Mutual Transformation of Flux-Controlled and Charge-Controlled Memristors

Two-ports for mutual transformation between flux-controlled memristors and charge-controlled memristors are proposed. Attention is paid to memristors with the region of negative differential memristance in the charge-flux constitutive relation, when such a transformation should be made carefully. Connections between this transformation, duality rules, and Chua’s table of higher-order elements are described. The proposed transforming cells can be made up of commercially available integrated circuits. Their proper operation is demonstrated via simulations and lab experiments with memristive oscillators.


I. INTRODUCTION
Today's resistive memory devices for non-volatile memories, memristive logic circuits or devices emulating neuromorphic systems are modelled as the so-called memristive systems [1], which are a generalization of the original concept of the memristor introduced in 1971 [2]. According to the current terminology, memristive systems are called extended memristors while the memristor from [1] is denoted as the ideal memristor [3]. In addition to the important role of memristors as models of existing memristive devices, the ideal or ideal generic memristors [3] are important building blocks of the so-called predictive modeling [4] whose key idea is to build complex nonlinear models of existing systems from a set of fundamental electrical elements, the ideal memristor being one of them.
The predictive model of the ideal memristor is its constitutive relation between the charge q and the flux , i.e. between the first time integrals of the memristor current and voltage [3]. The form of this relation predetermines the memristor behavior under arbitrary conditions. For passive memristors, the relation ( ) q     is strictly monotonic, and the corresponding inversion leads to the equivalent relation ( ) q q    . However, if the monotonicity does not hold, then at least one of the above constitutive relations does not exist as one-to-one correspondence. Then, according to the terminology from nonlinear electrical systems [5], the memristor with the unambiguous relation The laboratory experiments with memristive circuits can be performed either with memristors as existing solid-state devices or with their hardware emulators. The first case is rather rare since there are only a few possibilities of purchasing memristors [6], and the behavior and parameters of these devices may not correspond well to memristors with parameters required for specific experiment. That is why the popular method is to develop and build various kinds of memristor emulators.
Anyway, it can be useful to provide the QCM-to-FCM conversion if the available memristor is of the QCM type, and we request the FCM for a specific application. Similarly, analogous requests may appear for the FCM-to-QCM conversion. Such conversion circuits may be also interesting when the conversion is accompanied by modifying the shape of the original constitutive relation to the requested form. The simplest form of this shaping is the linear scaling.
The above conversion can be resolved by a combination of the concept of duality for memristors and higher-order elements [7] and the element transformation via gyrators [8].
The aim of this work is to present the circuit ideas of the FCM-to-QCM and QCM-to-FCM conversions. The duality and gyrator concepts are summarized in Section II. In Section III, the FCM-to-QCM and QCM-to-FCM converters are synthetized, with the conclusion that the latter is the same as the well-known OTA-based gyrator [9]. Suggested are implementations of these converters via off-the-shelf integrated circuits. Section IV demonstrates possible converter applications, computer simulations and laboratory experiments. Fig. 1 demonstrates the position of selected six fundamental circuit elements in Chua's table. Each element is defined by its constitutive relation between a pair of the variables (v (α) , i () ), where v (α) / i () is the time derivative or integral of element voltage / current of order α /  if α /  is positive or negative [10]. The quantities v (α) and i () are also referred to as generalized voltage and generalized current. Let us consider an element with the coordinates (α, ) defined by the unambiguous constitutive relation

