Abstract
Frequency generators are widely used in electronics. Here, we report the design and experimental realization of a memristive frequency generator employing a unique combination of only digital logic gates, a singlesupply voltage and a realistic thresholdtype memristive device. In our circuit, the oscillator frequency and duty cycle are defined by the switching characteristics of the memristive device and external resistors. We demonstrate the circuit operation both experimentally, using a memristor emulator, and theoretically, using a model memristive device with threshold. Importantly, nanoscale realizations of memristive devices offer smallsize alternatives to conventional quartzbased oscillators. In addition, the suggested approach can be used for mimicking some cyclic (Sisyphus) processes in nature, such as “dripping ants” or drops from leaky faucets.
Introduction
Cyclic evolutions and relaxation oscillators are ubiquitous in nature. One example of these is the leaky faucet [Fig. 1(a)], where the suspended fluid mass increases gradually, until it suddenly decreases, when the droplet tears of f^{1,2,3}. Other examples of relaxational dynamics can also be found in granular media^{4} as well as in mechanical^{5} and superconducting systems^{6}. Similar dynamics can be even found in wildlife. For example, some types of ants^{7,8} can continuously climb a rod, aggregate, and eventually drop, right after a critical mass is accumulated [see Fig. 1(b)].
Analogous phenomena in driven dissipative systems are known as Sisyphus processes. According to Greek mythology, King Sisyphus was doomed to repeatedly push a rock uphill, which would then roll back down [Fig. 1(c)]. Recently, Sisyphus processes^{9} were studied in electric circuits based on superconducting^{10,11,12,13} and normalstate^{14,15,16} systems. In these circuits, a driven artificial atom, a qubit, was coupled to either a mechanical or an electrical resonator. Depending on its detuning, the qubit was driven by the resonator either “uphill” (the usual Sisyphus process) or downhill (unusual, or “happy Sisyphus” process). The cycle was completed by relaxation to the ground state in the presence of an additional periodic signal.
Here we demonstrate a new miniaturized clock signal generator based on a memristive (memory resistive) device^{17} operating in a Sisyphuslike cycle. In modern electronics, the clock signal is most frequently produced by quartz generators and sometimes by RCbased circuits or other approaches. The quartz generators offer a high precision at the cost of their size. Less precise RCcircuits are more compact. Nanoscale memristive devices^{18} [combining their very small size with switching frequencies in a convenient range (e.g., hundreds of MHz)] are the core components of clock generators explored here.
In the literature, there are several known methods of using memristors in oscillating circuits. In particular, significant attention has been focused on nonlinear oscillators constructed from Chua’s oscillators, by replacing Chua’s diodes with memristors^{19,20}. The authors of ref. 21 proposed a programmable frequencyrelaxation oscillator. In such a circuit, a memristorbased digital potentiometer is used to set switching thresholds of a Schmitt trigger. Moreover, memristors can be employed to replace capacitors in relaxation oscillators resulting in compact reactanceless oscillators^{22,23,24,25,26}. Additionally, it was shown experimentally that some polymeric memory devices exhibit slow current oscillations when subjected to a constant voltage^{27}. Our work presents an advanced reactanceless oscillator design having a unique combination of only digital logic gates, a singlesupply voltage and a realistic thresholdtype memristive device.
In the proposed clock signal generator, the frequency is defined by the switching characteristics of a memristive device. Our circuit employs bipolar memristive devices^{18} operating in such a way that their memristances (memory resistances) increase/decrease at positive/negative voltages applied to the device, respectively. The effective circuits depicted in Fig. 1(d) show the two stages of its operation. In stage 1 (left circuit), the positive voltage applied to the memristor M causes an increase in its memristance. In the second stage, the device polarity is changed and the memristance decreases. In this way, the memristance oscillates between two values [see Fig. 1(e)], similarly to the oscillations of the boulder height above the ground level in the Sisyphus case [Fig. 1(c)]. Importantly, the circuit does not require any largesize components (such as a quartz resonator) typically used in conventional oscillator circuits.
Memristive Clock Signal Generator
Figure 2(a) presents the specific memristive clock generator circuit introduced in this work. Its components include the memristive system M, an OR gate with Schmitttrigger inputs (to the left), opendrain identity (upper) and NOT (lower) gates, three resistors and a capacitor. Note that the resistors R_{1} and R_{2} here play the same role as R_{1} and R_{2} in the effective circuits shown in Fig. 1(e). As will be readily observed, the output of the OR gate defines the operation stages: logical 0 corresponds to the increasing memristance stage 1, while logical 1 corresponds to the decreasing memristance stage 2. The hysteretic input levels of the Schmitttrigger inputs (denoted by V_{+} for the logic 1, and V_{−} for the logic 0, V_{−} < V_{+}) are employed as voltage thresholds triggering the stage changes. Figure 2(b) shows the calculated pinched hysteresis loops of the memristive system M described by Eqs (1, 2). In this calculation, R_{M}(t = 0) = 2 kΩ, V_{M}(t) = V_{0}sin(2πft), V_{0} = 2.9 V, and all other parameters of M are the same as in the memristor emulator specified below Eqs (1)–(2),.
