Exploring the Classic Perturbation Method for Obtaining Single and Multiple Solutions of Nonlinear Algebraic Problems with Application to Microelectronic Circuits

Exploring the Classic Perturbation Method for Obtaining Single and Multiple Solutions of Nonlinear Algebraic Problems with Application to Microelectronic Circuits

M. A. Sandoval-Hernandez1, O. Alvarez-Gasca2, A. D. Contreras-Hernandez2, J. E. Pretelin-Canela2,

1. E. Palma-Grayeb2, V. M. Jimenez-Fernandez2, U. Filobello-Nino2, D. Pereyra-Diaz2, S. F. Hernandez-Machuca2,
2. E. Sampieri-Gonzalez2, F. J. Gonzalez-Martinez2, R. Castaneda-Sheissa2, S. Hernandez-Mendez1,

J. Matias-Perez2, L. Cuellar-Hernandez2, C. Hoyos-Reyes2, J. Cervantes-Perez2, L. J. Varela-Lara2,

1. L. Vazquez-Aguirre2, L. Gil-Adalid2, J. L. Rocha-Fernandez2, N. Bagatella-Flores3, H. Vazquez-Leal2*1 Escuela de Ingeniería, Universidad de Xalapa, Km 2,
   Carretera Xalapa-Veracruz 341, 91190
   2 Facu ltad de Instrumentación Electrónica, Universidad Ver acr uzana,
   Circuito Gonzalo Aguirre Beltrán S/N, 91000, Xalapa, Veracruz, México.
   Xalapa, Veracruz, México
   3 Facultad de Física, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N,
   91000, Xalapa, Veracruz, México.

   Abstract It is well-known in electronics that establishing symbolic analytic expressions for the oper- ating point and transient analysis of circuits including diodes is not possible due to the fact that these devices are modelled using an exponential function. In general, to solve this problem, in this work, the Hybrid-Newton-Raphson Perturbation technique is proposed, which allows obtaining highly accurate and handy analytical approximate solution obtained by the use of the perturbation method. In turn, these solutions are used as a starting point for the Newton- Raphson method in order to improve its accuracy in few iterations. To verify this method, two case studies are solved. The first case is the solution for single operating point for direct current (DC) analysis of a circuit with two rectifier diodes. For the second case, the solution for multiple operating points in DC analysis of a tunnel diode, voltage source, and a resistor is carried out. Therefore, for a wide range of values for the voltage source and resistor, the proposed method is able to obtain valid and highly accurate solutions. To validate the results of the method we propose, we compared the results of the first case study against commercial simulators, while in the second case study, our results were compared against a homotopy continuation method, obtaining results in good agreement for both study cases.

   Keywords Perturbation method; Nonlinear circuits; Multiple operating points; Homotopy continuation method.

   1. INTRODUCTIONMany physical phenomena in science and engineering are modelled by nonlinear equation systems (NAES) because is the best and direct way to describe its behaviour, considering the laws that rule them. These NAES pose a problem because it is not possible to obtain, analytically, exact results [1] because the equations complexity is high. Therefore, to solve this kind of equations, numerical methods are employed; these methods allows us to formulate mathematical

      problems in a way that can be solved using arithmetic operations.

      Numerical methods allows solving equations and systems of equations by means of approaching, mathematically, the solutions of a problem expressed in mathematical terms.

      [2] details known numerical methods to solve differential equations like Eulers Method or Runge-Kutta Method; as for solving nonlinear algebraic equations, available methods are fixed point iteration [2], Secant Method [2], Brents Method [2], Newton-Raphson Method (NR) [2, 3], Homotopy Continuation [39], among others. Generally speaking, algebraic and differential problems are usually solved by the previous mentioned methods. Nevertheless, those techniques generally exhibit numerical instabilities, oscillations, false equilibrium states, among other problems [2]. Other aspect to be taken into account of numerical methods is that only provide solutions in the form of a set of numerical data. This data helps to visualize phenomena behaviour, leaving aside the possibility of performing analytical analysis that can only be performed when exact solution is able to be calculated. Nonlinear algebraic or differential problems commonly employed in engineering or applied physics usually do not possess known exact solution, however. As a response to this problem, approximative methods emerge, capable to provide analytic approximative solutions able to imitate the behaviour of the exact solution with a certain degree of accuracy and convergence domain.

      As for the solution of differential equations, approximative methods reported in literature are: iterative variational method (VIM) [10, 11], modified Taylor series method (MTSM) [12],

      Adomian [13, 14], power series [15], classic perturbation [16,

      17], homotopy perturbation method (HPM) [18,

      19], series expansion [20], rational homotopy perturbation method (RHPM), [21], homotopy fixed point [15], among others. Among all these methods, classic perturbation method

      [16] is the oldest and well-founded, but is restricted to problems containing, at least, one perturbative parameter

      << 1. Therefore, new derived methods from the classic method previously mentioned, like homotopy perturbation method (HPM) that avoids the restriction of the perturbative parameter and achieves more general solutions. Nevertheless, even this method may fail due to the high nonlinearity of diverse phenomena. [22] proposes the generalized homotopy method (GHM) as a generalization of the homotopy perturbation method (HPM), which is capable to provide precise solutions. Nevertheless, when a parameter of the perturbative type is available, it is always desirable to use the classic perturbation method. In addition, the classic perturbation method has the advantage that it can be applied to algebraic equations and differential equations, unlike other methods that can only be applied to differential equations.

