Memristive circuits simulate memcapacitors and meminductors
aa r X i v : . [ phy s i c s . i n s - d e t ] J a n Memristive circuits simulate memcapacitors andmeminductors
Yuriy V. Pershin and Massimiliano Di Ventra
Abstract —We suggest electronic circuits with memristors (re-sistors with memory) that operate as memcapacitors (capacitorswith memory) and meminductors (inductors with memory).Using a memristor emulator, the suggested circuits have beenbuilt and their operation has been demonstrated, showing a usefuland interesting connection between the three memory elements.
Index Terms —Memory, Analog circuits, Analog memories.
Introduction:
Memcapacitive and meminductive systems aretwo recently postulated classes of circuit elements with mem-ory [1] that complement the class of memristive systems [2],[3]. Their main characteristic is a hysteretic loop - which mayor may not pass through the origin [1] - in their constitutivevariables (charge-voltage for memcapacitors and current-fluxfor meminductors) when driven by a periodic input, and,unlike memristors, they can store energy. As of today, a fewsystems have been found to operate as memcapacitors andmeminductors (see [1] and references therein). However, theseare neither available on the market yet, nor their propertiescan be easily tuned to investigate their role in more complexcircuits. The same can be said about memristive systems.Therefore, electronic emulators of such memory elements thatcould be easily built and tuned would be highly desirable.We have previously designed and built a memristor emulator and shown its use in neuromorphic and programmable analogcircuits [4], [5]. In this Letter, we use such memristor emulatorto design and build memcapacitor and meminductor emulators ,and prove experimentally their main properties. Since all ofthese emulators can be built from inexpensive off-the-shelfcomponents we expect them to be extremely useful in thedesign, understanding and simulations of complex circuits withmemory.
Proposed circuits:
The proposed circuits of memcapacitorand meminductor emulators are shown in Fig. 1(b) and (c),respectively, together with the memristor emulator in Fig. 1(a).The memristor emulator is implemented as in Refs [4], [5]and consists of a digital potentiometer whose resistance iscontinuously updated by a microcontroller and determined bypre-programmed equations of current-controlled or voltage-controlled memristive systems. In our experiments, the mem-ristance R M is governed by an activation-type model [5], [6] Yu. V. Pershin is with the Department of Physics and Astronomy and USCNanocenter, University of South Carolina, Columbia, SC, 29208e-mail: [email protected]. Di Ventra is with the Department of Physics, University of California,San Diego, La Jolla, California 92093-0319e-mail: [email protected]. AW ADC V IN+ a = B V IN- a M MicrocontrollerMicrocontroller
R - A R+ A = M C C ( t )= R M ( t ) C /R b R - AC R+ A C = M L ( t )= RR M ( t ) C c Fig. 1. Circuits simulating (a) memristor, (b) memcapacitor and (c)meminductor. Their approximate equivalent circuits are shown on the right. defined by R M = x with ˙ x = ( βV M + 0 . α − β ) [ | V M + V T | − | V M − V T | ]) × θ ( x − R min ) θ ( R max − x ) , (1)where V M is the voltage across the memristor, and θ ( · ) isthe step function. We choose for this work the followingparameters α = 0 , β = 62 k Ω / (V · s) (used in the memca-pacitor emulator), β = 1 M Ω / (V · s) (used in the meminductoremulator), V T = 1 V, R min = 5 k Ω and R max = 10 k Ω .The memcapacitor emulator consists of this memristor M , acapacitor C and a resistor R connected to an operational am-plifier A as shown in Fig. 1b. Since the operational amplifierkeeps nearly equal voltages at its positive and negative inputs,the voltage on the capacitor C is applied to the right terminalof R . Therefore, we can think that an effective capacitorwith a time-dependent capacitance C ( t ) is connected to theright terminal of R , so that the relation RC ( t ) = R M ( t ) C holds. (Note that the voltage at the capacitor V C is equivalent to the voltage, V − , at the negative terminal of the opera-tional amplifier.) This allows us to determine the capacitanceas C ( t ) = R M ( t ) C /R = ( V in − V − ) / ( RdV − /dt ) since R M ( t ) = ( V in − V − ) /I = ( V in − V − ) / ( C dV − /dt ) . Inthe limit R ≪ R M , we obtain the approximate equivalentcircuit shown on the