Development of Modified Nodal Analysis into a Pedagogical Tool

IEEE TRANSACTIONS ON EDUCATION, VOL. E-28, NO. 1, FEBRUARY 1985

power, using (A9) and (AIO), is

S/N = -(( f(t))27 .(Al l)4eBIf we assume that Af(t) is a harmonic signal with maximumfrequency deviation Afi, we have

((TAf(t))2) = (TAfm)2/2 (A12)and

S/N = (TAf )2. (A13)

Asanexample, ifPo = 0.1 mW,B = 20kHz,T = 5 X 10 s,y = 0.25 A/ W, and Afm represents the maximut deviationof 75 kHz, we find

(S N)max = 1.44 X 10 (81.5 dB).This means, of course, that in practice, the S/ N ratio will bedetermined by the electronic noise of the preamplifier andthe postamplifier.

ACKNOWLEDGMENTWe would like to thank Dr. A. Korpel for his suggestions

and critical review. The device discussed here represents aneducational aspect of more fundamental research intoacoustooptics.

REFERENCES[1] R. Adler, "Interaction between light and sound," IEEE Spectrum, pp.

42-54, May 1967.[2] T. R. Bader, "Acousto-optics spectrum analysis," Appl. Opt., vol. 18,

pp. 1668-1672, June 1979.[3] A. Korpel, "Acousto-optics-A review offundamentals," Proc. IEEE,

vol. 69, pp. 48-53, Jan. 1981.[4] C. Drentea, "The Bragg-,cell receiver," Ham Radio, pp. 42-44, Feb.

1983.[5] T.-C. Poon and R. Pieper, "An optical FM receiver," Ham Radio, pp.

53-56, Nov. 1983.[6] Intra-Action Corp., 766 Foster Avenue, Bensenville, IL 60106.[7] Edmund Scientific, 101 E. Gloucester Pike, Barrington, NJ 08007.

[8] A. Korpel, "Acousto-optics," in Applied Solid State Science, vol. 3,R. Wolfe, Ed. New York: Academic, 1972.

[9] A. Korpel, R. Adler, and B. Alpiner. "Direct observation of opticallyinduced generation and amplification of sound," Appl. Phys. Lett.,vol. 5, pp. 86-88, Aug. 1964.

[10] A. Korpel, "Two-dimensional plane wave theory of strong acousto-optic interaction in isotropic media," J. Opt. Soc. Amer., vol. 69, pp.678-683, May 1979.

[11] A. Korpel and T.-C. Poon, "Explicit formalism for acousto-opticmultiple plane wave scattering," J. Opt. Soc. Amer., vol. 70, pp.817-820, July 1980.

[12] T. K. Gaylord and M. G. Moharam, "Thin and thick gratings:Terminology clarification," Appl. Opt., vol. 20, pp. 3271-3273, Oct.1981.

[13] R. L. Whitman and A. Korpel, "Probing ofacoustic surface perturba-tions by coherent light," AppL Opt., vol. 8, pp. 1567-1570, Aug. 1969.

[14] A. Yariv, Introduction to Optical Electronics. New York: Holt,Rinehart and Winston, 1976.

[15] J. W. Goodman, Introduction to Fourier Optics. New York:McGraw-Hill, 1968.

Ronald J. Pieper (S'79) received the B.S. degree inphysics from the University of Missouri, St. Louis,in 1974, the M.S. degree, also in physics, and theM.S.E.E. degree from the University of Wisconsin,Madison, in 1976 and 1979, respectively.He is currently working on the Ph.D. degree in

electrical and computer engineering in the area ofacoustooptics at the University of Iowa, Iowa City.

Ting-Chung Poon (M'83) was born in Hong Kongon March 19, 1955. He received the B.A. degree inphysics and mathematical sciences in 1977, and theM.S. and Ph.D. degrees in electrical and computerengineering from the University of Iowa, IowaCity, in 1979 and 1982, respectively.He was an the Faculty of the Department of

Electrical and Computer Engineering, the Univer-sity of Iowa, from 1982 to 1983. He is currentlywith the Department of Electrical Engineering, Vir-ginia Polytechnic Institute and State University,

Blacksburg. His current research interests include acoustooptics and opticalinformatiott processing.

Dr. Poon is a member of the Optical Society of America.

Development of Modified Nodal Analysis Into aPedagogical ToolALI M. RUSHDI, MEMBER, IEEE

Abstract-A presentation of the modified inodal analysis (MNA) ismade to students in an undergraduate course on circuit theory. Thispresentation improves students' understanding of the basic concepts ofanalysis and familiarizes them with the nature of techniques implementa-ble on the computer. In addition, students are introduced to the basictheorems of circuit theory through the unifying framework of MNA.

Manuscript received September 19, 1983; revised April 30, 1984.The author is with the Department of Electrical Engineering, King

Abdul Aziz University,Jeddah 21413, Saudi Arabia.

I. INTRODUCTIONA BASIC course in circuit analysis is perhaps the mostA important course in the electrical engineering curric-ulum. It has long enjoyed the role of a service course tomany of the subjects that follow in the curriculum. Asurvey of many recent textbooks for such a course revealsthe following.

