IEEE TRANSACTIONS ON EDUCATION, VOL. E-25, NO. 3, AUGUST 1982

This trend suggests the desirability of including a course cover-ing the fundamentals of teaching of science and engineering.

It is clear that the program is growing and is going beyondthe undergraduate level, and that the existence of a graduateprogram will be more than justified. Similar postgraduate pro-grams in England have proven to be encouraging experiences.Thus, for the future, the area of communications has plans

for reviewing undergraduate programs in light of a possiblegraduate program.

CONCLUSIONMore than ten years of experience in teaching communica-

tions at the undergraduate level has produced a dynamic pro-gram that joins theory, laboratory work, projects, and researchin a way that we consider appropriate for Venezuelan indus-trial and technological development. The academic com-munity and governmental and industrial areas are often sur-veyed, and they consider the program interesting and suitableto the country's needs.

REFERENCES[1] "First national plan for science and technology, Electronic and

telecommunications area," pp. 1-39, 1976-1980.[2] "Academic and statistic inform," Carabobo Univ., Nat. Council

Scientific Technolog. Res. (CONICIT), Valencia, Venezuela, no.4,pp. 280-465, 1977.

[3] K. Zaki and R. W. Newcomb, "A microwave-circuits laboratorywith multilevel educational objectives," IEEE Trans. Educ., vol.E-20, pp. 108-111, May 1977.

[4][51[6]

[7]

[8]

"Programs of study," Electrical Engineering School, CaraboboUniv., Valencia, Venezuela, 1968-1980."Special issue on engineering education," Proc. IEEE, vol. 66, pp.819-976, Aug. 1978.J. Brown, "Academic strategy," Proc. IEE, vol. 127, part A, no. 1,pp. 1-19, Jan. 1980.T. T. Woodson, "Accelerating the practice of engineering," IEEESpectrum, vol. 16, pp. 50-54, Sept. 1979.S. Waks, "Technological education as a synthesis of science andvocational education, implications on teacher training," inMELECOM'81 Conf Dig., sec. 6.3.3, pp. 1-4.

Aldo N. Bianchi was born in Buenos Aires,Argentina, in 1937. He received the ElectronicEngineer degree with honors from the Univer-sity of Buenos Aires in 1963 and the M.Sc.(EE) and Electrical Engineer degrees from theUniversity of Carabobo, Valencia, Venezuela,in 1973 and 1975, respectively.In 1965 he completed graduate studies on

microwaves at the Polytechnic Institute of! Brooklyn, Brooklyn, NY. From 1960 to 1968

he worked for the University of Buenos Airesin teaching and research. Since 1968 he has been a Professor at theUniversity of Carabobo, ranking now as Full Professor. In this institu-tion he has been Head of the Communications Laboratory, the Com-munications Area, and the Electronics and Communications Depart-ment. He is author of the textbooks Sistemas de Ondas Guiadas(Marcombo, Espafia, 1980), Introduccion a Microondas (Universidadde Carabobo, Venezuela), and several others. His areas of interestare applied electromagnetism (mircowaves, propagation) and engineer-ing education. Since 1968 he has been Counselor of the Carbobo Uni-versity Student Branch of the IEEE.

On Thevenin's and Norton's Equivalent CircuitsMOHAMED F. MOAD, SENIOR MEMBER, IEEE

Abstract-This paper provides a good look at the concept of equiva-lence in general and at Thevenin's and Norton's theorems in particular.The ain is to investigate simple, general, and comprehensive ways topresent and prove these theorems.

It is shown that linearity and the absence of external coupHing shouldnot be conditions for Thevenin's and Norton's theorems. It is alsoshown that the traditional open-circuit, short-circuit, and dead networkconditions can be replaced by any two distinct conditions at the ter-inmals. Two-ports and multiterminal networks are also included in thepresentation.

I. INTRODUCTIONTHE concept of equivalence such as two ports, Thevenin'sTequivalent circuit, and Norton's equivalent circuit pro-

vides a good insight into network behavior. Unfortunately,many engineers leave school with only a simple restricted ex-

Manuscript received November 13, 1981; revised April 19, 1982.The author is with the School of Electrical Engineering, Georgia In-

stitute of Technology, Atlanta, GA 30332.

posure to these important concepts. Some students do noteven see the proof of these theorems.The concept of equivalence is as simple in principle as that

of superposition, and it could be taught with the same sim-plicity. Yet, a survey of most textbooks that deal withThevenin's and Norton's theorems reveals the following.

