Research Article Modified Hyperspheres Algorithm to...

Research ArticleModified Hyperspheres Algorithm to Trace HomotopyCurves of Nonlinear Circuits Composed by Piecewise LinearModelled Devices

H Vazquez-Leal1 V M Jimenez-Fernandez1 B Benhammouda2 U Filobello-Nino1

A Sarmiento-Reyes3 A Ramirez-Pinero1 A Marin-Hernandez4 and J Huerta-Chua5

1 Electronic Instrumentation and Atmospheric Sciences School Universidad Veracruzana Cto Gonzalo Aguirre Beltran SN91000 Xalapa VER Mexico

2Higher Colleges of Technology Abu Dhabi Menrsquos College PO Box 25035 Abu Dhabi UAE3National Institute for Astrophysics Optics and Electronics Luis Enrique Erro No 1 Santa Marıa Tonantzintla72840 Puebla PUE Mexico

4Department of Artificial Intelligence Universidad Veracruzana Sebastian Camacho No 5 91000 Xalapa VER Mexico5 Facultad de Ingenierıa Civil Universidad Veracruzana Venustiano Carranza SN Colonia Revolucion93390 Poza Rica VER Mexico

Correspondence should be addressed to H Vazquez-Leal hvazquezuvmx

Received 22 April 2014 Accepted 2 July 2014 Published 11 August 2014

Academic Editor Ishak Altun

Copyright copy 2014 H Vazquez-Leal et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We present a homotopy continuation method (HCM) for finding multiple operating points of nonlinear circuits composed ofdevicesmodelled by using piecewise linear (PWL) representationsWe propose an adaptation of themodified spheres path trackingalgorithm to trace the homotopy trajectories of PWL circuits In order to assess the benefits of this proposal four nonlinear circuitscomposed of piecewise linear modelled devices are analysed to determine their multiple operating points The results show thatHCM can find multiple solutions within a single homotopy trajectory Furthermore we take advantage of the fact that homotopytrajectories are PWL curves meant to replace the multidimensional interpolation and fine tuning stages of the path trackingalgorithm with a simple and highly accurate procedure based on the parametric straight line equation

1 Introduction

The circuit simulation tools are constantly improved in orderto cope with the challenges due to the new fabricationtechnologies Among the circuit analysis methodologies thedirect current (DC) analysis is highlighted as one of themost important because it describes the static behaviour ofthe circuits As a result of the DC analysis of nonlinearcircuits one obtains a nonlinear algebraic equations system(NAES) The most common method applied to solve suchequations is the Newton-Raphson method (NRM) Howeverit is common that NRM fails due to its well-known problemsof convergence oscillation and divergence to infinity amongothers In fact NRM has a local convergence only which

means that if the starting point is not close enough to thesought solution the method will probably diverge What ismore if the circuit under analysis is multistable then NRMwill not be helpful because it can locate only one solutionper simulation ignoring the existence of more solutionsTherefore the homotopy continuationmethod (HCM) [1ndash37]arises as an alternative to NRM due to its characteristics tofind multiple operating points and better convergence [38]

In recent years the PWL modelling technique gainedpopularity as a tool for circuit simulation and other relatedareas [39 40] The basic idea is to replace traditional modelsby their piecewise linear (PWL) representations [41ndash44] Themain advantages are reduction of equations complexity thestraightforward inclusion of empirical models and potential

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 938598 11 pageshttpdxdoiorg1011552014938598

2 The Scientific World Journal

replacing of piecewise models by their unified PWL repre-sentation Several methodologies have been proposed to findmultiple solutions of PWL circuits [45ndash57]

However such methodologies exhibit some drawbackslike the requirement of several initial points to find multiplesolutions [53 54] the use of implicit PWL models [55 56]and the need of expressing the circuit equations in termsof the linear complementary problem (LCP) that impliescomputing model state variables [57] Therefore in orderto circumvent the aforementioned disadvantages we explorethe application of HCM methods in combination with anadaptation of the modified spheres algorithm (MSA) [37] forthe DC analysis of PWL circuits

This paper is organized as follows A brief descriptionof PWL modelling is presented in Section 2 In Section 3we introduce the proposed HCM and its path followingtechnique (MSA) In Section 4 four case studies of nonlinearcircuits are presented and solved by using a HCM methodNumerical simulations and a discussion about the results areprovided in Section 5 Finally a concluding remark is givenin Section 6

2 Brief Description of PWL Modelling

A mathematical model approach widely used in nonlinearcircuit analysis is the so-called piecewise linear (PWL)The aim of this kind of modeling is to approximate thenonlinear behavior of a circuit element by using a set oflinear mappingsThis means transforming a single nonlinearequation into a finite number of linear equations One of thefirst piecewise linear models was provided by Chua and Kangin [58] Another proposal was presented by Van Bokhovenin [59] Subsequent contributions were the extension of theChua model reported by Guzelis and Goknar in [60] andthe parametric proposal given by Vandenberghe et al in[61] among others While there are diverse proposals ofPWL models they can be classified into two classes Thefirst one contains explicit models For this class of modelsthe output vector can be obtained by just substituting theinput vector into the modelThe second one contains modelswhich are implicit In such models the output vector cannotbe obtained directly In contrast an algorithm has to beperformed by which the output vector is computed [62] Themore representative examples of explicit and implicit PWLdescriptions are the canonical model of Chua1 and the modelof Bokhoven1 respectively

The formal definition of the Chua1 model is expressed asfollows

Theorem 1 Any one-dimensional piecewise linear curve with119871 segments and 120590 break points 120573

1lt 1205732

lt sdot sdot sdot lt 120573120590can be

represented by the expression

119910 (119909) = 119886 + 119887119909 +

120590

sum

119894=1

119888119894

1003816100381610038161003816119909 minus 120573119894

1003816100381610038161003816 (1)

where the model parameters can be computed by

119886 = 119910 (0) minus

120590

sum

119894=1

119888119894

10038161003816100381610038161205731198941003816100381610038161003816

119887 =119869(1)

+ 119869(120590+1)

2

119888119894=

119869(119894+1)

minus 119869(119894)

2 119894 = 1 2 120590

(2)

with 119869(119894) denoting the slope of the 119894th constitutive segment in the

piecewise linear curve

Meanwhile the Bokhoven1 model is expressed by a statevariable system defined in formulation of LCP For furtherdetails about LCP the reader is referred to [63]

The main factor that motivates the use of PWL modelsis the simplicity of their structure which is linear in eachregion of the domainHowever in terms of circuit analysis theuse of piecewise linear models means transforming a singlenonlinear equation into several linear equations that couldeasily be solved by standard methods from linear algebraThe problem lies now in the extremely large number oflinear regions to be discarded to determine the entire set ofcircuit solutions Unfortunately this task requires enormouscomputational resources To overcome that problem severalmethodologies and algorithms have been proposed Forexample Chua and Ying [64] reported an efficient methodwhere the number of linear simultaneous equations to besolved could be decreased by a sign test The same idea isimproved by Yamamura and Ochiai in [65] where linearprogramming techniques are applied and a more efficientsign test algorithm is also reported Katzenelson presents analgorithm based on Newtonrsquos homotopy in [53] and morerecently Tadeusiewicz and Kuczynski offered a method thatcombines the homotopy concept and the theory known as alinear complementary problem [57]

3 The Proposed Homotopy SchemeThe equilibrium equation to describe the DC behaviour isobtained using the Kirchhoff laws resulting in

f (x) = 0 f isin R119899997888rarr R

119899 (3)

where x represents the electrical variables of the circuit and 119899

the number of variablesHomotopy methods are based on the fact that solutions

are connected by a curve denominated ldquosolution curverdquo orldquohomotopy curverdquo Such curve is induced by including anextra parameter in the original NAES resulting in

H (f (x) 120582) = 0 H isin R119899timesR 997888rarr R

119899 (4)

where 120582 is the homotopy parameter andHminus1(0) the family ofsolutions that conforms the homotopy path

An example of homotopy formulation is Newtonrsquos homo-topy

H (f (x) 120582) = f (x) + (120582 minus 1) f (x119894) = 0 (5)

where x119894is the starting point of the trajectory

The Scientific World Journal 3

(a) General homotopy curve (b) Piecewise linear homotopy curve

Figure 1 Solution curves with spheres [37]

This system has the following properties

(1) At the starting point 120582 = 0

H (f (x) 0) = f (x) minus f (x119894) = 0 (6)

where the homotopy system admits at least the solu-tion x

119894

(2) Thedeformation continues until crossing120582 = 1where

H (f (x) 1) = f (x) = 0 (7)

that is the homotopy is reduced to (3)

Thus the original problem becomes a numerical continu-ation problem [4 5 12 13 21 25ndash28] where the continuationvariable is the homotopy parameter 120582 The homotopy mapcreates a continuous line that crosses several times 120582 = 1

depending on the number of operating points A drawbackof the homotopy methods is that there is no generalizedmethodology to guarantee that a single homotopy pathpossesses all the operating points of any given nonlinearcircuit In contrastHCMcan locatemultiple operating pointsin comparison to NRM that can fail to find even a singleoperating point

31 Modified Spheres Algorithm Once the equilibrium equa-tion and homotopy map are constructed a new problememerges the homotopy trajectory should be traced in orderto detect the roots It is well known from the literature that ifthe path tracking algorithm is not correctly implemented thesimulation may fail to detect any root even though the rootsare in fact along the curve [4 5 12 13 21 25ndash28] For the caseof PWL circuits the problem for the path tracking algorithmlies in the prediction stage because most of the predictormechanisms are based on the tangent of the homotopy curveIf we consider that the derivative of PWL functions is notdefined at the break points then the tangent of the homotopycurve can not be evaluated at such points Therefore wepropose adapting the modified spheres algorithm (MSA) forthe path following of the homotopy curves of PWL circuitswhich is not based on the use of tangents of the trajectory

The homotopy formulation contains 119899 equations and(119899 + 1) variables where 119909

119894(119894 = 1 119899) represent the

variables of the system and 119909119899+1

is the homotopy parameter

120582 Nevertheless if we add the equation that describes a sphere[2 3 13 37 66] with center at 119888 (initial point of the trajectory)and radius 119903 expressed by

119878 (1199091 1199092 119909

119899+1) = (119909

1minus 1198881)2

+ (1199092minus 1198882)2

+ sdot sdot sdot + (119909119899+1

minus 119888119899+1

)2

minus 1199032= 0

(8)

then it is possible to apply a regular NRM to solve thehomotopy formulation

Therefore using (4) and (8) we formulate the augmentedsystem as

1198671(1198911(119909) 120582) = 0

1198672(1198912(119909) 120582) = 0

119867119899(119891119899(119909) 120582) = 0

119878 (1199091 1199092 119909

119899 120582) = 0

(9)

The solution curve can be traced by solving (9) for eachhypersphere and updating the center of the hypersphere ineach iteration stepThe hyperspheres (119878

1 1198782 ) are allocated

successively as shown in Figure 1(a) at each step the solutionobtained is used as the center of the new sphere In the samefashion Figure 1(b) depicts the application ofMSA algorithmfor the path tracking of PWL curves

The proposed adaptation of the MSA scheme [37] for theNewton homotopy applied to PWL circuits is described asfollows

(i) Predictor we use points 1198741and 119874

2to predict the

point 1198961 The next predictor point and successive

points are obtained as depicted in Figure 2(a)(ii) Corrector after calculating the point predictor (119896

