTA2,MAX gm2VOTA2,MAX VIN ≤ . (14) |YIN|

Apparently, the allowed VIN according to (14) changes with frequency. It reaches the minimum value at the lowest operating frequency where |YIN| is maximal. It can be concluded that the gyrator input voltage VIN must meet both the conditions (13) and (14) simultaneously when the linear operation of both OTAs is required. If the conditions for maximum gyrator input current IIN are required, they can be easily derived from (13) and (14) by substituting VIN = IIN/|YIN|.

2.3. Guidelines for Setting of Gyrator Parameters Practical guidelines for gyrator parameter design were compiled based on the above observations. By following these rules, in addition to the correct input admittance, we also ensure its highest possible frequency and magnitude range and prevent the active elements from non-linear operation and overloading. The starting point of the design is the required value of input admittance YIN specified, e.g., by FIN and α, or magnitude at specific frequency and phase, see (5) and (6). The first set of guidelines (labeled A) can be used if there is available any value of the fractance F of the FO capacitor loading the gyrator.

A.1. Choose the same gm1 and gm2 as the highest possible values for the correct functional- ity of the selected OTAs. The maximum transconductance values can be found in OTA documentation and are usually several hundreds of µS for CMOS and several tens of mS for BJT implementations. The choice of maximal transconductances extends the operating band to low frequencies. If this is not a priority, choose the transcon- ductances equal to the required input admittance magnitude at the commonly used frequency, e.g., at the geometric mean of the operating band. A.2. Calculate the value of fractance F from the magnitude part of (6). If the value of F turns out too high for implementation, then it is possible to use the next set of rules labeled B, where a lower value of F can be set at the beginning. If F turns out too small, the only solution is to choose OTAs with higher maximum transconductances. A.3. Check the frequency range of the correct functionality of the gyrator, according to (8), (9), or (11). If the range is not sufficient, consider reducing parasitic conductance(s) by the compensation described in Section 2.2.2. A.4. To prevent the OTAs from the non-linear operation, ensure that the magnitude of the gyrator input voltage VIN meets both the conditions (13) and (14) simultaneously in the operating frequency band. The next set of rules (labeled B) is applicable if a specific value of the fractance F should be set at the beginning.

B.1. Set a specific value of the fractance F. If a cut-off frequency ωGP1 should be reached without compensation of parasitic conductance GP1, the selected value F must be at least as determined from (8). B.2. Determine the required product gm1gm2 for the selected F and given FIN from the magnitude part of (6). Evaluate whether the product is achievable by the selected OTAs. If not, F must be reduced for the design to continue (regardless of fulfilling (8) in the previous step). B.3. Choose gm2 as the highest possible value for the proper functionality of the selected OTA2. B.4. Calculate gm1 from the magnitude part of (6). If the ratio gm1/gm2 turns out excessively low, then it is possible to decrease gm2 and an increase gm1 such that their product remains the same. However, keep in mind that decreasing gm2 may decrease the allowed gyrator input voltage, according to (14). Appl. Sci. 2021, 11, 291 9 of 19

B.5. Check the frequency range of the correct functionality of the gyrator, according to (8), (9), or (11). If the range is not sufficient, consider reducing parasitic conductance(s) by the compensation described in Section 2.2.2. B.6. To prevent the OTAs from the non-linear operation, ensure that the magnitude of the gyrator input voltage VIN meets both the conditions (13) and (14) simultaneously in the operating frequency band. To add more clarity, a flowchart of the above guidelines for the setting of gyrator parameters is enclosed in Figure7.

2.4. Differential (Floating) Version of the Gyrator

If the OTA2 with differential (balanced) output is available, it is possible to easily obtain a differential gyrator with floating input according to Figure8. The describing relations (4) to (6) remain valid; however, the parasitics GP2 and CP2 must be considered with halved values GP2/2 and CP2/2 in the analyses, as the OTA terminal parasitic elements are connected in series here (through ground). These reduced parasitic values improve the Appl. Sci. 2021, 11, x FOR PEER REVIEW 10 of 21 functionality of the gyrator at high frequencies. Another advantage of the floating gyrator is that the loading element Y remains grounded.

Start with required value of YIN (FIN, α)

Any Yes realizable value of F No of FO capacitor is available

Choose highest possible Set a realizable value of F gm1 = gm2 of OTAs

Determine the required product Calculate F from gm1gm2 from magnitude part of (6) magnitude part of (6)

Reduce F

Product gm1gm2 No Yes is achievable by F is too high for selected OTAs realization

Yes Choose OTAs with No higher maximum gm Choose the highest possible gm2 for OTA2

Yes F is too small for realization Calculate gm1 from magnitude part of (6) No

Yes The ratio gm1/gm2 is Compensation according excessively low to section 2.2.2

Decrease gm2 and increase No Frequency range No acc. to (8), (9), or gm1 such that their product (11) is sufficient remains the same

Yes

Ensure that VIN meets both (13) and (14)

End of the design

Figure 7. Flowchart of the setting of gyrator parameters. Figure 7. Flowchart of the setting of gyrator parameters.

OTA2 + gm2 _

OTA1 + YIN gm1 _ Y

Figure 8. A differential (floating) variant of the gyrator from Figure 2. Appl. Sci. 2021, 11, x FOR PEER REVIEW 10 of 21

Start with required value of YIN (FIN, α)

Any Yes realizable value of F No of FO capacitor is available

Choose highest possible Set a realizable value of F gm1 = gm2 of OTAs

Determine the required product Calculate F from gm1gm2 from magnitude part of (6) magnitude part of (6)

Reduce F

Product gm1gm2 No Yes is achievable by F is too high for selected OTAs realization

Yes Choose OTAs with No higher maximum gm Choose the highest possible gm2 for OTA2

Yes F is too small for realization Calculate gm1 from magnitude part of (6) No

Yes The ratio gm1/gm2 is Compensation according excessively low to section 2.2.2

Decrease gm2 and increase No Frequency range No acc. to (8), (9), or gm1 such that their product (11) is sufficient remains the same

Yes

Ensure that VIN meets both (13) and (14)

End of the design Appl. Sci. 2021, 11, 291 10 of 19

Figure 7. Flowchart of the setting of gyrator parameters.

