The paper reviews the properties of periodic and guiding structures which now play an important part in the operation of components, antennas, electron tubes and low-noise amplifiers. An account is first given of dispersive propagation in periodic-loaded lines, showing how the frequency characteristic breaks into pass and stop bands. The formation of forward- and backward-space harmonics and the effect of systematic modification of loading are examined. A description is then given of the various types of surface-wave structures including dielectric rods, dielectric-clad metals, and corrugated surfaces, as well as surface wave instruments and circuits. Practical slow-wave structures such as ladder lines, coupled cavities and helices are finally treated. The survey concludes with a bibliography.

Periodic and Guiding Structures at Microwave Frequencies

S. Mason's signal flow graph technique is applied to the analysis of ladder networks. Briefly, the flow graph of a ladder is developed using Ohm's and the two Kirchhoff's equations. The graph is then reciprocated so that it contains only forward paths. The transfer function, in the case of a simple ladder, is the reciprocal of the sum of all distinct paths from the output to the input node. In the case of ladders containing internal generators, independent or dependent, the transfer function can be found in a similar way with slight modifications. Other relations, such as the input impedance, transfer admittance, etc., can also be found directly from the flow graph. In essence, instead of writing mesh equations or applying recursion formulas of continuants, we construct a flow graph by inspection and analyze the network from the graph. In the last part of the paper, the striking similarities between the flow graph of a network and its analog computer simulation are pointed out. Indeed, the flow graph is the computer set-up in a schematic form.

Ladder Network Analysis by Signal-Flow Graph-Application to Analog Computer Programming

The driving-point properties of the distributed RC ( RC ) network are utilized for realizing an impedance approximating (jw) − 1 2 over a wide frequency range ω0 RC network and those of the terminating impedance. We show that with suitable termination, the value of ω0 can be reduced by a factor of ten from the value for open or short-circuit conditions. Thin film or semiconductor realization is possible for the resulting networks, but they can also be approximated by combinations of lumped R's and C's. Two types of lumped-element approximations are suggested: one employs a finite ladder to approximate the three-terminal RC network while the other is based on truncation of the partial-fraction expansion of the two terminal RC impedances. It is found that the former type of structure gives better results than the latter type.

Distributed and lumped RC realization of a constant argument impedance

Application of hypergeometric functions to analysis of linear, cascaded, identical fourpoles is investigated. Expressions for iterated network functions are derived in terms of this class of functions. Cascaded, isolated, singular fourpoles are examined in terms of the confluent hypergeometric function of several variables. Examples demonstrate how this function reduces to several special functions which are also useful in treating more general iterated networks. The Chebyshev polynomials and another hypergeometric function, called here the Jacobi-Chebyshev function, are used to analyze passive iterated networks. Extensions of the method lead to compound functions of hypergeometric functions. Expressions for current and voltage in transmission lines as a limiting case of infinitesimal fourpoles, together with an example of their application, are also included.

On Hypergeometric Functions in Iterated Networks

An examination of the properties of a pulse transmission matrix for a storage system is given in detail. An attempt is made to arrive at design criteria for a practical system. It appears that microstrip techniques afford an economic approach to the problem and that technically, impedances of the order of 50 ohms will present optimum over-all performance. Use of conductors of about 1 mm in width and electroformed earth shields separated by about 2 mm, will result in an access time less than 6 m? sec and a total physical size of less than a meter cube.

The Design Problems of a Megabit Storage Matrix for Use in a High-Speed Computer

The matrix algebra has a natural application to electric circuit theory, especially to the method of solution by means of substituted variables, such as symmetrical components. The solution of circuit equations by the method of diagonalizing the impedance matrix is developed in the paper, and applied to the solution of the difference equations of ladder networks. An extension to systems of difference equations follows. Diagonalizing is equivalent to the use of substituted currents, voltages and impedances such that there is no mutual coupling between the substituted networks. The substituted currents are then calculated without difficulty and the original circuit currents obtained by a simple transformation. The method has the additional advantage that, since the transformation is independent of the circuit constants, it may be applied to all circuits possessing the same degree of symmetry. Ladder networks may also be solved by regarding them as a series of four-terminal passive networks. The elements of matrix algebra are included for completeness.

R.E. Vowels

Papers

Matrix Methods in the Solution of Ladder Networks