II. ELEMENT-TO-ELEMENT TRANSFORMATION IN CHUA'S TABLE
and with the ambiguous constitutive relation Then such element is the (α, ) element controlled by the generalized voltage v (α) , or, in short, the voltage-controlled (α, ) element.
It is obvious that also the i () -controlled, or currentcontrolled (α, ) element with unambiguous constitutive relation (2) and ambiguous relation (1) can be defined accordingly. According to this notation, the charge-controlled memristors (QCM) and flux-controlled memristors (FCM) can also be denoted as the current-and voltage-controlled memristors.
It follows from the concept of electrical duality [11] that the dual element can be generated from the original element by interchanging the voltage and current. That is why the voltage-controlled (α, ) element with the constitutive relation (1) has its dual element in the form of currentcontrolled (, α) element with the constitutive relation It should be noted that every (α, ) element from Chua's table has its dual representation located in the table on the coordinates (, α), thus symmetrically to the position of the original element with respect to the diagonal with the parameter  -α = 0, drawn in Fig. 1 in red. Then, for example, the dual element to the inductor is the capacitor. Concerning the elements located on the above diagonal, for example, the resistor is changed to the conductor after interchanging the voltage and current. However, the nature of the elements on this diagonal is not changed due to duality transformation. As a particular result, the QCM and the FCM are dual elements. That is why the QCM-to-FCM and FCMto-QCM converters should provide the duality transformation. In general, such a transformation can be accomplished via the gyrator. Author Name: Preparation of Papers for IEEE Access (February 2017) VOLUME XX, 2017 chosen as in Fig. 2 (a), its classic equations [5] are rewritten as follows: 1 2 i Gv  ( 4 ) 2 1 i Gv  It is well-known that if the port 2 of the gyrator is connected to a current-controlled nonlinear resistor, then its port 1 behaves as a voltage-controlled resistor [5]. This observation can be generalized for the case of an arbitrary higher-order (α, α) element, i.e. an element located on the "red" diagonal in Chua's table in Fig. 1, connected to the port 2 as illustrated in Fig. 2 (b). Differentiating/integrating (4) and (5) α-times, α Z+ / Z− yields the gyrator equations for generalized voltages and currents It can be concluded from (6), (7) and from the corresponding Fig. 2 (b) that the classic gyrator can also be useful for transforming not only nonlinear controlled resistors but also an infinite number of other (α, α) elements, thus also memristors for α = -1. It will be shown in the following Section that the particular implementation of the gyrator should be different for the FCM-to-QCM and QCMto-FCM conversions. If the (α, α) element in Fig. 2 (b) is controlled by current, then it must be driven from the gyrator port 2 via a current source. It is in conformity with Eq. (7), which can be implemented as a transconductance amplifier with the transconductance G. In this particular case of converter, it can be therefore implemented via two OTAs (Operational Transconductance Amplifiers). However, if the (α, α) element being converted is controlled by voltage, then it requires voltage driving, and the gyrator equations should be rearranged in the form where R = 1/G. Such a transforming cell should be implemented via two current-to-voltage converters. It follows from the above analysis that the gyrator as a resistive two-port cannot be used for transforming between the voltage-controlled and current-controlled (α, ) elements for α  . Such transforming cells should contain blocks for time-domain integration or differentiation [12].

III. QCM-TO-FCM AND FCM-TO-QCM CONVERTERS
Consider a current-controlled nonlinear (α, α) element with the constitutive relation given on the right side of Fig. 3. The procedure of its conversion to a voltage-controlled (α, α) element with the left-side constitutive relation consists of the following steps: 1. Since the original element is controlled by current, it is driven by a current source that adjusts its proper operating point. The driving current is derived from the voltage of the transformed element through the proportionality factor G 2 . 2. The generalized voltage at the operating point of the original element, which follows from the unambiguous constitutive relation, is transferred into the generalized current of the transformed element with the proportionality factor G 1 .

Procedure of the synthesis of a circuit that transforms a current-controlled (α, α) element into a voltage-controlled (α, α) element. For α = -1, this circuit is the QCM-to-FCM converter.
It should be noted that the corresponding transforming cell in Fig. 3 can be considered a generalized gyrator described by Eqs. (6), (7), with two different gyration conductances G 1 , G 2 for ports 1 and 2, respectively. In the case of emulating a grounded memristor, it can be implemented by the wellknown circuit containing two OTAs in Fig. 4.

FIGURE 4. Implementation of the cell for QCM-to-FCM transformation via two gm stages.
A similar procedure, summarized in Fig. 5, can be applied to the synthesis of the FCM-to-QCM converter. The voltagecontrolled (α, α) element is driven by a voltage source controlled by a generalized current of the emulated element, with the proportionality factor R 2 . The current flowing through the original element is sensed and used for controlling the voltage across the port 1 with the proportionality factor R 2 . This way, the interchanging the voltage and current axes of the constitutive relations of both elements applies. It is obvious that the transforming cell is a generalized gyrator described by Eqs. (8) and (9) with two different gyration resistances R 1 and R 2 . An implementation of the FCM-to-QCM converter via a pair of positive and negative current conveyors is proposed in Fig. 6. This converter can be easily implemented by means of off-the-shelf integrated circuits. The CCII+ is available as AD844 and the CCII-can be made of two CCII+. The lowimpedance x terminal of the CCII+ drives the FCM while its (generalized) current is sensed and copied to the z terminal, which drives R 1 . The corresponding voltage drop on R 1 is copied to the x terminal of the CCII-for driving the port 1. The current flowing through this port is then mirrored to the z terminal of the CCII-, causing the voltage across R 2 . This voltage is then reflected directly to the port 2.   Before proceeding to laboratory measurements, the performance of designed circuits has been studied and verified via the SPICE simulations.

