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Perturbation method applied to a basic diode circuit

Revista Mexicana de Fisica

Because of the exponential characteristic of silicon diodes, exact solutions cannot be established when operating point and transient analysis are computed. To overcome that problem, the present work proposes a perturbation method which allows obtaining approximated analytic expressions of diode-based circuits. Simulation results show that numerical solutions obtained by using the proposed method are similar to those reported in literature, with the advantage of not requiring a user-selected arbitrary expansion point. Additionally, the method does not use the Lambert function W, reducing the proposed solution complexity, which makes it suitable for engineering applications.

RESEARCH Revista Mexicana de Fı́sica 61 (2015) 69–73 JANUARY-FEBRUARY 2015 Perturbation method applied to a basic diode circuit H. Vazquez-Leala,∗ , Y. Khanb , G. Fernandez-Anayac , U. Filobello-Ninoa , V.M. Jimenez-Fernandeza , A. Herrera-Mayd , A. Diaz-Sancheze , A. Marin-Hernandezf and J. Huerta-Chuag a Electronic Instrumentation School, Universidad Veracruzana, Cto. Gonzalo Aguirre Beltrán S/N, Xalapa 91000, Veracruz, Mexico, ∗ e-mail: hvazquez@uv.mx b Department of Mathematics, Zhejiang University, Hangzhou 310027, China. c Departamento de Fı́sica y Matemáticas, Universidad Iberoamericana, Prol. Paseo de la Reforma 880, 01219 D.F., Mexico. d Micro and Nanotechnology Research Center, Universidad Veracruzana, Calzada Ruiz Cortines 455, Boca del Rio 94292, Veracruz, Mexico. e National Institute for Astrophysics, Optics and Electronics, Luis Enrique Erro #1, Sta. Marı́a Tonantzintla 72840, Puebla, Mexico. f Department of Artificial Intelligence, Universidad Veracruzana, Sebastián Camacho No. 5, Xalapa 91000, Veracruz, Mexico. g Facultad de Ingenierı́a Civil, Universidad Veracruzana, Venustiano Carranza S/N, Col. Revolución, C.P. 93390, Poza Rica, Veracruz, Mexico. Received 3 December 2013; accepted 11 December 2014 Because of the exponential characteristic of silicon diodes, exact solutions cannot be established when operating point and transient analysis are computed. To overcome that problem, the present work proposes a perturbation method which allows obtaining approximated analytic expressions of diode-based circuits. Simulation results show that numerical solutions obtained by using the proposed method are similar to those reported in literature, with the advantage of not requiring a user-selected arbitrary expansion point. Additionally, the method does not use the Lambert function W , reducing the proposed solution complexity, which makes it suitable for engineering applications. Keywords: Circuit analysis; nonlinear circuits perturbation method. PACS: 07.50.Ek, 84.30.-r, 02.70.-c, 05.45.-a 1. Introduction In general terms, to find exact solutions of operating point and transient analysis of circuits containing diodes is not possible, mainly because the exponential characteristic of diodes. However, it is possible to obtain expressions that allow modeling diode behavior by using approximate methods. In Ref. 1, an approximate solution for transient response current in a serial circuit composed by an independent voltage source, an inductor, a resistance and a diode is proposed. Such solution have a good precision, but it requires a previous knowledge of the approximate steady state of the circuit in order to select an arbitrary expansion point. In Ref. 2, an explicit analytic expression for the DC current of a basic circuit containing a voltage source, a resistor, and a diode is achieved. Nevertheless, such expression is found in terms of Lambert W function, which increases the complexity making harder its application in engineering. In the same manner, in Ref. 1 two high-precision approximate analytical solutions for DC current in the diode circuit are proposed. What is more, to compute the first solution also requires to know a roughly approximate value for the expected DC current to select the expansion point, while the second solution requires the use of the Lambert W function mentioned before. In this work, a perturbation method [3-9] which do not need an arbitrary expansion point or the use of Lambert func- tion W , is proposed to obtain approximate solutions for DC and transient domains in diode-based circuits. Finally, some comparisons between the numerical approximations obtained using the proposed method, and the expressions reported in Ref. 1 and 2, will be shown. 