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By
Alper Ercan

^{1,a)}
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Affiliations:

a) Electronic mail: ae39@cornell.edu; Present address: Softkinetic, 1050 Brussels, Belgium.

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### Abstract

Analysis of electrical networks using variational techniques, while repeatedly mentioned in the literature, is not widely known or utilized. In this short communication, we emphasize the connection between Kirchhoff's network equations, energy conservation, and variational analysis techniques using a brief example.

© 2016 American Association of Physics Teachers

Received 20 May 2015
Accepted 07 December 2015

Acknowledgments:

I would like to thank the reviewers for their insightful feedback and also for bringing various references into my attention.

Article outline:

I. INTRODUCTION II. LUMPED NETWORKS AND ENERGY CONSERVATION III. FROM ENERGY BALANCE TO VARIATIONAL APPROACHES IV. CONCLUSIONS

I. INTRODUCTION II. LUMPED NETWORKS AND ENERGY CONSERVATION III. FROM ENERGY BALANCE TO VARIATIONAL APPROACHES IV. CONCLUSIONS

/content/aapt/journal/ajp/84/3/10.1119/1.4939576

1.

1. J. C. Maxwell, A Treatise on Electricity and Magnetism ( Clarendon Press, Oxford, 1892), pp. 407–408.

2.

2. J. Jeans, The Mathematical Theory of Electricity and Magnetism ( Cambridge U.P., Cambridge, 1925), p. 323, article 358.

3.

3. R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lecture on Physics ( Addison-Wesley, Reading, MA, 1964), Vol. 2, Chap. 19.

4.

4. D. A. Van Baak, “ Variational alternatives to Kirchhoff's loop theorem in dc circuits,” Am. J. Phys. 67, 36–44 (1999). A thermodynamic view of this approach is reported by Jose-Philippe Perez, “Thermodynamical interpretation of the variational Maxwell theorem in dc circuits,” Am. J. Phys. 68, 860–863 (2000).

5.

5. W. Millar, “ Some general theorems for non-linear systems possessing resistance,” Philos. Mag. Ser. 7 42, 1150–1160 (1951).

6.

6. F. L. Ryder, “ Network analysis by least powers theorems,” J. Franklin Inst. 254, 47–60 (1952).

7.

7. P. Penfield, Jr., R. Spence, and S. Duinker, Tellegen's Theorem and Electrical Networks ( MIT Press, Cambridge, MA, 1970), pp. 37–44.

8.

8. G. F. Oster and C. A. Doser, “ Tellegen's theorem and thermodynamic Inequalities,” J. Theor. Biol. 32, 219–241 (1971).

9.

9. C. Cherry, “ Some general theorems for non-linear systems possessing reactance,” Philos. Mag. Ser. 7 42, 1161–1177 (1951).

10.

10. S. Ramo, J. R. Whinnery, and T. V. Duzer, Fields and Waves in Communication Electronics ( John-Wiley and Sons, NJ, 1994), Chap. 4.

11.

11. R. M. Fano, L. J. Chu, and R. B. Adler, Electromagnetic Fields, Energy, and Forces ( MIT Press, Cambridge, MA, 1968), Chaps. 6 and 7.

12.

12. G. Kirchhoff, “ On the solution of the equations obtained from the investigation of the linear distribution of galvanic currents,” IRE transactions on circuit theory, March, 1958, pp. 4–7 (English translation by J. B. O'toole).

13.

13. R. Muller, “ A semiquantitative treatment of surface charges in DC circuits,” Am. J. Phys. 80(9), 782–788 (2012).

14.

14.See, for example, Ref. 10, Sec. 4.10, and Chaps. 5 and 11.

15.

15. L. O. Chua, C. A. Desoer, and E. S. Kuh, Linear and Non-linear Circuits ( McGraw-Hill, New York, 1987), Chap. 1.

16.

16. B. D. H. Tellegen, “ A general network theorem, with applications,” Philips Res. Rep. 7, 259–269 (1952); available at https://newcatalog.library.cornell.edu/catalog/371415.

17.

17. P. Penfield, Jr., R. Spence, and S. Duinker, “ A generalized form of Tellegen's theorem,” IEEE Trans. Circuit Theory 17(3), 302–305 (1970).

18.

18. For formal proofs as well as topological interpretations, see Ref. 7, pp. 21–22, or Ref. 15, pp. 29–34. In the electrical engineering literature, Eq. (1) is often presented as a special case of “Tellegen's Theorem,” which can be used to study various useful and interesting network relations (see Refs. 7 and 17).

19.

19. The functional S is first introduced by W. Millar (Ref. 5). He defined each term in the integral as the “co-content” of the component and proved the following: “if in an active non-reactive network the sum of co-contents of all elements is expressed in terms of the defining number of generalized current coordinates in the network, subject to only to the restriction of Kirchhoff's current law, then S is stationary for the actual distribution of currents.” He also showed that for networks with time-invariant components, the “co-content” will be an invariant of the motion. A more general discussion of function S and non-linear resistors can be found at Ref. 7, pp. 37–41.

20.

20.This can be observed by plotting the voltage vs current for the components and comparing the area difference δS for the variations around the operating point.

21.

21.See Ref. 5 or Ref. 7, pp. 37–44.

22.

22. A. A. P. Gibson and B. M. Dillon, “ The variational solution of electric and magnetic circuits,” IEEE Eng. Sci. Educ. J. 4(1), 5–10 (1995).

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On Kirchhoff's equations and variational approaches to electrical network analysis

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