**2. Foundation: the principle of controlling threshold parameters in memristor based oscillator circuits**

#### **2.1 Operating principles of reactance-less memristor based oscillators**

Oscillators without inductors and capacitors are the result of the memristor features applying. The self-excitation conditions are provided by the inertia of the resistance change of memristors when current flows through memristors devices. The absence of reactive elements allows to minimize the size of memristor based oscillators (MBO). The requirements to oscillator-based computing are met, in particular, by various variants of MBOs that differ in the number of memristor devices and the techniques of their coupling.

The schematic of typical reactance-less MBO is shown in **Figure 1a**. The circuit consists of memristor device *М* and a two-threshold comparator (TTC) with a current generator. The comparator converts the voltage *v* on the memristor to a binary output signal *vout* (**Figure 1b**). The current generator converts the output binary signal ("0" and "1") of *vout* to opposite corresponding currents *i v*ð Þ *(-I* and *+I*). The current input *iin* is conventional input for reactance-less MBO. The memristor is connected to the input of the comparator by anode.

The memristor resistance R is decreased at a positive voltage *v* at anode when a positive current *i* flows in. The transfer function of the comparator is shown in **Figure 1b**. The comparator output voltage is "0" at –*VM* <*v*<*Vm* and it is equal to

#### **Figure 1.**

*Typical illustrative graphs of behavior of reactance-less memristor based oscillator: (a) schematic of memristor based oscillator), (b) transfer function of comparator, (c) input function of comparator with current source, (d) waveforms of varying memristor resistor, (e) hysteresis loop for memristor resistor at phase plan.*

"1" otherwise. Here *VM* >*Vm*. The current generator in the negative feedback circuit of the comparator converts the binary output signal ("0", "1") into a negative current and a positive current through the memristor ð�*I*, þ *I*), respectively (**Figure 1c**). The input current *iin* is summed with the current *i v*ð Þ.

The memristor resistance can be considered as characteristic of oscillator state. Typical graph of varying memristor resistance in self-excitation mode of oscillator is given in **Figure 1d**. The phase plan (**Figure 1e**) illustrates the cycle of change of the memristor resistance R while oscillations as hysteresis loop.

Let us consider the cycle of periodic self-excitation mode of memristor oscillator (**Figure 1a**). Let's assume that for the initial moment of time *t*<sup>0</sup> the voltage value *v* is *v*>*Vm* (**Figure 1c**). In this case the current is positive *i* ¼ *I* and *vout* ¼ }1}. Therefore, the memristor resistance R and the memristor voltage are reduced. At time *t*<sup>1</sup> the voltage reaches the threshold value *v* ¼ *Vm*, the output voltage *vout* goes from state "1" to state "0". The value of memristor resistance is *Rm* ¼ *Vm=I* at this time point. Here *Rm* is lower threshold value of the memristor resistance. In this case current *i* and voltage *v* become negative: *i* ¼ �*I* and ¼ �*Vm* . The memristor resistance begin to increase, this leads to decreasing the negative voltage on the memristor. At time *t*<sup>2</sup> it reaches the value *v* ¼ �*VM*, the output of the comparator goes from "0" to "1", the current and voltage on the memristor become positive again: *i* ¼ *I*, *v* ¼ *VM* . At this time point, the resistance of the memristor achieves the value *RM* ¼ *VM=I* where *RM* is upper threshold resistance value. To provide periodicity of this process the following conditions must be satisfied

$$R\_{ON} < R\_m = \frac{V\_m}{I} < \frac{V\_M}{I} = R\_M < R\_{OFF}.\tag{1}$$

Here *RON* - is the minimal memristor resistance, *ROFF* – is the maximal memristor resistance. In this case, the memristor resistance will periodically change in the range from the lower threshold value *Rm* to the upper threshold resistance *RM* (**Figure 1d**). The change in resistance is triangular if the rate of change in the memristor resistance does not depend on its value. The rate of change is proportional to the current according to the drift-diffusion model approximation [3].

The input current impacts on the speed of memristor resistance change. The speed is increased at the same signs of the input current and the generator current and it is decreased in opposite case.