I) The proliferation of electronic devices which are

0018-9359/85/0200-0017 $01.00 1985 IEEE

17


described by models that contain dependent sources makesit essential to expand the set of basic circuit elements to beconsidered. The recent texts cover ten basic linear time-invariant elements, viz. four passive elements (G, C, L, andM), independent sources (CS and VS), and dependentsources (VCCS, CCCS, VCVS, and CCVS) [1]-[8]. More-over, more recent texts have started to treat the operationalamplifier or "op amp" as "just another circuit element"[6]-[8].

2) It is impossible to give an in-depth coverage of all themajor methods of analysis in the first course in the subject.Due to the importance of state variables, attempts havebeen made to use state variable formulation as the basic orsole method of analysis [1], [9]. However, most recent textsrecommend that the use of the traditional node and loopmethods be continued. One good feature of these methodsis that the circuit equations are easily written by inspectionin matrix form in the transformed (s or jwu) domain.However, these two methods are limited in their scope. Outof the ten basic elements, the node method can directlyhandle the five elements having i-v dependencies (i.e., G, C,L, CS, and VCCS), while the loop method can directlyhandle the six elements having v-i constraints (i.e., R, C, L,M, VS, and CCVS). In particular, neither node analysisnor loop analysis can directly handle all types of dependentsources. Consistent treatment of general circuits requiresthe use of a "mixed" method (see, e.g., [2], [10]-[12]) thatmay include the node and/ or loop method as a specialcase.

3) The computer is an indispensabie tool for analysisand design of larger and more complex circuits. Therefore,some recent texts [12] have emphasized computationalmethods and their computer implementations. However,many other texts [4], [13], [14] have recognized that it isdifficult to teach both the principles of circuit analysis andnumerical algorithms for circuit analysis in the first one-semester course. A compromise is to introduce the princi-ples and hand calculations of techniques that can beefficiently implemented on the computer and defer thenumerical aspects of these techniques to a more advancedcourse.

In view of the above, this author decided to present theprinciples of the modified nodal analysis (MNA) to hisstudents in the first undergraduate course on circuit analy-sis. This paper highlights the author's experience in teach-ing MNA during the past few semesters. MNA was used asa supplement rather than a replacement of node and loopanalyses. The choice of MNA for that role is justified byseveral reasons. MNA is quite general; it handles the tenbasic elements as well as other useful elements includingthe op amp, the nullator/ norator pair, and different two-port elements such as the gyrator, the ideal transformer,the negative-impedance converter (NIC), and the negative-impedance inverter (NII). The MNA matrix technique isvery simple, and is probably a less-error-prone techniquefor writing down the circuit equations. These equations aredirectly written by inspection, and are therefore easy toimplement on the computer. Students' learning is facili-tated by the MNA approach because of its generality,

conceptual clarity, and ready applicability in the study ofcertain network relations.

In the following, MNA is briefly outlined and thenillustrated by examples. Basic theorems and relations ofcircuit theory are discussed through the unifying frame-work of MNA. The dual to the MNA approach, viz.modified mesh analysis (MMA), is also hinted at. Inconclusion, certain pedagogical merits of MNA are reiter-ated.

II. BRIEF DESCRIPTION OF MNAIn 1975, Ho et al. [15] proposed MNA through a set of

self-consistent modifications to the nodal method thatretains its simplicity and other advantages, while removingits limitations. Since then, MNA has become the basis formany computer-aided circuit analysis programs [16]-[18].The following discussion of MNA is restricted to its use inthe transformed (s orjw) domain.

In matrix notation, the MNA equations are given byAX=S (1)

or by the equivalent partitioned form

EA2 A22 E uSx ES23 2where X = [Vn Iaux]T = vector of independent unknownvariables which are adequate for complete characterizationof the circuit, Vn - vector of nooe to datum voltages,Iaux=vector of auxiliary currents. (These are currents inelements with no i-v dependency and, possibly, currentsrequired as outputs. Current in the element L may also betaken as an auxiliary circuit variable to avoid numericalproblems in the dc limit in the jw domain or a steady-statelimit in the s domain, and to ensure uniformity of treat-ment when the circuit contains some mutual inductanceM.) S = [Sl, S2]T excitation vector whose entries arevalues of independent sources (including the initial-valuetransform sources in the s domain), S, = vector whoseentries are linear combinations of CS's, S2 = vector whoseentries are linear combinations of VS's, and A = matrixwhose entries depend solely on the circuit topology, fre-quency (s orjcf), and the parameters of elements other thanthe independent sources.

