Abstract:
The continuous-level memristor emulator is a circuit that uses off-the-shelf components to emulate a memristor. The circuit uses two current-feedback operational-amplifiers (CFOAs) and uses an operational transconductance amplifier (OTA)-based circuit in place of a diode resistive network to provide a continuous level of memristance instead of two binary states. The OTA is forced to work in its nonlinear region by the voltage V DC  applied to its positive input terminal. Thus, the transfer function of the OTA-based circuit will be a nonlinear function. Experimental testing shows that the continuous-level memristor emulator is operational as a memristor, and the emulator may be used, e.g., in place of a memristor in a multivibrator circuit.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to memristor emulators, and particularly to a continuous-level memristor emulator and its use in a multivibrator circuit. 
     2. Description of the Related Art 
     A memristor is a passive device that relates magnetic flux to current charge. Until 2008, the existence of the device was only theoretically postulated. In 2008, a team from Hewlett Packard claimed to have developed the device from a thin film of titanium dioxide. However, the device is not currently commercially available. There has been a great deal of interest in the device. Due to its unavailability, a great many circuits that emulate the properties of the device have been developed. The present inventors have developed memristor emulator circuits using current-feedback operational-amplifiers (CFOAs). However, these circuits have typically employed diode-resistive networks for implementing the required nonlinear resistances, and hence can provide only two values for the nonlinear resistances. Any type of binary memristor providing only two memresistance states is at a disadvantage. 
     Thus, a continuous-level memristor emulator solving the aforementioned problems is desired. 
     SUMMARY OF THE INVENTION 
     The continuous-level memristor emulator is a circuit that uses off-the-shelf components to emulate a memristor. The circuit uses two current-feedback operational-amplifiers (CFOAs) and uses an operational transconductance amplifier (OTA)-based circuit in place of a diode resistive network to provide a continuous level of memristance instead of two binary states. The OTA is forced to work in its nonlinear region by the voltage V DC  applied to its positive input terminal. Thus, the transfer function of the OTA-based circuit will be a nonlinear function. Experimental testing shows that the continuous-level memristor emulator is operational as a memristor, and the emulator may be used, e.g., in place of a memristor in a multivibrator circuit. 
     These and other features of the present invention will become readily apparent upon further review of the following specification and drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic diagram of a continuous-level memristor emulator according to the present invention. 
         FIG. 2A  is a schematic diagram of a memristor model that models input current, of a continuous-level memristor emulator according to the present invention, 
         FIG. 2B  is a schematic diagram of a memristor model that models output current, i R , of a continuous-level memristor emulator according to the present invention. 
         FIG. 3  is a plot showing the current (a) and the voltage (b) waveforms of the continuous-level memristor emulator according to the present invention. 
         FIG. 4  is a plot showing current-voltage characteristics of the continuous-level memristor emulator according to the present invention. 
         FIG. 5  is a schematic diagram of a multivibrator circuit using the continuous-level memristor emulator of  FIG. 1  in the feedback loop. 
         FIG. 6  is a schematic diagram showing a practical implementation of the AND gate of  FIG. 5 . 
         FIG. 7  is a plot showing typical voltage waveforms obtained from the multi-vibrator circuit of  FIG. 5  (at the arrow labeled “a”) and a voltage across the continuous-level memristor emulator in the circuit (at the arrow labelled “b”). 
         FIG. 8  is a schematic diagram of the multivibrator circuit of  FIG. 5 , but with a control voltage at the input to define a voltage-controlled multivibrator circuit. 
         FIG. 9  is a plot showing variation of the frequency of the output voltage of the multivibrator-VCO circuit of  FIG. 8 . 
         FIG. 10  is a plot showing variation of the duty cycle of the output voltage of the multivibrator-VCO circuit of  FIG. 8 . 
     
    
    
     Similar reference characters denote corresponding features consistently throughout the attached drawings. 
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     The continuous-level memristor emulator uses an operational transconductance amplifier (OTA)-based circuit connected to current feedback operational amplifiers (CFOAs), wherein the OTA is forced to work in its nonlinear region by the voltage V DC  applied to its positive input terminal. Thus, the transfer function of the OTA-based circuit will be a nonlinear function. The continuous level memristor emulator  100  of  FIG. 1  includes a first current feedback operational amplifier (CFOA 1 )  102   a , a second CFOA  102   b  (CFOA 2 ), an operational transconductance amplifier (OTA)  104  having a negative input, a positive input, and an output, the OTA negative input being connected to a w output terminal of the first CFOA  102   a , the OTA output being connected to the y input terminal of the second CFOA  102   b . A w terminal of second CFOA  102   b  is connected to the y terminal of the first CFOA  102   a . Resistor R 2  is connected between ground and a control input of OTA  104 . Resistor R 3  is connected between ground and the y terminal of the second CFOA  102   b . Resistor R 1  is connected between ground and the z terminal of the second CFOA  102   b . Capacitor C 1  is connected between ground and the z terminal of the first CFOA  102   a . Capacitor C 2  is connected between ground and the x terminal of the second CFOA  102   b . In the circuit of the present continuous level memristor emulator  100 , the input current i M  will be integrated by the capacitor C 1 . Thus, the voltage at the negative input of the OTA  104  will be given by: 
     
       
         
           
             
               
                 