OTA2 + gm2 _

OTA1 + YIN gm1 _ Y

FigureFigure 8. 8.A A differentialdifferential (floating)(floating) variant of of the the gyrator gyrator from from Figure Figure 2.2 . Appl. Sci. 2021, 11, x FOR PEER REVIEW 11 of 21 3. Simulation Results and Discussion To verify the presented theoretical studies and also demonstrate the design opti- mization,3. Simulation the computerResults and simulation Discussion of the gyrator behavior was carried out using OrCAD PSpiceTo software.verify the presented The commercially theoretical availablestudies and integrated also demonstrate circuit the LT1228 design [26 optimi-] was used as OTAzation, to the enable computer the readers simulation the of possibility the gyrator of behavior reproducing was carried the results. out using This OrCAD element pro- videsPSpice transconductance software. The commerciallygm controlled available by anintegrated external circuit DC currentLT1228 [26]ISET was; whereas, used as it holds gOTAm = 10to IenableSET. An the analysis readers the of thepossibility LT1228 of PSpice reproducing model the parasitic results. propertiesThis element revealed pro- that vides transconductance gm controlled by an external DC current ISET; whereas, it holds gm the conductance of input pins GP,IN is variable and depends on gm with the relation GP,IN = 10ISET. An analysis of the LT1228 PSpice model parasitic properties revealed that the ≈ 0.005gm. It is, therefore, necessary to take into account that the functionality of the conductance of input pins GP,IN is variable and depends on gm with the relation GP,IN ≈ gyrator will be more affected by the parasitic conductance at higher values of gm. The 0.005gm. It is, therefore, necessary to take into account that the functionality of the gyrator input parasitic conductance is not included in the LT1228 PSpice model, i.e., it is considered will be more affected by the parasitic conductance at higher values of gm. The input par- zero.asitic Theconductance parasitic is conductancenot included inG theP,OUT LT1228of output PSpice pinmodel, is approximately i.e., it is considered 0.23 zero. nS and can G g beThe neglected parasitic conductance in comparison GP,OUT with of outputP,IN forpin commonis approximately values 0.23 of nSm. and The can parasitic be ne- output capacitanceglected in co ismparison 5.2 pF. Thus,with G theP,IN modeledfor common parasitic values conductancesof gm. The parasiticGP1 and outputGP2 capaci-of the gyrator correspondtance is 5.2 pF. to 0.005Thusg, m2the andmodeled 0.005 parasiticgm1, respectively, conductances and G bothP1 and the GP2 parasitic of the gyrator capacitances cor- CP1 andrespondCP2 areto 0.005 5.2 pF.gm2 and 0.005gm1, respectively, and both the parasitic capacitances CP1 and CP2 are 5.2 pF. 3.1. Gyrator Simulation and Optimization 3.1. Gyrator Simulation and Optimization The input admittance magnitude of the gyrator chosen in this subsection is |YIN| = 10 mSThe atinput 1 kHz, admittance and the magnitude orders α ofare the 0.25, gyrator 0.5, chosen 0.75, 1, in i.e., this threesubsection of them is |Y representingIN| = FO10 mS and at one 1 kHz classic, and inductancethe orders αsimulator. are 0.25, 0.5, Two 0.75, sets 1, i.e., of elementthree of them parameters representing will beFO used for demonstration.and one classic inductance The first set simulator. (labeled Two set 1)sets will of element be designed parameters only towill provide be used the for required inputdemonstration. admittance, The regardless first set (labeled of the set previously 1) will be proposeddesigned only recommendations to provide the required concerning the input admittance, regardless of the previously proposed recommendations concerning wide frequency and magnitude range. Then, the elements of set 2 will be designed regard- the wide frequency and magnitude range. Then, the elements of set 2 will be designed ingregarding these recommendations these recommendations to demonstrate to demonstrate the optimizationthe optimization of admittanceof admittance characteristics. char- Theacteristics. schematic The ofschemat the gyratoric of the in gyrator the OrCAD in the OrCAD Capture Capture editor editor is depicted is depicted in Figure in Fig-9. Unless otherwiseure 9. Unless noted, otherwise the circuit noted, is the considered circuit is withoutconsidered the without compensation the compensation part indicated part by the dashedindicated box. by the dashed box.

Compensation of GP2

FigureFigure 9. OrCAD Capture Capture schematic schematic of the of thesimulated simulated gyrator. gyrator.

To show solely the influence of parasitic properties of the employed OTAs, ideal FO capacitors were used as gyrator load Y. These elements were modeled in OrCAD by GFREQ part representing voltage-controlled current source described by a table of fre- quency response in magnitude and phase domain [27]. Interpolation is performed be- tween the entries; whereas, magnitude is interpolated logarithmically and phase linearly. By interconnecting the input and output of the GFREQ part as displayed in Figure 9, the resulting element admittance is described by a series of (input frequency in Hz, magni- tude in dB, phase in degrees) triplets. Appl. Sci. 2021, 11, 291 11 of 19

To show solely the influence of parasitic properties of the employed OTAs, ideal FO capacitors were used as gyrator load Y. These elements were modeled in OrCAD by GFREQ part representing voltage-controlled current source described by a table of frequency response in magnitude and phase domain [27]. Interpolation is performed between the entries; whereas, magnitude is interpolated logarithmically and phase linearly. By interconnecting the input and output of the GFREQ part as displayed in Figure9, the resulting element admittance is described by a series of (input frequency in Hz, magnitude in dB, phase in degrees) triplets. The OTA transconductances are chosen gm1 = gm2 = 1 mS in the first set of circuit parameters. To obtain the required |YIN| = 10 mS at 1 kHz, the magnitude of loading admittance according to (4) is |Y| = 0.1 mS at 1 kHz. The orders of the loading admittance are α = 0.25, 0.5, 0.75, 1, i.e., three of them FO and one classic capacitor. The summary of this first set of element parameters is given in Table1, along with the expected lower and upper cut-off frequencies determined by the relations (8), (9), and (11).