2. Circuit analysis of a basic diode circuit Figure 1(a) shows a circuit containing a independent voltage source (V ), a resistor (R), an inductor (L), and a diode (D). The voltage drop at the diode is given by the expression µ ¶ i(t) VD = VT ln +1 , i(t) > 0 (1) Is where Is is the saturation current of the diode and VT is the thermal voltage. Now, we establish the nonlinear differential equation that describes the transient behavior for the circuit µ ¶ i(t) di(t) + VT ln + 1 − V = 0, Ri(t) + L dt Is i(0) = A, (2) where initial condition i(0) = A stands for the the initial current circulating through inductor L in t = 0. 70 H. VAZQUEZ-LEAL et al. Equation (2) does not have analytic solution due to the natural logarithm term from the diode model. Hence, Ec.(1) can be approximated as VD = VT ln µ i(t) Is ¶ = VT ln(i(t)) − VT ln(Is ). (3) Next, reformulating (2) using (3) Ri(t) + L di(t) + VT ln(i(t)) − VT ln(Is ) − V = 0, dt i(0) = A. (4) From perturbation theory [3-11], the current can be expressed in function of a power series of VT i(t) = i0 (t) + i1 (t)VT + i2 (t)VT2 + i3 (t)VT3 + i4 (t)VT4 + · · · (5) In order to keep solution simple, only the two first terms of (5) are substituted in (4). By regrouping terms µ ¶ di0 (t) di1 (t) + VT dt dt ¶ µ i1 (t) VT + VT ln(i0 (t)) + VT ln 1 + i0 (t) R(i0 (t) + i1 (t)VT ) + L − VT ln(Is ) − V = 0, (6) Now, by replacing the first term of the Taylor expansion of natural logarithm, we obtain R(i0 (t) + i1 (t)VT ) + L µ di0 (t) di1 (t) + VT dt dt + VT ln(i0 (t)) + VT ¶ i1 (t) VT i0 (t) − VT ln(Is ) − V = 0, (7) regrouping terms in function of VT , and equating to zero each coefficient, the following system is formed i0 (t) − V + Ri0 (t) = 0, i0 (0) = A, dt i1 (t) + Ri1 (t) − ln(Is ) + ln(i0 (t)) VT1 : L dt VT0 : L = 0, i1 (0) = 0. (8) F IGURE 1. a) t > 0, i(0) = A. Basic diode circuit, b) Transient for six sets of parameters E = [E1 , E2 , E3 , E4 , E5 , E6 ], as it is shown at Tables I and II. Numerical solution RK4 for (2) (circles) and its approximate solutions (11) (solid line) and (13) (solid circles) and c) Zoom to (b). Time is in Seconds and current in Ampers. Solving the system (8), we obtain ¶µ ¶ µ V V Rt i0 (t) = A− , + exp − R L R µ ¶ µ ¶ · 1 RA Rt ln(Is ) + G(t) + t exp − i1 (t) = R L L VL ¸ µ ¶ ln(Is ) A Rt − + ln(A) exp − , (9) R V L where µ · µ ¶¸ ¶ 1 A Rt G(t) = 1 − exp − R L V µ · µ ¶¸¶ Rt 1 × ln V + (RA − V ) exp − . (10) R L Rev. Mex. Fis. 61 (2015) 69–73 71 PERTURBATION METHOD APPLIED TO A BASIC DIODE CIRCUIT Using (5) and (9), we obtain the following first order expression for the transient current i(t) = i0 (t) + i1 (t)VT , µ ¶µ ¶ V Rt V = + exp − A− R L R µ ¶ µ ¶ · 1 RA Rt ln(Is ) + G(t) + t exp − + VT R L L VL ¸ µ ¶ ln(Is ) A Rt − + ln(A) exp − , (11) R V L The expression for steady state current (DC current) can be obtained calculating the limit (11) as idc 1 = lim i(t) = t→∞ R µ µ ¶¶ Is R V + VT ln , V V > 0. (12) 3. Another approximations In Ref. 1, the transient response for the same circuit was calculated, given the result ¶ µ (RB + VT )t , (13) i(t) = idc + (A − idc ) exp − BL where the DC current idc is idc = B(V − VT ln (B/Is ) + VT ) , RB + VT V ≥ VT ln(B/Is ) − VT . (14) where B is the is the current at the expansion point. After several algebraic manipulations of (14), the need of an expansion point for B [1] is not longer required, resulting µ µ ¶¶¶ µ Is R V V , (15) −W exp idc = Is exp VT VT VT where W represents the Lambert W function [12-22]. TABLE I. Transient (2) RK4 and its approximate solutions (11) and (13); DC exact solution (16) and its approximations (12), (14) and (15); for 3 different sets of E =[V (Volts), R (Ω), L (H), A (Ampers)]. Transient Time (Sec) E1 =[3,5,0.1,0] E2 =[4,10,0.2,0.1] E3 =[5,15,1,0.2] This work (11) 0.00 0.0000000000 0.1000000000 0.2000000000 RK4 0.00 0.0000000000 0.1000000000 0.2000000000 (13) [1] 0.00 0.0000000000 0.1000000000 0.2000000000 This