Two Mathematical Comments on the Thevenin Theorem…gluskin/pdf/...on_Thevenin_Theorem.E.Gluskin.pdf · Two Mathematical Comments on the Thevenin Theorem: An ‘‘Algebraic Ideal’’

Research ArticleTwo Mathematical Comments on the Thevenin TheoremAn lsquolsquoAlgebraic Idealrsquorsquo and the lsquolsquoAffine Nonlinearityrsquorsquo

Emanuel Gluskin

Kinneret College on the Sea of Galilee 15132 Jordan Valley Israel

Correspondence should be addressed to Emanuel Gluskin gluskineebguacil

Received 10 March 2015 Accepted 26 May 2015

Academic Editor Tito Busani

Copyright copy 2015 Emanuel Gluskin This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We discuss the most important and simple concept of basic circuit theorymdashthe concept of the unideal sourcemdashor the Thevenincircuit It is explained firstly how the Thevenin circuit can be interpreted in the algebraic sense Then we critically consider thecommon opinion that it is a linear circuit showing that linearity (or nonlinearity) depends on the use of the port The differencebetween the cases of a source being an input or an internal element (as it is inTheveninrsquos circuit) is important hereThe distinction inthe definition of linear operator in algebra (here in system theory) and in geometry is also important for the subject and we suggestthe wide use of the concept of ldquoaffine nonlinearityrdquoThis kind of nonlinearity should be relevant for the development of complicatedcircuitry (perhaps in a biological modeling context) with nonprescribed definition of subsystems when the interpretation of a portas input or output can become dependent on the local intensity of a process

1 Introduction

For the first time the equivalent circuit was suggested byHerman von Helmholtz in 1853 Later it was rediscovered(and proved for complicated linear 1-ports) by Leon CharlesThevenin (in 1883) and then by Edward Lawry Norton andHans FerdinandMayer (both in 1926) [1] Since passing fromthe series circuit of Thevenin including a voltage source tothe parallel Norton version including a current source is animmediate application of the equivalent generator theoremwe will always speak about ldquoThevenin theoremrdquo or justldquotheoremrdquo meaning the series circuit

Though the rigor proof (in [2 3] using the basic substi-tution theorem) of the theorem is nontrivial the very resultis not surprising Indeed when having some voltage at theport of a circuit one can naturally try to use the circuit asa voltage source Since furthermore it is difficult to create agood voltage source the idea of an equivalent circuit with aninternal resistor (or an internal impedance119885(119904) or119885(119895120596) butwe will use the simplest classical model with a usual resistor)providing the dependence of the output voltage on the loadthat is the nonideality of the source naturally appears

What is really surprising is not the technical but thelogical side namely the fact that Helmholtz who was able to

formulate in 1847 the law of conservation of energy for bothmechanical and electrical systems suggested in 1853 thetheorem only for electrical systems while it is not difficult toreplace the voltage source by a source of a mechanical forceand find mechanical equivalent for the 1-port circuit ThusHelmholtz (who of course knew the circuit equations sug-gested by Kirchhoff in 1847 and could expect intensive devel-opment of circuit theory) saw in electrical engineering some-thing relevant to the theorem which is not found in mechan-ics It is clear today that this ldquosomethingrdquo is associatedwith theconcept of port (input output) involved in the formulation ofthe theorem because this concept is flexible not inmechanicsbut in electrical engineering that easily creates very com-plicated structures that can be seen as composed of somesubsystems relevant to the theorem That is the point istechnology but we will stress the theoretical side See also [4]

The closely associated question to be asked is that ofwhether a subsystem is an active one having a load or by itselfis a load of a stronger circuit The first case is more commonin the applications of the Thevenin theorem but the ldquoaffinenonlinearityrdquo with which we will be concerned is better seenin the second case and it is methodologically important thatthe question about linearity or nonlinearity of a subsystem

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 743189 7 pageshttpdxdoiorg1011552015743189

2 Mathematical Problems in Engineering

(a)

Linear active

circuit (1-port)

with only

internal

controls

a

b

(b)

+

b

i

a

+

vE(ETh)

R(RTh)

Theveninrsquos equivalent 1-port

Figure 1 (b) The commonly usedThevenin equivalent of a linear active circuit (a) The rigorous proof of (a) rarr (b) is found in [2 3]

(or of the whole system) can be influenced by the thus-seendegree of activity of the circuit This circumstance perhapsunexpected for many is in fact not surprising because wealways consider linearity (or nonlinearity) of a certain systemand any correct definition of a systemmust include definitionof its ports (See also [4] and references there for a develop-ment of this outlook showing in particular that nonlinearitydoes not just mean a curviness of a characteristic)

