基于预定拉格朗日量的电路综合
Circuit Synthesis based on Prescribed Lagrangian
摘要：
本文提出了一种基于预定二次拉格朗日量的电路综合算法。即电路演化方程等价于相应的欧拉-拉格朗日方程。建议的综合是一个系统的方法，允许实现任何有限维的物理系统描述的拉格朗日在一个无损的电路，使他们的演化方程是等价的。合成电路由（i）电容和电感各自为正值或负值的电容器和电感器，以及（ii）回转器组成。电路拓扑设计是基于一组LC基本回路（f-回路），这些回路由串联的回转器、电容器和电感器组成的GLC链路耦合。基本拉格朗日的独立变量的集合被确定为与f-环电荷定义为相应的电流的时间积分。所有f-环的EL方程说明基尔霍夫电压定律，而基尔霍夫电流定律自然成立，因为耦合的f-环和相应的电荷和电流的设置。所提出的合成特别地提供了在电路中有效实现期望的光谱特性。该合成还提供了一种通过具有正或负电容和电感的基本电容器和电感器来实现任意互电容和电感的方法。
英文摘要：
We advance here an algorithm of a synthesis of an electric circuit based on prescribed quadratic Lagrangian. That is the circuit evolution equations are equivalent to the relevant Euler-Lagrange equations. The proposed synthesis is a systematic approach that allows to realize any finite dimensional physical system described by a Lagrangian in a lossless electric circuit so that their evolution equations are equivalent. The synthesized circuit is composed of (i) capacitors and inductors of positive or negative values for the respective capacitances and inductances, and (ii) gyrators. The circuit topological design is based on the set of LC fundamental loops (f-loops) that are coupled by GLC-links each of which is a serially connected gyrator, capacitor and inductor. The set of independent variables of the underlying Lagrangian is identified with f-loop charges defined as the time integrals of the corresponding currents. The EL equations for all f-loops account for the Kirchhoff voltage law whereas the Kirchhoff current law holds naturally as consequence of the setup of the coupled f-loops and the corresponding charges and currents. The proposed synthesis in particular provides for efficient implementation of the desired spectral properties in an electric circuit. The synthesis provides also a way to realize arbitrary mutual capacitances and inductances through elementary capacitors and inductors of positive or negative respective capacitances and inductances.
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