%0 Journal Article %1 howitt1931group %A Howitt, Nathan %D 1931 %I American Physical Society %J Phys. Rev. %K electrical_networks equivalent_networks identifiability resistance_distance symmetry_group %N 12 %P 1583--1595 %R 10.1103/PhysRev.37.1583 %T Group Theory and the Electric Circuit %U http://link.aps.org/doi/10.1103/PhysRev.37.1583 %V 37 %X Electrical networks consisting of inductances, resistances, an. d capacitances form a group with the impedance function as an absolute invariant. That is, to a given impedance function there corresponds an infinite number of networks, any one of which can be obtained from any other by a special linear transformation of the in- stantaneous mesh currents and charges of the network. In this manner one may arrive at the complete infinite set of networks equivalent to a given network of any number of meshes. This is done by writing down the three fundamental quadratic forms of the network. Then a linear affine transformation of the instantaneous mesh currents and charges of the network results in the formation of new quadratic forms, the matrices of the coefficients of which represent a member of the group, i.e. , an equivalent network. Instead of performing the substitutions, the three matrix multi- plications C A C are used, one for each quadratic form, where A represents the original matrix, C the transformation matrix, and C' its conjugate. It may be possible to ex- tend this theory to include continuous systems where the quadratic forms become integrals or infinite series and one deals with infinite matrices and infinite transfor- mations.

@article{howitt1931group, abstract = {Electrical networks consisting of inductances, resistances, an. d capacitances form a group with the impedance function as an absolute invariant. That is, to a given impedance function there corresponds an infinite number of networks, any one of which can be obtained from any other by a special linear transformation of the in- stantaneous mesh currents and charges of the network. In this manner one may arrive at the complete infinite set of networks equivalent to a given network of any number of meshes. This is done by writing down the three fundamental quadratic forms of the network. Then a linear affine transformation of the instantaneous mesh currents and charges of the network results in the formation of new quadratic forms, the matrices of the coefficients of which represent a member of the group, i.e. , an equivalent network. Instead of performing the substitutions, the three matrix multi- plications C A C are used, one for each quadratic form, where A represents the original matrix, C the transformation matrix, and C' its conjugate. It may be possible to ex- tend this theory to include continuous systems where the quadratic forms become integrals or infinite series and one deals with infinite matrices and infinite transfor- mations.}, added-at = {2016-08-08T19:51:50.000+0200}, author = {Howitt, Nathan}, biburl = {https://www.bibsonomy.org/bibtex/2b1939f1b25f447d4359fd7f4527a8e73/peter.ralph}, doi = {10.1103/PhysRev.37.1583}, interhash = {32a68f934186b82c1a659e2d778e081d}, intrahash = {b1939f1b25f447d4359fd7f4527a8e73}, journal = {Phys. Rev.}, keywords = {electrical_networks equivalent_networks identifiability resistance_distance symmetry_group}, month = jun, number = 12, numpages = {0}, pages = {1583--1595}, publisher = {American Physical Society}, timestamp = {2016-08-08T19:55:43.000+0200}, title = {Group Theory and the Electric Circuit}, url = {http://link.aps.org/doi/10.1103/PhysRev.37.1583}, volume = 37, year = 1931 }