{"title":"Circuit synthesis based on a prescribed Lagrangian","authors":"A. Figotin","doi":"10.1090/spmj/1801","DOIUrl":null,"url":null,"abstract":"<p>On the basis of a prescribed quadratic Lagrangian, an algorithm of synthesis for an electric circuit is suggested here. That is, the circuit evolution equations are equivalent to the relevant Euler–Lagrange equations. The proposed synthesis is a systematic approach that allows one to realize any finite-dimensional physical system described by a quadratic 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 <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L upper C\"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mi>C</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">LC</mml:annotation> </mml:semantics> </mml:math> </inline-formula> fundamental loops (f-loops) that are coupled by <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G upper L upper C\"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mi>L</mml:mi> <mml:mi>C</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">GLC</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-links each of which is a serially connected gyrator, capacitor, or 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 is fulfilled naturally as a consequence of the setup of the coupled f-loops and the corresponding charges and currents. In particular, the proposed synthesis 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.</p>","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"St Petersburg Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/spmj/1801","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
On the basis of a prescribed quadratic Lagrangian, an algorithm of synthesis for an electric circuit is suggested here. That is, the circuit evolution equations are equivalent to the relevant Euler–Lagrange equations. The proposed synthesis is a systematic approach that allows one to realize any finite-dimensional physical system described by a quadratic 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 LCLC fundamental loops (f-loops) that are coupled by GLCGLC-links each of which is a serially connected gyrator, capacitor, or 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 is fulfilled naturally as a consequence of the setup of the coupled f-loops and the corresponding charges and currents. In particular, the proposed synthesis 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.
在规定的二次拉格朗日的基础上,这里提出了一种电路合成算法。也就是说,电路演化方程等价于相关的欧拉-拉格朗日方程。所提出的合成是一种系统方法,可以将二次拉格朗日描述的任何有限维物理系统实现为无损电路,从而使它们的演化方程等效。合成电路由以下两部分组成:(i) 电容和电感的正值或负值;(ii) 回旋器。电路拓扑设计基于一组 L C LC 基本回路(f-loop),这些回路通过 G L C GLC 链接耦合,每个链接都是一个串联的回旋器、电容器或电感器。基本拉格朗日的自变量集与 f 环电荷相一致,定义为相应电流的时间积分。所有 floop 的 EL 方程都考虑了基尔霍夫电压定律,而基尔霍夫电流定律则因耦合 floop 的设置以及相应的电荷和电流而自然实现。特别是,所提出的合成方法可在电路中有效实现所需的频谱特性。此外,该合成法还提供了一种方法,通过电容和电感各自为正或负的基本电容和电感,实现任意的互容和互感。
期刊介绍:
This journal is a cover-to-cover translation into English of Algebra i Analiz, published six times a year by the mathematics section of the Russian Academy of Sciences.