Dawei Ding, Zongzhi Li and Nian Wang

(School of Electronics and Information Engineering,Anhui University,Hefei 230601,China)

Abstract:According to the fact that the actual inductor and actual capacitor are fractional, the mathematical and state-space averaging models of fractional order Buck converters in continuous conduction mode(CCM) are constructed by using fractional calculus theory. Firstly, the parameter conditions that ensure that the converter working in CCM is given and transfer functions are derived. Also, the inductor current and the output voltage are analyzed. Then the difference between the mathematical model and the circuit model are analyzed, and the effect of fractional order is studied by comparing the integer order with fractional order model. Finally, the dynamic behavior of the current-controlled Buck converter is investigated. Simulation experiments are achieved via the use of Matlab/Simulink. The experimental results verify the correctness of theoretical analysis, the order should be taken as a significant parameter. When the order is taken as a bifurcation parameter, the dynamic behavior of the converter will be affected and bifurcation points will be changed as order varies.

Keywords:Buck converter; fractional order; continuous conduction mode(CCM); modeling; dynamic analysis; bifurcation

1 Introduction

The Buck converter is an important component of a DC-DC converter, its working principle is different from a Boost converter because it has the ability to reduce direct-current (DC) voltage, and has been extensively used in many applications, such as power electronic technology and computer technology[1]. Modeling is a significant procedure for devising converters to meet actual demands, for example, the accuracy of the model has a crucial influence on capability of the final design. Thus a lot of effort has been put into establishing a proper model and find some better analysis methods. In order to research Buck converters, some appropriate models and analysis methods have been proposed in recent years[1-8]. For instance, it is assumed that its characteristic frequency is much smaller than its switching frequency, and the averaged model, which is used to derive the small signal model, can be obtained by averaging the circuit variable during the switching period. This is also helpful to study the dynamic characteristics of Buck converter in the low frequency domain[1-4]. In addition, the averaged model can not be used to analyze the dynamic characteristics of high frequency region, so the discrete model has been proposed to analyze the overall dynamic behavior. This model is mainly used to solve the differential equations that are different under different operational conditions, and then circuit variables will be sampled and collected at a certain moment. Researchers usually use the discrete model to identify nonlinear behavior and complete the accurate simulation of the Buck converter[5-8]. All the above research results are a good reference for researchers to design an accurate Buck converter. However, the above findings gotten under this assumption is that both actual inductor and actual capacitor are integer order[1-8].

With the rapid development of integer calculus and its wide range of applications[9-11], researchers gradually turned their attention to fractional calculus applications, and a lot of meaningful results have been obtained[12-20]. Most noteworthy, the research results of actual inductor and actual capacitor prove that they are all actually fractional order and must be analyzed by means of fractional correlation theory[15-20]. For instance, an actual inductor was proved actually fractional order by Westerlund and he also constructed its fractional order model[15]. The order of fractional order capacitor were also measured and obtained under different dielectrics[16]. It was pointed out the integer order capacitor cannot exist in reality since its impedance modality would go against mathematical principles[17]. The fractional Chua’s circuit was successfully realized by using the fractional order model of inductor and capacitor, and the rationality of theoretical analysis were indicated[18]. Fractional order capacitors with different orders were also developed by using diverse fractal structures[19]. It was pointed out that an inductor with a different order would be produced because of skin effect shown by Tenreiro Machado[20]. So, the fractional order model of inductor and capacitor should be established to research true electrical characteristics. So, a problem was subsequently produced: why was it possible to approximate the actual circuit by using the data of integer order model? A primary reason for this problem is that the fractional order is approximately equal to 1, and accordingly their dynamic behavior can be approximately described in terms of the integer order model. This approximation will cause error results if the fractional order is a little far from 1, such as the capacitor with order of 0.59 and inductor with different order[19-20]. Therefore, a fractional order model of the Buck converter should be constructed. So far, the research on fractional order systems is becoming more and more mature, and a lot of research results on fractional order modeling of DC-DC converters have been generated[21-27]. For instance, the fractional order Buck-Boost DC/DC converter were constructed by using the state average modeling method, and the following two modes were also included: pseudo continuous conduction mode(PCCM) and discontinuous conduction mode(DCM)[22-24]. The fractional order modeling problem of the Boost converter was also solved by the same method[25-27]. The above research results and methods are also useful for modeling of Buck converter.

The contents are as follows.Some basic fractional theories on fractional order inductors and capacitors are given in Section 2. The fractional order state-space averaging model of a Buck converter in CCM is established and the transfer functions are provided in Section 3. The parameter condition that ensures converter work in CCM is also given. Both numerical simulation and circuit experiments are completed to verify the theoretical analysis, and some dynamic behaviors of the current-controlled Buck converter are investigated in Section 4. Finally, some conclusions are given.

