Study on Nonlinear Phenomena in Buck-Boost Converter with Switched-Inductor Structure

Abstract

The switched-inductor structure can be inserted into a traditional Buck-Boost converter to get a high voltage conversion ratio. Nonlinear phenomena may occur in this new converter, which might well lead the system to be unstable. In this paper, a discrete iterated mapping model is established when the new Buck-Boost converter is working at continuous conduction current-controlled mode. On the basis of the discrete model, the bifurcation diagrams and Poincare sections are drawn and then used to analyze the effects of the circuit parameters on the performances. It can be seen clearly that various kinds of nonlinear phenomena are easy to occur in this new converter, including period-doubling bifurcation, border collision bifurcation, tangent bifurcation, and intermittent chaos. Value range of the circuit parameters that may cause bifurcations and chaos are also discussed. Finally, the time-domain waveforms, phase portraits, and power spectrum are obtained by using Matlab/Simulink, which validates the theoretical analysis results.

1. Introduction

In recent years, much attention was paid to the switched-inductor structure because of its several unique characteristics. One of the most remarkable feature of the switched-inductor structure is that it can be combined with the traditional DC-DC converters to provide new converters with a steep voltage conversion ratio [1, 2]. The DC-DC converter is strongly nonlinear system, and it has rich irregular phenomena like subharmonic oscillations, time-bifurcation, quasi-periodicity, chaos, and so on [3–8].

The nonlinear phenomena in the new Buck-Boost converter may be rather complex than that in the traditional converters, which will deteriorate the performance of the converter to some extent. The bifurcation and the chaos in traditional DC-DC converters were studied extensively [9–14]. It was demonstrated that there was various kinds of nonlinear phenomena in the traditional Buck-Boost converter [15–17]. However, very little attention was paid to complex behaviors in Buck-Boost converter combined with the switched-inductor structure.

For this reason, the nonlinear phenomena in a current-mode controlled Buck-Boost converter with switched-inductor structure are studied in this paper. The discrete iterated mapping model under continuous conduction mode (CCM) is established. The bifurcation diagrams and Poincare sections are drawn based on the discrete-time model, which intuitively reflect the influence of the circuit parameters on the system performance. It can be shown clearly that many kinds of nonlinear phenomena existing in this new converter, such as period-doubling bifurcation, border collision bifurcation, tangent bifurcation, and intermittent chaos. At last, the time-domain waveform, phase portraits, and power spectrum under various load resistance are obtained by Matlab/Simulink, which validates the conclusions that came from the theoretical analysis. The research result provides an important reference for engineering design and performance analysis.

2. Operation Principle and Modeling of the Converter

2.1. Operation Principle

The schematic of the new current-mode controlled Buck-Boost converter with switched-inductor structure is shown in Figure 1. In general, the value of 𝐿₁ equals 𝐿₂ in this 𝐿-switching structure according to [1], and all elements are assumed to be ideal ones. Let current flowing through 𝐿₁ (i.e., 𝑖_𝐿1) be the control signal, operation process of this system is described as follows. A feedback path composed of an RS trigger and a comparator was used to control switch 𝑆. In the comparator, the current 𝑖_𝐿1 is compared with a reference current 𝐼_ref to generate on-off signal. At the beginning of a cycle, switch 𝑆 is turned on. It remains on state so that the 𝑖_𝐿1 increases, until it reaches the value 𝐼_ref. Then, the switch 𝑆 is turned off, and it remains off state until the next cycle begins. Thus, the converter toggles between two linear configurations when it operates in CCM. The switching topologies are shown in Figure 2.

Figure 1 

Current-mode controlled buck-boost converter with switched-inductor structure.

(a)

(b)

Figure 2 

Switching topologies of the converter. (a) Switch is on; (b) switch is off.