Table 1. First set of element parameters for simulation of the circuit in Figure9 and its expected cut-off frequencies.

g = g F f min{f , f 0 } m1 m2 α GFREQ TABLE GP1 GP2 CP2 (mS) (F·secα–1) (Hz) (Hz) 1 0.25 11.23 µ (10m,-105,22.5) (100meg,-55,22.5) 6.25 m 24.5 M 1 0.5 1.262 µ (10m,-130,45) (100meg,-30,45) 2.5 4.54 M 1 0.75 141.7 n (10m,-155,67.5) (100meg,-5,67.5) 18.4 1.36 M 1 1 15.92 n Classic capacitor 15.92 nF instead of GFREQ 50 553 k

The second set of gyrator element parameters is designed following rules A from Section 2.3, which should provide optimization of both the frequency and dynamic range of the gyrator input admittance. The maximum transconductances of LT1228 OTA are chosen based on the datasheet gm1 = gm2 = 20 mS. To obtain the required |YIN| = 10 mS at 1 kHz, the magnitude of loading admittance is |Y| = 40 mS at 1 kHz. The summary of this second set of element parameters with the expected cut-off frequency values is given in Table2.

Table 2. The second set of element parameters for simulation of the circuit in Figure9 and its expected cut-off frequencies.

g = g F f min{f , f 0 } m1 m2 α GFREQ TABLE GP1 GP2 CP2 (mS) (F·secα–1) (Hz) (Hz) 20 0.25 4.493 m (10m,-52.96,22.5) (100meg,-2.96,22.5) 39.1 n 24.5 M 20 0.5 504.6 µ (10m,-77.96,45) (100meg,22.04,45) 6.25 m 4.54 M 20 0.75 56.68 µ (10m,-102.96,67.5) (100meg,47.04,67.5) 339 m 464 k (918 k) 20 1 6.366 µ Classic capacitor 6.366 µF instead of GFREQ 2.5 100 k (391 k)

The values of lower cut-off frequency f GP1 for the second set are lower than for the first set, as expected. As seen in the last column of Table2 (values that are not in parentheses), the upper cut-off frequencies decreased for α = 0.75 and 1 compared with Table1. This bandwidth degradation is caused by the parasitic conductance GP2, which raised from 5 µS for the first element set to 100 µS for the second set, due to increasing the transconductance gm1 (it holds GP2 = 0.005gm1). The parasitic conductance GP2 can be decreased as described in Section 2.2.2, resulting in an increase of the upper cut-off frequency. The compensation circuit with negative admittance utilized for this purpose is marked with the dashed box in Figure9. The transconductance gmC of the OTA in the compensation circuit was set to 99.5 µS, which leads to GP2 ≈ 1 µS and that is an even better value than for the first set of elements. It should be taken into account that the compensation circuit causes a doubling of the parasitic capacitance CP2, therefore, the results after compensation have improved, but still do not reach the values as for the first set, see the frequencies in the last column of Table2 in parentheses. Appl. Sci. 2021, 11, x FOR PEER REVIEW 13 of 21

The solid black lines are valid for the first set of circuit parameters, the dotted green lines for the second set, and the dashed blue lines indicate ideal results. It is worth to mention again that the simulations in this subsection are performed with ideal FO capacitor loading the gyrator to solely illustrate the effect of OTA non-ideal properties. Common FO capacitors emulated by RC structures of reasonable complexity have limited accuracy in a limited frequency band [28], which would unnecessarily degrade the results, and it would not be clear whether the distortion is caused by non-idealities of OTA or FO ca- pacitor. The simulated admittance characteristics in Figures 10–13 are in accordance with the Bode plots in Figure 5, and the important asymptotic magnitude and phase values are confirmed. The limited frequency band of the correct functionality of the gyrator can be observed, and the simulated magnitude cut-off frequencies correspond to the expected values from Tables 1 and 2. Considering the maximum phase error ±2 degrees, the useful frequency ranges for the designed FO inductance simulators are listed in Table 3.

Table 3. Comparison of frequency ranges of FO inductance simulators for maximum phase error ±2 Appl. Sci. 2021, 11, 291 degrees. 12 of 19

Set α = 0.25 α = 0.5 α = 0.75 α = 1 1 62 Hz–395 kHz 950–1 MHz 1.44 kHz–226 kHz 1.45 kHz–72 kHz 2The 0.4 simulated mHz–2.43 magnitudeMHz 2.4 Hz and–24.6 phase-frequency kHz 26 Hz– characteristics173 kHz 71 ofHz the–148 gyrator kHz input admittance are depicted in Figures 10–13 for the α values 0.25, 0.5, 0.75, 1, respectively. The solid blackFurthermore, lines are the valid simulations for the first confirm set of that circuit the parameters,admittance magnitude the dotted at greenlow fre- lines for quencies is limited to the value gm1gm2/GP1, which is 200 mS for the first set of elements the second set, and the dashed blue lines indicate ideal results. It is worth to mention again and 4 S for the second one. The constant magnitude section between ωGP2 and ωCP2 is not that the simulations in this subsection are performed with ideal FO capacitor loading the apparent in the simulations as the value GP2 is too low to take effect. This section would gyrator to solely illustrate the effect of OTA non-ideal properties. Common FO capacitors be visible for the set 2 magnitudes with α values 0.75 and 1 if no GP2 compensation were emulatedapplied. byThe RC effect structures of the additional of reasonable capacitance complexity of the compensation have limited circui accuracyt is demon- in a limited frequencystrated in band Figures [28 12a], which and 13 woulda by increasing unnecessarily the magnitude degrade at thehigh results, frequencies and itcompared would not be clearto set whether 1 results. the The distortion simulated is causedresults also by non-idealitiesconfirm that the of bandwidth OTA or FO of capacitor.correct gyrator operation increases as α decreases. 1000 α = 0.25 set 1 100 100 α = 0.25 set 2 80

α = 0.25 ideal ) 60 mS ) ( } deg ( IN 10 40 } IN

{Y 20 1 e 0 Phas

Magnitude{ Y −20 0.1 −40

0.01 −60 0.00001 0.001 0.1 10 1000 100,000 0.00001 0.001 0.1 10 1000 100,000 Frequency (kHz) Frequency (kHz) (a) (b)

FigureFigure 10. Simulated 10. Simulated magnitude magnitude (a )(a and) and phase phase ((bb)) characteristicscharacteristics of of gyrator gyrator input input admittance admittance for α for = 0.25.α = 0.25.

