volume | PIER Journals

Abstract

We present a general theory of linear continuous circuits (microwave networks, waveguides, transmission lines, etc.) based on Lie theory. It is shown that the fundamental relationship between the low- and high-frequency circuits can be fully understood via the machinery of Lie groups. By identifying classes of distributed-parameter circuits with matrix (Lie) groups, we manage to derive the most general differential equation of the n-port network, in which its low-frequency (infinitesimal) circuit turns out to be the associated Lie algebra. This equation is based on identifying a circuit Hamiltonian derived by heavily exploiting the Lie-group-theoretic structure of continuous circuits. The solution of the equation yields the circuit propagator and is formally expressed in terms of ordered exponential operators similar to the quantum field theory's formula of perturbation theory (Dyson expansion). Moreover, the infinitesimal operators generating the per-unit-length lumped element local circuit approximation appear to correspond to operators (such as observables) in quantum theory. This analogy between quantum theory and circuit theory through a shared Hamiltonian and propagator structure is expected to be beneficial for the two separate disciplines both conceptually and computationally. Several applications are presented in the field of microwave network analysis where we introduce and study the Lie algebras of important generic classes of circuits, such as lossless, reciprocal, and nonreciprocal networks. Applications to the problems of generalized matching and representation theorems in terms of uniform transmission lines are also outlined using topological methods derived from our Lie-theoretic formulation and exact theorems on continuous matching are obtained to illustrate the potential practical use of the theory.

1. Hasan, M. Z. and C. L. Kane, "Colloquium: Topological insulators," Rev. Mod. Phys., Vol. 82, 3045-3067, Nov. 2010.
doi:10.1103/RevModPhys.82.3045

2. Ozawa, T., H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, "Topological photonics," Rev. Mod. Phys., Vol. 91, 015006, Mar. 2019.
doi:10.1103/RevModPhys.91.015006

3. Schwarz, A. S., Topology for Physicists, Springer-Verlag, 1994.
doi:10.1007/978-3-662-02998-5

4. Penrose, R., Techniques of Differential Topology in Relativity, Society for Industrial and Applied Mathematics, 1972.
doi:10.1137/1.9781611970609

5. Ranada, A. F., "Topological electromagnetism," Journal of Physics A: Mathematical and General, Vol. 25, No. 6, 1621-1641, Mar. 1992.
doi:10.1088/0305-4470/25/6/020

6. Mikki, S. and Y. Antar, "A topological approach for the analysis of the structure of electromagnetic flow in the antenna near-field zone," 2013 IEEE Antennas and Propagation Society International Symposium (APSURSI), 1772-1773, Jul. 2013.

7. Mikki, S. M. and Y. M. Antar, "Morphogenesis of electromagnetic radiation in the near-field zone," Asia Pacific Radio Science Conference (URSI), Taipei, Taiwan, Sep. 2–7, 2013.

8. Mikki, S. and Y. Antar, New Foundations for Applied Electromagnetics: The Spatial Structure of Fields, Artech House, 2016.

9. Lie, S., Theory of Transformation Groups I: General Properties of Continuous Transformation Groups. A Contemporary Approach and Translation, Springer, 2015.

10. Weyl, H., The Theory of Groups and Quantum Mechanics, Martino Publishing, 2014.

11. Penrose, R., The Road to Reality: A Complete Guide to the Laws of the Universe, Vintage Books, 2007.

12. Schwinger, J., et al., Classical Electrodynamics, Perseus Books, 1998.

13. Collin, R., Foundations for Microwave Engineering, IEEE Press, 2001.
doi:10.1109/9780470544662

14. Chew, W. C., Waves and Fields in Inhomogenous Media, Wiley-IEEE, 1999.
doi:10.1109/9780470547052

15. Felsen, L., Radiation and Scattering of Waves, IEEE Press, 1994.
doi:10.1109/9780470546307

16. Zeidler, E., Quantum Field Theory III: Gauge Theory, Springer, 2011.
doi:10.1007/978-3-642-22421-8

17. Thyssen, P. and A. Ceulemans, Shattered Symmetry: Group Theory from the Eightfold Way to the Periodic Table, Oxford University Press, 2017.

