This paper implements two kinds of memristor-based colpitts oscillators, namely, the circuit where the memristor is added into the feedback network of the oscillator in parallel and series, respectively. First, a MULTISIM simulation circuit for the memristive colpitts oscillator is built, where an emulator constructed by some off-the-shelf components is utilized to replace the memristor. Then the physical system is implemented in terms of the MULTISIM simulation circuit. Circuit simulation and experimental study show that this memristive colpitts oscillator can exhibit periodic, quasi-periodic, and chaotic behaviors with certain parameter’s variances. Besides, in a sense, the circuit is robust with circuit parameters and device types.

Much attention has been attracted to the investigation of memristor after a long period of expectation and stagnation.^{1–4} However, memristor is not commercially available so far. Thus various techniques and circuits have been reported by utilizing the off-the-shelf components to emulate a real memristor’s characteristics, such as the technique by controlling the cubic polynomial function implemented by two multipliers,^{5–7} technique based on complementary MOS field effect transistors and one multiplier,^{8} based on diode bridge circuit,^{9,10} the current feedback operational amplifier (CFOA),^{11} the light dependent resistor,^{12} and so on. On the other hand, colpitts oscillator, for its simple circuit structure and potential applications in secure communication, has been studied by many researchers for decades. It is worthy of mentioning M. P. Kennedy and his group.^{13–15} Their works involve multi-aspects of the colpitts oscillator, including configuration, equilibrium points, attractors, bifurcation analysis and Lyapunov exponents (LEs), and applications to design, etc. For example, in one of their literatures,^{13} they proposed a generic chaotic colpitts oscillator model, where it was composed of a NPN bipolar junction transistor as a gain amplifier, one bias resistor *R*_{EE} used to construct the bias circuit, and a feedback network with one resistor *R*_{L}, one inductor *L*, and two capacitors *C*_{1}, *C*_{2}. They specifically noticed the effect of resistor *R*_{L} on this classical colpitts circuit: the state equations of the circuit model indicated that the oscillation would not encircle the operating point if the *R*_{L} is zero.^{13} And in the generic colpitts circuits, the *R*_{L} is usually realized by a constant resistor. Meanwhile, memristor, as the fourth basic element, has many new characteristics which are different from those of constant resistors, such as the properties of nonlinearity, time variance, memory, pinched hysteresis looped *v*-*i* characteristic and sensitive dependence on its initial polarity and position.^{3,8,16,17} Therefore, in this paper we utilize the memristor which has variable memristance with the variable current-voltage slope to add to the generic colpitts oscillator and aim to study its dynamic property. In fact, we have carried out some work on the memristive colpitts oscillator.^{16} and in Ref. 16, we constructed the dynamic model based on the basic characteristics and investigated the phase attractor, bifurcation and chaos of this oscillator mainly in a mathematical sense and by numerical simulations. As a further and follow-up study, in this paper we use MULTISIM circuit simulation software and ordinary hardware simultaneously to implement the memristive colpitts oscillator. In this circuit, the memristor is substituted by a small system called memristor emulator, which is built from the off-the-shelf devices and shows the characteristics of hysteresis loop of a real memristor. As an example of combing a memristor emulator with a colpitts oscillator, Ref. 9 proposed a memristor-based fourth-order shinriki circuit, where the memristor is realized by a diode bridge circuit following with a RC filter. And it analyzed the dynamic property of the memristive chaotic circuit. On the other hand, Ref. 8 proposed another memristor emulator circuit, which is a small system consisting of some off-the-shelf components: such as multiplier, operational amplifiers, some MOS field effect transistors and some common resistors, capacitors. And it was verified that this emulator can be a promising alternative candidate of memristor. So in this paper, the properties and dynamical behaviors of the colpitts oscillating circuit are investigated by introducing the memristor emulator circuit based on Ref. 8 into the feedback network of the colpitts oscillator in two ways, namely, in parallel and series connections, and through both MULTISIM simulation and physical implementation, respectively. In contrast to the normalized dimensionless numerical simulation, the MULTISIM circuit simulation can reflect the effect of the elements’ parameters on the characteristics of the circuit more intuitively. Therefore, this paper is focused on the following things: at first, the memristor’s pinched hysteresis loop and the periodic, chaotic behaviors of the memristive colpitts oscillators are verified via the MULTISIM simulation. Secondly, the oscillating characteristics are analyzed after the memristor is connected in parallel and series with the colpitts circuit’s feedback network, respectively. Ultimately, the memristor-based colpitts circuits are designed by physical experiment and the test properties with the real oscilloscope are demonstrated to be in agreement with the circuit simulation results.

