IEEE TRANSACTIONS ON EDUCATION, VOL. E-28, NO. 3, AUGUST 1985

A Network-Oriented Approach to the Teachingof Electromagnetics

ISTVAN PALOCZ, SENIOR MEMBER, IEEE, AND NATHAN MARCUVITZ, LIFE FELLOW, IEEE

Abstract-In this paper, electromagnetic theory is presented by ex-panding upon ideas that electrical engineering students already pos-sess. The equivalent netwotk concept, well known for lumped circuits(ordinary differential equations), is generalized to electromagnetic fieldsin free space (partial differential equations). The network, rigorouslyequivalent to Maxwell's equations, is suitable for all problems in freespace. One illustrative antenna problem is presented and its relevanceto lumped network calculations is also shown.

I. INTRODUCTIONI LECTROMAGNETIC theory poses difficulties to

any electrical engineering students. The topic is ab-stract and seetns to be unrelated to previous experience.The pedagogy used today traces back to the time when

electromagnetic theory was taught to only physics majorswho had a thorough background in potential theory. Dueto other important developments (quantum theory, solid-state physics, etc.), a course in potential theory is nolonger given to physicists; we are told that present physicsstudents also have difficulties in mastering electromag-netic theory, just as their counterparts in electrical engi-neering, The present authors attempt to present electro-magnetic theory by expanding upon ideas that electricalengineering students already possess.

In this paper, to show you the flavor of a forthcomingbook, we first generalize the equivalent network concept,well known for lumped circuits (ordinary differential equa-tions), to fields (linear partial differential equations). Werestrict ourselves to electromagnetism in free space; theequivalent network will be rigorously equivalent to Max-well's equations in free space; i.e., anything that can bederived from Maxwell's equations can also be obtainedfrom the network and vice versa. As an illustrative ex-ample of the network, we shall treat an antenna problem;in so doing, we will not introduce new concepts to engi-neers (such as vector potential, etc.) that many authorsconsider indispensable.

II. EQuIVALENT NETWORK OF FREE SPACEConsider Maxwell's equations in free space

VxEr,t) ---AOaH(ir, t)-V x E(r, t) =M Z dat

aE(r)t)V xH(r,t) =e 'oa + J(r, t)

(la)

V * E(r, t) = - p(r, t)

V *H(r, t) = 0.

Let us first consider the source-free case

J(r, t) = 0

p(r, t)= 0.

(Ic)

(Id)

(2a)

(2b)

Students in EM theory should understand clearly that theabsence of sources does not necessarily imply a quiescentsystem. They encountered an example in lumped circuittheory, where a lossless LC circuit can oscillate forever atits free-resonant frequency if energy is present initially.We know from the theory of linear differential equations

that the homogeneous equations (1) and (2) are satisfiedby plane waves traveling in the k direction

E(r, t) = EeJ(w,- k r)

H(r, t) = Hej(t-k r).

(3a)

(3b)

On the right-hand side of (3), the complex amplitudes Eand H are neither functions of time t nor space r; theydepend, in general, on the temporal and spatial frequencyparameters X and k. Let us substitute (3) into (1) in theabsence of sources (2). From their courses in lumped cit-cuit theory, students know that for a harmonic solution(3), the time derivative operator a/at is equivalent to amultiplication by jw. Similarly, the spatial operator V canbe replaced by a multiplication by the vector -jk. Thus,we get

a

V - -jk

Vx -* -jk x.

Hence, in the absence of sources, Maxwell's equations infree space take the form

k x E = wtiLHH x k = WcEE(lb)

(5a)

(Sb)

Manuscript received July 19, 1984; revised December 4, 1984.The authors are with the Department of Electrical Engineering and

Computer Science, Polytechnic Institute of New York, Brooklyn, NY 11201.

k- E=0

H k=0.

