Abstract
The memristor is a nonlinear device with a particular memory function and is widely used in various circuit researches. This work studies the peak current mode controlled (PCMC) buck converter with the memristive load at the continuous current mode (CCM). Firstly, a state equation for a buck converter with the memristive load is derived and a generic voltagecontrolled memristor simulator is constructed by using a nonlinear function model; Secondly, facing the system chaos caused by changing bifurcation parameters, we introduce ramp compensation to stabilize the system at period1. The chaos is effectively suppressed, this provides a guide for parameters choosing in buck converters with nonlinear loads in practical applications. The simulation is implemented by using MATLAB and PSIM.
Introduction
Since the birth of switching devices in the 1960s, switching converters have developed exceptionally rapidly [1]. The DCDC converter is the essential component of building a switching power supply and has been widely used in various industries [2]. In fact, the switching converter is a strong nonlinear dynamic system, and some irregular phenomena often occur during operation, which affects the regular operation of the system [3].
In 1984, Brockett and Wood first confirmed chaos in a buck converter [4]. So far, the research of DCDC converters has ushered in new development, where the rich nonlinear phenomena have been discovered, including period bifurcation, boundary collision bifurcation, NeimarkSacker bifurcation, coexistence attractor, etc. [5, 6]. These phenomena hinder the normal stability of the system. In most practical situations, in order to ensure the stable operation, the occurrence of nonlinear phenomena should be considered to suppress in the DCDC converter.
In many nonlinear studies of DCDC converters, most of the studied load types are resistive loads. However, the change in load type can indeed affect the nonlinear behavior of the DCDC converter. The literature [7] found that the inductive load in a buckboost converter can improve the stability; The writings [8, 9] analyzed the dynamic change of the DCDC converter with a battery load; The research [10] studied the peak current mode control of the current source load in a buck converter.
With the successful preparation of nanoscale solidstate memristors by HP Labs in 2008 [11], memristors have rapidly become the research focus in recent years. It has applied in the fields of neuromorphic computing, hardware security applications, access devices, analog circuits, etc., and memristors have a broad application prospect in the future [12, 13]. Some study found that the memristive characteristic exists in the switch resistance used as the access devices with a memory effect [14,15,16]. Recently, a memristor convolutional neural network composed entirely of hardware has been implemented [17].
Previously, Zhang et al. studied the nonlinear dynamics of boost converters with memristive loads [18]. Bao et al. concluded that the memristive load will not affect the bifurcation structure, but will widen the normal operating area and cause the output voltage dropping in buckboost converters [19]. Ma et al. studied the slowscale instability in voltagemode controlled HBridge inverter with the memristive load [20]. As the most basic DCDC converter, The buck converter’s nonlinear dynamic behavior with memristive load has not been studied and analyzed.
This study focuses on the nonlinear dynamics of the peak current mode controlled (PCMC) buck converter with the memristive load. The concrete contents include:

(1)
Construct a generic voltagecontrolled memristor simulator based on a nonlinear function model and analyze the stability under stable DC.

(2)
Observe the dynamic behavior of the converter with the memristive load by changing the parameter reference current.

(3)
Ramp compensation is used to suppress the occurrence of chaos and make the system stable at period1.
The implementation of memristor simulator
The essential characteristics of memristors
The memristor was proposed by Professor Chua in 1971 from the perspective of circuit symmetry [21]. Now it has been used as the fourth basic element of the circuit except the resistance, the inductance and the capacitance, shown in Fig. 1. Assume that i_{m} and v_{m} denote as the input current and voltage of the memristor, the memristance (M) is defined as the relationship between charge q and magnetic flux ϕ:
According to the definition of current and Faraday law of electromagnetic induction: the relationships between current and charge, voltage and magnetic flux are, respectively:
From Eqs. (1) and (2), we can get:
where W(ϕ) is the memductance.
Equation (3) represents the constitutive relationships of the chargecontrolled memristor and the magneticcontrolled memristor, respectively. The memristor can be regarded as a basic passive twoterminal element. It can remember the amount of charge flowing through it and has three essential characteristics [22]:

(1)
When driven by a bipolar periodic signal, the voltagecurrent relationship of the device is a pinched hysteresis loop that shrinks at the origin.

(2)
The hysteresis lobe area decreases monotonously as the excitation frequency increases from the critical frequency.