Equations (2) consist of the two sets of equationsAIIVn +Ai2Iaux =Si (3)A 21 Vn + A 22Iaux =S2. (4)

These are simply a convenient arrangement of the circuitequations. The KVL equations are implicitly satisfied bythe assumption of the existence of nodal voltages. Equa-tion (3) represents KCL applied at all the nodes excludingthe datum. Elements whose currents are not declared asauxiliary variables have their constitutive relations in-cluded in (3). For other elements, one auxiliary equationrepresenting the constitutive relation of each element isincluded in (4) so that the number of equations remainsequal to the number of unknowns in (1).The contribution of each circuit element to the MNA

equations is considered separately, since it is represented

18

RUSHDI: MODIFIED NODAL ANALYSIS

VS-C gv fi' C2 9g2 Is

Fig. 1. A circuit that contains all types of dependent and independentsources.

by the so-called "element stamp" [15]. A table containingthe different element stamps for both the s and jw domainsis easily prepared and made available to the students.Formulation of the MNA equations for a given circuit isstarted by arbitrary labeling of the node to datum voltagesand the required auxiliary currents. Subsequently, anempty matrix A and an empty vector S, both of properdimensions, are prepared. Then the elements of the circuitare processed one by one. The contribution of each elementis read from the stamps table and stamped into A and Saccording to the labels of the element's nodes and, possi-bly, its current. When all of the elements are processed, theMNA equations are complete. At this stage, the studentsare able to check their work by confirming that the first setof the MNA equations is really statements of KCL at theappropriate nodes.

III. EXAMPLESExample 1The circuit shown in Fig. 1 contains all the types of

dependent and independent sources. Therefore, it cannotbe analyzed directly by either the node or the loop method.The unknowns in the MNA formulation for it are the fournodal voltages V,, V2, V3, and V4 and the three auxiliary

[ V,(s) VA(S)KCL (1) g1 + S(Cl C3) -SC3KCL (2) -Scq g2 + S(C2 + C3)

Aux. (Ia) 1 0Aux. (I,,) 0 1

Fig. 2. A circuit that contains a mutual inductance.

Fig. 3. A delay equalizer circuit.

For convenience, the vector of unknowns X in (5) is notplaced to the right of the matrix A, but X' is placed aboveA instead. Equations (5) are written by using the standardelement stamps. Nevertheless, abbreviated descriptions ofthese equations are written on the left side to serve as aquick check.

Example 2The circuit shown in Fig. 2 is initially energized so that

all initial capacitor voltages and inductor currents aregenerally nonzero. This circuit cannot be handled directlyby the node method. In addition, the circuit equationsobtained by loop analysis are not as concise as thoseobtained by MNA. In the s domain, the MNA equationsare

Ia(S) Ib(s)]I 0 I(s) + C,v,(O ) + C3VI2(O )0 1 C2V2(0-) - C3V12(O-)

-sL, -sM -Llija(O )- Mi(O )-sM

-sL2_j -Mia(O-)-L2ih(O-)currents Ia (in the VS), Ih (in the CCVS), and I,. (in theVCVS). No auxiliary current is needed in this case for theCCCS since its controlling current Ia is one of the auxiliarycurrents. The MNA equations for this example are givenby (5).

Example 3The circuit shown in Fig. 3 is the general form of a delay

equalizer [19], and it contains one op amp. With the outputcurrent of the op amp taken as an auxiliary current, the

V2 PV3-gl 0

g1 0

14

09"

0 j(w) C2 -jw'U C2

Ia Ib II]1 0 00 1 0/3 -1 I 1=

0 -jwUC2 g2+jw(OC2 0 0 -l

0 C

1 -1'

-JA 1

0 0 0 0

0

-1

rm 0 0

0 0 0O

-o

00

'S

VS0

0

[ VIgi +jwC,

O

KCL (1)KCL (2)KCL (3)KCL (4)Aux. ('a)Aux. (Ib)Aux. (lI)

0

O0

0

(5)

19


INA equations are

KCL (1)KCL (2)KCL (3)KCL (4)Aux. ('a)

[ VI V2

YI + Y2 -YI

V3

Y2

YI + Y4 0y1

Y20

-

0

0

y4

Y2 + Y3

0

V4 I.]

0 ol-Y4 01

0 0I=

Y4 1

-1

0

0

0

(7)

The above formulation is similar to that suggested in[20] where a dummy node connected to the output node ofthe op amp through a (-1) conductance is introduced.Both formulations ensure that the op amp has a zerooutput impedance. However, the introduction of an auxil-iary current or a dummy node can be avoided by deletingthe KCL equation at the op amp output node. In this case,(7) reduces to

[ VI V2KCL(1) Y+ Y2 -YKCL (2)

-Y, Y, +

KCL (3)-Y2 0

Aux. () L 0 1

Equations (8) do not follow the general form of MNA, butthey are more convenient than (7) for obtaining transferfunctions or driving-point impedances.

IV. CIRCUIT RELATIONS IN MNA PERSPECTIVEIn the following, some circuit relations are discussed

through the unifying framework of MNA. Since MNA isjust a combination of topological relations and elementcharacteristics, the forthcoming discussion should be un-derstood to be based directly on these fundamental circuitrelations and not to rely on any particulars of MNA.