      In the absence of an exact solution, an approximative solution could provide valuable information about the general behaviour of the dynamic system in terms of the parameters that make it up; information that would be impossible to obtain otherwise. For the case of electronic circuits, the transient behaviour is modelled using differential equations; usually, commercial simulators just provide a numerical solution like a curve or table of values. In fact, circuit designers usually resort to empirical rules based on try and failure tests to optimize or characterize their designs. A tangible benefit for applying approximative methods in electronics [16, 23] is the possibility to analyze the transient behaviour of the circuits, that is, it is possible to identify which parameters have the most influence on the establishing time on the circuit or which parameter combination may extend or shorten the transient, potentially allowing the creation of analytical schemes to reduce power consumption, control setting times, among other aspects. In general, find the exact solution on direct current (DC) circuits that include diodes is not possible given the exponential characteristic of these devices. Nevertheless, it is possible to model the behaviour of a diode in DC circuits using approximative methods [24]. Therefore, the perturbation method can be employed to propose accurate analytical solutions and avoid the use of Newton-Raphson.

      An approximative solution is introduced for the transient and steady state (DC analysis) of a simple series circuit, which has one independent voltage source, an inductor, a resistor, and a diode [1]. The provided solutions have good accuracy, but require previous knowledge of the steady state of the circuit in order to select an adequate expansion point. On the other hand, [25] proposes an exact expression for the DC analysis of the same circuit. Nevertheless, the obtained expression is expressed in terms of the Lambert W function, which is little known and not included in standard libraries of many mathematical programs. To this problem is added the fact that the methodology proposed by [25] cannot be generalized to circuits containing multiple diodes given the high nonlinearities. Likewise, [23] provides an approximative solution for the transient and DC analysis for the same circuit, which was obtained using the perturbation method [2631].

      The perturbation method from [23] was applied to a series circuit containing resistor, inductor, and diode obtaining an algebraic expression for the current in the circuit. Two case studies are proposed. The first case will obtain an approximative analytical expression by the perturbation method for the electrical current in a circuit containing a voltage source, a resistor, and two different diodes avoiding the application of NR. This circuit, but with a single diode, have been analysed in analogue electronic books, obtaining the operating point in a graphical and numerical way [32, 33]. For the second case, the goal is to obtain an approximative analytical expression for multiple operating points of a circuit with a tunnel diode, a resistor, and a voltage source. The operating points will be located using hybrid technique that includes the classic perturbation method and the Newton-Raphson method.

      The hybrid perturbation Newton-Raphson technique is explored as an alternative to the homo- topy continuation method [4,6,8,34,35] because it requires deep understanding of several specialized numerical methods regularly adapted to the problem under study [36]. Continuation methods also exhibit numerical problems, especially with tracing methods since the corrective step may show convergence issues as the number of equations increases. As it is well- known, continuation meth- ods provide only numerical solutions, while perturbation methods provide approximative analytical solutions simple to evaluate in terms of the parameters under study.

      This work is organized as follows. Section 2 introduces the classical perturbation method. In Section 3 provides a brief description of the Newton-Raphson numerical method. A case study that proposes an approximative solution for single and multiple operating points of the current in a circuit with two diodes is presented in Section 5. In Section 6, a case study featuring the use of the hybrid homotopy perturbation method to calculate multiple operating points for a tunnel diode is presented. Finally, Section 7 provides conclusions of this work and suggests possible future work.

   2. CLASSIC PERTURBATION METHOD Nonlinear equations that model phenomena, whether theyare physical, chemical, among others, usually do not possess exact solutions. Exact solutions [37] to these problems are not always available, thus the need of using approximative methods to obtain a solution. Approximative analytic solutions for nonlinear equations obtained using classic perturbation should be compared against the numerical solution of the problem through the use of known numerical methods [2, 3]. In the absence of an exact solution, the approximative solution can provide valuable information about the general behaviour of the static or dynamic system in terms of their parameters; this information would be impossible to obtain by any other way. Procedure of the perturbation method is provided by [24, 38]; following the guidelines, now consider solving the following nonlinear equation

      () = , (1)

      also, consider that

      () = , (2)

      is an auxiliary equation that possess an explicit solution known in the form

      = (), (3)

      is the unique solution of (2). () is the linear operator, that is, complies with the principles of superposition and homogeneity; this is

      (11 + 22) = 1(1) + 2(2), (4)

      for the case of two functions 1 and 2, scalars 1 and 2, we can then rewrite (1) in the following form

      () + (() ()) = . (5) Introducing function () = () (), (5) becomes

      () = + (). (6)

      To ease the discussion of particular solutions for (6), the introduction of a small parameter is considered, where 1; therefore, the equation is written as

      () = + (). (7)

      considering the power series in , with independent coefficients of , becoming

      = 0 + 1 + 22 + , (8) by doing this can be seen that term 0, obtained when =

      0, is a linear solution of

      () = . (9)

      This solution can be written as 0 = ().

      and so on. It is important to note that is an infinite system and can be solved recursively. Obtaining requires to know

      when 0 1. Therefore, the first equation is given as follows

      0 = (). (13) For the second, we obtain

      1 = ((0)) = ((())). (14)

      Continuing, we can express each in terms of . By doing this way we can state that when a perturbation parameter is available, it is always convenient consider the use of the classic perturbation method. Besides, this method has the advantage that can be applied to algebraic equations and differential equations as well [24].