work (11) 0.02 0.2940427361 0.2469713801 0.2228947813 RK4 0.02 0.2948394355 0.2471752991 0.2229078118 (13) [1] 0.02 0.2932367519 0.2466090191 0.2227823932 This work (11) 0.05 0.4235142569 0.3124639698 0.2465129412 RK4 0.05 0.4247266709 0.3128637547 0.2465683401 (13) [1] 0.05 0.4242469063 0.3125006693 0.2463499002 This work (11) 0.10 0.4568994517 0.3294586538 0.2683204993 RK4 0.10 0.4582435276 0.3299344440 0.2684483344 (13) [1] 0.10 0.4581825403 0.3297197849 0.2681875195 This work (11) 0.15 0.4595585919 0.3308233101 0.2785541171 RK4 0.15 0.4609175439 0.3313083307 0.2787337170 (13) [1] 0.15 0.4608970616 0.3311150649 0.2784762497 This work (11) 0.30 0.4597884740 0.3309423605 0.2866721322 RK4 0.30 0.4611492802 0.3314284941 0.2869128044 (13) [1] 0.30 0.4611329550 0.3312380295 0.2866836894 DC E1 =[3,5,0.1,0] E2 =[4,10,0.2,0.1] E3 =[5,15,1,0.2] This work (12) 0.459788589306 0.330942421957 0.287609148788 Exact (16) [2] 0.461149372833 0.331428537242 0.287861896561 Aprox. (14) [1] 0.461133075751 0.331238094926 0.287642323377 Aprox. (15) [1] 0.461149372834 0.331428537241 0.287861896560 Rev. Mex. Fis. 61 (2015) 69–73 72 H. VAZQUEZ-LEAL et al. As in Ref. 2, an exact expression for the current in the circuit shown in Fig. 1(a) is formulated ¶¶ µ µ V + Is R Is R VT . (16) exp W idc = −Is + R VT VT 4. Numerical Simulation and Discussion It was considered for all numerical examples that Is = 1E-12 A and VT = 25.85 mV just as reported in Ref. 1. Some representative points from the circuit transient analysis, for six sets of parameters E = [E1 , E2 , E3 , E4 , E5 , E6 ], are shown in Table I and Table II. Previously reported numerical solutions, such as the approximation (13) reported in Ref. 1 (Employing B = 0.5 as an expansion point), and the numerical curve obtained by using a fourth order Range-Kutta numerical method (RK4), are compared with the approximation obtained in this work (11). Therefore, Table I and Table II show the DC steady current value obtained by using the proposed approximation (12), the exact solutions (16) obtained by [2], and the approximations (14) and (15) reported by [1]. It can be noticed that expression (15) is the most exact, followed by (14) and (12), which have similar precision. For instance, for the case E6 , relative errors of DC analysis are 8.2E-12, 1.5E-4, and 3.1E-4 for (15), (14), and (12), respectively. Besides, from both Tables, it can be observed that the proposed approximation (11) achieved a good accuracy, similar to the obtained by (13). For instance, for case E6 , the average relative error of transient analysis for the selected points is 5.7E-5 and 1.9E-5 for (11) and (13) respectively. However, (11) and (12) have the advantage of not requiring any arbitrary expansion point B, which implies to know the approximate value for DC current in the circuit (see Fig. 1(a)). For that reason, the proposed method provides a more general analytic approximation for both transient current and DC expression. In addition, Eq. (12), besides its simplicity, is just expressed in terms of exponentials and natural logarithms, and the Lambert W function is not required as in the exact (16) and approximate expressions (15), which reduces its complexity and makes it suitable for engineering applications. Finally, Figs. 1(b) and 1(c) are a graphical comparison of numerical data presented in Table I and Table II, where it can be observed the high accuracy of the proposed approximations. Based on those results, further works can be focused TABLE II. Transient (2) RK4 and its approximate solutions (11) and (13); DC exact solution (16) and its approximations (12), (14) and (15); for 3 different sets of E =[V (Volts), R (Ω), L (H), A (Ampers)]. Transient Time (Sec) E4 =[6,50,0.5,0.3] E5 =[7,30,2,0.4] E6 =[8,20,2.5,0.5] This work (11) 0.00 0.3000000000 0.4000000000 0.5000000000 RK4 0.00 0.3000000000 0.4000000000 0.5000000000 (13) [1] 0.00 0.3000000000 0.4000000000 0.5000000000 This work (11) 0.02 0.1327916582 0.3508506734 0.4800694461 RK4 0.02 0.1328176289 0.3508526347 0.4800704783 (13) [1] 0.02 0.1326159952 0.3508437863 0.4800704512 This work (11) 0.05 0.1080892378 0.2999834489 0.4555697682 RK4 0.05 0.1081446676 0.2999938148 0.4555755750 (13) [1] 0.05 0.1077772223 