A. Superposition PrincipleIt is a fact that a circuit made up entirely of linear

elements is itself linear. This fact constitutes the proof ofthe superposition principle. Nevertheless, this principle canbe exhibited in a particularly useful and lucid fashion withthe aid of MNA.The solution to (1) may be found by Cramer's rule. If A

is the determinant ofA and Aj, is the cofactor of element aof A, then a typical unknown component Xi is given by

Xi= A E Aji Sj. (9)If the independent sources of both types are denoted by Skwhere k =1 2, then since each Sj is a summation ofeither CS or VS values, it may be expressed as

S = L Cjk Skk

(10)

where Cjk -1, 0, or 1, and hence upon substituting (10)in (9) and interchanging the order of summations, the

expression for Xi becomes

x,i= ( I E C,' 'Ai Skk=EkFk Sk = E xi (II)k k

which means that the value of Xi (whether it is a nodalvoltage or an auxiliary current) is a linear combination of

V3 V4]-Y2 s

Y4 0-Y4 0 (8)

Y2 + Y3

-1 0iLthe values of the independent sources (whether they areCS's or VS's). The value of Xi may be calculated as thealgebraic sum of the individual responses [k) k = 1, 2,

where )yk) is caused by, and is proportional to, theindependent source Sk acting alone, i.e., with all otherindependent sources killed (set to zero). The same appliesto any particular current or voltage in the circuit sincethese are linear combinations of the Xi's.The present proof of superposition is more general than

similar proofs in available textbooks (see, e.g., [3]) whichare usually restricted to circuits having one type of inde-pendent sources only, or may otherwise request that exist-ing sources of one type be transformed to the other type. Inaddition, since the formulation in (1), and hence thesolution in (11), encompasses all four types of dependentsources, it becomes easier for students to differentiatebetween the natures of the two types of sources; theindependent sources are "true" sources giving rise toexcitations that can be superposed in the linear case, whilethe dependent sources contribute to the circuit equations ina way similar to that of the elements G,C,L, and M in thesense that both of these elements and the dependentsources are manifest only in the matrix A in (1) [andsubsequently only in A and Aji in (11)]. In fact, a point thatnormally causes uneasiness for students is clarified, name-ly, that dependent sources should always stay (i.e., never beset to zero) in the individual circuits obtained whensuperposition is applied.B. Network FunctionsThe use of MNA makes it possible to deduce general

formulas for many network functions Fik where

20


TABLE IRECIPROCITY PROPERTIES

Connections at the Ports of NNo. Port I Port r Port i Port r Property

I Cs oc oc Cs V,I,I /,2 VS SC SC VS Ih VI = Id/ V3 VS OC SC CS Vr/ V, = -11,4 CS SC OC VS 1r7 it VV,/V

Ib)Fig. 4. A two-port network N under two different excitations. Standardreference directions are indicated for the ports voltages and currents.

x iFlk=- A(12)Sk |5 =OVjkk

may denote a driving-point or transfer immittance or atransfer ratio, depending on the natures of Xi and Sk. Sinceany entry in A is, in general, a first-degree polynomial in sorjw, both A and Aji in (12) are polynomials, and hence Fikis a rational function of s or ji).C. ReciprocityThe linear time-invariant network N shown in Fig. 4 is

reciprocal if [4], [21]b

24(Vk ik -Vk I) 0 (13)k= 1

for all transform variables { Vk, Ik} and I Vk, Ikl that satisfythe circuit equations. It is easily seen from (13) that out ofthe elements referred to in the Introduction, only theelements G, C, L, M, NII, and the ideal transformer arereciprocal elements in the jwc domain. The same elementsare the reciprocal ones in the s domain also provided thatC, L, and M are initially unenergized. Hence, a networkconsisting of these elements only is a reciprocal network.Viewing the stamps of the different elements in the MNArepresentation, it is found that

1) reciprocal elements have symmetric stamps in A, butno entries in S,

2) independent sources (including the initial-conditiontransform sources) have symmetric stamps in A besidesome entries in S as well, and

3) other elements have stamps that destroy the symme-try of A.

Therefore, the network N can be said to be reciprocal ifit contains no independent sources, and its matrix A issymmetric, i.e., if

used when the network N contains independent sources,provided that the result is interpreted as dealing only withthe component of the response due to an external source[22, p. 471]. The first two properties in Table I indicate thatthe transfer immittance is invariant to an interchange ofthe positions of the input and output ports. Property 1 canbe proved for a network with symmetric Y [4] and Prop-erty 2 can be proved for a network with symmetric Z [22].All four properties can be easily proved for a network withsymmetric A. As an illustration of this point, the proof ofProperty 4 is considered. In this case, the MNA equationsfor the network N excited by a CS at its i-port and havingan SC at its r-port are

KCL ({i})KCL (1,)KCL (12)KCL (r,)KCL (r2)Aux. ({]})Aux. (r)

[Vi}VV V kI I~ V2

A

0 0

I" Vr ILIj Ir III

O]1i1

1-10,

so that the current I, is given by

ir , (Ar Ai2r).

0-

II

1,

0

0

0

0

(15)

(16)

When the network N is excited by a VS at its r-port and hasan OC at its i-port, the MNA equations (15) remain thesame, except that the unknowns X are replaced by theirhatted versions X and that the excitation vector is replacedby

[O 0 0 0 0 0 V]T.

