OPERATION OF THERMIONIC VACUUM TUBE CIRCUITS 459 



The relations of (57) and (58) have been given many times, and are 

 included here only in order to illustrate the ease with which simple 

 problems may be solved from fundamental relations. 



Application of the Theory 



The illustrations will serve to give a sufficiently comprehensive view 

 of the methods of applying the general equations to special cases. 



Inasmuch as the derivation of the equations requires no assump- 

 tions other than that the static curves of grid current-grid potential, 

 and plate current-plate potential of the tube remain constant, the 

 accuracy with which a given problem may be calculated depends 

 only upon the ability to determine the effective differential coefficients 

 required by the Taylor's series expansions, and the number of terms 

 of the series included. Practically, the component of current of a 

 given frequency resulting from any higher order term is entirely 

 negligible with respect to the component of the same frequency 

 resulting from lower order terms. For precise results in a general 

 case the calculations are necessarih' tedious, since the physical proc- 

 esses are quite complex. However, in any given special case one of 

 the respective approximations indicated is usually allowable, which 

 greatly simplifies matters. In the event that any question arises 

 concerning the proper phase angles for the complex impedances, the 

 correct result may always be arrived at by writing the voltages in 

 full complex form, as illustrated in the mathematical digression. The 

 impedances will then take care of themselves. 



While it is difficult to show mathematically the convergence of the 

 series of (31), experience has shown that the convergence is so rapid 

 that higher order terms may be neglected, unless new frequencies 

 developed by them are under investigation. In these cases, the 

 conditions of the problem are often such that simplifying assumptions 

 may be made at the outset. If familiarity with the complex im- 

 pedances has been attained, it will, in many cases, be sufficient to 

 derive all equations on the basis of resistance onl\-, and then intro- 

 duce the complex impedances in the manner indicated by the analogy 

 between these and the general equations. 



The higher order coefficients are given below for the special case 

 where resistances, only, are considered, and where the voltage, Cg, 

 is known. It is found more convenient to use the P's, equation (4), 

 in their derivative form than to attempt to express them in terms of 

 M and rp, so referring to the expansion 



ip=aieg-\-a.ieg--\-aieg'-{-aieg'^-\-aieg=-{-_ , 