right of Fig. 1b. On the other hand, thememinductor emulator is similar to the design of a gyrator witha memristor replacing a resistor, and the equivalent inductance L ( t ) = RR M ( t ) C , as it is evident from Fig. 1c. In bothcases, the time dependence of the equivalent capacitance, C ,and inductance, L , is due to the time dependence of R M . V - V in V o lt a g e ( V ) Time (s) ω =8Hz a b C ( µ F ) V C (V) ω =4Hz ω =8Hz Fig. 2. Memcapacitor emulator response to a square wave signal. We usedthe circuit shown in Fig. 1b with R = 480Ω and C = 10 µ F. a Time-dependence of the input voltage signal V in and voltage at the negative input ofthe operational amplifier A . b . The equivalent capacitance C ( t ) numericallyextracted from V in and V − signals as described in the text. a V o lt a g e ( V ) Time (s) V in ; exp(-( t -0.17) ∗ V - ; exp(-( t -0.36) ∗ b V L (V) L ( H ) Fig. 3. Meminductor emulator (Fig. 1c with R = 480Ω and C = 10 µ F)response to a square wave signal. a Time-dependence of the input voltagesignal V in and voltage V − = V L at the negative input of the operationalamplifier A . b Schematics of meminductor hysteresis loop drawn with theinductance L obtained using exponential fits to V − signals as shown in a . In order to prove that these circuits emulate the behaviorof memcapacitors and meminductors, we have analyzed theirresponse under the application of a square wave signal. Thisis shown in Fig. 2, where, on the left panel, we show boththe input voltage V in and the voltage at the negative terminalof the operational amplifier V − , and, on the right panel, theequivalent capacitance of the memcapacitor emulator at twovalues of frequency of the square wave signal. Clear hysteresis loops are visible in the capacitance as a function of the voltageat the capacitor V C = V − . We also note that the capacitancehysteresis is frequency dependent: the loop is much smallerat the higher frequency of 8Hz. This is a manifestation of atypical property of memory circuit elements [1] that at highfrequencies behave as linear elements. The fluctuations of C in Fig. 2b are related to the limited resolution of our dataacquisition system and some noise in the circuit.Similar considerations apply to the meminductor emulatoras demonstrated in Fig. 3. Here, it is clearly seen that theshape of the V − signal (which in this case is equal tothe voltage on the equivalent inductor V L ) depends on thepolarity of applied voltage. We extracted numerically theequivalent inductance L from V in and V − signals and foundthat it contains a considerable amount of noise. Less noisyestimation of equivalent inductance is obtained using a fit of V − signal by exponentially decaying curves as demonstratedin Fig. 3a giving a decaying time τ = L/R , from which wehave extracted L . Since the memristor emulator state in thememinductor emulator changes fast (its parameters are givenbelow Eq. (1)), the equivalent inductance L switches betweentwo limiting values as shown schematically in Fig. 3b.Finally, we would like to mention that although the sug-gested emulators reproduce the essential features of real mem-capacitors and meminductors, certain aspects are different. Inparticular, the designed emulators are active devices requiringa power source for their operation. More importantly, theseemulators do not actually store energy, which might be alimitation in specific applications. However, from the pointof view of circuit response, almost any kind of memcapacitorand meminductor operation model can be realized using anappropriate memristor-emulator operation algorithm. Conclusions:
We have demonstrated that simple circuitswith memristors can exhibit both memcapacitive and memin-ductive behavior. Memcapacitor and meminductor emulatorshave been designed and built using the previously suggestedmemristor emulator [4], [5] since solid-state memristors arenot available yet. These emulators can be created from inex-pensive off-the-shelf components, and as such they providepowerful tools to understand the different functionalities ofthese newly suggested memory elements without the needof expensive material fabrication facilities. We thus expectthey will be of use in diverse areas ranging from non-volatilememory applications to neuromorphic circuits.
Acknowledgment:
This work has been partially funded bythe NSF grant No. DMR-0802830.R
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