As shown in Figure 2, 𝐿₁ and 𝐿₂ are being charged in parallel when the switch is on; while they are being discharged in series when the switch is off. Two parameters are defined as follows: 𝐿_eq is denoted as the equivalent inductor of 𝐿₁ and 𝐿₂, and 𝑖_eq is denoted as the current that flows through the 𝐿_eq. And an assumption is made that the switch 𝑆 is on during [𝑡₀,𝑡₁] and is off during [𝑡₁,𝑡₂]. Therefore, both of these two configurations can be characterized by a dynamic matrix and an input matrix, say 𝐴₁, 𝐴₂ and 𝐵₁, 𝐵₂, respectively.

Let 𝑥=[𝑖_eq𝑢_𝐶]^𝑇 be a vector of state variables, the differential equation of the configuration can be written as follows during [𝑡₀,𝑡₁]: 𝑥̇=𝐴₁𝑥+𝐵₁𝐸,𝑡₀≤𝑡≤𝑡₁, (1)

where 𝐴₁=1000−𝑅𝐶,𝐵₁=1𝐿_eq0. (2)

Similarly, the differential equation can be written as follows during [𝑡₁,𝑡₂]: 𝑥̇=𝐴₂𝑥+𝐵₂𝐸,𝑡₁≤𝑡≤𝑡₂, (3)

where 𝐴₂=10−𝐿_eq1𝐶−1𝑅𝐶,𝐵₂=00. (4)

Noting that there is a conversion between 𝐿₁ and 𝐿₂ from parallel connection to series connection, the value of 𝐿_eq and 𝑖_eq in (1) and (3) are not equal. Taking 𝐿₁=𝐿₂ into consideration, we can get 𝐿_eq and 𝑖_eq in (1) as follows: 𝐿_eq=𝐿₁×𝐿₂𝐿₁+𝐿₂=12𝐿₁,𝑖_eq=𝑖_𝐿1+𝑖_𝐿2=2𝑖_𝐿1. (5)

Also, we can get 𝐿_eq and 𝑖_eq in (3) as follows: 𝐿_eq=𝐿₁+𝐿₂=2𝐿₁,𝑖_eq=𝑖_𝐿1=𝑖_𝐿2. (6)

Considering that 𝑖_𝐿1 is the control signal, we redefine 𝑥^′=[𝑖_L1𝑢_𝐶]^𝑇 as a new vector in order to simplify the derivation of the discrete iterated mapping model. And the corresponding differential equations can be obtained according to (1)–(6), as shown by 𝑥̇^′=𝐴^′₁𝑥^′+𝐵^′₁𝐸,𝑡₀≤𝑡≤𝑡₁,𝑥̇^′=𝐴^′₂𝑥^′+𝐵^′₂𝐸,𝑡₁≤𝑡≤𝑡₂, (7)(8)

where 𝐴^′₁=⁰⁰_{0−1/𝑅𝐶}, 𝐴^′₂=^{0−1/(2𝐿1)}_{1/𝐶−1/𝑅𝐶}, 𝐵^′₁=^1/𝐿1₀, and 𝐵^′₂=⁰₀.

According to the control theory, solutions of (7) and (8) can be written as 𝑥^′𝑡₁=𝑓₁𝑥^′𝑡₀,𝑡₁=𝑒^{𝐴′1(𝑡1−𝑡0)}𝑥^′𝑡₀+^𝑡1_𝑡0𝑒^{𝐴′1(𝑡0−𝜏)}𝐵^′₁𝐸𝑑𝜏=𝜙₁𝑡₁−𝑡₀𝑥^′𝑡₀+^𝑡1_𝑡0𝜙₁𝑡₀−𝜏𝐵^′₁,𝑥𝐸𝑑𝜏_′𝑡₂=𝑓₂𝑥^′𝑡₁,𝑡₂=𝑒^{𝐴′2(𝑡2−𝑡1)}𝑥^′𝑡₁+^𝑡2_𝑡1𝑒^{𝐴′2(𝑡1−𝜏)}𝐵^′₂𝐸𝑑𝜏=𝜙₂𝑡₂−𝑡₁𝑥^′𝑡₁+^𝑡2_𝑡1𝜙₂𝑡₁−𝜏𝐵^′₂𝐸𝑑𝜏, (9)

where 𝜙_𝑖(𝑡)=𝑒^{𝐴′𝑖𝑡}, 𝑖=1,2; 𝜙_𝑖(𝑡) is state transition matrix of 𝐴^′_𝑖.

