Research Article  Open Access
A Novel Generalized Memristor Based on ThreePhase Diode Bridge Rectifier
Abstract
Memristive characteristics in threephase diode bridge rectifier circuit are proposed in this paper. The conduction of the diodes is discussed and the characteristics of the pinched hysteresis loop are analyzed by both numerical simulations and circuit simulations. The hysteresis loops of each phase not only are pinched at the origin but also have the other two intersection points in the first quadrant and the third quadrant when threephase bridge rectifier circuit is running under normal operation. Other conditions are also discussed when a variety of faults conditions occur. The simulation results verify that the threephase bridge rectifier circuit can be described as a generalized memristor element during several operation states.
1. Introduction
It has been proved that memristive characteristics exist in many systems [1, 2]. Since the discovery of the memristor, it has been widely used in neuromorphic circuits [3], nonvolatile information storage [4], chaotic systems and oscillators [5, 6], and other applications. In recent years, the research of generalized memristor has aroused wide interest of the scholars. A class of diode bridge circuits has been analyzed because of the memristive characteristics [7–14]. A simple electronic circuit which only consists of four diodes and a secondorder RLC filter was proved to have memory properties [7]. By replacing the secondorder RLC filter with a firstorder parallel RC filter, another equivalent realization circuit of a generalized memristor was implemented, which decreases a fundamental circuit elementinductor [8, 9]. Meanwhile, a diode bridge circuit with a series first order RL filter also satisfies the definition of an ideal memristor; as a result, it can be described as a generalized memristor as well [10]. After that, a modified diode bridge circuit cascaded with a secondorder filter containing an inductor and a capacitor was proposed and was proved to constitute a generalized memristor [11]. Recently, an improved memristive diode bridge circuit was put forward comprising four diodes and an inductor, which has a much simpler circuit realization [12, 13]. On the basis of the fractional calculus theory, a fractionalorder capacitorbased diode bridge circuit was proposed, in which the fractional order capacitor is circuit implemented utilizing Oustaloup approximation technique [14].
Since a singlephase diode bridge rectifier circuit can express the memristive features, does a threephase diode bridge rectifier circuit have the same characteristics? In this paper, a diode bridge circuit which contains six diodes and a parallel first order RC filter is proposed, and the input voltage of such circuit is threephase voltage which has the same amplitude and frequency; phase difference of each phase is 120 degrees. In order to study whether a threephase diode bridge rectifier circuit can be regarded as a generalized memristor under several operating conditions, in Section 2, we theoretically analyze the conduction conditions of six diodes in threephase diode bridge rectifier circuit under different operating conditions. And the relation curves between input voltages and input currents are given by numerical simulations and circuit simulations. Conclusions are given in Section 3.
2. ThreePhase Bridge Rectifier Circuit Running under Several Conditions
Equipment can be described as a voltagecontrolled ideal memristor when a periodic voltage is applied to two terminals of the equipment, and the current response is periodic and has the same frequency [15]. Besides, the locus in the vi plane invariably passes through the origin; the current flowing through the device is always zero when the input voltage is zero, as shown in Figure 1(a). The other two essential characteristics of a memristor are the facts that the areas of the hysteresis loops decrease monotonically as the frequency increases when frequency is greater than a critical value and the pinched hysteresis loop will shrink to a nonlinear singlevalued function when the frequency tends to infinite [16].
(a)
(b)
2.1. Running under Normal Condition
Consider six diodes and a parallel RC filter constituted a threephase diode bridge rectifier circuit, which is depicted in Figure 2. When the circuit is running under normal condition, the input voltages are given as , , and , and suppose the conduction voltage of the diode is zero; the conduction situations of six diodes in one cycle are shown in Table 1.

It can be seen that only two diodes are conductive at the same time, which is similar to singlephase diode bridge circuit proposed in [8]. As a result, the input current of a phase can be derived from the mathematical model of the singlephase diode bridge circuit, which is written aswhere ; , , and indicate the reverse saturation current, emission coefficient, and thermal voltage of the diode, respectively. expresses the input voltage of circuit under different circumstances, which can be shown asAnd represents the voltage across capacitor C, which can be calculated byThus, the mathematical model of threephase diode bridge circuit is established, and numerical simulation can be carried out in MATLAB. Circuit parameters used for numerical simulation are given in Table 2. Figure 3 represents the input current of a phase with time , and the relation curve between input voltage and input current is shown in Figure 4. It can be seen that the locus in plane is hysteresis loop and is pinched at the origin. Besides, the trajectory is intersecting in the first quadrant and the third quadrant.