The simulated admittance characteristics in Figures 10–13 are in accordance with the Bode plots in Figure5, and the important asymptotic magnitude and phase values are confirmed. The limited frequency band of the correct functionality of the gyrator can be observed, and the simulated magnitude cut-off frequencies correspond to the expected values from Tables1 and2. Considering the maximum phase error ±2 degrees, the useful frequency ranges for the designed FO inductance simulators are listed in Table3. Furthermore, the simulations confirm that the admittance magnitude at low frequen- cies is limited to the value gm1gm2/GP1, which is 200 mS for the first set of elements and 4 S for the second one. The constant magnitude section between ωGP2 and ωCP2 is not apparent in the simulations as the value GP2 is too low to take effect. This section would be visible for the set 2 magnitudes with α values 0.75 and 1 if no GP2 compensation were applied. The effect of the additional capacitance of the compensation circuit is demonstrated in Figures 12a and 13a by increasing the magnitude at high frequencies compared to set Appl. Sci. 2021, 11, x FOR PEER REVIEW 14 of 21 1 results. The simulated results also confirm that the bandwidth of correct gyrator opera- tion increases as α decreases.

1000 100 α = 0.5 set 1 80 100 α = 0.5 set 2 60

α = 0.5 ideal ) mS ) ( }

deg 40 (

IN 10 }

IN 20 {Y e 1 0 Phas

Magnitude{ Y −20 0.1 −40

0.01 −60 0.0001 0.001 0.01 0.1 1 10 100 1000 10,000 100,000 0.0001 0.001 0.01 0.1 1 10 100 1000 10,000 100,000 Frequency (kHz) Frequency (kHz) (a) (b)

FigureFigure 11. Simulated 11. Simulated magnitude magnitude (a) and(a) and phase phase (b) ( characteristicsb) characteristics of of gyrator gyrator input input admittance admittance forfor α = 0.5. 0.5.

10,000 100 α = 0.75 set 1 80 1000 α = 0.75 set 2 60

α = 0.75 ideal ) mS ) ( 100 40 } deg ( IN } 20 10 IN {Y 0 e 1 −20 Phas

Magnitude{ Y −40 0.1 −60 0.01 −80 0.0001 0.001 0.01 0.1 1 10 100 1000 10,000 100,000 0.0001 0.001 0.01 0.1 1 10 100 1000 10,000 100,000 Frequency (kHz) Frequency (kHz) (a) (b) Figure 12. Simulated magnitude (a) and phase (b) characteristics of gyrator input admittance for α = 0.75.

10,000 100 α = 1 set 1 80 1000 α = 1 set 2 60

100 α = 1 ideal )

mS ) 40 ( } deg (

IN 20

10 } IN 0 {Y

1 e −20

0.1 Phas −40 Magnitude{ Y −60 0.01 −80 0.001 −100 0.0001 0.001 0.01 0.1 1 10 100 1000 10,000 100,000 0.0001 0.001 0.01 0.1 1 10 100 1000 10,000 100,000 Frequency (kHz) Frequency (kHz) (a) (b) Figure 13. Simulated magnitude (a) and phase (b) characteristics of gyrator input admittance for α = 1.

3.2. Inductance Simulator for Fractional-Order Respiratory Model Fractional-order electrical circuit models are useful in modeling the mechanical impedance of the respiratory system [11,12]. Mechanical impedance of the human res- piratory system can be represented by an equivalent model given in Figure 14. Appl. Sci. 2021, 11, x FOR PEER REVIEW 14 of 21 Appl. Sci. 2021, 11, x FOR PEER REVIEW 14 of 21

1000 100 1000 α = 0.5 set 1 100 α = 0.5 set 1 80 100 α = 0.5 set 2 80 100 α = 0.5 set 2 60 α = 0.5 ideal ) mS )

( 60

α = 0.5 ideal ) } mS ) deg 40 ( (

IN 10 } }

deg 40 ( IN 10 IN } 20 {Y IN

e 20 1 {Y 0 1 e

Phas 0

Magnitude{ Y −20 0.1 Phas Magnitude{ Y −20 0.1 −40 −40 0.01 −60 Appl. Sci. 2021, 11, 291 13 of 19 0.010.0001 0.001 0.01 0.1 1 10 100 1000 10,000 100,000 −600.0001 0.001 0.01 0.1 1 10 100 1000 10,000 100,000 0.0001 0.001 0.01 0.1Frequency1 10 (kHz)100 1000 10,000 100,000 0.0001 0.001 0.01 0.1Frequency1 10 (kHz)100 1000 10,000 100,000 Frequency(a) (kHz) Frequency(b) (kHz) (a) (b) Figure 11. Simulated magnitude (a) and phase (b) characteristics of gyrator input admittance for α = 0.5. Figure 11. Simulated magnitude (a) and phase (b) characteristics of gyrator input admittance for α = 0.5. 10,000 100 10,000 100 α = 0.75 set 1 80 α = 0.75 set 1 1000 α = 0.75 set 2 80 1000 α = 0.75 set 2 60

α = 0.75 ideal ) 60 mS )