1) a

corrected point (1198743) is calculated by solving (9) This

procedure is detailed in [37] Nonetheless if we con-sider thatmdashfor this workmdashthe homotopy trajectoryis described as a PWL curve then the corrector stepwill require most of the time one iteration to correctthe prediction over straight lines except at the breakpoints where it will require more steps to correct thecurve (see Figure 2(b))


S1

S2

S3

o2

o3

o1

k1

k2

Homotopy trajectory

(a) General homotopy trajectory

S1

S2

S3

S4

o4

o2

o3

o1

k1

k2

Homotopy trajectory

(b) Piecewise linear homotopy trajectory

Figure 2 Spheres algorithm [37]

(iii) There is a potential issue called reversion phenom-enon that provokes a backward tracing In [37] a strat-egy based on gradients and angles of the intersectionof the sphere along the trajectory is proposed

(iv) Find zero strategy [12 22] the finding zero strategyshould start after the trajectory crosses 120582 = 1 Thisprocedure requires detecting the two points (A andB) before and after 120582 = 1 as depicted in Figure 3

(v) Interpolation of operating points [12 22] traditionalschemes of path tracking algorithms require the appli-cation of complicatedmultidimensional interpolationalgorithms as those reported in [37] Nonethelessas we will show in the cases study section thehomotopy trajectory of PWL circuits is also a PWLcurve Therefore we propose using the formula ofa parametric straight line to interpolate the solutionat 120582 = 1 Using the points 119860 and 119861 we create twovectors A and B respectively resulting in the follow-ing equation

B + 119905 (B minus A) = 0 (10)where 119905 is the parameter that describes the 119899 + 1-dimensional straight line To perform the interpola-tion we obtain the value of 119905 that induces 120582 = 1 andupdate the rest of the equations to obtain thesought solution 119878

lowast(see Figure 3) This process can be

repeated each time the homotopy trajectory crosses120582 = 1

(vi) Improving accuracy for final solutions also known asfine tuning [22] traditional path following schemesincluding the ones reported for the MSA scheme[13 37] require extra steps of NRM to improve theaccuracy of the interpolated solutions However theaforementioned interpolation step can theoreticallyobtain a highly accurate solution The reason relieson the fact that the homotopy curve crosses exactlyover the roots of the equilibrium equation then thestraight line (10) also crosses over the exact solution

0 05 1 15

x

120582

4

2

0

minus2

Slowast

A

B

Figure 3 Interpolation procedure using a parametric straight line

4 Cases Study

In the present section we will solve four case studies [64]to show the usefulness of the proposed method to performthe DC analysis of nonlinear circuits composed of devicesmodelled using the explicit PWL model (1) For all the casesrsquostudy we use a constant radius 119903 = 01 for the hyperspheres

41 Circuit with Two Nonlinear Resistors The following casestudy shows a simple circuit composed of two nonlinearresistors as depicted in Figure 4 The models of the resistors1198771and 119877

2are

1198771 1198941= minus

125

8+

9

8V1+

7

8

1003816100381610038161003816V1 + 11003816100381610038161003816 minus

3

2

1003816100381610038161003816V1 minus 21003816100381610038161003816 +

3

4

1003816100381610038161003816V1 minus 51003816100381610038161003816

minus1

8

1003816100381610038161003816V1 minus 111003816100381610038161003816 minus

9

8

1003816100381610038161003816V1 minus 131003816100381610038161003816 + 2

1003816100381610038161003816V1 minus 151003816100381610038161003816


Table 1 Numerical solutions for (12)

Solution Iteration V1

V2

MSE = (1198912

1+ 1198912

2)2

1198781

168 149999999999 149999999998 01198782

196 400000000122 0999999999476 222119890 minus 18

1198783

214 566666666670 0666666666664 212119890 minus 20

+

minus

9V

R1

R2

V1

V2

i1

i2

+

minus

+

minus

2Ω

Figure 4 Two nonlinear resistor circuits

1198772 1198942=

29

4+

3

2V2minus

3

2

1003816100381610038161003816V2 + 81003816100381610038161003816 +

3

2

1003816100381610038161003816V2 + 51003816100381610038161003816 minus

3

2

1003816100381610038161003816V2 + 31003816100381610038161003816

+3

2

1003816100381610038161003816V2 + 11003816100381610038161003816 minus

3

4

1003816100381610038161003816V2 minus 31003816100381610038161003816 minus

5

4

1003816100381610038161003816V2 minus 81003816100381610038161003816 +

3

2

1003816100381610038161003816V2 minus 101003816100381610038161003816

+1003816100381610038161003816V2 minus 13

1003816100381610038161003816 minus5

4

1003816100381610038161003816V2 minus 161003816100381610038161003816 +

1

4

1003816100381610038161003816V2 minus 181003816100381610038161003816

(11)

described by 7 and 11 PWL segments respectivelyUsing Kirchhoff laws we obtain

1198911(V1 V2) = V1+ V2+ 21198941minus 9 = 0

1198912(V1 V2) = V1+ V2+ 21198942minus 9 = 0

(12)

Applying the Newton homotopy to (12) combined with MSAyields

1198671(V1 V2 120582) = 119891

1(V1 V2) + (120582 minus 1) 119891

1(V10

V20

) = 0

1198672(V1 V2 120582) = 119891

2(V1 V2) + (120582 minus 1) 119891

2(V10

V20

) = 0

119878 (V1 V2 120582) = (V

1minus 1198881)2

+ (V2minus 1198882)2

+ (120582 minus 1198883)2

minus 1199032= 0

(13)

where V10

= minus5 and V20

= minus4 are the initial point of thehomotopy at 120582

0= 0 and 119878(V

1 V2 120582) is the equation of the

hypersphere whose center will be updated at each iteration ofthe method

For the first hypersphere the center is located at 1198881= V10

1198882

= V20 and 119888

3= 1205820 The centers of the successive hyper-

spheres are obtained using the aforementioned procedure inSection 31 As a result of MSA algorithm the three operatingpoints of the circuit have been located (see Figure 5) Inaddition Table 1 shows the computed solutions iterationsand the mean square error (MSE)

42 Circuit withThreeNonlinear Resistors The following casestudy shows a circuit composed of three nonlinear resistors asdepicted in Figure 6 The models of 119877

1 1198772 and 119877

3resistors

are

1198771 1198941=

5

6

1003816100381610038161003816V1 + 61003816100381610038161003816 minus

5

6

1003816100381610038161003816V1 minus 61003816100381610038161003816

1198772 V2=

1

6

10038161003816100381610038161198942 + 11003816100381610038161003816 minus

1

6

10038161003816100381610038161198942 minus 51003816100381610038161003816

1198773 1198943= V3minus

5

4

1003816100381610038161003816V3 minus 11003816100381610038161003816 + 2

1003816100381610038161003816V3 minus 21003816100381610038161003816 minus

1003816100381610038161003816V3 minus 31003816100381610038161003816

(14)

described by 3 3 and 4 PWL segments respectivelyUsing Kirchhoff laws [64] we obtain

1198911(V1 V2 V3) = V1+ 1198942+ V3minus 1198941minus 5 = 0

1198912(V1 V2 V3) = 1198942+ V3minus V2minus 5 = 0

1198913(V1 V2 V3) = minusV

3minus 1198943+ 5 = 0

(15)

Next we apply the Newton homotopy to (15) as done forthe first case study using V

10= 15 119894

20= minus1 and V

30= 15

as the initial point of the homotopy As a result of tracing thehomotopy path the three operating points of the circuit havebeen located (see Figure 7) In addition Table 2 shows thefound solutions iterations and themean square error (MSE)

43 Schmitt Trigger Circuit Consider the Schmitt trigger cir-cuit of Figure 8(a) where the bipolar transistors aremodelledusing the simplified Ebers-Moll (see Figure 8(b)) model ofNPN transistors as depicted in Figure 8(c) The PWL modelof five segments of the diodes of all transistors is

119894119889(V119889) = minus 005486777833 + 01482755558V

119889

+ 0011577793181003816100381610038161003816V119889 minus 0306

1003816100381610038161003816

+ 0011818697881003816100381610038161003816V119889 minus 03375

1003816100381610038161003816

+ 0049045369221003816100381610038161003816V119889 minus 0366

1003816100381610038161003816

+ 0075833695151003816100381610038161003816V119889 minus 03875

1003816100381610038161003816

(16)




1198942

V3

MSE1198781

137 99333333333 220000000000 286666666667 171119890 minus 22

1198782

382 00999999999 220000000000 286666666666 145119890 minus 22

1198783

610 minus100666666666 220000000000 286666666667 386119890 minus 23

S1

S2

S3

1

6

4

2

0

minus2

minus4

0 02 04 06 08 1

120582

(a) Projection 120582-V1

S1

S2

S3

2

2

1

0

minus1

minus2

minus3

minus4

0 05 1 15

120582

(b) Projection 120582-V2

Figure 5 Homotopy path for (13)

R1

R2

R3minus

minusminus

minus

+

minus

+

++

+

minus

+5V11

2

3

i2

i3

i3

2i1

22

i1

2Ω1Ω 1Ω 1Ω

Figure 6 Three nonlinear resistor circuits

S1

S2

S3

15

10

5

0

minus5

minus10

1

0 05 1 15 2

120582

Figure 7 Projection 120582-V1of homotopy path for (15)

Using Kirchhoff laws we obtain

1198911(V1 V2) = 333 minus 1000 (119894

119889(V1) + 119894119889(V2)) minus V

1= 0

1198912(V1 V2) = 4 minus 1392119894

119889(V1) minus 1096119894

119889(V2) minus V2= 0

(17)

Then Newton homotopy is applied in the same fashionas in the first example using as starting point V

10= minus5 and

V20

= minus2 at 120582 = 0 The results show that the homotopy tra-jectory crosses for the three operating points of the Schmitttrigger circuit as depicted in Figure 9 and Table 3

44 Chuarsquos Circuit with Nine Solutions Consider Chuarsquos cir-cuit of Figure 10 where the bipolar transistors are modelledusing the simplified Ebers-Moll (see Figure 8(b)) model ofNPN transistors The PWL model for the diodes of alltransistors is (16)


1198911(V1 V2 V3 V4) = 436634V

2+ 6103168119894

119889(V1)

+ 2863168119894119889(V2) minus 12 = 0

1198912(V1 V2 V3 V4) = 54V

1+ V3+ 3580119894

119889(V1) + 6620119894

119889(V2)

+ 700119894119889(V3) + 500119894

119889(V4) minus 22 = 0

1198913(V1 V2 V3 V4) = 436634V

4+ 6103168119894

119889(V3)

+ 2863168119894119889(V4) minus 12 = 0

1198914(V1 V2 V3 V4) = V1+ 54V

3+ 700119894

119889(V1)

+ 3580119894119889(V3) + 6620119894

119889(V4) minus 22 = 0

(18)

The Newton homotopy is applied to (18) in the sameway as in the first case study We trace two trajectories withthe following starting points 119876

1= [minus7 minus1 8 1] and 119876

2=

[0 minus7 0 0] After using the adaptedMSA algorithm the ninesolutions of the circuit were found (see Figure 11) In additionTable 4 shows the found solutions iterations and the meansquare error (MSE)




