electronics

Article Memristors as Candidates for Replacing Digital Potentiometers in Electric Circuits

Ivo Markovi´c,Milka Potrebi´c* and Dejan Toši´c

School of Electrical Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, P.O. Box 35-54, 11120 Belgrade, Serbia; [email protected] (I.M.); [email protected] (D.T.) * Correspondence: [email protected]

Abstract: Digital potentiometers are substantial components for the design of many mixed-signal electronic circuits and systems. Their capability to program resistance value almost instantly provides hardware designers an additional level of freedom. Unfortunately, this feature is limited to DC and lower frequencies, due to parasitic effects. Nowadays, memristors as continuously tunable resistors are becoming candidates for potentiometer successors. Memristors are two-terminal non-volatile devices which have less significant parasitic effects and a wide resistance range. The memristance value can be changed on the fly. Using nanotechnology, memristor implementation has a nanoscale footprint with nanosecond transition between resistive states. In this paper, we present a comparison between the frequency characteristics of digital potentiometers and the only commercially available memristors. Memristor parasitic effects dominate at higher frequencies which extends the bandwidth. In order to present the advantages of memristive circuits, we have analyzed and implemented tunable circuits such as a voltage divider, an inverting amplifier, a high-pass filter, and a phase shifter. A commercially available memristor by KnowM Inc. is used for this purpose. Experimental results obtained by the measurements verify that a memristor has equal or better characteristics than a digital potentiometer. Memristive realizations of voltage dividers and inverting amplifiers have a



wider bandwidth, while filters and phase shifters with a memristor have almost identical frequency
 characteristics as the corresponding realizations with a digital potentiometer.

Citation: Markovi´c,I.; Potrebi´c,M.; Keywords: digital potentiometer; frequency characteristics; memristive circuit; memristor; tunable circuit Toši´c,D. Memristors as Candidates for Replacing Digital Potentiometers in Electric Circuits. Electronics 2021, 10, 181. https://doi.org/ 10.3390/electronics10020181 1. Introduction The memristor is the fourth fundamental circuit element. It was envisioned and Received: 13 December 2020 postulated by Chua [1]. The four fundamental two-terminal (one-port) elements of electric Accepted: 12 January 2021 circuits can be axiomatically defined via a constitutive relation between a pair of circuit Published: 15 January 2021 variables [2]. The four basic circuit variables are current i(t), voltage v(t), flux ϕ(t), and charge q(t). By definition, Publisher’s Note: MDPI stays neu- dq(t) i(t) = (1) tral with regard to jurisdictional clai- dt ms in published maps and institutio- t t nal affiliations. Z Z q(t) = i(τ)dτ = q0 + i(τ)dτ (2)

−∞ t0

t0 Copyright: © 2021 by the authors. Li- Z censee MDPI, Basel, Switzerland. q0 = i(τ)dτ (3) This article is an open access article −∞ distributed under the terms and con- q0 is the initial state of q(t) at the initial time t = t0, ditions of the Creative Commons At- tribution (CC BY) license (https:// dϕ(t) v(t) = (4) creativecommons.org/licenses/by/ dt 4.0/).

Electronics 2021, 10, 181. https://doi.org/10.3390/electronics10020181 https://www.mdpi.com/journal/electronics Electronics 2021, 10, x FOR PEER REVIEW 2 of 20

dq(t) i(t)  (1) dt t t q(t)  i()d  q0  i()d (2)  t0

t0 q0  i()d (3) 

q0 is the initial state of q(t) at the initial time t  t0 , d(t) v(t)  (4) dt Electronics 2021, 10, 181 2 of 18 t t (t)  v()d  0  v()d (5) t t0t Z Z ϕ(t) = v(τ)dτ = ϕ + v(τ)dτ (5) t0 0 −∞ t 0  v()d 0 (6) t Z0 ϕ0 = v(τ)dτ (6)  is the initial state of (t) at the initial time . 0 −∞ The variables q(t) and (t) are time integrals of the variables i(t) and v(t) , ϕ0 is the initial state of ϕ(t) at the initial time t = t0. respectively;The variables they needq(t) notand beϕ (associatedt) are time with integrals a real of physical the variables chargei( tor) anda realv( tphysical), respec- flux.tively; they need not be associated with a real physical charge or a real physical flux. TheThe axiomatic axiomatic definition definition of of the the fundamental fundamental electric electric circuit circuit elements elements is is shown shown in in FigFigureure 1.1 .

Voltage R Current v(t) i(t)

FR (v,i)  0 Resistor

FC (q,v)  0 C Capacitor Inductor L

FL (,i)  0

Memristor

FM (,q)  0 q(t) (t) Charge M Flux

FigureFigure 1. 1. FundamentalFundamental electric electric circuit circuit elements: elements: resistor, resistor, capacitor, capacitor, inductor, inductor, and and memristor. memristor. The The elementelement symbols symbols are are enclose enclosedd by by a a rectangle rectangle with with a dark band to distinguish thethe referencereference polaritypolarity of ofeach each element. element.

The ideal charge-controlled memristor is defined by the constitutive relation

ϕ = Φ(q) (7)

via the q-dependent Ohm’s law

dΦ(q) v = M(q) i, M(q) = , (8) dq

where M(q) is called the memristance at q. Φ(q) is a continuous and piecewise-differentiable function with bounded slopes. The memristance at any time depends on the entire past history of the element’s current. Probably, the key feature of the memristor is that it exhibits analog non-volatile memory: when a memristor is opened/short-circuited, or when the excitation/power is switched off, the memristor holds its memristance/state. Practically, the memristor can be identified by its distinctive “fingerprint”: (1) Assume arbitrary initial conditions. (2) Excite the device by a periodic bipolar voltage or current with both positive and negative values (the mean value should be zero, no DC component). Electronics 2021, 10, x FOR PEER REVIEW 3 of 20

The ideal charge-controlled memristor is defined by the constitutive relation

  (q) (7) via the q -dependent Ohm’s law d(q) v  M (q) i , M (q)  , (8) dq

where M (q) is called the memristance at q . (q) is a continuous and piece- wise-differentiable function with bounded slopes. The memristance at any time depends on the entire past history of the element’s current. Probably, the key feature of the memristor is that it exhibits analog non-volatile memory: when a memristor is opened/short-circuited, or when the excitation/power is switched off, the memristor holds its memristance/state. Practically, the memristor can be identified by its distinctive “fingerprint”: (1) Assume arbitrary initial conditions. Electronics 2021, 10, 181 (2) Excite the device by a periodic bipolar voltage or current with both positive3 of and 18 negative values (the mean value should be zero, no DC component). (3) Observe a pinched hysteresis loop, a double-valued Lissajous figure passing (3) throughObserve the a pinched origin, hysteresisconfined to loop, the afirst double-valued and the third Lissajous quadrants figure of the passing i - v throughplane. (4) Increasethe origin, the confined excitation to thefrequency first and (theoretically the third quadrants to infinity) of theandi- observev plane. that the loop (4) shrinksIncrease continuously the excitation tofrequency a straight (theoretically line whose to infinity) slope depends and observe on the that excitation the loop waveform.shrinks continuously to a straight line whose slope depends on the excitation waveform. According to KnowMKnowM Inc.Inc. (Santa(Santa Fe, NM,NM, USA),USA), eacheach memristormemristor shouldshould bebe formedformed (conditioned)(conditioned) priorprior toto furtherfurther utilization.utilization. The formingforming isis performedperformed byby excitingexciting aa discretediscrete memristor with a sinusoidalsinusoidal waveformwaveform for severalseveral secondsseconds (for(for atat leastleast 2020 cycles):cycles): atat aa frequencyfrequency ofof 100 Hz, with an amplitude gradually increasingincreasing fromfrom 0.10.1 VV toto 2.5 V,V, untiluntil thethe expectedexpected pinched hysteresishysteresis looploop (Lissajous(Lissajous figure)figure) appears.appears. The memristormemristor forming circuit is shownshown inin FigureFigure2 2.. The The current-limiting current-limiting resistor resistor is is 47 kkΩ.Ω. After forming the device, the current-limitingcurrent-limiting resistorresistor waswas changedchanged toto 1010 kkΩΩ andand thethe generatorgenerator frequency waswas varied, from 1 Hz to 100 Hz,Hz, inin orderorder toto produceproduce aa groupgroup ofof Lissajous figures.figures. The “memristor fingerprintfingerprint test” explained above isis illustratedillustrated byby thethe measurements ofof aa KnowMKnowM memristor,memristor, andand thethe correspondingcorresponding hysteresishysteresis loops,loops,Figure Figure3 .3.

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Figure 2. Memristor forming circuit.

FigureFigure 3. 3.LissajousLissajous figures figures of of the the memristor memristor from from Figure Figure 22 for for various various excitation excitation frequencies. frequencies. “If “If it’s it’s pinchedpinched it’s it’s a amemristor” memristor” [3] [3.].