The classical theorem related to the most basic circuit-theory concept of nonideal source is taught at the very begin-ning of the standard electrical engineering education butrevising just the most basic concepts is most useful and thisrelates also to the Thevenin theorem In the relatively recentworks [5 6] a criterion is suggested not appearing in the clas-sical theory defining the conditions when a 1-port includinga dependent source is as a whole an ideal or a nonidealsource and here we make some new steps

Regarding the use here of the ldquounpopularrdquo concepts ofaffine nonlinearity and algebraic ideal it should be noted thatin [4] a state of the practically very important fluorescentlamp circuit is classified using the concept of affine nonlin-earity Without this interpretation the (actually very strong)nonlinearity of the lamp circuit is not obvious The fact thatthe circuit specialists should pay more attention to the con-cept of affine nonlinearity is not only because of theThevenincircuit

The present completion of the conceptual frame canattract mathematical students to circuit theory and on thecontrary the electrical engineering students to some addi-tional mathematics and the vision of a complicated systemsuggested in Section 42 may be relevant for a biologicalmodeling Hopefully our discussion will finally motivatesome applications that would become some ldquotools in handrdquofor an ordinary electrical engineer

2 The Circuit

Consider the well-known Thevenin circuit to which manylinear 1-port circuits are reduced [1ndash3] It is an equivalent cir-cuit whichmeans that it influences the external circuit just asthe original circuit does

For simplicity we will consider the Thevenin circuit as itis usually introduced that is for the simple resistive circuitsSee Figure 1

For a circuit including inductors and capacitors we canpass on to the domain of the Laplace-variable 119904 in whichall the linear circuits become ldquoalgebraicrdquo just as the resistivecircuits directly are in the time domain Then the Theveninequivalent includes some 119881Th(119904) instead of VTh(119905) and some119885Th(119904) instead of119877Th As is well known [2 3] dynamic (in thetime domain) linear circuits can be treated in the 119904-domainusing all of the network theorems amongwhich theThevenintheorem is extremely important and the transfer to the 119904-domain does not change the circuitrsquos topology

The circuit of Figure 1(b) interests us from two points ofview First of all (Section 3) using the mathematical conceptof ideal we observe the universal simplicity of this circuitthat can be extended in a linear form by adding more linearelements and also sources and then again contracted to thissimple form See Figure 2 The possibility of the contractiongives this simple topology the ldquoabsorption featurerdquo typical foran algebraic ldquoidealrdquo

Then in Section 4 we consider the 1-port nature of theequivalent circuit which allows one to use the circuit as a loadfor a stronger circuit and argue that the fact (associated withthe 1-port nature) that the circuitrsquos source is an internal onemeans a nonlinearity

Not yet coming to the point of the nonlinearity we speakin Section 3mdashas it is traditionally donemdashonly about linearcircuit

3 An Algebraic Outlook

For transferring to themathematical point of view associatedwith the concept of ideal [7ndash9] it is worthwhile first to tryto understand why the standard mathematical education ofelectrical engineers includes only linear algebra and not thegeneral algebra where ideals rings and groups are taught

Electrical engineering students and specialists are used toworking with expressions of the type

11988611199091 + 11988621199092 + sdot sdot sdot (1)

Mathematical Problems in Engineering 3

+ + +

RTh1 R

Th2 R

Thn

ETh1 E

Th2 E

Thn

middot middot middot


(a)

+

R0

E0

(b)

Figure 2 (a) Any such ldquoladder-typerdquo circuit and even a much more complicated 1-port can be equivalently (for the external circuitry)reduced to (b) 119864

0and 119877

0are easily calculated [1 2] So (b) is the ldquoidealrdquo considering additions of linear elements to such a circuit as (b) as

an interaction (ie a binary operation) with (b) we obtain the form of (b) again

where 119886119896 are numbers from a field [7] 119860 and 119909

119896 are the

vectors (functions) of interest One says that the set of thevectors is given over field 119860 This scheme of the linear algebraincludes only multiplication of the type 119886119909 Such mathe-matical objects as ideals rings and groups where the 119909119909-type multiplication arises are not included in the educationof the electrical specialists

Group theory demonstrates some very important appli-cations of the ldquo119909119909-operationrdquo to physics and chemistry [10ndash16] reflecting some physical symmetry of a real system Nogeometric symmetry is the present point but in mathematicsitself groups reflect preservation of some properties of theelements of a set which is close to the circuit situation underthe discussion because every linear ldquoautonomousrdquo 1-port hasthe feature of the simpleThevenin circuit and preservation ofthis feature over the set of linear 1-ports (or under the linearconstruction operations) is our focus