V2

MSE1198781

59 0022145787125 0374592098924 611119890 minus 18

1198782

63 0348923957274 0358608185920 170119890 minus 21

1198783

68 0372176159314 minus0117291118129 5119890 minus 21

333V+

minus

1kΩ

11kΩ

8kΩ

1kΩ

1kΩ

minus

+

10V

T1

T2

(a) Original circuit

098idCollector (C)

Base (B)

Emitter (E)

+

minus

idd

B

E

C

(b) Simplified Embers-Moll model

333V+

minus

+

minus

+

minus

+

minus

d1 id1

1kΩ

1kΩ

11kΩ

8kΩd2 id2

1kΩ

098id2

098id1

10V

(c) Schmitt trigger circuit using Embers-Moll model

Figure 8 Schmitt trigger circuit

5 Numerical Simulation and Discussion

All case studies were successfully solved using the proposedmethodology For the first three case studies it was possibleto find within a single trajectory the three operating pointsof each problem and for the last case study we find thenine solutions of Chuarsquos circuit using two starting pointsThehigh accuracy of the located operating points shows that thesimple interpolation algorithm based on straight lines is apowerful tool and is simple to implement (see Tables 1ndash4)Besides the accuracy of the interpolate solutions allows usto discard the stage of applying NRM extra steps to increaseaccuracy usually required by path tracking algorithms [4 512 13 21 25ndash28] It is important to remark that the variety ofsolved circuits exhibits the high potential of HCM combinedwith MSA to solve multistable nonlinear circuits integratedby devices modelled with explicit PWL representations

In [53 54] methods based on the Newton homotopy arereported which are capable of locating only one solution per

simulation Therefore if user requires to find more solutionsit is necessary to propose some random initial points to per-form more simulations Instead the proposed methodologyis capable of locatingmultiple operating points within a singlepath or simulation

Methods reported in [55 56] use implicit PWL modelsThis implies that the number of linear regions explodes due tothe diode synthesis Besides compared to the explicit modelsimplicit PWL models require a more complex algorithm tocompute the model state variables The proposed methodol-ogy uses an explicit model representation easy to implement

In [57] a methodology that depends on the specificcircuit topology description of multiport with extracted idealdiodes is reported In such methodology circuit equationsare expressed in terms of the LCP which implies computingmodel state variablesThe proposedmethodology is based ona straightforward methodology based on the traditional cir-cuit analysis tools used to build commercial circuit simulatorsand a simple path tracking algorithm easy to implement



Path Solution Iteration V1

V2

V3

V4

MSE1198761

1198781

105 minus0734973033383 0376723496304 0318166142629 0372621919721 114119890 minus 17

1198761

1198782

111 minus0650249656974 0376723495848 minus0239265377696 0376723494677 664119890 minus 14

1198761

1198783

235 0326931131382 0369666979551 minus048305281023 0376723495005 361119890 minus 16

1198761

1198784

243 0329725453371 0368724929946 0324331786706 0370543296483 627119890 minus 19

1198761

1198785

312 0386520101863 minus429094430670 0337714498625 0365872023364 640119890 minus 17

1198761

1198786

328 0388146243604 minus475726420472 minus112762657849 0376723498380 569119890 minus 17

1198762

1198787

58 0383283219902 minus363542706051 0383283217035 minus363542647848 173119890 minus 16

1198762

1198788

193 0338139358469 0364994969864 0387969158453 minus468386026807 286119890 minus 19

1198762

1198789

215 minus119554608083 0376723498781 0389904592568 minus548612114477 242119890 minus 16

S1

S2S3

1

0

minus1

minus2

minus3

minus4

minus5

0 02 04 06 08 1 12

120582

(a) Homotopy path

S1

S2

S3

1

120582

095 1 105 110

04

03

02

01

00

minus01

minus02

(b) Zoom to the solutions region

Figure 9 Homotopy path for (17) projected over 120582-V1

10k

10Vminus

+

4k30k

12V+

minus 4k

i1 i2

+

minus2 30k

minus

+2V

30k

1

+

minus 05 k 101 k5k

1

+

minus

i305 k 101 k

i4

30k+

minus

1

4k 4k

+

minus

12V

Figure 10 Chuarsquos circuit with nine solutions

Further research should be addressed in the followingtopics

(i) Implement a strategy to use the fact that the homo-topy curves are straight lines to accelerate the homo-topy simulation

(ii) Implement a circuit simulator to solve high densitytransistor circuits modelled by the PWL technique

(iii) Replace the Newton homotopy by other methodsas the fixed point homotopy [14] double boundedhomotopy [12 37] double bounded polynomialhomotopy [11 36] Newton fixed-point homotopy[67] 119889-homotopy [68] and multiparameter homo-topy [13 17] among others This research can leadto proposal of better homotopy schemes with betterresults in aspects like number of found solutions CPUtime and global convergence among others

(iv) Theoretically obtain the position of the break pointsof the PWL homotopy curve significantly decreasingthe number of steps Such research can conduct to avery fast path tracking scheme


3

120582

8

6

4

2

0

minus2

0 02 04 06 08 1

(a) Homotopy path for initial point1198761

S1

S2

S3

S6

S4 S5

3

09999 1 10001 10003

120582

minus05

minus1

minus15

0

(b) Zoom for (a)

3

120582

05

04

03

02

01

0

0 05 1 15 2 25

(c) Homotopy path for initial point1198762

S9

S8

S7

3

120582

090 095 1 105 110 115 120

03950

03925

03900

03875

03850

03825

03800

(d) Zoom for (c)

Figure 11 Homotopy paths for (18) projected over 120582-V3

(v) Propose a methodology to obtain an optimal initialpoint for the homotopy simulationThis golden start-ing point will possess the characteristic of producinga minimum number of iterations and a maximumnumber of found solutions or all solutions

6 Conclusions

In this work we presented a homotopy scheme based onthe Newton homotopy and a modified MSA path trackingalgorithm applied to the DC simulation of nonlinear cir-cuits composed of devices modelled by PWL techniquesThe effectiveness and power of the proposed scheme wereexhibited by the successful solution of all the operatingpoints of several circuits including devices as nonlinearresistors diodes transistors and transactors among oth-ers In addition the high accuracy of the solutions wasreached by applying a simple interpolation technique thatdiscards the use of Newton-Raphson extra steps to increasethe accuracy of the interpolated solutions Finally furtherresearch should be performed to extend the applicationof the proposed scheme to very large integrated circuits(VLSI)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to acknowledge Alejandra Diaz-Armendariz Roberto Ruiz-Gomez and Rogelio-AlejandroCallejas-Molina for their contribution to this project HVazquez-Leal gratefully acknowledges the financial supportprovided by theNational Council for Science andTechnologyof Mexico (CONACyT) through Grant CB-2010-01 157024

References

[1] H Jimenez-Islas G M Martinez-Gonzalez J L Navarrete-Bolanos J E Botello-Alvarez and J Manuel Oliveros-MunozldquoNonlinear homotopic continuation methods a chemical engi-neering perspective reviewrdquo Industrial amp Engineering ChemistryResearch vol 52 no 42 pp 14729ndash14742 2013

[2] J M Oliveros-Munoz and H Jimenez-Islas ldquoHypersphericalpath trackingmethodology as correction step in homotopiccontinuation methodsrdquo Chemical Engineering Science vol 97pp 413ndash429 2013


[3] H Jimenez-Islas ldquoSehpe programa para la solucion de sistemasde ecuaciones no lineales mediante metodo homotopico conseguimiento hiperesfericordquo Avances en Ingenierıa Quımica vol6 pp 174ndash179 1996

[4] D J Bates J D Hauenstein A J Sommese and I WamplerldquoAdaptive multiprecision path trackingrdquo SIAM Journal onNumerical Analysis vol 46 no 2 pp 722ndash746 2008

[5] D-J Bates J-D Hauenstein A-J Sommese and C-W Wam-pler ldquoStepsize control for adaptivemultiprecision path track-ingrdquo Contemporary Mathematics vol 496 pp 21ndash31 2009

[6] A Morgan and A Sommese ldquoComputing all solutions to pol-ynomial systems using homotopy continuationrdquo Applied Math-ematics and Computation vol 24 no 2 pp 115ndash138 1987

[7] HVazquez-Leal R Castaneda-Sheissa A Yildirim et al ldquoBipa-rameter homotopy-based direct current simulation of multi-stable circuitsrdquo Journal ofMathematics amp Computer Science vol2 no 3 pp 137ndash150 2012

[8] A Ushida Y Yamagami Y Nishio I Kinouchi and Y InoueldquoAn efficient algorithm for finding multiple DC solutionsbased on the SPICE-orientedNewton homotopymethodrdquo IEEETransactions on Computer-Aided Design of Integrated Circuitsand Systems vol 21 no 3 pp 337ndash348 2002

[9] K Ahuja L T Watson and S C Billups ldquoProbability-onehomotopy maps for mixed complementarity problemsrdquo Com-putational Optimization andApplications vol 41 no 3 pp 363ndash375 2008

[10] R C Melville L Trajkovic S Fang and L TWatson ldquoArtificialparameter homotopy methods for the DC operating pointproblemrdquo IEEE Transactions on Computer-Aided Design ofIntegrated Circuits and Systems vol 12 no 6 pp 861ndash877 1993

[11] H Vazquez-Leal L Hernandez-Martınez A Sarmiento-ReyesR Castaneda-Sheissa and A Gallardo-Del-Angel ldquoHomotopymethod with a formal stop criterion applied to circuit simu-lationrdquo IEICE Electronics Express vol 8 no 21 pp 1808ndash18152011

[12] H Vazquez-Leal L Hernandez-Martınez A Sarmiento-Reyesand R Castaneda-Sheissa ldquoNumerical continuation scheme fortracing the double bounded homotopy for analysing nonlinearcircuitsrdquo in Proceedings of the International Conference onCommunications Circuits and Systems pp 1122ndash1126 HongKong China May 2005

[13] H Vazquez-Leal R Castaneda-Sheissa F Rabago-Bernal ASarmiento-Reyes and U Filobello-Nino ldquoPowering multipa-rameter homotopy-based simulation with a fast path-followingtechniquerdquo ISRN Applied Mathematics vol 2011 Article ID610637 7 pages 2011

[14] K Yamamura T Sekiguchi and Y Inoue ldquoA fixed-point homo-topymethod for solvingmodified nodal equationsrdquo IEEETrans-actions on Circuits and Systems I Fundamental Theory andApplications vol 46 no 6 pp 654ndash665 1999

[15] L Vandenberghe and J Vandewalle ldquoVariable dimension algo-rithms for solving resistive circuitsrdquo International Journal ofCircuitTheory andApplications vol 18 no 5 pp 443ndash474 1990

[16] R Geoghegan J C Lagarias and R C Melville ldquoThreadinghomotopies and dc operating points of nonlinear circuitsrdquoSIAM Journal on Optimization vol 9 no 1 pp 159ndash178 1999

[17] D M Wolf and S R Sanders ldquoMultiparameter homotopymethods for finding dc operating points of nonlinear circuitsrdquoIEEE Transactions on Circuits and Systems I FundamentalTheory and Applications vol 43 no 10 pp 824ndash838 1996

[18] L T Watson and R T Haftka ldquoModern homotopy methodsin optimizationrdquo Computer Methods in Applied Mechanics andEngineering vol 74 no 3 pp 289ndash305 1989

[19] V M Perez J E Renaud and L T Watson ldquoHomotopycurve tracking in approximate interior point optimizationrdquoOptimization and Engineering vol 10 no 1 pp 91ndash108 2009