1) Almost all textbooks [1]-[8] limit the presentation ofthese theorems to one-port networks.2) They exclude networks that are not linear.3) They exclude networks with external magnetic and/or

electric coupling.4) Some textbooks [1] introduce the concepts and the

proof simultaneously, while others [2] separate them by afew hundred pages.5) Some textbooks [3] leave the proof to the Appendix.6) Some textbooks [4] precede the proof by substituting

the load with a current source and using the substitutiontheorem.

0018-9359/82/0800-0099$00.75 i 1982 IEEE

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IEEE TRANSACTIONS ON EDUCATION, VOL. E-25, NO. 3, AUGUST 1982

7) Some textbooks [5] obtain the parameters by the elim-ination of variables, while almost all others [1] -[4], [6]-[8]advocate the separate evaluation of two individual parameters.The presentation of the two-port concept by most textbooks

has similar restrictions: linearity, no external coupling, and nointernal excitation. The evaluation of the four parameters isusually achieved by two, and sometimes four, separateevaluations.In short, textbooks use several methods of introducing and

finding equivalent circuits. The aim of this paper is to identifythese methods and introduce them in a simple, general, com-prehensive, and efficient way.

II. EQUIVALENCELet us define equivalence in the most general form, and then

modify this definition as needed.Definition 1 (or Theorem 1): Two networks are equivalent

(interchangeable) in their external behavior if they have identi-cal terminal characteristic, i.e.,

Y =f(X,u). (1)

The set u represents all the network's internal excitations in-cluding the initial stored energy (initial conditions), x repre-sents one half of all of the terminal independent variables(voltages and/or currents), and y represents the other half.There are no restrictions on the number of terminals nor onthe type of network. The terminals could be divided intogroups that may not be interconnected, i.e., each group canbe connected to one network (port) only. The choice of thesets of variables x and y is arbitrary. However, sometimes it isdesirable to be able to excite the network through either oneof these sets. In such a case, it is advisable not to include avoltage and a current of any one terminal in one set.

III. METHODS OF FINDING EQUIVALENT CIRCUITSA. The One-Step ApproachUse any method of analysis to find the relations among x, y,

and the necessary internal variables.' Eliminate the internalvariables resulting in (1). Model the results. To be more spe-cific, we have the following.

1) For a two-terminal network, write a set of equations interms of v (the terminal voltage), i (the terminal current), andinternal variables (voltages and/or currents). Eliminate the in-ternal variables. Model the result as a sourceless network inseries with a voltage source (or in parallel with a currentsource).

2) For a two-port network (with or without internal excita-tion), use a similar approach with the following results.

a) Mesh analysis gives the open-circuit parameters andmodel.

b) Node analysis gives the short-circuit parameters andmodel.

1To avoid confusion with open circuits, a box could be placed at theterminals. The same equations will be obtained if the box is replacedby voltage sources for mesh analysis or current sources for node analy-sis. The process could be simplified if we start with the equations thathave the external variables (y) on the left side, followed by a set ofequations with zeros on the left side.

(a) (b)Fig. 1. (a) A nonlinear network. (b) Its Thevenin model.

c) The ladder approach or the sequential method [9] givesthe transmission equations directly.This simple one-step approach would be useful in introduc-

ing the concept of equivalence without the need for a proof.Definition 1 is itself the proof. It has no restrictions. It isapplicable to nonlinear networks as well as to networks withexternal coupling. In addition, it could save up to 50 percentof the time required for obtaining equivalent circuits or two-port parameters by other methods. To illustrate the generalnature of this approach, let us find an equivalent circuit forthe network shown in Fig. 1(a). This network is nonlinear,and it is coupled to another network by the external cur-rent io.For this network,

v = i + i, + u (external loop)

v=i+(i- i +4io)+2i2 (right mesh).

Eliminating the internal variable i, gives

v=-2 +i2 + - + 2io.2 2

The equivalent circuit, Thevenin's model of (4), is shown inFig. 1(b). The Norton's model can be obtained from (4) as

. 2 2.2 au 4i3 3 3 3

(2)(3)

(4)

(5)

B. The Two-Step ApproachFor a network that satisfies the separation property only

(let us call such a network quasi-linear), (1) can be obtained as

y =f(x) +g(u) (6)where y and g(u) are column matrices. This equation showsthat two such networks are equivalent if they have the samef(x) (the network terminal characteristic with u = 0) and thesame g(u) (the response at the terminals when x = 0). Sinceg(u) is a column matrix, we may draw the same conclusion ifthe two networks have the samef(x) plus the same terminalvoltages and currents under only one set of terminal condi-tions (any set, not necessarily x = 0). From this we may con-clude the following.Theorem 2: Two linear or quasi-linear networks are equiv-

alent in their external behavior (interchangeable) if they satisfythe following two conditions: a) they have equivalent deadnetworks (the networks with the internal excitations madezero), and b) they have identical terminal voltages and currentsunder only one set (any desired set) of terminal conditions.