One can notice from Fig. 2(a) that indeed, the increasing memristance stage 1 switches to the decreasing memristance stage 2 as soon as the voltage level V_{M} reaches the V_{+} threshold. At this instance in time, the output of the OR gate changes to 1, grounding the bottom terminal of M and setting the identity gate into the high impedance state. Assuming that V_{M,2} > V_{−} right after the switching (below we discuss all the related requirements), the circuit will remain in the decreasing memristance stage 2 while V_{M,2} > V_{−}. The transition from stage 2 to stage 1 occurs in a similar way (as soon as V_{M,2} < V_{−}, and thus both inputs of OR become logical zeros). In Fig. 2(a), the R_{3}C circuit introduces a short time delay to ensure proper switching between the phases. The generator output frequency is not influenced by this delay.
Next, we discuss the experimental implementation of the clock signal generator. In our experiments, the memristive system M is realized with a digital memristor emulator^{21,28}. Its main parts include a microcontroller, analogtodigital converter and digital potentiometer. The operation of the digital memristor emulator is straightforward: using the analogtodigital converter, the microcontroller cyclically measures the voltage applied to the digital potentiometer, calculates an updated value of the memristance (using preprogrammed equations of voltagecontrolled or currentcontrolled memristive system^{17}), and writes the updated value of the memristance into the digital potentiometer. Figure 5(b) of ref. 28 presents a photograph of the specific digital memristor emulator realization employed in the present paper. More details regarding the emulator design can be found in refs 29,30.
The memristor emulator was preprogrammed with a model of voltagecontrolled memristive systems with threshold^{29,30}
where I and V_{M} are the current through and the voltage across the memristive system, respectively, and x is the internal state variable playing the role of the memristance. Here R_{M}(x) ≡ x, β is a positive switching constant characterizing the intrinsic rate of memristance change when V_{M} > V_{t}, V_{t} is the threshold voltage, and the + or − sign is selected according to the device connection polarity. Additionally, it is assumed that the memristance is limited to the interval [R_{on}, R_{off}] (note that R_{on} < R_{off}). The specific model parameters used in our emulator are β = 62 kΩ/V ⋅ s, V_{t} = 1.2 V, R_{on} = 1 kΩ, and R_{off} = 10 kΩ, and R_{M}(t = 0) = (R_{on} + R_{off})/2. We built the circuit shown in Fig. 2(b) using a TI SN74HC7032 positiveOR gate with Schmitttrigger inputs, CD74AC05 inverters with opendrain outputs, R_{1} = 4.7 kΩ, R_{2} = 2.7 kΩ, R_{3} = 10 kΩ, and C = 1.35 nF.
Our circuit is fully reproducible within the specifications of the circuit components. Assuming that the future memristor devices will be available with well characterized characteristics (similarly, for example, to usual resistors or capacitors), we expect that the suggested circuit based on real memristors will be reproducible too. In addition, our measurements and simulations did not display any significant dispersion of the memristive properties as well of the output signal produced by the circuit. Basically, the memristor emulator operates deterministically and thus the circuit also operates deterministically. As deterministic models are frequently used to describe the response of real memristive devices and, in many cases, show a very good agreement with experiments, we believe that the selected approach (based on the emulator) illustrates our idea in a very realistic manner.
Figure 3(a) presents results of our measurements. While the V_{M,1} and V_{M,2} curves clearly demonstrate two stages of the circuit operation, the output V_{out} [displayed in Fig. 2(a)] shows a stable clock signal with a period of about 0.194 s and a duty cycle of about 27%. With the knowledge of R_{1} and R_{2}, the V_{M,1} and V_{M,2} curves were used to extract the time dependence of the memristance R_{M} depicted in Fig. 3(b). Its variations are similar to the desired variations of R_{M} sketched in Fig. 1(e).
Circuit Analysis
In Fig. 3(b), the memristance R_{M} of M changes periodically from to , and back (see also Fig. 1(e)). Analytically, one can find the corresponding durations of the time intervals, τ_{1} and τ_{2}. In what follows, we focus on the effective circuit models depicted in Fig. 2(d) to find τ_{1} and τ_{2} analytically.