[20] M Kuno and J-D Seader ldquoComputing all real solutionsto systems of nonlinear equations with a global fixed-pointhomotopyrdquo Industrial amp Engineering Chemistry Research vol27 no 7 pp 1320ndash1329 1988

[21] L T Watson ldquoNumerical linear algebra aspects of globallyconvergent homotopy methodsrdquo SIAM Review vol 28 no 4pp 529ndash545 1986

[22] M Sosonkina L TWatson andD E Stewart ldquoNote on the endgame in homotopy zero curve trackingrdquo ACM Transactions onMathematical Software vol 22 no 3 pp 281ndash287 1996

[23] L T Watson S M Holzer and M C Hansen ldquoTrackingnonlinear equilibrium paths by a homotopymethodrdquoNonlinearAnalysisTheory Methods amp Applications vol 7 no 11 pp 1271ndash1282 1983

[24] L T Watson ldquoGlobally convergent homotopy methodsrdquo inEncyclopedia of Optimization C A Floudas and P M PardalosEds pp 1272ndash1277 Springer New York NY USA 2009

[25] L T Watson S C Billups and A P Morgan ldquoAlgorithm 652hompack a suite of codes for globally convergent homotopyalgorithmsrdquoAssociation for ComputingMachinery Transactionson Mathematical Software vol 13 no 3 pp 281ndash310 1987

[26] D J Bates J D Hauenstein and A J Sommese ldquoEfficient pathtrackingmethodsrdquoNumerical Algorithms vol 58 no 4 pp 451ndash459 2011

[27] E L Allgower and K Georg Numerical Path Following 1994[28] E L Allgower and K Georg Continuation and Path Following

1992[29] K S Gritton J D Seader and W Lin ldquoGlobal homotopy

continuation procedures for seeking all roots of a nonlinearequationrdquo Computers amp Chemical Engineering vol 25 no 7-8pp 1003ndash1019 2001

[30] L Trajkovic R-C Melville and S-C Fang ldquoPassivity and no-gain properties establish global convergence of a homotopymethod for DC operating pointsrdquo in Proceedings of the IEEEInternational Symposium on Circuits and Systems vol 2 pp914ndash917 New Orleans La USA May 1990

[31] J Verschelde ldquoPolynomial homotopy continuation with phc-packrdquo ACM Communications in Computer Algebra vol 44 no3-4 pp 217ndash220 2011

[32] D J Bates J D Hauenstein A J Sommese and C W Wam-pler ldquoBertini software for numerical algebraic geometryrdquohttpsbertinindedu

[33] T L Lee T Y Li and C H Tsai ldquoHOM4PS-20 a softwarepackage for solving polynomial systems by the polyhedralhomotopy continuation methodrdquo Computing vol 83 no 2-3pp 109ndash133 2008

[34] A J Sommese and C W Wampler II The Numerical Solutionof Systems of Polynomials Arising in Engineering and ScienceWorld Scientific Singapore 2005

[35] T Gunji S Kim M a Kojima K Fujisawa and T Mizu-tani ldquoPHoMmdasha polyhedral homotopy continuation methodfor polynomial systemsrdquo Computing Archives for ScientificComputing vol 73 no 1 pp 57ndash77 2004

[36] H Vazquez-Leal A Sarmiento-Reyes Y Khan et al ldquoNewaspects of double bounded polynomial homotopyrdquo British


Journal of Mathematics amp Computer Science vol 3 no 4 pp549ndash566 2013

[37] D Torres-Munoz H Vazquez-Leal L Hernandez-Martinezand A Sarmiento-Reyes ldquoImproved spherical continuationalgorithm with application to the double-bounded homotopy(dbh)rdquo Computational and Applied Mathematics vol 33 no 1pp 147ndash161 2014

[38] L T Watson ldquoGlobally convergent homotopy algorithms fornonlinear systems of equationsrdquoNonlinear Dynamics vol 1 no2 pp 143ndash191 1990

[39] H Vazquez-Leal ldquoPiece-wise-polynomial methodrdquo Computa-tional amp Applied Mathematics vol 33 no 2 pp 289ndash299 2014

[40] I Guerra-Gomez T McConaghy and E Tlelo-Cuautle ldquoOper-ating-point driven formulation for analog computer-aideddesignrdquoAnalog Integrated Circuits and Signal Processing vol 74no 2 pp 345ndash353 2013

[41] R Trejo-Guerra E Tlelo-Cuautle V H Carbajal-Gomez andG Rodriguez-Gomez ldquoA survey on the integrated design ofchaotic oscillatorsrdquoAppliedMathematics and Computation vol219 no 10 pp 5113ndash5122 2013

[42] R Trejo-Guerra E Tlelo-Cuautle J M Jimenez-Fuentes et alldquoIntegrated circuit generating 3- and 5-scroll attractorsrdquo Com-munications inNonlinear Science andNumerical Simulation vol17 no 11 pp 4328ndash4335 2012

[43] V M Jimenez-Fernandez C Reyes-Betanzo M Angelica-Cerdan Z J Hernandez-Paxtian H Vazquez-Leal and AItzmoyotl-Toxqui ldquoPrediction of silicon dry etching using apiecewise linear algorithmrdquo Journal of the Chinese Institute ofEngineers vol 36 no 7 pp 941ndash950 2013

[44] V M Jimenez-Fernandez E Munoz-Aguirre H Vazquez-Lealet al ldquoA piecewise linear fitting technique for multivalued two-dimensional pathsrdquo Journal of Applied Research and Technologyvol 11 no 5 pp 636ndash640 2013

[45] S Pastore ldquoFast and efficient search for all DC solutions of PWLcircuits bymeans of oversized polyhedrardquo IEEE Transactions onCircuits and Systems I Regular Papers vol 56 no 10 pp 2270ndash2279 2009

[46] K Yamamura and K Yomogita ldquoFinding all solutions of pie-cewise-linear resistive circuits using an LP testrdquo IEEE Trans-actions on Circuits and Systems I Fundamental Theory andApplications vol 47 no 7 pp 1115ndash1120 2000

[47] L Ying S Wang and H Xiaolin ldquoFinding all solutions ofpiecewise-linear circuits usingmixed linear programming algo-rithmrdquo in Proceedings of the Control and Decision Conference(CCDC rsquo08) pp 4204ndash4208 2008 (Chinese)

[48] K Yamamura and T Ohshima ldquoFinding all solutions of pie-cewise-linear resistive circuits using linear programmingrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 45 no 4 pp 434ndash445 1998

[49] K Yamamura ldquoFinding all solutions of piecewise-linear resis-tive circuits using simple sign testsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 40 no 8 pp 546ndash551 1993

[50] K Yamamura and S Tanaka ldquoFinding all solutions of piecewise-linear resistive circuits using the dual simplex methodrdquo inProceedings of the IEEE Internaitonal Symposium onCircuits andSystems (ISCAS rsquo00) vol 4 pp 165ndash168 Geneva SwitzerlandMay 2000

[51] M Tadeusiewicz and S Halgas ldquoFinding all the DC solutionsof a certain class of piecewise-linear circuitsrdquo Circuits Systemsand Signal Processing vol 18 no 2 pp 89ndash110 1999

[52] S Pastore and A Premoli ldquoPolyhedral elements a new algo-rithm for capturing all the equilibrium points of piecewise-linear circuitsrdquo IEEE Transactions on Circuits and Systems IFundamental Theory and Applications vol 40 no 2 pp 124ndash131 1993

[53] J Katzenelson ldquoAn algorithmfor solving nonlinear resistornetworksrdquo The Bell System Technical Journal vol 44 pp 1605ndash1620 1965

[54] K Yamamura and K Horiuchi ldquoGlobally and quadraticallyconvergent algorithm for solving nonlinear resistive networksrdquoIEEE Transactions on Computer-Aided Design of IntegratedCircuits and Systems vol 9 no 5 pp 487ndash499 1990

[55] S N Stevens and P M Lin ldquoAnalysis of piecewise-linearresistive networks using complementary pivot theoryrdquo IEEETransactions on Circuits and Systems vol 28 pp 429ndash441 1981

[56] J T J van Eijndhoven ldquoSolving the linear complementarityproblem in circuit simulationrdquo SIAM Journal on Control andOptimization vol 24 no 5 pp 1050ndash1062 1986

[57] M Tadeusiewicz and A Kuczynski ldquoA very fast method for theDC analysis of diode-transistor circuitsrdquo Circuits Systems andSignal Processing vol 32 no 2 pp 433ndash451 2013

[58] L O Chua and S M Kang ldquoCanonical piecewise linearmodelingrdquo IEEE Transactions on Circuits and Systems vol 33no 5 pp 511ndash525 1984

[59] W Van Bokhoven Piecewise Linear Modeling and AnalysisKluwer Technische Boeken Deventer The Netherlands 1981

[60] C Guzelis and I C Goknar ldquoA canonical representation forpiecewise-affine maps and its applications to circuit analysisrdquoIEEE Transactions on Circuits and Systems vol 38 no 11 pp1342ndash1354 1991

[61] L Vandenberghe B L de Moor and J Vandewalle ldquoThe gen-eralized linear complementarity problem applied to the com-plete analysis of resistive piecewise-linear circuitsrdquo IEEE Trans-actions on Circuits and Systems vol 36 no 11 pp 1382ndash13911989

[62] T A M Kevenaar and D M W Leenaerts ldquoA comparisonof piecewise linearmodel descriptionsrdquo IEEE Transactions onCircuits and Systems I Fundamental Theory and Applicationsvol 38 no 12 pp 996ndash1004 1992

[63] C van de Panne ldquoA complementary variant of Lemkersquos methodfor the linear complementary problemrdquoMathematical Program-ming vol 7 pp 283ndash310 1974

[64] L O Chua and R L P Ying ldquoFinding all solutions of piecewise-linear circuitsrdquo International Journal of Circuit Theory andApplications vol 10 no 3 pp 201ndash229 1982

[65] K Yamamura and M Ochiai ldquoAn efficient algorithm forfinding all solutions of piecewise-linear resistive circuitsrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 39 no 3 pp 213ndash221 1992

[66] K Yamamura ldquoSimple algorithms for tracing solution curvesrdquoIEEE Transactions on Circuits and Systems I FundamentalTheory and Applications vol 40 no 8 pp 537ndash541 1993

[67] D Niu X Wu Z Jin and Y Inoue ldquoAn effective and globallyconvergent newton fixedpoint homotopy method for mostransistor circuitsrdquo IEICE TRANSACTIONS on Fundamentalsof Electronics Communications and Computer Sciences vol 96no 9 pp 1848ndash1856 2013

[68] J Lee and H Chiang ldquoConstructive homotopy methods forfinding all or multiple dc operating points of nonlinear circuitsand systemsrdquo IEEE Transactions on Circuits and Systems IFundamental Theory and Applications vol 48 no 1 pp 35ndash502001

International Journal of

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Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



replacing of piecewise models by their unified PWL repre-sentation Several methodologies have been proposed to findmultiple solutions of PWL circuits [45ndash57]

However such methodologies exhibit some drawbackslike the requirement of several initial points to find multiplesolutions [53 54] the use of implicit PWL models [55 56]and the need of expressing the circuit equations in termsof the linear complementary problem (LCP) that impliescomputing model state variables [57] Therefore in orderto circumvent the aforementioned disadvantages we explorethe application of HCM methods in combination with anadaptation of the modified spheres algorithm (MSA) [37] forthe DC analysis of PWL circuits