   3. THE NEWTON-RAPHSON METHODThe Newton-Raphson method exhibits local convergence, thus, is necessary to propose an initial value close enough to the desired root. The relative closeness of the starting point to the root is closely related to the nature of the function itself. Once this step is performed, the method linearise the function to a tangent line at the initial point. The abscissa at the origin of the line will be, accordingly to the method, a better approximation to the root than the previous value. Successive iterations are performed until the method converges close enough to the desired root (see Figure 1).

      Be () a real domain continuous function in the interval [, ], with defined derivative in the interval [, ]. Starting with an initial value , an approximation +1 is obtained closer to the root, given by the formula

      ( )

      In order to obtain the following terms 1, 2, , systematically, we substitute given by (8) in (7) and equate terms, obtaining

      (0 + 1 + 22 + ) = + (0 + 1 +

      22 + ). (10)

      Because the term is a linear operator, the left side of the equality can be written as

      (0) + (1) + 2(2) + . (11)

      Assuming that () is analytical in , we can expand the right side of (10) in power series of , achieving the expression

      (0 + 1 + 22 + ) = (0) +

      1(0, 1) + 22(0, 1, 2) + , (12)

      where each coefficient only depends of the quantities

      0, 1, 2,, . Combining the expansions of (11) and (12), equalling coefficients of , equation (7) gives rise to an infinite system of equations.

      +1

      =

      ()

      , = 0,1, (15)

      (0) =

      (1) = (0)

      (2) = 1(0, 1)

      (+1) = (0, 1, , ),

      (0) =

      (1) = (0)

      (2) = 1(0, 1)

      (+1) = (0, 1, , ),

       

      Figure 1: Geometric interpretation of the Newton-Raphson

      method.

      The iterative process is repeated until the -intercept of the tangent line matches the equation root or when the difference between two successive approximations fulfils an already defined tolerance error. The Newton-Raphson method exhibits quadratic convergence given by the fact that

      significant numbers double on each iteration [6]. lthough the NR method is efficient, if the equation possess multiple roots, multiple turning points or a steep slope in the vicinity of the root, then, the probabilities of the algorithm to diverge or oscillate increase [3, 39].

   4. HOMOTOPY CONTINUATION METHOD. Obtaining multiple solutions using local convergencemethods is complicated since they can obtain unique solutions based on the initial condition established at the beginning of the process. Besides, these methods do not provide information about the existence of additional solutions. Nevertheless, homotopy continuation methods allows solving a system of nonlinear equations like this:

      () = 0, : , (16) for multiple solutions. The original system can be

      expressed as:

   5. SINGLE OPERATION POINT CASE.The computer setup to perform analysis on this paper is as follows: Intel 3 CPU, speed of 2.40GHz, 3 of RAM, Windows 7 Ultimate 64 bits operating system, and the software environment (Maple 17 mathematical and simulation software).

      (, ) = () + (1 )(), (17)

      where (()) = (, ): +1 , , [0,1]. Function () is a auxiliar function that has known, or trivial solution. If = 0, then a known, or trivial, solution for

      () is obtained. If = 1, solutions of the original system

      () are found. By this method, homotopy continuation generates a set of solutions 1(0) for the system () = 0. All the available solutions to solve the system during the continuous deformation are found from = 0 to = 1. Figure 2 shows an schematic diagram of the Homotopy continuation method procedure. This is described step by step as follows:

      Step 1. Define the nonlinear equation that models the problem to solve. It has the form (16), where represents the

      variable(s) of the problem. Then, we define 0, the point where the homotopy begins.

      Step 2. The starting point of the algorithm. is the maximum number of roots to search in the system, is the homotopic parameter and is the counter loop for algorithm. Therefore, we assign 0 = 0, = 0, = 0, = 0, = 0,

      Step 3. This step in the algorithm involves two stages. First, Euler method is applied in order to perform a prediction; starting from (, ) an Euler based prediction is

      made under a tangential scheme to obtain (, ). The second part of this step consists in applying an iterative method to correct the estimation obtained from Euler method and thus to obtain a point over ( + 1, +1) trajectory, see Fig. 2.

      Step 4. In case a crossing at = 1 is detected, then, a jump to step 5 is performed. Otherwise, step 3 is repeated.

      Step 5. Once a crossing at = 1 is detected, points 1 y

      2 are utilized to interpolate 3 over = 1 [35].

      Figure 2: Homotopy continuation method procedure.

      On electronics, the rectifier diode is one of the most important fundamental semiconductor devices; one of their main applications is to perform half-wave rectification of a single-phase supply. This process allows converting alternating current (AC) to direct current (DC) [32, 40]. Now it is proposed, as case study, a series circuit containing a single independent voltage source, one resistor and two diodes; assuming that each diode has different saturation current . The circuit is depicted in Figure 3.

      Step 6. The interpolated 3 point is used as the initial point of NR method; this procedure is done to improve the root precision [23].