TheThe firstly firstly produced produced HP HP memristor memristor [4 [4]] consists consists of of a two-layer a two-layer thin thin film film (TiO 2 (TiO), sand-2), sandwichedwiched between between two tw platinumo platinum electrodes. electrodes. The The first first layer layer is doped is doped with oxygenwith oxygen vacancies, va- cancies,so it behaves so it behaves as a semiconductor. as a semiconductor. The otherThe other region region is undoped is undoped and and has anhas insulating an insu- latingproperty. property. Memristance Memristance value value is changed is changed by changing by changing the width the width of the of two the regions: two regions: as the asdoped the doped region region expands expands in the in undopedthe undoped region, region, the totalthe total resistance resistance is being is being reduced, reduced, and andvice vice versa. versa. The The HP HP memristor memristor is characterized is characterized with with resistance resistance values values in ON in (aroundON (around 1 kΩ ) 1 andkΩ) OFFand OFF (around (around 100 k 100Ω) states.kΩ) states. HP isHP profiling is profiling memristors memristors for AI for and AI computing.and computing. TheThe only only com commerciallymercially available available memristors memristors by KnowM Inc.Inc. [[5]5] areare based based on on work work of Prof. Campbell [6]. The KnowM memristors are metal ion-conducting devices. They rely on of Prof. Campbell [6]. The KnowM memristors are metal ion-conducting devices. They the movement of silver ions (Ag+) into channels within the active layer to change the device rely on the movement of silver ions (Ag+) into channels within the active layer to change resistance. Permanent conductive channels that contain silver (Ag) agglomeration sites the device resistance. Permanent conductive channels that contain silver (Ag) agglomer- are generated by metalcatalyzed reactions within the device’s active layer. The memristor ation sites are generated by metalcatalyzed reactions within the device’s active layer. The memristor resistance is determined by the amount of silver (Ag) within the channel. The reported dimensions of a single memristor cell are around 1 µm. The memristance in the OFF state is at least 1 MΩ (if it was not irreversibly reduced to around 100 kΩ by the ap- plication of a current greater than 1 mA) and in the ON state it is around 400 Ω. Power handling is around 3 mW (5 dBm). KnowM is profiling their memristors for AI usage, but they seem to be suitable for electronic circuits as well. The first RF memristor [7] was fabricated on an intrinsic silicon wafer with a 380 nm thick thermally grown silicon dioxide layer. A silver (Ag) layer was used as a terminal for one side, and a gold (Au) with a thin titanium (Ti) adhesion layer as a terminal for the other side. The terminals are separated by a 35 nm wide air gap. The formation/rupture of multiple conductive filaments between two electrodes is dictating memristor states. In the ON state, there is always at least one conductive filament, which leads to the low re- sisting state. In the OFF state, ruptures are dominant, so this state is dominantly defined by the capacitances of the gaps between the filaments. The Pi’s memristor is accurately modelled with a small resistance value (around 3.6 Ω) in the ON state, and a capacitor in the OFF state (around 1.4 fF). The memristor is operational at frequencies from 10 MHz to 110 GHz, with insertion loss of around 0.3 dB in the ON state, and isolation of around 30 dB in the OFF state. Power handling is around 20 dBm. More recent RF memristors (switches) are being developed by a group from the University of Austin. They reported [8] the observation of stable non-volatile resistance switching in single-layer atomic sheets sandwiched between metal electrodes. The switching mechanism is attributed to ionic diffusion, filament, and interfacial redox in bulk oxides and electrolytes. The improved solution, based on monolayer hBN, was

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resistance is determined by the amount of silver (Ag) within the channel. The reported dimensions of a single memristor cell are around 1 µm. The memristance in the OFF state is at least 1 MΩ (if it was not irreversibly reduced to around 100 kΩ by the application of a current greater than 1 mA) and in the ON state it is around 400 Ω. Power handling is around 3 mW (5 dBm). KnowM is profiling their memristors for AI usage, but they seem to be suitable for electronic circuits as well. The first RF memristor [7] was fabricated on an intrinsic silicon wafer with a 380 nm thick thermally grown silicon dioxide layer. A silver (Ag) layer was used as a terminal for one side, and a gold (Au) with a thin titanium (Ti) adhesion layer as a terminal for the other side. The terminals are separated by a 35 nm wide air gap. The formation/rupture of multiple conductive filaments between two electrodes is dictating memristor states. In the ON state, there is always at least one conductive filament, which leads to the low resisting state. In the OFF state, ruptures are dominant, so this state is dominantly defined by the capacitances of the gaps between the filaments. The Pi’s memristor is accurately modelled with a small resistance value (around 3.6 Ω) in the ON state, and a capacitor in the OFF state (around 1.4 fF). The memristor is operational at frequencies from 10 MHz to 110 GHz, with insertion loss of around 0.3 dB in the ON state, and isolation of around 30 dB in the OFF state. Power handling is around 20 dBm. More recent RF memristors (switches) are being developed by a group from the University of Austin. They reported [8] the observation of stable non-volatile resistance switching in single-layer atomic sheets sandwiched between metal electrodes. The switch- ing mechanism is attributed to ionic diffusion, filament, and interfacial redox in bulk oxides and electrolytes. The improved solution, based on monolayer hBN, was published in 2019 [9]. The structure is rated to ~0.33 nm length, has an insertion loss of no more than 0.2 dB and isolation of no less than 15 dB for frequencies up to 110 GHz. Power handling is around 20 dBm. Switching time is less than 15 ns with applied voltage of around 1 V. A memristor is a two-terminal element that is able to change its state “on the fly”. A memristor’s resistance, known as memristance, is determined by the history of applied voltage and/or current. The key features of a memristor are that it is a non-volatile and nanoscale element with tunable memristance in a wide resistance range. These features encouraged scientists and engineers to propose a lot of potential applications of memristors. Some of the fabricated memristors have the capability to switch between low and high memristances (ON and OFF states), thus acting as electronic switches. In other cases, the memristance value can be tuned continuously between ON and OFF states [10]. Memristors with binary states are widely used in artificial intelligence. A Chi- nese group reported a fully hardware-implemented memristor convolutional neural net- work [11]. Memristors are components compatible with CMOS technology [12] which can be used to enhance processor performance and realize applications based on artificial intelligence. The fact that memristors are one-port devices enables them to be used as memory and processing components at the same time. Memristive RF switches [7,9] enable reconfigurability of microwave devices. There are a lot of applications where memristors are used in RF/microwave circuitries [13,14] such as filters [15,16] and tunable inductors [17,18]. In [19], authors proposed a reconfigurable phase shifter based on a memristor instead of a traditional switch as a PIN diode. This memristive realization reduces the power consumption while preserving the functionality. Unfortunately, RF memristors are not commercially available. Digital potentiometers (called just potentiometers in the rest of the paper) have a wide application in mixed-signal electronic circuits and systems such as tunable amplifiers [20,21] and sensors [22,23]. In the last decade, memristive applications were being proposed for the realization of tunable amplifiers [24] and programmable filters [24,25]. Circuit dynamics with memristors was analyzed and a memristor based oscillator was proposed in [26]. Memristance programming is one of the major difficulties in analog memristor operation. It is not easy to obtain the target resistance value quickly and accurately at the same time. Memristance setup is analyzed in some recent researches [27–30]. Electronics 2021, 10, 181 5 of 18

In this paper, we outline the advantages of memristors compared to potentiometers. We use self-directed-channel memristors with tungsten (W) dopant [6], packaged in an encapsulated edge board. The encapsulated edge board is placed in a PCI-E 36 break- out board [31]. The KnowM memristor is compared to the best in class potentiometer from Analog Devices [32]. Limitations of potentiometers are taken as a starting point. Potentiometers have shunt parasitic capacitances that reduce its operating frequency range. Memristors are smaller in size and do not need a constant power supply. Potentiometers have a more restricted range of resistance values compared to memristors. The memristor impedance was measured and compared to that of a potentiometer. The results were discussed along with important differences between the inherent properties of the two devices. Moreover, to further the comparative analysis, several memristor based implementations are provided: a voltage divider, an inverting amplifier, a high pass filter, and a phase shifter.