2 Fundamental Theory

2.1 Fractional Calculus

(1)

whereγ(γ∈) is the order,aandtrepresent the bounds.

With respect to fractional calculus,there are three main definitions: Grunwald-Letnikov(GL) definition, Riemann-Liouville (RL) definition, and Caputo definition[22].

Caputo definition is:

(2)

where Γ( ) is the Gamma function andn∈Nis the first integer which is not less thanγ,n-1<γ

It meets the following condition by means of Laplace transform:

(3)

It satisfies the following form for zero initial condition by means of Laplace transform:

(4)

Theoretical calculations are implemented based on the Caputo derivative and the more specific initial conditionx(0). Hartley and Lorenzo discuss the error arising from using the Laplace transform of Caputo derivative[28]. It is now pretty well established that the initial state of a fractional differential equation can be accurately represented by the weighted integral ofz(ω,0)[29]. Nevertheless, for convenience, the Caputo derivative and the initial conditionx(0) are adopted on the basis of the acceptable error.

2.2 The Fractional Order Model of Capacitor and Inductor

Westerlund[15]proposed a new theory about capacitor based on fractional calculus in 1994, the current corresponding to the input voltage is:

(5)

whereCfandβrepresents the capacitance and the order, and they are related to the type of dielectric and the loss of the capacitor respectively. Capacitor materials were also provided by Westerlund.

Westerlund[16]also created a fractional order model of inductor, which has the following form:

(6)

whereLfandαrepresents the inductance and the order of inductor.

3 Modeling of Fractional Order Buck Converter

3.1 The Mathematical Model

The circuit structure is given in Fig.1(a). It works as follows:Pwis a period signal, when a high levelPwappears, switchSis closed, there is no conductive diode, this mode will last for a time interval (0, dT),dandTrepresent the duty cycle and switching period respectively. Then when a low levelPwappears, switchSis off, there is a conductive diode, this mode will last for a time interval (dT,T).iL,v0, andvinrepresent the inductor current, output voltage and input voltage respectively. Due to the Buck converter operating in CCM, there are only two switching modes, and the time domain waveform of its inductance current is given in Fig.1(b).The state equations are as follows.

WhenSis closed,

(7)

WhenSis off,

(8)

So the mathematical model of the fractional order Buck converter described by Eq.(7) and (8) is affected byαandβ.

(a) Circuit diagram

(b) Time domain diagram

3.2 State-Space Averaging Model

Due to the distinctive working characteristics, circuit variables, such as inductor currentiLand output voltagev0, have been affected by the high frequency switching harmonics. To eliminate these effects, a circuit variable should be averaged over one switching period, i.e.,

(9)

wherexrepresent a random circuit variable.

According to the properties of fractional calculus, its fractional-order form is obtained:

(10)

whereγis the order and 0<γ<1.

Above all, state-space averaging model of fractional order Buck converter in CCM can be created.

Average values 〈iL〉, 〈v0〉, 〈vin〉 anddcan be defined as the following forms:

(11)

State-space averaging model is as follows:

(12)

Eq.(12) can be rewritten:

(13)

DC components in Eq.(13) are as follows:

(14)

The quiescent operation point is as follows:

(15)

The voltage ratio is defined as follows:

(16)

AC components in Eq.(13) are as follows:

(17)

(18)

Using Caputo fractional calculus definition and treating the input voltageVinas a constant, then the inductor current rippleΔiLin (0,DT) can be obtained as

(19)

where Γ( ) is the Gamma function. The duty cycleD, the input voltageVin, the switching periodT, the inductanceL, and the order of inductorα, above all of these will affect the inductor current rippleΔiL. Whenαis increased,ΔiLis reduced.

The maximum and minimum values of inductor current are

(20)

Using Caputo definition and Adomian decomposition method, then the output voltage rippleΔv0can be obtained as

(21)

whereEβ( ) is the Mittag-Leffler function andVOTis the value of output voltage when the switch is on.

The approximate expression ofVOTis

(22)

So,

(23)

Substituting Eq.(15) and (23) into inequality(21), the following inequality is obtained:

(24)

The duty cycleD, the input voltageVin, the switching periodT, the capacitorC, load resistanceR, and the order of capacitorβ, above all of these will affect the output voltage rippleΔv0. Whenβis increased,Δv0is reduced.

To keep the Buck converter operating in CCM, the inductor current must be greater than 0, i.e.,

(25)

Substituting Eq.(15) and (19) into inequality(25), the following inequality is obtained:

(26)

To ensure the Buck converter operates in CCM, the inductanceL, load resistanceR, the switching periodT, duty cycleD, and the order of inductorαshould be all considered. Working conditions of inductorLand load resistanceRcan be clearly gained. It is also obtained: when the order of inductorαbecomes larger, the Buck converter is easy to operate in CCM. Particularly, whenα=1, Eq.(26) is the same as the result of the integer order model.