      Step 7. If the maximum number of roots is achieved, then it is proceeded to store the found roots and the trajectory is plotted; otherwise, the cycle is repeated at step 3.

      Figure 3: Series circuit with two diodes.

      Using Shockley diode equation [32, 33], that approximatives the behaviour of the diode as follows

      three equations using NR, the circuit current is

      0.3605448941.

      =

      (exp () 1), (18)

      where is the diode current, in Amperes; is the voltage across the diode, in Volts; is the reverse bias saturation current (or scale current); is the magnitude of the charge of an electron, its value is 1.602 × 1019 Coulomb; is the absolute temperature of the – junction; is the Boltzmann constant, its value is 1.380649 × 10231; is the ideality factor, also known as the quality factor or sometimes emission coefficient, depends on the fabrication process, its value is 1 for germanium and 2 for silicon. The

      quotient = / is the thermal voltage, is approximatively 25.85 at 300. Therefore, solving for

      Figure 4: Circuit operating point with two diodes.

   6. SOLUTION USING THE PERTURBATION METHOD.To solve (25), the current is expressed in terms of a power series for [17, 41], given that << 1 and in agreement to the classic perturbation [24, 38]

      = 0 + 1 + 22 + 33 + . (26)

      , the voltage drop for both diodes, considering = 1, the expression becomes

      Taking the first four terms of (26), an approximation of level 3 is obtained; substituting in (25), we obtain

      = ln ( + 1) , > 0, (19) 2 3

      (0 + 1 + 2 + 3 ) + 2ln(0 + 1 +

      22 + 33) ln(2) ln(1) = 0.

      where is the reverse saturation current; is the diode

      current, and is the thermal voltage. Applying Kirchhoff laws, the algebraic equation to describe the operating point is

      Rearranging, (27) can be expressed as

      (27)

      + 1 + 2 = 0.

      (0 + 1 + 22 + 33) + 2ln(0) +

      (20) 2 3

      2 ln (1 + 1+2 +3 ) ln(

      ) ln(

      ) = 0.

      Substituting (19) in (20)

      + ln (1 +

      ) = 0. (21)

      0

      2

      1

      (28)

      Considering that

      2

      1, the complexity of (19) can be

      Replacing the first term of the Taylor expansion of the natural logarithm, we obtain

      reduced by approximating the logarithmic term as follows

      (0 + 1 + 22 + 33) + 2 (ln(0) +

      1+22+33

      = ln (

      ) = ln() ln(). (22)

      0

      ) ln(2) ln(1) = 0. (29)

      Therefore, for each diode we obtain

      Regrouping terms based on and equalling to zero each coefficient, we create the following system of equations

      1 = ln (

      1

      ) = ln() ln(1), (23)

      = ln ( ) = ln() ln(

      ). (24)

      2

      2

      2

      Rewriting (21) using (23) and (24), the mesh equation is obtained

      + 2ln() ln(1) ln(2) = 0. (25)

      The operating point of the circuit, modelled by (20), (23), and (24), is the intersection of three surfaces (see Figure 4). For this example, the proposed values are = 3, = 5,

      1 = 1 12, and 2 = 1 9. Solving the system of

       

      0: 0 = 0,

       

      1: 1 + 2ln(0) ln(2) ln(2) = 0,

      diodes with different saturation currents. Table 1 and Table 2 show results obtained using the following saturation currents

      1 = 1 12 and 2 = 1 9. As can be seen, the results show that our method is general, whether saturation currents are different or equal. Also, it can be noticed the

      2

      2

       

      2:

      3:

      + 2 (1) = 0,

      0

      + 2 (2) = 0.

      circuit current was calculated for different supply voltages and resistor values; the results are in good agreement between the different approximation orders and the exclusively numerical solutions given by Orcad or LTspice (commercial

      3

      0

      (30)

      simulators).

      It is important to notice that the order 1 of the analytic approximation is a compact analytic expression, simple to

      Solving (30), it is possible to obtain approximative analytic expressions for the operating point current

      valuate, accurate compared to orders 2 and 3 which provide higher accuracy, closer to the results obtained from electronic circuit simulators. The downside of these higher order

      1 2

      expressions is the large number of terms.

      =

      ( + ln ( 1 2 )) , > 0, > 0,

       

      2

      (31)

      As have seen in the tables, we tested different supply voltages and different resistor values, this allowed to

      2 2 empirically conclude that the valid region of the bias voltage

      = + 1 ( 2 ) ln (12 ) , > 0, > 0, where approximations have good accuracy is about >

      2

      (32)

      1.355 Volts. In particular, it would be interesting to broaden the proposed method to nonlinear circuits that include MOS

      2 3 2

      2 3 2

       

       

       

       

      = + 1 ( 2 + 4 ) ln (12 ) , > 0, > 0.

      transistors and/or bipolar transistors; using the modified

      2

      2

      (33)

      nodal analysis as a tool to obtain the equilibrium equation of the circuit.