0018-9359/85/0800-0150$01.00 1985 IEEE

(5c)(Sd)

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(4)

PALOCZ AND MARCUVITZ: TEACHING ELECTROMAGNETICS

It is appropriate to point out that the vector equations (5)are a set of algebraic vector equations; thus, we have re-duced a system of linear partial differential equations toalgebraic equations; this can be accomplished for any lin-ear partial differential equation and represents a general-ization of what is done for the ordinary differential equa-tions of lumped network theory where k = 0. Equations(5) are in coordinate-free form. Let us introduce a right-handed orthogonal vector basis k, TO, TO. As shown inFig. 1, ko is a unit vector in the direction of wave travel k;TO and TOT are unit vectors perpendicular to k,. The com-plex vector amplitudes for the electric and magnetic fieldsE and H are decomposed into longitudinal (subscript L)and transverse (subscript 1) components as follows:

E = ko EL + TXET' + TO" ET"

H = kOHL + THT, + To"HT'Since

k = kok

zo

o0

Fig. 1. Characteristic vector basis T' is in the plane ko and the axis z,. T"is perpendicular to the paper and points toward you.

a) Resistor

vltVI R

(6a) V= RI

(6b)

b) Inductor

t4Ivt L

V= jwLI

c) Capacitor

vtTCI jS C\

d) Transformer-.,II - I2

t 2

a:b

VI - I2 - aV2 II b

Fig. 2. Linear network elements and basic relations.

(6c)

substituting (6) into (5a) we get

0 = jcu4o( -jHL)

-kET" = J"'/-o( -jHT')

kET' = jwoA(HT").By duality [or alternately decomposing (Sb)], we get

0 = jLE0EL

-k(-jHT",) = jkEET'

+k(-jHT') = jE0ET"'

(7a)

(7b)

(7c)

-jHL LO Et| cXo

K(-jHHrL) p

iJHT" KET't ET' t Co°ETI {I

K3. Hdo P

-jHT'j KE T"I E0 T t (°

K:lI

Fig. 3. Network diagram of free space.(7d)

(7e)

(7f)Equations (7) are the algebraic steady-state Maxwell'sequations for free space: (7a), respectively, (7d) containsthe same information as (5d), respectively, (Sc).We now introduce a network to schematize the algebraic

equations (7). Let us pictorialize (7) as circuit relationsand adopt the following nomenclature. The circuit will becomposed of "inductance parameters" Au, "capacitanceparameters" E0; in the circuits we have "voltages" E and"currents" (-jH). In the following, we omit the quotationmarks for inductance, voltage, etc. We schematize in-ductance, capacitance, and ideal transformers in the usualmanner; this is shown in Fig. 2. All the relati ns in (7)are equivalently contained in the networks of Fig. 3. Wesee four independent networks in the figure. In the upper-left corner, a current -jHL flows in an inductor tL, hence,the voltage drop is jcoji(-jHL). Since the inductance isshorted, the voltage drop is zero. This statement is rep-resentative of (7a). In the upper-right corner of Fig. 3, a

capacitor E, is shown, its admittance (jWE,) times the volt-age (EL) gives a zero current since the circuit is open.This is representative of (7d). In the middle portion of Fig.3, we see an inductor 1k9 in the primary and a capacitor e,

in the secondary; these are coupled by an ideal trans-former of turn ratio k to one. This network is rigorously

equivalent to (7c) and (7e). In the primary, we see a volt-age kET', which equals the inductive impedance jwgo timesthe current -jHT, [see (7c)]. In the secondary, the voltageis ET' since the turn ratio is k to one. The capacitive ad-mittance jwEo, times the voltage ET equals the current -k(-jHT,,) flowing from P to P' through the capacitor. Thisis just what (7e) says. The lower part of Fig. 3 shows thesame network as the middle one except that the prime sub-script is replaced by a double prime and vice versa, alsothe direction of the currents is reversed; the relations ob-tained from this network are identical to (7b) and (7f).Thus, we conclude that the network in Fig. 3 is rigorouslyequivalent to (7); the information contained in Fig. 3 isalso obtainable from (7) and conversely; thus, conclusionsderivable from (7) also follow from the free-space equiv-alent network shown in Fig. 3.As an illustration of the use of the above free-space net:

work, let us evaluate the power radiated by an antenna infree space. Accordingly, we excite the network by a cur-rent generator ig(t) (the antenna current) and determinethe voltage response of the network from which the aver-age radiated power can be calculated. We cxpect that thereal power absorbed by the network is just the power ra-diated by the antenna. Here we appear to encounter a par-adox! Since no dissipative element appears to be present,

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IEEE TRANSACTIONS ON EDUCATION, VOL. E-28, NO. 3, AUGUST 1985

see Fig. 3, we wonder how the network can absorb powerfrom the applied electric current source.

