On the Energy or Power Efficiency of a Linear System

A linear time-invariant (LTI) system is usually characterized by its gains, which are ratios of various output and input signal magnitudes. When the main function of the system is energy conversion, one is more interested in the power or energy gain or loss between input and output terminals. Such a "gain" represents the efficiency of the system as an energy conversion device. In this paper, we show that the power or energy "gain" of an LTI system can be determined as a nonlinear function of three signal gains of the system. This results in an explicit formula for the power (energy) gain in terms of the system parameters. This can then be optimized over the design parameters to maximize the gain and thus the system energy efficiency. A detailed DC circuit example is included for illustration. The energy efficiency of an LTI system operating in a steady state in response to a sinusoidal signal is also determined.