      The approximative solution of the circuit operating point in Figure 3 obtained using classic perturbation uses two

      Current i	E1[3V, 5]	E2[4V, 10]	E3[5V, 15]	E4 [6V, 20]	E5 [12V, 20]	E6 [20V, 20]
      Order 1 (31)	0.3195771786	0.2618848439	0.2418849642	0.2316860801	0.5298942946	0.9285738105
      Iterations PNR	2	2	2	2	2	1
      Order 2 (32)	0.3244097985	0.2636699823	0.2428305403	0.2322747184	0.5301963334	0.9287584471
      Iterations PNR	2	2	2	2	1	1
      Order 3 (33)	0.3243265163	0.2636469094	0.2428207632	0.2322696463	0.5301950321	0.9287579698
      Iterations PNR	2	2	2	1	1	1
      Orcad 16.6	0.3258754939	0.2640259201	0.2429704051	0.2323434187	0.5302105407	0.9287613770
      LTspice IV	0.3257340000	0.2639550000	0.2429230000	0.2323080000	0.5301740000	0.9287240000

      Table 1: Equations (31), (32), and (33) with 1 = 2 = 1 12, =[(Volts), ()], versus commercial simulators.

      Current i	E1[3V, 5]	E2[4V, 10]	E3[5V, 15]	E4 [6V, 20]	E5 [12V, 20]	E6 [20V, 20]
      Order 1 (31)	0.3910033682	0.2975979387	0.2656936941	0.2495426275	0.5477508420	0.9464303578
      Iterations PNR	2	2	2	2	2	1
      Order 2 (32)	0.3946050768	0.2989214854	0.2663930879	0.2499774019	0.5479759489	0.9465688353
      Iterations PNR	2	2	2	2	1	1
      Order 3 (33)	0.3945430073	0.2989043786	0.2663858562	0.2499736556	0.5479749791	0.9465684773
      Iterations PNR	2	2	1	2	1	1
      Orcad 16.6	0.3953079029	0.299095941	0.2664621786	0.2500113734	0.5479827285	0.9465696966
      LTspice IV	0.3952020000	0.2990430000	0.2664270000	0.2499850000	0.5479550000	0.9465410000

      Table 2: Equations (31), (32), and (33) with 1 = 2 = 1 9, =[(Volts), ()], versus commercial simulators.

      Current i	E1[3V, 5]	E2[4V, 10]	E3[5V, 15]	E4 [6V, 20]	E5 [12V, 20]	E6 [20V, 20]
      Order 1 (31)	0.3552902734	0.2797413913	0.2537893292	0.2406143538	0.5388225683	0.9375020841
      Iterations PNR	2	2	2	2	2	1
      Order 2 (32)	0.3595074377	0.2812957338	0.2546118141	0.2411260601	0.5390861412	0.9376636412
      Iterations PNR	2	2	2	1	1	1
      Order 3 (33)	0.3594347618	0.2812756439	0.2546033097	0.2411216509	0.5390850056	0.9376632236
      Iterations PNR	2	2	1	1	1	1
      Orcad 16.6	0.3605448941	0.2815510671	0.2547126722	0.2411756791	0.5390962851	0.9376654206
      LTspice IV	0.3604220000	0.28114890000	0.2546720000	0.2411450000	0.5390640000	0.9376330000

      Table 3: Equations (31), (32), and (33) with 1 = 1 12, 2 = 1 9, =[(Volts), ()], versus commercial simulators.

   7. MULTIPLE OPERATION POINT CASE.It is known, based on semiconductor literature, that the characteristic curve of tunnel diode, even in the simplest circuit, may have multiple operating points (see Figure 5a). For this study case, multiple operating points are calculated in a circuit with such device. One of the common applications of a tunnel diode is in high frequency oscillators [32]. Figure 5b depicts the circuit under study, include voltage source, one resistor, and a tunnel diode all connected in series. [6, 42] proposes a model for tunnel diode, given as

      is Boltzmann constant, and is temperature given in Kelvin. The current is a function of the intrinsic constants of diode [42].

      Applying Kirchhoff voltage law, the mesh equation of the circuit is

      = . (35)

      Substituting (35) in (34), we obtain

      + ( )exp (1 ) + exp ( ( )) = 0,

      = exp (1 ) + exp ( ), (34)

      (36)

      equation (36) of the circuit can not be solved using conventional algebraic methods [43].

      where is the voltage between terminals, given in Volts; peak tension, given in Volts; peak current, given in Amperes; is the current flowing through the diode, in

      Amperes. The quotient is the reciprocal of the thermal

      voltage defined by the

      elemental charge, given in Volts 1;

   8. p>SOLVING METHOD BY COMPUTER.From the classic perturbation theory [17, 24, 38, 41], when a nonlinear equation has a parameter much less than 1, it is possible to apply the methodology mentioned above to obtain an approximative solution. For this case, given that << 1, isFigure 5: Tunnel diode circuit.

      feasible to apply classic perturbation. Thus, the circuit current may be expressed as power series. For this study case

      = 0 + 11 + 22 + 33 + . (37)

      2 3

      Substituting (37) in (36) and expanding using Taylor series of order 2, we obtain

      + () exp (1 ) +

      exp(1

      )()

       

      + (exp ( ( )) exp ( ( )) ( )) = 0.