( 100 α = 0.75 ideal ) 40 } mS ) deg ( (

IN 100 40 } } deg 20 ( IN IN 10 } 20 {Y IN 0

10 e {Y 0 1 e −20 1 Phas −20 Magnitude{ Y Phas −40 Magnitude{ Y 0.1 −40 0.1 −60 −60 0.01 −80 0.010.0001 0.001 0.01 0.1 1 10 100 1000 10,000 100,000 −800.0001 0.001 0.01 0.1 1 10 100 1000 10,000 100,000 0.0001 0.001 0.01 0.1Frequency1 10 (kHz)100 1000 10,000 100,000 0.0001 0.001 0.01 0.1Frequency1 10 (kHz)100 1000 10,000 100,000 Frequency(a) (kHz) Frequency(b) (kHz) (a) (b) FigureFigure 12. 12.Simulated Simulated magnitude magnitude (a) ( anda) and phase phase (b) ( characteristicsb) characteristics of gyratorof gyrator input input admittance admittance for forα = α 0.75. = 0.75. Figure 12. Simulated magnitude (a) and phase (b) characteristics of gyrator input admittance for α = 0.75. 10,000 100 10,000 100 α = 1 set 1 80 1000 α = 1 set 1 80 1000 α = 1 set 2 60 α = 1 set 2 60 100 α = 1 ideal )

mS ) 40 (

α = 1 ideal )

} 100 mS ) deg 40 ( (

IN 20 } 10 } deg ( IN IN 20 10 } 0 {Y IN

1 e 0 {Y −20

1 e −20 0.1 Phas −40 Magnitude{ Y Phas −40 0.1 −60 Magnitude{ Y 0.01 −60 0.01 −80 −80 0.001 −100 0.0010.0001 0.001 0.01 0.1 1 10 100 1000 10,000 100,000 −1000.0001 0.001 0.01 0.1 1 10 100 1000 10,000 100,000 0.0001 0.001 0.01 0.1Frequency1 10 (kHz)100 1000 10,000 100,000 0.0001 0.001 0.01 0.1Frequency1 10 (kHz)100 1000 10,000 100,000 Frequency(a) (kHz) Frequency(b) (kHz) (a) (b) Figure 13. Simulated magnitude (a) and phase (b) characteristics of gyrator input admittance for α = 1. FigureFigure 13. 13.Simulated Simulated magnitude magnitude (a )(a and) and phase phase (b )(b characteristics) characteristics of gyratorof gyrator input input admittance admittance for α for= 1.α = 1. 3.2. Inductance Simulator for Fractional-Order Respiratory Model 3.2. Inductance Simulator for Fractional-Order Respiratory Model Table 3.FractionalComparison-order of frequency electrical ranges circuit of FOmodels inductance are useful simulators in formodeling maximum the phase mechanical error Fractional-order electrical circuit models are useful in modeling the mechanical ±2impedance degrees. of the respiratory system [11,12]. Mechanical impedance of the human res- impedancepiratory system of the can respirator be representedy system by [11,12].an equivalent Mechanical model impedance given in Figure of the 14. human res- piratorySet system can beα = represented 0.25 by anα = equivalent 0.5 modelα = 0.75given in Figure α14.= 1 1.44 kHz–226 1 62 Hz–395 kHz 950–1 MHz 1.45 kHz–72 kHz kHz 0.4 mHz–2.43 2 2.4 Hz–24.6 kHz 26 Hz–173 kHz 71 Hz–148 kHz MHz

3.2. Inductance Simulator for Fractional-Order Respiratory Model Fractional-order electrical circuit models are useful in modeling the mechanical impedance of the respiratory system [11,12]. Mechanical impedance of the human respira- tory system can be represented by an equivalent model given in Figure 14. Appl. Sci. 2021, 11, 291 14 of 19 Appl. Sci. 2021, 11, x FOR PEER REVIEW 16 of 21 Appl. Sci. 2021, 11, x FOR PEER REVIEW 16 of 21

1 α  s L R  sαL R sC

Figure 14. FO impedance model of the human respiratory system [11]. FigureFigure 14. 14.FO FO impedance impedance model model of of the the human human respiratory respiratory system system [11 [11].]. The impedance of this system is expressed as TheThe impedance impedance of of this this system system is is expressed expressed as as

