Appendix B: Continuity Theorems in Network Analysis – Pulse and Digital Circuits

Appendix B: Continuity Theorems in Network Analysis

Constant Flux Linkage Theorem: The constant flux linkage theorem states that flux linkages in an inductor cannot change abruptly.

Proof: Let flux linkages change abruptly. Consequently, in keeping with Faraday’s laws, infinite voltage has to be induced. As infinity is not practical, the assumption of abrupt change is wrong. (This logic is called the logic of “reductio-ad-absurdum or the logic of contradiction.”) If L is constant, this theorem is called the constant current theorem.

Constant Charge or Constant Voltage Theorem: These theorems state that the charge or voltage in a capacitor cannot change abruptly. Both of these theorems can be proved using the same logic described for the previous theorem.

Limitations: If infinity could be accommodated in terms of an impulse function, all the three theorems can be violated. Consider the three examples described below:

Example 1: What happens when a constant dc voltage, V is connected to an uncharged capacitor at t = 0? A current of VCδ(t) is established. VC is the impulse strength. In other words, the capacitor is open after t = 0+. At t = 0, the impulse current exists.

Example 2: What happens when a capacitor is connected to a constant current source? Here we are forcing a current through the capacitance. Hence, there is no question of whether the capacitor can carry direct current. Thus, when connected to a constant current source, a linearly increasing voltage is established across the capacitor, which may eventually puncture the dielectric medium. A capacitance is an open circuit, designed only for constant voltage across it but not for constant current. (If a capacitor is forced into a constant current electronic circuit, it is liable to be punctured.)

Example 3: A dual example can be framed for an inductor. It is a short circuit for constant current only but not for constant voltage across it.

An Impulse Function

An impulse function (delta function), shown in Fig. 1, is defined as:

Also note that here, and L[δ(t)] = 1. In a practical sense, an infinitely large quantity spanned over an infinitesimally small period constitutes an impulse.*

FIGURE B.1 A delta function or impulse function.