# A Novel Generalized Memristor Based on Three-Phase Diode Bridge Rectifier

Authors: Chaojun Wu; Ningning Yang; Cheng Xu; Rong Jia; Chongxin Liu
Journal: Complexity
 (2019)

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## 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.


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## Body

## 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 Section2, 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 thev-iplane 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].Figure 1
The definition of a voltage-controlled ideal memristor. (a) Hysteresis loop of a memristor. (b) The symbol of a memristor.
(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 Figure2. When the circuit is running under normal condition, the input voltages are given as ua=8sin⁡(2πft), ub=8sin⁡(2πft-2π/3), and uc=8sin⁡(2πft-4π/3), and suppose the conduction voltage of the diode is zero; the conduction situations of six diodes in one cycle are shown in Table 1.Table 1
Conduction situations of six diodes.
Time interval I II III IV V VI Conducting diodes V D 1 , V D 6 V D 1 , V D 2 V D 3 , V D 2 V D 3 , V D 4 V D 5 , V D 4 V D 5 , V D 6  Input voltage relation u a > u c > u b u a > u b > u c u b > u a > u c u b > u c > u a u c > u b > u a u c > u a > u bFigure 2
Three-phase diode bridge rectifier circuit.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 ofa phase can be derived from the mathematical model of the single-phase diode bridge circuit, which is written as(1)ia=2ISe-ρudsinh⁡ρuinwhere ρ=1/(2nVT); IS, n, and VT indicate the reverse saturation current, emission coefficient, and thermal voltage of the diode, respectively. uin expresses the input voltage of circuit under different circumstances, which can be shown as(2)uin=ua-ub:ua>uc>ubua-uc:ua>ub>ucub-ua:ub>uc>uauc-ua:uc>ub>ua0:uc>ua>ub0:ub>ua>ucAnd ud represents the voltage across capacitorC, which can be calculated by(3)duddt=2ISe-ρudcosh⁡ρuin-1C-udRCThus, 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 ofaphase with time t, and the relation curve between input voltage ua and input current ia is shown in Figure 4. It can be seen that the locus in ua-ia plane is hysteresis loop and is pinched at the origin. Besides, the trajectory is intersecting in the first quadrant and the third quadrant.Table 2
Circuit parameters for numerical simulation.
Parameters Significations Values I S Reverse saturation current of diode 2.682 nA  n Emission coefficient of diode 1.836  V T Thermal voltage of diode 25 mV  R Resistance 10 kΩ  C Capacitance 1 uF  f Frequency 50 HzFigure 3
Input current of a phase with time t.Figure 4
Locus inua-ia plane.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 Table2. Figure 5 shows the relation curves of input voltage and input current ina 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 f=5kHz. 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 inb phase andc phase are the same asa 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).Figure 5
Curves of input voltageua and input current ia.


### 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 Figure6. The voltage betweena phase andb phase is depicted as uab and uC+ indicates the voltage value at the positive terminal of the capacitorC. In Figure 6,a phase andb phase are running in normal operation, whilec phase is connected to the ground because of short circuit. Set the input voltage as ua=8sin⁡(2πft), ub=8sin⁡(2πft-2π/3), and assume the conduction voltage of the diode is zero; the conduction situations of six diodes in one cycle are shown in Table 3.Table 3
Conduction situations of six diodes.
Time interval I II III IV Conducting diodes V D 1 , V D 6 / V D 3 , V D 4 /  Input voltage relation u a > u C + > u b u C + ≥ u a   ( o r   u b ) u b > u C + > u a u C + ≥ u b   ( o r   u a )Figure 6
Three-phase bridge rectifier circuit running under single-phase short-circuit condition.It can be seen that, at the beginning, the input voltage charges to the capacitor, diodesVD1 and VD6 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 VD2 and VD5 are turned off all the time; the current flowing throughc phase is zero. Therefore, the mathematical model of input current ina 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 as(4)ia=2ISe-ρudsinh⁡ρuabTake the same parameters as in Table 2; the input current ofa phase versus time t can be described in Figure 7, and the loci of ua vs. ia inv-iplane with different frequencies can be depicted in Figure 8. Whenfis set to 5 kHz, to 10 kHz, and to 20 kHz, the curves of input voltage ua and input current ia 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.Figure 7
Input current ofa phase versus timet.Figure 8
Loci inv-i plane.Then we considera phase to be the input terminal andb phase to be the output terminal, and the voltage difference betweena phase andb phase can be calculated as uab=ua-ub=16sin⁡π/3cos⁡(2πft-π/3). 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 VD2 and VD5 are turned off all the time, soc 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 f as 50Hz, as 200Hz, and as 500Hz; theI-V curves of input voltage uab and input current ia 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 ofc phase short-circuit, the circuit betweena terminal andb terminal can be described as a generalized memristor.Figure 9
Equivalent circuit of three-phase bridge rectifier circuit running under single-phase short-circuit condition.Figure 10
Curves of input voltageuab and input current ia.