                   
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     This voltage will be processed by the nonlinear scalar formed of the OTA-based circuit. Thus, the output current of the OTA  104  will be given by: 
                     i   R     =       F   ⁡     (     v   R     )       =         v   R       R   eq       .               (   2   )               
In equation (2) F is a nonlinear function representing the input-output relationship of the OTA-based circuit comprising the OTA  104 , resistors R 2  and R 3 , and the DC bias voltage V DC . In order for the function F to be nonlinear, it is necessary to force the OTA  104  to work in its nonlinear region. This can be achieved by applying a relatively large bias voltage V DC  at the positive input terminal of the OTA  104 . In equation (2) R eq  is the equivalent nonlinear resistance represented by the function F(v R ). The voltage at terminal y of the CFOA  102   b  will be given by:
 
v y =i R R 3 .  (3)
 
This voltage will be differentiated by the capacitor C 2  to produce a voltage v M  given by:
 
                     v   M     =       R   1     ⁢     R   3     ⁢     C   2     ⁢         ⅆ     i   R         ⅆ   t       .               (   4   )               
Equations (1) and (4) can be represented by models  200   a  and  200   b , as shown in  FIGS. 2A and 2B . This is equivalent to transferring a current-controlled resistor into a flux-controlled memristor. If the input current i M  is a sinusoidal current of the form i M =I m  sin ωt, then using equations (1), (2) and (4), it is easy to show that the equivalent resistance of the memristor will be given by:
 
                   M   =           C   2     ⁢     R   1     ⁢     R   3           C   1     ⁢     R   eq         .             (   5   )               
Inspection of equations (2) and (5) shows that the memristance can acquire multiple values, so long as the function F is a continuous nonlinear function, which is the case.
 
     The present continuous-level memristor emulator circuit  100  shown in  FIG. 1  was experimentally tested using an off-the-shelf LM3080AN OTA and AD844 CFOAs. The results obtained with C 1 =2.2 μF, R 2 =100 kΩ, R 3 =20 kΩ, V DC =11.7V, C 2 =2.2 μF, R 1 =10 kΩ, and DC supply voltages=±12V are shown in plots  300  and  400  of  FIGS. 3 and 4 , respectively. These results confirm the operation of the continuous-level memristor emulator circuit  100  with the classical bow-tie shown in plot  400  of  FIG. 4 . In order to block possible high frequency oscillations, a capacitance of 1 nF may be connected in parallel with R 1 . 
     The functionality of the present emulator circuit  100  was also tested by using it in a practical implementation of a multivibrator circuit  500 , as shown in  FIG. 5 . The multivibrator circuit  500  is a complete circuit, including multivibrator current-feed operational amplifier CFOA 2 , multivibrator current-feed operational amplifier CFOA 1 , comparator  1  (Comp  1 ), and comparator  2  (Comp 2 ) connected in a feedback circuit via AND gate  505  for oscillation. The continuous-level memristor emulator  100  is connected from ground to the z terminal of CFOA 1 . The proposed implementation uses AP358 comparators. More specifically, a resistor R m1  is connected to the x terminal of multivibrator amplifier CFOA 2 , and as shown in  FIG. 5 , resistor R m1  is connected from ground to the x terminal of multivibrator amplifier CFOA 2 . The z terminal of multivibrator amplifier CFOA 2  is connected to the x terminal of multivibrator amplifier CFOA 1 . The memristor emulator  100  is connected from ground to the z terminal of multivibrator amplifier CFOA  1 . The y terminal of multivibrator amplifier CFOA 1  is connected to ground. The w terminal of multivibrator amplifier CFOA 1  is connected to the inverting input of comparator Comp 1  and to the non-inverting input of comparator Comp 2 . The positive (non-inverting input) terminal of comparator Comp 1  has a positive reference voltage v p  applied. The negative terminal (inverting input) of comparator Comp 2  has a negative reference voltage v n  applied. The outputs of comparators Comp 1  and Comp 2  feed respective inputs of the AND gate  505 . The output of the AND gate  505  is connected to they terminal of multivibrator amplifier CFOA 2 . 
     The AND gate  505  of multivibrator circuit  500  is realized using two 2N7000 NMOS transistors, two VP2106 PMOS transistors, and a UA741CN operational amplifier configured as a comparator, as shown in  FIG. 6 . With R 1 =5.6kΩ, V p =2.5V, V n =−0.73V, R 1  (of circuit  100 )=60 kΩ and V=−V + =12V, the waveforms of the output voltage and the voltage across the memristor emulator are shown in plot  700  of  FIG. 7 . Inspection of plot  700  clearly shows that the circuit of  FIG. 5  is acting as a multivibrator circuit generating a rectangular waveform. The duty cycle of this rectangular waveform can be easily controlled by changing V p  and/or, and/or R m1 , and/or the nonlinear operating point of circuit  100  of  FIG. 1 . As shown in  FIG. 8 , a control voltage instead of ground can be connected to R m1  of voltage-controlled multivibrator circuit  500 . Plots  900  and  1000  of  FIGS. 9 and 10  show the variation of the duty cycle of the output rectangular waveform with the control voltage V C . Inspection of plot  900  shows that frequencies up to 2 kHz can be obtained and inspection of plot  1000  shows that it is possible to obtain 50% duty cycle that is a square wave output voltage when the control voltage is around 0.8 V. 
     It is to be understood that the present invention is not limited to the embodiments described above, but encompasses any and all embodiments within the scope of the following claims.