2. Comparison between Frequency Characteristics of the Potentiometer and Memristor

Several companies develop potentiometers. Apparently, only Analog Devices [33] Electronics 2021, 10, x FOR PEER REVIEWis capable of providing devices with a nominal resistor tolerance error lower than6 20%. of 20

Analog Devices claims that the maximal tolerance for some of their potentiometers is 1%. This fact qualifies them as the best in class, so their model of potentiometer was used in this research. The AC model of the potentiometer is presented in Figure 4 [34], whereas its 3 dB The AC model of the potentiometer is presented in Figure4[ 34], whereas its 3 dB bandwidth (BW) can be determined using an extended schema, represented with dotted bandwidth (BW) can be determined using an extended schema, represented with dotted lines. The bandwidth of the potentiometer is dependent on configuration. For the ex- lines. The bandwidth of the potentiometer is dependent on configuration. For the experi- perimental test circuit in Figure 4, capacitor CA can be neglected because it is in parallel mental test circuit in Figure4, capacitor CA can be neglected because it is in parallel with with an ideal voltage generator. The effect of capacitor CB can be neglected when terminal an ideal voltage generator. The effect of capacitor C can be neglected when terminal B B is grounded. For the circuit presented in Figure 4, ifB the A terminal is the input with the is grounded. For the circuit presented in Figure4, if the A terminal is the input with the grounded B terminal, then the W terminal is the output, and the bandwidth is equal to grounded B terminal, then the W terminal is the output, and the bandwidth is equal to 1 BWP  1 . (9) BWP = 2πRRCWA || WB W. (9) 2π(RWA||RWB)CW

FigureFigure 4. AC model of potentiometer.

Most potentiometers with a 1% precision,precision, suchsuch asas AD5270,AD5270, havehave thethe followingfollowing shuntshunt parasiticparasitic capacitances CA == CCB B= =90 90 pF, pF, CWC W= 40= 40pF. pF. Those Those are are also also the the lowest lowest parasitic parasitic ca- capacitancespacitances that that Analog Analog Devices Devices reports reports for for 1% 1% precis precisionion potentiometers. potentiometers. InIn orderorder to to analyze analyze parasitic parasitic effects effects of the of the potentiometer, potentiometer, we connect we connect the potentiometer the potenti- intoometer a voltage into a divider.voltage divider. We analyzed We analyzed two test circuitstwo test (configurations) circuits (configurations) as shown asin shown Figure 5in. ForFigure the first5. For configuration the first configuration in Figure5a, in we Fig findure that 5a, thewe potentiometerfind that the potentiometer impedance is equal imped- to ance is equal to a 1 ZP( jω) = RWAk , (10) a jωC1A ZRP(jω) WA || , (10) jωCA

where f represents the operating frequency, and ω = 2πf. Parameter RWA presents the re- sistance between terminals W and A. For the second configuration in Figure 5b, the po- tentiometer impedance is equal to

b 1 ZRRP(jω) ( WA || WB ) || , (11) jωCW

where RWB presents the resistance between terminals W and B. For the two analyzed circuit configurations, the second one (Figure 5b) has the wider bandwidth because capacitance CW is less than half of capacitance CA. For the circuit in Figure 5b, resistors RWA and RWB are connected in parallel, which implies a twice lower resistance. In the rest of the paper, the potentiometer is modeled with a variable resistor b (RP = RWA||RWB) and a parasitic capacitor (CP = CW = 40 pF), i.e., ZZPP .

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where f represents the operating frequency, and ω = 2πf. Parameter RWA presents the resistance between terminals W and A. For the second configuration in Figure5b, the potentiometer impedance is equal to

1 Electronics 2021, 10, x FOR PEER REVIEW b 7 of 20 ZP( jω) = (RWAkRWB)k , (11) jωCW

where RWB presents the resistance between terminals W and B.

Electronics 2021, 10, x FOR PEER REVIEW 7 of 20

(a) (b)

FigureFigure 5. 5.Two Two configurations configurations of of the the voltage voltage divider, divider, with with a grounded a grounded potentiometer potentiometer with with common common terminal terminal (a) A,(a) and A, and (b) W.(b) W. For the two analyzed circuit configurations, the second one (Figure5b) has the wider

bandwidthIn the becauseopen literature, capacitance we haveCW isnot less found than adequate half of capacitance simulation C modelsA. For the of KnowM’s circuit in (a) (b) Figurememristors5b, resistors that wouldRWA fitand ourR WBexperimentalare connected setup. in parallel,Memristor which model impliess that arealize twice lowertransi- Figure 5. Two configurationsresistance.tions of betweenthe voltage In the states divider, rest are of with the presented a paper,grounded the in potentiometer [3 potentiometer5,36] for with frequencies common is modeled terminal up to with 100(a) A, a Hz, variableand whereas(b) resistor a bi- W. b (naryRP = memristorRWA||RWB is) andmodeled a parasitic in [37]. capacitor Accordingly, (CP = anCW experimental= 40 pF), i.e., approachZP = ZP. is undertaken to findIn thethe openimpedance literature, characteristics we have not of foundthe memristor. adequate Memr simulationistance models programming of KnowM’s is re- In the open literature, we have not found adequate simulation models of KnowM’s memristorsalized using that Digilent would Analog fit our experimental Discovery 2 setup.(AD2) Memristor instrumentation models tool that [3 realize8] together transitions with memristors that would fit our experimental setup. Memristor models that realize transi- between states are presented in [35,36] for frequencies up to 100 Hz, whereas a binary KnowMtions between’s software states package are presented for memristor in [35,36] for analysis frequencies [39]. upHowever, to 100 Hz, the whereas memristor a bi- setup memristorcircuitrynary memristor is isnot modeled in is the modeled focus in [37 inof]. [3this Accordingly,7]. Accordingly,research. an an experimental experimental approach approach is isundertaken undertaken to findto thefindWe impedance the used impedance a vol characteristicstage characteristics divider of to theof determine the memristor. memristor. the Memristance Memr memristoristance programmingprogramming impedance (Fig is is re- realizedure 6a). usingMeasurementsalized Digilent using AnalogDigilent using DiscoveryAnalog an oscilloscope Discovery 2 (AD2) introduce2 instrumentation(AD2) instrumentation additional tool unknowns[ 38tool] together [38] together—probes’ with with KnowM’s imped- softwareances.KnowM The package’s probe’ssoftware for impedance package memristor for canmemristor analysis be neglected [analysis39]. However, at [ 3the9]. However,circuit’s the memristor inputthe memristor because setup setup circuitryit is in par- is notallelcircuitry in with the focus anis not ideal ofin thisthe voltag focus research.e ofgenerator, this research. whereas this impedance at the output remains. In this Wecircuit,We used used there a voltage a are vol tagetwo divider remaining divider to to determine determineunknown the thevariables: memristor memristor the impedance impedance (Figure (Fig of theure 6memristor6a).a). Mea- surementsandMeasurements the parasitic using using animpedance oscilloscope an oscilloscope of the introduce probe introduce connected additional additional at unknowns—probes’the unknowns output of— probes’the test imped-impedances.circuit. First, ances. The probe’s impedance can be neglected at the circuit’s input because it is in par- Thethe probe’sparasitic impedance impedance can of the be neglected probe should at the be circuit’s determined input from because Figure it is 6 inb. parallel with anallel ideal with voltage an ideal generator, voltage whereasgenerator, this whereas impedance this impedance at the output at the output remains. remains. In this In circuit, therethis are circuit, two there remaining are two unknownremaining unknown variables: variables: the impedance the impedance of the of memristorthe memristor and the and the parasitic impedance of the probe connected at the output of the test circuit. First, parasitic impedance of the probe connected at the output of the test circuit. First, the the parasitic impedance of the probe should be determined from Figure 6b. parasitic impedance of the probe should be determined from Figure6b.

(a) (b)

Figure 6. Circuit for determination of (a) memristor impedance and (b) impedance of oscilloscope probe. (a) (b)

FigureFigure 6. Circuit 6. Circuit for for determination determinationThe voltage of of (a ()dividera) memristor memristor has impedance a constant and and frequency(b ()b impedance) impedance response of oscilloscope of oscilloscope in the probe. analyzed probe. frequency band The voltage divider has a constant frequency response in the analyzed frequency band R2 HVD (jω) . (12) RR R212 HVD (jω) . (12) When we use an oscilloscope, the frequencyRR12 response becomes When we use an oscilloscope, the frequency response becomes

Electronics 2021, 10, 181 7 of 18

The voltage divider has a constant frequency response in the analyzed frequency band

R2 HVD(jω) = . (12) R1 + R2 When we use an oscilloscope, the frequency response becomes

probe R2kZprobe( jω) HVD ( jω) =   . (13) R1 + R2kZprobe( jω)

Using the measured frequency response, we can find the impedance of the oscilloscope probe as probe R R H ( jω) Z ( ω) = 1 2 VD probe j probe . (14) R2 − (R1 + R2)HVD ( jω)

For a determined probe impedance Zprobe, we can identify the memristor impedance ZM. From Figure6a, we find the frequency response of the voltage divider as ( ω)k ( ω) M ZM j Zprobe j HVD( jω) =   . (15) R1 + ZM( jω)kZprobe( jω)