Figs. 1(a) and (b) show the model of the colpitts oscillator with the memristor added in parallel and series, respectively. And Fig. 2 describes the schematic diagram of the colpitts oscillator with parallel connected memristor in detail, where the memristor is replaced by the emulator circuit based on Ref. 8.

where $RON$ and $ROFF$ are the OPEN-resistance and OFF-resistance, respectively. *W* denotes the dopant region of the HP (Hewlett-Packard) type of memristor.^{3} *D* denotes the depth of the sandwiched area for the memristor.

In this paper, the memristor is chosen to be the decremental configuration based on Ref. 8 (see Fig. 2). So its memristance characteristics are described as follows:^{8}

where *M* denotes the equivalent memristance, *R*_{T} and *C* are the resistor and capacitor which are connected to the two inputs of the multiplier AD633AN as shown in Fig. 2, respectively. And $qC(t)$ is the charge on the capacitor *C*.

where $iin$ is the current flowing through the memristor, and it is outputted by the current mirrors composed of four pairs of MOSFETs.

Figs. 3(a) and (b) show the oscillating waveform of the general colpitts oscillator in time domain and phase trajectory by MULTISIM simulation. Next, by adding the memristor emulator in parallel into the colpitts circuit based on the schematic diagram as shown in Fig. 2, the oscillating waveform and the phase trajectory are shown in Figs. 4(a) and (b), respectively. And the current-voltage characteristics of the memristor in the independent condition and in parallel condition are shown in Figs. 5(a) and (b), respectively. And they are shown that the memristance changes with the variance of the slope of the voltage-current, where the current (the vertical axis in Figs. 5(a) and (b)), which has a proportional relationship with the voltage on *R*_{T}(*R*_{T} =4kΩ), is represented by this voltage. And the input signal *v*_{i}=0.03Vrms (*f*=500Hz) in Fig. 5(a). In all the following MULTISIM simulations, the model parameters are: *L*=0.22mH, *C*_{1}=*C*_{2}=100nF, *R*_{L}=35Ω, *R*_{EE}=390Ω, 2N2222 is chosen as the gain amplifier. *R*_{S} =10kΩ, NMOS, PMOS FET transistors are 2N7000, BST110, respectively.

In Fig. 2, by combining the memristor emulator with the feedback network of the colpitts circuit in parallel, it allows the memristor to be grounded, and has less effect on the structure of the colpitts oscillator. Next, we discuss the situation of the memristor connected in series with the resistor *R*_{L}, inductor *L*.

In the series connection as shown in Fig. 1(b), the memristor usually needs a floating operation. But in fact, it does not start up with the floating ground based on Ref. 8. In the test of physical circuit, it shows the average current in the series connection is only 0.025∼0.15mA, but in the independent colpitts circuit, the average current is about 12∼14mA. So the current in the direct cascading is not enough to start up the oscillator. Then the floating operation is replaced by the series connection of *R*_{L}, *M* and *L* through a virtual ground or a reference voltage. And the oscillating waveform, the phase trajectory, and the current-voltage characteristics of the memristor are shown in Figs. 6 and 7, respectively.

After the investigation through MULTISIM circuit simulation, the physical circuit is realized to further verify the property of the memristive colpitts oscillator. In the aforementioned simulations, the memristor is replaced by the emulator circuit. Therefore, we also use the ready-made components (like amplifiers, multiplier, resistors, capacitors and MOS transistors, etc.) to replace the memristor in the physical circuit due to its commercial unavailability. So the whole circuit is mainly built by the components at hand, which are little different from the ones in the MULTISIM simulation. For example, three TL082CP chips, one AD633JN are taken as the operational amplifiers, multiplier, respectively. And six 2N7000, six VP0300LA are the NMOS and PMOS transistors, respectively. The oscillating transistor is chosen to be 2N2219A, etc. The circuit is realized as shown in Fig. 8 for series situation (it only needs a small change for the parallel connection). The waveforms are recorded by Tektronix digital oscilloscope TDS1002B. The oscilloscope images show the chaotic behavior by the observed time domain waveform and the phase trajectory (the lissajous figure) in XY mode, as shown in Fig. 9. It is in accordance with the simulation result. In addition, we can get more dynamical behaviors such as the limit cycle, the multi-periods and so on (see Fig. 10). Also, the test results for the series connection are shown in Figs. 11 and 12, respectively.