      (38)

      Grouping terms of order zero from (38), we have

      2exp(+1) 2

      exp( +1)

      exp( +1)

      = ((

      )2 +

      + exp ( + 1) 1)) (

      +

      )) + ( ) +

      2

      0 ( 2

      0 2 1

      exp()

      exp()

      2

      2

       

      ( )2 + .

      exp()

      exp()

      (39)

      Successive terms of (39) are not shown. Taking the coefficient of order zero from perturbative sum (39), which is a second degree trinomial in terms of 0, we are able to solve it using elemental algebra

       

      = ( 1 ) (22exp(2)( )2(exp())2 22 exp( + + ( + 1)) + 4(exp( ))2 + (2 + ( + )) exp( + 1) 2exp( ),

      2( 2exp(+1))

      and

      (40)

       

      = ( 1 ) (22exp(2)( )2(exp())2 22 exp( + + ( + 1)) + 4(exp( ))2 + (2 + ( + )) exp( + 1) 2exp( ),

      2( 2exp(+1))

      (41)

      where is the expansion point for the Taylor series. In consequence, (40) and (41) are expressed in terms of the circuit parameters. Values for the voltage source, resistor, and saturation current of the tunnel diode are proposed as follows: =

      50 3, = 0.3, = 4,

      = 100 3, = 1 9,

      = 401. Figure 6 shows the operating points for the circuit

      in Figure 5b using the values mentioned above.

      Figure 6: Operating point with loading line and tunnel diode curve.

      In this way, the hybrid technique PNR applies classic perturbation to (36) proposing a loop current given by (37), the result are equations (40) and (41). Arbitrary expansion points employed to find the operating points of the circuit are

      = 0.05 and = 0.00. Using the proposed values in the circuit and performing the algebraic procedures, allows obtaining 01, 02, 03, 04; where suffix refers to perturbation. Therefore, to calculate the bias point 1 in Figure 6, 01 is selected as initial point for NR the estimation of 01. In consequence, fast converge of NR is guaranteed to find every operating point (36). The first two columns to the

      left of Table 4 show all the approximative roots calculated using the perturbation method. Notice that solution 04 has been discarded because is not within the operating point region, as it can be seen in Figure 6.

      Applying the hybrid scheme PNR on the NR process can be seen that using current values from the first two columns of Table 4 as starting points, the number of iterations are 7, 5, and 3 while taking 0.261, 0.218, and 0.140, respectively; the error criteria is about 1 10 in Maple 17.

      The advantage of this hybrid NPR technique lies on being a symbolic method, simple to evaluate avoiding the use of homotopy continuation. From the literature about homotopy continuation methods is known that this algorithm is built from fundamental blocks, these are: predictive step; NR corrective step, which is the most expensive block in computational terms; = 1 crossing detection; interpolation and fine-tuning [23, 35], among the most important. Using homotopy continuation took 362 iterations to reach = 1 and 16091 iterations of the NR corrector, taking a total of 115.924 to obtain all operating points. As already shown for the case of multiple operating points, the number of NR iterations exceeded by 16076, being this the main cause for the computation time difference between these two techniques. In consequence, the advantage of using the Hybrid PNR technique, in computational terms, to calculate operating points of the circuit in Figure 6 is noticeable since took just 15 NR iterations and 0.619 total time.

      Figure 7 and Figure 8 depict the homotopy trajectories

      1. of solution to determine the different equilibrium states of the circuit with the three operating points when = 1. The operating points calculated using continuation homotopy are located in the penultimate column in Table 4.
         

         	Perturbation a = 0.05	Perturbation a = 0.00	PNR	PNR Iteration	Homotopy	Iteration
         	ip01 = 0.06495043272A		i1 = 0.07028795173A	7	i01 = 0.070208925515A	335
         	ip02 = 0.04330225198A		i2 = 0.03450748128A	5	i02 = 0.034456395350A	347
         		ip03 = 0.005364498003A	i3 = 0.006118858757A	3	i03 = 0.006053597072A	362
         		ip04 = 0.1747600592A

         	Perturbation a = 0.05	Perturbation a = 0.00	PNR	PNR Iteration	Homotopy	Iteration
         	ip01 = 0.06495043272A		i1 = 0.07028795173A	7	i01 = 0.070208925515A	335
         	ip02 = 0.04330225198A		i2 = 0.03450748128A	5	i02 = 0.034456395350A	347
         		ip03 = 0.005364498003A	i3 = 0.006118858757A	3	i03 = 0.006053597072A	362
         		ip04 = 0.1747600592A

          

         Table 4: Comparison between operating points obtained by different solving methods.

         Figure 7: Homotopy trajectory of the circuit current for multiple

         operating points.

         Figure 8: Homotopy trajectory of voltage for multiple operating points.

         .

   9. CONCLUSIONS.

The first case study proposed an approximative solution using classic perturbation of the operating point for a circuit with two diodes, independent voltage source, and resistor. The solution is an analytical expression in terms of the parameters in the circuit, such as bias value, thermal voltage, saturation current, and resistance. The analytical solution mentioned above has the advantage of being derivable and integrable in terms of all its parameters. For the second case study, the proposal included the use of a hybrid PNR scheme as a mathematical strategy to obtain an approximative result for the multiple solutions that a circuit with nonlinear elements, afterwards the solution will be used as initial point for the NR convergence method in order to increase the accuracy of the solutions for all operating points. Another aspect of the hybrid PNR techniqueis that it avoids the use of random start points on the NR algorithm. The comparison between the hybrid PNR method and homotopy methods showed that the proposed scheme improves the easiness to implement because its fast convergence in a few iterations, thus, computation time is significantly reduced.