Equation (15) can be rearranged into

M R1Zprobe( jω)HVD( jω) ZM( jω) = , (16)   M Zprobe( jω) − R1 + Zprobe( jω) HVD( jω)

which gives us the memristor impedance. For this analysis, we used R1 = 98.4 kΩ and R2 = 98.4 kΩ, whereas the specified frequency range is from 10 kHz to 5 MHz. The memristor impedance has been obtained by using experimental results and Equations (14) and (16) The impedance of the potentiometer was obtained using the same methodology. Figure7 illustrates the comparison between the impedance magnitudes of the memristor and potentiometer. To get a better insight into the comparison between memristor and potentiometer, we have divided the results according to initial impedance magnitudes at 10 kHz. At frequencies below 1 kHz, the i-v characteristic of the memristor is a pinched hysteresis loop, which means that the memristor passes through multiple conductance states. Accordingly, the memristor should not be used for DC up to around 10 kHz. Over 10 kHz, the memristor’s hysteresis loop degenerates to a straight line, which corresponds to a purely resistive element. It can be noticed that the magnitude of the memristor impedance is flatter compared to the potentiometer. At higher frequencies, the parasitic capacitance of the potentiometer is dominant over the specified resistance. For example, when the potentiometer resistance is set to 100 kΩ, parasitic capacitance restricted the maximal value of impedance magnitude to 50 kΩ at 60 kHz. We propose an AC model of the memristor using its measured values of impedance. For lower memristances up to around 15 kΩ, the memristor can be modeled as a resistor with a parasitic shunt capacitor of 8 pF. This extracted parasitic capacitance is a consequence of the memristor itself, as well as the influence of the encapsulated edge board and the PCI-E 36 breakout board. Therefore, the memristor’s parasitic capacitance is lower than 8 pF. For memristances higher than 15 kΩ, a constant shunt capacitance is not properly modeling parasitic effects. For memristances above 15 kΩ, shunt capacitance is slightly lower than 8 pF for frequencies below 300 kHz, and slightly higher than 8 pF for frequencies higher than 300 kHz. For memristance value of 1 MΩ the model is not good since the capacitive part becomes dominant due to extremely high memristance. In the real device, Electronics 2021, 10, x FOR PEER REVIEW 8 of 20

probe RZ2|| probe (jω) HVD (jω) . (13) RRZ1  2|| probe (jω)

Using the measured frequency response, we can find the impedance of the oscillo- scope probe as

RRH probe (jω) Z (jω) 12 VD . probe probe (14) RRRH2 1 2  VD (jω)

For a determined probe impedance Zprobe, we can identify the memristor impedance ZM. From Figure 6a, we find the frequency response of the voltage divider as

M ZZM(jω) || probe (jω) HVD (jω) . (15) RZZ1  M(jω) || probe (jω)

Equation (15) can be rearranged into

RZH(jω)M (jω) Z (jω) 1 probe VD , M M (16) ZRZHprobe(jω) 1 probe (jω) VD (jω)

which gives us the memristor impedance. For this analysis, we used R1 = 98.4 kΩ and R2 = 98.4 kΩ, whereas the specified fre- quency range is from 10 kHz to 5 MHz. The memristor impedance has been obtained by using experimental results and Equations (14) and (16) The impedance of the potenti- ometer was obtained using the same methodology. Figure 7 illustrates the comparison between the impedance magnitudes of the memristor and potentiometer. To get a better insight into the comparison between memristor and potentiometer, we have divided the Electronics 2021, 10, 181 results according to initial impedance magnitudes at 10 kHz. 8 of 18 Electronics 2021, 10, x FOR PEER REVIEW 9 of 20 At frequencies below 1 kHz, the i-v characteristic of the memristor is a pinched hysteresis loop, which means that the memristor passes through multiple conductance states. Accordingly, the memristor should not be used for DC up to around 10 kHz. Over 10 kHz, the memristor’s hysteresis loop degenerates to a straight line, which corresponds the capacitanceto a purely isresistive not that element. significant. Comparison of simulation using proposed AC model and measurement results is illustrated in Figure8.

Electronics 2021, 10, x FOR PEER REVIEW 9 of 20 (c)

Figure 7. Comparison between impedance magnitudes of memristor and potentiometer. Ranges of initial impedance magnitude at 10 kHz: (a) 400 Ω–4500 Ω, (b) 400 Ω–15 kΩ, (c) 1 kΩ–1 MΩ for memristor and 1 kΩ–100 kΩ for potentiom- eter. Impedances were measured(a) at 10 kHz for memristors and at DC for the potentiometer.(b)

It can be noticed that the magnitude of the memristor impedance is flatter compared to the potentiometer. At higher frequencies, the parasitic capacitance of the potentiome- ter is dominant over the specified resistance. For example, when the potentiometer re- sistance is set to 100 kΩ, parasitic capacitance restricted the maximal value of impedance magnitude to 50 kΩ at 60 kHz. We propose an AC model of the memristor using its measured values of impedance. For lower memristances up to around 15 kΩ, the memristor can be modeled as a resistor with a parasitic shunt capacitor of 8 pF. This extracted parasitic capacitance is a conse- quence of the memristor itself, as well as the influence of the encapsulated edge board and the PCI-E 36 breakout board. Therefore, the memristor’s parasitic capacitance is lower than 8 pF. For memristances higher than 15 kΩ, a constant shunt capacitance is not properly modeling parasitic effects. For memristances above 15 kΩ, shunt capacitance is (c) slightly lower than 8 pF for frequencies below 300 kHz, and slightly higher than 8 pF for Figure 7. ComparisonFigure 7. Comp betweenarison between impedancefrequencies impedance magnitudeshigher magnitudes than 300 of ofkHz. memristor memristor For memristance and and potentiometer. potentiometer. value of Ranges 1 MΩ Ranges ofthe initial model of impedance initialis not good impedance magnitude at 10 kHz: (a) 400since Ω– 4500the capacitiveΩ, (b) 400 Ω part–15 kΩ, becomes (c) 1 kΩ dominant–1 MΩ for duememristor to extremely and 1 kΩ high–100 kΩmemristance. for potentiom- In the magnitude at 10 kHz: (a) 400 Ω–4500 Ω,(b) 400 Ω–15 kΩ,(c) 1 kΩ–1 MΩ for memristor and 1 kΩ–100 kΩ for potentiometer. eter. Impedances were measuredreal device, at 10 kHz the for capacitance memristors is and not at that DC forsignificant. the potentiometer. Comparison of simulation using pro- Impedances were measured at 10 kHzposed for AC memristors model and measure and at DCmen fort results the potentiometer. is illustrated in Figure 8. It can be noticed that the magnitude of the memristor impedance is flatter compared to the potentiometer. At higher frequencies, the parasitic capacitance of the potentiome- ter is dominant over the specified resistance. For example, when the potentiometer re- sistance is set to 100 kΩ, parasitic capacitance restricted the maximal value of impedance magnitude to 50 kΩ at 60 kHz. We propose an AC model of the memristor using its measured values of impedance. For lower memristances up to around 15 kΩ, the memristor can be modeled as a resistor with a parasitic shunt capacitor of 8 pF. This extracted parasitic capacitance is a conse- quence of the memristor itself, as well as the influence of the encapsulated edge board and the PCI-E 36 breakout board. Therefore, the memristor’s parasitic capacitance is lower than 8 pF. For memristances higher than 15 kΩ, a constant shunt capacitance is not properly modeling parasitic effects. For memristances above 15 kΩ, shunt capacitance is Electronics 2021, 10, x FOR PEER REVIEWslightly lower than 8 pF for frequencies below 300 kHz, and slightly higher than 8 10pF of for 20

frequencies higher than 300 kHz. For memristance value of 1 MΩ the model is not good

since the capacitive part becomes dominant due to extremely high memristance. In the real(a) device, the capacitance is not that significant. Comparison(b) of simulation using pro- posed AC model and measurement results is illustrated in Figure 8.

(c)

Figure 8. Comparison between simulation and experimental results for impedance magnitudes of memristor. Ranges of (a) (b) initial impedanceFigure magnitude8. Comparison at between 10 kHz: simulation (a) 400 andΩ–4500 experimentalΩ;(b) results 400 Ω for–15 impedance kΩ;(c) 1magnitudes kΩ–125 k ofΩ memristor.. Ranges of initial impedance magnitude at 10 kHz: (a) 400 Ω–4500 Ω; (b) 400 Ω–15 kΩ; (c) 1 kΩ–125 kΩ.