### 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 Figure11, whereb phase andc phase are short circuit to ground. Set the input voltage as ua=8sin⁡(2πft), ub=uc=0, 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 VD1, VD2, and VD6 are conductive when input voltage ua is larger than the voltage at the positive terminal of the capacitorC, and the current flowing through the diode VD2 is equal to that of diode VD6 and equal to half of the current flowing through diode VD1.Table 4
Conduction situations of six diodes.
Time interval I II III IV Conducting diodes V D 1 , V D 2 , V D 6 / V D 3 , V D 4 , V D 5 /  Input voltage relation u a > u C + > 0 u C + ≥ u a and uC+≥0 u a < u C + < 0 u C + ≥ u a and uC+≥0Figure 11
Three-phase bridge rectifier circuit running under two-phase short-circuit condition.When input voltageua is less than the capacitor voltage uC+ and uC+≥0, capacitorC 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 VD3 is equal to that of diode VD5 and equal to half of the current flowing through diode VD4.For the three-phase bridge rectifier circuit running under the condition of two-phase short-circuit, the current ina phase cannot be deduced simply from the single-phase diode bridge circuit because there is three-diode conduction at the same time. The current ia can be written as follows applying Kirchhoff’s current law:(5)ia=i1-i4By the same way, we can get(6)i1+i3+i5=iC+iRAnd from above analysis we know that(7)i4=2i3=2i5And(8)i1=2i2=2i6From equations (5), (6), and (7), we get(9)2i1=ia+iC+iR=ia+Cduddt+udRIn addition, the constitutive relations of diodes VDk (k=1~6) can be described as(10)ik=ISe2ρuk-1where uk represents the voltage across the corresponding diodes VDk. Considering u3=u1-ua and substituting equations (7) and (10) into equation (5) yield(11)ia=ISe2ρu1-1-2ISe2ρu3-1=ISe2ρu1-2ISe2ρu1-ua+IS=ISe2ρu11-2e-2ρua+ISWhen the three-phase diode bridge rectifier circuit is running in time interval I, diodes VD1, VD2, and VD6 break over; applying Kirchhoff’s voltage law, we get(12)ua=u1+ud+u2The relationship between u1 and u2 can be obtained by substituting equation (10) into equation (8), which can be described as (13)e2ρu1=2e2ρu2-1Thus, the mathematical model ofa 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 ofa 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:(14)ua=-u4-ud-u5(15)ia=ISe2ρu42e2ρua-1-IS(16)ia+2i4=Cduddt+udR(17)e2ρu4=2e2ρu5-1(18)i4=ISe2ρu4-1Take the same parameters as in Table 2; numerical simulation can be carried out, and the current ofa phase can be obtained by summing the ia of the two time intervals. Figure 12 indicates the input current ofa phase versus timet, and the locus inv-i plane is given in Figure 13.Figure 12
Input current ofa phase versus timet.Figure 13
Locus inv-i plane.Setf= 50 Hz, 100 Hz, and 500Hz; circuit simulation results show that the loci ofa phase in theI-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. TheI-V characteristic curves satisfy the substitutive characteristics of a memristive element and can be considered as a generalized memristor froma phase when two-phase short-circuit occurs inb phase andc phase.Figure 14
Curves of input voltageua and input current ia.


