分数阶电容的串联与并联分析

分数阶电容的串联与并联分析 ANALYSIS OF FRACTIONAL ORDER CAPACITOR IN SERIES AND PARALLEL CONNECTIONS 5 WANG Faqiang, MA Xikui (State Key Laboratary of Electrical Insulation and Power Equipment, School of Electrical Engineering, Xi'an Jiaotong University, Xi'an 710049) Abstract: Based on fractional calculus and the fractional order nature of the capacitor, the models for the fractional order capacitor in series and parallel connections are established and analyzed. The influence of the varied fractional order on the characteristic of the fractional order capacitor in series 10 and parallel connections are given. Finally, the realization form of the fractional order capacitor is designed and the correctness of theoretical analysis is confirmed by using the circuit simulations from power electronics simulator (PSIM). Key words: Fractional order capacitor; series connection; parallel connection. 15 0 Introduction As an important circuit element, the capacitor has been widely used in the field of electrical and electronics engineering and attracts many researchers and engineers to concern it [1-2]. Especially, since the application of the capacitor in engineering is influenced by the precision of 20 its model seriously, modeling and analysis of the capacitor is a hot issue all the time and many good results have been obtained in the existing literatures, such as, how to obtain the equivalent circuit when the capacitor operates in low or high frequency region, how to derive the equivalent [1-2]. But, many good results capacitor when the capacitors are in series or parallel connections have been presented under the condition that the capacitor is integer order. However, the existing research results about the model for the capacitor show that the integer order model of the real 25 capacitor is inaccuracy because it is fractional order in nature [3-9]. For example, Westerlund et al pointed out that the real capacitor is fractional order and measured the order of the fractional order capacitor under different dielectric from the experiment, such as the fractional order of the capacitor is 0.9776 for polyvinylidenefluoride, 0.9821 for metalized paper, and 0.9978 for polycarbonate [3]. Jonscher and Bohannan also pointed out that the integer order capacitor can not 30 exist in nature since its impedance would violate causality [4-5]. What’s more important, Jesus et al proposed that the different orders of the fractional order capacitor can be fabricated by choosing the different fractal structures, such as the curve of Koch, carpet of Sierpinski, and curves of Hilbert [6]. Therefore, the real capacitor should be modeled by using fractional calculus to describe their real electricity characteristics. 35 From the classical circuit theory [1], one can see that the series and parallel connections are two basic and important structures, and the rule for calculation or analysis of these two cases are vital for designing the circuit in engineering applications. Also, an important conclusion can be obtained, that is, the equivalent result of the capacitors in series or parallel connections is still the capacitor. However, the above important conclusions are obtained under the condition that the 40 capacitor is integer order. Naturally, some questions happen to the researchers: for the fractional order capacitor, does the same conclusion can be obtained? i.e., does the equivalent result of the fractional order capacitor in serie s or parallel connections be also the capacitor? If it does, what Foundations: the National Natural Science Foundation of China under Grant (No. 51007068), the Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant (No. 20100201120028), the Fundamental Research Funds for the Central Universities of China under Grant (No. 2012jdgz09) Brief author introduction:WANG Faqiang (1980), Associate Professor,Modeling and control of power electronics. eecjob@126.com -1- condition should be satisfied? Therefore, it is necessary to analyze the rule for the fractional order 45 capacitors in series or parallel connections to clarify the above basic questions and these are studied in this paper. 