Then we consider using PSpice to carry out circuit simulations in order to verify whether one phase can be described as a generalized memristor. Diodes in the threephase rectifier bridge are chosen as 1N4148 and parameters of the resistor and the capacitor in first order parallel RC filter are set as in Table 2. Figure 5 shows the relation curves of input voltage and input current in a phase with the different threephase input voltage frequencies. Because the input voltage is threephase input voltage, the frequency of each phase is changed at the same time. When the frequency is greater than the critical value, the areas of the hysteresis loops decrease monotonically as the frequency increases and what we can see in Figure 5 is that the hysteresis loop almost shrinks to a nonlinear singlevalued function when . In addition, it can be seen that the hysteresis loops not only are pinched at the origin but also have the other two intersection points in the first quadrant and the third quadrant. The described hysteresis loops satisfy the three essential features of a memristor; as a result, it can be regarded as a generalized memristor. Because the input voltage has only difference in phase, the curves of input voltage and input current in b phase and c phase are the same as a phase; each phase of the threephase diode bridge rectifier circuit can be known as an ideal memristor when the circuit is running under normal operation and input voltage is threephase voltage (threephase voltage has the same frequency and phase difference is 120 degrees).
2.2. Running under SinglePhase ShortCircuit Condition
Consider threephase bridge rectifier circuit is running under singlephase shortcircuit condition; the equivalent circuit can be described as in Figure 6. The voltage between a phase and b phase is depicted as and indicates the voltage value at the positive terminal of the capacitor C. In Figure 6, a phase and b phase are running in normal operation, while c phase is connected to the ground because of short circuit. Set the input voltage as , , and assume the conduction voltage of the diode is zero; the conduction situations of six diodes in one cycle are shown in Table 3.

It can be seen that, at the beginning, the input voltage charges to the capacitor, diodes and break over, and when the input voltage is less than the capacitor voltage value at the positive terminal, the capacitor discharges, the six diodes switch off, and the currents that flow through six diodes are zero. By the way, diodes and are turned off all the time; the current flowing through c phase is zero. Therefore, the mathematical model of input current in a phase when the threephase diode bridge circuit is running under singlephase shortcircuit condition is similar to that of the singlephase diode bridge circuit proposed in [8], which can be written asTake the same parameters as in Table 2; the input current of a phase versus time can be described in Figure 7, and the loci of vs. in vi plane with different frequencies can be depicted in Figure 8. When f is set to 5 kHz, to 10 kHz, and to 20 kHz, the curves of input voltage and input current are almost coincident, which do not accord with the three fingerprints of a memristor; as a result, it cannot be considered as a generalized memristor.
Then we consider a phase to be the input terminal and b phase to be the output terminal, and the voltage difference between a phase and b phase can be calculated as . The voltage difference between the input terminal and output terminal can be equivalent to an input voltage in input terminal, and output terminal is connected to the ground, as shown in Figure 9. We have mentioned above that diodes and are turned off all the time, so c phase has no current. The circuit characteristics of Figure 9 are the same as Figure 6. Circuit simulation software Pspice is utilized to verify the above conjectures. Set frequencies as 50Hz, as 200Hz, and as 500Hz; the IV curves of input voltage and input current are shown in Figure 10. As seen in Figure 10, the hysteresis loops are pinched at origin and the areas of hysteresis loop gradually decrease with the increase of the frequency. The characteristics of the pinched hysteresis loop conform to the three fingerprints of the generalized memristor. Accordingly, when threephase bridge rectifier circuit is running under the condition of c phase shortcircuit, the circuit between a terminal and b terminal can be described as a generalized memristor.
2.3. Running under TwoPhase ShortCircuit Condition
When the threephase bridge rectifier circuit is running under the condition of twophase shortcircuit, the circuit diagram can be seen as Figure 11, where b phase and c phase are short circuit to ground. Set the input voltage as , , and make the same assumption that the conduction voltage of the diode is zero; the conduction situations of six diodes are shown in Table 4. Diodes , , and are conductive when input voltage is larger than the voltage at the positive terminal of the capacitor C, and the current flowing through the diode is equal to that of diode and equal to half of the current flowing through diode .

When input voltage is less than the capacitor voltage and , capacitor C discharges and diodes switch off. Conduction situations of diodes in time intervals III and IV are similar to those in time intervals I and II, respectively, where, in time interval III, the current flowing through the diode is equal to that of diode and equal to half of the current flowing through diode .
