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1 Memristive circuits simulate memcapacitors and meminductors Yuriy V. Pershin and Massimiliano Di Ventra 0 Abstract—We suggest electronic circuits with memristors (re- 1 sistors with memory) that operate as memcapacitors (capacitors 20 wUistihngmaemmoermyr)isatonrdemmuemlaitnodr,utchtoerssu(gignedsutecdtorcsircwuiitths hmaveemobreye)n. aa WA VINA+DC n builtandtheiroperationhasbeendemonstrated,showingauseful B VIN- = andinterestingconnectionbetween thethreememory elements. a M J Index Terms—Memory, Analog circuits, Analog memories. 8 MMiiccrrooccoonnttrroolllleerr 1 Introduction:Memcapacitiveandmeminductivesystemsare two recently postulated classes of circuit elementswith mem- ] t ory [1] that complement the class of memristive systems [2], e R d [3]. Theirmain characteristicis ahystereticloop-whichmay - or may not pass through the origin [1] - in their constitutive - R s AA n variables (charge-voltage for memcapacitors and current-flux + 1 = i for meminductors) when driven by a periodic input, and, . M s unlike memristors, they can store energy. As of today, a few ic systems have been found to operate as memcapacitors and b C1 CR(t)(=t)C /R s meminductors(see[1]andreferencestherein).However,these M 1 y h are neither available on the market yet, nor their properties R p can be easily tuned to investigate their role in more complex [ cTihrceureitfso.reT,heelecstarmoneicceamnublaetosrasiodfasbuochutmmememorryisteilveemseynststetmhast. CC1 -+AA1 = R 2 v could be easily built and tuned would be highly desirable. 3 We have previously designed and built a memristor emulator M L(t)= c 8 and shown its use in neuromorphicand programmableanalog RR (t)C 5 M 1 circuits[4],[5].InthisLetter,weusesuchmemristoremulator 1 . todesignandbuildmemcapacitorandmeminductoremulators, Fig. 1. Circuits simulating (a) memristor, (b) memcapacitor and (c) 0 and prove experimentally their main properties. Since all of meminductor. Theirapproximate equivalent circuits areshownontheright. 1 these emulators can be built from inexpensive off-the-shelf 9 components we expect them to be extremely useful in the 0 : design,understandingandsimulationsofcomplexcircuitswith defined by RM =x with v memory. i X x˙ = (βVM +0.5(α−β)[|VM +VT|−|VM −VT|]) Proposed circuits: The proposed circuits of memcapacitor r and meminductor emulators are shown in Fig. 1(b) and (c), ×θ(x−Rmin)θ(Rmax−x), (1) a respectively,togetherwiththememristoremulatorinFig.1(a). where VM is the voltage across the memristor, and θ(·) is The memristor emulator is implemented as in Refs [4], [5] the step function. We choose for this work the following and consists of a digital potentiometer whose resistance is parameters α = 0, β = 62kΩ/(V·s) (used in the memca- continuouslyupdatedby a microcontrollerand determinedby pacitor emulator), β =1MΩ/(V·s) (used in the meminductor pre-programmed equations of current-controlled or voltage- emulator), VT =1V, Rmin =5kΩ and Rmax =10kΩ. controlled memristive systems. In our experiments, the mem- ThememcapacitoremulatorconsistsofthismemristorM,a ristance RM is governed by an activation-type model [5], [6] capacitorC1 andaresistorRconnectedtoanoperationalam- plifier A1 asshownin Fig. 1b.Since the operationalamplifier keepsnearlyequalvoltagesatitspositiveandnegativeinputs, Yu.V.PershiniswiththeDepartmentofPhysicsandAstronomyandUSC thevoltageonthecapacitorC1 isappliedtotherightterminal Nanocenter, University ofSouthCarolina, Columbia, SC,29208 of R. Therefore, we can think that an effective capacitor e-mail:[email protected]. with a time-dependent capacitance C(t) is connected to the M.DiVentraiswiththeDepartment ofPhysics,University ofCalifornia, SanDiego,LaJolla, California 92093-0319 right terminal of R, so that the relation RC(t) = RM(t)C1 e-mail:[email protected]. holds. (Note that the voltage at the capacitor VC is equivalent 2 to the voltage, V−, at the negative terminal of the opera- loopsarevisibleinthecapacitanceasafunctionofthevoltage tional amplifier.) This allows us to determine the capacitance at the capacitor VC = V−. We also note that the capacitance as C(t) = RM(t)C1/R = (Vin − V−)/(RdV−/dt) since hysteresis is frequency dependent: the loop is much smaller RM(t) = (Vin − V−)/I = (Vin − V−)/(C1dV−/dt). In at the higher frequency of 8Hz. This is a manifestation of a the limit R ≪ RM, we obtain the approximate equivalent typical property of memory circuit elements [1] that at high circuit shown on the right of Fig. 1b. On the other hand, the frequencies behave as linear elements. The fluctuations of C meminductoremulatorissimilartothedesignofagyratorwith in