146 IX. The Theory of Wave Filters containing a finite number of Sections. By HAROLD A. WHEELER, B.S. in Phys., and FRANCIS D. MURNAGHAN, M.A., Ph.D., Associate Professor of Applied Mathematics, Johns Hopkins University*. I. INTRODUCTION. HE simple electric wave filter is built up of a number in fig. 1. When a steady alternating voltage E, is impressed at one end of the filter, the resulting voltage E, at the other end bears a ratio to the impressed voltage which is a function of the frequency of alternation. In the present uses to which filters are put, it is desirable that this response ratio be constant in certain predetermined frequency bands-the transmission (pass) bands-and as small as possible in the remaining suppression (stop) bands. The degree to which this condition can be realized determines the merit of any given filter. The location of the frequency bands depends on the structure of the recurrent sections. The response-ratio functions of the frequency can be expressed in terms of determinants. The simplest determinant of this type-Dn in Part V. of this paper-was evaluated fifty years ago by Rayleigh † in connexion with the problem of determining natural frequencies of vibration of a loaded string. The complete determinant, D,(a, b), was given by Pupin, and was evaluated by a different method. The most extensive study of the wave filter has been made in connexion with loaded telephone lines, and is based on Campbell's solution of the infinite line §, i. e., the filter with an infinite number of sections. The real distinction between the infinite line and the finite line is the presence of reflexions at the terminations of the latter. The behaviour of a finite line is similar to that of the same number of sections in an infinite line, when the terminating impedances are so chosen as nearly to avoid reflexions, so that the infinite line solution has been found very useful. In the present paper response formulæ are developed for finite Campbell filters, so that reflexions are taken account of without having to make special adjustment of the end conditions, automatically taking account of reflexions. *Communicated by the Authors. + Theory of Sound,' vol. i. p. 120 (1877). § Bell System Tech. Jour. p. 1 (Nov. 1922). Particular attention is given to the non-dissipative constant-K type filter, terminated by any values of resistance. The M-derived type, due to Zobel, is also treated, and its advantages over the former type are shown. These response formulæ are useful in that they show the deviations from the ideal performance and from the approximate solution based on the infinite line. The solution for the latter ideal case is then obtained as a limiting case of the solution for the actual finite line, and the expressions for the iterative impedance and propagation exponent are derived. The natural frequencies of finite, conservative lines of recurrent structure are of theoretical interest, and bear a close relationship to the filter properties of the same lines. On applying our method to this problem, a convenient graphical solution, which is applicable to lines of any recurrent structure, however complex, is obtained. In this Wave filter with mid-series terminations, (n-1) sections. connexion two variations from the recurrent line will be described. The first may be called an exponential line, and is built up of sections whose impedances change in geometric progression from section to section. The second is an alternating line with two recurrent sections in alternating succession. Throughout our paper the terminology and notation generally used in papers dealing with the electric wave filter are adhered to. The term "frequency" denotes the number f of cycles per second; the "angular frequency" is the number of radians per seçond, so that w=2πf. II. WAVE-FILTER CIRCUITS, MID-SERIES TERMINATION. 1. The Solution for a Finite Line.-Fig. 1 shows a Campbell (loc. cit.) filter circuit built up of recurrent * Bell System Tech, Jour. ii. p. 16 (1923). sections (Z1, Z2) and terminated by input and output circuits (Za, Zo) connected at mid-series terminations. This network of (n-1) sections gives rise to a system of n linear equations in I, I, ... In, the currents in the successive meshes of the network. Eo denotes the applied alternating voltage, and E1, E2... En denote the resulting voltages at successive pairs of terminals along the filter. Z denotes an electric impedance of any character, in general a complex function of the frequency. The equations for the currents in the successive meshes are 0= (In-In-1)Z2+In(÷Z1+Zo). It is convenient to introduce the following notation so that I, w, wa, and we are functions of the frequency Our equations take, then, the form The determinant of the coefficients is Dn(a, b), Part V., in which the cofactor of the element in the first row and last column is unity, so that We may define a response function G, as the ratio of output voltage En to applied voltage E. E=ZIn, we have Noting that for a filter of (n-1) sections. The corresponding ratio with no filter sections interposed (n-1=0) is In the response solutions to be developed, it is most convenient to solve for the relative response ratio which shows immediately the result of interposing the (n-1) filter sections between the terminating impedances. The reciprocal of this ratio is given by Eqs. (14) and (15) for the general case. 2. Iterative Impedance of Infinite Line.-Fig. 2 shows a segment of (n-1) sections removed from an infinite line. Wave filter with mid-series terminations, arranged to assimilate a segment of an infinite line. The remainder of the line on the left (input) side has been replaced by its equivalent impedance - K1, and the remainder of the line on the right (output) side is replaced by its impedance K1. This equivalent impedance is known as the iterative impedance of the infinite line. In applying the analysis of the preceding paragraph to the present case, we note that E=0 (absent), so that Dn(a, b) must vanish if the other voltages are to exist in the line. Since we desire a solution independent of n, we note in Eq. (13) for D,(a, b) that the even powers of (n-1) are concentrated in cosh (n-1) and the odd powers in sinh (n-1)г. The Any generator circuit connected to an impedance, Z=E/I, may be represented as the negative impedance, -Z-E/I, since the terminal voltage E of the generator circuit is equal to that of the impedance, and the current into the former is -I,; the reverse of the current into the latter. www expression vanishes for all values of n when the two respective coefficients are equated to zero : wa+ws=0, wawa+sinh2 T=0. It is convenient to substitute for this case Then the two coefficients vanish when w=sinh T. Since coshr 1+ = ᏎᏃ ' 3. Propagation Exponent of Infinite Line.-The finite line behaves as the same number of sections in an infinite line when either or both of the terminating impedances are made equal to the iterative impedance, thereby avoiding repeated terminal reflexions. We have Za=K, a=2x=sinh T ; and (or) Z=K, ==sinhT, so that Eq. (15) reduces to the simple form The latter equation defines the propagation exponent T in terms of the voltage ratio across a single section, while the former equation gives the voltage ratio across (n-1) sections, in an infinite line. The propagation exponent is complex in general, and we write TA+iB, where the attenuation exponent A accounts for changes in magnitude of the voltage from section to section, and the phase angle B accounts for the corresponding phase displacements. *The usual term "propagation constant" is misleading, since the quantity is not constant but a function of the frequency. In such cases the terms "coefficient," " factor," etc., are to be preferred, since they are descriptive, and do not imply constancy. |