 1 Z s s L  R 1 1 . (18) ( ) = α + +   Z s Zs sL s LR R sC .. (15)(18) sβsCC The experimentally extracted values of (18) for human respiratory system are L = TheThe experimentally experimentally extractedextracted values of (18) (15) for for human human respiratory respiratory system system are are L = 0.0193 H·sec−0.5313− , α = 0.4687, R = 0.0147 Ω, C = 0.9921 F·sec−0.2729,− β = 0.7271 and the fre- L 0.0193= 0.0193 H· Hsec·sec−0.53130.5313, α =, 0.4687,α = 0.4687, R = 0.0147R = 0.0147 Ω, CΩ =, 0.9921C = 0.9921 F·sec F−·0.2729sec, β0.2729 = 0.7271, β = 0.7271and the and fre- quency range of the model validity is 4 Hz to 48 Hz [12]. The FO inductor with imped- thequency frequency range range of the of model the model validity validity is 4 Hz is 4to Hz48 toHz 48 [12]. Hz The [12]. FO The inductor FO inductor with imped- with ance sαL includedα in this model will be implemented by the discussed OTA-based gyrator impedanceance sαL includeds L included in this in model this model will be will implemented be implemented by the bydiscussed the discussed OTA-ba OTA-basedsed gyrator in this section as an example of its practical utilization. The grounded version of the gyratorin this in section this section as an as example an example of its of practical its practical utilization. utilization. The The grounded grounded version version of of the model will be demonstrated, as the floating version requires OTA with differential out- themodel model will will be bedemonstrated, demonstrated, as the as the floating floating version version requires requires OTA OTA with with differential differential out- put (see Figure 8), which is not the case of the used integrated circuit LT1228. The value outputput (see (see Figure Figure 8),8), which which is not the casecase ofof thethe usedused integrated integrated circuit circuit LT1228. LT1228. The The value value of FIN in accordance with (5) and (18) is equal to 1/L = 51.81 H−1·−sec1 0.53130.5313. This value repre- ofofF INFINin in accordanceaccordance withwith (5)(5) andand (18) (15) is equal equal to to 1/ 1/LL = =51.81 51.81 H H−1·sec·0.5313sec . This. Thisvalue value repre- sents a very high admittance magnitude of 11.43 S at the corner frequency of 4 Hz, and representssents a very a very high high admittance admittance magnitude magnitude of 11.43 of 11.43 S at S the at thecorner corner frequency frequency of 4 of Hz 4, Hz, and the implementation of the inductor is therefore challenging. andthe the implementation implementation of the of the inductor inductor is therefore is therefore challenging. challenging. Rules A, in Section 2.3, will be used to design the gyrator element parameters. The RulesRules A, A in, in Section Section 2.3 2.3, will, will be be used used to to design design the the gyrator gyrator element element parameters. parameters. The The transconductances are chosen gm1 = gm2 = 20 mS again as the maximum possible values for transconductancestransconductances are are chosen choseng gm1m1 == gm2 == 20 20 mS mS again again as as the the maximum maximum possible possible values values for LT1228. To obtain the required |YIN|Y = 11.43 S at 4 Hz, the magnitude of loading admit- forLT1228. LT1228. To Toobtain obtain the the required required |YIN || IN= 11.43| = 11.43 S at 4 S Hz, at 4 the Hz, magnitude the magnitude of loading of loading admit- tance is |Y| = 34.987 μS at 4 Hz, its order is α = 0.4687, and F = 7.72 μF·sec−0.5313. Thanks−0.5313 to admittancetance is |Y is| = | 34.987Y| = 34.987 μS at 4µ SHz, at 4its Hz, order its orderis α = 0 is.4687,α = 0.4687, and F = and 7.72F μ=F·sec 7.72−0.5313µF·sec. Thanks to. the narrow frequency band (4 Hz to 48 Hz) compared to the simulations in the previous Thanksthe narrow to the frequency narrow frequency band (4 Hz band to 48 (4 Hz Hz) compared to 48 Hz) comparedto the simulations to the simulations in the previous in subsection and to implement it by available components, the FO capacitor loading the thesubse previousction and subsection to implement and to implementit by available it by components, available components, the FO capacitor the FO loading capacitor the gyrator was emulated by the Valsa structure [29] containing standard resistors and ca- loadinggyrator the was gyrator emulated was emulatedby the Valsa by the structure Valsa structure [29] containing [29] containing standard standard resistors resistors and ca- pacitors, as shown in Figure 15. Another reason to employ this RC emulator is that the andpacitors capacitors,, as shown as shown in Figure in Figure 15. Another15. Another reason reason to employ to employ this this RC RC emulator emulator is that is that the theGFRE GFREQQ part part cannot cannot be beused used in intransient transient analysis analysis,, which which will will demonstrate demonstrate dynamic dynamic op- GFREQ part cannot be used in transient analysis, which will demonstrate dynamic op- optimizationtimization of of the the gyrator gyrator at at the end of thisthis section.section. TheThe elementelement valuesvaluesof of the the Valsa Valsa RC RC timization of the gyrator at the end of this section. The element values of the Valsa RC circuitcircuit computed computed by by the the method method described described in Referencein Reference [29 [29]] for for the the mentioned mentioned values valuesα, F ,α, circuit computed by the method described in Reference [29] for the mentioned values α, maximumF, maximum phase phase error error±0.6 ±0.6 degrees, degrees, and and the the frequency frequency range range 0.8 0.8 Hz Hz to to 240 240 Hz Hz (to (to have have F, maximum phase error ±0.6 degrees, and the frequency range 0.8 Hz to 240 Hz (to have sufficientsufficient margin) margin) are are listed listed in in Table Table4. 4. sufficient margin) are listed in Table 4.

R1 R2 R3 R4 C0 R1 R2 R3 R4 0 R0 C R0 C1 C2 C3 C4 C1 C2 C3 C4

Figure 15. Valsa RC circuit to emulate FO capacitor. Figure 15. Valsa RC circuit to emulate FO capacitor. Figure 15. Valsa RC circuit to emulate FO capacitor. Table 4. Resistances and capacitances in the circuit from Figure 15 (α = 0.4687, F = 7.72 μF·sec−0.5313). Table 4. Resistances and capacitances in the circuit from Figure 15 (α = 0.4687, F = 7.72 μF·sec−0.5313). Table 4. Resistances and capacitances in the circuit from Figure 15( α = 0.4687, F = 7.72 µF·sec−0.5313). R0 (kΩ) R1 (kΩ) R2 (kΩ) R3 (kΩ) R4 (kΩ) R0 (kΩ) R1 (kΩ) R2 (kΩ) R3 (kΩ) R4 (kΩ) R0 (k150Ω) R1 (k100Ω) R2 (k39Ω ) R3 (k16Ω ) R4 (k6.8Ω) 150 100 39 16 6.8 C1500 (nF) C 1001 (μF) C 392 (nF) C 163 (nF) C 6.84 (nF) C0 (nF) C1 (μF) C2 (nF) C3 (nF) C4 (nF) C0 (nF)56 C1 (µ2F) C2 (nF)750 C3 (nF)270 C4 (nF)100 5656 22 750750 270270 100100 The admittance magnitude and phase characteristics of the emulated FO capacitor The admittance magnitude and phase characteristics of the emulated FO capacitor (solid black lines) and the ideal values (dashed blue lines) are presented in Figure 16. It (solidThe black admittance lines) and magnitude the ideal and values phase (dashed characteristics blue lines) of the are emulated presented FO capacitor in Figure (solid 16. It can be observed that the circuit operates according to the above-mentioned required pa- blackcan be lines) observed and the that ideal the values circuit (dashed operates blue according lines) are to the presented above- inmentioned Figure 16 required. It can bepa- rameters. observedrameters. that the circuit operates according to the above-mentioned required parameters. Appl. Sci. 2021, 11, 291 15 of 19

The gyrator lower cut-off frequency computed by (8) is f GP1 = 37.6 Hz, which is not sufficiently low, as the FO inductor should operate from 4 Hz. This limitation is caused by the parasitic conductance GP1 = 100 µS, which will be decreased by the circuit with negative admittance. Thus, now the compensation circuit in the dashed box in Figure9 is replaced and connected in parallel with the FO capacitor loading the gyrator. The transconductance gmC is 97 µS, and it leads to GP1 ≈ 3.5 µS. The expected cut-off frequency after compensation is f GP1 ≈ 0.03 Hz and maximum obtainable input admittance magnitude gm1gm2/GP1 ≈ 115 S, which is ten times higher than the required admittance magnitude of the implemented FO inductor at 4 Hz. Note that these values are valid for the gyrator loaded with an ideal

Appl. Sci. 2021, 11, x FOR PEER REVIEWFO capacitor. It can be expected that the results will be degraded after connecting17 of 21 the emulator of FO capacitor, which has limited accuracy of its admittance.