According to Eq.(18),the Laplace transform of converter equations under the condition of zero initial value is obtained as follows:

(27)

(28)

(29)

(30)

(31)

It is easy to find orders of the inductor and capacitor that have an effect on transfer functions. That means fractional orders can be considered as extra parameters. Integer order transfer functions can be derived when orders are 1, and integer order model can be seen as a special case of fractional order model.

4 Experimental Research and Dynamic Analysis

The theoretical analysis is verified by the numerical and circuit simulation experiments, and dynamic analysis is used to research how the order affects dynamic behavior of a current-mode controlled fractional order Buck converter.

4.1 Approximation Circuit Models of Fractional Order Capacitor and Inductor

Approximation circuit models of capacitor and inductor are obtained based on analog fractance circuit and the improved Oustaloup approximation algorithm[26-27].Fractional order circuit elements are approximately achieved by choosing the chain structure as the fractance unit. The approximation circuit models are shown in Fig.2(a) and (b).

(a) Fractional order inductor

(b) Fractional order capacitor

WhenL=3 mH,α=0.8, the respective resistance values are:

RL1= 7.16 kΩ,RL2= 340.84 Ω

RL3= 34.25 Ω,RL4= 3.54 Ω

RL5= 367 mΩ,RL6= 38 mΩ

RL7= 4 mΩ,RL8= 0.4 mΩ

RL9= 42 μΩ,RL10= 5 μΩ

The respective inductance values are:

L1= 95 μH,L2= 77 μH,L3= 131.6 μH

L4= 231.6 μH,L5= 408 μH,L6= 719.4 μH

L7= 1.268 mH,L8= 2.235 mH,L9= 3.934 mH

WhenC=100 μH,β=0.8, the respective resistance values are:

RC1= 20 mΩ,RC2= 160 mΩ

RC3= 1.5 Ω,RC4= 14.6 Ω

RC5= 141 Ω,RC6= 1.36 kΩ

RC7= 13.131 kΩ,RC8= 126.742 kΩ

RC9=1.222 MΩ,RC10= 102.85 MΩ

The respective capacitance values are:

C1= 6.5 μF,C2= 13.98 μF

C3= 24.5 μF,C4= 43.2 μF

C5= 76.2 μF,C6= 134.2 μF

C7= 236.6 μF,C8= 417 μF

C9= 736 μF,C10= 560 μF

4.2 Simulation Results of Buck Converter

4.2.1SimulationresultsoffractionalorderBuckconverter

The mathematical model of fractional order Buck converter is built by the improved Oustaloup approximation algorithm in Fig.3(a). Fractional Ints-αis a fractional integral unit, and its internal structure is given[27]. The switching frequency is selected asf=25 kHz and the corresponding rotational frequency isω=2πf=1.57×105rad/s, other parameters are:ωh=2×105rad/s,ωb= 5×10-6rad/s,N=8. Simulation parameters are as follows:vin=20 V,L=3 mH,C=100 μF,d=0.6,f= 25 kHz,α=0.8,β=0.8.

Whenα=0.8,β=0.8, according to Eq. (26), the load resistance value of fractional order Buck converter operating in the critical state is obtained:R=41.634 5 Ω. WhenR<41.634 5 Ω, the converter works in CCM, so the load resistance value is chosen as:R=30 Ω. The circuit simulation model is shown in Fig.3(b).

The time domain waveforms of inductor currentiLand output voltagev0are shown in Fig.4. Clearly, the Buck converter works in CCM. The experimental and theoretical values are shown in Table 1. It can be found that results of numerical simulation and theoretical analysis are consistent, so the theoretical analysis is correct.

(a) The mathematical model

(b) The circuit model

Fig.3ModelsoffractionalorderBuckconverter

(a) Inductor currentiL

(b) Output voltagev0

Fig.4ThewaveformoffractionalorderBuckconverter

Table 1 Results of fractional order Buck converter

The comparison between the mathematical model and the circuit model is given in Fig.5. Due to non-ideal circuit elements, the experimental result of the circuit model is slightly different from the numerical simulation. The correctness of the theoretical analysis is further verified.

(a)Inductor currentiL

(b) Output voltagev0

Fig.5Comparisonresultsoffractionalordermathematicalmodelandcircuitmodel

4.2.2SimulationresultsofintegerorderBuckconverter

When the fractional integral element is changed into an integral unit of integral order, an integer order mathematical model and a circuit model can be obtained.