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

Assuming the line impedanceRa=Rb=Rc=0.1Ω, the schematic diagram of three-phase bridge rectifier circuit running under interphase short-circuit condition is shown in Figure 15, where ua, ub, uc and ia, ib, ic are input voltages and input currents, respectively, and ua′, ub′, and uc′ are the positive voltages of diodes VD1, VD3, and VD5. Set the three-phase input voltage as ua=8sin⁡(2πft), ub=8sin⁡(2πft-2π/3), and uc=8sin⁡(2πft-4π/3), and also suppose the conduction voltage of the diode is zero; the conduction situations of six diodes are shown in Table 5.Table 5
Conduction situations of six diodes.
Time interval I II III IV Conducting diodes V D 1 , V D 2 , V D 6 / V D 3 , V D 4 , V D 5 /  Input voltage relation u a ′ > u C + > u b ′ = u c ′ u C + ≥ u a , u b , u c u b ′ = u c ′ > u C + > u a ′ u C + ≥ u a , u b , u cFigure 15
Three-phase bridge rectifier circuit running under interphase short-circuit condition.It can be seen that the voltage ofb phase equals that ofc phase because of the short circuit betweenb phase andc phase. As a result, diodes VD2 and VD6 switch on at the same time and share the current of diode VD1. In the same way, diodes VD3 and VD5 share the current of diode VD4 in time interval III. The mathematical model ofa phase current is similar to that ofa phase current when three-phase diode bridge rectifier circuit is running under two-phase short-circuit condition; the only difference is to change ua in equations (11), (12), (14), and (15) into ua′. In order to validate whether the input voltage and the input current ofa 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 theI-V characteristic curves are depicted in Figure 16.Figure 16
Curves of input voltageua and input current ia.It can be seen that areas of the loci in theI-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 froma phase when interphase short circuit occurs betweenb phase andc 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 Figure17. Due to the voltage drop of the resistor, the voltage of A terminal is always higher than that of B terminal; hence the diodes VD2 and VD5 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 ofa phase current is the same as equation (4). Set the same parameters as the condition under single-phase short-circuit; theI-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 betweena terminal andb terminal can be described as a generalized memristor.Figure 17
Three-phase bridge rectifier circuit running under single-phase open circuit condition.


## 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 Figure2. When the circuit is running under normal condition, the input voltages are given as ua=8sin⁡(2πft), ub=8sin⁡(2πft-2π/3), and uc=8sin⁡(2πft-4π/3), and suppose the conduction voltage of the diode is zero; the conduction situations of six diodes in one cycle are shown in Table 1.Table 1
Conduction situations of six diodes.
Time interval I II III IV V VI Conducting diodes V D 1 , V D 6 V D 1 , V D 2 V D 3 , V D 2 V D 3 , V D 4 V D 5 , V D 4 V D 5 , V D 6  Input voltage relation u a > u c > u b u a > u b > u c u b > u a > u c u b > u c > u a u c > u b > u a u c > u a > u bFigure 2
Three-phase diode bridge rectifier circuit.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 ofa phase can be derived from the mathematical model of the single-phase diode bridge circuit, which is written as(1)ia=2ISe-ρudsinh⁡ρuinwhere ρ=1/(2nVT); IS, n, and VT indicate the reverse saturation current, emission coefficient, and thermal voltage of the diode, respectively. uin expresses the input voltage of circuit under different circumstances, which can be shown as(2)uin=ua-ub:ua>uc>ubua-uc:ua>ub>ucub-ua:ub>uc>uauc-ua:uc>ub>ua0:uc>ua>ub0:ub>ua>ucAnd ud represents the voltage across capacitorC, which can be calculated by(3)duddt=2ISe-ρudcosh⁡ρuin-1C-udRCThus, 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 ofaphase with time t, and the relation curve between input voltage ua and input current ia is shown in Figure 4. It can be seen that the locus in ua-ia plane is hysteresis loop and is pinched at the origin. Besides, the trajectory is intersecting in the first quadrant and the third quadrant.Table 2
Circuit parameters for numerical simulation.
Parameters Significations Values I S Reverse saturation current of diode 2.682 nA  n Emission coefficient of diode 1.836  V T Thermal voltage of diode 25 mV  R Resistance 10 kΩ  C Capacitance 1 uF  f Frequency 50 HzFigure 3
Input current of a phase with time t.Figure 4
Locus inua-ia plane.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 Table2. Figure 5 shows the relation curves of input voltage and input current ina 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 f=5kHz. 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 inb phase andc phase are the same asa 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).Figure 5
Curves of input voltageua and input current ia.