3. Evaluation of Memristor Tunability and Persistence Some of the biggest concerns related to memristors are variability, tunability, reten- tion and endurance. Variability is primarily related to the minimum and maximum memristance values. These values are fabrication dependent and cannot be significantly extended by programming procedures. Another pair of constrains are (1) repeatability of memristors programming states during multiple cycles within one memristor, and (2) repeatability of target memristance value with different memristors. These two con- strains are tackled by programming techniques. When it comes to memristor programming, our work relies on results [27–30] pub- lished by the team lead by Vourkas. To the best of our knowledge, the best programming results of KnowM memristors were published by his team. In [28] the team compares programming techniques using a voltage divider with a memristor and a so-called cur- rent sensing technique with an operational amplifier that introduces a virtual ground. In both cases, the memristance target value was measured using ADC. In theory, the num- ber of states of a memristive device depends on the precision of the used ADC, which is, for modern ADCs, around 216 ≈ 65,000 or even 224 ≈ 16.5 × 106 states. In [27] the authors describe that there are two major phases of memristance programming: (1) large ampli- tude programming pulses that are used to quickly reach the target resistance in a short time, but with cruder precision (±250 Ω), and (2) small amplitude signals of different polarity for fine-tuning the target memristance. Results provided in [28] were obtained by repeating the same programming pulses for 50 iterations, for each state. They achieved great repeatability which differs no more than 4% (around ±150 Ω at 1.8 kΩ) of the target memristance. The team’s results show how states within the range of1 kΩ to around 100 kΩ were achieved using programming pulses in less than 100 ms. In addition to Vourkas’s team, we experimented with the memristors’ extreme values, and obtained results of 1 MΩ and 450 Ω by directly applying a DC voltage to a memristor, without a current limiting resistor. In order to examine retention and endurance of memristor states, we conducted a series of experiments. We excited memristor–resistor voltage divider by a sinusoidal waveform and tracked memristance values. Frequency was set to 10 kHz as it is the most critical frequency in the defined operational range of a memristor. The signal amplitude was varied. The duration of each test was 10 min. In Figure 9 we display results for dif- ferent states of the memristor, where memristance values are in range from 800 Ω to 180 kΩ.

Electronics 2021, 10, 181 9 of 18

3. Evaluation of Memristor Tunability and Persistence Some of the biggest concerns related to memristors are variability, tunability, retention and endurance. Variability is primarily related to the minimum and maximum memristance values. These values are fabrication dependent and cannot be significantly extended by programming procedures. Another pair of constrains are (1) repeatability of memristors programming states during multiple cycles within one memristor, and (2) repeatability of target memristance value with different memristors. These two constrains are tackled by programming techniques. When it comes to memristor programming, our work relies on results [27–30] pub- lished by the team lead by Vourkas. To the best of our knowledge, the best programming results of KnowM memristors were published by his team. In [28] the team compares programming techniques using a voltage divider with a memristor and a so-called current sensing technique with an operational amplifier that introduces a virtual ground. In both cases, the memristance target value was measured using ADC. In theory, the number of states of a memristive device depends on the precision of the used ADC, which is, for modern ADCs, around 216 ≈ 65,000 or even 224 ≈ 16.5 × 106 states. In [27] the authors describe that there are two major phases of memristance programming: (1) large amplitude programming pulses that are used to quickly reach the target resistance in a short time, but with cruder precision (±250 Ω), and (2) small amplitude signals of different polarity for fine-tuning the target memristance. Results provided in [28] were obtained by repeating the same programming pulses for 50 iterations, for each state. They achieved great repeata- bility which differs no more than 4% (around ±150 Ω at 1.8 kΩ) of the target memristance. The team’s results show how states within the range of 1 kΩ to around 100 kΩ were achieved using programming pulses in less than 100 ms. In addition to Vourkas’s team, we experimented with the memristors’ extreme values, and obtained results of 1 MΩ and 450 Ω by directly applying a DC voltage to a memristor, without a current limiting resistor. In order to examine retention and endurance of memristor states, we conducted a series of experiments. We excited memristor–resistor voltage divider by a sinusoidal waveform and tracked memristance values. Frequency was set to 10 kHz as it is the most critical frequency in the defined operational range of a memristor. The signal amplitude Electronics 2021, 10, x FOR PEER REVIEWwas varied. The duration of each test was 10 min. In Figure9 we display results11 forof 20 different states of the memristor, where memristance values are in range from 800 Ω to 180 kΩ.

Figure 9. Retention and endurance tests of memristor states.

When the excitationexcitation signal amplitude was 0.5 V oror below,below, thethe memristormemristor preservedpreserved stable states.states. ForFor amplitudesamplitudes of of around around 1 V,1 V, the the memristance memristance had had not not changed changed dramatically dramati- duringcally during the time, the but time the, sinusoidalbut the sinusoidal waveform waveform was degenerated. was degenerated We noticed. We that noticed memristors that arememristors sensitive are to DC sensitive currents to evenDC currents if they were even of if the they order were of aof couple the order of microamperes. of a couple of This mi- factcroamperes. can be interpreted This fact as can a downside be interpreted due to as oversensitivity. a downside Withdue to proper oversensitivity. compensation With of proper compensation of electric circuits, this problem can be eliminated. On the other side, high sensitivity can be used as a tool for the slow gradual change of memristance value during operation.

4. Comparison between Key Features of Memristors and Potentiometers We have analyzed key differences between memristors and potentiometers, as well as advantages of one component over another. In this research, the KnowM memristor and AD2750 Analog Design potentiometer have been compared, whereas Table 1 pre- sents a summary view of this comparison. Potentiometers operate on DC and low frequencies, whereas a memristor cannot operate on frequencies lower than 10 kHz because it passes through multiple conduct- ance states. Memristor programming is based on the application of small amplitude voltage pulses. In comparison to a potentiometer, one of the memristors’ advantages is a flatter impedance magnitude at higher frequencies. For example, if we set the initial re- sistance value to 10 kΩ, the memristor’s bandwidth is around 1 MHz, whereas the po- tentiometer has a bandwidth of around 400 kHz. The memristor’s bandwidth is wider even for higher initial resistance values. Parasitic capacitances of both elements are dominant at high frequencies, but the parasitic effects are more significant for the poten- tiometer.

Table 1. Comparison: KnowM memristor vs. Potentiometer AD5270.

Feature KnowM Memristor Potentiometer AD5270 Minimal (starting) 10 kHz DC operating frequency Bandwidth for Rinit = 1 kΩ >5 MHz 5 MHz Bandwidth for Rinit = 10 kΩ ≈1 MHz 400 kHz Bandwidth for Rinit = 100 kΩ ≈120 kHz 60 kHz Bandwidth for Rinit = 1 MΩ ≈55 kHz Not applicable Resistance range 400 Ω–1 MΩ 100 Ω–100 kΩ Parasitic capacitance <8 pF <40 pF Non-volatile Yes 50 cycles Footprint 1 µm × 1 µm 1 mm × 1 mm Power handling 3 mW 10 mW

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electric circuits, this problem can be eliminated. On the other side, high sensitivity can be used as a tool for the slow gradual change of memristance value during operation.

4. Comparison between Key Features of Memristors and Potentiometers We have analyzed key differences between memristors and potentiometers, as well as advantages of one component over another. In this research, the KnowM memristor and AD2750 Analog Design potentiometer have been compared, whereas Table1 presents a summary view of this comparison. Potentiometers operate on DC and low frequencies, whereas a memristor cannot operate on frequencies lower than 10 kHz because it passes through multiple conductance states. Memristor programming is based on the application of small amplitude voltage pulses. In comparison to a potentiometer, one of the memristors’ advantages is a flatter impedance magnitude at higher frequencies. For example, if we set the initial resistance value to 10 kΩ, the memristor’s bandwidth is around 1 MHz, whereas the potentiometer has a bandwidth of around 400 kHz. The memristor’s bandwidth is wider even for higher initial resistance values. Parasitic capacitances of both elements are dominant at high frequencies, but the parasitic effects are more significant for the potentiometer.

Table 1. Comparison: KnowM memristor vs. Potentiometer AD5270.

Feature KnowM Memristor Potentiometer AD5270 Minimal (starting) 10 kHz DC operating frequency Bandwidth for Rinit = 1 kΩ >5 MHz 5 MHz Bandwidth for Rinit = 10 kΩ ≈1 MHz 400 kHz Bandwidth for Rinit = 100 kΩ ≈120 kHz 60 kHz Bandwidth for Rinit = 1 MΩ ≈55 kHz Not applicable Resistance range 400 Ω–1 MΩ 100 Ω–100 kΩ Parasitic capacitance <8 pF <40 pF Non-volatile Yes 50 cycles Footprint 1 µm × 1 µm 1 mm × 1 mm Power handling 3 mW 10 mW Programming time ms µs Complexity of programming Medium Low On the fly switching Yes No Number of resistance states Any value in the range 1024 Depends on the Resistance accuracy ±1% programming circuit External circuit for resistance control Required Required Constant power supply Not required Required Leaking current No 50 nA

The resistance range of KnowM memristors is from around 400 Ω to around 1 MΩ, whereas the range of AD5270 is from 100 Ω to 100 kΩ. The memristor can be theoretically set to any resistance value from the range, whereas the potentiometer has 1024 resistance states. The potentiometer has wiper resistance which is around 35 Ω. The memristor is non-volatile because its state is expected to remain stable without any DC voltage across the element. On the other hand, the potentiometer can store resistance value into non-volatile on-chip memory (wiper memory) up to 50 times, and the value from wiper memory is set on power up. During operating regime, a resistance value of the potentiometer can be adjusted an unlimited number of times. The footprint of the analyzed potentiometer package is around 3 mm × 3 mm. We assume that the chip itself is not smaller than 1 mm × 1 mm. Fabricated KnowM memris- tors are around 1 µm × 1 µm, which is reported in [31]. The potentiometer can handle a signal with a power level of around 10 mW, whereas the memristor is limited to around 3 mW. Both components require an external circuit for the resistance control. The mem- ristor is expected to continuously change its state during the operation mode i.e., on the Electronics 2021, 10, 181 11 of 18

fly, whereas the potentiometer is not in a defined state when switching between states. Potentiometers require a constant power supply during operation mode and have a low leakage current. Memristors do not require a constant power supply, and therefore do not have leaking current. A fundamental characteristic of the potentiometer is the microseconds switching time between resistance states with high accuracy. For the potentiometer, maximum nominal resistor tolerance error is ±1%. The memristor’s programming time depends on the target accuracy and the differ- ence between the previous and the next memristance state. As a consequence of different conditions, a memristor’s programming time is of the order of millisecond [27]. The com- parative analysis shows that the memristor brings some advantages comparing to poten- tiometers. However, memristor programming is still an open area for further research and improvement.

