Maximally-flat Filters in Waveguide 



By W. W. MUMFORD 



Microwave radio relay repeaters require the use of band-pass filters which 

 match closely the impedances of the interconnecting transmission lines and which 

 suppress adjacent channels adequately. A type of structure called a Maximally- 

 Flat filter meets these requirements. 



The ladder network which gives a maximally-flat insertion loss characteristic 

 is discussed and several methods of achieving its counterpart in microwave trans- 

 mission lines are presented. Resonant cavities are used to simulate tuned circuits 

 and the necessary formulas relative to this approximate equivalence are given. 



Experimental data confirm the theory and show that this technique yields 

 remarkable impedance matches. 



Introduction 



"1 T /"E USUALLY associated the word filter with any device which is selec- 

 ^ ^ tive. The electric wave filter has that property which enables it to 

 transmit energy in one band or bands of frequencies and to inhibit energy in 

 other bands. Selectivity is the result of either selective absorption^ • ^* 

 or selective reflection. This paper discusses a special case of the classical 

 lossless transducer which derives its selective properties entirely from selec- 

 tive reflection. The insertion loss of this type of filter can be analyzed in 

 terms of the input reflection coefficient and the input standing wave ratio. 



In many applications of lossless filters it is desirable to obtain a character- 

 istic such that the insertion loss, and hence the reflection coefficient, is small 

 over as wide a band as possible. A special case described here, referred to 

 as a maximally-flat filter, has a loss characteristic such that a maximum 

 number of its derivatives are zero at midband. While the maximally-flat 

 type of characteristic does not give the smallest possible reflections over a 

 finite pass band, it does give small reflections, and has added advantages of 

 simplicity in design and in many cases less transient distortion than filters 

 giving smaller reflections. 



The desirable characteristics of maximally-flat filters have long been 

 realized.^-' Mr. W. R. Bennett^ of these Laboratories derived the con-, 

 stants for a maximally-flat ladder network in the late 20's, and gave simple 

 expressions for the element values. Butterworth,^ Landon^ and Wallman^ 

 have treated maximally-flat filter-amplifiers in which the filter sections are 

 separated by amplifier tubes. Darlington* has considered the general case 

 of four terminal filters which have insertion loss characteristics that can be 

 prescribed, but he places the emphasis more on filters that have tolerable 



* A list of selected references appears at the end of the paper. 



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