10,000 Emulated Ideal

) 1000

S

μ

( Y} 100

Magnitude{ 10

1 0.01 0.1 1 10 100 1000 10,000 Frequency (Hz) (a)

80

) 60

deg

( {Y}

e 40 Phas

20

0 0.01 0.1 1 10 100 1000 10,000 Frequency (Hz) (b)

Figure 16. 16. AdmittanceAdmittance magnitude magnitude (a) ( aand) and phase phase (b) (characteristicsb) characteristics of the of approximated the approximated FO capac- FO capacitor itor from Figure 15. from Figure 15.

The gyrator lower cut-off frequency computed by (8) is fGP1 = 37.6 Hz, which is not The simulated magnitude and phase frequency characteristics of the gyrator input sufficiently low, as the FO inductor should operate from 4 Hz. This limitation is caused admittance are shown in Figure 17. The solid black lines represent simulated results, and by the parasitic conductance GP1 = 100 μS, which will be decreased by the circuit with thenegative blue admittance. lines indicate Thus ideal, now values the compensation with the solid circuit part in the denoting dashed thebox frequencyin Figure 9 is band of interest,replaced i.e.,and 4connected Hz to 48 in Hz. parallel with the FO capacitor loading the gyrator. The trans- conductanceIn this frequencygmC is 97 μS band,, and it theleads maximum to GP1 ≈ 3.5 absolute μS. The expected magnitude cut- erroroff frequency is 1.5 S, after and phase errorcompensation is 3.1 degrees is fGP1 when ≈ 0.03 comparing Hz and maximum the simulated obtainable results input with admittance the ideal ones. magnitude It is apparent thatgm1gm2 the/GP1 error ≈ 115 increases S, which at is the ten lowertimes boundhigher than of the the frequency required band.admittance The magnitudemagnitude errorof can bethe easilyimplemented corrected FO by inductor fine increasing at 4 Hz. Note the product that thesegm1 valuesgm2, whichare valid can for be the set gyrator electronically. Theloaded phase with deviation an ideal FO can capacitor. be reduced It can by be fine-tuning expected that the the transconductance results will be degradedgmC; however, theafter demands connecting on the the emulator accuracy of FO of capacitor setting g, mCwhichincrease, has limited and accuracy it is important of its admit- to prevent overcompensatingtance. GP1 to a negative value. TheThe simu dynamiclated magnitude optimization and was phase also frequency performed characteristics based on theof the conditions gyrator input introduced inadmittance Section 2.2.3are shown. When in Figure following 17. The these solid conditions, black lines represent only allowed simulated voltages results and, and currents the blue lines indicate ideal values with the solid part denoting the frequency band of interest, i.e., 4 Hz to 48 Hz. Appl. Sci. 2021, 11, 291 16 of 19

will appear in the gyrator structure, and the amplifiers will operate in linear range with- out overloading. The maximum input voltage amplitude of both OTAs for their linear operation is considered VOTA,MAX = 50 mV. The maximum input voltage of the gyrator can be determined as the maximum value of VIN that meets both conditions (13) and (14) simultaneously. The maximum input current can then be computed using IIN = VIN|YIN|. The following Table5 shows the maximum allowed amplitudes of voltages and currents at the input (VIN, IIN) of the gyrator, ensuring linear operation of both OTAs for three frequency points, in the beginning, inside (at 25 Hz), and at the end of the operating band. Appl. Sci. 2021, 11, x FOR PEER REVIEWThe computed amplitudes of voltages and currents at the gyrator load (V , I 18) of are 21 LOAD LOAD also stated.

100 Simulated

Ideal

)

S (

} 10

IN Y

1 Magnitude{

0.1 0.1 1 10 100 1000 Frequency (Hz) (a) −10

−20

) deg

( −30

}

IN {Y

e −40 Phas −50

−60 0.1 1 10 100 1000 Frequency (Hz) (b) Figure 17. Simulated magnitude (a) and phase (b) characteristics of gyrator input admittance im- Figure 17. Simulated magnitude (a) and phase (b) characteristics of gyrator input admittance plementing FO inductor for the respiratory model. implementing FO inductor for the respiratory model. In this frequency band, the maximum absolute magnitude error is 1.5 S, and phase Tableerror 5.isMaximum 3.1 degrees allowed when voltages comparing and currents the simulated in the gyrator results structure. with the ideal ones. It is ap- parent that the error increases at the lower bound of the frequency band. The magnitude f (Hz) VIN (µV) IIN (mA) VLOAD (mV) ILOAD (µA) error can be easily corrected by fine increasing the product gm1gm2, which can be set elec- 4 87.5 1 50 1.75 tronically. The phase deviation can be reduced by fine-tuning the transconductance gmC; 25 206 1 50 4.13 however, the demands on the accuracy of setting gmC increase, and it is important to 48 280 1 50 5.61 prevent overcompensating GP1 to a negative value. The dynamic optimization was also performed based on the conditions introduced in SectionApplying 2.2.3. the When rules for following dynamic these optimization, conditions, the only transient allowed simulation voltages with andthe currents maxi- mumwill appear allowed in input the gyratorcurrent amplitude structure, IandIN = the 1 mA amplifiers and frequency will operate 25 Hz was in linear carried range out, andwithout the resulting overloading. current The and maximum voltage waveforms input voltage are presented amplitude in of Figure both 18 OTAs. The voltagesfor their are relative to ground, and currents are considered positive when flowing into the gyrator linear operation is considered VOTA,MAX = 50 mV. The maximum input voltage of the gy- (IIN) and into the loading FO capacitor (ILOAD). rator can be determined as the maximum value of VIN that meets both conditions (13) and (14) simultaneously. The maximum input current can then be computed using IIN = VIN|YIN|. The following Table 5 shows the maximum allowed amplitudes of voltages and currents at the input (VIN, IIN) of the gyrator, ensuring linear operation of both OTAs for three frequency points, in the beginning, inside (at 25 Hz), and at the end of the operating band. The computed amplitudes of voltages and currents at the gyrator load (VLOAD, ILOAD) are also stated.