III. A LUMPED NETWORK PROBLEMWe encounter the same situation in a lossless LC net-

work problem. Since EE students are more familiar withthis simple lumped network, let us discuss it first. Givenan inductor L and a capacitor C connected as shown inFig. 4, let us first consider the circuit in an unexcited state,i.e., the current and voltage are zero in the circuit for neg-ative values of time; then at some time, which we call t =0, we apply to the circuit an impulsive generator ig(t) =Q b(t) and ask: what is the energy W supplied to the cir-cuit?The voltage on the capacitor jumps to QIC at t = 0 and

for all values of time it is given by

v(t) = C COS Oat U(t) (8)

where u(t) is the unit step function (its value is unity fort > 0, and zero for t < 0), and wa = Ii/ZC. To calculatethe energy W supplied to the LC network, we note that thevoltage at the moment when the delta function acts is theunit step function and the delta function is the derivativeof the unit-step function, therefore,

W = Vv(t)ig(t) dt = -X cos Wat u(t) b(t) dt

= Q X u(t) d()dt

Q2 co du2(t) d 2 1C J-,dt 2 dt=C2' (9a)

A deeper insight is gained by recalculating the energy sup-plied in the transform domain. According to Parseval'stheorem, well known to EE students from their circuit the-ory course, the energy delivered for all values of time toa circuit of resistance R by a current source Ig(w) can becalculated in the w domain as

L Cc

3L v c 19e

Fig. 4. LC circuit excited by a current generator.

To facilitate writing, we replace w - wa, (respectively,Co) + CWa) by w; this may be viewed as shifting the coordi-nate system so that the origin of the coordinate system anda pole coincide. To proceed, we allow w to have a smallimaginary part and then take the limit when the imaginarypart goes to zero

lim . = lim 2 2 + 2 2O-* co jo a-- CO + wX + aj

(lOb)

and use either the upper or the lower sign on both sides ofthe last equation. The real (respectively, imaginary) partof (lOb) is plotted in Fig. 5(a), [respectively, 5(b)] for agiven small value of ca. We see in Fig. 5(a) that wl(w2 +a2) is zero at X = 0, otherwise, it behaves in the limit as1/w; Fig. 5(b) pictures the imaginary part limit as propor-tional to a delta function. The area of the latter [i.e.,l -W adwl(w2 + a2)] is -X for any positive a value. Thus,we write

I for X * Olima-O w F ja ±+jwr6(w) for co = 0.

(1 la)

In the present case, we use the upper sign in (Ila) to en-sure convergence (exponential damping factor) in the de-fining equation of the Fourier integral for F(jw). Thus, atthe poles, the expression in the square bracket of (lOa)becomes imaginary; therefore, (lOa) becomes real at thepoles

(9b)

Here, IIg(w) 2, the square of the transform of ig(t) = Q6(t),is simply Q . The generator sees a capacitive admittancejwC and an inductive admittance l/jwL; these are in par-allel (see Fig. 6), hence, they add up. Thus, the impedanceseen is

Z=R + jX= 1jwC +jL

I2CLw-wa~+ wj(10a)j2C _co coa co + (tZa- awhere wa is the free resonant frequency 1iiiiC. At firstglance, it seems that the impedance has only an imaginarypart. In (lOa), we have poles at w = +wa and, hence,before evaluating the integral in (9b) we need to know theimpedance contribution from these poles.

R = ± [b(w - Wa) + 6(w + wa)]2C (lOc)

and so the energy supplied to the circuit is

7rQ2CO dw =2

w 7r [6(w a- Wa) +6(w + cWJa] 27 =c2C J- 6w w) ~ wn2ir 2C'

(9c)This type of calculation is, in general, applicable to alllinear fields. Dirac used such a calculation in quantummechanics in 1947. We shall show that this method is suit-able for calculation of the energy supplied by a radiatingsource (antenna) and stored (radiated) in the direction pat-tern of the antenna.