2.2. Discrete Iterated Mapping Model

In general, the discrete time model can be established by means of stroboscopic map, S-switching map, or A-switching map. The stroboscopic map was widely used to analyze the nonlinear phenomena in the switching converters [18–20]. In this work, the discrete time model of the Buck-Boost converter with switched-inductor is obtained by using the stroboscopic map.

Let 𝑇 be the switching period of the switch 𝑆, 𝑖_1,𝑛 be the instantaneous value of 𝑖_𝐿1 at the beginning of 𝑛th cycle 𝑡=𝑛𝑇, and 𝑢_𝑛 be the instantaneous value of 𝑢_𝐶, then the solution of the state equation (7) can be given as 𝑖_𝐿1(𝑡)=𝑖_1,𝑛+𝐸𝑡𝐿₁,𝑢_𝐶(𝑡)=𝑢_𝑛𝑒^2𝑘𝑡, (10)

where 𝑘=−1/2𝑅𝐶.

Also, when the converter is working during “off” period, the solution of 𝑖_𝐿1 and 𝑢_𝐶 in state equation (8) can be given as 𝑖_𝐿1(𝑡,)=𝑒^𝑘𝑡𝑐₁cos𝑤𝑡+𝑐₂,𝑢sin𝑤𝑡_𝐶(𝑡)=−𝐿₁+𝐿₂𝑒^𝑘𝑡×𝑘𝑐₁+𝑐₂𝑤cos𝑤𝑡+𝑘𝑐₂−𝑐₁𝑤sin𝑤𝑡, (11)

where 1𝑅>2𝐿₁+𝐿₂𝐶,𝑐₁=𝐼_ref𝑤=1𝐿₁+𝐿₂𝐶−(−𝑘)², (12)

and 𝑐₂=1𝑤−𝑢_𝑛𝑒^{2𝑘𝑡𝑛}𝐿₁+𝐿₂−𝑘𝑐₁. (13)

Based on the operating principle of the converter, the switching condition of the system can be defined as 𝐼_ref−𝑖_𝐿1𝑡_𝑛=0. (14)

According to (10) and (14), the on-time 𝑡_𝑛 can be obtained by 𝑡_𝑛=𝐿₁×𝐼_ref−𝑖_1,𝑛𝐸. (15)

If 𝑡_𝑛≥𝑇 and the converter works in “on” state during the whole 𝑛𝑇∼(𝑛+1)𝑇, the values of 𝑢_𝑛 and 𝑖_1,𝑛 at the next clock instant are given by 𝑖_1,𝑛+1=𝑖_1,𝑛+𝐸𝑇𝐿₁,𝑢_𝑛+1=𝑢_𝑛𝑒^2𝑘𝑇. (16)

If 𝑡_𝑛<𝑇, the converter works in “on” state during 𝑛𝑇∼(𝑛𝑇+𝑡_𝑛) and “off” state during (𝑛𝑇+𝑡_𝑛)∼(𝑛+1)𝑇, the mapping are given by 𝑖_1,𝑛+1=𝑒^𝑘𝑡𝑚𝑐₁cos𝑤𝑡_𝑚+𝑐₂sin𝑤𝑡_𝑚,𝑢_𝑛+1=−𝐿₁+𝐿₂𝑒^𝑘𝑡𝑚×𝑘𝑐₁+𝑐₂𝑤cos𝑤𝑡_𝑚+𝑘𝑐₂−𝑐₁𝑤sin𝑤𝑡_𝑚, (17)

where 𝑡_𝑚 is the off-time, and it can be calculated by 𝑡_𝑚𝑡=𝑇×1−^𝑛𝑇mod1. (18)