18. Chirikjian, G. and A. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis, CRC Press, 2001.

19. Chirikjian, G., Stochastic Models, Information Theory, and Lie Groups, Birkhauser, 2009.
doi:10.1007/978-0-8176-4803-9

20. Chevalley, C., Theory of Lie Groups, Dover Publications, Inc., 2018.

21. Godement, R., Introduction to the Theory of Lie Groups, Springer, 2017.
doi:10.1007/978-3-319-54375-8

22. Weyl, H., The Classical Groups: Their Invariants and Representations, Princeton University Press, 1946.

23. Sudarshan, E. C. G. and N. Mukunda, Classical Dynamics: A Modern Perspective, World Scientific, 2016.

24. Collin, R. E., Field Theory of Guided Waves, Wiley-IEEE Press, 1991.

25. Cohn, P. M., Lie Groups, University Press, 1957.

26. Stillwell, J., Naive Lie Theory, Springer, 2008.
doi:10.1007/978-0-387-78214-0

27. Gilmore, R., Lie Groups, Physics, and Geometry: An Introduction for Physicists, Engineers and Chemists, Cambridge University Press, 2008.
doi:10.1017/CBO9780511791390

28. Baker, A., Matrix Groups: An Introduction to Lie Group Theory, Springer, 2002.

29. Hall, B., Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer, 2015.
doi:10.1007/978-3-319-13467-3

30. Loewner, C., Theory of Continuous Groups, Dover Publications, 2008.

31. Zeidler, E., Quantum Field Theory I: Basics in Mathematics and Physics, Springer, 2009.

32. Mikki, S. and Y. Antar, A rigorous approach to mutual coupling in general antenna systems through perturbation theory, Vol. 14, 115-118, IEEE Antennas and Wireless Communication Letters, 2015.

33. Mikki, S. M. and Y. Antar, "Aspects of generalized electromagnetic energy exchange in antenna systems: A new approach to mutual coupling," EuCap 2015, 1-5, Apr. 2015.

34. Hassani, S., Mathematical Physics: A Modern Introduction to Its Foundations, Springer, 2013.

35. Cui, T., et al., Metamaterials: Beyond Crystals, Noncrystals, and Quasicrystals, CRC Press, 2016.
doi:10.1201/b21590

36. Mikki, S. M. and A. A. Kishk, "Electromagnetic wave propagation in nonlocal media: Negative group velocity and beyond," Progress In Electromagnetics Research B, Vol. 14, 149-174, 2009.
doi:10.2528/PIERB09031911

37. Mikki, S. M. and A. A. Kishk, "Nonlocal electromagnetic media: A paradigm for material engineering," Passive Microwave Components and Antennas, InTech, Apr. 2010.

38. Jackson, J., Classical Electrodynamics, Wiley, 1999.

39. Erdmann, K. and M. J. Wildon, Introduction to Lie Algebras, Springer, 2006.
doi:10.1007/1-84628-490-2

40. Munkres, J., Topology, Pearson, 2018.

41. Schwede, S., Global Homotopy Theory, Cambridge University Press, 2018.
doi:10.1017/9781108349161

42. Mosher, R. and M. C. Tangora, Cohomology Operations and Applications in Homotopy Theory, Dover Publications, 2008.

43. Kelley, J., General Topology, Dover Publications, Inc., 2017.

44. Godement, R., "Analysis I: Convergence, Elementary Functions," Springer, Berlin New York, 2004.

45. Godement, R., Analysis II: Differential and Integral Calculus, Fourier Series, Holomorphic Functions, Springer-Verlag, 2005.

46. Lee, J., Introduction to Smooth Manifolds, Springer, 2012.
doi:10.1007/978-1-4419-9982-5

47. Lang, S., Introduction to Differentiable Manifolds, Interscience, 1962.

48. Pontryagin, L. S., Topological Groups, Gordon and Breach Science Publishers, 1986.

49. Montgomery, D. and L. Zippin, Topological Transformation Groups, Dover Publications, Inc., 2018.

50. Husain, T., Introduction to Topological Groups, Dover Publications, 2018.

Prof. Said Mikki