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## Synthesis of a Distributed-Parameter Multi-Port Network Applied to Designing a Microwave Filter

In the field of theoretical study on distributed parameter multi-port circuits, Japanese researchers has played a leading role. In early stage of the study, they attempted to construct microwave filters only of parallel and cascade connections of unit elements (each of which is a coaxial transmission line sections of respective length

*L*and charateristic impedance

*W*) without use of ideal transformers and series connections of unit elements, Within the restrictions as above, characterization and also realizability of the distributed parameter circuit function were discussed.

At that time, a number of new papers presented by Japanese researchers had attracted world-wide attention. One of them, for example, is “Baseband Equalizer with Distributed Parameter Elements” (Nishizeki & Saito) published in the Trans. IECE in 1972. At the practical stand of view, no use of series connection and ideal transformer, while , the use of multi-line elements (coupled transmission line sections of characteristic impedance

*W*and of length

*L*) newly introduced into microwave filter design, has provided an exellent solution to many fundamental problems in the synthesis of resiprocal, lossless, isometric, distributed parameter networks.

First, we demonstrated that the necessary and sufficient conditon for the symmetrical matrix inv[

*W*] of real number (‘inv’ indicates the inverse matrix) to be the characteristic adomittance matrix of an ideal lossless multi-conductor transmission line is that it be a non-singular hyper-dominant one. A non-singular hyper-dominant one says a non-singular matrix in which diagonal elements are all positive, others negative, and also the sum of all of elements in each line is positive (Fig.1). Here, it is noted that, in practice, a shielded multi-conductor line consisting of the exterior conductor enclosing other conductors is utilized as a multi-line element, and the voltage of each conductor is measured with respect to the exterior.

From the fact that a non-singular hyper-dominant matrix can be converted into a non-singular diagonal mtrix with positive diagonal elements by linear transformation, we showed that the transmission along a (

*n*+1)-conductor multi-line is devided into of

*n*independent single-phase transmissions whose characteristic admittances are taken to be values of the diagonal elements, and that the winding ratio of the ideal transformer bank used to combine linearly the above single-phase transmissions corresponds to the vector utilized to the transfomation. It follows immediately from this fact that the equivalent of a distributed parameter networks composed of multi-line elements can be illustrated by combination of transfomer banks and single-phase line elements. For example, as shown in Fig.2, a 2-port network comprising a single-phase element and a 3-conductor element is equivalent to the 2-port comprising 4 unit single-phase elements.

One of the most valuable merits introduced by use of a multi-line elements comes from such coupling between conducters in the multi-line, that is controllable easily. The function of the coupling corresponds to that of a mutual inductance or a coupling capacitance in a lumped parameter netwoks, and no single-phase element has the similar function.

In high frequency band such as needed the use of a distributed parameter element to design a microwave filter, narrow bandpass or bandstop characteristics are often required. But it is not always easy to obtain narrow-band characteristics under the construction comprising only single-phase elements, because of which characteristic admittances are restricted to be producible in practice within a limited range of quantity. The narrow bandpass characteristics, neverseless, becoms to be realizable easily, by making the coupling looser between conductors located on the incident and the transmission sides of the coupled multi-line element.

We manufactured a prototype narrow bandpass filter (in 150MHz) in trial (Fig. 3), and set it in rear of the receiving antenna of the automobile communication system of newspaper companies, in order to examine it succeed to prevent the mixed mudulation happening occasionally. One section of 4-conductor line and one section of singl-phase line are interconnected into the structure with input and output conductors were coupled loosely to the resonator. The loose coupling leads to narrow-band characteristics of the filter.

Next we proposed the synthesis method by means of separating in cascade a multi-conductor line element, and proved its validity associated with Richards matrix theorem. It was demonstrated that the residue

*Y**(

*p*) is also a positive-real matrix after extracting a multi-conductor line element with characteristic admittance matrix

*Y*(1) from a positive-real matrix

*Y*(

*p*) with respect to the complex frequency

*p*=jtan(

*kfL*) (where

*k*is a constant,

*f*the real frequency and

*L*length of the conductors in a single-phase line or a multi-line element) (Fig. 4). This procedure has given an important basic synthesis method.

Here,

*Y**(

*p*) is given by

*Y**(

*p*)=inv[

*E*-

*Y*(

*p*)inv[

*Y*(1)]

*p*] [

*Y*(

*p*)-

*Y*(1)]

where

*E*is the unit matrix. Structure of the filter circuit manufactured in trial by means of the above synthesis method is shown in Fig. 5. And also, the attenuation characteristics of the filter measured is shown in the same figure. It should be noted, however, that the characteristic admittance matrix

*Y*(1) separated from

*Y*(

*p*) in cascade must be hyper-dominant ( note also that allowing the use of ideal transformers this restriction imposed on

*Y*(1) disappear).

Furthermore, the application of unit multi-line elements in the synthesis of a positive-real

*Y*(

*p*) is another one of the most usefull progress in the field of study on a microwave filter design. Such method is called the cascade synthesis, through which each zero point of the even part of

*Y*(

*p*), Ev[

*Y*]={

*Y*(

*p*)+

*Y*(-

*p*)}/2 appearing on the complex frequency

*p*plane, is realized by respective section interconnected in cascade in front of the load resistance. It had been proven that any combinations of single-phase elements make it realizable that even part zero points of a positive-real function

*Y*(

*p*), except the zero points located between —1 and +1 on the real axis of

*p*plane. So we demonstrated that such zero point also becomes realizable by the introduction of a multi-line element into a filter circuit, in which (for example, shown in Fig.6) a 4-conductor element ( with 3 conductors folded) and several number of single-phase elements are interconnected in cascade. Thus, the cascade synthesis of a distributed parameter positive-real rational function was completed.

The series of fundamental research focused to systemaize the distributed parameter network synthesis, as stated above, has brought many aspects into the contributions in this field. For that accomplishment, Nobuji Saito (Department of

Communication Engineering, Faculty of Engineering at Tohoku University) received the Accomplishment Award and the Kobayashi Memorial Achievement Award from IEICE in 1990, and was elected to IEEE Fellow in 1993.

#### Publications

[1] N. Saito、Couple-Line Filters、1970、Ch.7, in Theory and Design of Microwave Filters and Circuits (ed. A. Matumoto), Academic Press[2] N. Saito, H. Uchida, K. Nagai、Mode Analysis of the Three-Connductor Transmission Line by Transforming the Characteristic Resistance Circuit、1963、Report of Research Inst. Of Elect. Comm., Tohoku Univ., SCI.REP.RITS, B-(Elect.Comm.), 15, 3

[3] N. Saito, K. Nagai、Sufficient Condition for Adomittance Matrix Realizable in Characteristic Adomittance of Multiline、1965、Report of Research Inst. Of Elect. Comm., Tohoku Univ., SCI.REP.RITS, B-(Elect.Comm.), 16, 4

[4] N. Saito, K. Nagai、Equivalent Network of a Four-Conductor Line Section、1965、Report of Reserch Inst. Of Elect. Comm., Tohoku Univ., SCI.REP.RITU, B-(Elect.Comm.), 16, 4

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Electricity & Electric Power

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### Events in World

- 1990
- East and West Germany was reunified.
- 1990
- National Center Test for University Admissions started.