V.2 No 1

47

Mismatched ladder filters

5. Conclusions

We have analysed the solutions for finite ladder filters obtained on the basis of the exact analytical solutions for corresponding mechanical elastic lumped lines and original methods of the dynamical electromechanical analogy DEMA. We have ascertained that finite ladder filters cannot be considered as a simple assemblage of the elementary two-ports, summing their delay phases and multiplying the transmission coefficients. As a result of multiple reflections from the ends in finite filters, in the amplitude-frequency and phase-frequency characteristics of the filter there arise the resonance phenomena effecting on the transmission coefficient and total delay phase. It reflects also on the filter input impedance characteristic inequal to the impedance and not active at the pass band in general case. The progressive wave can arise in the filter only in case of reciprocal trigonometric dependence of the load phase on frequency, which is unrealisable with the currently existing circuitry.

By way of r_LLC filter as an example we have proved theoretically and corroborated experimentally that with the filter active load variation the main transformations of the amplitude-frequency and phase-frequency characteristics take place in the low and middle domains of the pass band. With the growing impedance of the load up to the value close to that characteristical, the resonance peaks amplitude falls. With the further increase of impedance, the first peak frequency vanishes and the number of peaks diminishes by an unity. With it the peak amplitude gradually grows again. In case of the active-capacitive load, with the diminishing capacity the resonance peaks amplitudes gradually grow, but the number of peaks does not change, only a newly arisen peak merges with the first resonance peak of the filter.

The results presented here can be extended to more complicated ladder filters.

Respond to the review by Dr J.O. Scanlan, Chief Editor of International Journal of Circuit Theory and Applications

1.      Bonch-Bruevich, A. M. Radio electronics in experimental physics. Nauka, Moscow, 1966, 768 pp. (Russian)

2.      Zernov, N. Z. and Karpov, V. G. Theory of radio engineering circuits. Energy, Moscow – Leningrad, 1965, 886 pp. (Russian)

3.      Gurevich, I. V. Computational foundations of radio engineering networks (Linear networks under harmonic action). Svyaz’, Moscow, 1975, 366 pp. (Russian)

4.      Kugushev, A. M. and Golubeva, N. S. Foundations of radio rlectronics. Energia, Moscow, 1969, 880 pp. (Russian)

5.      Hu, A.S., Lam, F.W. and Lin, C. Recursive formulas of a multiple-sectioned transmission line. IEEE Transactions on circuits and systems, CAS-21, 5 (September 1974), pp.640-642

6.      Karavashkin, S.B. Refined method of electric long lumped-parameters line calculation on the basis of exact analytical solutions for mechanical elastic lines. Control of Oscillations and Chaos 2000. Proceedings of the conference, vol.1, p.154. St.Petersburg, Russia, 2000.

7.      Gardner, M. F. and Barns, J. L. Transients in linear systems with lumped parameters. Inostrannaya literatura, Moscow, 1961, 570 pp. (Russian; original edition: M. F. Gardner and J. L. Barns. Transients in Linear Systems, vol.1: Lumped-constant systems. John Wiley & Sons Inc., NY, and Chapman & Hall ltd, London, 1942)

8.      Karplus, W. J. Analog simulation solution of the field problems. Inostrannaya literatura, Moscow, 1962, 488 pp. (Russian; original edition: W.J. Karplus. Analog Simulation Solution of Field Problems. McGraw-Hill Book company Inc., New York – Toronto – London, 1958).

9.      Atkinson, F. V. Discrete and continuous boundary problems. Mir, Moscow, 1968, 750 pp. (Russian; original edition: Atkinson, F. V. Discrete and Continuous Boundary Problems. Academic Press, New York – London, 1964)

10.  Karavashkin, S.B. Exact analytical solution on infinite one-dimensional elastic lumped-parameters line vibration. Materials, Technologies, Tools. The Journal of National Academy of Sciences of Belarus, 4 (1999), 3, pp.15 – 23 (Russian)

11.  Karavashkin, S.B. Exact analytical solution for 1D elastic homogeneous finite lumped line vibration.. Materials, Technologies, Tools. The Journal of National Academy of Sciences of Belarus, 4 (1999), 4, pp.5 – 14 (Russian)

12.  Gutenmaer, L. I. Introduction to the Russian edition of the book by W. J. Karplus [8].

13.  Physical encyclopaedia, vol. 5. Sovetskaya Encyclopedia, Moscow, 1966.