36 IEEE TRANSACTIONS ON EDUCATION, VOL. E-29, NO. 1, FEBRUARY 1986
presence of students in the lab nearly any time of the day or nightthat the instructor happened by. z,
The vehicle itself, that of a traffic intersection with which thestudents could all relate intimately, gave them an easy tie to realityfor their introduction to using a multitasking operating system. In , ZTz v Connectionaddition, the project was used as a display within the School of (a) Z.,', networkEngineering for a University-wide open house and Student Day. Atime constraint was placed on the teams by the open house date. BHaving this externally imposed deadline enhanced the sense of Z8Breality to the exercise. As an aside, for those departments associ-ated with schools which have traffic departments, this specific ex-ercise could be expanded to become an "educationally useful" de-sign project, as discussed by Niederjohn and Schmitz [2]. Y,
The expertise among this group varied considerably. The non-EE students ranged from professional programmers to one who had - Cdone virtually no programming prior to this course sequence. Since (b) { Connectionthe students were grouped in twos and threes, there was ample op-portunity for slower programmers to learn from more experiencedones. Thus, the group, or team, approach was an integral part of 4Y8 Bthe learning process.
Riley [3] has identified seven phases encountered by a teamproject. These are: I) enthusiasm, II) data gathering, III) group Fig. Definition(a)ofZ(b) ofY;theconnectionnetworkcontainsonlydivergence, IV) group convergence, V) group panic, VI) group ef- wires.fort, and VII) group accomplishment. Because of the time con-straints and the limited scope of this project, it appeared that most 1. INTRODUCTIONof the groups involved maintained their enthusiasm (I) through thedata gathering (II), group convergence (IV), and group effort (VI) At least since 1949 (see [1] and, e.g., [2]-[6]), the "admittanceto arrive at the group accomplishment (VII) without spending much ratio summation theorem" has been known for networks withouttime/effort in group divergence (III) or group panic (V). Had the mutual couplings. Together with the dual theorem, it can be statedproject been of such a scope as to have spanned a semester, the as the "immittance ratio summation theorem," as follows.students might have experienced the full range of Riley's phases For a network consisting of B branches with branch impedancesand the instructor might have had a more serious motivation and Zr, the equationmanagement problem. Keeping the project well defined, as simple Bas possible, and reasonably short helps a great deal with such prob- E Z /Z' = L (la)lems. v=
This project has vigorously reinforced my belief that engineeringis best taught when the concepts are firmly attached to practicalities wto which students can relate. Although there is nothing unique about driving point impedances seen by breaking into the branch withtraffic light simulation, it is one excellent vehicle for the demon- impedanceZcn See Fig. (a) ("pliers method," see [71); or for astration of handling several independent tasks with a multitasking network consisting of B branches with branch admittances Yn =operating system. 1IZ^, the equation
B
REFERENCES Z YV/Yl = N (lb)p= I
[1] J. L. Kozikowski, "Team design in the classroom-A case study," IEEETrans. Educ., vol. E-20, pp. 106-108, May 1977. is valid, where N is the number of nodes minus the number of sep-
[2] R. J. Niederjohn and R. J. Schmitz, "The case for project-oriented arate parts of the network, mostly one, and Yp' are the driving pointcourses with 'educationally useful' student design projects," IEEE admittances measured across the two nodes of Yp, see Fig. 1(b)Trans. Educ., vol. E-25, pp. 65-70, May 1982. ("soldering iron method," see [7]).
[3] M. W. Riley, "Phases encountered by project teams," IEEE Trans. In recent publications, this theorem has been rediscovered orEduc., vol. E-23, pp. 212-213, Nov. 1980. generalized [8]-[1l]. The purpose of the present paper is a twofold
one: we will give a new and very short proof of the above theoremand discuss energy or power interpretations of it.
II. PROOF
We start with the supposition that each branch impedance Zv hasNotes on the Immittance Ratio Summation Theorem a series voltage source E,. With E we denote the voltage vector
with elements El, E2, * * *, EB, and with I the vector with the branchEDUARD SCHWARTZ currents II, I2, ,IB The relation between these vectors is given
by
Abstract-A new and very short proof and energy or power inter- I Y E (2)pretations of the "immittance ratio summation theorem" are dis- where Y is the (short circuit) admittance matrix. Y can be calcu-cussed. lated on the 1oop analysis base with (see, e.g., [6] or [12])
Y = n(nTZdn)-lnT. (3)Manuscript received July 27, 1984.The author was with the University of Braunschweig, Braunschweig, Zd iS the B X B diagonal matrix with Z1, Z2, * , ZB as diagonal
West Germany. He is now deceased. elements; n is the B X L matrix relating the L independent loop
IEEE TRANSACTIONS ON EDUCATION, VOL. E-29, NO. 1, FEBRUARY 1986 37
__________V_________ B B
RI E VE = 4TPaZ* RIRV. (8)
On the other hand, we can use the "orthonormalized mode equiv-(a) l~ r R, Connection alent circuit" from [12] for Fig. 2(a) which is illustrated in Fig.U(a), 'tR' v network 2(b). It has only L = B - N equal resistances R, where L and N
have the same meaning as in Section I. The ideal transformer net-B work between the B voltage sources and the L equal resistances is