This paper is organized as follows A brief descriptionof PWL modelling is presented in Section 2 In Section 3we introduce the proposed HCM and its path followingtechnique (MSA) In Section 4 four case studies of nonlinearcircuits are presented and solved by using a HCM methodNumerical simulations and a discussion about the results areprovided in Section 5 Finally a concluding remark is givenin Section 6

2 Brief Description of PWL Modelling

A mathematical model approach widely used in nonlinearcircuit analysis is the so-called piecewise linear (PWL)The aim of this kind of modeling is to approximate thenonlinear behavior of a circuit element by using a set oflinear mappingsThis means transforming a single nonlinearequation into a finite number of linear equations One of thefirst piecewise linear models was provided by Chua and Kangin [58] Another proposal was presented by Van Bokhovenin [59] Subsequent contributions were the extension of theChua model reported by Guzelis and Goknar in [60] andthe parametric proposal given by Vandenberghe et al in[61] among others While there are diverse proposals ofPWL models they can be classified into two classes Thefirst one contains explicit models For this class of modelsthe output vector can be obtained by just substituting theinput vector into the modelThe second one contains modelswhich are implicit In such models the output vector cannotbe obtained directly In contrast an algorithm has to beperformed by which the output vector is computed [62] Themore representative examples of explicit and implicit PWLdescriptions are the canonical model of Chua1 and the modelof Bokhoven1 respectively

The formal definition of the Chua1 model is expressed asfollows

Theorem 1 Any one-dimensional piecewise linear curve with119871 segments and 120590 break points 120573

1lt 1205732

lt sdot sdot sdot lt 120573120590can be

represented by the expression

119910 (119909) = 119886 + 119887119909 +

120590

sum

119894=1

119888119894

1003816100381610038161003816119909 minus 120573119894

1003816100381610038161003816 (1)

where the model parameters can be computed by

119886 = 119910 (0) minus

120590

sum

119894=1

119888119894

10038161003816100381610038161205731198941003816100381610038161003816

119887 =119869(1)

+ 119869(120590+1)

2

119888119894=

119869(119894+1)

minus 119869(119894)

2 119894 = 1 2 120590

(2)

with 119869(119894) denoting the slope of the 119894th constitutive segment in the

piecewise linear curve

Meanwhile the Bokhoven1 model is expressed by a statevariable system defined in formulation of LCP For furtherdetails about LCP the reader is referred to [63]

The main factor that motivates the use of PWL modelsis the simplicity of their structure which is linear in eachregion of the domainHowever in terms of circuit analysis theuse of piecewise linear models means transforming a singlenonlinear equation into several linear equations that couldeasily be solved by standard methods from linear algebraThe problem lies now in the extremely large number oflinear regions to be discarded to determine the entire set ofcircuit solutions Unfortunately this task requires enormouscomputational resources To overcome that problem severalmethodologies and algorithms have been proposed Forexample Chua and Ying [64] reported an efficient methodwhere the number of linear simultaneous equations to besolved could be decreased by a sign test The same idea isimproved by Yamamura and Ochiai in [65] where linearprogramming techniques are applied and a more efficientsign test algorithm is also reported Katzenelson presents analgorithm based on Newtonrsquos homotopy in [53] and morerecently Tadeusiewicz and Kuczynski offered a method thatcombines the homotopy concept and the theory known as alinear complementary problem [57]

3 The Proposed Homotopy SchemeThe equilibrium equation to describe the DC behaviour isobtained using the Kirchhoff laws resulting in

f (x) = 0 f isin R119899997888rarr R

119899 (3)

where x represents the electrical variables of the circuit and 119899

the number of variablesHomotopy methods are based on the fact that solutions

are connected by a curve denominated ldquosolution curverdquo orldquohomotopy curverdquo Such curve is induced by including anextra parameter in the original NAES resulting in

H (f (x) 120582) = 0 H isin R119899timesR 997888rarr R

119899 (4)

where 120582 is the homotopy parameter andHminus1(0) the family ofsolutions that conforms the homotopy path

An example of homotopy formulation is Newtonrsquos homo-topy

H (f (x) 120582) = f (x) + (120582 minus 1) f (x119894) = 0 (5)

where x119894is the starting point of the trajectory






H (f (x) 0) = f (x) minus f (x119894) = 0 (6)


119894


H (f (x) 1) = f (x) = 0 (7)






119894(119894 = 1 119899) represent the




119878 (1199091 1199092 119909

119899+1) = (119909

1minus 1198881)2

+ (1199092minus 1198882)2

+ sdot sdot sdot + (119909119899+1

minus 119888119899+1

)2

minus 1199032= 0

(8)



1198671(1198911(119909) 120582) = 0

1198672(1198912(119909) 120582) = 0

119867119899(119891119899(119909) 120582) = 0

119878 (1199091 1199092 119909

119899 120582) = 0

(9)






2to predict the



1) a




S1

S2

S3

o2

o3

o1

k1

k2

Homotopy trajectory


S1

S2

S3

S4

o4

o2

o3

o1

k1

k2

Homotopy trajectory










0 05 1 15

x

120582

4

2

0

minus2

Slowast

A

B


4 Cases Study



2are

1198771 1198941= minus

125

8+

9

8V1+

7

8

1003816100381610038161003816V1 + 11003816100381610038161003816 minus

3

2

1003816100381610038161003816V1 minus 21003816100381610038161003816 +

3

4

1003816100381610038161003816V1 minus 51003816100381610038161003816

minus1

8

1003816100381610038161003816V1 minus 111003816100381610038161003816 minus

9

8

1003816100381610038161003816V1 minus 131003816100381610038161003816 + 2

1003816100381610038161003816V1 minus 151003816100381610038161003816




V2

MSE = (1198912

1+ 1198912

2)2

1198781

168 149999999999 149999999998 01198782

196 400000000122 0999999999476 222119890 minus 18

1198783

214 566666666670 0666666666664 212119890 minus 20

+

minus

9V

R1

R2

V1

V2

i1

i2

+

minus

+

minus

2Ω


1198772 1198942=

29

4+

3

2V2minus

3

2

1003816100381610038161003816V2 + 81003816100381610038161003816 +

3

2

1003816100381610038161003816V2 + 51003816100381610038161003816 minus

3

2

1003816100381610038161003816V2 + 31003816100381610038161003816

+3

2

1003816100381610038161003816V2 + 11003816100381610038161003816 minus

3

4

1003816100381610038161003816V2 minus 31003816100381610038161003816 minus

5

4

1003816100381610038161003816V2 minus 81003816100381610038161003816 +

3

2

1003816100381610038161003816V2 minus 101003816100381610038161003816

+1003816100381610038161003816V2 minus 13

1003816100381610038161003816 minus5

4

1003816100381610038161003816V2 minus 161003816100381610038161003816 +

1

4

1003816100381610038161003816V2 minus 181003816100381610038161003816

(11)


1198911(V1 V2) = V1+ V2+ 21198941minus 9 = 0

1198912(V1 V2) = V1+ V2+ 21198942minus 9 = 0

(12)


1198671(V1 V2 120582) = 119891

1(V1 V2) + (120582 minus 1) 119891

1(V10

V20

) = 0

1198672(V1 V2 120582) = 119891

2(V1 V2) + (120582 minus 1) 119891

2(V10

V20

) = 0

119878 (V1 V2 120582) = (V

1minus 1198881)2

+ (V2minus 1198882)2

+ (120582 minus 1198883)2

minus 1199032= 0

(13)

where V10

= minus5 and V20


0= 0 and 119878(V




1198882

= V20 and 119888




1 1198772 and 119877

3resistors

are

1198771 1198941=

5

6

1003816100381610038161003816V1 + 61003816100381610038161003816 minus

5

6

1003816100381610038161003816V1 minus 61003816100381610038161003816

1198772 V2=

1

6

10038161003816100381610038161198942 + 11003816100381610038161003816 minus

1

6

10038161003816100381610038161198942 minus 51003816100381610038161003816

1198773 1198943= V3minus

5

4

1003816100381610038161003816V3 minus 11003816100381610038161003816 + 2

1003816100381610038161003816V3 minus 21003816100381610038161003816 minus

1003816100381610038161003816V3 minus 31003816100381610038161003816

(14)




1198913(V1 V2 V3) = minusV

3minus 1198943+ 5 = 0

(15)


10= 15 119894

20= minus1 and V

30= 15



119894119889(V119889) = minus 005486777833 + 01482755558V

119889

+ 0011577793181003816100381610038161003816V119889 minus 0306

1003816100381610038161003816

+ 0011818697881003816100381610038161003816V119889 minus 03375

1003816100381610038161003816

+ 0049045369221003816100381610038161003816V119889 minus 0366

1003816100381610038161003816

+ 0075833695151003816100381610038161003816V119889 minus 03875

1003816100381610038161003816

(16)




1198942

V3

MSE1198781

137 99333333333 220000000000 286666666667 171119890 minus 22

1198782

382 00999999999 220000000000 286666666666 145119890 minus 22

1198783

610 minus100666666666 220000000000 286666666667 386119890 minus 23

S1

S2

S3

1

6

4

2

0

minus2

minus4

0 02 04 06 08 1

120582


S1

S2

S3

2

2

1

0

minus1

minus2

minus3

minus4

0 05 1 15

120582



R1

R2

R3minus

minusminus

minus

+

minus

+

++

+

minus

+5V11

2

3

i2

i3

i3

2i1

22

i1

2Ω1Ω 1Ω 1Ω


S1

S2

S3

15

10

5

0

minus5

minus10

1

0 05 1 15 2

120582



1198911(V1 V2) = 333 minus 1000 (119894

119889(V1) + 119894119889(V2)) minus V

1= 0

1198912(V1 V2) = 4 minus 1392119894

119889(V1) minus 1096119894

119889(V2) minus V2= 0

(17)


10= minus5 and

V20




1198911(V1 V2 V3 V4) = 436634V

2+ 6103168119894

119889(V1)

+ 2863168119894119889(V2) minus 12 = 0

1198912(V1 V2 V3 V4) = 54V

1+ V3+ 3580119894

119889(V1) + 6620119894

119889(V2)

+ 700119894119889(V3) + 500119894

119889(V4) minus 22 = 0

1198913(V1 V2 V3 V4) = 436634V

4+ 6103168119894

119889(V3)

+ 2863168119894119889(V4) minus 12 = 0

1198914(V1 V2 V3 V4) = V1+ 54V

3+ 700119894

119889(V1)

+ 3580119894119889(V3) + 6620119894

119889(V4) minus 22 = 0

(18)


1= [minus7 minus1 8 1] and 119876

2=





V2

MSE1198781

59 0022145787125 0374592098924 611119890 minus 18

1198782

63 0348923957274 0358608185920 170119890 minus 21

1198783

68 0372176159314 minus0117291118129 5119890 minus 21

333V+

minus

1kΩ

11kΩ

8kΩ

1kΩ

1kΩ

minus

+

10V

T1

T2


098idCollector (C)

Base (B)

Emitter (E)

+

minus

idd

B

E

C


333V+

minus

+

minus

+

minus

+

minus

d1 id1

1kΩ

1kΩ

11kΩ

8kΩd2 id2

1kΩ

098id2

098id1

10V












V2

V3

V4

MSE1198761

1198781

105 minus0734973033383 0376723496304 0318166142629 0372621919721 114119890 minus 17