By the formal definition ideal 119868 is a subset of a set119872 sup 119868with elements 119892

119896 so that for all 119898 isin 119872 and for all 119892 isin 119868

119898119892 isin 119868 It is the specific ldquoabsorptionrdquo property of 119868 and onesays that ldquo119868 is an ideal of119872rdquo Since it can be in particular that119898 isin 119868 every ideal is also a group but the specific for the idealfeature of the ldquoabsorptionrdquo is exhibited only for119898 notin 119868

If we consider the operation of the circuit simplificationthat is of replacing a circuit by its Thevenin equivalent asa here-relevant operation (however far it is from arithmeticmultiplication) thenThevenin 1-port is the ldquoidealrdquo of all of thelinear 1-ports See Figure 2 again

Note also that the ldquoabsorptionrdquo nature of the ideal hassome recursive feature as it is with the specific fractal featureof a (any) 1-port discussed in [17] (Every branch of a 1-port isalso a 1-port and it is possible to repeat the given structure ofthe whole 1-port in every of its branches and to continue thisprocedure recursively)

Applications of modern algebra to system theory madeby professionalmathematicians for example [18] are too dif-ficult for engineers even to circuit theorists and some com-promise of the ldquologisticrdquo of these applications has to be foundClassical works for instance [19 20] do not include ourpoints as is necessary That is the material is too difficult onthe one hand and too simple (weak) on the other

An immediate reason for this omission in the educationof the EE specialists is seen if we pay attention to the problemwith the physical dimensions Consider for instance the

basic condition defining a group 119866 if we multiply two ele-ments of the group 1198921 and 1198922 then the result 1198923 = 11989211198922 alsobelongs to119866This is not a simple requirement for an engineersince if the elements of 119866 are measured say in volts (V) thatis

[1198921] = [1198922] = 119881 (2)

then

[1198923] = [11989211198922] = [1198921] [1198922] = 1198812 (3)

that is 1198923 has a different dimension (units) and should notbelong to119866 One can overcome this problem using togetherwith the multiplication a special factor 119896 having the dimen-sion 1119881 or including radic119896 as a factor in each element thusldquonormalizingrdquo the elements in the dimensional sense Then

1198923 = 11989611989211198922 isin 119866 (4)

without any dimensional problem but one faces then theproblem of the physical sense of 119896 this parameter is to bea universal one so that (4) would be correct for all of theelements of 119866

However the algebraic ldquomultiplicationrdquo appearing in thedefinition of the group need not be the simple arithmeticmultiplication related to numbers the statement is just thatfor two elements of a set one finds by some rule the third ele-ment that belongs to a prescribed set It is even said in [21] thatin some cases one can consider addition being such a ldquomulti-plicationrdquoThen for instance ldquowalkingrdquo along a straight line issuch an addition of the distance which can be seen as a groupThis example can be extended furthermore to any transla-tion symmetry

In the theory of symmetry groups have wide applicationswithout any dimensional problemThis problem is avoided atleast seemingly if we use for instance the operators acting onphases (angles of a symmetric-obstacle position) as a group ofrotations

11989011989412057911198901198941205792 = 119890

119894(1205791+1205792) (5)

even though there still can be some amplitudes (numericallytaken as 1) having a special dimension


Thus when we have the elements of a group as an opera-tion that only interests us as in the case of the rotation then (4)is legitimized for 119896 = 1 However themain constructive pointfor the rotations and shifts and also for the circuit problemin mind is that the result of the operation that is 119892

3 is as

simple as the involved operations 1198921and 119892

2

For the circuit application the concept that is unusual formathematics is that of ldquoconstructionrdquo that is of realization ofthe simplified circuit However any application ofmathemat-ics to reality requires some constraints and idealizations andnobody asks to the point for instance howphysically difficultis it to rotate a symmetricmassive or amicroscopic body andwhat instruments are necessary for doing this Similarly tothe known in statistical physics image of ldquoMaxwellrsquos Demonrdquowho just does his job one could suggest speaking aboutldquoThevenin (Helmholtz) Technicianrdquo simplifying the circuitsas needed who would be a quite realistic figure though thedetails of his job would be similarly irrelevant to the mathe-maticians

4 Linear and Nonlinear

Consider now another point certainly practically importantalso missed regarding Thevenin equivalent in the classicaltheory at least as this theory is traditionally taught andpresented in the popular textbooks

