Abstract:
The floating memristor emulator is based on a circuit implementation that uses grounded capacitors and CFOAs in addition to combinations of diodes and resistors to provide the required nonlinearity and time constants. This circuit results in low power consumption, cost reduction and ease of implementation because it avoids the use of multipliers, ADCs and RDACs. The present circuit is used in an FM demodulator, which exploits the frequency-dependence of the memristance. Successful use in the FM demodulator confirmed the functionality of the present floating memristor emulator circuit.

Description:
BACKGROUND OF THE INVENTION 
       [0001]    1. Field of the Invention 
         [0002]    The present invention relates to memristors, and particularly to a floating memristor emulator that can be used in frequency-to-voltage conversion. 
         [0003]    2. Description of the Related Art 
         [0004]    Since its inception, several emulators have been presented for the grounded memristor. However, only few floating memristor emulators are available in the literature. Those few designs have numerous components, which present size and power consumption problems. 
         [0005]    Thus, a floating memristor emulator solving the aforementioned problems is desired. 
       SUMMARY OF THE INVENTION 
       [0006]    The floating memristor emulator is based on a circuit implementation that uses grounded capacitors and CFOAs in addition to combinations of diodes and resistors to provide the required nonlinearity and time constants. This circuit results in low power consumption, cost reduction and ease of implementation because it avoids the use of multipliers, ADCs and RDACs. The present circuit may be used in an FM demodulator, which exploits the frequency-dependence of the memristance. Successful use in the FM demodulator confirmed the functionality of the present floating memristor emulator circuit. 
         [0007]    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 
         [0008]      FIG. 1  is a schematic diagram of a floating memristor emulator according to the present invention. 
           [0009]      FIG. 2A  is a schematic diagram showing an I M  model in terms of input current for a floating memristor emulator according to the present invention. 
           [0010]      FIG. 2B  is a schematic diagram showing an I R  model in terms of emulator current for a floating memristor emulator according to the present invention. 
           [0011]      FIG. 3A  is a plot showing current and voltage waveform characteristics of the floating memristor emulator according to the present invention. 
           [0012]      FIG. 3B  is a plot showing current-voltage characteristics with a wide difference in the resistance values of the floating memristor emulator according to the present invention. 
           [0013]      FIG. 4A  is a plot showing current and voltage waveform characteristics of the floating memristor emulator according to the present invention at 2.9 kHz. 
           [0014]      FIG. 4B  is a plot showing current-voltage characteristics with narrow difference in resistance values at 2.9 kHz of the floating memristor emulator according to the present invention. 
           [0015]      FIG. 5  is a plot showing behavior at 6.0 kHz of the floating memristor emulator according to the present invention. 
           [0016]      FIG. 6  is a schematic diagram showing a test circuit for the floating memristor emulator according to the present invention. 
           [0017]      FIG. 7  is a schematic diagram showing a FM demodulator using the floating memristor emulator according to the present invention. 
           [0018]      FIG. 8  is a plot showing FM and the converted AM of the FM demodulator using the floating memristor emulator according to the present invention. 
           [0019]      FIG. 9  is a plot showing the converted AM signal and the output demodulated signal of the FM demodulator using the floating memristor emulator according to the present invention. 
           [0020]      FIG. 10  is a plot showing the input FM signal and the output modulating signal at the output of the low pass filter connected to the FM demodulator using the floating memristor emulator according to the present invention. 
       
    
    