For the threephase bridge rectifier circuit running under the condition of twophase shortcircuit, the current in a phase cannot be deduced simply from the singlephase diode bridge circuit because there is threediode conduction at the same time. The current can be written as follows applying Kirchhoff’s current law:By the same way, we can getAnd from above analysis we know thatAndFrom equations (5), (6), and (7), we getIn addition, the constitutive relations of diodes (k=1~6) can be described aswhere represents the voltage across the corresponding diodes . Considering and substituting equations (7) and (10) into equation (5) yieldWhen the threephase diode bridge rectifier circuit is running in time interval I, diodes , , and break over; applying Kirchhoff’s voltage law, we getThe relationship between and can be obtained by substituting equation (10) into equation (8), which can be described as Thus, the mathematical model of a phase current when threephase diode bridge rectifier circuit is running under twophase shortcircuit condition in time interval I can be calculated by equations (9)~(13). In the same way, we can obtain the mathematical model of a phase current when threephase diode bridge rectifier circuit is running under twophase shortcircuit condition in time interval II, which can be computed by the following equations:Take the same parameters as in Table 2; numerical simulation can be carried out, and the current of a phase can be obtained by summing the of the two time intervals. Figure 12 indicates the input current of a phase versus time t, and the locus in vi plane is given in Figure 13.
Set f = 50 Hz, 100 Hz, and 500Hz; circuit simulation results show that the loci of a phase in the IV plane are hysteresis loops and are pinched at the origin, as depicted in Figure 14. Same as the phenomenon like singlephase shortcircuit, the area of the pinched hysteresis loop decreases as the frequency increases and the hysteresis loop shrinks to a nonlinear singlevalued function when frequency increases to large enough. The IV characteristic curves satisfy the substitutive characteristics of a memristive element and can be considered as a generalized memristor from a phase when twophase shortcircuit occurs in b phase and c phase.
2.4. Running under Interphase ShortCircuit Condition
Assuming the line impedance , the schematic diagram of threephase bridge rectifier circuit running under interphase shortcircuit condition is shown in Figure 15, where , , and , , are input voltages and input currents, respectively, and , , and are the positive voltages of diodes , , and . Set the threephase input voltage as , , and , and also suppose the conduction voltage of the diode is zero; the conduction situations of six diodes are shown in Table 5.

It can be seen that the voltage of b phase equals that of c phase because of the short circuit between b phase and c phase. As a result, diodes and switch on at the same time and share the current of diode . In the same way, diodes and share the current of diode in time interval III. The mathematical model of a phase current is similar to that of a phase current when threephase diode bridge rectifier circuit is running under twophase shortcircuit condition; the only difference is to change in equations (11), (12), (14), and (15) into . In order to validate whether the input voltage and the input current of a phase accord with the pinched hysteresis loop, circuit simulation is carried out when the frequencies of the input voltage are set as 50 Hz, 200 Hz, and 500Hz, and the IV characteristic curves are depicted in Figure 16.
It can be seen that areas of the loci in the IV plane decrease as the frequency increases and finally reduce to zero, and the hysteresis loops are pinched at the origin. Samely, it can be named as a generalized memristor from a phase when interphase short circuit occurs between b phase and c phase.
2.5. Running under SinglePhase OpenCircuit Condition
The circuit structure of threephase bridge rectifier circuit running under singlephase opencircuit condition is depicted in Figure 17. Due to the voltage drop of the resistor, the voltage of A terminal is always higher than that of B terminal; hence the diodes and are turned off all the time, the same situation as the threephase bridge rectifier circuit running under the singlephase shortcircuit condition. Therefore, the mathematical model of a phase current is the same as equation (4). Set the same parameters as the condition under singlephase shortcircuit; the IV characteristic curves are the same as that of the threephase bridge rectifier circuit during singlephase shortcircuit, which is illustrated in Figure 10. As a result, the circuit between a terminal and b terminal can be described as a generalized memristor.
3. Conclusions
The analyses of whether a threephase diode bridge rectifier circuit can be called as a generalized memristor are discussed in this paper. The discussion is carried out when threephase bridge rectifier circuit is running under normal operation or singlephase shortcircuit, twophase shortcircuit, interphase shortcircuit, and singlephase opencircuit conditions. Assume that the tube voltage drop of diode is zero; the conduction situations of six diodes are investigated and circuit operation conditions are analyzed. The pinched hysteresis loops are numerically simulated in MATLAB and validated by circuit simulation software PSpice. The results confirm that the threephase diode bridge rectifier circuit has memristive characteristics during several working conditions.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant No. 51507134), Scientific Research Program Funded by Shaanxi Provincial Department of Water Resources (Grant No. 2017slkj15), Natural Science Foundation of Shaanxi Province (Grant No. 2018JM5068), Key Project of Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2018ZDXMGY169), and Xi'an Science and Technology Innovation Project (Grant No. 201805037(21)).