1 Review of the fractional order capacitor According to Petráš’s work [7], the fractional order capacitor is described by fractional calculus [10-11] and the expression about the relationship between its voltage across (vC(t)) and its current through (iC(t)) in the time domain is 50 , d vC (t) (1) iC (t) , C, dt, Obviously, if ,=1, the above equations are the same as the integer order model of the capacitor. By using the Laplace transform of fractional calculus on (1), the expression of the fractional order capacitor about the relationship between its voltage across (vC(s)) and its current through 55 (iC(s)) in the complex frequency domain is 1 vC (s) , (2) GC (s) , C, s, iC (s) Here, the fractional order capacitor is chosen as 200μF with ,=0.8, and ,=0.4, respectively, and the Bode diagram of their impedance are plotted in Fig.1. One can see that the phase of each impedance, which shows the phase difference between the voltage across and the current through 60 , under the fractional order capacitor, are the constant negative value with certain fractional order any frequency, such as the phase is -72? when ,=0.8, and -36? when ,=0.4. So, this point can be used to identify the fractional order capacitor, i.e., if the above condition is satisfied, it is a fractional order capacitor. Otherwise, it is not. Especially, the phase is -90? when ,=1 and this is well known in the classical circuit theory [1]. Consequently, the fractional order capacitors in series connections are concerned. 65 Bode Diagram 100 ,=0.8 80 ,=0.4 60 40 20 0 Magnitude (dB) Phase (deg) -20 0 -30 -60 -90 1 2 3 4 10 10 10 10 Frequency (Hz) Fig.1. Bode diagram of GC(s) with 200μF under different order ,. 2 Fractional order capacitor in series connections [1] According to the circuit theory , the series connected circuit elements carry the same -2- 70 current and the whole voltage is the sum of the voltage across each elements. So, the equations in the time domain in this case can be derived as follows ,n ,1 ,2 , d v1 (t) d v2 (t) n (t) d v , ... , C, n ,iC (t ,) C, 1 , C, 2 ,1 dt,n (3) dt dt,2 , ,vC (t) , v1 (t) ， v2 (t) ， ... ， vn (t) , where vn(t) is the voltage across the fractional order capacitor C,n. Thus, the corresponding equations in the complex frequency domain can be derived by using the Laplace transform on (3) ,1 , 2 , n ,,iC (s) , C, 1s v1 (s) , C, 2 s v2 (s) , ... , C, n s vn (s) 75 (4) , (s) ， v ,,vC (s) , v1 2 (s) ， ... ， vn (s) Accordingly, the equivalent impedance of the series connections can be derived by calculating vC(s)/iC(s). 1 1 1 vC (s) , ， ， ... ， (5) Geqs (s) , ,1 iC (s) C, 1s C, 2s,2 C, n s,n If ,1=,2=…=,n, the equivalent impedance Geqs(s) of the series connections can be derived as 80 following simple formulation 1 1 1 1 ， ， ... ， (6) ) Geqs (s) , ( C, n s,1 C, 1 C, 2 Thus, the equivalent capacitor Ceqs can be obtained as follows. 1 1 1 1 , ， ， ... ， (7) Ceqs C, 1 C, 2 C, n Note that, the fractional order of this equivalent capacitor Ceqs also equals ,1. Obviously, the 85 [1]. formulation about (7) is the same as in the classical circuit theory But, if the above conditions (,1=,2=…=,n) can not be satisfied, the equivalent impedance shows that there is no any fractional order capacitor can be equivalent to this series connections. In other words, the fractional order capacitor in series connection in this case is no longer being the capacitor. For example, two fractional order capacitors in series connections are concerned. Firstly, we 90 take C,1=300μF with ,1=0.8 and C,2=200μF with ,2=0.8 and combine with them in series connections. The equivalent capacitor can be easily calculated by using (7), i.e., Ceqs=120μF with the same order as ,1 and ,2. Moreover, by plotting the Bode diagram of Geqs(s) and showing in Fig.2, it is also found the phase is -72? under any frequency, which implies that the equivalent of these two fractional order capacitors in series connections is still the fractional order capacitor 95 with the same order. However, if we choose C,2=200μF with ,2=0.8 and C,3=1000μF with ,2=0.4 and combine with them in series connections, the equivalent impedance which can be easily derived by using (5) and the corresponding Bode diagram is also shown in Fig.2. One can observe that the phase is changed with the frequency, such as for f=54Hz, the phase equals -47.6?, but for f=906Hz, the 100 phase equals -40.7?. Therefore, there is no fractional order capacitor can be considered as the equivalent capacitor of these series connections. In other words, the fractional order capacitors with different orders in series connections can not be equivalent to any fractional order capacitor. -3- Bode Diagram 80 C,1 with C,2 60 C,2 with C,3 40 20 Phase (deg) Magnitude (dB) 0 0 -30 -60 -90 1 2 3 4 10 10 10 10 Frequency (Hz) 105 Fig.2. Bode diagrams of Geqs(s) about C,1 and C,2 in series connection (solid) and C,2 and C,3 in series connection (dashed). 