       [0021]    Similar reference characters denote corresponding features consistently throughout the attached drawings. 
       DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
       [0022]    The present floating memristor emulator circuit includes four current feedback operational amplifiers (CFOA&#39;s  102   a ,  102   b ,  102   c , and  102   d ), configured as shown in  FIG. 1 . The first  102   a , second  102   c , third  102   b , and fourth  102   d  current feedback operational amplifiers (CFOAs), each have y, x, z, and w terminals. The y terminal of first CFOA 1   102   a  is connected two the z terminal of the second CFOA 2   102   c . The y terminal of the third CFOA 3   102   b  is connected to the z terminal of the fourth CFOA 4   102   d . A differential voltage input, v inp , v inn  is formed from the y terminals of the first and third CFOAs ( 102   a ,  102   b ). The x terminals of CFOA 1   102   a  and CFOA 3   102   b  are in operable communication with each other. For example, a potentiometer R 1  may be connected between the x terminals of CFOA 1   102   a  and CFOA 3   102   b  (the wiper portion being connected to CFOA 3   102   b ). Grounded capacitors C 1  through C 4  are connected to their respective CFOAs ( 102   a ,  102   c ,  102   b , and  102   d ). A parallel combination (R 3  and D 1 ) has a cathode portion of D) connected to the w terminal of COFA 1   102   a . The R 3 , D 1  combination is connected in series with the upper part of the potentiometer R 5  which is connected to the y terminal of CFOA 4   102   d . The wiper portion of potentiometer R 5  is connected to ground. A parallel combination (R 2  and D 2 ) has an anode portion of D 2  connected to the w terminal of COFA 3   102   b . The R 2 , D 2  combination is connected in series with the lower part of the potentiometer R 5 , which is connected to they terminal of CFOA 2   102   c . The input voltage produces a current through the resistance R 1  given by: 
         [0000]        i   R     1   =( v   inp   −v   inn   /R   1 .  (1)
 
         [0023]    This current will flow outward from terminal x of CFOA 1  ( 102   a ) and inward into terminal x of CFOA 3   102   b . This current will be induced in terminal z of CFOA 1  ( 102   a ), where it will be integrated by the capacitor C 1  to produce a voltage given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     v 
                     
                       R 
                       p 
                     
                   
                   = 
                   
                     
                       1 
                       
                         C 
                         1 
                       
                     
                      
                     
                       ∫ 
                       
                         
                           
                             
                               v 
                               inp 
                             
                             - 
                             
                               v 
                               inn 
                             
                           
                           
                             R 
                             1 
                           
                         
                          
                         
                           
                              
                             t 
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
         [0024]    This voltage will be induced on terminal w of CFOA 1  ( 102   a ) and will produce an outward current from terminal w of CFOA 1  ( 102   a ), i Rp  through the parallel combination of R 3  and D 1  in series with the upper part of the potentiometer R 5 . This current can be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     i 
                     
                       R 
                       p 
                     
                   
                   = 
                   
                     
                       
                         v 
                         
                           R 
                           p 
                         
                       
                       
                         
                           R 
                           
                             5 
                              
                             upper 
                           
                         
                         + 
                         
                           R 
                           
                             eq 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0025]    In equation (3), R 5upper  is the resistance of the upper part of the potentiometer R 5  and R eq1  is a nonlinear resistance that depends on the status of the diode D 1 . The voltage at terminal y of the CFOA  4  ( 102   d ) will depend on the status of the diode D 1 . This voltage can be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     v 
                     1 
                   
                   = 
                   
                     
                       
                         
                           v 
                           
                             R 
                             p 
                           
                         
                          
                         
                             
                         
                          
                         
                           R 
                           
                             5 
                              
                             upper 
                           
                         
                       
                       
                         
                           R 
                           
                             5 
                              
                             upper 
                           
                         
                         + 
                         
                           R 
                           
                             eq 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
         [0026]    The voltage v 1  will be induced on terminal x of the CFOA 4  ( 102   d ) and will be differentiated by the capacitor C 4 . Thus, the outward current in the lower input terminal will be given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     i 
                     inn 
                   
                   = 
                   
                     
                       C 
                       4 
                     
                      
                     
                       
                         
                            
                           
                             v 
                             1 
                           
                         
                         
                            
                           t 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
         [0027]    In a similar way the current i R     1    will be induced in the terminal z of CFOA 3  ( 102   b ) and will be integrated by the capacitor C 3  to produce a voltage given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     v 
                     
                       R 
                       n 
                     
                   
                   = 
                   
                     
                       
                         - 
                         1 
                       
                       
                         C 
                         3 
                       
                     
                      
                     
                       ∫ 
                       
                         
                           
                             
                               v 
                               inp 
                             
                             - 
                             
                               v 
                               inn 
                             
                           
                           