1198761

1198782


1198761

1198783

235 0326931131382 0369666979551 minus048305281023 0376723495005 361119890 minus 16

1198761

1198784

243 0329725453371 0368724929946 0324331786706 0370543296483 627119890 minus 19

1198761

1198785

312 0386520101863 minus429094430670 0337714498625 0365872023364 640119890 minus 17

1198761

1198786


1198762

1198787


1198762

1198788

193 0338139358469 0364994969864 0387969158453 minus468386026807 286119890 minus 19

1198762

1198789


S1

S2S3

1

0

minus1

minus2

minus3

minus4

minus5

0 02 04 06 08 1 12

120582

(a) Homotopy path

S1

S2

S3

1

120582

095 1 105 110

04

03

02

01

00

minus01

minus02



10k

10Vminus

+

4k30k

12V+

minus 4k

i1 i2

+

minus2 30k

minus

+2V

30k

1

+

minus 05 k 101 k5k

1

+

minus

i305 k 101 k

i4

30k+

minus

1

4k 4k

+

minus

12V








3

120582

8

6

4

2

0

minus2

0 02 04 06 08 1


S1

S2

S3

S6

S4 S5

3

09999 1 10001 10003

120582

minus05

minus1

minus15

0

(b) Zoom for (a)

3

120582

05

04

03

02

01

0

0 05 1 15 2 25


S9

S8

S7

3

120582

090 095 1 105 110 115 120

03950

03925

03900

03875

03850

03825

03800

(d) Zoom for (c)



6 Conclusions




Acknowledgments


References










































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














H (f (x) 0) = f (x) minus f (x119894) = 0 (6)


119894


H (f (x) 1) = f (x) = 0 (7)






119894(119894 = 1 119899) represent the




119878 (1199091 1199092 119909

119899+1) = (119909

1minus 1198881)2

+ (1199092minus 1198882)2

+ sdot sdot sdot + (119909119899+1

minus 119888119899+1

)2

minus 1199032= 0

(8)



1198671(1198911(119909) 120582) = 0

1198672(1198912(119909) 120582) = 0

119867119899(119891119899(119909) 120582) = 0

119878 (1199091 1199092 119909

119899 120582) = 0

(9)






2to predict the



1) a




S1

S2

S3

o2

o3

o1

k1

k2

Homotopy trajectory


S1

S2

S3

S4

o4

o2

o3

o1

k1

k2

Homotopy trajectory










0 05 1 15

x

120582

4

2

0

minus2

Slowast

A

B


4 Cases Study



2are

1198771 1198941= minus

125

8+

9

8V1+

7

8

1003816100381610038161003816V1 + 11003816100381610038161003816 minus

3

2

1003816100381610038161003816V1 minus 21003816100381610038161003816 +

3

4

1003816100381610038161003816V1 minus 51003816100381610038161003816

minus1

8

1003816100381610038161003816V1 minus 111003816100381610038161003816 minus

9

8

1003816100381610038161003816V1 minus 131003816100381610038161003816 + 2

1003816100381610038161003816V1 minus 151003816100381610038161003816




V2

MSE = (1198912

1+ 1198912

2)2

1198781

168 149999999999 149999999998 01198782

196 400000000122 0999999999476 222119890 minus 18

1198783

214 566666666670 0666666666664 212119890 minus 20

+

minus

9V

R1

R2

V1

V2

i1

i2

+

minus

+

minus

2Ω


1198772 1198942=

29

4+

3

2V2minus

3

2

1003816100381610038161003816V2 + 81003816100381610038161003816 +

3

2

1003816100381610038161003816V2 + 51003816100381610038161003816 minus

3

2

1003816100381610038161003816V2 + 31003816100381610038161003816

+3

2

1003816100381610038161003816V2 + 11003816100381610038161003816 minus

3

4

1003816100381610038161003816V2 minus 31003816100381610038161003816 minus

5

4

1003816100381610038161003816V2 minus 81003816100381610038161003816 +

3

2

1003816100381610038161003816V2 minus 101003816100381610038161003816

+1003816100381610038161003816V2 minus 13

1003816100381610038161003816 minus5

4

1003816100381610038161003816V2 minus 161003816100381610038161003816 +

1

4

1003816100381610038161003816V2 minus 181003816100381610038161003816

(11)


1198911(V1 V2) = V1+ V2+ 21198941minus 9 = 0

1198912(V1 V2) = V1+ V2+ 21198942minus 9 = 0

(12)


1198671(V1 V2 120582) = 119891

1(V1 V2) + (120582 minus 1) 119891

1(V10

V20

) = 0

1198672(V1 V2 120582) = 119891

2(V1 V2) + (120582 minus 1) 119891

2(V10

V20

) = 0

119878 (V1 V2 120582) = (V

1minus 1198881)2

+ (V2minus 1198882)2

+ (120582 minus 1198883)2

minus 1199032= 0

(13)

where V10

= minus5 and V20


0= 0 and 119878(V




1198882

= V20 and 119888




1 1198772 and 119877

3resistors

are

1198771 1198941=

5

6

1003816100381610038161003816V1 + 61003816100381610038161003816 minus

5

6

1003816100381610038161003816V1 minus 61003816100381610038161003816

1198772 V2=

1

6

10038161003816100381610038161198942 + 11003816100381610038161003816 minus

1

6

10038161003816100381610038161198942 minus 51003816100381610038161003816

1198773 1198943= V3minus

5

4

1003816100381610038161003816V3 minus 11003816100381610038161003816 + 2

1003816100381610038161003816V3 minus 21003816100381610038161003816 minus

1003816100381610038161003816V3 minus 31003816100381610038161003816

(14)




1198913(V1 V2 V3) = minusV

3minus 1198943+ 5 = 0

(15)


10= 15 119894

20= minus1 and V

30= 15



119894119889(V119889) = minus 005486777833 + 01482755558V

119889

+ 0011577793181003816100381610038161003816V119889 minus 0306

1003816100381610038161003816

+ 0011818697881003816100381610038161003816V119889 minus 03375

1003816100381610038161003816

+ 0049045369221003816100381610038161003816V119889 minus 0366

1003816100381610038161003816

+ 0075833695151003816100381610038161003816V119889 minus 03875

1003816100381610038161003816

(16)




1198942

V3

MSE1198781

137 99333333333 220000000000 286666666667 171119890 minus 22

1198782

382 00999999999 220000000000 286666666666 145119890 minus 22

1198783

610 minus100666666666 220000000000 286666666667 386119890 minus 23

S1

S2

S3

1

6

4

2

0

minus2

minus4

0 02 04 06 08 1

120582


S1

S2

S3

2

2

1

0

minus1

minus2

minus3

minus4

0 05 1 15

120582



R1

R2

R3minus

minusminus

minus

+

minus

+

++

+

minus

+5V11

2

3

i2

i3

i3

2i1

22

i1

2Ω1Ω 1Ω 1Ω


S1

S2

S3

15

10

5

0

minus5

minus10

1

0 05 1 15 2

120582



1198911(V1 V2) = 333 minus 1000 (119894

119889(V1) + 119894119889(V2)) minus V

1= 0

1198912(V1 V2) = 4 minus 1392119894

119889(V1) minus 1096119894

119889(V2) minus V2= 0

(17)


10= minus5 and

V20




1198911(V1 V2 V3 V4) = 436634V

2+ 6103168119894

119889(V1)

+ 2863168119894119889(V2) minus 12 = 0

1198912(V1 V2 V3 V4) = 54V

1+ V3+ 3580119894

119889(V1) + 6620119894

119889(V2)

+ 700119894119889(V3) + 500119894

119889(V4) minus 22 = 0

1198913(V1 V2 V3 V4) = 436634V

4+ 6103168119894

119889(V3)

+ 2863168119894119889(V4) minus 12 = 0

1198914(V1 V2 V3 V4) = V1+ 54V

3+ 700119894

119889(V1)

+ 3580119894119889(V3) + 6620119894

119889(V4) minus 22 = 0

(18)


1= [minus7 minus1 8 1] and 119876

2=





V2

MSE1198781

59 0022145787125 0374592098924 611119890 minus 18

1198782

63 0348923957274 0358608185920 170119890 minus 21

1198783

68 0372176159314 minus0117291118129 5119890 minus 21

333V+

minus

1kΩ

11kΩ

8kΩ

1kΩ

1kΩ

minus

+

10V

T1

T2


098idCollector (C)

Base (B)

Emitter (E)

+

minus

idd

B

E

C


333V+

minus

+

minus

+

minus

+

minus

d1 id1

1kΩ

1kΩ

11kΩ

8kΩd2 id2

1kΩ

098id2

098id1

10V












V2

V3

V4

MSE1198761

1198781

105 minus0734973033383 0376723496304 0318166142629 0372621919721 114119890 minus 17

1198761

1198782


1198761

1198783

235 0326931131382 0369666979551 minus048305281023 0376723495005 361119890 minus 16

1198761

1198784

243 0329725453371 0368724929946 0324331786706 0370543296483 627119890 minus 19

1198761

1198785

312 0386520101863 minus429094430670 0337714498625 0365872023364 640119890 minus 17

1198761

1198786


1198762

1198787


1198762

1198788

193 0338139358469 0364994969864 0387969158453 minus468386026807 286119890 minus 19

1198762

1198789


S1

S2S3

1

0

minus1

minus2

minus3

minus4

minus5

0 02 04 06 08 1 12

120582

(a) Homotopy path

S1

S2

S3

1

120582

095 1 105 110

04

03

02

01

00

minus01

minus02



10k

10Vminus

+

4k30k

12V+

minus 4k

i1 i2

+

minus2 30k

minus

+2V

30k

1

+

minus 05 k 101 k5k

1

+

minus

i305 k 101 k

i4

30k+

minus

1

4k 4k

+

minus

12V








3

120582

8

6

4

2

0

minus2

0 02 04 06 08 1


S1

S2

S3

S6

S4 S5

3

09999 1 10001 10003

120582

minus05

minus1

minus15

0

(b) Zoom for (a)

3

120582

05

04

03

02

01

0

0 05 1 15 2 25


S9

S8

S7

3

120582

090 095 1 105 110 115 120

03950

03925

03900

03875

03850

03825

03800

(d) Zoom for (c)



6 Conclusions




Acknowledgments


References










































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










S1

S2

S3

o2

o3

o1

k1

k2

Homotopy trajectory


S1

S2

S3

S4

o4

o2

o3

o1

k1

k2

Homotopy trajectory










0 05 1 15

x

120582

4

2

0

minus2

Slowast

A

B


4 Cases Study



2are

1198771 1198941= minus

125

8+

9

8V1+

7

8

1003816100381610038161003816V1 + 11003816100381610038161003816 minus

3

2

1003816100381610038161003816V1 minus 21003816100381610038161003816 +

3

4

1003816100381610038161003816V1 minus 51003816100381610038161003816

minus1

8

1003816100381610038161003816V1 minus 111003816100381610038161003816 minus

9

8

1003816100381610038161003816V1 minus 131003816100381610038161003816 + 2

1003816100381610038161003816V1 minus 151003816100381610038161003816




V2

MSE = (1198912

1+ 1198912

2)2

1198781

168 149999999999 149999999998 01198782

196 400000000122 0999999999476 222119890 minus 18

1198783

214 566666666670 0666666666664 212119890 minus 20

+

minus

9V

R1

R2

V1

V2

i1

i2

+

minus

+

minus

2Ω


1198772 1198942=

29

4+

3

2V2minus

3

2

1003816100381610038161003816V2 + 81003816100381610038161003816 +

3

2

1003816100381610038161003816V2 + 51003816100381610038161003816 minus

3

2

1003816100381610038161003816V2 + 31003816100381610038161003816

+3

2

1003816100381610038161003816V2 + 11003816100381610038161003816 minus

3

4

1003816100381610038161003816V2 minus 31003816100381610038161003816 minus

5

4

1003816100381610038161003816V2 minus 81003816100381610038161003816 +

3

2

1003816100381610038161003816V2 minus 101003816100381610038161003816

+1003816100381610038161003816V2 minus 13

1003816100381610038161003816 minus5

4

1003816100381610038161003816V2 minus 161003816100381610038161003816 +

1

4

1003816100381610038161003816V2 minus 181003816100381610038161003816

(11)