## 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 Figure6. The voltage betweena phase andb phase is depicted as uab and uC+ indicates the voltage value at the positive terminal of the capacitorC. In Figure 6,a phase andb phase are running in normal operation, whilec phase is connected to the ground because of short circuit. Set the input voltage as ua=8sin⁡(2πft), ub=8sin⁡(2πft-2π/3), and assume the conduction voltage of the diode is zero; the conduction situations of six diodes in one cycle are shown in Table 3.Table 3
Conduction situations of six diodes.
Time interval I II III IV Conducting diodes V D 1 , V D 6 / V D 3 , V D 4 /  Input voltage relation u a > u C + > u b u C + ≥ u a   ( o r   u b ) u b > u C + > u a u C + ≥ u b   ( o r   u a )Figure 6
Three-phase bridge rectifier circuit running under single-phase short-circuit condition.It can be seen that, at the beginning, the input voltage charges to the capacitor, diodesVD1 and VD6 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 VD2 and VD5 are turned off all the time; the current flowing throughc phase is zero. Therefore, the mathematical model of input current ina 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 as(4)ia=2ISe-ρudsinh⁡ρuabTake the same parameters as in Table 2; the input current ofa phase versus time t can be described in Figure 7, and the loci of ua vs. ia inv-iplane with different frequencies can be depicted in Figure 8. Whenfis set to 5 kHz, to 10 kHz, and to 20 kHz, the curves of input voltage ua and input current ia 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.Figure 7
Input current ofa phase versus timet.Figure 8
Loci inv-i plane.Then we considera phase to be the input terminal andb phase to be the output terminal, and the voltage difference betweena phase andb phase can be calculated as uab=ua-ub=16sin⁡π/3cos⁡(2πft-π/3). 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 VD2 and VD5 are turned off all the time, soc 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 f as 50Hz, as 200Hz, and as 500Hz; theI-V curves of input voltage uab and input current ia 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 ofc phase short-circuit, the circuit betweena terminal andb terminal can be described as a generalized memristor.Figure 9
Equivalent circuit of three-phase bridge rectifier circuit running under single-phase short-circuit condition.Figure 10
Curves of input voltageuab and input current ia.