                             R 
                             1 
                           
                         
                          
                         
                           
                              
                             t 
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0028]    In equations (2) and (6), the voltage v M =v inp −v inn  is the differential input voltage. The voltage v Rn  will be induced on terminal w of CFOA 3  ( 102   b ) and will produce an inward current i Rn  through the parallel combination of R 2  and D 2  in series with the lower part of the potentiometer R 5 . This current can be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     i 
                     
                       R 
                       n 
                     
                   
                   = 
                   
                     
                       
                         - 
                         
                           v 
                           
                             R 
                             n 
                           
                         
                       
                       
                         
                           R 
                           
                             5 
                              
                             lower 
                           
                         
                         + 
                         
                           R 
                           
                             eq 
                              
                             
                                 
                             
                              
                             2 
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0029]    In equation (6) R 5lower  is the resistance of the lower part of the potentiometer R 5  and R eq2  is a nonlinear resistance that depends on the status of the diode D 2 . The voltage at terminal y of CFOA 2  ( 102   c ) can be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     v 
                     2 
                   
                   = 
                   
                     
                       
                         
                           v 
                           
                             R 
                             n 
                           
                         
                          
                         
                             
                         
                          
                         
                           R 
                           
                             5 
                              
                             lower 
                           
                         
                       
                       
                         
                           R 
                           
                             5 
                              
                             lower 
                           
                         
                         + 
                         
                           R 
                           
                             eq 
                              
                             
                                 
                             
                              
                             2 
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0030]    In equation (8), R 5lower  is the resistance of the lower part of the potentiometer R 5  and R eq2  is a nonlinear resistance that depends on the status of the diode. This voltage will be induced on terminal x of CFOA 2  ( 102   c ) and will be differentiated by the capacitor C 2 . Thus, the inward current in the upper input terminal will be given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     i 
                     inp 
                   
                   = 
                   
                     
                       - 
                       
                         C 
                         2 
                       
                     
                      
                     
                       
                         
                            
                           
                             v 
                             2 
                           
                         
                         
                            
                           t 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0031]    Assuming that the diodes D 1  and D 2  are identical, C 1 =C 3 =C i , C 2 =C 4 =C d , R 2 =R 3 , and the potentiometer R 5  is midway with 
         [0000]    
       
         
           
             
               
                 R 
                 
                   5 
                    
                   upper 
                 
               
               = 
               
                 
                   R 
                   
                     5 
                      
                     lower 
                   
                 
                 = 
                 
                   
                     1 
                     2 
                   
                    
                   
                     R 
                     5 
                   
                 
               
             
             , 
           
         
       
     
         [0000]    then R eq1 =R eq2 =R eq , 
         [0000]    
       
         
           
             
               
                 v 
                 
                   R 
                   n 
                 
               
               = 
               
                 
                   - 
                   
                     v 
                     
                       R 
                       p 
                     
                   
                 
                 = 
                 
                   
                     - 
                     
                       1 
                       2 
                     
                   
                    
                   
                     v 
                     R 
                   
                 
               
             
             , 
           
         
       
     
         [0000]    i Rn =i Rp =i R  and v 2 =−v 1 . Combining equations (1) and (6), the voltage v R =v Rp −v Rn  can be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     v 
                     R 
                   
                   = 
                   
                     
                       
                         v 
                         
                           R 
                           p 
                         
                       
                       - 
                       
                         v 
                         
                           R 
                           n 
                         
                       
                     
                     = 
                     
                       
                         2 
                         
                           
                             C 
                             i 
                           
                            
                           
                             R 
                             1 
                           
                         
                       
                        
                       
                         ∫ 
                         
                           
                             v 
                             m 
                           
                            
                           
                             
                                
                               t 
                             
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
         [0032]    Using equations (2), (3), (6) and (7) the current i R =i Rp =i Rn  can be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     i 
                     R 
                   
                   = 
                   
                     
                       1 
                       
                         k 
                         1 
                       
                     
                      
                     
                       ∫ 
                       
                         
                           v 
                           m 
                         
                          
                         
                           
                              