(3)
When the frequency approaches infinity, the pinched hysteresis loop shrinks into a singlevalued function.
The memristor simulator based on a nonlinear function model
The voltagecontrolled memristor simulator
There exists various type of nonliner models [23,24,25]. Due to the memristor not being manufactured commercially, this study uses a memristor simulator to study its nonlinear dynamic phenomenon. The currently recognized physical models of memristors are mainly divided into linear impurity drift models, window function models, and nonlinear function models [13, 26]. Nonlinear function models compose of operational amplifiers and multipliers, which can quickly and conveniently realize generic memristors that described as follows [27, 28].
where x is the internal state variable of the system.
A voltagecontrolled memristor simulator is constructed by using a nonlinear function model. As shown in Fig. 2, the memristor simulator includes OpAmps Op_{1} and Op_{2}, resistors R_{1}, R_{2}, R_{3}, multipliers mul_{1} and mul_{2}, and a capacitance C_{1}. The Op_{1} works as a voltage follower, and Op_{2}, R_{1}, R_{2}, and C_{1} as an integrator.
Assume that i_{m} and v_{m} are the input current and voltage of the memristor simulator, the circuit relationship of the memristor simulator can be expressed as:
where g_{1} and g_{2} are the gains of Mul_{1} and Mul_{2}.
According to (4) and (5), we have:
Let, R_{1} = 1 kΩ, R_{2} = 1 kΩ, R_{3} = 15 Ω, C_{1} = 22 nF, g_{1} = − 0.1, g_{2} = 1.5, v_{m} = 5sin(40000 πt) or v_{m} = 5sin(20000 πt).
The analysis of the memristor simulator
The VI curve under periodic AC of the memristor simulator is displayed as a pinched hysteresis loop through the origin and shrinks as the frequency increases, as shown in Fig. 3(a). The simulation results show that the memristor simulator has three essential characteristics. It can replace the real memristor to participate in the circuit experiment.
Because the buck converter converts a DC power supply from one voltage level to another, the memristor simulator is necessary to study the DC characteristics. Under stable DC, dv_{p}/dt = 0, Eq. (5) can be expressed as
The DC VI curve presents a cubic function relationship. It can be seen in Fig. 3(b) that the input voltage should be controlled as small as possible to avoid a sudden current increase caused by excessive voltage.
Study the stability of the memristor. Figure 4 shows the dynamic path of the memristor according to the state Eq. (6). It can be seen that (R_{2}*v_{m}/R_{1}, 0) on the xaxis is the fixed point of the system. The flow goes to the left when dv_{p}/dt < 0 and to the right when dv_{p}/dt > 0. The flow meets at the fixed point from all initial conditions, so the system is globally stable.
The PCMC buck converter with the memristive load
Referring to Fig. 5, the PCMC buck converter with the memristive load composes of a switch S, a diode D, an inductance L, a capacitance C, an RS flipflop and a comparator. Through the periodic on and off the switch S, the voltage E provided by the DC power source is converted into a voltage v, which is less than and in the same direction as E, to power the memristive load.
In order to make the switch S turn on and off periodically, and make the output stable, it is necessary to introduce the PWM control. The clock signal connects to the SET terminal of the RS flipflop, and the switch S is controlled to turn on. The comparator compares the inductance current i with the reference current Iref, its output connects to the RESET terminal of the RS flipflop, and controls the switch to turn off.
According to whether the current i on the inductance L is continuous, the working mode of the buck converter can be divided into a continuous current mode (CCM) and a discontinuous current mode (DCM). Assume that the duration of each cycle is T, two operating states in CCM are as follows:

When the clock pulse signal comes with state 1, the RS flipflop is set, the switch S turns on, the diode D is subjected to reverse voltage to close, and the inductance current i rises for a duration of t_{on}.