## 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 Figure11, whereb phase andc phase are short circuit to ground. Set the input voltage as ua=8sin⁡(2πft), ub=uc=0, 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 VD1, VD2, and VD6 are conductive when input voltage ua is larger than the voltage at the positive terminal of the capacitorC, and the current flowing through the diode VD2 is equal to that of diode VD6 and equal to half of the current flowing through diode VD1.Table 4
Conduction situations of six diodes.
Time interval I II III IV Conducting diodes V D 1 , V D 2 , V D 6 / V D 3 , V D 4 , V D 5 /  Input voltage relation u a > u C + > 0 u C + ≥ u a and uC+≥0 u a < u C + < 0 u C + ≥ u a and uC+≥0Figure 11
Three-phase bridge rectifier circuit running under two-phase short-circuit condition.When input voltageua is less than the capacitor voltage uC+ and uC+≥0, capacitorC 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 VD3 is equal to that of diode VD5 and equal to half of the current flowing through diode VD4.For the three-phase bridge rectifier circuit running under the condition of two-phase short-circuit, the current ina phase cannot be deduced simply from the single-phase diode bridge circuit because there is three-diode conduction at the same time. The current ia can be written as follows applying Kirchhoff’s current law:(5)ia=i1-i4By the same way, we can get(6)i1+i3+i5=iC+iRAnd from above analysis we know that(7)i4=2i3=2i5And(8)i1=2i2=2i6From equations (5), (6), and (7), we get(9)2i1=ia+iC+iR=ia+Cduddt+udRIn addition, the constitutive relations of diodes VDk (k=1~6) can be described as(10)ik=ISe2ρuk-1where uk represents the voltage across the corresponding diodes VDk. Considering u3=u1-ua and substituting equations (7) and (10) into equation (5) yield(11)ia=ISe2ρu1-1-2ISe2ρu3-1=ISe2ρu1-2ISe2ρu1-ua+IS=ISe2ρu11-2e-2ρua+ISWhen the three-phase diode bridge rectifier circuit is running in time interval I, diodes VD1, VD2, and VD6 break over; applying Kirchhoff’s voltage law, we get(12)ua=u1+ud+u2The relationship between u1 and u2 can be obtained by substituting equation (10) into equation (8), which can be described as (13)e2ρu1=2e2ρu2-1Thus, the mathematical model ofa 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 ofa 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:(14)ua=-u4-ud-u5(15)ia=ISe2ρu42e2ρua-1-IS(16)ia+2i4=Cduddt+udR(17)e2ρu4=2e2ρu5-1(18)i4=ISe2ρu4-1Take the same parameters as in Table 2; numerical simulation can be carried out, and the current ofa phase can be obtained by summing the ia of the two time intervals. Figure 12 indicates the input current ofa phase versus timet, and the locus inv-i plane is given in Figure 13.Figure 12
Input current ofa phase versus timet.Figure 13
Locus inv-i plane.Setf= 50 Hz, 100 Hz, and 500Hz; circuit simulation results show that the loci ofa phase in theI-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. TheI-V characteristic curves satisfy the substitutive characteristics of a memristive element and can be considered as a generalized memristor froma phase when two-phase short-circuit occurs inb phase andc phase.Figure 14
Curves of input voltageua and input current ia.


## 2.4. Running under Interphase Short-Circuit Condition

Assuming the line impedanceRa=Rb=Rc=0.1Ω, the schematic diagram of three-phase bridge rectifier circuit running under interphase short-circuit condition is shown in Figure 15, where ua, ub, uc and ia, ib, ic are input voltages and input currents, respectively, and ua′, ub′, and uc′ are the positive voltages of diodes VD1, VD3, and VD5. Set the three-phase input voltage as ua=8sin⁡(2πft), ub=8sin⁡(2πft-2π/3), and uc=8sin⁡(2πft-4π/3), and also suppose the conduction voltage of the diode is zero; the conduction situations of six diodes are shown in Table 5.Table 5
Conduction situations of six diodes.
Time interval I II III IV Conducting diodes V D 1 , V D 2 , V D 6 / V D 3 , V D 4 , V D 5 /  Input voltage relation u a ′ > u C + > u b ′ = u c ′ u C + ≥ u a , u b , u c u b ′ = u c ′ > u C + > u a ′ u C + ≥ u a , u b , u cFigure 15
Three-phase bridge rectifier circuit running under interphase short-circuit condition.It can be seen that the voltage ofb phase equals that ofc phase because of the short circuit betweenb phase andc phase. As a result, diodes VD2 and VD6 switch on at the same time and share the current of diode VD1. In the same way, diodes VD3 and VD5 share the current of diode VD4 in time interval III. The mathematical model ofa phase current is similar to that ofa phase current when three-phase diode bridge rectifier circuit is running under two-phase short-circuit condition; the only difference is to change ua in equations (11), (12), (14), and (15) into ua′. In order to validate whether the input voltage and the input current ofa 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 theI-V characteristic curves are depicted in Figure 16.Figure 16
Curves of input voltageua and input current ia.It can be seen that areas of the loci in theI-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 froma phase when interphase short circuit occurs betweenb phase andc 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 Figure17. Due to the voltage drop of the resistor, the voltage of A terminal is always higher than that of B terminal; hence the diodes VD2 and VD5 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 ofa phase current is the same as equation (4). Set the same parameters as the condition under single-phase short-circuit; theI-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 betweena terminal andb terminal can be described as a generalized memristor.Figure 17
Three-phase bridge rectifier circuit running under single-phase open circuit condition.


## 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.



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