134 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1951 



If a flat transmission is required over a given band of frequencies we may 

 replace the limits of integration by coi and C02 , these quantities representing 

 the edges of the useful band, which yields 



2CRo 

 or 



£2 2r 1 



(12) 



El/iRo C(C02 - COi) C(/2 - /l) 



Translating this expression into the equivalent acoustical system, where C 

 represents the compliance of the human ear cavity, it may be shown* that, 



Vc 



by substituting C = ~^ , replacing the voltage E by the pressure p de- 



veloped in the cavity, and by using the factor of 10^ to convert from practical 

 to c.g.s. power units the following equation results 



This is the expression for the available power response of an ideal receiver 

 which is assumed to have a flat response over the band of frequencies ex- 

 tending from /i to /2 . 



The plot of this equation expressed in decibels, and marked "Locus of Per- 

 formance Limit" is shown in Fig. 16(b) taking Vo as 6 cc and /i as zero. 

 From this curve, an ideal receiver having a bandwidth of 3500 cps would 

 have a response level of 88.3 db. It is also evident that the low frequency 

 analysis given in the preceding section corresponds to a receiver with a 

 range of approximately 1600 cps, while for wider ranges the response level 

 would be lower, corresponding to the three other bands shown in the figure. 



From the analysis given above, it is clear that the 88.3 db level limit super- 

 sedes the 91.7 db value based on the low frequency losses alone, because the 

 former value takes account of the frequency range over which a receiver is 

 designed to operate. A complete loss theory would undoubtedly arrive at the 

 lower limit. However, because of the reactive load, it has not been possible 

 to derive a suitable formula which includes the dependence on frequency 

 range, and at the same time shows the character of the losses. The utility of 

 the low frequency analysis lies in the fact that it shows the relative impor- 

 tance of the various losses, and where the most opportunity for improve- 

 ments lies, and their likely magnitudes. It must be realized that although 

 ring armature receivers may be built in the laboratory, which have smaller 

 losses than the receiver discussed, the present design is a compromise chosen 

 to be most suitable for use in the subscriber's telephone set. 



•Unpublished work by T. J. Pope, Bell Telephone Laboratories, Inc. 