Whenα=1,β=1, the time domain waveforms are shown in Fig.6. Clearly, the Buck converter operates in CCM. The experimental and theoretical values of integer order model are shown in Table 2. It can be found that results of numerical simulation are equal to the theoretical analysis, so the correctness of the theoretical analysis is verified.

Table 2 Results of integer order Buck converter

(a) Inductor currentiL

(b) Output voltagev0

The comparison between the mathematical model and the circuit model is also given in Fig.7. The experimental result of the circuit model is almost the same as the numerical simulation, therefore the theoretical analysis is correct.

It can be obtained that ripples, the time of rise, the time of delay, the time of adjustment, the peak time and the overshoot in the dynamic response have changed greatly by comparing Fig.5 and Fig.7. The influence of the fractional order on the converter is obvious, so that the order should be considered as an extra parameter. This shows that if the integer order model is used to describe the converter approximately, an erroneous result will be obtained. In order to describe real dynamics characteristics of the Buck converter in CCM, a fractional order model should be used according to the fact that actual inductor and actual capacitor are both actually fractional.

4.3 Dynamic Analysis

Some dynamic behaviors of the current-controlled Buck converter will be analyzed. The purpose of this part is to discuss the effect of the fractional orderαandβon dynamic behavior. The circuit model is shown in Fig.8.

(a) Inductor currentiL

(b) Output voltagev0

Fig.7Comparisonresultsofintegerordermathematicalmodelandcircuitmodel

Fig.8 The current-controlled Buck converter

Reference currentIrefand input voltagevinare used as the bifurcation parameter. Simulation parameters are as follows:vin=3.3 V,L=4.7 μH,C=10 μF,R=1.2 Ω,d=0.6,f= 1 000 kHz. Whenα=1 andβ=1, the bifurcation diagram with reference currentIrefof integer order model is shown in Fig.9(a). Whenα=0.8 andβ=0.8, the bifurcation diagram with reference currentIrefof fractional order model is shown in Fig.9(b).

(a) The result of integer order converter

(b) The result of fractional order converter

As shown in Fig.9 (a) and the related enlarged diagrams in Fig.10 (a): whenIref<1.1 A, the converter operates in a stable period-1 state; whenIref=1.1 A, bifurcation occurs and the converter operates in period-2 state; untilIref=1.35 A and increaseIref, the converter gradually enter chaotic state. As shown in Fig.9 (b) and the related enlarged diagrams in Fig.10 (b): whenIref<1.47 A, the converter operates in a stable period-1 state; whenIref=1.47 A, bifurcation occurs and the converter operates in period-2 state; whenIref= 1.55 A, the converter operates in period-4 state; untilIref=1.6 A and increaseIref, the converter gradually enter chaotic state.

WhenIref= 1 A, the bifurcation diagrams with the input voltagevinare shown in Fig.11.

As shown in Fig.11(a) and the related enlarged diagrams in Fig.12(a): the converter initially operates in a chaotic state, whenvin=2.35 V, the converter enters the period-3 state; whenvin=2.4 V, inverse bifurcation occurs and the converter operates in period-2 state; whenvin=2.7 V, inverse bifurcation occurs again and the converter operates in a stable period-1 state. As shown in Fig.11 (b) and the related enlarged diagrams in Fig.12 (b): the converter initially operates in a chaotic state, whenvin=2.08 V, the converter enters the period-4 state; whenvin=2.13 V, inverse bifurcation occurs and the converter operates in period-2 state; whenvin=2.25 V, inverse bifurcation occurs again and the converter operates in a stable period-1 state.

(a) The result of integer order converter

(b) The result of fractional order converter

(a) The result of integer order converter

(b) The result of fractional order converter

(a) The result of integer order converter

(b) The result of fractional order converter

From the above results, they all go through double period bifurcation, finally resulting in chaos for varyingIrefwhether the fractional order or the integer order converter. However, they have different routes to chaos and bifurcation points are moved backward whenαandβvary. For the input voltagevin, their chaotic paths are almost identical, but bifurcation points are moved forward asαandβvaries. The fractional order is the direct cause of the difference in dynamic behavior.

5 Conclusions

The fractional order model can really reflect the real situation of Buck converter. The mathematical and state-space averaging models of fractional order Buck converter in CCM is established based on fractional calculus in this paper. The parameter condition that ensures the converter work in CCM and the transfer functions are given through theoretical analysis. Then numerical simulation results and circuit experimental results are compared, and the correctness of theoretical analysis is verified. Finally, its some dynamic behaviors are analyzed. Theoretical analysis and experimental results show that the inductor current, output voltage, ripples, the time of rise, the time of delay, the time of adjustment, the peak time, the overshoot and transfer functions are all affected by the fractional order, so the fractional order should be considered as an extra parameter. According to dynamic analysis, the bifurcation point will be changed and the converter has different chaotic paths as the order varying.