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MOAD: THEVENIN'S AND NORTON'S EQUIVALENT CIRCUITS

This may be interpreted as a general form of Thevenin'sand Norton's theorems for two-terminal as well as multi-terminal networks that could be time-varying, quasi-linear, andexternally coupled. One way of obtaining an equivalent net-work with this theorem is as follows.

1) Remove all internal excitations and simplify or modifythis dead network by any equivalence technique desired.2) Insert sources (voltage sources in series and current

sources in parallel) in any number and manner desired as longas the terminal voltages and currents are identical to those ofthe original network under only one set of termination. Thisset of termination may consist of any mixture of open circuits,short circuits, resistors, sources, etc.The two-step approach applied to Fig. 1(a) does give the

equivalent circuit of Fig. l(b) even though the network is non-linear. Notice that any controlled source can always be treatedas an independent source provided we do not lose its con-trolling variable (or a related quantity) in part 1) above andthat it is not ignored or altered in part 2).

C. he N + I Condition Approach

Both approaches discussed so far rely on finding the rela-tionship between the voltages and currents at the terminals.The question we want to explore with the third approach is:how many and what kind of tests do we need to perform ontwo linear networks to determine if they are equivalent?Equation (6) represents a set of linear N X N + 1 equations

where N is the number of x's ory's, i.e., one-half the numberof independent terminal variables. Two such sets of equationswill be identical if they satisfy only N + 1 independent condi-tions. The only restriction on the conditions is that each con-dition produces a set of equations that is independent of anyof those obtained from the previous conditions. With this inmind we may conclude the following.Theorem 3: Two linear multiterminal networks (that could

be time varying) are equivalent (interchangeable) in their ex-ternal behavior if they satisfy any N + 1 independent terminalconditions (N only if the networks have no internal excitation).With this approach, each condition can be specified by N

terminal variables, while the rest of the variables must befound from the network.For example, (6) for a linear, time-invariant, two-port net-

work (in the S domain) could be written in the form

E1 `Z111 +z12I2+E1oo

E2 - Z21I1 +Z22I2+E200 (7)

Suppose we let the (') represent one condition, the (") a secondcondition, etc. Equation (7) can be written under these con-ditions as

Iflz +I'z12 +E100 =E?

I'Z j +I2z12 +E100 =El

Fig. 2. Example of equivalence by one step and two conditions.

and

Ilz21 +Iz22 +E2o =;El

I1Z21 +I2Z22 +E2 =ElI1z21 +I2z22 +E20 =E2- (9)

It is easily seen that the parameters z11,z12, Z21, Z22,E1, andE200 can be uniquely determined from (8) and (9) by Cramer'srule, provided that the deterninant

I1 I2 1

i'1 2'2 1

I't I2"

(10)

is not singular. Therefore, two-port networks with intemalexcitation will be equivalent if they satisfy only three inde-pendent terminal conditions (two if the networks have no in-ternal excitation). The same number of independent condi-tions are applicable in the time domain. For a lineartwo-terminal network, for example, the step response [thevalue ofy in (6) with x = u (t)] and the response to x = 2u (t)will be sufficient to evaluate g(u) and f(x).For a linear two-terminal network, the N + 1 condition ap-

proach will be the two-condition approach (including openand short circuits). If such a network is time-invariant, theterminal characteristics (in the S domain) has the form

V=ZI+ Vo. (11)If condition 1 gives V1 and I, and condition 2 gives V2 andI2, i.e.,

(12)V1 =Z1 + VOand

V2 =ZI2 + VOit follows that

V- V1 V2- V1 ZI-I1 I2'I

or

'= VI1(I-II)+Vi.

(13)

(14)

As an example, let us use the one-step and two-condition ap-proaches to find Thevenin's equivalent circuit of the networkof Fig. 2 at the AB terninals.Solution:The One-Step Approach:

(8) V= 1 - 2i2 (right mesh)

101

ttt ifif +E =Efit

ilzil 2Z12 100 I (I 5)

IEEE TRANSACTIONS ON EDUCATION, VOL. E-25, NO. 3, AUGUST 1982

0=I+3i2 +3i, (node)

0 = 1 + 2i2 - il (outside mesi

Eliminating il and then i2 from the:

The Two-Condition Approach:

Condition l: V=1 + i2=0°,

Condition 2: V=2-'i =0,

Therefore,

V-1 2-1 2+3 3 39

2

or

V= 9I+ 3.9 3.'

Therefore, by either approach, Theiconsists of a 2/9 S2 resistor in seriessource. Notice that for two-terminmat the terminals could be replaced b(l = 0, for example).TT-1... -fhr -epnsr..AR s11 +L.

