A General Theory of Electric Wave Filters * 



By H. W. BODE 



THE growth of the field of electric wave filters since their original 

 discovery by Dr. G. A. Campbell shows the filtering action by no 

 means inheres in any particular physical configuration. Filters have 

 been built, for example, as recurrent or non-recurrent ladder structures, 

 as lattices, as bridged-T's, and in a variety of combinations of trans- 

 formers with ordinary elements. No general theory uniting all these 

 configurations has, however, been developed. Each structure has 

 been treated by methods which are primarily adapted to that con- 

 figuration alone. In consequence, such questions as the relations 

 between filters of different types and the possibilities of securing im- 

 proved characteristics by going to a still wider variety of configura- 

 tions have remained unsettled. 



The present paper is an attempt to develop a general filter theory 

 independent of any particular structure, by means of which these 

 questions can be answered. For the sake of a rigorous discussion, 

 the term "filter" has been used to signify a four- terminal network of 

 ordinary lumped elements which, when terminated in its image 

 impedances, transmits freely one continuous band of real frequencies 

 and attenuates all other real frequencies. Since in actual operation 

 the distinction between free transmission and attenuation is always 

 more or less obscured by terminal effects and parasitic dissipation, 

 this definition is necessarily somewhat arbitrary. It agrees, however, 

 with common usage except in its rejection of multiple band-pass filters, 

 which are rarely used in practice. 



It follows from the definition that a "filter" can include only re- 

 active elements. Otherwise, however, the structure considered may 

 be an arbitrary four-terminal network and may include transformers as 

 well as ordinary inductances and capacitances. The analysis is based 

 upon a combination of the ordinary image parameter method of 

 analyzing networks and the normal coordinate method familiar in the 

 dynamics of vibrating systems. It is found that the conditions for 

 filtering action can be expressed by means of relations between the set 

 of normal coordinates of the network when it is short circuited at both 



* This paper is a summary of a recent article with the same title appearing in the 

 Journal of Mathematics and Physics of. Massachusetts Institute of Technology, Novem- 

 ber, 1934. It is included largely for its value in connection with the accompanying 

 paper on "Ideal Wave Filters." 



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