5. Possible Application of Memristors in Adjustable Circuits According to experimental results, the memristor has better characteristics at higher frequencies compared to the potentiometer. This fact opens up the possibility that memris- tors can be used instead of potentiometers. We analyze various circuit designs based on the memristor and potentiometer, such as a voltage divider, an inverting amplifier, a high pass filter, and a phase shifter. We implemented laboratory prototypes of these memristor-based circuits in order to present potential applications. In order to better observe the advantages of memristors, we select to analyze analog circuits with typical functionalities.

Electronics 2021, 10, x FOR PEER REVIEW 13 of 20 5.1. Voltage Divider The first example is a voltage divider which uses a potentiometer to control signal attenuation. Realization of this circuit with a memristor is proposed in Figure6a. For this analysis,When wethe selected voltage adivider resistor isR realized1 = 10 kΩ with. a memristor or a potentiometer, the volt- age gainWhen function the voltage is divider is realized with a memristor or a potentiometer, the voltage gain function is p i Zi p(j ) G (j ) Z (jω) , i  M,P, (17) i VD ω p i GVD(j ) = ZRip(j ) 1 , i = M, P, (17) Zi (jω) + R1 p where Zi p(j ) is equal to ZZi || probe . where Zi (jω) is equal to Zi||Zprobe . InIn order order to to illustrate illustrate the the comparison comparison between between these these two two realizations, realizations, we we present present the the voltagevoltage gain gain function function in in Fig Figureure 10 10. .We We measured measured responses responses of of both both potentiometer potentiometer and and memrmemristiveistive implementation. implementation. From From Figure Figure 10 10, it, itcan can be be noticed noticed that that the the re realizationalization with with thethe memristor memristor has has a ahigher higher cutoff cutoff frequency. frequency. As As the the initial initial resistance resistance value value increases, increases, the the differencedifference between between potentiometer potentiometer and and memristor memristor becomes becomes more more obvious. obvious.

FigureFigure 10. 10. VoltageVoltage gain gain magnitude magnitude of the of voltage the voltage divider divider with memristor with memristor and potentiometer and potentiometer.. Im- pedancesImpedances were were measured measured at 10 at kHz 10 kHzfor memristors for memristors and andat DC at for DC the for potentiometer the potentiometer..

5.2. Inverting Amplifier The next example is an inverting amplifier which uses a potentiometer to control the voltage gain. Application of memristors in amplifiers was theoretically proposed [18,24,40]. Some authors propose memristor usage in variable gain amplifier (VGA) [41] and automatic gain control (AGC) circuits [42]. Our realization of this circuit with a memristor instead of a potentiometer is pro- posed in Figure 6a. For this realization we selected the following resistor values: R1 = 46.4 kΩ, R2 = 29.3 kΩ, R3 = 45.4 kΩ, R4 = 45.7 kΩ, R5 = 1.95 kΩ, R6 = 1.93 kΩ. The operational amplifier used is a dual low-noise operational amplifier NA5532AP [43] from Texas In- struments [44], with a typical unity-gain bandwidth of 10 MHz. For this experimental analysis, the impedances of the oscilloscope probes can be neglected. At the circuit's in- put the oscilloscope probe is connected in parallel with a voltage generator, whereas the resistor value R6 is chosen to be significantly lower than the probe’s impedance at the circuit's output. In order to minimize the influence of input bias current through the memristor, we add the capacitor C1 (1 µF) in series with the memristor (Figure 11a). The capacitance C1 was chosen to have no effect on frequency response in the desired fre- quency band and the voltage gain function can be approximated as

M RR33 RR44 GIA (j )     . (18) RRRZR2 2 5 M(j ) 2 On the other hand, this circuit realization with a potentiometer is presented in Fig- ure 11b, and its voltage gain function is

Electronics 2021, 10, 181 12 of 18

5.2. Inverting Amplifier The next example is an inverting amplifier which uses a potentiometer to control the voltage gain. Application of memristors in amplifiers was theoretically proposed [18,24,40]. Some authors propose memristor usage in variable gain amplifier (VGA) [41] and automatic Electronics 2021, 10, x FOR PEER REVIEW 14 of 20 gain control (AGC) circuits [42]. Our realization of this circuit with a memristor instead of a potentiometer is proposed in Figure6a. For this realization we selected the following resistor values: R1 = 46.4 kΩ, R2 = 29.3 kΩ, R3 = 45.4 kΩ, R4 = 45.7 kΩ, R5 = 1.95 kΩ, R6 = 1.93 kΩ. The operational amplifier used is a dual low-noiseP operationalRR33 amplifierRR44 NA5532AP [43] from Texas In- GIA (j )     . (19) struments [44], with a typical unity-gain bandwidthRRRZR2 2 5 of P 10(j MHz. ) 2 For this experimental analysis, the impedances of the oscilloscope probes can be neglected. At the circuit’s input the oscilloscopeIn order probe to illustrate is connected the comparison in parallel withbetween a voltage these generator,two realizations, whereas we the present resistor the voltage gain function of both cases in Figure 12. We measured responses of the circuit value R6 is chosen to be significantly lower than the probe’s impedance at the circuit’s output.with Ina potentiometer order to minimize as well the influenceas with memristive of input bias implementation. current through Analyzing the memristor, the results we addfrom the capacitor Figure 12C, 1 we(1 µ noticedF) in series that with the realizationthe memristor with (Figure the potentiometer 11a). The capacitance has dominantC1 wasparasitic chosen toeffects have for no lower effect onvalues frequency of voltage response gain inmagnitude the desired whi frequencych corresponds band and to higher the voltageresistance gain functionvalues. On can the be other approximated hand, the as memristive circuit has a lower cutoff frequency for lower resistance values which corresponds to a higher value of voltage gain magni-   tude. M R3 R3 R4 R4 GIA(jω) ≈ − + + . (18) R2 R2 R5 + ZM(jω) R2

(a) (b)

FigureFigure 11. 11.Inverting Inverting amplifier amplifier based based on realizationon realization with: with: (a) memristor;(a) memristor; (b) potentiometer.(b) potentiometer.

OnTo the have other a hand, better this insight circuit into realization this comparison, with a potentiometer we present is the presented responsein at Figure the lower 11b, andpassband its voltage frequencies gain function from is 10 kHz up to 200 kHz, as shown in Figure 12b. The memrsi- tive circuit has almost the same cutoff frequency for all values of voltage gain. However, the circuit with a potentiometerP hasR 3a higherR3 cutoffR4 frequencyR4 for higher values of voltage GIA(jω) = − + + . (19) gain and vice versa. The voltage Rgain2 magnitudeR2 R5 + ZP (ofjω a) memristiveR2 circuit is almost flat within the whole analyzed frequency range with a maximal deviation of ±1 dB. For real- In order to illustrate the comparison between these two realizations, we present the ization with the potentiometer, the maximal deviation is ±8 dB. It may be concluded that voltage gain function of both cases in Figure 12. We measured responses of the circuit with the memristor is a valid candidate to replace the potentiometer for higher resistance a potentiometer as well as with memristive implementation. Analyzing the results from Figurevalues. 12, we noticed that the realization with the potentiometer has dominant parasitic effects forIn lowerorder valuesto better of identify voltage gainthe nature magnitude of peaks which around corresponds 500 kHz to of higher voltage resistance gain mag- values.nitude, On we the have other analyzed hand, the the memristive comparison circuit between has a lowermemristor cutoff and frequency an ideal forresistor lower (see resistanceFigure 13 values). The which realization corresponds with an to ideal a higher resistor value does of not voltage have gain the peak magnitude. of voltage gain at around 500 kHz, which means that this peak is a result of parasitic capacitances of the potentiometer and memristor.