For the sake of convenience, let us consider the system dynamics from t = 0 in both stages and perform calculations based on the memristive device model employed in our emulator [Eqs (1, 2)]. Then, using
where i = 1, 2 denotes the two time intervals, we integrate Eq. (2) with appropriate initial and final conditions for both intervals, and accounting for the memristor polarity. As a result, one can find the following expression for the duration of the two stages
The oscillation period is given by
The boundary values of the memristance, and , can be expressed through the Schmitttrigger input thresholds as
The expression for the period T can be written in a simple form assuming that R_{1} = R_{2} and V_{p} ≫ V_{t} (and also so that ). Then we obtain
This together with Eq. (6) gives
The above formulas are plotted in Fig. 4. There, it is assumed that all the circuit parameters (except R_{1} and R_{2}) are fixed. This figure shows that both the period T and the duty cycle τ_{2}/T can be tuned in a certain range by varying R_{1} and/or R_{2}. In Fig. 4, the solid curve was plotted for the same value of R_{2} as used in the experiment, while the other curves demonstrate the changes assuming equal resistances, R_{2} = R_{1}, for several values of the memristor threshold voltage V_{t}. In particular, Fig. 4 demonstrates that the formula (8), which is valid at V_{t} = 0 and R_{2} = R_{1}, provides a good estimation for the period T for small values of the threshold voltage V_{t} when R_{2} = R_{1}. Moreover, Fig. 4 also shows several experimentally measured periods and corresponding duty cycles that exhibit a very good agreement with our analytical results.
Finally, we consider the limitations imposed on the clock signal generator components required for its proper operation. The numerical estimations provided below employ parameter values that are close to those of our experimental implementation of the signal generator. For the convenience of readers, here we list the values of these parameters: V_{p} = 5 V, V_{+} = 3 V, V_{−} = 1.8 V, V_{t} = 1.2 V, R_{on} = 1 kΩ, and R_{off} = 10 kΩ.
1. In the stage 1, the minimum (maximum) values of V_{M} must be below (above) V_{+}; namely, V_{M}(R_{on}) < V_{+} and V_{M}(R_{off}) > V_{+}. Consequently,
Numerically, 0.667 kΩ < R_{1} < 6.67 kΩ.
2. In the stage 2, the minimum (maximum) values of V_{M} must be below (above) V_{−}, namely, V_{M}(R_{on}) < V_{−} and V_{M}(R_{off}) > V_{−}. Consequently,
Numerically, 1.778 kΩ < R_{2} < 17.78 kΩ.
3. Moreover, right after the switching from stage 1 to stage 2, the voltage across M should stay above V_{−}, namely, , where is given by Eq. (6). It follows that
and, numerically, R_{2}/R_{1} < 2.67. We note that the requirement that in the transition from stage 2 to stage 1 the voltage across M stays below V_{+}, is also given by Eq. (11). For the same transition, one can also require that in stage 1, however, this requirement is weaker than the criterion 4.
4. In order to start oscillations from any initial condition, the voltage across M (in the worstcase limit R_{M} = R_{on}) should exceed V_{t}. This results in
Using the abovementioned parameters, we obtain R_{1} < 3.167 kΩ.
5. Lastly, we mention the obvious requirement on V_{−}, namely, V_{t} < V_{−}.
One can easily notice that in our experimental implementation of the circuit shown in Fig. 2(b), the selected values of R_{1} = 4.7 kΩ and R_{2} = 2.7 kΩ satisfy the criteria in Eqs (9, 10, 11). Regarding the criterion (12), we were able to use a large value of R_{1}, because the initial value of the emulator memristance was selected above R_{on}.
Conclusion
We have proposed and analyzed the design of a memristive clock signal generator. Assuming a realistic threshold model of a memristive device, we experimentally demonstrated the operation of a memristive frequency generator. While our demonstration is based on a slow memristor emulator, the real memristive devices can result in frequencies in the industrially important MHzGHz range. Moreover, the specific proposed circuit for frequency generation offers frequency and duty cycle tunability. These theoretical considerations together with our experimental emulation shows the potential of such circuits for very compact frequency generators.
Additional Information
How to cite this article: Pershin, Y. V. et al. Memristive Sisyphus circuit for clock signal generation. Sci. Rep. 6, 26155; doi: 10.1038/srep26155 (2016).
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Acknowledgements
This work has been supported by the NSF grant No. ECCS1202383, USC Smart State Center for Experimental Nanoscale Physics, the RIKEN iTHES Project, MURI Center for Dynamic MagnetoOptics, and a GrantinAid for Scientific Research (A), and Russian Scientific Foundation grant No. 151320021.
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Y.V.P. conducted the experiment, Y.V.P. and S.N.S. did the calculations. All the authors discussed the results and cowrote the manuscript.
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Pershin, Y., Shevchenko, S. & Nori, F. Memristive Sisyphus circuit for clock signal generation. Sci Rep 6, 26155 (2016). https://doi.org/10.1038/srep26155
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