504 BELL SYSTEM TECHNICAL JOURNAL 



Appendix II 

 Proofs of Linear Transducer Theorems 



Theorem I: Any passive network whose attenuation constant is 

 zero at all frequencies is a limiting case of a physical wave-filter wherein 

 the transmitting band extends over the entire frequency range. The 

 proof that the phase constant increases with frequency in the trans- 

 mitting band of any wave-filter has already been given by the writer 

 in the paper, "Theory and Design of Uniform and Composite Electric 

 Wave-Filters," B. S. T. J., January, 1923, pages 37-38. In the present 

 case, therefore, the phase constant increases throughout the frequency 

 range. 



The proof relating to the iterative impedance will be given in two 

 steps which comprise essentially the proofs of two impedance theorems. 

 From the first of these it will follow immediately that the transducer 

 under consideration has everywhere a real iterative impedance because 

 of symmetry and a transmitting band extending over the entire fre- 

 quency range; from the second, this real iterative impedance is a con- 

 stant resistance throughout the frequency range. 



Wave-Filter Impedance Theorem: In all transmitting hands the iterative 

 impedances of a recurrent section of any electric wave-filter are conjugate 

 impedances. If the section is symmetrical, they are equal and real 

 without a reactance component. 



From the general formulae on page 617 of -B. 5. T. J., October, 1924, 

 we may write the iterative impedances as: 





I = i((^a + X,) tanh r ± (X„ - X,)), (93) 



where Xa and Xh are the open-circuit driving-point impedances at the 

 ends a and h of the transducer. In a wave-filter recurrent section which 

 is made up of non-dissipative reactance elements the impedances Xa 

 and Xft have only reactance components. Also, in a transmitting 

 band the attenuation constant is zero, so that here F = iB and tanh V 

 = i tan B. From this, it follows readily that in any transmitting 

 band the first term of the right-hand member of (93) represents a 

 positive resistance component and the second term a reactance com- 

 ponent. Hence, the resistance components of iv„ and Kh are identical 

 while their reactance components differ only in sign; that is, Ka and 

 Kh are conjugate impedances in all transmitting bands. 

 As results of the above we may state parenthetically: 



Corollary I: The absolute values of the iterative impedances of a 

 wave-filter recurrent section are equal at any frequency in all 

 transmitting bands; and 