IV. POWER RADIATED BY A DIPOLE ANTENNAThe previous calculation for the lumped network can be

generalized to antennas. Let us excite free space by a cur-

152

00 dwW = RlIgl2-

00 27r'

PALOCZ AND MARCUVITZ: TEACHING ELECTROMAGNETICS

a

2 2w + a

w

(b)

-iH JELI JL

K(-jHT') p

-jHTI~ KETI tET' I 'IO 4JTI

KIl K(-jHT) P

-jHT,I KET EI I 0 i

K:lFig. 5. (a) Real part. (b) Imaginary part of 11(o - jct) for a given small Fig. 6. Electromagnetic network of free with applied current gener-

value of a. ator.

rent generator (antenna); denoting the components of thecurrent density by JL, JT', JT", we observe that these haveto be added to the displacement current densities j1wE(EL,ET', ET") in (7d)-(7f), respectively,

0 = jwoEEL + JL

-k(-jHT7,) = jOEOET' + JT'

k(-jHT') = jPEOET" + JT".

(7g)

(7h)

(7i)

Equations (7a)-(7c) are of course unaltered. We can in-clude the current generator in the equivalent network offree space, see Fig. 6. This network is valid for all non-

zero values of w. If X equals zero, we have a static (timeindependent) situation; in this latter instance electrostaticsand magnetostatics are not coupled.The network picture enables us to treat problems -in free

space. Due to lack of space, we illustrate the networkmethod by solving only one problem: the calculation ofthe total power radiated by a small dipole antenna. Theantenna is of length 1 and oriented along the z, direction.We let the origin of our coordinate system coincide withthe antenna; the time dependence is harmonic (e(t) so thecurrent density is given by

J(r, t) = d1 J(r)e"wt = Jd Ilz0 6(r)eiwt. (12)

Here, I is the root-mean-square current. Note that the di-mension of 6(r) is m-3 since the volume integral of 6(r) isdimensionless, hence, the dimension of J(r) is A/m2.We define the three-dimensional Fourier-Laplace trans-

form pair by

J(k) = j|J(r) ek rd 3r (13a)(all r)

d3kJ(r) = i J(k)e-jk r 3. (13b)

(27r)~(all)

k

Since liz0 is a constant factor and the transform of 6(r) isunity, the transform of J(r) = Ilz0 6(r) is J(k) = Hlz,. Wedecompose J(k) into two components, one is along ko andthe other along To; the latter is in the plane spanned by koand z0 and is perpendicular to ko (see Fig. 7). Thus, J(k)has no component in the Tot direction; JT" equals zero inFig. 6, and the network in the bottom of this figure is notexcited. Hence, it plays no part in this radiation problem.

K0

Fig. 7. Connection between JT, J(k) and the angle 0.

The network at the top of Fig. 6 is excited, but the gen-

erator JL always sees an imaginary impedance 1Ij.c-, forall given nonzero values of omega, hence, it too does notcontribute to radiation for X i 0.The only relevant network for the radiation problem is

located in the middle of Fig. 6. The generator -JT' sees

an admittance jwE, in parallel with an inductance ,u, whoseadmittance seen through the 1:k transformer is k2Ilj.,These admittances are in parallel, hence, they add; thus,the impedance as seen at PP' in Fig. 6 is

ZT' = RT' + JXT=2Icoco +

k

J(SZO

=69yo 2k k + K | (14a)

In the last expression, K0 is an abbreviation for

From our previous experience with the lumped LC cir-cuit, we are not puzzled by the fact that the impedance in(14a) seems to be purely imaginary. We observe, of course,the pole at k = KO. As a matter of fact, KO is complex witha positive real part and a small negative imaginary partdue to dissipation (always present physically). As thewavenumber k varies from zero to infinity, we have a sin-gle pole at k = KO = KO - jx, where

lima,O k + ja- KO

for k- KO * 0

-jr6(k - KO) fork - KO = 0

(llb)

w

2 2w +0a

w

(a)

153

J Le-

zo

IEEE TRANSACTIONS ON EDUCATION, VOL. E-28, NO. 3, AUGUST 1985

and so()1/2

6k

RT' = \ _2-Ko) = 62)6(k.-Ko) (14b)

P', the total time average real power radiated, can be eas-ily calculated by extending Parseval's theorem (9b) tothree dimensions

60Xr2(11)2(2Xr)3r1, = i ' g0,

Rr lJT | d3k (S d)* ( sin' O )(X k k - Ko) dk)

= 20(IK.)-2.