Fig. 2b are related to the limited resolution of our data amemristorreplacingaresistor,andtheequivalentinductance acquisition system and some noise in the circuit. L(t) = RRM(t)C1, as it is evident from Fig. 1c. In both Similar considerations apply to the meminductor emulator cases, the time dependence of the equivalent capacitance, C, as demonstrated in Fig. 3. Here, it is clearly seen that the and inductance, L, is due to the time dependence of RM. shape of the V− signal (which in this case is equal to the voltage on the equivalent inductor VL) depends on the a 300 polarity of applied voltage. We extracted numerically the 2 b equivalent inductance L from Vin and V− signals and found that it contains a considerable amount of noise. Less noisy 250 ω=4Hz 1 estimation of equivalent inductance is obtained using a fit of ω=8Hz V) V− signal by exponentially decaying curves as demonstrated ge ( 0 F)µ 200 in Fig. 3a giving a decaying time τ = L/R, from which we Volta V C ( 150 hmaevmeienxdturcatcoterdemLu.lSationrcechtahnegmesefmasritst(oitrsepmarualmateotrersstaatreeignivthene -1 - below Eq. (1)), the equivalentinductanceL switches between V 100 two limiting values as shown schematically in Fig. 3b. -2 in ω=8Hz Finally, we would like to mention that although the sug- 0.00 0.25 0.50 -1 0 1 gestedemulatorsreproducetheessentialfeaturesofrealmem- Time (s) VC (V) capacitors and meminductors, certain aspects are different. In particular,the designedemulatorsare active devices requiring Fig.2. Memcapacitor emulator responsetoasquarewavesignal. Weused the circuit shown in Fig. 1b with R = 480Ω and C1 = 10µF. a Time- a power source for their operation. More importantly, these dependenceoftheinputvoltagesignalVinandvoltageatthenegativeinputof emulators do not actually store energy, which might be a theoperationalamplifierA1.b.TheequivalentcapacitanceC(t)numerically limitation in specific applications. However, from the point extracted fromVin andV− signalsasdescribed inthetext. of view of circuit response, almost any kind of memcapacitor and meminductor operation model can be realized using an appropriate memristor-emulator operation algorithm. 3 Conclusions: We have demonstrated that simple circuits a 45 b 2 with memristors can exhibit both memcapacitive and memin- ductive behavior. Memcapacitor and meminductor emulators 1 40 have been designed and built using the previously suggested ) e (V 0 H) 35 memristor emulator [4], [5] since solid-state memristors are ag-1 L ( not available yet. These emulators can be created from inex- olt 30 pensive off-the-shelf components, and as such they provide V-2 powerful tools to understand the different functionalities of -3 25 these newly suggested memory elements without the need V; exp(-(t-0.17)∗11) -4 Vin ; exp(-(t-0.36)∗20) of expensive material fabrication facilities. We thus expect - 20 they will be of use in diverse areas ranging from non-volatile 0.0 0.2 0.4 0.6 0.8 1.0 -2 0 2 Time (s) V (V) memory applications to neuromorphic circuits. L Acknowledgment: This work has been partially funded by Fig. 3. Meminductor emulator (Fig. 1cwith R=480Ω andC1 =10µF) the NSF grant No. DMR-0802830. response to a square wave signal. a Time-dependence of the input voltage signal Vin and voltage V− = VL at the negative input of the operational amplifier A1. b Schematics of meminductor hysteresis loop drawn with the REFERENCES inductance Lobtained usingexponential fitstoV− signals asshownina. [1] M. Di Ventra, Y. V. Pershin, and L. O. Chua, “Circuit elements with memory: memristors, memcapacitors and meminductors,” Proc. IEEE, vol.97,pp.1717–1724,2009. In order to prove that these circuits emulate the behavior [2] L. O. Chua and S. M. Kang, “Memristive devices and systems,” Proc. of memcapacitors and meminductors, we have analyzed their IEEE,vol.64,no.2,pp.209–223,1976. [3] L. O. Chua, “Memristor - the missing circuit element,” IEEE Trans. response under the application of a square wave signal. This CircuitTheory,vol.18,no.5,pp.507–519,1971. is shown in Fig. 2, where, on the left panel, we show both [4] Y.V.PershinandM.DiVentra,“Experimentaldemonstrationofassocia- the input voltage Vin and the voltage at the negative terminal tivememorywithmemristiveneural networks,”arXiv:0905.2935, 2009. [5] ——,“Practicalapproachtoprogrammableanalogcircuitswithmemris- of the operational amplifier V−, and, on the right panel, the tors,”arXiv:0908.3162, 2009. equivalent capacitance of the memcapacitor emulator at two [6] Y.V.Pershin, S.LaFontaine, andM.DiVentra, “Memristive model of valuesoffrequencyofthesquarewavesignal.Clearhysteresis amoeba’s learning,” Phys.Rev.E,vol.80,p.021926,2009.