On the Approach to Steady State of a Linear Variable Network Containing One Reactance

It is often desirable to determine the manner in which a circuit approaches steady-state operation after a periodic input is applied. More precisely, an answer is sought to the question: "After how many cycles of input may the voltages and currents within the circuit be considered periodic, to within a certain error?" If the circuit is linear, an answer can always be obtained by the ordinary methods of mathematical attack. All too often, however, such procedures are little more than tours de force of tedious calculations. This is particularly true when some of the circuit component values are variable with time, as is sometimes the case when relays, tubes, switches, and the like are present. In this paper a method will be developed for answering the question posed above for a restricted, but still quite general, class of linear circuits. Practically speaking, the main restriction is that there be only one reactive component in the circuit. The analysis proceeds by development of a linear, first-order difference equation, whose solution characterizes the circuit's transition from quiescence to transient to steady operation.