Hence, the voltage V, is given by/- vrVl = VIl, VI2 / (Ari rl2) (17)

S=O and A =AT (14)The above definition parallels the presentation available inmost texts in which a network is shown to be reciprocal itits matrix Z in loop analysis, or dually, its matrix Y in nodeanalysis, are symmetric.

Table I shows certain properties that a reciprocal net-work N has [4]. The reciprocity properties in Table I can be

From (16) and (17), it is seen that Property 4 is satisfied ifA - A and A = A , i.e., if A is symmetric.

ri, I,r r12 12r'D. Two-Port Relations

If the linear time-invariant network N in Fig. 4(a)contains no independent sources, then two homogeneousrelationships can be written involving the four variables II,

VI Vr

(a)

Vr

21


TABLE IITWO-PORT RELATIONS

Function ExcitationsIndependent Dependent

No. Name Variables variables i-port r-portI OC impedance I,, I, V,, VI CS Cs2 SC admittance V,, VI I,jI, VS VS3 Hybrid II, V, V,,I, CS VS4 Inverse hybrid V,l,I I,,V, VS CS5 Chain (trans- V,,I, V,J, Norator CS

mission) shuntedby a nul-lator inserieswith aVS

6 Inverse chain V,I, J,Vrir CS Noratorshuntedby a nul-lator inserieswith aVS

VI, Ir and Vr of the two ports. The interrelationshipsamong these four variables may be expressed in six differ-ent ways, depending on which two of the variables areregarded as independent and which two are regarded asdependent [3], [23]. Table II shows these six possibilities,together with the appropriate excitations at the /- and r-ports necessary for deriving them. In some texts (see, e.g.,[3]), node analysis is used for the derivation of the OCimpedance relations, while loop analysis is used for thederivation of the SC admittance relations.MNA can be used, easily and directly, to derive all six

sets of relations in Table II. In other words, all six sets maybe viewed as special cases of MNA relations. For example,if the excitations indicated in line 6 of Table II are used, theMNA equations take the form

KCL ({i})KCL (11)KCL (12)KCL (r,)KCL (r2)Aux. ({j})Aux. (I)

[{V } V1 V12 Vr Vr2 {IJ} Ir!

|A I01

-1

0 1 -1 0 0 0' 0

-II

00

I(n-m)a

Linear +Network Vm Load

0

'(n-rn)Fig. 5. Network under consideration partitioned into a linear network N

and a load D.

the use of special circuits, sources, or network equations.He also realized that these particulars are useful in under-standing the physical implications of the theorem. In thefollowing, a general and constructive proof based on MNAis given for both Thevenin's and Norton's theorems. Theproof exhibits useful details required in the application ofthese theorems, and may be considered as an extension toan earlier proof [26] based on node and loop analyses.

Let two parts of the network under consideration bedistinguished as network N and load D (Fig. 5), connectedthrough one port ao. Without loss of generality, node o istaken as the datum, and assuming the number of othernodes in N to be m, node a is arbitrarily labeled as the lastamong these so that Vap, is labeled Vp,. If the network N ischaracterized by the MNA equations (1), and the load D ischaracterized by the general linear relation

CIVm + C2I( - m) = Sr, (19)then MNA can be applied to the combination of N and D,with D treated as a single element (although it may be anetwork of great complexity). The resulting equations are

AeXe = Se

which can be expanded as

---------

--L-- --m A 1A

n 2 f l ' O C0

(20)

S

-LiSI'(21)

0 In (20), the dimension of Ae is denoted by n > m, so thatthe number of the auxiliary currents is considered to be

Vjj (n -m) and the last of these currents is the current at port(18) ao. From (20), the unknowns V,, and I(;np 11) are obtained

via Cramer's rule asfrom which it is easy to deduce linear expressions forVr= VrV-rV and Ir in terms of VI and II. These expressionscan be identified as the inverse chain relations for thenetwork N.E. Thevenin's and Norton's Theorems

Different proofs of Thevenin's and Norton's theoremsare available in many texts [24]. Recently, Haley [25] gavean elegant proof of Thevenin's theorem that follows di-rectly from the definition of linearity and does not resort to

n

Vm = -, E 'Ai.l SiA 111=

I nItn m) =- VE Ain S

(22)

(23)where A is the determinant of Ae and Ai,, and z\i, arecofactors of obvious meanings. It is clear from (21) that

A - Cl 'Anm + C2 A2n

22

(24a)


Aim C2 (Aim)(n- 1)(n- I) for i =1 25 - * (n - 1)(24b)

Ain Cl (A,in)(n- I)m- C1 (Aim)(n- 1)(n -1) for i 1, 2, (n - )

(24c)Ann = (Anm)m(n- 1) =- (Anim)(n I)(n- 1) =-(A/nn)mm (24d)Now, expressions for the three pertinent parameters, viz.

the open-circuit voltage V0f, the short-circuit current Is,and the input impedance Zin are derived with the aid of theMNA matrix equations and then interrelated to yieldThevenin's and Norton's relations.

