Showing papers on "Topology (electrical circuits) published in 1970"

On the Corona of Two Graphs.

Abstract: Our object in this note is to construct a new and simple operation on two graphs Gt and G2, called their corona, with the property that the group of the new graph is in general isomorphic with the wreath product of the groups of Ga and of G2. Consider two permutat ion groups A and B of order m and n respectively acting on objects sets X = { x 1, x2 ..... Xd} and Y={Yl,Yz,. . . ,Y~}. By their composition or wreath product A [B] we will mean that permutat ion group of order mn a acting on X × Y in which each permutat ion c~A and each sequence (/3~,/~2 . . . . . /~d) of permutations in B induce the permutat ion y =(~;/~1, f12 . . . . . /3a) such that Y(xi, y j) =(c~xi,/~iyj). We write A = B to mean that two permutat ion groups A and B are not only isomorphic but also permutat ionally equivalent. More specifically let h : A ~ B be an isomorphism. To define A = B, we also require another 1 1 map f : X~--~Y between the objects such that for all x in X and c~ in A, f (~x )=h(~) f (x ) . Let graphs 2) G 1 and G 2 have point sets 1/1 and V 2. Then GI[G2], their lexicographic product, has V1 x V2 as its set of points, with two of its points u = (u~, uz) and v =(v l , vz) adjacent whenever ul is adjacent to v~ in Gi, or Ux = v l and u2 is adjacent to v2 in G2. Let A~ and Az be the groups of graphs G~ and G2; then necessary and sufficient condit ions for A ~ CA/] to be permutationally equivalent with the group of GI [G2] were found by Sabidussi [2]. In order to state this result, we recall from [1] that F(G) denotes the group of graph G and (7 the complement of G, that the neighborhood N(v) of a point v of G is the set of points adjacent with v, and the closed neighborhood of v is N(v) u {v}.

Solvability and analysis of linear active networks by use of the state equations

Abstract: A study of the reduction process used to obtain the state equations for a network containing capacitors, resistors, inductors, and controlled and incontrolled sources of all types, gives the necessary and sufficient conditions for such a network to possess a solution. These conditions involve both the topology of the network and its element values. The formation of the state equations also depends both on the network's topology and its element values. It is shown that the state variables can be chosen from the variables associated with the controlled sources, and any set of state variables chosen for the passive network, formed by removing all the sources. A topological restriction on the active network, which ensures that the maximum order of complexity of the active network is the same as that of this passive network, is given.

On the synthesis of any RC-realizable rational transfer function using a nonuniform RC distributed circuit

Abstract: A synthesis procedure that can realize any RCrealizable rational transfer function as the open-circuit-voltage transfer ratio of a nonuniform, passive RC distributed circuit is presented. The resulting circuits have a simple, standardized topology and can be readily constructed by thin-film techniques. The synthesis procedure has been completely programmed on a digital computer. From the specification of the transfer function and the electrical properties of the construction materials, the computer produces the complete geometry for the construction of the distributed circuit.

Edge T-matrix in network theory

Abstract: : Theory of the edge T-matrix is developed and applied to derive algorithms for solving basic problems of network theory such as the determination of a fundamental loop or cut-set matrix, the path matrix, the circuit matrix, the se matrix and the tree summation calculation. Application of the Tree summation to the determination of sensitivity functions without actual derivative operation will also be investigated. Formulas for determining topologically the sensitivity function will be proved. (Author)