Given the success of the methodology, further work includes to extend it in circuits with higher number of components, that includes more diodes and decide if the resultant expressions can provide analytical information. It is possible to improve the understanding about the behaviour of such circuits. A potential application of the analytical expressions is to apply them on stability or frequency analysis [40, 44] in circuits with nonlinear elements.

ACKNOWLEDGMENT

Authors thank M.Sc. Roberto Ruíz Gomez and M.TN. César Colocia García for their technical support and the financial support of Secretaría de Educación Pública México (SEP) through grant 2703 E476300.0275652.

REFERENCES

1. 1. C. Henry Edwards and D. E. Penney. Ecuaciones Diferenciales y problemas con valores en la frontera, cómputo y modelado, cuarta edición. Pearson Educación Prentice Hall, 2009.
   2. S. C. Chapra and R. P. Canale. Métodos númericos para ingenieros, Sexta edición. Mc Graw Hill, 2011.
   3. Richard L. Burden and J. Douglas Faires. Análisis numérico, novena edición,. Cenage Learning, 2011.
   4. Mario Arley Vidal-Geronimo and Gamaliel Ble-Gonzalez. Homotopia de punto fijo. Revista de Ciencias Basicas UJAT, 7(1):314, 2015.
   5. Libin Jiao, Bo Dong, and Bo Yu. Efficiently counting affine roots of mixed trigonometric polynomial systems. Journal of Systems Science and Complexity, 30(4):967982, 2017.
   6. Hector Vázquez-Leal, Luis. Hernández-Martínez, Arturo Sarmiento- Reyes, and Roberto Castaneda-Sheissa. Numerical continuation scheme for tracing the double bounded homotopy for analysing nonlinear circuits. International Conference on Communications, Circuits and Systems, Hong Kong, China, pages pp. 11221126, May 2005.
   7. Shinya Suzumura, Kohei Ogawa, Masashi Sugiyama, Masayuki Karasuyama, and Ichiro Takeuchi. Homotopy continuation approaches for robust sv classification and regression. Machine Learning, 106(7):10091038, 2017.
   8. Delia Torres-Munoz, Luis Hernandez-Martinez, and Hector Vazquez-Leal. Spherical continuation algorithm with spheres of variable radius to trace homotopy curves. International Journal of Applied and Computational Mathematics, pages 113, 2015.
   9. Naser Mostashiri, Alireza Akbarzadeh, and Amir Rezaei. Implementing the homotopy continuation method in a hybrid approach to solve the kinematics problem of spatial parallel robots. Intelligent Service Robotics, pages 114, 2017.
   10. Y. Khan, H. Vázquez-Leal, L. Hernández-Martínez, and N. Faraz. Variational iteration algorithm-ii for solving linear and non-linear odes. International Journal of the Physical Sciences, Vol. 7(25):30994002, 29 June, 2012.
   11. Amina Kassim Hussain, Fadhel Subhi Fadhel, Zainor Ridzuan Yahya, and Nursalasawati Rusli. Variational iteration method (vim) for solving partial integro-differential equations. Journal of Theoretical and Applied Information Technology, 88(2):367, 2016.
   12. H. Vázquez-Leal, B. Benhammouda, U. Filobello-Nino, A. Sarmiento-Reyes, V. M. Jimenez-Fernández, A. Marin-Hernández,
       1. Herrera-May, A. Díaz-Sanchez, and J. Huerta-Chua. Modified taylor series method for solving nonlinear differential equations with mixed boundary conditions defined on finite intervals. SpringerPlus, 3:160, 2014.
   13. Y. Khan, H. Vázquez-Leal, and N. Faraz. An auxiliary parameter method using adomian polynomials and laplace transformation for nonlinear differential equations. Applied Mathematical Modelling, Volume 37, , Issue 5:27022708, 1 March 2013.
   14. MA Banaja, AA Al Qarni, HO Bakodah, Qin Zhou, Seithuti P Moshokoa, and Anjan Biswas. The investigate of optical solitons in cascaded system by improved adomian decomposition scheme. Optik-International Journal for Light and Electron Optics, 130:11071114, 2017.
   15. H. Vázquez-Leal, Y. Khan, U. Filobello-Nino, A. Sarmiento-Reyes,
       1. Díaz-Sánchez, , and L. F. Cisneros-Sinencio. Fixed-term homotopy. Journal of Applied Mathematics, vol. 2013, Article ID 972704:11 pages, 2013.
   16. Hector Vazquez-Leal, Uriel Filobello-Nino, Yildirim Ahmet, Luis Hernandez-Martinez, Roberto Castaneda-Sheissa, Jesus Sanchez- Orea, J. E. Molinar-Solis, and Alejandro Diaz-Sanchez. Transient and dc approximate expressions for diode circuits. ieice electronics express. IEICE Electronics Express, 9(6):522530, 2012.
   17. U. Filobello-Nino, H. Vázquez-Leal, A. Yildirim Y. Khan, V.M. Jimenez-Fernández, A.L. Herrera-May, R. Castaneda-Sheissa, and J. Cervantes-Perez. Perturbation method and laplace-padé approximation to solve nonlinear problems. Miskoic Mathematical Notes, 14(1):89101, 2013.
