Patent GB-423924-A (GB)
Title: Four-terminal electrical networks
IPC: H03H7/00
Filed: 1933-05-05
Abstract: 423,924. Impedance networks. GEWERTZ, C. M., Massachusetts Institute of Technology, Cambridge, Massachusetts, U.S.A. May 5, 1933, No. 13119. Convention date, May 5, 1932. [Class 40 (iii)] A general method of synthesis is given whereby four-terminal impedance networks for use in communication or power circuits can be realized exactly with a finite number of elements when the driving- point and transfer short-circuit admittances or open-circuit impedances are prescribed in the form of polynomial complex functions satisfying certain restrictions. The networks may be dissipative or purely reactive and may contain negative resistances. They may be used for joining two communication circuits in a reflectionlessmanner, for modifying the impedance of a circuit, for simulating a transmission line, for modifying wave-form, or for other like purposes. For any quadripole one short-circuit driving-point admittance (i.e. the admittance facing one pair of terminals when the other pair is short-circuited) is denoted by alpha11; the short-circuit admittance for the opposite pair of terminals is alphann, and the corresponding short-circuit transfer admittance (i.e. the ratio of output current to input voltage under short-circuit conditions) is alpha1n. For open-circuit conditions the open-circuit impedances #11' #nn, #1n are defined in an analogous manner. The alphas when expressed in terms of the #s all have the determinant as denominator so that poles and zeros of the determinant correspond to zeros and poles of the alphas ; and the converse is true of the #s when they are expressed in terms of alphas. Such a transformation is referred to as an inversion. It is assumed that when the required network has been prescribed by means of its two characteristic impedances and its propagation characteristic, the above quantities can be expressed in a form of which the following is a typical example while alpha1n and alphann are equal to expressions having the same denominator as alpha11 but different cubic functions of # for their numerators, where #=(gamma+j#), # is 2# times the frequency and gamma is real. The starting point of the calculation is a set of alphas or a set of #s defined by polynomial fractions of the above type. The Specification examines the conditions that have to be satisfied in order that a " network representation " of such expressions may be realizable : thus alpha11, alphann, #11' #nn must be " positive real functions " of #-i.e. they must be real when # is real, and their real parts must be positive or zero when y is positive or zero-and alpha1nand #1n must satisfy a certain determinant condition given in the Specification. A set of alphas or #s is regarded as constituting a matrix having the form of the determinant set out above. When it conforms to the requirements just stated it constitutes a " positive real matrix " and a network realizing it can be designed in accordance with the invention. Sections or " component networks." When there is a constant ratio between two of the elements the matrix can be realized as a simple section. Thus, if alpha11¸alpha1n is a finite real constant, the alpha matrix can be realized as shown in Fig. 5, where Y1=alpha<2>1n + alpha11; Y2=(alpha11αnn-alpha<2>1n)alpha1n/alpha<2>11; and the ratio D of the transformer T=alpha11+alpha1n; or the same matrix can be realized in another equivalent section, Fig. 6, (not shown). If alpha11¸alphann is a real constant the matrix can be realized as a lattice section with a transformer, Figs. 10, 11, (not shown) ; and the matrix can be realized in a further manner when alphann¸alpha1n is a finite real constant, Fig. 8, (not shown). Under similar conditions the # matrices can be similarly realized. If alpha1n, (alpha11 - alpha1n) and (alphann-alpha1n) are positive real functions, the alpha matrix can be realized as a # section, Fig. 15, (not shown) ; and if the #s have a similar property they can be realized as a T-section or a Z-section, Fig. 15, (not shown). If alpha11=alphann=alpha1n, the section becomes a simple series element, Fig. 16, (not shown) ; if #11=#nn=#1n it becomes a simple shunt element, Fig. 17, (not shown). Multi-section networks. When the matrix cannot be realized as a single section it has to be transformed into the sum of a number of simple matrices, and the corresponding sections are combined to form a group of component networks. If the calculation is based on short-circuit admittances alpha the sections are coupled in parallel, Fig. 18, (not shown), but if it is based on open-circuit impedances # the sections are coupled in series, Fig. 19, (not shown), the coupling being effected by means of transformers except in certain special cases. The Specification discusses in considerable detail the method by which this analysis can be effected in various cases, according to the location of the poles and zeros of the alphas or #s in the complex plane. The analysis consists generally in successively removing poles from the given functions, each alpha or # being thus reduced to a sum of simple terms. A set of homologous terms, one from each alpha or one from each #, is called a " family " and constitutes a matrix which can be realized as described above, except that the matrix formed by the last family may require further reduction. In this case it is necessary to " invert " this last component matrix by expressing the alphas in terms of the corresponding #s, or vice versa, so as to obtain a matrix of lower order in # after common factors have been cancelled and poles on the boundary have been extracted ; and this operation is repeated as often as may be necessary upon the successive matrices so obtained. It may be necessary to extract from alpha11, alphann, #11 or #nn a real positive constant part r<1> which must be selected according to a rule that is described with reference to Figs. 27, 28, 31, 32, 37, 38 (not shown). Instead of being constant, however, the extracted part may be a positive real function Y, so chosen that a zero value is imparted to the determinant composed of the real parts of the remaining two alphas or #s and of the residue of the alpha or # from which Y has been extracted. In this case repeated inversion yields a succession of matrices having zero determinants and the final matrix is composed of real constants. Numerical examples of multisection networks designed as described are given in the Specification and illustrated in Figs. 20, 26, 29, 30, 33 .. 38, 39 (not shown). Fig. 44 shows the realization of a matrix in which while Fig. 45 shows the corresponding realization when #11' #nn, #1n have those values. Fig. 46 is a schematic representation of Fig. 44 and shows a network realizing a set of alphas and comprising a section a1 together with a group of further sections, a1 and a2 being connected in shunt. Group a2 realizes the # matrix obtained by inversion of the unrealizable residue of the initial alpha matrix and comprises components b1, b2 connected in series with a further group b3 which realizes the alpha matrix obtained by inversion from the unrealizable part of the preceding # matrix. Group b3 comprises sections c1, C2 connected in parallel with a group C3 which realizes the final # matrix and comprises sections d1, d2, d3 connected in series. The component Y shunted across the whole filter represents the matrix which was extracted from the original matrix so as to leave a reduced matrix having a zero determinant. Fig. 47 represents Fig. 45 and shows the manner in which the component networks are grouped when the calculation starts from a # matrix.