On the most energy-efficient input functions and the lowest input energy of passive linear circuits

Energy-efficient has been an interested research for decades. For fundamentals circuits, resistors in a circuit always dissipate energy when they have current flow and it decreases energy efficiency. When the resistive circuits are in transient state, energy dissipation is usually inevitable. However, it can be minimized. Either energy dissipation or input energy are dependent with voltage and current functions in the circuits. Therefore, there are the most energy-efficient functions, the functions that minimize the input energy (or the energy dissipation). The functions can be obtained by applying calculus of variations, a powerful mathematics theory for functional optimization. Once the most energy-efficient functions are obtained, the lowest input energy can be calculated. For a series RC circuit, one of the simplest first order circuit, the most energy- efficient input current function and input voltage function are derived and proved to be a constant function and a raised ramp function respectively. The optimization by using calculus of variations is expanded to any first order passive circuits. It turns out that the most energy-efficient function for general first order circuits is hyperbolic family. The lowest input energy is calculated and stated as corollaries. The corollaries are essentials to provide insight into transient duration analysis. The transient duration analysis is performed for three cases of increasing, decreasing and unchanging total stored energy. All these cases are analyzed comprehensively. In each case, it is further divided into two sub cases of (un)changing without and with zero crossing. The analysis reveals two important circuit behaviors which are speed and energy trade-off and energy flow. It is shown that the speed and energy trade-off may be violated under certain conditions. The energy flow of the circuit is discussed thoroughly and a zero-energy transient duration that is defined as a transient duration with total input energy being zero is derived. All the analysis can be summarized into a single digram called the transient duration diagram. The digram reveals important relationships between the circuit parameters in the transient duration analysis. Not only first order circuit that are optimized, but also second order circuit. Series LC circuit and parallel LC circuit are chosen to be optimized because they are typical second order circuits. The most energy-efficient functions of the circuits can be catego- rized into three modes which are (i) hyperbolic mode, (ii) hyperbolic and linear mode and (iii) hyperbolic and trigonometric mode. Each mode are discussed and theirs tran- sient responses are illustrated. The lowest input energy are evaluated and the transient duration analysis are done rigorously. Finally, generalization of the most energy-efficient functions and the lowest input energy for any order circuits is drawn. A dummy function is introduced which is essential for generalization. The most energy-efficient input functions in terms of the dummy function are proposed as well as the lowest input energy. General formulas or general procedure for determining the most energy-efficient functions and the lowest input energy for any orders circuits is explained. Using of the general formulas is discussed. Applying the general formulas of RC chain circuit is explained thoroughly.Doctor of Philosoph