   18. H. Vázquez-Leal, R. Castaneda-Sheissa, U. Filobello-Nino, A. Sarmiento-Reyes, and J. Sánchez Orea. High accurate simple approximation of normal distribution integral. Mathematical Problems in Engineering, vol. 2012, Article ID 124029,:22 pages, 2012.
   19. H. Vázquez-Leal and K. Boubaker. Approximate solutions for the model of evolution of cocaine consumption in spain using hpm and bpes methods. Nova Scientia, Number 12, Vol. 6(2), 2014.
   20. H. Vázquez-Leal and A. Sarmiento-Reyes. Power series extender method. Mathematical Problems in Engineering, EN PRENSA, 2014.
   21. H. Vázquez-Leal. Rational homotopy perturbation method. Journal of Applied Mathematics, vol. 2012, Article ID 490342:11 pages, 2012.
   22. H. Vázquez-Leal. Generalized homotopy method for solving nonlinear differential equations. Computational and Applied Mathematics, 33, Issue 1:275288, 2014.
   23. Hector Vazquez-Leal, Yasir Khan, G. Fernandez-Anaya, Uriel Filobello-Nino, V. M. Jimenez-Fernandez, A. Herrera-May, Alejandro Diaz-Sanchez, A. Marin-Hernandez, and Jesus Huerta- Chua. Perturbation method applied to a basic diode circuit. Revista Mexiana de F13053fsica, 61(1):6973, 2015.
   24. Mark H. Holmes. Introduction to Perturbation Methods . Springer, 1995.
   25. T. C. Banwell. Bipolar transistor circuit analysis using the lambert w-function. IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, 47(11):16211633, 2000.
   26. E. Di-Costanzo and A. Marasco. Approximate analytic solution of the dirichlet problems for laplaces equation in planar domains by a perturbation method. Computers and Mathematics with Applications, 63(1):6067, 2012.
   27. Abd G. M. El-Latif. On a problem of modified Lindstedt-Poincare for certain strongly non-linear oscillators. Applied Mathematics and Computation, 152(3):821836., 2004.
   28. A. Doroudi and A. Rezaeian-Asl. Calculation of secular axial frequencies in a nonlinear ion trap with hexapole and octopole and decapole superpositions by a modified Lindstedt-Poincare method. International Journal of Mass Spectrometry, 309(0):104108, 2012.
   29. R. Pusenjak. Extended Lindstedt-Poncaire, Method for non- stationary resonances of dynamical systems with cubic nonlinearities. Journal of Sound and Vibration, 314(1-2):194216, 2008.
   30. Liu Hong-Mei. Approximate period of nonlinear oscillators with discontinuities by modified Lindstedt-Poincare method. Chaos, Solitons and Fractals, 23(2):577579, 2014.
   31. Paolo Amore and Alfredo Aranda. Improved Lindstedt Poincare method for the solution of nonlinear problems. Journal of Sound and Vibration, 283(3):11151136, 2005.
   32. Robert L Boylestad and Louis Nashelsky. Electronic Devices and Circuit Theory. Prentice Hall, 2012.
   33. Shilling Belove. Electronic Circuits. McGraw Hill, 1993.
   34. K. Yamamura. Simple algorithms for tracking curves. IEEE transactions on circuts and systems. I. Fundamental theory and applications, Vol. 40 no. 8:537541, Aug. 1993.
   35. M. Sosonkina, L. T. Watson, and D. E. Stewart. Note on the end game in homotopy zero curve tracking. ACM Transactions On Matehematical Software, Vol. 22 no. 3:281287, September. 1996.
   36. Eugene L Allgower and Kurt Georg. Numerical continuation methods. Springer-Verlag, 1990.
   37. U. Filobello-Nino, H. Vázquez-Leal, Y. Khan, A. Perez-Sesma, A. Díaz-Sánchez, A. Herrera-May, D. Pereyra-Díaz, R. Castaneda- Sheissa, V.M. Jimenez-Fernández, and J. Cervantes-Perez. A handy exact solution for flow due to a stretching boundary with partial slip. Revista Mexicana de F13053fsica, E 59:5155, 2012.
   38. Richard H. Bellman. Perturbation techniques in mathematics, engineering and physics. Dover Books Publications, 2003.
   39. Tjalling J. Ypma. Historical development of the newton raphson method. SIAM Review, Vol. 37(4):531551, 1995.
   40. Adel S Sedra and Kenneth Carless Smith. Microelectronic circuits. New York: Oxford University Press, 1998.
   41. Uriel Filobello-Nino, Hector Vázquez-Leal, K. Boubaker, Agustin Perez-Sesma Yasir Khan, Arturo Sarmiento-Reyes, Victor Manuel Jimenez-Fernández, Alejandro Diaz-Sánchez, A. Herrera-May, Jesus Sánchez-Orea, and K. Pereyra-Castro. Perturbation method as a powerful tool to solve highly nonlinear problems: The case of gelfands equation. Asian Journal of Mathematics and Statistics, 6:7682, 2013.
   42. M. Shur. Introduction to electronic devices. J. Wiley, 2005.
   43. J. V. Uspensky. TeorÃa de ecuaciones. Limusa- Noriega S.A. de C.V., 1997.
   44. Katsuhiko Ogata. Modern control engineering. Prentice-Hall, 1996.