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Derivation of the Most Energy-Efficient Source Functions by Using Calculus of Variations

This paper presents the derivation of the most energy-efficient source functions by using calculus of variations. The fundamental theorem of calculus of variations for finding the functions that are local minima or local maxima of the integral is first explained. The theorem is then applied to derive analytically the source functions that minimize energy dissipation in circuits during transient. In particular, the most energy-efficient input current function and voltage source function for charging series R-C circuit are derived and proved to be a constant function and a raised ramp function respectively. The theorem of calculus of variations is extended for other common first order linear circuits as well. Their most energy-efficient source functions and lowest energy dissipation expressions are summarized in compact parameters involving speed and energy trade off limits. Numerical results are illustrated to depict the upper limit of speed-energy trade off and to justify the consistency with the analytical results. Application of the most energy-efficient source function for charging and discharging the capacitor of supercapacitor and adiabatic circuits is discussed along with comparison with other functions.