3 Fractional order capacitor in parallel connections From the circuit theory [1], one can know that the parallel connected circuit elements have the same voltage across their terminals and the whole current is the sum of the current through each branch. Thus, the equations in the time domain in this case can be derived as follows. 110 , ,n vC (td) ,in (t) , C, n (8) , dt,n ,iC (t) , i1 (t) ， i2 (t) ， ... ， in (t) , where in(t) is the current through the fractional order capacitor C,n. Thus, based on the Laplace transform of the fractional calculus, the corresponding equations in the complex frequency domain can be obtained , n s v ,,in (s) , C, n C (s) 115 (9) , (s) ， ,,iC (s) , i1 2i (s) ， ... ， in (s) So, the equivalent impedance of the parallel connections can be obtained by calculating vC(s)/iC(s). 1 vC (s) , (10) Geqp (s) , CiC (s) , 1s,1 ， C, 2s,2 ， ... ， C, n s,n If ,1=,2=…=,n, the equivalent impedance Geqp(s) of the parallel connections can be derived 120 as following simple formulation 1 (11) Geqp (s) , (C, 1 ， C, 2 ， ... ， C, n )s,1 Thus, the equivalent capacitor Ceqp can be obtained as follows. Ceqp , C, 1 ， C, 2 ， ... ， C, n (12) Note that, the fractional order of this equivalent capacitor Ceqp equals ,1. Obviously, the -4- [1] 125 formulation (12) is also the same as in the classical circuit theory . The example of the Bode diagrams about the C,1=300μF with ,1=0.8 and C,2=200μF with ,2=0.8 in parallel connections, which is shown in Fig.3, presents this point clearly. Therefore, the equivalent of the fractional order capacitors in parallel connections is also still the fractional order capacitor with the same order. 130 ,1=,2=…=,n) can not be satisfied, there is also no any the However, if the above conditions ( fractional order capacitor can be equivalent to this parallel connections. The example of the Bode diagrams about the C,2=200μF with ,1=0.8 and C,3=1000μF with ,3=0.4 in parallel connections, which is shown in Fig.3, presents this point clearly. It is found that the phase is changed with the frequency, such as for f=54Hz, the phase equals -60.4?, but for f=906Hz, the phase equals -67.3?. Therefore, the fractional order capacitor with different orders in parallel connections is also no 135 longer being the capacitor. Bode Diagram 100 C,1 with C,2 80 C,2 with C,3 60 40 20 Magnitude (dB) 0 Phase (deg) -20 0 -30 -60 -90 1 2 3 4 10 10 10 10 Frequency (Hz) Fig.3. Bode diagrams of Geqp(s) about C,1 with C,2 in parallel connection (solid) and C,2 with C,3 in parallel connection (dashed). 140 4 PSIM simulations In order to confirm the above theoretical results, the circuit simulations from power electronics simulator (PSIM) are given. From the existing literatures, one can know that the PSIM is a good software for simulating the circuit to plot the Bode diagram [12-14]. Note that, the fractional order capacitor can not be found in the market and in the PSIM. Fortunately, it can be realized by using the approximate method about the chain fractance [10, 15-17] with the Oustaloup’s 145 approximation [18]. The approximated circuit for the fractional order capacitor is shown in Fig.4. For C,1=300μF with ,1=0.8, the resistors and capacitors in Fig. 4 are chosen as R11=6.52m,, R12=52m,, R13=0.5,, R14=4.9,, R15=47,, R16=453.5,, R17=4377,, R18=42.2k,, R19=407k,, R110=34M,, C11=20,F, C12=42,F, C13=73.5,F, C14=130,F, C15=228,F, C16=403,F, 150 C17=710,F,C18=1251μF, C19=2208μF, C110=1680μF. For C,2=200μF with ,2=0.8, the resistors and capacitors in Fig. 4 are chosen as R21=10m,, R22=78m,, R23=0.75,, R24=7.3,, R25=70.4,, R26=680,, R27=6.6k,, R28=63k,, R29=611k,, R210=51.4M,, C21=13,F, C22=28,F, C23=49,F, C24=86,F, C25=152,F, C26=268.4,F, C27=473,F, C28=834,F, C29=1472,F, C210=1120,F. -5- 155 For C,3=1000μF with ,3=0.4, the