A. FCM AND QCM WITH NEGATIVE DIFFERENTIAL MEMRISTANCES
The memristors with non-monotonic constitutive relations can be easily implemented as popular and in the literature well described memristors with piece-wise-linear characteristics [3], [13]. Such locally active memristors can be useful for constructing nonlinear oscillators, artificial neurons, and networks working on the edge of chaos [14], [15], while passive memristors are useful for mimicking the synapses [16]. Emulators of the active memristors are used in this work for generating nonlinear oscillations via two benchmark circuits, denoted above as the applications App1 and App2. These memristive oscillators fall into the class of the oscillators introduced in [13] by Itoh and Chua.
The memristive emulators published so far can be classified as analogue and hybrid [17], [18], the latter containing the analogue memristive port, the analog-todigital interface, and a part of digital signal processing. Since the analogue emulators do not provide a flexible modification of the nonlinear characteristics of the emulated memristors, the hybrid approach was used. The memristors with piece-wise-linear constitutive relations shown in Fig. 9 were simulated as current and voltage sources controlled by time integrals of the memristor voltage and current, respectively. For lab experiments, the memristors were emulated by the hybrid emulator described in [19]. In general, the memristive port can be emulated via one of three modules, namely the digital potentiometer (Rmodule), the controlled voltage source (V-module), and the controlled current source (I-module). These modules are controlled by a program according to the specific forms of the port and state equations of the emulated memristor.
Since the emulated memristors exhibit negative differential memristance, the R-module should be avoided. Finally, the FCM was emulated by the I-module and the QCM by the V-module according to Figs. 9 (a) and (b), respectively.
In addition to the maximum achievable sampling frequency of 100 kHz, there are also some voltage and current limitations associated with the V and I modules that must be taken into account, namely abs(v) < 3 V and abs(i) < 3 mA [19].

