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