If the load D is replaced by an OC, its constraintbecomes

I = (n m)which is a special case of (19) with

C, =0, C21 I,and S1=O. (26)Hence, in view of (22), (24a), and (24b), the open-circuitvoltage at port ao is given in the superposition form

I (n- 1)VOC= {Vm} 9 E (Afm)n -)(n I)Si (27)

nn i=

Voc { Vm}CI =Sn= 0, C2 =I Cl = Sn = 0, C2 = 1.Similarly, if the load D is replaced by an SC, its

constraint becomesm 0 (28)

which is a special case of (19) withC, = 1, C2 =0, and Sn =0. (29)Hence, in view of (23), (24a), and (24c), the short-circuitcurrent at port ao is given in the superposition form

-

(n -I)I5C =tIAn- m)}J= A (Aim)(n- 1) (n- I)Si (30)

'nm i-=X

{'i= II(n1-in)iS,=2=0, CI = = Sn = C2 = 0, Cl = 1.The input impedance Zin at port ao is the ratio between

the voltage Vm at that port and the current (-I(n_m)) drivenat it into network N when that network is excited (eitherby a VS or a CS) at port ao only, i.e., when Sn # 0, allindependent sources in N are set to zero (S = 0 for i = 1,2, * * , (n - 1)), and all other elements remain in N. Hence,in view of (22), (23), and (24d), Zin is given byZi= 1=-Anm1 Ann = element mm of A -' (31)Comparing (27), (30), and (31), it is noted that Zin as

defined above is given byZin = V"1, / I, (32)Expression (32) above is undefined if the network N

contains no independent sources, i.e., if Si = Ofor i = 1, 2,* *, (n 1), since in this case, both VO,. and J,, are equal tozero.

Now, (22) can be rewritten with the aid of (24a), (24b),and (19) as

(n- 1)(Cl Anm + C2 Ann) Vm = C2 Ei (Aim)(n-1) (n- I)Si

i =

+ 'An"I Sn- C2Ann VVc + Anm (CiVm+ C2I1(n- m)).

Hence, it reduces, with the aid of (31) and for finite Zin, toVm = Voc ZinIZnVZ Im1 01 in l(n m)- (33)Similar straightforward manipulations of (23), (24a),

(24c), and (19) yield the following dual equation for finite Yin:(25) IAn - m) = Ist- Yin Vm (34)

Equation (34) can be obtained also by substituting (32) into(33). Each of the two relations (33) and (34) identifies thelinear terminal behavior of the linear network N. Equation(33) yields Thevenin's equivalent circuit, while (34) yieldsits dual, Norton's equivalent circuit.

In conclusion, it is noted that the assumption of loadlinearity is not a necessary one. If the load is nonlinear,then the substitution theorem can be used [27] to replacethe load at its operating point by a linear one prior toapplication of the MNA equations.

F. Natural FrequenciesFor any one-port network, there are two sets of natural

frequencies at which signals can exist in the networkwithout the necessity of supplying driving power [23]. Forthe network N in Fig. 5, the SC natural frequencies of thenetwork with respect to port ao are the poles of the driving-point admittance Z`' (s), while the OC natural frequenciesare the poles of the port's driving-point impedance Zjn(s).In view of (31), these are given, respectively, by the roots of

cofactor mm of A = ((35)(36)determinant of A 0O

where A is the s-domain MNA matrix for N. It is clearfrom (35) and (36) that the form of the natural response ofa network depends only on A and not on S.

V. THE DUAL METHODLoop and node analyses may be thought of as dual

mnethods. The exact dual of node analysis is a particularsubset of loop analysis, viz. mesh analysis [28], which isapplicable to planar circuits only, and in which the meshesor windowpanes of the circuit are selected as its independ-ent loops, with all loop currents taking one direction only(either clockwise or counterclockwise). The dual to MNAis modified mesh analysis (MMA) in which the unknowncircuit variables are the mesh currents together with auxil-iary branch voltages for elements lacking v-i dependencies.In a way similar to that of MNA equations, the MMAequations are readily written in matrix form in terms ofstandard element stamps. Modified general loop analvsis

23


M

-

-LI

v

Fig. 6. A circuit that is easier to solve by MMA.

can also be conceived. However, it lacks the simplicity ofhaving standard element stamps.

Acquainting students with the idea of MMA proveduseful in stressing the concept of duality, and in providingan alternative approach that might be simpler for the handcalculations for some particular circuits. For example, thecircuit in Fig. 6 is better solved via MMA since, in thiscase, only four unknowns are needed, namely, the meshcurrents II, I2, and I3 together with the branch voltage Vaacross the VCVS. The circuit MMA equations in the sdomain are

[I (s)KVL (1)

KVL (2)

KVL (3)

Aux. (V0)

I2(S)R2+ sL, -R2

-R2 RI+ R2 + IC

sM

13(s)

sM

[3] M. E. Van Valkenburg, Network Analysis, 3rd ed. EnglewoodCliffs, NJ: Prentice-Hall, 1974.