0.2999615267 0.4555749149 This work (11) 0.10 0.1068193756 0.2528235002 0.4258116764 RK4 0.10 0.1068791939 0.2528539440 0.4258306310 (13) [1] 0.10 0.1064887143 0.2527677354 0.4258266654 This work (11) 0.15 0.1068109940 0.2305930709 0.4058814021 RK4 0.15 0.1068708896 0.2306427324 0.4059161317 (13) [1] 0.15 0.1064800771 0.2305037616 0.4059064256 This work (11) 0.30 0.1068109383 0.2128518485 0.3776080977 RK4 0.30 0.1068708682 0.2129315062 0.3776843143 (13) [1] 0.30 0.1064800188 0.2127081260 0.3776537615 DC E4 =[6,50,0.5,0.3] E5 =[7,30,2,0.4] E6 =[8,20,2.5,0.5] This work (12) 0.106810938331 0.210778575971 0.365471210979 Exact (16) [2] 0.106870843487 0.210865816323 0.365587482915 Aprox. (14) [1] 0.106480018836 0.210620562523 0.365530401850 Aprox. (15) [1] 0.106870843487 0.210865816323 0.365587482912 Rev. Mex. Fis. 61 (2015) 69–73 PERTURBATION METHOD APPLIED TO A BASIC DIODE CIRCUIT 73 to find an equivalent circuit based on Ec. (12), which allows the analysis of larger circuit in DC domain. In the same fashion, applications of Ecs. (11) and (12) can be extended to explore their possibilities of being used to calculate other analytical characteristics, such as transitory power consumption, symbolic small signal analysis [23] and symbolic sensitivity analysis, among many others. which similar works reported in literature, but with a reduced complexity derived from the fact of not requiring: a previously known expansion arbitrary point and the use of the Lambert function W . Finally, the obtained expression for the steady state current or DC current is easier to use than other solutions reported in recent literature, with no significant loss in accuracy. 5. Acknowledgments Conclusions In this work, two approximate solutions for a basic diode, both in DC domain and transient analysis, were proposed. Several simulations results allowed us to conclude that proposed approximations have a similar accuracy in comparison The authors wish to acknowledge to Rogelio Alejandro Callejas-Molina and Roberto Ruiz-Gomez for their technical support. Besides, this work has been supported by CONACYT México research project CB-2010-01 #157024. 1. H. Vazquez-Leal et al., IEICE Electronics Express, 9 (2012) 522-530. 13. V. Hariharan, J. Vasi, and V. Ramgopal Rao, Solid- State Electronics, 53 (2009) 218-224. 2. T. C. Banwell, Fundamental Theory and Applications 47 (2000) 1621-1633. 14. D.C. Allan, Journal of Non-Crystalline Solids 358 (2012) 440442. 3. V. Marinca and N. Herisanu, Nonlinear dynamical systems in engineering: Some approximate approaches. (Springer, 2011). 15. D. Brkič, Applied Mathematics Letters 24 (2011) 1379-1383. 4. E. Di Costanzo and A. Marasco, Computers and Mathematics with Applications 63 (2012) 60-67. 5. G.M. Abd EL-Latif, Applied Mathematics and Computation 152 (2004) 821-836. 16. W. Jung and M. Guziewicz, Materials Science and Engineering: B 165 (2009) 57-59. 17. Z. You, R.R. Huilgol, and E. Mitsoulis, International Journal of Engineering Science 46 (2008) 799-808. 6. A. Doroudi and A. Rezaeian Asl, International Journal of Mass Spectrometry 309 (2012) 104-108. 18. J. Ding and R. Radhakrishnan, Solar Energy Materials and Solar Cells 92 (2008) 1566-1569. 7. R.R. Pušenjak, Journal of Sound and Vibration, 314 (2008) 194-216. 19. F. Ghani and M. Duke, Solar Energy 85 (2011) 2386-2394. 8. Hong-Mei Liu, Solitons and Fractals 23 (2005) 577-579. 20. S.M. Disney and Roger D.H. Warburton, International Journal of Production Economics 140 (2011) 756-764. 9. P. Amore and Alfredo Aranda, Journal of Sound and Vibration 283 (2005) 1115-1136. 21. D. Veberič, Computer Physics Communications 183 (2012) 2622-2628. 10. U. Filobello-Nino et al., Miskolc Mathematical Notes, 14 (2013) 89-101. 22. Marko Goli¡cnik. On the lambert w function and its utility in biochemical kinetics. Biochemical Engineering Journal, 63 (2012) 116-123. 11. U. Filobello-Nino et al., Asian Journal of Mathematics & Statistics 6 (2013) 76-82. 12. Yuan Tian, Kaibiao Sun, and Lansun Chen, Mathematics and Computers in Simulation 82 (2011) 318-331. 23. C. Sanchez-Lopez, F.V. Fernandez, E. Tlelo-Cuautle, and S. XD. Tan, IEEE Transactions on Circuits and Systems I: Regular Papers, 58 (2011) 1382-1395. Rev. Mex. Fis. 61 (2015) 69–73