References
 R. Tetzlaff, Memristors and Memristive Systems, Springer, 2013.
 M. Li, H. Deng, M. Wei et al., “The recent advances of research on memristor and memristive characteristic devices,” Advanced Materials Research, vol. 685, pp. 201–206, 2013. View at: Publisher Site  Google Scholar
 S. Vaidyanathan, C. K. Volos, I. M. Kyprianidis, I. N. Stouboulos, and E. TleloCuautle, “Memristor: a new concept in synchronization of coupled neuromorphic circuits,” Journal of Engineering Science and Technology Review, vol. 8, no. 2, pp. 157–173, 2015. View at: Publisher Site  Google Scholar
 Y. Ho, G. M. Huang, and P. Li, “Nonvolatile memristor memory: device characteristics and design implications,” in Proceedings of the 2009 IEEE/ACM International Conference on ComputerAided Design, ICCAD 2009, pp. 485–490, USA, November 2009. View at: Google Scholar
 D. Cafagna and G. Grassi, “On the simplest fractionalorder memristorbased chaotic system,” Nonlinear Dynamics, vol. 70, no. 2, pp. 1185–1197, 2012. View at: Publisher Site  Google Scholar  MathSciNet
 J. Sun, Y. Shen, Q. Yin, and C. Xu, “Compound synchronization of four memristor chaotic oscillator systems and secure communication,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 23, no. 1, Article ID 013140, p. 821, 2013. View at: Publisher Site  Google Scholar
 F. Corinto and A. Ascoli, “Memristive diode bridge with LCR filter,” IEEE Electronics Letters, vol. 48, no. 14, pp. 824825, 2012. View at: Publisher Site  Google Scholar
 B. Bao, J. Yu, F. Hu, and Z. Liu, “Generalized memristor consisting of diode bridge with first order parallel RC filter,” International Journal of Bifurcation & Chaos, vol. 24, no. 11, p. 3008, 2014. View at: Google Scholar
 M. Chen, M. Li, Q. Yu, B. Bao, Q. Xu, and J. Wang, “Dynamics of selfexcited attractors and hidden attractors in generalized memristorbased Chua's circuit,” Nonlinear Dynamics, vol. 81, no. 12, pp. 215–226, 2015. View at: Publisher Site  Google Scholar  MathSciNet
 H. G. Wu, B. C. Bao, and Q. Xu, “First order generalized memristor emulator based on diode bridge and series RL filter,” Sexualities, vol. 16, no. 16, pp. 651–664, 2015. View at: Google Scholar
 H. Wu, B. Bao, Z. Liu, Q. Xu, and P. Jiang, “Chaotic and periodic bursting phenomena in a memristive Wienbridge oscillator,” Nonlinear Dynamics, vol. 83, no. 12, pp. 893–903, 2016. View at: Publisher Site  Google Scholar  MathSciNet
 B.C. Bao, P. Wu, H. Bao, M. Chen, and Q. Xu, “Chaotic bursting in memristive diode bridgecoupled SallenKey lowpass filter,” IEEE Electronics Letters, vol. 53, no. 16, pp. 11041105, 2017. View at: Publisher Site  Google Scholar
 Q. Xu, Q. Zhang, N. Wang, H. Wu, and B. Bao, “An improved memristive diode bridgebased band pass filter chaotic circuit,” Mathematical Problems in Engineering, vol. 2017, no. 4, pp. 1–11, 2017. View at: Google Scholar
 N. Yang, C. Xu, C. Wu, R. Jia, and C. Liu, “Modeling and analysis of a fractionalorder generalized memristorbased chaotic system and circuit implementation,” International Journal of Bifurcation & Chaos, vol. 27, no. 13, Article ID 1750199, 2018. View at: Google Scholar
 L. Chua, “Everything you wish to know about memristors but are afraid to ask,” Radioengineering, vol. 24, no. 2, pp. 319–368, 2015. View at: Publisher Site  Google Scholar
 S. P. Adhikari, M. P. Sah, H. Kim, and L. O. Chua, “Three fingerprints of memristor,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 60, no. 11, pp. 3008–3021, 2013. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2019 Chaojun Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.