[4] T. N. Trick, Introduction to Circuit Analysis. New York: Wiley,1977.

[5] W. Hayt and J. Kemmerly, Engineering Circuit Analysis, 3rded. New York: McGraw-Hill, 1978.

[6] D. E. Johnson, J. L. Hilburn, and J. R. Johnson, Basic ElectricCircuit Analysis. Englewood Cliffs, NJ: Prentice-Hall, 1978.

[7] M. E. Van Valkenbiurg and B. K. Kinariwala, Linear Circuits.Englewood Cliffs, NJ: Prentice-Hall, 1982.

[8] J. W. Nilsson, Electric Circuits. Reading, MA: Addison-Wesley,1983.

[9] R. A. Rohrer, Circuit Theory: An Introduction to the State VariableApproach. New York: McGraw-Hill, 1970.

[10] T. A. Bickart, "Network equations," IEEE Trans. Educ., vol. E-7,pp.9- I, Mar. 1964.

[I 1] F. H. Branin, Jr., "Computer method of network analysis," Proc.IEEE, vol. 55, pp. 1787-1801, Nov. 1967.

[12] S. W. Director, Circuit Theory: A Computational Approach. NewYork: Wiley, 1975.

[13] L. 0. Chua and P. M. Lin, Computer-Aided Analysis of ElectronicCircuits: Algorithms & Computational Techniques. EnglewoodCliffs, NJ: Prentice-Hall, 1975.

[14] P. R. Adby, Applied Circuit Theory: Matrix and Computer Meth-ods. Chichester, England: Ellis Horwood, 1980.

[15] C. Ho, A. E. Ruehli, and P. A. Brennan, "The modified nodal

Va(s)]

I_

SC

r0,- I R3+ sL2 +-SC ~~SC

0

0

-1

0

V(s) + L,i, (O-) + Mi3 (O-)

-s vc(O )

-S-vc(0-) + Mil (O )+ L2i3 (O )

0

(37)

VI. CONCLUSIONSThe MNA manual procedure described here is essen-

tially what is done by computer programs based on MNAfor formulating circuit equations. Therefore, by learningMNA, students not only improved their understanding oftopological relations, element characteristics, circuit theo-rems, duality, and other concepts of circuit analysis, butthey also were exposed to the nature of techniques that canbe implemented on the computer. The MNA method waswell received by students, and their performance in examquestions requiring MNA was consistently good. It wasclear that MNA gave many students confidence to attack,and to correctly solve, relatively large circuit problemsand/ or problems containing dependent sources or opamps.

ACKNOWLEDG MENTThe author first learned MNA from Prof. T. N. Trick

when he was his student in a graduate course at theUniversity of Illinois. The author is also thankful to thereviewers for their constructive comments.

REFERENCES[1] T. S. Huang and R. R. Parker, Network Theory: An Introductory

Course. Reading, MA: Addison-Wesley, 1971.[2] R. D. Strum and J. R. Ward, Electric Circuits and Networks. New

York: Quantum, 1973.

approach to network analysis," IEEE Trans. Circuits Syst., vol.CAS-22, pp. 504-509, June 1975.

[16] 1. N. Hajj, P. Yang, and T. N. Trick, "Avoiding zero pivots in themodified nodal approach," IEEE Trans. Circuits Syst., vol. CAS-28,pp. 271-279, Apr. 1981.

[17] A. E. Ruehli and G. S. Ditlow, "Circuit analysis, logic simulation,and design verification for VLSI," Proc. IEEE, vol. 71, pp. 34-48, Jan.1983.

[18] M. L. Liou, Y. Kuo, and C. F. Lee, "A tutorial on computer-aidedanalysis of switched-capacitor circuits," Proc. IEEE, vol. 71, pp.987-1005, Aug. 1983.

[19] J. Millman and C. C. Halkias, Integrated Electronics: Analog andDigital Circuits and Systems. New York: McGraw-Hill, 1972.

[20] S. K. Mitra, Analysis and Synthesis of Linear Active Networks.New York: Wiley, 1969.

[21] C. A. Desoer and E. S. Kuh, Basic Circuit Theory. New York:McGraw-Hill, 1969.

[22] E. Brenner and M. Javid, Analysis of Electric Circuits, 2nd ed.New York: McGraw-Hill, 1967.

[23] S. S. Haykin, Active Network Theory. Reading, MA: Addison-Wesley, 1970.

[24] M. F. Moad, "On Thevenin's and Norton's equivalent circuits,"IEEE Trans. Educ., vol. E-25, pp. 99-102, Aug. 1982.

[25] S. B. Haley, "The Thevenin circuit theorem and its generalization tolinear algebraic systems," IEEE Trans. Educ., vol. E-26, pp. 34-36,Feb. 1983.

[261 P. Sankaran, K. S. Rao, V. V. B. Rao, and V. G. K. Murti, "Onnetwork theorems," IEEE Trans. Educ., vol. E-l 1, pp. 154-155, June1968.

[27] F. F. Judd and P. M. Chirlian, "The application of the compensationtheorem in the proof of Thevenin's and Norton's theorems," IEEETrans. Educ., vol. E-13, pp. 87-88, Aug. 1970.