I* (15), the volume dement in k space is given by

d3k = k2 sin 0d# dO dk.

sociate Professor and was promoted to the rank of Professor in 1967. In1973, when the Engineering School of New York University merged withthat of Polytechnic Institute of Brooklyn to form the Polytechnic Instituteof New York, he became a Professor of Electrical Engineering and Electro-physics in the newly formed Institute where he is currently working. He hasgained industrial experience in Hungary's largest electronics firm Tungs-ram Co., Budapest, from 1945 to 1950 and was retained as a Consultantfrom 1950 to 1956. He was a member of the Research Staff of IBM from1957 to 1965, first at the Watson Research Laboratory at Columbia Uni-versity, New York, NY, and later at IBM's Research Center in YorktownHeights, NY. He spent part of a sabbatical year at the Hansen Laboratoryof Stanford University, Stanford, CA, and part at INPE, Sao Jost, Brazil.

Dr. Pal6cz was the recipient of the Hungarian National Award for Ex-cellent Teaching and is a member of Tau Beta Pi, Eta Kappa Nu, Sigma Xi.

(16)Note that we caklulated the radiated power without find-ing the spatial behavior of the electric and magnetic fields.If, on the other hand, the explicit determination of the spa-tial behavior of the fields is of interest, it can be calculatedby first finding the fields as seen by the generator at PP'(see Fig. 6) as a function of k; then transforming theminto the r dornain. This three-dimensional Fourier trans-fom tion is a generalization of what the students havelearned in one dimension. One observes that this methodis applicable to any antenna or antenna array in free space;RTJ in (14b) is the same for any current distribution. Afterfinding IJT'I, (15) gives the power radiated. In class, weusually calculate a number of relevant problems (for ex-ample, half-wave dipole, arrays, etc.). No new concept isneeded for these, just straightforward calculations.

Istv6n PaK6cz (SM'59) received the Engineeringl degree from the Technical University Budapest,

HRngary, in 1945 and became Docent at the sameUniversity in 1954. He received the Ph.D. degreein electrophysics from the Polytechnic Institute ofBrooklyn, Brooklyn, NY, in 1962.

He joined the Faculty of the Technical Univer-sity Budapest on a full-time basis in 1950 and wasan Associate Professor from 1954 to 1956. In 1965he joined the Department of Electrical Engineer-ing of New York University, New York, as an As-

Nathan Marcuvitz (S'36-A'37-M'55-SM'57-F'58-LF'79) received the B.E.E. (1935), M.E.E.(1941), and D.E.E. (1947) degrees from the Po-lytechnic Institute of Brooklyn, Brooklyn, NY.

From 1935 to 1940 he was a Development En-gineer with Radio Corporation of America. He wasa staff member of the Radiation Laboratory at theMassachusetts Institute of Technology, Cam-bridge, from 1942 to 1946. On the PolytechnicFaculty from 1946 to 1966, he served as an Assis-tant Professor of Electrical Engineering from 1946

to 1949; Associate Professor, 1949-1951; Professor, 1951-1965; Directorof the Microwave Research Institute, 1957-1961; Professor of Electrophys-ics, 1961-1965; Vice President of Research and Acting Dean of the Grad-uate Center, 1961-1963. On leave from the Polytechnic Institute of Brook-lyn during 1963-1964, he served as an Assistant Director of the DefenseResearch and Engineering (Research), Department of Defense, Washing-ton, DC. From 1964 to 1965, he was the Dean of Research and the Deanof the Graduate Center at Polytechnic Institute of Brooklyn prior to hisappointment as the Polytechnic's first Institute Professor from 1965 to 1966.In 1966 he accepted an appointment as Professor of Applied Physics at theSchool of Engineering and Science, New York University, New York, andserved in that capacity until 1973 when he returned to the newly mergedPolytechnic Institute of New York as a Professor of Applied Physics, andcurrently as an Institute Professor.

Dr. Marcuvitz is a member of the National Academy of Engineering, theAmerican Physical Society, Eta Kappa Nu, Sigma Xi, and Tau Beta Pi.

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