Appl. Sci. 2021, 11, 291 17 of 19

The delay between input current and voltage found by the simulation is 4.53 msec; the corresponding α is 0.453 which is close to the required value of 0.4687. The amplitude of the simulated input voltage 224 µV is slightly higher than the value 206 µV according to Table5. This matches the fact that the gyrator input admittance magnitude is lower than the target value, which is also observed in Figure 17a. The current through the gyrator load reaches 4.42 µA, also slightly higher than the expected 4.13 µA. The simulated load voltage of 50.8 mV is almost equal to the value of 50 mV in Table5. It is verified that the voltage and current amplitudes of the gyrator are optimized and do not cause the non-linear function Appl. Sci. 2021, 11, x FOR PEER REVIEW 19 of 21 of active elements. It should be noted here that this particular FO inductor for the respiratory model is characterized by uncommonly high admittance magnitude at low frequency. Therefore, whenTable 5. it Maximum is excited allowed by a current voltages ofand 1 currents mA, a veryin the smallgyrator voltage structure. of less than 300 µV appears at

its terminals.f (Hz) Moreover,VIN (μV) the gyratorIIN (mA) load currentV isLOAD very (mV) small, of theI orderLOAD (μA) of microamperes. Thus,4 in the case87.5 of real implementation1 and measurement,50 these small1.75 electrical signals would25 probably be206 disturbed by1 noise. In this case,50 it would be more4.13 appropriate to use OTAs48 with higher280 transconductance 1 and maximum50 output current or at5.61 least with a higher maximum voltage and current at the same transconductance. In general, it is suitable for the gyratorApplying to select the rules OTAs for with dynamic transconductance optimization, the values transient comparable simulation to the with required the input admittancemaximum allowed magnitude input atcurrent the operating amplitude frequency.IIN = 1 mA and This frequency will provide 25 Hz comparable was carried voltage amplitudesout, and the atresulting the input current and and load voltage of the waveforms gyrator, as are well presented as comparable in Figure 18. currents The at the samevoltages points. are relative to ground, and currents are considered positive when flowing into the gyrator (IIN) and into the loading FO capacitor (ILOAD).

0.3 Input voltage Input current 1.5

0.2 1

V) mA)

m 0.1 0.5

0 0

−0.1 −0.5

Input Input current ( Input Input voltage ( −0.2 −1

−0.3 −1.5 0 10 20 30 40 50 60 70 80 Time (msec) (a) 60 Load voltage Load current 6 5 40 4

3

A) V)

20 2 μ m 1 0 0 −1

−20 −2 Load current ( Loadvoltage ( −3 −40 −4 −5 −60 −6 0 10 20 30 40 50 60 70 80

Time (msec) (b) Figure 18. Optimized voltage and current waveforms at the input (a) and load (b) of gyrator ex- Figure 18. Optimized voltage and current waveforms at the input (a) and load (b) of gyrator excited cited with 1 mA current amplitude. with 1 mA current amplitude. The delay between input current and voltage found by the simulation is 4.53 msec; the corresponding α is 0.453 which is close to the required value of 0.4687. The amplitude of the simulated input voltage 224 μV is slightly higher than the value 206 μV according to Table 5. This matches the fact that the gyrator input admittance magnitude is lower than the target value, which is also observed in Figure 17a. The current through the gy- rator load reaches 4.42 μA, also slightly higher than the expected 4.13 μA. The simulated load voltage of 50.8 mV is almost equal to the value of 50 mV in Table 5. It is verified that Appl. Sci. 2021, 11, 291 18 of 19

4. Conclusions In this article, we presented a comprehensive analysis of the OTA-based gyrator implementing FO inductor to identify and evaluate the limitations caused by the active element real properties, particularly affecting the operational frequency bandwidth, range of the obtainable input admittance, and signal dynamic range. Based on this analysis, design recommendations were provided to overcome or minimize these limitations caused by the real properties of the assumed active elements. During the gyrator design, it is important to set sufficiently large OTA transconductances that are preferably comparable with the required gyrator input admittance magnitude. This also leads to sufficiently large fractance F, which reduces the circuit sensitivity to the parasitics GP1 and CP1. To further improve the properties of the gyrator, it is possible to suppress the parasitic conductances by connecting a shunt active compensation circuit with negative conductance also designed with OTA. It is important to take into account that the FO capacitor connected to the gyrator also has limited accuracy and frequency bandwidth. Therefore, it is advisable to match the operating frequency bands of the FO capacitor and the gyrator to avoid unnecessary degradation of the FO inductor function. The proposed optimization steps were verified by OrCAD PSpice simulations while designing FO inductors with different orders; whereas, the FO inductor 0.0193 H·sec−0.5313 with order 0.4687 being part of fractional-order respiratory model operating in the frequency range 4 Hz to 48 Hz was also successfully emulated.

Author Contributions: Conceptualization, N.H.; Formal analysis, D.K. and J.K.; Funding acquisition, J.K.; Investigation, D.K., N.H. and R.S.; Methodology, D.K.; Validation, J.D. and R.S.; Visualization, J.D.; Writing—original draft, D.K. and J.K. All authors have read and agreed to the published version of the manuscript. Funding: This research and the APC were funded by The Czech Science Foundation, grant number 19-24585S. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Data is contained within the article. Acknowledgments: This article is based upon work from COST Action CA15225, a network sup- ported by COST (European Cooperation in Science and Technology). Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

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