The well-known definition of the linear operator actingon some time-functions 119891

119896(119905) relevant to system theory and

algebra is

[

119873

sum

1119886119896119891119896] =

119873

sum

1119886119896 [119891119896] (6)

In particular

[119886119891] = 119886 [119891] (6a)

and setting 119886 = 0 we have

[0] = 0 (7)

This algebraic definition of linearity requiring 0 rarr 0differs from the definition of linearity in geometry where ashift can be added and it is important that the system-theorydefinition leaves the affine dependence that is the geometriclinearity (an obvious generalization of 119910 = 119886119909 + 119887)

[

119873

sum

1119886119896119891119896(119905)] =

119873

sum

1119886119896 [119891119896(119905)] + 120593 (119905) (8)

where 120593(119905) is a known function as a possibility of system non-linearity This simple nonlinearity becomes just trivial if weassume theThevenin circuit to be a load for a stronger circuit(which is a quite natural use for a 1-port) and compare thecircuit of Figure 1(b) with the circuit of Figure 3 in which theThevenin source is replaced by a voltage hard-limiter

+

b

i

a

+

v

RTh

EThz =

Figure 3 An equivalent circuit with Zener-diode if the externalcircuit with a stronger battery (applying its voltage as shown theplus up) uses ourThevenin circuit as a load (Notice the direction ofthe current)

It is a very simple comparisonsituation but it is unusualfor one to realize that when a source ceases to be applied toa circuit via a special input and becomes an internal circuitelement playing a role of a constant term of the type of thefunction 120593(119905) in (8) then this source becomes nothing elsebut a strongly nonlinear element (Notice that 120593(119905) in (8) isnot any input)

The problem of the definition of circuit inputs becomesthus the crucial one in the classification of a circuit as linearor nonlinear Consider also Figure 4

Figure 5(b) compares the affine nonlinearity of the circuitwith the linearity of the possible simple load (a passive resistor119877119871) for the circuit we need both lines to find the work-point

However if the Thevenin circuit is a load for such strongercircuits we work with two affine lines

In order to see the importance of the affine nonlinearitylet us consider two examples

41 The Characteristic of a Solar Cell It is worth noting thatthe curviness of the typical solar cell characteristic [22 23]shown in Figure 6 is not the main reason for the nonlinearityof the characteristic If this characteristic were the straightline of the type

119894 (V) = 119894 (0) minus 119896V 119896 gt 0 119894 (0) = 0 (9)

similar to the line with the negative slope in Figure 5(b) wewould still have the active element and in fact not the curvi-ness but the closeness of the characteristic to affine nonlinearityis the main part of the nonlinearity of the cell

42 A Model of a Living Medium with Growing Sources Theschematic model of Figure 7 also effectively demonstrates therole of the affine nonlinearity but this time in view of thesupposition that complicated circuitry may arise in the mod-eling of biological systemsWe consider amedium inwhich avoltage source can grow up thus decreasing its internal resis-tance that is becoming a stronger (more ideal) sourceThusthe sources can compete with each other


The chosenoutput

4-port

E1

E2

E3

(a)

The sameoutput

1-port

ETh

(b)

Figure 4 (a) A linear circuit withmany inputs (may be some sensed physical influences eg the sunrsquos radiation) (b)TheThevenin equivalentof (a) which can be seen as an affine-nonlinear 1-port

+

b

i

a

+

vE(ETh)

R(RTh)

ldquoTheveninrsquos 1-portrdquo

(a)

i

v

A

0

E

Affine

Linear

(i) = minusRi + E

RLi

(b)

Figure 5The affine action (a)The circuit that can either have a load or be by itself a load for a stronger circuit (eg for a similar circuit witha stronger source orand a smaller internal resistance) (b) The relevant graphs

v0

i()

i = RL

Figure 6 A schematic characteristic of a solar cellThe fact that 119894(0)is nonzero ismost important for the very fact of the nonlinearity andfor the power application

Such observations are relevant to complicated circuitrywhere definition of subsystems is not simple

Another interesting example of affine nonlinearitydescribed in [4] gives a theory of fluorescent lamp circuits