ll lossless. Thus, the total energy E delivered by the B voltage sourcesfor the B experiments is equal to the total energy absorbed by theL equal resistances
r E = (nil + * ni, + ***+ n IL) UIl12T12R +**LilIli 1 1 n1, n,~ + (np1+ + n + +nL) IU,12T12R +
,. l lWith (5), this lengthy expression can be condensed to
| j 1i¢;na,i1(({nBY| 1;(gnB E = 4TPaR' trace {nTRdn}, (10)I where Rd is (analogously to Zd in section II) the B X B diagonalI ... .. L matrix with RI, R2, , RB as diagonal elements, and n is the B
EJ C- xo X L matrix with the ideal transformer turns ratios arranged just asR R R in the circuit of Fig. 2(b). But from [12] it is known that, due to
Fig. 2. Illustration of B experiments (v = 1 to B): (a) original network, (b) orthonormalization,orthonormalized equivalent circuit. nTRdn = R * (11)
where IL is the L x L unit matrix. From (10) and (11), we cancurrents and the B branch currents; T denotes the transpose of a conclude thatmatrix. From (3) we find E = 4TPaL. (12)
trace {Y * Zd} = trace {n(nTZdn)lnT * Zd) Finally, comparing (12) and (8), we have
or with trace {A * B} = trace {B - A} (see, e.g., [13]) BEI4TPa = R,IR' = L. (13)
Obviously, the right-hand-side matrix is the L x L unit matrix and That means we have found a proof of (la) (with resistances insteadthe trace of it is L. The left-hand-side matrix is the product of the of impedances) with the aid of energy considerations or an energy(short circuit) admittance matrix and the diagonal matrix with ele- interpretation of (la). The total energy E for the B experimentsments Z1, Z2, ..* , ZB, the trace of which is Y I1Z1 + Y22Z2 + . . . under consideration, normalized with 4TPa (this is the energy of+ YBBZB; but the Y, are the 1/Z' of Fig. l(a) and (la). Thus, the any of the B voltage sources delivered to R, alone for a time interval
f () is c.S w, t of duration T), is equal to the sum of resistance ratios R,IR' andproof of (la) aS completed. Starting witotis equatnon, tle most oPsimple proof of (lb) can be made with some simple manipulations equal to the number of independent loops.as known, e.g., from [4]. The treatment of (lb) is completely dual and needs no discussion
here. But it may be worth mentioning that the total energy in theIII. ENERGY OR POWER INTERPRETATIONS dual case is, in general, not the same as above, and that the reason
for this lies in the fact that the equivalence transformation of voltageNow we consider a network consisting of B positive resistances sources with internal resistances into current sources with internal
R, instead of the impedances Z, above. With this network, we per- conductances are only equivalent with respect to the external pow-form B experiments as shown in Fig. 2(a): For v = 1 to B, we ers but not with regard to the internal powers.introduce successively B ac sources with complex amplitudes U, in Finally, it may be mentioned that analogous proofs can be givenseries with the branch resistances R, assuming that the B available with B simultaneously existing ac sources with different frequen-powers cies, or with B simultaneously existing uncorrelated noise sources;
P= U12/8R, = . IUV218RV = ... = IBUI2/8RB (5) in these cases, the total power may be considered instead of thetotal energy above.
are equal. The durations T of these B successive experiments arealso equal and we suppose that T is very great compared to theperiods of the ac voltages. Consequently, the energy delivered by The author wishes to thank Dipl.-Phys. W. Mathis for valuablethe voltage source Up for any of the B experiments is given by discussions.
FV = TI U,j2I2Rp (6) REFERENCES
where the Rp are defined analogously to the Zt above [see Fig. 2(a)] . [1] R. M. Foster, "sThe average impedance of an electrical network," inFrom (6) and (5) we find Reissner Anniversary Volume, Contributions to Applied Mechan-ics. Ann Arbor, MI: Edwards, 1949, pp. 333-340.
F, = 4TPaR,IR;. (7) [2] L. Weinberg, "Kirchhoff's third and fourth laws," IRE Trans. CircuitTheory, vol. CT-5, pp. 8-30, Mar. 1958.
The total energy delivered by the B voltage sources for the B ex- [3] R. M. Foster, "An extension of a network theorem," IRE Trans. Cir-periments is cuit Theory, vol. CT-8, pp. 75-76, Mar. 1961.
38 IEEE TRANSACTIONS ON EDUCATION, VOL. E-29, NO. 1, FEBRUARY 1986
[4] L. Weinberg, Network Analysis and Synthesis. New York: McGraw- and transfer network impedances," Proc. IEEE, vol. 72, pp. 395-396,Hill, 1962, pp. 170-172. Mar. 1984.
[5] P. Penfield, R. Spence, and S. Duinker, Tellegen's Theorem and Elec- [10] W.-K. Chen, "A theorem on the summation of return differences andtrical Networks. Cambridge, MA: M.I.T. Press, 1970, pp. 74-76. some consequences," Proc. IEEE, vol. 72, pp. 386-397, Mar. 1984.
[6] W. H. Kim and H. E. Meadows, Modern Network Analysis. New [11] H. J. Butterweck, "Alternative formulation of Roytman's networkYork: Wiley, 1971, pp. 117, 321. theorem and a new proof," to be published.
[7] E. A. Guillemin, Introductory Network Theory, 6th ed. New York: [12] E. Schwartz, "A new interpretation of the Kirchhoff modes," ArchivWiley, 1960, p. 92. far Elektronik und Ubertragungstechnik, vol. 28, pp. 420-423, Oct.
[8] K. Thulasiraman, R. Jayakumar, and M. N. S. Swamy, "Graph-the- 1974.oretic proof of a network theorem and some consequences," Proc. [13] S. Barnett and C. Storey, Matrix Methods in Stability Theory.IEEE, vol. 71, pp. 771-772, June 1983. London, England: Nelson, 1970, p. 8.
[9] I. N. Hajj, "A note on a theorem on the summation of driving-point