B. BENCHMARK CIRCUITS -NONLINEAR MEMRISTIVE OSCILLATORS
The circuit in Fig. 10 (a) can be considered a parallel combination of linear capacitor and the FCM with nonlinear flux-to-charge constitutive relation in Fig. 9 (a), driven via a pulse voltage source in series with a linear inductor L. The corresponding dual circuit in Fig. 10 (b) forms the serial connection of the inductor and the QCM with nonlinear charge-to-flux constitutive relation in Fig. 9 (b), driven by a current pulse source in parallel with a linear capacitor.
The circuit in Fig. 10 (a) resembles the well-known oscillator employing the tunnel diode, thus a voltagecontrolled nonlinear resistor with negative differential resistance whose operating point is set into the region of negative resistance via a DC voltage source. The oscillator in Fig. 10 (a) uses the FCM instead of the tunnel diode. The operating point of the FCM is set via a pulse source v pul . In fact, the circuit in Fig. 10 (a) is obtained from this classic circuit via applying the so-called "shift-type" duality transformation, where the nonlinear resistors are transformed into memristors, the voltage sources are replaced by flux sources, while the linear elements are preserved [20]. Then the strength of the narrow pulse v pul should correspond to the voltage of the original DC source. The purpose of this pulse is to deliver the flux to the memristor, which moves its operating point into the region of its negative memristance.
The oscillations in the dual circuit in Fig. 10 (b) are initiated by a pulse from the current source i pul . The amount of the corresponding charge injected into the circuit moves the operating point of the QCM into the interval q L , q H of the flux-charge constitutive relation of the QCM in Fig. 9 (b).
The technical limits of the emulator have been taken into account when designing the oscillator parameters. Since the sampling frequency used by the digital part of the emulator is 100 kHz, the oscillation frequency should be designed below 1 kHz, thus eliminating the accuracy issue caused by a limited number of samples within one repeating period. A low oscillating frequency leads to large inductance with high ESR (Equivalent Series Resistance), the latter affecting the oscillating frequency and potentially damping the oscillation. The impedance level, i.e. the oscillator reactances at given frequency and the emulator memristances, should be also designed with regard to the current and voltage limits associated with the V-and I-modules of the emulator.
Note that the oscillator in Fig. 10 (b) was designed from the oscillator in Fig. 10 (a) as a dual circuit, applying the following voltage and current scaling that optimize the dynamic range of voltages and currents: where v, i / vʹ, iʹ are the voltages and currents of the original / dual circuit, and k i and k v are scaling coefficients. Then, according to the well-known duality rules, the resistor R is changed to conductor Gʹ = Rk i /k v , the inductor L to capacitor This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022.3186281 simulations in Figs. 11 (a), (b) for the designed parameters of ideal components confirm dual waveforms with an oscillation frequency of ca 665 Hz. The simulations also confirm the hypothesis that the oscillations can be accomplished without the triggering pulses via setting proper initial values of the memristor state. For the oscillators in Figs. 10 (a) and (b), the initial scaled flux and charge should be set to 1 Vs and 1 mAs, respectively. It reduces the above oscillating circuits to three elements in parallel and in series.
Figs. 11 (c), (d) demonstrate the oscillator sensitivity to parasitic losses in the circuit, namely to the serial resistance R L of the inductor. This effect is clearly visible for rather high resistance R L = 200  which is comparable to the memristances R m1 , R m2 , and R m3 , in decreasing levels and modified shapes of the generated waveforms. It is also evident that the operating point of the memristor is slightly moved from the center of the region of negative memristance, which causes an asymmetry in the v-i pinched hysteresis loops. It is worth noting that such parasitic effects also disturb the waveform duality.
Finally, the oscillators from Figs. 10 (a), (b) were constructed via the hybrid emulators of the FCM and QCM with the parameters given in Section B. The oscillations were provided by setting the initial values of q sini = 1 mAs and  sini = 1 Vs. Special attention was paid to the algorithm of numerical integration of the memristor voltage and current in order to compute their state variables. The simple Euler method amplifies undesired low-frequency fluctuations and, concurrently, it does not provide an exact phase shift of 90 degrees, which is necessary for a proper generation of v-i pinched hysteresis loops of memristors, The best results were achieved with a lossy bilinear integrator with a cutoff frequency of 10 Hz. Fig. 12 shows the waveforms and pinched hysteresis loops generated by specimens of the oscillators. The initial conditions were slightly modified in order to achieve symmetric hysteresis. The measurements are in good agreement with SPICE simulations.

C. QCM-TO-FCM AND FCM-TO-QCM CONVERTERS
The QCM-to-FCM converter in Fig. 4 is based on two OTAs. Since the commercial OTAs such as LM13700 provide low input resistance and also low range of input voltage for linear operation, this converter was finally implemented by three AD844 transimpedance amplifiers as shown in Fig. 13 (a). The amplifier No. 1 together with G 1 form a transconductance amplifier with the transconductance G 1 . The amplifier No. 2 serves as the current inverter for driving the port No. 1 whereas the amplifier No. 3, whose high-impedance input is driven also from the port No. 1, operates as OTA with the transconductance G 2 . Its highimpedance output drives the port No. 2. Since the voltage outputs of the amplifiers No. 1 and 3 provide buffered signals appearing on inverting inputs, they can be used for sensing the voltages across the ports No. 2 and 1, respectively.
The implementation of the FCM-to-QCM converter from Fig. 6 via three transimpedance amplifiers is shown in Fig.  13 (b). The schematics is similar to the QCM-to-FCM converter in Fig. 13 (a), but now, the amplifiers play the role of current conveyors according to Fig. 6. The amplifier No. 1 forms the conveyor CCII+ whereas the amplifiers 3 and 2 work as the CCII-.