5 One More Important Application ofthe Thevenin Theorem

Our final comment again in simple terms relates to an unu-sual application of the concept of nonideal sourceWe usuallysee the nonideality of a source as its disadvantage but in thisexample it is an advantage because in a dangerous faulty sit-uation we have to ldquofightrdquo against the sourceThis topic relatesto electrical safety Consider a faulty generator having itsmetallic body electrified and a student who (while makingan experiment with this generator in the power laboratory)touches this body See Figure 8 A good grounding can savethe studentrsquos life The resistance of the grounding must be sosmall that the voltage division between the internal genera-torrsquos and the grounding resistances (impedances) ensures thevoltage on the body of the generator to be sufficiently low Itis not easy to technically realize a small value of the groundingresistor (this may require a special grounding for the labora-tory) but this argument is important because it clearly showsthe necessity in the good grounding for a power laboratory inwhich the students check the working power equipment

Since a more powerful generator (ie a more ideal volt-ages source) has smaller 119885eq electrical safety either requires


a c

b d

+ +R

In a feeding medium

V1 V2

(a)

c

d

+ +R

Nonlinear load

V1 V2 V2 is stronger

(b)

a

b

+ +R

Nonlinear load

V1 V2V1 is stronger

(c)

Figure 7 (a) We have two options for the sources ldquogrowingrdquo in a feeding medium namely one of the sources may become dominant (b)and (c) one of the sources has grown stronger and we can see in it the ideal source with zero internal resistance and to consider this sourceto be the input for the rest of the circuit The other source is weak that is its internal resistance (or impedancemdashsee discussion of Figure 1)is significant it is the (affine) nonlinear side of the circuit which will have a response of type (8) Notice that in (b) and (c) we have only oneinput

Theload of

the

The machine

Powerunit

++

The fault

0

What is the voltageof the metallic body of

the machine with

E2E1

Zeq

respect to ldquo0rdquo

ldquopower unitrdquo

Rman ≫ |Zgrounding|

Zgrounding


(a)

+

0

The touch

E2

Zeq

(b)

Figure 8 The situation of the electrical fault Instead of being connected to the load of the power unit 119885eq appeared to be connected to themetallic body of the generator (machine) Understanding the generator as a nonideal voltage source having internal impedance 119885eq makes itobvious that the ratio 119885grounding119885eq must be sufficiently small


making 119885grounding sufficiently small or the power of theequipment to be tested by the students be sufficiently low

Denoting as1198810the highest permitted for touching voltage

and as 119881119892the voltage of the generator we have for the ratio

119885grounding119885eq the condition100381610038161003816100381610038161003816100381610038161003816

119881119892

119885grounding

119885eq

100381610038161003816100381610038161003816100381610038161003816

lt 1198810 (10)

For the typical values of 1198810= 30Vrms and 119881

119892= 220Vrms

this gives |119885grounding119885eq| lt 0136It remains to connect 119885eq with the nominal power of the

generator This is done in [24]Obviously such argument is relevant to organizing any

new studentsrsquo laboratory and on behalf of the pedagogicalside we have here an introduction to the generally importanttopic of electrical safety [3 25ndash27] using the simple equiva-lent-circuit theorem

6 Conclusions and Final Remark

The concept ofThevenin theorem and the associated conceptof nonideal source are among the most basic concepts ofelectrical circuit theory We have shown here and also in[5 6] that these concepts still are unexplored for the analyticalstudy

The procedure of deriving Thevenin equivalent is inter-preted as an operation related to the algebraic concept ofideal This connection can stimulate a mathematician tobecome interested in the circuit theory and an electrical engi-neering specialist (student) to take interest in the generalalgebra

We also strongly argued for the importance of observingaffine nonlinearity in circuit theory (see also [4]) which isclosely associated with the role of the choice of inputs whichwas presumably predicted by Helmholtz 162 years ago How-ever simple and natural the theorem in focus is it is perhapsthe most important contribution to the circuit theory madeafter Gustav Robert Kirchhoff (a colleague of Helmholtzin BerlinUniversity and the other outstanding teacher ofMaxPlank) introduced his circuits laws

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The author is grateful to Yael Nemirovsky for making thematerial of Section 5 relevant for him and to Raul Rabinovicifor a discussion of this material The author is also gratefulto the unknown reviewers for their helpful comments andto Gady Golan Doron Shmilovitz Jacob Bear and MichaelWerner for their kind attention to his research efforts

References

[1] httpenwikipediaorgwikiThC3A9venin27s theorem

[2] C A Desoer and E S Kuh Basic Circuit Theory McGraw-Hill1969

[3] J Irwin and R M Nelms Basic Engineering Circuit AnalysisJohn Wiley amp Sons 2008

[4] E Gluskin ldquoStructure nonlinearity and system theoryrdquo Inter-national Journal of Circuit Theory and Applications vol 43 no4 pp 524ndash543 2015 Section 62