1198911(V1 V2) = V1+ V2+ 21198941minus 9 = 0

1198912(V1 V2) = V1+ V2+ 21198942minus 9 = 0

(12)


1198671(V1 V2 120582) = 119891

1(V1 V2) + (120582 minus 1) 119891

1(V10

V20

) = 0

1198672(V1 V2 120582) = 119891

2(V1 V2) + (120582 minus 1) 119891

2(V10

V20

) = 0

119878 (V1 V2 120582) = (V

1minus 1198881)2

+ (V2minus 1198882)2

+ (120582 minus 1198883)2

minus 1199032= 0

(13)

where V10

= minus5 and V20


0= 0 and 119878(V




1198882

= V20 and 119888




1 1198772 and 119877

3resistors

are

1198771 1198941=

5

6

1003816100381610038161003816V1 + 61003816100381610038161003816 minus

5

6

1003816100381610038161003816V1 minus 61003816100381610038161003816

1198772 V2=

1

6

10038161003816100381610038161198942 + 11003816100381610038161003816 minus

1

6

10038161003816100381610038161198942 minus 51003816100381610038161003816

1198773 1198943= V3minus

5

4

1003816100381610038161003816V3 minus 11003816100381610038161003816 + 2

1003816100381610038161003816V3 minus 21003816100381610038161003816 minus

1003816100381610038161003816V3 minus 31003816100381610038161003816

(14)




1198913(V1 V2 V3) = minusV

3minus 1198943+ 5 = 0

(15)


10= 15 119894

20= minus1 and V

30= 15



119894119889(V119889) = minus 005486777833 + 01482755558V

119889

+ 0011577793181003816100381610038161003816V119889 minus 0306

1003816100381610038161003816

+ 0011818697881003816100381610038161003816V119889 minus 03375

1003816100381610038161003816

+ 0049045369221003816100381610038161003816V119889 minus 0366

1003816100381610038161003816

+ 0075833695151003816100381610038161003816V119889 minus 03875

1003816100381610038161003816

(16)




1198942

V3

MSE1198781

137 99333333333 220000000000 286666666667 171119890 minus 22

1198782

382 00999999999 220000000000 286666666666 145119890 minus 22

1198783

610 minus100666666666 220000000000 286666666667 386119890 minus 23

S1

S2

S3

1

6

4

2

0

minus2

minus4

0 02 04 06 08 1

120582


S1

S2

S3

2

2

1

0

minus1

minus2

minus3

minus4

0 05 1 15

120582



R1

R2

R3minus

minusminus

minus

+

minus

+

++

+

minus

+5V11

2

3

i2

i3

i3

2i1

22

i1

2Ω1Ω 1Ω 1Ω


S1

S2

S3

15

10

5

0

minus5

minus10

1

0 05 1 15 2

120582



1198911(V1 V2) = 333 minus 1000 (119894

119889(V1) + 119894119889(V2)) minus V

1= 0

1198912(V1 V2) = 4 minus 1392119894

119889(V1) minus 1096119894

119889(V2) minus V2= 0

(17)


10= minus5 and

V20




1198911(V1 V2 V3 V4) = 436634V

2+ 6103168119894

119889(V1)

+ 2863168119894119889(V2) minus 12 = 0

1198912(V1 V2 V3 V4) = 54V

1+ V3+ 3580119894

119889(V1) + 6620119894

119889(V2)

+ 700119894119889(V3) + 500119894

119889(V4) minus 22 = 0

1198913(V1 V2 V3 V4) = 436634V

4+ 6103168119894

119889(V3)

+ 2863168119894119889(V4) minus 12 = 0

1198914(V1 V2 V3 V4) = V1+ 54V

3+ 700119894

119889(V1)

+ 3580119894119889(V3) + 6620119894

119889(V4) minus 22 = 0

(18)


1= [minus7 minus1 8 1] and 119876

2=





V2

MSE1198781

59 0022145787125 0374592098924 611119890 minus 18

1198782

63 0348923957274 0358608185920 170119890 minus 21

1198783

68 0372176159314 minus0117291118129 5119890 minus 21

333V+

minus

1kΩ

11kΩ

8kΩ

1kΩ

1kΩ

minus

+

10V

T1

T2


098idCollector (C)

Base (B)

Emitter (E)

+

minus

idd

B

E

C


333V+

minus

+

minus

+

minus

+

minus

d1 id1

1kΩ

1kΩ

11kΩ

8kΩd2 id2

1kΩ

098id2

098id1

10V












V2

V3

V4

MSE1198761

1198781

105 minus0734973033383 0376723496304 0318166142629 0372621919721 114119890 minus 17

1198761

1198782


1198761

1198783

235 0326931131382 0369666979551 minus048305281023 0376723495005 361119890 minus 16

1198761

1198784

243 0329725453371 0368724929946 0324331786706 0370543296483 627119890 minus 19

1198761

1198785

312 0386520101863 minus429094430670 0337714498625 0365872023364 640119890 minus 17

1198761

1198786


1198762

1198787


1198762

1198788

193 0338139358469 0364994969864 0387969158453 minus468386026807 286119890 minus 19

1198762

1198789


S1

S2S3

1

0

minus1

minus2

minus3

minus4

minus5

0 02 04 06 08 1 12

120582

(a) Homotopy path

S1

S2

S3

1

120582

095 1 105 110

04

03

02

01

00

minus01

minus02



10k

10Vminus

+

4k30k

12V+

minus 4k

i1 i2

+

minus2 30k

minus

+2V

30k

1

+

minus 05 k 101 k5k

1

+

minus

i305 k 101 k

i4

30k+

minus

1

4k 4k

+

minus

12V








3

120582

8

6

4

2

0

minus2

0 02 04 06 08 1


S1

S2

S3

S6

S4 S5

3

09999 1 10001 10003

120582

minus05

minus1

minus15

0

(b) Zoom for (a)

3

120582

05

04

03

02

01

0

0 05 1 15 2 25


S9

S8

S7

3

120582

090 095 1 105 110 115 120

03950

03925

03900

03875

03850

03825

03800

(d) Zoom for (c)



6 Conclusions




Acknowledgments


References










































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












V2

MSE = (1198912

1+ 1198912

2)2

1198781

168 149999999999 149999999998 01198782

196 400000000122 0999999999476 222119890 minus 18

1198783

214 566666666670 0666666666664 212119890 minus 20

+

minus

9V

R1

R2

V1

V2

i1

i2

+

minus

+

minus

2Ω


1198772 1198942=

29

4+

3

2V2minus

3

2

1003816100381610038161003816V2 + 81003816100381610038161003816 +

3

2

1003816100381610038161003816V2 + 51003816100381610038161003816 minus

3

2

1003816100381610038161003816V2 + 31003816100381610038161003816

+3

2

1003816100381610038161003816V2 + 11003816100381610038161003816 minus

3

4

1003816100381610038161003816V2 minus 31003816100381610038161003816 minus

5

4

1003816100381610038161003816V2 minus 81003816100381610038161003816 +

3

2

1003816100381610038161003816V2 minus 101003816100381610038161003816

+1003816100381610038161003816V2 minus 13

1003816100381610038161003816 minus5

4

1003816100381610038161003816V2 minus 161003816100381610038161003816 +

1

4

1003816100381610038161003816V2 minus 181003816100381610038161003816

(11)


1198911(V1 V2) = V1+ V2+ 21198941minus 9 = 0

1198912(V1 V2) = V1+ V2+ 21198942minus 9 = 0

(12)


1198671(V1 V2 120582) = 119891

1(V1 V2) + (120582 minus 1) 119891

1(V10

V20

) = 0

1198672(V1 V2 120582) = 119891

2(V1 V2) + (120582 minus 1) 119891

2(V10

V20

) = 0

119878 (V1 V2 120582) = (V

1minus 1198881)2

+ (V2minus 1198882)2

+ (120582 minus 1198883)2

minus 1199032= 0

(13)

where V10

= minus5 and V20


0= 0 and 119878(V




1198882

= V20 and 119888




1 1198772 and 119877

3resistors

are

1198771 1198941=

5

6

1003816100381610038161003816V1 + 61003816100381610038161003816 minus

5

6

1003816100381610038161003816V1 minus 61003816100381610038161003816

1198772 V2=

1

6

10038161003816100381610038161198942 + 11003816100381610038161003816 minus

1

6

10038161003816100381610038161198942 minus 51003816100381610038161003816

1198773 1198943= V3minus

5

4

1003816100381610038161003816V3 minus 11003816100381610038161003816 + 2

1003816100381610038161003816V3 minus 21003816100381610038161003816 minus

1003816100381610038161003816V3 minus 31003816100381610038161003816

(14)




1198913(V1 V2 V3) = minusV

3minus 1198943+ 5 = 0

(15)


10= 15 119894

20= minus1 and V

30= 15



119894119889(V119889) = minus 005486777833 + 01482755558V

119889

+ 0011577793181003816100381610038161003816V119889 minus 0306

1003816100381610038161003816

+ 0011818697881003816100381610038161003816V119889 minus 03375

1003816100381610038161003816

+ 0049045369221003816100381610038161003816V119889 minus 0366

1003816100381610038161003816

+ 0075833695151003816100381610038161003816V119889 minus 03875

1003816100381610038161003816

(16)




1198942

V3

MSE1198781

137 99333333333 220000000000 286666666667 171119890 minus 22

1198782

382 00999999999 220000000000 286666666666 145119890 minus 22

1198783

610 minus100666666666 220000000000 286666666667 386119890 minus 23

S1

S2

S3

1

6

4

2

0

minus2

minus4

0 02 04 06 08 1

120582


S1

S2

S3

2

2

1

0

minus1

minus2

minus3

minus4

0 05 1 15

120582



R1

R2

R3minus

minusminus

minus

+

minus

+

++

+

minus

+5V11

2

3

i2

i3

i3

2i1

22

i1

2Ω1Ω 1Ω 1Ω


S1

S2

S3

15

10

5

0

minus5

minus10

1

0 05 1 15 2

120582



1198911(V1 V2) = 333 minus 1000 (119894

119889(V1) + 119894119889(V2)) minus V

1= 0

1198912(V1 V2) = 4 minus 1392119894

119889(V1) minus 1096119894

119889(V2) minus V2= 0

(17)


10= minus5 and

V20




1198911(V1 V2 V3 V4) = 436634V

2+ 6103168119894

119889(V1)

+ 2863168119894119889(V2) minus 12 = 0

1198912(V1 V2 V3 V4) = 54V

1+ V3+ 3580119894

119889(V1) + 6620119894

119889(V2)

