TABLES OF PHASE 881 



It will also be noted that in the illustration given the characteristic is 

 approximated, commencing at zero frequency, by a series of semi-infinite 

 slopes, each of which is a constant times the characteristic of Fig. 1 of the 

 basic paper, for which Tables I to IV were computed. The characteristic 

 could have been approximated just as well with a series of semi-infinite 

 constant slopes, commencing at / = oo and going down in frequency, each 

 having a flat attenuation above a critical frequency /„ and constant slope 

 at frequencies below. In summing the phase for such an approximation 

 Tables I to IV may be used by reading the angles for ///„ from the /o// 

 tables and vice versa as indicated in the basic paper. 



As an illustration of the above procedure, consider the determination of 

 the phase associated with the characteristic given by 20 log ] Z [ shown in 

 Fig. 5. The characteristic is first approximated by a series of straight lines 

 as shown in dotted form. The critical frequencies and values of ^ = 20 

 log I Z I at these critical frequencies are then read from the straight line 

 approximation^ and the slopes of the various straight line segments deter- 

 mined as illustrated in Table V. 



Having determined the slopes of the various segments of the straight line 

 approximation, the phase at any desired frequency is summed as illustrated 

 in Table VI where the phase for/ = 1.5 is summed. 



The mesh computed value of d for the network in question is plotted in 

 Fig. 6 and it will be noted that the phase summation of Table VI checks the 

 true value to within the accuracy to which the phase can be read from the 

 curve. The identical procedure is followed in determining the phase at 

 any other frequency. As an illustration of the accuracy of the method, the 

 phase was determined at a considerable number of frequencies and the results 

 shown as individual points in Fig. 6. The straight line approximation to 

 20 log I Z I of Fig. 5 was of the order of ± .25 db and, in accordance with the 

 estimated accuracy of the method given above, the maximum departure of 

 the phase summation from the true phase is approximately ± 1.5°. 



A much simpler approximation than that of Fig. 5 may be used without a 

 great loss in accuracy. For instance, a five-line approximation determined 

 by the critical frequencies of Table VII will match 20 log | Z | to within 

 approximately ± .5 rfZ* and therefore should give a phase summation 

 within ± 3° of the true phase. The phase was actually summed at 12 fre- 

 quencies chosen at random for this five-line approximation and the maxi- 

 mum departure of the summed phase from the true phase was 3.2°. With 

 experience in use of the method, simpler approximations can be used and 

 ; the phase determined more accurately than the limits of accuracy of the 

 summation at individual frequencies by plotting the individual summations 



* The original plot was expanded and had much greater scale detail than can be shown 

 , with clarity on a single page plate. 