2.3. Voltage Conversion Ratio of the System

Let 𝑈_𝑖 and 𝑈_𝑜 be value of input voltage and output voltage at steady-state, respectively. Ignoring the resistance of the inductors and taking 𝐿₁=𝐿₂ into consideration, the voltage drop on 𝐿₁ during the “on” period can be written as 𝑈_𝐿1=𝑈_𝑖. (19)

The voltage drop on 𝐿₁ during the “off” period can be written as 2𝑈_𝐿1=−𝑈_𝑜. (20)

Thus, a voltage-second balance on 𝐿₁ can be written as 𝐷⋅𝑈_𝑖+()⋅−11−𝐷2𝑈_𝑜=0 (21)

giving 𝑈_𝑜=2𝐷𝑈1−𝐷^𝑖, (22)

where 𝐷 is the duty cycle at steady-state.

Similarly, for a classical Buck-Boost converter, we can easily get that 𝑈_𝑜=𝐷𝑈1−𝐷^𝑖. (23)

Comparing (22) with (23), it can be seen that the voltage conversion ratio of the new system is double that of the classical one.

3. Analysis of Dynamic Characteristics

For the power converters, many kinds of methods like phase portraits, bifurcation diagrams, and time-domain waveform can be used to analyze the nonlinear phenomenon in the system. In this work, the bifurcation diagrams and Poincare sections are drawn based on the discrete iterated mapping model. Time-domain waveform, phase portraits, and power spectrum are obtained by building simulation module in Matlab/Simulink, which validates the theoretical analysis results.

3.1. Bifurcation Diagrams

It can be seen clearly from Figure 1 that there are six major parameters in the system: the reference current 𝐼_ref, inductance 𝐿₁ and 𝐿₂, load resistance 𝑅, input voltage 𝐸, and capacitance 𝐶. It is impossible to draw all the bifurcation diagrams for all possible parameter values and analyze them in detail in just one paper, so we present only a few typical cases. In order to facilitate the analysis of the complex dynamic behavior of the converter, (24) is given to calculate the borderline of 𝑖_𝐿1, which is denoted by 𝐼_𝑏. Consider the following: 𝐼_𝑏=𝐼_ref−𝐸𝑇𝐿₁. (24)

Let reference current 𝐼_ref vary from 2 A to 13 A with a step of 0.02 A while other parameters are fixed at 𝐸=24 V, 𝐿₁=1 mH, 𝐿₂=1 mH, 𝐶=4 μF, 𝑅=20 Ω, and 𝑓=10 KHz. Figure 3(a) shows the bifurcation diagram with 𝐼_ref as parameter, the dotted line in Figure 3(a) is the borderline of 𝑖_𝐿1 with the value of 𝐼_𝑏=𝐼_ref−2.4 according to (24). This diagram shows clearly period doubling bifurcations at 𝐼_ref=3.25 A. When 𝐼_ref is equal to 5.25 A, the period-2 orbit collides with the borderline and turns to 4-periodic. Therefore, this period doubling phenomenon is not due to a standard pitchfork bifurcation, but because of a border collision bifurcation. The period-4 orbit goes through a normal period doubling bifurcation at 𝐼_ref=6 A and a period-8 orbit is created. Then, this orbit collides with the borderline again and turns into a chaotic orbit. Note that the chaotic state degenerate to 3-periodic when 𝐼_ref is nearly equal to 11.5 A. This special phenomenon is called tangent bifurcation, and the chaos that happened before tangent bifurcation is called intermittent chaos.

(a)

(b)

(c)

Figure 3 

Bifurcation diagram with different circuit parameters. (a) 𝐼ref as parameter; (b) 𝑅 as parameter; (c) 𝐸 as parameter.

