#### Abstract

Memristive characteristics in three-phase 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 three-phase 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 three-phase 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 second-order RLC filter was proved to have memory properties [7]. By replacing the second-order RLC filter with a first-order parallel RC filter, another equivalent realization circuit of a generalized memristor was implemented, which decreases a fundamental circuit element-inductor [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 second-order 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 fractional-order capacitor-based diode bridge circuit was proposed, in which the fractional order capacitor is circuit implemented utilizing Oustaloup approximation technique [14].

Since a single-phase diode bridge rectifier circuit can express the memristive features, does a three-phase 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 three-phase voltage which has the same amplitude and frequency; phase difference of each phase is 120 degrees. In order to study whether a three-phase 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 three-phase 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. Three-Phase Bridge Rectifier Circuit Running under Several Conditions

Equipment can be described as a voltage-controlled 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* v-i *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 single-valued 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 three-phase 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 single-phase diode bridge circuit proposed in [8]. As a result, the input current of** a** phase can be derived from the mathematical model of the single-phase 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 three-phase 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 three-phase 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 three-phase input voltage frequencies. Because the input voltage is three-phase 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 single-valued 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 three-phase diode bridge rectifier circuit can be known as an ideal memristor when the circuit is running under normal operation and input voltage is three-phase voltage (three-phase voltage has the same frequency and phase difference is 120 degrees).

##### 2.2. Running under Single-Phase Short-Circuit Condition

Consider three-phase bridge rectifier circuit is running under single-phase short-circuit 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 three-phase diode bridge circuit is running under single-phase short-circuit condition is similar to that of the single-phase 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* v*-*i *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* I*-*V* 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 three-phase bridge rectifier circuit is running under the condition of** c** phase short-circuit, the circuit between** a** terminal and** b** terminal can be described as a generalized memristor.

##### 2.3. Running under Two-Phase Short-Circuit Condition

When the three-phase bridge rectifier circuit is running under the condition of two-phase short-circuit, 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 three-phase bridge rectifier circuit running under the condition of two-phase short-circuit, the current in** a** phase cannot be deduced simply from the single-phase diode bridge circuit because there is three-diode 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 three-phase 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 three-phase diode bridge rectifier circuit is running under two-phase short-circuit 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 three-phase diode bridge rectifier circuit is running under two-phase short-circuit 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* v*-*i* 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* I*-*V* plane are hysteresis loops and are pinched at the origin, as depicted in Figure 14. Same as the phenomenon like single-phase short-circuit, the area of the pinched hysteresis loop decreases as the frequency increases and the hysteresis loop shrinks to a nonlinear single-valued function when frequency increases to large enough. The* I*-*V* characteristic curves satisfy the substitutive characteristics of a memristive element and can be considered as a generalized memristor from** a** phase when two-phase short-circuit occurs in** b** phase and** c** phase.

##### 2.4. Running under Interphase Short-Circuit Condition

Assuming the line impedance , the schematic diagram of three-phase bridge rectifier circuit running under interphase short-circuit 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 three-phase 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 three-phase diode bridge rectifier circuit is running under two-phase short-circuit 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* I*-*V* characteristic curves are depicted in Figure 16.

It can be seen that areas of the loci in the* I*-*V* 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 Single-Phase Open-Circuit Condition

The circuit structure of three-phase bridge rectifier circuit running under single-phase open-circuit 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 three-phase bridge rectifier circuit running under the single-phase short-circuit condition. Therefore, the mathematical model of** a** phase current is the same as equation (4). Set the same parameters as the condition under single-phase short-circuit; the* I-V* characteristic curves are the same as that of the three-phase bridge rectifier circuit during single-phase short-circuit, 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 three-phase diode bridge rectifier circuit can be called as a generalized memristor are discussed in this paper. The discussion is carried out when three-phase bridge rectifier circuit is running under normal operation or single-phase short-circuit, two-phase short-circuit, interphase short-circuit, and single-phase open-circuit 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 three-phase 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. 2017slkj-15), Natural Science Foundation of Shaanxi Province (Grant No. 2018JM5068), Key Project of Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2018ZDXM-GY-169), and Xi'an Science and Technology Innovation Project (Grant No. 201805037(21)).