Passivity (engineering) - Thermodynamic Passivity

Thermodynamic Passivity

In control systems and circuit network theory, a passive component or circuit is one that consumes energy, but does not produce energy. Under this methodology, voltage and current sources are considered active, while resistors, transistors, tunnel diodes, glow tubes, capacitors, metamaterials and other dissipative and energy-neutral components are considered passive. Circuit designers will sometimes refer to this class of components as dissipative, or thermodynamically passive.

While many books give definitions for passivity, many of these contain subtle errors in how initial conditions are treated (and, occasionally, the definitions do not generalize to all types of nonlinear time-varying systems with memory). Below is a correct, formal definition, taken from Wyatt et al. (which also explains the problems with many other definitions). Given an n-port R with a state representation S, and initial state x, define available energy E_A as:

where the notation sup_x→T≥0 indicates that the supremum is taken over all T ≥ 0 and all admissible pairs {v(·), i(·)} with the fixed initial state x (e.g., all voltage–current trajectories for a given initial condition of the system). A system is considered passive if E_A is finite for all initial states x. Otherwise, the system is considered active. Roughly speaking, the inner product is the instantaneous power (e.g., the product of voltage and current), and E_A is the upper bound on the integral of the instantaneous power (i.e., energy). This upper bound (taken over all T ≥ 0) is the available energy in the system for the particular initial condition x. If, for all possible initial states of the system, the energy available is finite, then the system is called passive.