The bifurcation diagram with load resistance 𝑅 as parameter is shown in Figure 3(b). The load resistance 𝑅 is varied from 10 Ω to 55 Ω with a step of 0.05 Ω while other parameters are fixed at 𝐸=24  V, 𝐿₁=1 mH, 𝐿₂=1 mH, 𝐶=4 μF, 𝐼_ref=4 A, and 𝑓=10 KHz. Since the borderline is independent of 𝑅 according to (24),it is shown as a horizontal straight line with 𝐼_𝑏=1.6 A in this figure. And it can be seen clearly that the period doubling bifurcation occurs at 𝑅=14 Ω, then the period-2 orbit collides with the borderline at 𝑅=30 Ω and a stable period-4 orbit is generated. The period-4 orbit goes through a normal period doubling bifurcation at 𝑅=39 Ω and a period-8 orbit is created, which collides with the borderline at 𝑅=40.5 Ω and finally turns into a chaotic orbit. It can be noted that there are no periodic windows within the chaotic region.

The bifurcation diagram with input voltage 𝐸 as parameter is shown in Figure 3(c). The input voltage 𝐸 is varied from 5 V to 30 V with a step of 0.05 V while other parameters are fixed at 𝑅=20 Ω, 𝐿₁=1 mH, 𝐿₂=1 mH, 𝐶=4 μF, 𝐼_ref=4 A, and 𝑓=10 KHz. Thus, the borderline of 𝑖_𝐿1 can be obtained by 𝐼_𝑏=4−0.1 𝐸 according to (24). It can be seen clearly from Figure 3(c) that with the change of the input voltage 𝐸, its route to chaos is similar with Figure 3(a), including period-doubling bifurcation, border collision bifurcation, more than once tangent bifurcation, and intermittent chaos. At the same time, the diagram reveals that the bifurcation occurs with the decrease of 𝐸, which is opposite to the change of 𝐼_ref and 𝑅. And it is worth noting that the 3-periodic window is the most obvious periodic window in the intermittent chaotic region.

3.2. Poincare Sections

Poincare sections can be an effective method to analyze the dynamic characteristics of nonlinear systems. According to nonlinear dynamics theory, the performance of the system can be judged by observing the number of the cutoff points on the Poincare sections. One point or a few discrete points indicates that the converter is working in periodical state, while the cutoff point with a fractal structure reveals a chaotic behavior.

To prove that when the reference current 𝐼_ref changes from 2 A to 13 A, the converter exists; the nonlinear phenomena shown is in Figure 3(a), Figure 4 gives the Poincare sections under different value of 𝐼_ref.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 4 

The Poincare sections under different value of reference current. (a) 𝐼ref=2 A; (b) 𝐼ref=4  A; (c) 𝐼ref=5.8 A; (d) 𝐼ref=6.1 A; (e) 𝐼ref=6.7 A; (f) 𝐼ref=9 A; (g) 𝐼ref=10.5 A; (h) 𝐼ref=11.1 A.

It can be seen clearly from Figures 4(a)–4(d) that the points on the Poincare sections are discrete when 𝐼_ref is equal to 2 A, 4 A, 5.8 A, and 6.1 A, respectively. The converter is working in period-1, period-2, period-4, and period-8 successively according to the number of the points. For Figures 4(e) and 4(f), the cutoff points act as a banded structure, indicating that the converter is operating in chaotic state when 𝐼_ref is equal to 6.7 A or 9 A. Figures 4(g) and 4(h) show that there is a 3-periodic window in the intermittent chaotic region, and the 3-periodic orbit turns to period-6 before it comes into chaos. The above consequences are coinciding with the results shown in Figure 3(a). Therefore, the Poincare sections further prove the correctness of the discrete-time mapping model and reflect the operating state of the converter under different value of 𝐼_ref more vividly.

3.3. Time-Domain Waveform and Phase Portraits

The bifurcation diagram in Figure 3(b) shows that the converter goes through period-1, period-2, period-4, and period-8 and finally turns into intermittent chaos when the load resistance increases from 10 Ω to 55 Ω. In order to prove the existence of the above phenomena, we set up a simulation model by Matlab/Simulink in this section according to Figure 1, and we select different value of 𝑅 in this simulation model; the corresponding time-domain waveforms and phase portraits are shown in Figures 5, 6, 7, 8, and 9.