(16) requires m - 1 (n) equations. The short-circuit current re-quires m (n - 1) equations.

h). (17) 5) The one-step approach handles degenerate cases (such asse equations gives an infinite open-circuit voltage) better than the other two.

6) Any of the three methods can be used to directly handle(18) equivalent circuits for transfer (input-output) relations. In

(6), treat x as the input variables andy as the output vari-ables. The number of variables here does not have to be the

l1, I 3 (19) 7) Any of the three methods can be used to show that in ai2= -1/2, -I = 1.5. (20) dc circuit with or without controlled sources, the plot of any

(response) variable versus any other (excitation) variable is astraight line, and that the slope of this line depends only onthe dead network. This important property can be found inseveral texts, but unfortunately, it is ignored by many.8) Finally, the two-condition approach has a practical aspect

that cannot be matched by open- and short-circuit conditions.In checking for equivalence by open- and short-circuit mea-surements, one needs a voltmeter that draws no current and an

(21) ammeter that has zero voltage drop. Such meters do not exist.With the result presented here, any meter can be used to per-

with a 5/3 V volta form two readings under any two different conditions to es-wtnet/orks,Vachvoltatnge tablish the equivalence. Another practical advantage of they anintetrka condition two-condition approach is that one does not need to measure

an extreme quantity (such as the short-circuit current) thatcould damage the device.

UlliNva VUWIIWIOV OFIIIvU, an WlJ %UIvn'UL F VOILLVItU IWIV

are supposed to be in the time domain. They can also be usedin the S domain or the frequency domain, as shown by (7).However, if we want the equivalence to be valid for all fre-quencies, we must specify any condition in all frequencies byusing the Fourier transform or the Laplace transform.

IV. CONCLUSIONSEach of the three methods discussed here has its own ad-

vantages. The following guidelines could be helpful in choos-ing a specific method.

1) The one-step approach is useful in presenting the conceptof equivalence since it is general and it hardly requires anyproof.22) The two-step approach gives meaning to the individual

parameters.3) The most suitable method of evaluating one-port3 pa-

rameters isa) the two-step approach for simple networks with no

controlled sources where answers are obtained by inspection;b) the two-condition approach for simple networks with

controlled sources where answers are obtained almost byinspection;

c) the one-step approach otherwise.For a clearer division, one might count the number of equa-

tions needed. The one-step approach requires m mesh equa-tions (n node equations). The dead network impedance re-quires the same number of equations. The open-circuit voltage

2The one-step approach has no restrictions at alL The two-step ap-proach is restricted to quasi-linear networks. TheN + 1 conditions ap-proach is restricted to linear networks.3 Sirnlar conclusions can be obtained for two ports.

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REFERENCESJ. B. Cruz and M. E. Van Valkenburg, Signals in Linear Circuits.Boston, MA: Houghton Mifflin, 1974, pp. 143-152.Bose and Stevens,lntroductoryNetwork Theory. New York:Harper and Row, 1965, pp. 33, 301-306.K. L. Su, Fundamental of Circuits, Electronics, and Signal Analy-sis. Boston, MA: Houghton Mifflin, 1978.C. A. Desoer and E. S. Kuh, Basic Circuit Theory. New York:McGraw-Hill, 1969, p. 676.T. N. Trick, Introduction to Circuit Analysis. New York: Wiley,1977, pp. 125-128, 339-341.G. H. Hostetter,FundamentalsofNetworkAnalysis. Cambridge,MA: Harper and Row, 1980.A. B. Carlson and D. G. Griser, Electrical Engineering. Reading,MA: Addison-Wesley, 1981.L. S. Borrow, Elementary Linear Circuit Analysis. New York:Holt, Rinehart and Winston, 1981.M. F. Moad, "A sequential method of network analysis," IEEETrans. Circuit Theory, vol. CT-17, pp. 99-104, Feb. 1970.

Mohamed F. Moad (SM'65) was born in Da-mascus, Syria, on September 23, 1928. He re-ceived the B.E.E. and M.S. degrees in 1957 andthe Ph.D. degree in 1961 from the Georgia In-stitute of Technology, Atlanta, all in electricalengineering.He was appointed an Instructor in the School

of Electrical Engineering, Georgia Institute ofTechnology, from 1957 to 1961. From 1961to 1963 he was in charge of the Department ofStudies and Television Studios with the Syrian

Broadcasting Service. He returned to the Georgia Institute of Tech-nology in 1963 as an Assistant Professor, and became an Associate Pro-fessor in 1968. His main interest lies in the field of networks andsystems.Dr. Moad is a member of Sigma Xi and Eta Kappa Nu.

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