+ 700119894119889(V3) + 500119894

119889(V4) minus 22 = 0

1198913(V1 V2 V3 V4) = 436634V

4+ 6103168119894

119889(V3)

+ 2863168119894119889(V4) minus 12 = 0

1198914(V1 V2 V3 V4) = V1+ 54V

3+ 700119894

119889(V1)

+ 3580119894119889(V3) + 6620119894

119889(V4) minus 22 = 0

(18)


1= [minus7 minus1 8 1] and 119876

2=





V2

MSE1198781

59 0022145787125 0374592098924 611119890 minus 18

1198782

63 0348923957274 0358608185920 170119890 minus 21

1198783

68 0372176159314 minus0117291118129 5119890 minus 21

333V+

minus

1kΩ

11kΩ

8kΩ

1kΩ

1kΩ

minus

+

10V

T1

T2


098idCollector (C)

Base (B)

Emitter (E)

+

minus

idd

B

E

C


333V+

minus

+

minus

+

minus

+

minus

d1 id1

1kΩ

1kΩ

11kΩ

8kΩd2 id2

1kΩ

098id2

098id1

10V












V2

V3

V4

MSE1198761

1198781

105 minus0734973033383 0376723496304 0318166142629 0372621919721 114119890 minus 17

1198761

1198782


1198761

1198783

235 0326931131382 0369666979551 minus048305281023 0376723495005 361119890 minus 16

1198761

1198784

243 0329725453371 0368724929946 0324331786706 0370543296483 627119890 minus 19

1198761

1198785

312 0386520101863 minus429094430670 0337714498625 0365872023364 640119890 minus 17

1198761

1198786


1198762

1198787


1198762

1198788

193 0338139358469 0364994969864 0387969158453 minus468386026807 286119890 minus 19

1198762

1198789


S1

S2S3

1

0

minus1

minus2

minus3

minus4

minus5

0 02 04 06 08 1 12

120582

(a) Homotopy path

S1

S2

S3

1

120582

095 1 105 110

04

03

02

01

00

minus01

minus02



10k

10Vminus

+

4k30k

12V+

minus 4k

i1 i2

+

minus2 30k

minus

+2V

30k

1

+

minus 05 k 101 k5k

1

+

minus

i305 k 101 k

i4

30k+

minus

1

4k 4k

+

minus

12V








3

120582

8

6

4

2

0

minus2

0 02 04 06 08 1


S1

S2

S3

S6

S4 S5

3

09999 1 10001 10003

120582

minus05

minus1

minus15

0

(b) Zoom for (a)

3

120582

05

04

03

02

01

0

0 05 1 15 2 25


S9

S8

S7

3

120582

090 095 1 105 110 115 120

03950

03925

03900

03875

03850

03825

03800

(d) Zoom for (c)



6 Conclusions




Acknowledgments


References










































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












1198942

V3

MSE1198781

137 99333333333 220000000000 286666666667 171119890 minus 22

1198782

382 00999999999 220000000000 286666666666 145119890 minus 22

1198783

610 minus100666666666 220000000000 286666666667 386119890 minus 23

S1

S2

S3

1

6

4

2

0

minus2

minus4

0 02 04 06 08 1

120582


S1

S2

S3

2

2

1

0

minus1

minus2

minus3

minus4

0 05 1 15

120582



R1

R2

R3minus

minusminus

minus

+

minus

+

++

+

minus

+5V11

2

3

i2

i3

i3

2i1

22

i1

2Ω1Ω 1Ω 1Ω


S1

S2

S3

15

10

5

0

minus5

minus10

1

0 05 1 15 2

120582



1198911(V1 V2) = 333 minus 1000 (119894

119889(V1) + 119894119889(V2)) minus V

1= 0

1198912(V1 V2) = 4 minus 1392119894

119889(V1) minus 1096119894

119889(V2) minus V2= 0

(17)


10= minus5 and

V20




1198911(V1 V2 V3 V4) = 436634V

2+ 6103168119894

119889(V1)

+ 2863168119894119889(V2) minus 12 = 0

1198912(V1 V2 V3 V4) = 54V

1+ V3+ 3580119894

119889(V1) + 6620119894

119889(V2)

+ 700119894119889(V3) + 500119894

119889(V4) minus 22 = 0

1198913(V1 V2 V3 V4) = 436634V

4+ 6103168119894

119889(V3)

+ 2863168119894119889(V4) minus 12 = 0

1198914(V1 V2 V3 V4) = V1+ 54V

3+ 700119894

119889(V1)

+ 3580119894119889(V3) + 6620119894

119889(V4) minus 22 = 0

(18)


1= [minus7 minus1 8 1] and 119876

2=





V2

MSE1198781

59 0022145787125 0374592098924 611119890 minus 18

1198782

63 0348923957274 0358608185920 170119890 minus 21

1198783

68 0372176159314 minus0117291118129 5119890 minus 21

333V+

minus

1kΩ

11kΩ

8kΩ

1kΩ

1kΩ

minus

+

10V

T1

T2


098idCollector (C)

Base (B)

Emitter (E)

+

minus

idd

B

E

C


333V+

minus

+

minus

+

minus

+

minus

d1 id1

1kΩ

1kΩ

11kΩ

8kΩd2 id2

1kΩ

098id2

098id1

10V












V2

V3

V4

MSE1198761

1198781

105 minus0734973033383 0376723496304 0318166142629 0372621919721 114119890 minus 17

1198761

1198782


1198761

1198783

235 0326931131382 0369666979551 minus048305281023 0376723495005 361119890 minus 16

1198761

1198784

243 0329725453371 0368724929946 0324331786706 0370543296483 627119890 minus 19

1198761

1198785

312 0386520101863 minus429094430670 0337714498625 0365872023364 640119890 minus 17

1198761

1198786


1198762

1198787


1198762

1198788

193 0338139358469 0364994969864 0387969158453 minus468386026807 286119890 minus 19

1198762

1198789


S1

S2S3

1

0

minus1

minus2

minus3

minus4

minus5

0 02 04 06 08 1 12

120582

(a) Homotopy path

S1

S2

S3

1

120582

095 1 105 110

04

03

02

01

00

minus01

minus02



10k

10Vminus

+

4k30k

12V+

minus 4k

i1 i2

+

minus2 30k

minus

+2V

30k

1

+

minus 05 k 101 k5k

1

+

minus

i305 k 101 k

i4

30k+

minus

1

4k 4k

+

minus

12V








3

120582

8

6

4

2

0

minus2

0 02 04 06 08 1


S1

S2

S3

S6

S4 S5

3

09999 1 10001 10003

120582

minus05

minus1

minus15

0

(b) Zoom for (a)

3

120582

05

04

03

02

01

0

0 05 1 15 2 25


S9

S8

S7

3

120582

090 095 1 105 110 115 120

03950

03925

03900

03875

03850

03825

03800

(d) Zoom for (c)



6 Conclusions




Acknowledgments


References










































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












V2

MSE1198781

59 0022145787125 0374592098924 611119890 minus 18

1198782

63 0348923957274 0358608185920 170119890 minus 21

1198783

68 0372176159314 minus0117291118129 5119890 minus 21

333V+

minus

1kΩ

11kΩ

8kΩ

1kΩ

1kΩ

minus

+

10V

T1

T2


098idCollector (C)

Base (B)

Emitter (E)

+

minus

idd

B

E

C


333V+

minus

+

minus

+

minus

+

minus

d1 id1

1kΩ

1kΩ

11kΩ

8kΩd2 id2

1kΩ

098id2

098id1

10V












V2

V3

V4

MSE1198761

1198781

105 minus0734973033383 0376723496304 0318166142629 0372621919721 114119890 minus 17

1198761

1198782


1198761

1198783

235 0326931131382 0369666979551 minus048305281023 0376723495005 361119890 minus 16

1198761

1198784

243 0329725453371 0368724929946 0324331786706 0370543296483 627119890 minus 19

1198761

1198785

312 0386520101863 minus429094430670 0337714498625 0365872023364 640119890 minus 17

1198761

1198786


1198762

1198787


1198762

1198788

193 0338139358469 0364994969864 0387969158453 minus468386026807 286119890 minus 19

1198762

1198789


S1

S2S3

1

0

minus1

minus2

minus3

minus4

minus5

0 02 04 06 08 1 12

120582

(a) Homotopy path

S1

S2

S3

1

120582

095 1 105 110

04

03

02

01

00

minus01

minus02



10k

10Vminus

+

4k30k

12V+

minus 4k

i1 i2

+

minus2 30k

minus

+2V

30k

1

+

minus 05 k 101 k5k

1

+

minus

i305 k 101 k

i4

30k+

minus

1

4k 4k

+

minus

12V








3

120582

8

6

4

2

0

minus2

0 02 04 06 08 1


S1

S2

S3

S6

S4 S5

3

09999 1 10001 10003

120582

minus05

minus1

minus15

0

(b) Zoom for (a)

3

120582

05

04

03

02

01

0

0 05 1 15 2 25


S9

S8

S7

3

120582

090 095 1 105 110 115 120

03950

03925

03900

03875

03850

03825

03800

(d) Zoom for (c)



6 Conclusions




Acknowledgments


References










































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












V2

V3

V4

MSE1198761

1198781

105 minus0734973033383 0376723496304 0318166142629 0372621919721 114119890 minus 17

1198761

1198782


1198761

1198783

235 0326931131382 0369666979551 minus048305281023 0376723495005 361119890 minus 16

1198761

1198784

243 0329725453371 0368724929946 0324331786706 0370543296483 627119890 minus 19

1198761

1198785

312 0386520101863 minus429094430670 0337714498625 0365872023364 640119890 minus 17

1198761

1198786


1198762

1198787


1198762

1198788

193 0338139358469 0364994969864 0387969158453 minus468386026807 286119890 minus 19

1198762

1198789


S1

S2S3

1

0

minus1

minus2

minus3

minus4

minus5

0 02 04 06 08 1 12

120582

(a) Homotopy path

S1

S2

S3

1

120582

095 1 105 110

04

03

02

01

00

minus01

minus02



10k

10Vminus

+

4k30k

12V+

minus 4k

i1 i2

+

minus2 30k

minus

+2V

30k

1

+

minus 05 k 101 k5k

1

+

minus

i305 k 101 k

i4

30k+

minus

1

4k 4k

+

minus

12V








3

120582

8

6

4

2

0

minus2

0 02 04 06 08 1


S1

S2

S3

S6

S4 S5

3

09999 1 10001 10003

120582

minus05

minus1

minus15

0

(b) Zoom for (a)

3

120582

05

04

03

02

01

0

0 05 1 15 2 25


S9

S8

S7

3

120582

090 095 1 105 110 115 120

03950

03925

03900

03875

03850

03825

03800

(d) Zoom for (c)



6 Conclusions




Acknowledgments


References










































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










3

120582

8

6

4

2

0

minus2

0 02 04 06 08 1


S1

S2

S3

S6

S4 S5

3

09999 1 10001 10003

120582

minus05

minus1

minus15

0

(b) Zoom for (a)

3

120582

05

04

03

02

01

0

0 05 1 15 2 25


S9

S8

S7

3

120582

090 095 1 105 110 115 120

03950

03925

03900

03875

03850

03825

03800

(d) Zoom for (c)



6 Conclusions




Acknowledgments


References










































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article Modified Hyperspheres Algorithm to...

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