D. MEMRISTIVE OSCILLATORS WITH QCM-TO-FCM AND FCM-TO-QCM CONVERTERS
In order to check the idea of the QCM-to-FCM and the FCM-to-QCM conversions via circuits from Figs. 4 and 6, SPICE simulations and laboratory experiments were accomplished according to Figs. 14 (a) and (b).
As expected, the parallel LC cell in Fig. 14 (a) provided the autonomous oscillations with the FCM but not with the QCM, and the serial LC cell in Fig. 14 (b) worked well with the QCM but not with the FCM. In other words, the circuits in Fig. 13 cannot oscillate after removing the converters from the networks. The parallel configuration can oscillate with the memristor of the N-type flux-to-charge characteristic, and the series configuration with the memristor of the N-type charge-to-flux characteristic. For experiments, the SPICE models and specimens of the conversion circuits from Fig. 13 were constructed. Since the converters form an interface between the original and dual circuitry, their transforming port equations should be in the form of Eqs. (10), (11). That is why the resistances 1/G 1 and 1/G 2 in the QCM-to-FCM converter and the resistances R 1 and R 2 in the FCM-to-QCM converter in Fig. 13 were designed as 1/G 1 = R 1 = k v = 178.6 , 1/G 2 = R 2 = 1/k i = 1k.
In addition to simple behavioral SPICE models based on ideal conversion equations between voltages and currents, the converters from Figs For this value, the simulated waveforms correspond well with the laboratory measurements given below in Fig. 16.
The mechanism of how R AD affects the oscillator behavior is different for the QCM-to-FCM and for the FCM-to-QCM converters.
Observing the QCM-to-FCM implementation in Fig.  13 (a), it is obvious that the input resistances R AD of the integrated circuits No. 1 and 3 act in series with R 1 = 1/G 1 and R 2 = 1/G 2 . Then the corresponding scaling coefficients k v and k i , which determine the impedance transformations, are changed. This results in modifying the impedance conditions in the circuit.
For the FCM-to-QCM converter in Fig. 13 (b), these coefficients are not affected, but the R AD now operates in series with both converter ports. As the negative consequence for the oscillator in Fig. 14 (b), R AD is added to the lossy resistor R L and also to the memristance of the FCM, thus increasing their values. Since the R AD is rather high, it affects the memristor voltage and current waveforms as shown in Figs. 15 (c), (d).
The v-i pinched hysteresis loop of the FCM in the FCMto-QCM oscillator exhibits a specific "sloping distortion". An analysis reveals that the cause consists in the voltage drop on the R AD which operates in series with the memristor. The experiments with the SPICE model also demonstrate a high sensitivity of the shapes of generated signals to the initial state of the memristor (the initial charge of the QCM and the initial flux of the FCM). For example, the waveforms of the QCM-to-FCM oscillator in Fig. 15  the hysteresis lobes were strongly asymmetric, because the mean value of the charge waveform was shifted down due to changing the impedance conditions in the circuit. The measurements confirmed the above error analysis. The waveforms from Fig. 16 were obtained after fine-tuning the initial conditions q ini and  ini .
Although it may be beyond the main idea of this article, it is worth mentioning that the negative effects of parasitic resistances on the oscillator behavior can be partially compensated by modifying the resistances acting in series, namely the resistors in transforming circuits and the memristor resistances R m1 , R m2 , and R m3 .

V. CONCLUSIONS
The results presented in this work can be summarized as follows: 1. The demand for the charge-controlled memristor when only its flux-controlled counterpart is available and vice versa can be settled by simple transforming two-ports.
2. If a flux-controlled memristor with unambiguous flux-tocharge characteristic and ambiguous charge-to-flux characteristic needs to be converted into a chargecontrolled memristor, then the transforming cell is a resistive two-port consisting of two current-controlled voltage sources. It can be implemented via commercial current conveyors.
3. If a charge-controlled memristor with unambiguous charge-to-flux characteristic and ambiguous flux-tocharge characteristic needs to be converted into a fluxcontrolled memristor, then the transforming cell is a resistive two-port consisting of two voltage-controlled current sources. It can be implemented via commercial operational transconductance amplifiers (OTAs).
4. The two-ports from Items 2 and 3 can also be used for mutual conversions between the voltage-controlled and current-controlled nonlinear (α, α) higher-order elements from Chua's table, whose special case for α = -1 is just the memristor.
5. The transforming cells from Items 2 and 3 provide the duality transformation. That is why the QCM-to-FCM converter can be generally used for the conversion from