(a)

(b)

Figure 5 

The time-domain waveform and phase portrait when 𝑅=10 Ω. (a) Time-domain waveform; (b) phase portrait.

(a)

(b)

Figure 6 

The time-domain waveform and phase portrait when 𝑅=20 Ω. (a) Time-domain waveform; (b) phase portrait.

(a)

(b)

Figure 7 

The time-domain waveform and phase portrait when 𝑅=35 Ω. (a) Time-domain waveform; (b) phase portrait.

(a)

(b)

(c)

Figure 8 

The time-domain waveform and phase portrait when 𝑅=39.5 Ω. (a) Time-domain waveform; (b) phase portrait; (c) the partially enlarged view of phase portrait.

(a)

(b)

Figure 9 

The time-domain waveform and phase portrait when 𝑅=55 Ω. (a) Time-domain waveform; (b) phase portrait.

From Figures 5–8, we can easily find that when the load resistance is equal to 10 Ω, 20 Ω, 35 Ω, and 39.5 Ω, the state of the system is corresponding to period-1, period-2, period-4, and period-8, respectively. Note that the phase portrait in Figure 8(b) cannot reflect the periodicity of the converter clearly, so we give the enlarged view of it in Figure 8(c) in order to illustrate the number of the cycles better.

The time-domain waveform and phase portrait when the load resistance is equal to 55 Ω are shown in Figure 9. It can be observed that the time-domain waveform appears to be random, and there is a strange attractor in the phase portrait, which indicates that the converter is working in chaotic state.

According to the time-domain waveforms and phase portraits from Figure 5 to Figure 9, it can be seen clearly that the conclusions obtained by Matlab/Simulink are in good agreement with the results got by bifurcation diagram, which confirms the validity of the theoretical analysis better.

3.4. Power Spectrum

The power spectrum is also an effective way to analyze the stability of the circuit. When the circuit operates in the cycle 𝑛, the peak of the waveform will only appear when the frequency is equal to 𝑖/𝑛 times of the operating frequency (𝑖 is a positive integer). When the circuit is in a chaotic state, the power spectrum manifests as a continuous spectrum, which contains a peak corresponding to the periodic motion. Figures 10(a)–10(d) clearly reveal that when the load resistance is equal to 10 Ω, 20 Ω, 35 Ω, and 55 Ω, the converter works in period-1, period-2, period-4, and chaotic state, respectively. The results are in good agreement with the previous analysis, which further verify the correctness of the results obtained by theoretical analysis.

(a)

(b)

(c)

(d)

Figure 10 

The power spectrum under different value of load resistance. (a) 𝑅=10 Ω; (b) 𝑅=20 Ω; (c) 𝑅=35 Ω; (d) 𝑅=55 Ω.

4. Conclusion

Inserting the switched-inductor structure into traditional Buck-Boost converter can increase its voltage conversion ratio significantly. However, it will lead to a conversion of the inductors from series connection to parallel connection, which makes it easier for the system to be affected by the circuit parameters. In this paper, the nonlinear behaviors of the Buck-Boost converter with switched-inductor structure have been studied systematically. The discrete iterated mapping model under CCM is established. The effects of circuit parameters on system performance are analyzed using bifurcation diagrams and Poincare sections. Then, the corresponding time-domain waveforms, phase portraits, and power spectrum are obtained by Matlab/Simulink. The research results show that various kinds of nonlinear phenomena are easy to come in this new converter, including period-doubling bifurcation, border collision bifurcation, tangent bifurcation, and intermittent chaos. The results between the theoretical analysis and the simulation are in general agreement with each other. According to the analysis above, the Buck-Boost converter with switched-inductor structure belongs to strongly nonlinear system, and its performance can be easily affected when the circuit parameters are varied. Therefore, the parameters should be chosen appropriately according to the research results. The analysis methods and research findings will possess an important reference value to engineering design and performance analysis.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (no. 51107016), the National Key Basic Research Program of China (973 Program) under Grant (2013CB035605), and the Postdoctoral science-research developmental foundation of Heilongjiang province (no. LHB-Q12086).