Finally, the numerical analysis is done for the illustration of the above-mentioned simulation and experiment results. In this paper the drift model considering nonlinearity Ref. 3 is applied to the memristor as depicted in the first expression of Eq. (1). And the state equations of the whole system (for series connection) are shown in the expressions of Eq. (4).

where the state variables $x,uC1,uC2$ represent the dopant region of the memristor, the voltages on capacitances $C1$, $C2$, respectively. And $iL$ is the current flowing through the inductance *L*. The other parameters $D,RON,ROFF$ denote the thickness of sandwiched area, OPEN-resistance and OFF-resistance in the memristor, respectively. $VCC$ and $VEE$ represent the positive and negative power supply voltages, respectively. Obviously, it satisfies the voltage relationships as shown in Eq. (4), where $ube,uce$ denote the voltages on BE and CE junctions of BJT, respectively. And the sgn(.) function and the current *i*_{e} of the gain amplifying transistor in equation (4) satisfy the following conditions as shown in Equations (5) and (6), respectively.

where *V*_{T}*$\u2248$* 0.026V in the common room temperature, and *I*_{ss} is the reverse saturation current.

In Equation (4), the collector current of the NPN transistor *i*_{c} is as follows:

In the basis of state equation (4), the LET toolbox^{18} is used to solve the Lyapunov exponents: the four LE exponents are 0.5724, -25669.0421, -1.2978e+13, -1.4707e+18, respectively. Assume that the initial values of the four state variables are all 1s. It verified the chaotic behavior for the model parameters simulation as shown in Table I.

Parameter
. | Signification
. | Value
. |
---|---|---|

$VCC$ | Positive power supply voltage | 5V |

$VEE$ | Negative power supply voltage | -5V |

$\mu v$ | dopant mobility coefficient | 10^{−8} |

$D$ | thickness of sandwiched area in the memristor | 10nm |

$RON$ | OPEN resistance of memristor | 1kΩ |

$ROFF$ | OFF resistance of memristor | 150kΩ |

L | inductance | 0.22mH |

$C1$ | capacitance | 100nF |

$C2$ | capacitance | 100nF |

$REE$ | resistance | 390Ω |

Parameter
. | Signification
. | Value
. |
---|---|---|

$VCC$ | Positive power supply voltage | 5V |

$VEE$ | Negative power supply voltage | -5V |

$\mu v$ | dopant mobility coefficient | 10^{−8} |

$D$ | thickness of sandwiched area in the memristor | 10nm |

$RON$ | OPEN resistance of memristor | 1kΩ |

$ROFF$ | OFF resistance of memristor | 150kΩ |

L | inductance | 0.22mH |

$C1$ | capacitance | 100nF |

$C2$ | capacitance | 100nF |

$REE$ | resistance | 390Ω |

In comparison with the traditional colpitts oscillator^{13–15} (for the length of the paper, the results in Refs. 13–15 are not shown here), there are several different aspects in this paper:

The memristance in this oscillatory circuit changes dynamically, while the resistance *R*_{L} in conventional colpitts oscillator is constant. So it is more flexible and may contain more complex changes of dynamic properties in the memristor based colpitts oscillator.

In summary, a memristor-based colpitts oscillatory circuit with BJT is designed in this paper, where the memristor is applied to the classical colpitts circuit in parallel and in series with the original coupling resistor *R*_{L}, respectively. Circuit simulation by MULTISIM software exhibit the nonlinear dynamic properties of the memristor-based circuit including the variable impedance characteristics of the memristor, phase trajectory, time evolution property of oscillatory waveform. Finally, and most importantly, the physical implementation and experimental test verify the simulation results, such as limit cycle, quasi-periodic and chaotic behaviors of the memristive system with proper choice of circuit parameters.

The authors wish to thank all of the editors and anonymous referees for their valuable comments and recommendations. This work was supported by the National Natural Science Foundation under Grant 61174025.