A theory of nonlinear networks. I
Authors:
R. K. Brayton and J. K. Moser
Journal:
Quart. Appl. Math. 22 (1964), 1-33
DOI:
https://doi.org/10.1090/qam/169746
MathSciNet review:
169746
Full-text PDF Free Access
Abstract |
References |
Additional Information
Abstract: This report describes a new approach to nonlinear RLC-networks which is based on the fact that the system of differential equations for such networks has the special form \[ L\left ( i \right )\frac {{di}}{{dt}} = \frac {{\partial P\left ( {i,v} \right )}}{{\partial i}},C\left ( v \right )\frac {{dv}}{{dt}} = - \frac {{\partial P\left ( {i,v} \right )}}{{\partial v}}.\] The function, $P\left ( {i,v} \right )$, called the mixed potential function, can be used to construct Liapounov-type functions to prove stability under certain conditions. Several theorems on the stability of circuits are derived and examples are given to illustrate the results. A procedure is given to construct the mixed potential function directly from the circuit. The concepts of a complete set of mixed variables and a complete circuit are defined.
W. Bode, Network analysis and feedback amplifier design, D. Van Nostrand Co., Inc., Princeton, N. J., 1945
- Ernst A. Guillemin, Synthesis of passive networks. Theory and methods appropriate to the realization and approximation problems, John wiley and Sons, Inc., New York; Chapman and Hall, Ltd., London, 1958. MR 0137480
E. J. Cartan, Leçons sur les invariants integraux, A. Hermann et Fils, Paris, 1922
- R. J. Duffin, Nonlinear networks. IIb, Bull. Amer. Math. Soc. 54 (1948), 119–127. MR 24497, DOI https://doi.org/10.1090/S0002-9904-1948-08964-9
E. Goto et al., Esaki diode high-speed logical circuits, IRE Trans. on Electronic Computers EC-9 (1960) 25
- Jürgen Moser, Bistable systems of differential equations, Symposium on the numerical treatment of ordinary differential equations, integral and integro-differential equations (Rome, 1960) Birkhäuser, Basel, 1960, pp. 320–329. MR 0123065
- J. K. Moser, Bistable systems of differential equations with applications to tunnel diode circuits, IBM J. Res. Develop. 5 (1961), 226–240. MR 128575, DOI https://doi.org/10.1147/rd.53.0226
- B. D. H. Tellegen, A general network theorem, with applications, Philips Research Rep. 7 (1952), 259–269. MR 51142
L. Esaki, New phenomenon in narrow Ge p-n junctions, Phys. Rev. 109 (1958) 603
- William Millar, Some general theorems for non-linear systems possessing resistance, Philos. Mag. (7) 42 (1951), 1150–1160. MR 44364
- Colin Cherry, Some general theorems for non-linear systems possessing reactance, Philos. Mag. (7) 42 (1951), 1161–1177. MR 44365
- Joseph LaSalle and Solomon Lefschetz, Stability by Liapunov’s direct method, with applications, Mathematics in Science and Engineering, Vol. 4, Academic Press, New York-London, 1961. MR 0132876
N. G. Chetayev, Stability of motion, Moscow, 1946
W. Bode, Network analysis and feedback amplifier design, D. Van Nostrand Co., Inc., Princeton, N. J., 1945
E. A. Guillemin, Introductory circuit theory, J. Wiley and Sons, Inc., N. Y., 1958
E. J. Cartan, Leçons sur les invariants integraux, A. Hermann et Fils, Paris, 1922
R. Duffin, Nonlinear networks III, Bull. Amer. Math. Soc. 54, (1948) 119
E. Goto et al., Esaki diode high-speed logical circuits, IRE Trans. on Electronic Computers EC-9 (1960) 25
J. Moser, Bistable systems of differential equations, Proc. of the Rome Symposium, Provisional International Computation Centre, Birkhäuser Verlag, 1960, pp. 320–329
J. Moser, Bistable systems with applications to tunnel diodes, IBM J. Res. Dev. 5 (1961) 226
B. D. H. Tellegen, A general network theorem with applications, Phillips Research Reports 7 (1952) 259
L. Esaki, New phenomenon in narrow Ge p-n junctions, Phys. Rev. 109 (1958) 603
W. Millar, Some general theorems for nonlinear systems possessing resistance, Phil. Mag. 42 (1951) 1150
C. Cherry, Some general theorems for nonlinear systems possessing reactance, Phil. Mag. 42 (1951) 1161
J. LaSalle and S. Lefschetz, Stability by Liapunov’s direct method with applications, Academic Press, New York, London, 1961, p. 66
N. G. Chetayev, Stability of motion, Moscow, 1946
Additional Information
Article copyright:
© Copyright 1964
American Mathematical Society