Electronics 2021, 10, x FOR PEER REVIEW 15 of 20

Electronics 2021, 10, 181 13 of 18

Electronics 2021, 10, x FOR PEER REVIEW 15 of 20

(a) (b)

Figure 12. Voltage gain magnitude of the inverting amplifier amplifier based on memristor and potentiometer potentiometer.. Analyzed frequency ranges: ( (aa)) 10 10 kHz kHz–1–1 MHz; MHz; (b (b) )10 10 kHz kHz–200–200 kHz kHz.. Impedances Impedances were were measured measured at 10 at 10kHz kHz for formemristors memristors and and at DC at DCfor the for potentiometerthe potentiometer..

To have a better insight into this comparison, we present the response at the lower passband frequencies from 10 kHz up to 200 kHz, as shown in Figure 12b. The memrsitive circuit has almost the same cutoff frequency for all values of voltage gain. However, the circuit with a potentiometer has a higher cutoff frequency for higher values of voltage gain and vice versa. The voltage gain magnitude of a memristive circuit is almost flat within the whole analyzed frequency range with a maximal deviation of ±1 dB. For realization with the potentiometer, the maximal deviation is ±8 dB. It may be concluded that the memristor is a valid candidate to replace the potentiometer for higher resistance values. (a) In order to better identify the nature of peaks around(b) 500 kHz of voltage gain mag- nitude, we have analyzed the comparison between a memristor and an ideal resistor Figure 12. Voltage gain magnitude of the inverting amplifier based on memristor and potentiometer. Analyzed frequency (see Figure 13). The realization with an ideal resistor does not have the peak of voltage ranges: (a) 10 kHz–1 MHz; (b) 10 kHz–200 kHz. Impedances were measured at 10 kHz for memristors and at DC for the gain at around 500 kHz, which means that this peak is a result of parasitic capacitances of potentiometer. the potentiometer and memristor. Figure 13. Voltage gain magnitude of the inverting amplifier based on memristor and ideal resistor: simulated response of the circuit with ideal resistor, and measured response of memristive im- plementation.

5.3. High-Pass Filter One of the examples is a high pass filter which uses a potentiometer to control the cutoff frequency. This group of devices were also proposed, and theoretical results are available in the open literature [24,25,40]. Our realization of this circuit with a memristor instead of a potentiometer is pro- posed in Figure 14, where the selected capacitor is C1=2 nF. For this experimental analy- sis, the impedances of oscilloscope probes can be neglected. At the circuit's input the os- cilloscope probe is connected in parallel with a voltage generator, whereas the capacitor value C1 is chosen to be significantly higher than the probe impedance at the circuit's output.Figure 13. 13. Than VoltageVoltage we gainan gainalyzed magnitude magnitude this filterof the of withinverting the invertinga potentiometer amplifier amplifier based in basedon order memristor on to memristorbetter and noticeideal and resistor: a ideal dif- ferencesimulatedresistor: in simulatedresponse relation ofto responsethe a memristor. circuit of with the ideal circuit resistor, with and ideal measured resistor, response and measured of memristive response im- of plementation.memristiveThe high implementation. -pass filter with a memristor (potentiometer) has the cutoff frequency, ap- proximately 5.3. High-Pass Filter 5.3. High-Pass Filter i fC1 (2  Re(Z (j  )) ) , i  M,P. (20) One of the examples is ci a,HPF high pass filterfilter which 1 uses aa potentiometerpotentiometer to control the cutoff frequency. This group of devices were also proposed, and theoretical results are cutoffParasitic frequency capacitors. This group of the of memristordevices were and also potentiometer proposed, and have theoretical been neglected results be-are available in the open literature [24,25,40]. causeavailable their in capacitances the open literature have a [much24,25 ,lower40]. value comparing to capacitor C1. Our realizationrealization ofof this this circuit circuit with with a memristora memristor instead instead of a of potentiometer a potentiometer is proposed is pro- in Figure 14, where the selected capacitor is C =2 nF. For this experimental analysis, the posed in Figure 14, where the selected capacitor1 is C1=2 nF. For this experimental analy- sis,impedances the impedances of oscilloscope of oscillo probesscope can probes be neglected. can be neglected. At the circuit’s At the inputcircuit's the input oscilloscope the os- cilloscope probe is connected in parallel with a voltage generator, whereas the capacitor value C1 is chosen to be significantly higher than the probe impedance at the circuit's output. Than we analyzed this filter with a potentiometer in order to better notice a dif- ference in relation to a memristor. The high-pass filter with a memristor (potentiometer) has the cutoff frequency, ap- proximately

i fCci,HPF1 (2  Re(Z (j  )) 1 ) , i  M,P. (20)

Parasitic capacitors of the memristor and potentiometer have been neglected be- cause their capacitances have a much lower value comparing to capacitor C1.

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probe is connected in parallel with a voltage generator, whereas the capacitor value C1 is chosen to be significantly higher than the probe impedance at the circuit’s output. Than we Electronics 2021, 10, x FOR PEER REVIEW 16 of 20 analyzed this filter with a potentiometer in order to better notice a difference in relation to a memristor.

Electronics 2021, 10, x FOR PEER REVIEW 16 of 20

FigureFigure 14.14. High-passHigh-pass filterfilter withwith memristor.memristor.

TheIn order high-pass to illustrate filter with the a memristor comparison (potentiometer) between thes hase thetwo cutoff realizations, frequency,approximately we present the magnitude response in Figure 15. For both implementations, with the potentiometer and f i =∼ ( (Z ( ω))C ) i = with the memristor, the magnitudec ,HPF 1/ response2πRe i jhas been1 , experimentallyM, P. verified. It can (20) be seen that the magnitude responses are almost the same for both filter realizations. FigureParasitic 14. High capacitors-pass filter with of the memristor. memristor and potentiometer have been neglected because their capacitances have a much lower value comparing to capacitor C1. InIn order to illustrate the comparison between thes thesee two realizations, we present the magnitude response in Fig Figureure 1515. For both implementations, with the potentiometer and with the memristor memristor,, the magnitude response has been experimentally verified.verified. It can be seen that the magnitude responses are almost the same for both filterfilter realizations.realizations.

Figure 15. Magnitude response of the high-pass filter based on memristor and potentiometer. Im- pedances were measured at 10 kHz for memristors and at DC for the potentiometer.

5.4. Phase Shifter A phase shifter is the next example which uses a potentiometer to control the phase Figureshift value. 15. 15. MagnitudeMagnitude Theoretical response response examples of ofthe the ofhigh realizations high-pass-pass filter filter basedare basedava onilable memristor on memristorin the and open potentiometer and literature potentiometer. .[ 4Im-5]. pedancesImpedancesOur were realization were measured measured of at the 10 at 10phasekHz kHz for shifter formemristors memristors with and a andmemristor at DC at DC for forthe is the proposedpotentiometer potentiometer. in .Figure 16. For this realization we selected resistors R1 = 47.8 kΩ and R2 = 47.8 kΩ, and capacitor C1 = 1.2 5.4.nF. Phase Phase The S Shifter selectedhifter operational amplifier is a dual low-noise operational amplifier NA5532APA phase [ 4shifter 3]. For isthe the experimental n nextext example verification, which uses the a potentiometer impedances of to oscilloscope control the probes phase shiftcan bevalue. negle Theoretical Theoreticalcted. At the examples examples circuit's inputof realizations the oscilloscope are ava available probeilable inis connected the open literature in parallel [454 with5].]. a voltageOur realization generator, of ofwhereas thethe phase phase the shifter shifteroutput with with impedance a memristora memristor of isthe proposedis operational proposed in Figurein amplifier Figure 16. For1 6is. thisForsig- realization we selected resistors R = 47.8 kΩ and R = 47.8 kΩ, and capacitor C = 1.2 nF. thisnificantly realization lower w thane selected the probe resistors impedance1 R1 = 47.8 at kΩ the and circuit's2 R2 = 47.8 output. kΩ, andThereafter, capacitor 1the C 1phase = 1.2 nF.Theshifter selected The realization selected operational with operational the amplifier potentiometer amplifier is a dual is low-noise is analyzed. a dual operational low-noise amplifier operational NA5532AP amplifier [43]. NA5532APFor theWhen experimental [the43]. phase For the verification, shifter experimental is realized the impedancesverification, with a memristor ofthe oscilloscope impedances (potentiometer), probes of oscilloscope can the be neglected.frequency probes canAtresponse the be circuit’snegle is cted. input At the oscilloscopecircuit's input probe the oscilloscope is connected probe in parallel is connected with a voltage in parallel generator, with whereas the output impedance of the operational amplifier is significantly lower than the a voltage generator, whereas the outputj impedanceCZ (j  )  1 of the operational amplifier is sig- probe impedance at the circuit’si output. Thereafter,1 i the phase shifter realization with the nificantly lower than the probeHPS impedance(j ) at the circuit's, i  M,P.output. Thereafter, the phase(21) potentiometer is analyzed. jCZ1 i (j  )  1 shifter realization with the potentiometer is analyzed. and Whenthe phase the phaseshift at shifter the desired is realized frequen withcy a can memristor be read (potentiometer),off directly from the the frequency phase re- responsesponse curve. is

i jCZ1 i (j  )  1 HPS (j ) , i  M,P. (21) jCZ1 i (j  )  1 and the phase shift at the desired frequency can be read off directly from the phase re- sponse curve.