resistors and capacitors in Fig. 4 are chosen as R31=1.8,, R32=3.7,, R33=11.5,, R34=36,, R35=111.5,, R36=346,, R37=1076,, R38=3344,, R39=10374,, R310=150k,, C31=40nF, C32=337nF, C33=1.8,F, C34=10,F, C35=55,F, C36=299,F, C37=1638,F, C38=8968,F, C39=49174,F, C310=218509,F. Rn1 Rn 4 Rn 2 Rn3 Rn 5 Cn1 Cn 4 Cn 2 Cn3 Cn5 C, n , Rn10 Rn9 Rn8 Rn 7 Rn 6 Cn10 Cn9 Cn8 Cn 7 Cn 6 Fig.4. Approximated model for the fractional order capacitor 160 Here, only the comparisons between the PSIM simulations and theoretical analysis about C,1 case are concerned and shown in Fig.5. Obviously, the PSIM simulations are in good agreement with the theoretical analysis. So, it is effective to use this method to substitute the fractional order capacitor approximately. Bode Diagram 80 PSIM simulations Theoretical analysis 60 40 20 Phase (deg) Magnitude (dB) 0 0 -30 -60 -90 1 2 3 4 10 10 10 10 Frequency (Hz) 165 Fig.5 Bode diagram of GC(s) about 300μF with ,1=0.8 from the PSIM simulations and theoretical analysis. Additionally, the comparisons between the PSIM simulations and the theoretical analysis about the Bode diagrams for the impedance of C,1 with C,2 in series connections and C,2 with C,3 in series connections are shown in Fig.6. Also, the case about the parallel connections of C,1 with C,2 and C,2 with C,3 are shown in Fig.7. It is obvious that the PSIM simulations are also in good 170 agreement with the theoretical analysis about these two cases, and accordingly the correctness of the above theoretical analysis is confirmed. -6- Bode Diagram 80 60 ,2 with C,3 C40 20 Phase (deg) Magnitude (dB) Phase (deg) C,1 with C,2 Magnitude (dB) 0 0 C,2 with C,3 -30 C,1 with C,2 -60 -90 1 2 3 4 10 10 10 10 Frequency (Hz) Fig.6. Bode diagrams of Geqp(s) about C,1 with C,2 in series connections and C,2 with C,3 in series connections from the PSIM simulations (dashed) and theoretical analysis (solid). 175 Bode Diagram 60 40 ,2 with C,3 C20 C,1 with C,2 0 -20 0 -30 ,2 with C,3 C -60 C,1 with C,2 -90 1 3 2 4 10 10 10 10 Frequency (Hz) Fig.7. Bode diagrams of Geqp(s) about C,1 with C,2 in parallel connections and C,2 with C,3 in parallel connections from the PSIM simulations (dashed) and theoretical analysis (solid). 5 Conclusions 180 In summary, the conclusions can be obtained from the theoretical analysis and PSIM simulations, which are shown as follows. (1) The fractional order capacitors with the same order are in series or parallel connections can be equivalent to the fractional order capacitor and this point is consistent with the classical circuit theory. (2) If the fractional order capacitors with different orders are in series or parallel connections, 185 the corresponding impedance can not meet any impedance of the fractional order capacitor. Therefore, in engineering applications, if we obtain the fractional order capacitor from the fractional order capacitors in series or parallel connections, we must choose the fractional order capacitors with the same orders. Otherwise, the mistake will be done. -7- 190 References [1] QIU G Y. Electric circuits [M]. Beijing: Higher Education Press, 2006. [2] CHEN Y Z. Capacitors and its applications [M]. Bejing: Science Press, 2005. [3] WESTERLUND S, EKSTAM L. Capacitor theory [J]. IEEE Transactions on Dielectrics and Electrical Insulation, 1994, 1(5):826-839. [4] JONSCHER A K. Dielectric relaxation in solids [M]. London, Chelsea Dielectric Press, 1983. 195 [5] BOHANNAN G W. Analog realization of a fractional control element-revisited [A]. Proc. of the 41st IEEE Int. Conf. on Decision and Control, Tutorial Workshop 2: Fractional Calculus Applications in Automatic Control and Robotics, 2002. 1-4. [6] JESUS I S, TENREIRO MACHADO J A. Development of fractional order capacitors based on electrolyte processes[J]. Nonlinear Dynamics, 2009, 56: 45-55. 200 [7] PETRAS I. A note on the fractional-order Chua's system[J]. Chaos, Solitons and Fractals, 2008, 38: 140-147. 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IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, 2000, 47(1): 25-39. 225 分数阶电容的串联与并联分析 王发强，马西奎 (电力设备电气绝缘国家重点实验室, 电气学院, 西安交通大学, 西安 710049) 230 摘要:基于分数阶微积分理论及电容的分数阶本质，建立并分析了分数阶电容串联和并联的 模型。给出了分数阶电容阶数对分数阶电容的串联以及并联的影响。最后，了分数阶电 容的电路实现形式，以 PSIM 电路仿真实验结果验证了理论分析的正确性。 关键词:分数阶电容，串联，并联。 中图分类号:TM132 235 -8-