Solving the two capacitor paradox through a new asymptotic approach

摘要

Closing an ideal switch between two passive linear lumped time-invariant capacitors with different initial voltages gives rise to an apparent paradox. A part of the total energy stored in the two capacitors suddenly vanishes, but, seemingly, the switch cannot account for this loss. Hence, the energy conservation law appears to be violated. The author reconsiders this circuit, and gives a complete explanation of it based on a new embedding of the switch model. The embedding consists of a family of smooth switches with a finite transition time, and therefore is much more general and more realistic than the standard model. Furthermore, the asymptotic analysis carried out not only shows that the switch accounts for the energy loss (and thus the energy conservation law is not violated), but also explains the mechanisms underlying the behaviour of the circuit.