                             t 
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0000]    In equation (11) the parameter k 1  is given by, 
         [0000]    
       
         
           
             
               
                 
                   
                     k 
                     1 
                   
                   = 
                   
                     
                       
                         
                           ( 
                           
                             
                               R 
                               5 
                             
                             + 
                             
                               2 
                                
                               
                                 R 
                                 eq 
                               
                             
                           
                           ) 
                         
                          
                         
                           C 
                           i 
                         
                          
                         
                           R 
                           1 
                         
                       
                       2 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Also combining equations (5) and (9) the input current can be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     i 
                     M 
                   
                   = 
                   
                     
                       i 
                       inp 
                     
                     = 
                     
                       
                         i 
                         inn 
                       
                       = 
                       
                         
                           k 
                           2 
                         
                          
                         
                           
                             
                                
                               
                                 v 
                                 R 
                               
                             
                             
                                
                               t 
                             
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
         [0000]    In equation (13) the parameter k 2  is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     k 
                     2 
                   
                   = 
                   
                     
                       
                         
                           C 
                           d 
                         
                          
                         
                           R 
                           5 
                         
                       
                       
                         
                           R 
                           5 
                         
                         + 
                         
                           2 
                            
                           
                             R 
                             eq 
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
         [0033]    Equations (11) and (13) can be represented by models  200   a  and  200   b  of  FIGS. 2A and 2B , respectively. Models  200   a  and  200   b  correspond to a voltage-controlled memristor, where the voltage exciting the memristor v M  is integrated in the form of a current i R . This current is converted via a nonlinear resistor to voltage v R , and the voltage is transformed by differentiation to the memristor current i M . As stated supra, the present memristor emulator circuit uses four CFOAs. They are of type AD844. Simple Germanium (Ge) diodes in the circuit provide the necessary nonlinear function. Four equal-valued, grounded capacitors (47 nF) complete the z and x terminal connections for memristor circuit  100 . Two equal-valued resistors (3 kΩ) complete the w and y terminal connections for memristor circuit  100 . The variability of the resistor connections, wherein the equal-valued 3 kΩ resistors are interconnected by a 1 kΩ potentiometer, allows for compensation for any mismatch between the capacitors (C 1 , C 2 , C 3 , and C 4 ). 
         [0034]    Experimental results of the floating memristor emulator circuit  100  are shown in plots  300   a ,  300   b ,  400   a ,  400   b , and  500  of  FIGS. 3A, 3B, 4A, 4B, and 5 , respectively. Inspection of the plots clearly shows the frequency dependence of the memristance. As the frequency increases, the memristor emulator tends to behave as a normal resistor. 
         [0035]    The functionality of the present floating memristor emulator circuit  100  of  FIG. 1  was tested by using it in FM-to-AM conversion. The FM-AM conversion circuit  600  shown in  FIG. 6  is a simple frequency dependent, variable-gain inverting amplifier exploiting to advantage the frequency dependence of the memristance to form an FM discriminator circuit that is used in the first stage of the FM demodulator  700  shown in  FIG. 7 . The FM-AM conversion circuit  600  was tested using an FM signal formed of a carrier of frequency=2 kHz, a modulating frequency=100 Hz and frequency deviation=900 Hz. As shown in  FIG. 7 , the output of FM-to-AM converter (discriminator)  600  circuit of  FIG. 6  was applied to an envelope detector of the FM demodulator  700 , which fully demodulates an FM signal input to the FM discriminator. A low pass filter follows the envelope detector. The first stage of the FM demodulator uses the floating memristor emulator  100  connected to the negative input of an operational amplifier OA 1  with resistive negative feedback (R 1 ). The positive input of OA 1  is connected to ground. Operational amplifier OA 1 &#39;s output feeds a second stage (envelope detector) of the FM demodulator. The results obtained are shown in plots  800 - 1000  of  FIGS. 8-10 , respectively. Inspection of plots  800  through  1000  clearly shows that the present FM-to-AM converter works as expected and exploits to advantage the frequency dependence of the floating memristor emulator  100  of  FIG. 1 . 
         [0036]    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.