[28] J. R. O'Malley, Circuit Analysis. Englewood Cliffs, NJ: Prentice-Hall, 1980.

24

l~


Ali M. Rushdi (S'73-M'81) was born in Port-Said, Egypt, on May 24, 1951. He received theB.Sc. degree with honors in electrical engineering(electronics and communications) from CairoUniversity, Cairo, Egypt, in 1974 and the M.S.and Ph.D. degrees in electrical engineering fromthe University of Illinois, Urbana-Champaign, in1977 and 1980, respectively.

In 1974 he was appointed Demonstrator andInstructor at the Department of Electrical Engi-neering, Cairo University. From 1976 to 1980 he

was a Research Assistant at the Electrom,agnetics Laboratory, University

of Illinois. Since 1980 he has been with King Abdul Aziz University,Jeddah, Saudi Arabia, as Assistant Professor of Electrical and ComputerEngineering. At King Abdul Aziz University, he has structured and taughta variety of undergraduate courses and senior projects, and he has activelyparticipated in the establishment of the newly founded M.S. program. Hehas published in the areas of electromagnetics, switching theory, reliabil-ity analysis, and circuit theory. He is an author of the textbookPrinciples of Engineering Measurements (Jeddah: KAAU Press, 1984, inArabic), and a contributor to the forthcoming Annotated ArabicDictionary of ihe Technical Terms of Electrical Engineering.

Dr. Rushdi is a member of the Engineers Syndicate of Egypt, AMSE,nine IEEE societies, Eta Kappa Nu, and Phi Kappa Phi.

Enhanced Capabilities for a Student-Oriented FiniteElement Electrostatic Potential Program

JAMES L. DAVIS AND JAMES F. HOBURG, MEMBER, IEEE

Abstract-Several enhancements to FINEL, a student-oriented finiteelement program for electrostatk potential problems, and PLOTR, anassociated plotting program, are described. These include incorporationof specified charge density within the solution region, specification of ma-terial regions with differing relative permittivities, "ballooning" for de-scription of open boundary conditions, energy storage computation, andseveral convenience features.

I. INTRODUCTIONTHE uses of FINEL, a student-oriented finite element pro-Tgram for electrostatic potential problems, and PLOTR, an

associated plotting program, were described in a recent paper[3]. The programs provide an exposure to finite element mod-eling for students in "Field Analysis and Engineering," acourse for juniors in Electrical and Computer Engineering atCarnegie-Mellon University. Students are able to experimentwith relatively complex geometric structures, and thereby gaininsights which might not otherwise "sink in." As well, theygain some experience with a numerical method which is cur-rently undergoing very rapid development and widespread en-gineering use.

Students in "Field Analysis and Engineering" have accessto FINEL and PLOTR by way of accounts on University main-frame machines. They are asked to use these tools on one ortwo of a total of eleven or twelve weekly problem sets. One ortwo additional problem sets typically involve some use of nu-merical methods. For example, students might be asked towrite and apply their own finite difference (relaxation method)programs and/or to program numerical integrations overknown source distributions to determine electrostatic potentialvalues at specified points and to compare them to analytic so-lutions.

This paper describes several enhancements to FINEL and

Manuscript received March 19, 1984; revised May 7, 1984.The authors are with the Department of Electrical and Computer Engi-

neering, Carnegie-Mellon University, Pittsburgh, PA 15213.

PLOTR which provide additional capabilities for use of theprograms as educational tools. FINEL and PLOTR are For-tran programs, implemented at Carnegie-Mellon University ina DEC System 20 environment. PLOTR employs DISSPLA,a product of ISSCO, to generate plots on a Tektronix plottingterminal or CALCOMP plotter. The second author should becontacted for information on obtaining the user guide andsource codes for FINEL and PLOTR.

II. ENHANCEMENTSThe following specific developments distinguish FINEL and

PLOTR from their ancestors.

A. Incorporation of Specified Charge Density Within theSolution Region

Originally, FINEL solved Laplace's equation, V2k = 0,within a given region. The program now solves Poisson'sequation, V2$ = -ple, with normalized charge density p/eOspecified at nodal points in the finite element grid.

This capability provides the student with an opportunity toexplore specific consequences of Gauss' law in realistic engi-neering problems. Thus, for example, the potential structurewithin an electrostatic precipitator with grounded walls, high-voltage corona wire, and known "space charge" density in theinterior provides fertile ground for the establishment of in-sights. (Students whose interests lead them to question howthe "given" charge density is actually determined are set ontheir way toward more challenging analyses and issues [2].)The finite element modeling serves as a different point of viewupon problems whose analytic solutions involve decompositionof the potential solution into particular and homogeneous parts.B. Specification ofMaterial Regions with Differing RelativePermittivitiesFINEL now permits specification of relative permittivities

for groups of elements. Thus, the student is able to experiment

0018-9359/85/0200-0025$01.00 1985 IEEE

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Development of Modified Nodal Analysis into a Pedagogical Tool

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