[5] E Gluskin and A Patlakh ldquoAn ideal source as an equivalentone-portrdquo Far East Journal of Electronics and Communicationvol 5 no 2 pp 79ndash89 2010

[6] E Gluskin ldquoAn extended frame for applications of theHelmholtz-Thevenin-Norton Theoremrdquo Journal of ElectricalEngineering amp Electronic Technology vol 2 no 2 pp 1ndash4 2013

[7] B L van der Waerden Modern Algebra Frederick UngarPublishing New York NY USA 1953

[8] S Leng Algebra Springer New York NY USA 2005[9] M Artin Algebra Prentice Hall 1991[10] HWeylTheTheory of Groups and QuantumMechanics Dover

1950[11] M TinkhamGroupTheory and QuantumMechanics McGraw-

Hill New York NY USA 1964[12] R Hermann Lie Groups for Physicists W A Benjamin 1966[13] P Ramond Group Theory A Physicistrsquos Survey Cambridge

University Press Cambridge UK 2010[14] H Eyring J Walter and G E Kimball Quantum Chemistry

Wiley New York NY USA 1960[15] C Kittel Quantum Theory of Solids John Wiley amp Sons New

York NY USA 1963[16] A VincentMolecular Symmetry and GroupTheory Wiley New

York NY USA 1988[17] E Gluskin ldquoOn the symmetry features of some electrical cir-

cuitsrdquo International Journal of Circuit Theory and Applicationsvol 34 no 6 pp 637ndash644 2006

[18] O Pfante andNAy ldquoOperator-theoretic identification of closedsub-systems of dynamical systemsrdquoDiscontinuity Nonlinearityand Complexity vol 4 no 1 pp 91ndash109 2015

[19] L O Chua C A Desoer and E S Kuh Linear and NonlinearCircuits McGraw-Hill 1987

[20] B J Leon andD J Schaefer ldquoVolterra series and Picard iterationfor nonlinear circuits and systemsrdquo IEEE Transactions on Cir-cuits and Systems vol 25 no 9 pp 789ndash793 1978

[21] I M Gelfand Lectures on Linear Algebra Dover New York1989

[22] TMarkvart Solar Electricity JohnWileyamp Sons NewYorkNYUSA 2000

[23] E Lorenzo Solar Electricity Engineering of Photovoltaic Sys-tems Progensa 1994

[24] E Gluskin and R Rabinovici ldquoSome simple formulae forgrounding in a laboratory and respective definition of lsquostrongcurrentrsquo a gate into electrical safetyrdquo httpvixraorgpdf15060074v2pdf

[25] W F Cooper Electrical Safety Engineering Newnes-Butter-worth London UK 1978

[26] L B Gordon ldquoElectrical hazards in the high energy laboratoryrdquoIEEE Transactions on Education vol 34 no 3 pp 231ndash242 1991

[27] E Gluskin Betichut Hashmalit Lecture Notes on ElectricalSafety The British Library Shelfmark HEC 1998 (Hebrew)

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Stochastic AnalysisInternational Journal of


(a)

Linear active

circuit (1-port)

with only

internal

controls

a

b

(b)

+

b

i

a

+

vE(ETh)

R(RTh)

Theveninrsquos equivalent 1-port

Figure 1 (b) The commonly usedThevenin equivalent of a linear active circuit (a) The rigorous proof of (a) rarr (b) is found in [2 3]

(or of the whole system) can be influenced by the thus-seendegree of activity of the circuit This circumstance perhapsunexpected for many is in fact not surprising because wealways consider linearity (or nonlinearity) of a certain systemand any correct definition of a systemmust include definitionof its ports (See also [4] and references there for a develop-ment of this outlook showing in particular that nonlinearitydoes not just mean a curviness of a characteristic)

The classical theorem related to the most basic circuit-theory concept of nonideal source is taught at the very begin-ning of the standard electrical engineering education butrevising just the most basic concepts is most useful and thisrelates also to the Thevenin theorem In the relatively recentworks [5 6] a criterion is suggested not appearing in the clas-sical theory defining the conditions when a 1-port includinga dependent source is as a whole an ideal or a nonidealsource and here we make some new steps

Regarding the use here of the ldquounpopularrdquo concepts ofaffine nonlinearity and algebraic ideal it should be noted thatin [4] a state of the practically very important fluorescentlamp circuit is classified using the concept of affine nonlin-earity Without this interpretation the (actually very strong)nonlinearity of the lamp circuit is not obvious The fact thatthe circuit specialists should pay more attention to the con-cept of affine nonlinearity is not only because of theThevenincircuit

