RESISTANCES AND CAPACITANCES 



and by the usual process the phase shift turns out to ht 



4 



(j> = tan~^ 



coCR — 



1 



(oCR 



(Graph 19) 



Relationship between transmission characteristic and phase shift 



It often happens that apparatus contains filters, either of the R-C type 

 we have been discussing or of the R-C-L type to be dealt with in Chapter 5, 

 whose precise natures are not known. If the transmission characteristic is 

 known, or can be measured, then it is possible* to compute the phase shift. 



If on the transmission characteristic we are considering a region of the 

 curve which is reasonably straight, then the phase shift ^ at frequency a> is 

 given simply by 



^" = 15 (^)„ 



where (d^/dM)^^ is the slope of the characteristic in dB's per octave at 

 frequency co. Thus a tapered 3-section low-pass R-C filter, well above the 

 turn-over frequency, has a slope of 18 dB's per octave and the phase shift is 



^0, = 7^ X 18 or 270 degrees 



If the transfer characteristic is rather less simple, perhaps a low-pass type of 

 curve but containing a 'bump', as shown in Figure 3.55, then in the region 



Frequency 



Figure 3.55 



of the bump the phase shift is modified and it is necessary to add a correcting 

 term to the expression already given. The equation which follows is due to 

 Bode, and is quoted by F. E. Terman {Radio Engineer's Handbook). If the 

 slope of the curve at the frequency m where the phase shift is required is 

 {dAjdu)^, and the general slope of the part of the characteristic upon which 

 the bump is superposed is (d^)/(dw), as before, then 



Uu) ~ 



, 77 (dA\ 1 f 



loge COth 



2r" 



where u = loge (frequency /frequency co). 

 * Provided there are no 'all pass' sections present, e.g. 3.44 or 3.45. 



52 