When i = Iref, the state 2 reset the RS flipflop, the switch S is turned off, and the diode D turns on under a forward voltage for a duration of t_{off}.
Until the next clock pulse signal comes, the switch S turns on again.
According to the two Kirchhoff's laws, the state equations of the PCMC buck converter with the memristive load are written as follows:
State 1:
State 2:
The inductance L can be selected according to the critical load current I_{OB} = v(1D)(2Lf)^{−1}, and the capacitance C can be selected according to the output voltage ripple Δv = v(1D)(8LCf^{2})^{−1}[1], where D is the duty cycle, and f is the frequency of the buck converter.
In order to facilitate quantitative analysis and calculation, we need normalize Eqs. (9) and (10). Let,
The switch S condition can be expressed as:
With the above preparation, normalizing Eqs. (9) and (10) gives
Through the Eqs. (11), (12) and (13), the numerical simulation of the PCMC buck converter with the memristive load can be performed.
Numerical Simulation
To implement the numerical simulation by using Matlab, the Euler method is used to discretize the differential Eqs. (12), (13). The discrete rule is described as:
The circuit parameters are specified as Table 1.
To ensure the relative calculation accuracy, a smaller calculation step Δt = 7.58e6 should be used. The initial states of x, y, z in the state equations are defined as zero.
Selecting reference current Iref as bifurcation variable. Figure 6 shows perioddoubling bifurcation behavior of the PCMC buck converter with the memristive load. It can be seen that the inductance current i has a perioddoubling as Iref increases. The first bifurcation occurs near Iref = 1.684 A, then there is a narrow chaos window. At Iref = 1.778A, the inductance current i returns to perioddoubling, then quickly enters quasi period, finally comes chaos.
Because the output voltage of the buck converter is a pulsation DC voltage greater than zero, the v_{p} is always less than or equal to zero in the actual operation of the memristor simulator. For convenience of analysis, the v_{p} value represented by z is treated as the opposite number, denoted as z.
Take the reference currents Iref = 1.5A, 1.85A, 2.6A, respectively, to get phase portraits shown in Fig. 7.
When Iref = 1.5A, as shown in Fig. 7(a)(d), Fig. 7(a) shows parameter i changing in the same interval, Fig. 7(b)(d) show phase portraits being a single ring, and the system is in a stable period1.
When Iref = 1.85A, periodic motion occurs, the system changes from period1 to period2, Fig. 7(e) shows that parameter i becomes two intervals, and Fig. 7(f)(h) show that phase planes are a double ring, resulting in perioddoubling bifurcation.
When Iref = 2.6A, the Fig. 7(i) shows that parameter i is irregular, the phase plane forms many uncountable tori, and the system enters chaos.
Figure 8 shows the Poincaré map under chaos. Obviously, the Poincaré section is a dense point set with layers.
The discrete iterative model of the buck converter [29] is adopted to draw the bifurcation diagram, shown in Fig. 9, it can be seen that bifurcation occurs when Iref = 0.814 under the resistance load. Period1 to period2 firstly, then to period4, and the system enters chaos finally. Comparing with Fig. 5, it can be concluded that the memristive load produces a perioddoubling bifurcation that is consistent with the resistive load, and the addition of the memristive load delays the bifurcation point of the reference current as the bifurcation parameter.
Simulation under ramp compensation
Ramp compensation can effectively suppress chaos in currentcontrolled PWM converters [30]. Figure 10(a) shows that the inductance current waveform without ramp compensation. Let D =m_{d}/(m_{d}m_{r}), referring to Fig. 10(a), when the duty ratio D > 0.5, the falling slope m_{d} is greater than the rising slope m_{r} [31], the converter disturbance current Δi_{L}' increases, thereby causing system instability.
Introducing the slope compensation method, let m be the slope of the ramp compensation, as shown in Fig. 10(b), the disturbance current Δi_{L}' has been reduced. Let ΔiL' = Δi_{L}(mm_{d})/(mm_{r}), when ΔiL' < ΔiL, the system is stable, it must meet:
The relationship between the compensated current Icon and the reference current Iref is:
The PCMC buck converter with ramp compensation is shown in Fig. 11. Choosing a fixed slope m =  1500, and also choosing Iref = 2.6A, it can be clearly observed that there is no chaos in the converter at this time, and the inductance current i is stabilized at the period1 in Fig. 12, indicating that chaos has been suppressed.
PSIM circuit simulation
The circuit diagram for the PCMC buck converter is constructed by using the PSIM simulation software, shown in Fig. 13.
The selection of component parameters is consistent with the numerical simulation in Table 1. Comparing the simulation waveform in Fig. 14 with that in Fig. 6(b), (h), and (k), it can be seen that the phase portrait are basically the same, and the error does not exceed 0.1, which proves the effectiveness of the model.
Select Iref = 2.6A and introduce ramp compensation after 7 ms. It is observed from Fig. 15 that the inductance current is gradually suppressed from chaos to stable, and it has been stabilized at 10 ms. The output voltage changes from a large oscillation to a small amplitude change after the ramp compensation is introduced. The added ramp compensation effectively suppresses the chaotic behavior of the PCMC buck converter with the memristive load.
Conclusion
This paper focuses on the nonlinear behavior of the PCMC buck converter with the memristive load. Through the numerical analysis of its state equations, bifurcation diagrams, phase portraits, and Poincaré mappings have been made. It is found that as the bifurcation parameter (Iref) increasing, the system exhibits productive nonlinear behaviors such as perioddoubling bifurcation, periodic motion, and chaos. Compared with the buck converter with the resistive load, it is found that the memristive load does not affect the bifurcation structure, and can delay the bifurcation point. Secondly, the introduced ramp compensations suppress the chaotic system to period1, so that the stability of the system is improved. Finally, the correctness of theoretical and numerical analysis is verified by PSIM simulation. This research provides parameters choosing for buck switching power supply design in practical applications. The explored method may be extended to other power electronic systems including supercapacitors or batteries [32,33,34,35] and nonlinear circuits [36,37,38].
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (No. 61873138), and in part by the Taishan Scholar Project Fund of Shandong Province of China.
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Zhu, B., Fan, Q., Li, G. et al. Chaos suppression for a Buck converter with the memristive load. Analog Integr Circ Sig Process (2021). https://doi.org/10.1007/s1047002101799x
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Keywords
 Buck converter
 Memristive load
 Ramp compensation
 Chaos