EElectronicslectronics 20212021, ,1010, ,x 181FOR PEER REVIEW 1715 of of 20 18

Electronics 2021, 10, x FOR PEER REVIEW 17 of 20

FigureFigure 16. 16. PhasePhase shifter shifter with with memristor memristor..

InWhen order the to present phase shifter comparison is realized between with these a memristor two realizations, (potentiometer), we show the the frequency phase responseresponse in is Figure 17. The phase responses of the phase shifter based on potentiometer and on memristor were experimentallyi jω verified.C1Zi(jω From) − 1 Figure 17, it can be seen that phase HPS(jω) = , i = M, P. (21) responses are almost the same for bothj ω realizations.C1Zi(jω) + 1The phase response decreases with frequencyFigureand the16. Phase phasedue to shifter shiftconstant with at the memristorcapacitor desired .value frequency C1. can be read off directly from the phase response curve. InIn order order to to present present comparison comparison between between these these two two realizations, realizations, we we show show the the phase phase responseresponse in in Fig Figureure 1 177. .The The phase phase response responsess of of the the phase phase shifter shifter based based on on potentiometer potentiometer andand on on memristor memristor were were ex experimentallyperimentally verified. verified. From From Figure Figure 1 177, ,it it can can be be seen seen that that phase phase responsesresponses are are almost almost the the same same for for both both realizations. realizations. The The phase phase response response decreases decreases with with frequency due to constant capacitor value C1. frequency due to constant capacitor value C1.

Figure 17. Phase response of the phase shifter based on memristor and potentiometer. Impedances were measured at 10 kHz for memristors and at DC for the potentiometer.

6. Conclusions In this paper, we considered the possibility of replacing potentiometers with memristors in some typical electric circuits. Our motivation lies in the key features of the memristorFigureFigure 17. 17. Phase Phase as a response non response-volatile of of the the onephase phase port shifter shifter nanoscale based based on on devicememristor memristor that and and can potentiometer. potentiometer. be tuned to Impedances Impedancesa specific memristancewerewere measured measured in at ata 10 10wide kHz kHz resistancefor for memristors memristors range. and and Potentiometers at at DC DC for for the the potentiometer. potentiometer. are widely used in mixed-signal ele6.ctronic Conclusions circuits and systems, but the application is limited to low frequencies due to parasitic6. Conclusions effects. We compared a commercially available KnowM memristor with a In this paper, we considered the possibility of replacing potentiometers with memris- tungstenIn this (W) paper,dopant wewith considered the best in class the possibilitypotentiometer of replacingfrom Analog potentiometers Devices. with tors in some typical electric circuits. Our motivation lies in the key features of the memristor memristorsOne of ourin some goals typical was to electric verify thatcircuits. the memristor Our motivation has less lies significant in the key parasitic features effects of the as a non-volatile one port nanoscale device that can be tuned to a specific memristance in a thanmemristor the potentiometer. as a non-volatile We analyzed one port the nanoscale frequency device range that from can 10 be kHz tuned to to5 MHz a specific and wide resistance range. Potentiometers are widely used in mixed-signal electronic circuits observedmemristance the in impedances a wide resistance of both range. components. Potentiometers Impedance are widelyanalysis used confirmed in mixed that-signal the and systems, but the application is limited to low frequencies due to parasitic effects. We memristorelectronic circuitshas a wider and systems,bandwidth but than the theapplication potentiometer, is limited because to low the frequencies memristor’s due par- to compared a commercially available KnowM memristor with a tungsten (W) dopant with asiticparasitic effects effects. dominate We comparedat higher frequencies. a commercially It was available shown KnowMthat the resistance memristor range with is a the best in class potentiometer from Analog Devices. widertungsten in c(W)ase dopantof the memristorwith the best (400 in Ωclass–1 MΩ) potentiometer in comparison from Analogto the potentiometer Devices. (100 One of our goals was to verify that the memristor has less significant parasitic effects Ω–100One kΩ). of our We g proalsoposed was to an verify AC modelthat the for memristor the KnowM has less memristor, significant based parasitic on experi- effects than the potentiometer. We analyzed the frequency range from 10 kHz to 5 MHz and mentalthan the results. potentiometer. For memristance We analyzed values thebelow frequency 15 kΩ, therange memristor from 10 can kHz be tomodeled 5 MHz with and observed the impedances of both components. Impedance analysis confirmed that the observed the impedances of both components. Impedance analysis confirmed that the a memristorvariable resistor has a wider and a bandwidth shunt capacitance than the potentiometer,of 8 nF, whereas because for values the memristor’s above 15 parasitickΩ the memristor has a wider bandwidth than the potentiometer, because the memristor’s par- shunteffects capacitance dominate at varie highers with frequencies. frequency. It wasFor frequencies shown that the below resistance 300 kHz, range the is parasitic wider in asitic effects dominate at higher frequencies. It was shown that the resistance range is capacitancecase of the is memristor slightly less (400 thanΩ–1 8 MpF,Ω )whereas in comparison above 300 to the kHz, potentiometer it is more than (100 8 nF.Ω–100 kΩ). wider in case of the memristor (400 Ω–1 MΩ) in comparison to the potentiometer (100 WeThe proposed memristor an AC has model a significantly for the KnowM smaller memristor, footprint. based An onexternal experimental circuit for results. the re- For Ω–100 kΩ). We proposed an AC model for the KnowM memristor, based on experi- sistancememristance control values is requi belowred for 15 both kΩ, thecomponents. memristor The can memristor be modeled is expected with a variable to change resistor its mental results. For memristance values below 15 kΩ, the memristor can be modeled with a variable resistor and a shunt capacitance of 8 nF, whereas for values above 15 kΩ the shunt capacitance varies with frequency. For frequencies below 300 kHz, the parasitic capacitance is slightly less than 8 pF, whereas above 300 kHz, it is more than 8 nF. The memristor has a significantly smaller footprint. An external circuit for the re- sistance control is required for both components. The memristor is expected to change its

Electronics 2021, 10, 181 16 of 18

and a shunt capacitance of 8 nF, whereas for values above 15 kΩ the shunt capacitance varies with frequency. For frequencies below 300 kHz, the parasitic capacitance is slightly less than 8 pF, whereas above 300 kHz, it is more than 8 nF. The memristor has a significantly smaller footprint. An external circuit for the re- sistance control is required for both components. The memristor is expected to change its state on the fly, whereas the potentiometer is not operative during switching. The potentiometer resistance state switching is of the order of microseconds, whereas mem- ristors need milliseconds. The memristor is expected to be tuned to any memristance value in the whole resistance range, whereas the potentiometer has 1024 resistance states. The potentiometer has non-volatile memory that can store a resistance value for 50 times, whereas the memristor itself is non-volatile and does not require a constant power supply to be operational. In order to illustrate the differences between the memristor and the potentiometer, we implemented laboratory prototypes of memristor-based circuits with typical functionalities. Experimental results showed that the memristor has equal or better characteristics than the potentiometer. Memristor parasitics are less pronounced than the potentiometer parasitics consequently memristor based resistive circuits, amplifier/divider, has wider frequency bandwidth. Filters and phase shifters contain dynamic elements. The capacitors in this circuit absorbs parasitic capacitances and minimizes their influences. Effectively, small parasitic capacitors are connecting in parallel with much larger capacitors. The memristor has a wider resistance range compared to the potentiometer, so more resistance values can be obtained in all analyzed circuits. Memristance tuning is one of the major challenges in the design of memristive circuits. Switching time of binary state memristors is of the order of nanoseconds, while the switch- ing energy is around a picojoule. Unfortunately, there is still no appropriate mechanism for memristance tuning that is rapid and accurate at the same time. Some recent research results about the programming of commercially available memristors are very promising, but this is still an open area for further study and improvement.

Author Contributions: Conceptualization, I.M. and M.P.; methodology, D.T.; validation, I.M., M.P., and D.T.; formal analysis, D.T.; investigation, I.M.; resources, I.M.; writing—original draft preparation, I.M.; writing—review and editing, M.P.; visualization, I.M. and M.P.; supervision, M.P. and D.T.; project administration, M.P.; funding acquisition, I.M. and M.P. All authors have read and agreed to the published version of the manuscript. Funding: This work was supported in part by the Ministry of Education, Science and Technological Development of the Republic of Serbia. Data Availability Statement: Data is contained within the article. The data presented in this study are available in this article. Conflicts of Interest: The authors declare no conflict of interest.

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