The present completion of the conceptual frame canattract mathematical students to circuit theory and on thecontrary the electrical engineering students to some addi-tional mathematics and the vision of a complicated systemsuggested in Section 42 may be relevant for a biologicalmodeling Hopefully our discussion will finally motivatesome applications that would become some ldquotools in handrdquofor an ordinary electrical engineer

2 The Circuit

Consider the well-known Thevenin circuit to which manylinear 1-port circuits are reduced [1ndash3] It is an equivalent cir-cuit whichmeans that it influences the external circuit just asthe original circuit does

For simplicity we will consider the Thevenin circuit as itis usually introduced that is for the simple resistive circuitsSee Figure 1

For a circuit including inductors and capacitors we canpass on to the domain of the Laplace-variable 119904 in whichall the linear circuits become ldquoalgebraicrdquo just as the resistivecircuits directly are in the time domain Then the Theveninequivalent includes some 119881Th(119904) instead of VTh(119905) and some119885Th(119904) instead of119877Th As is well known [2 3] dynamic (in thetime domain) linear circuits can be treated in the 119904-domainusing all of the network theorems amongwhich theThevenintheorem is extremely important and the transfer to the 119904-domain does not change the circuitrsquos topology

The circuit of Figure 1(b) interests us from two points ofview First of all (Section 3) using the mathematical conceptof ideal we observe the universal simplicity of this circuitthat can be extended in a linear form by adding more linearelements and also sources and then again contracted to thissimple form See Figure 2 The possibility of the contractiongives this simple topology the ldquoabsorption featurerdquo typical foran algebraic ldquoidealrdquo

Then in Section 4 we consider the 1-port nature of theequivalent circuit which allows one to use the circuit as a loadfor a stronger circuit and argue that the fact (associated withthe 1-port nature) that the circuitrsquos source is an internal onemeans a nonlinearity

Not yet coming to the point of the nonlinearity we speakin Section 3mdashas it is traditionally donemdashonly about linearcircuit

3 An Algebraic Outlook

For transferring to themathematical point of view associatedwith the concept of ideal [7ndash9] it is worthwhile first to tryto understand why the standard mathematical education ofelectrical engineers includes only linear algebra and not thegeneral algebra where ideals rings and groups are taught

Electrical engineering students and specialists are used toworking with expressions of the type

11988611199091 + 11988621199092 + sdot sdot sdot (1)


+ + +

RTh1 R

Th2 R

Thn

ETh1 E

Th2 E

Thn



(a)

+

R0

E0

(b)


0and 119877




119896 are the











[1198921] = [1198922] = 119881 (2)

then

[1198923] = [11989211198922] = [1198921] [1198922] = 1198812 (3)


1198923 = 11989611989211198922 isin 119866 (4)




11989011989412057911198901198941205792 = 119890

119894(1205791+1205792) (5)




3 is as


2






algebra is

[

119873

sum

1119886119896119891119896] =

119873

sum

1119886119896 [119891119896] (6)

In particular

[119886119891] = 119886 [119891] (6a)


[0] = 0 (7)


[

119873

sum

1119886119896119891119896(119905)] =

119873

sum

1119886119896 [119891119896(119905)] + 120593 (119905) (8)


+

b

i

a

+

v

RTh

EThz =








119894 (V) = 119894 (0) minus 119896V 119896 gt 0 119894 (0) = 0 (9)




The chosenoutput

4-port

E1

E2

E3

(a)

The sameoutput

1-port

ETh

(b)


+

b

i

a

+

vE(ETh)

R(RTh)


(a)

i

v

A

0

E

Affine

Linear

(i) = minusRi + E

RLi

(b)


v0

i()

i = RL








a c

b d

+ +R

In a feeding medium

V1 V2

(a)

c

d

+ +R

Nonlinear load


(b)

a

b

+ +R

Nonlinear load

V1 V2V1 is stronger

(c)


Theload of

the

The machine

Powerunit

++

The fault

0


the machine with

E2E1

Zeq




Zgrounding


(a)

+

0

The touch

E2

Zeq

(b)







119881119892

119885grounding

119885eq

100381610038161003816100381610038161003816100381610038161003816

lt 1198810 (10)


119892= 220Vrms










Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+ + +

RTh1 R

Th2 R

Thn

ETh1 E

Th2 E

Thn



(a)

+

R0

E0

(b)


0and 119877




119896 are the











[1198921] = [1198922] = 119881 (2)

then

[1198923] = [11989211198922] = [1198921] [1198922] = 1198812 (3)


1198923 = 11989611989211198922 isin 119866 (4)




11989011989412057911198901198941205792 = 119890

119894(1205791+1205792) (5)




3 is as


2






algebra is

[

119873

sum

1119886119896119891119896] =

119873

sum

1119886119896 [119891119896] (6)

In particular

[119886119891] = 119886 [119891] (6a)


[0] = 0 (7)


[

119873

sum

1119886119896119891119896(119905)] =

119873

sum

1119886119896 [119891119896(119905)] + 120593 (119905) (8)


+

b

i

a

+

v

RTh

EThz =








119894 (V) = 119894 (0) minus 119896V 119896 gt 0 119894 (0) = 0 (9)




The chosenoutput

4-port

E1

E2

E3

(a)

The sameoutput

1-port

ETh

(b)


+

b

i

a

+

vE(ETh)

R(RTh)


(a)

i

v

A

0

E

Affine

Linear

(i) = minusRi + E

RLi

(b)


v0

i()

i = RL








a c

b d

+ +R

In a feeding medium

V1 V2

(a)

c

d

+ +R

Nonlinear load


(b)

a

b

+ +R

Nonlinear load

V1 V2V1 is stronger

(c)


Theload of

the

The machine

Powerunit

++

The fault

0


the machine with

E2E1

Zeq




Zgrounding


(a)

+

0

The touch

E2

Zeq

(b)







119881119892

119885grounding

119885eq

100381610038161003816100381610038161003816100381610038161003816

lt 1198810 (10)


119892= 220Vrms










Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











3 is as


2






algebra is

[

119873

sum

1119886119896119891119896] =

119873

sum

1119886119896 [119891119896] (6)

In particular

[119886119891] = 119886 [119891] (6a)


[0] = 0 (7)


[

119873

sum

1119886119896119891119896(119905)] =

119873

sum

1119886119896 [119891119896(119905)] + 120593 (119905) (8)


+

b

i

a

+

v

RTh

EThz =








119894 (V) = 119894 (0) minus 119896V 119896 gt 0 119894 (0) = 0 (9)




The chosenoutput

4-port

E1

E2

E3

(a)

The sameoutput

1-port

ETh

(b)


+

b

i

a

+

vE(ETh)

R(RTh)


(a)

i

v

A

0

E

Affine

Linear

(i) = minusRi + E

RLi

(b)


v0

i()

i = RL








a c

b d

+ +R

In a feeding medium

V1 V2

(a)

c

d

+ +R

Nonlinear load


(b)

a

b

+ +R

Nonlinear load

V1 V2V1 is stronger

(c)


Theload of

the

The machine

Powerunit

++

The fault

0


the machine with

E2E1

Zeq




Zgrounding


(a)

+

0

The touch

E2

Zeq

(b)







119881119892

119885grounding

119885eq

100381610038161003816100381610038161003816100381610038161003816

lt 1198810 (10)


119892= 220Vrms










Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










The chosenoutput

4-port

E1

E2

E3

(a)

The sameoutput

1-port

ETh

(b)


+

b

i

a

+

vE(ETh)

R(RTh)


(a)

i

v

A

0

E

Affine

Linear

(i) = minusRi + E

RLi

(b)


v0

i()

i = RL








a c

b d

+ +R

In a feeding medium

V1 V2

(a)

c

d

+ +R

Nonlinear load


(b)

a

b

+ +R

Nonlinear load

V1 V2V1 is stronger

(c)


Theload of

the

The machine

Powerunit

++

The fault

0


the machine with

E2E1

Zeq




Zgrounding


(a)

+

0

The touch

E2

Zeq

(b)







119881119892

119885grounding

119885eq

100381610038161003816100381610038161003816100381610038161003816

lt 1198810 (10)


119892= 220Vrms










Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










a c

b d

+ +R

In a feeding medium

V1 V2

(a)

c

d

+ +R

Nonlinear load


(b)

a

b

+ +R

Nonlinear load

V1 V2V1 is stronger

(c)


Theload of

the

The machine

Powerunit

++

The fault

0


the machine with

E2E1

Zeq




Zgrounding


(a)

+

0

The touch

E2

Zeq

(b)







119881119892

119885grounding

119885eq

100381610038161003816100381610038161003816100381610038161003816

lt 1198810 (10)


119892= 220Vrms










Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119881119892

119885grounding

119885eq

100381610038161003816100381610038161003816100381610038161003816

lt 1198810 (10)


119892= 220Vrms










Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

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