RESISTANCE AND CAPACITANCE IN SERIES 



instead of resistive. The expression for the transmission factor is similar to 

 that for the all-resistance potentiometer, but it contains some terms labelled 



by -;• 



The resistance-capacitance high-pass filter — With the arrangement of 

 Figure 3.23, the transmission factor 



Kout 



R 



in 



R-jXc 



Now Xq 



1 



(oC 



Kin 



R 



1 



R — 



J_ 

 oiC 



(oCR 



C 



© Kn ^i K 



Figure 3.23 



Figure 3.24 



The resistance-capacitance low-pass filter — This is the arrangement of 

 Figure 3.24. By a similar process to the above 



Kout 



Vm 



{1 + (coCRfY'^ 



The moduli for the transmission factors for these two filters are plotted as 

 functions of oj on hnear scales in Graph 8, which effectively conceals the 

 essential symmetry in the behaviour of the two circuits. Replotting on 

 scales of log frequency and log [transmission factor] (i.e. a hnear scale in 

 dB's) presents the much more satisfactory picture of Graph 9. From Graph 9 

 it is clear why the devices are called 'filters'. In the high-pass case, for example, 

 evidently all frequencies much above co — \\CR are passed without signifi- 

 cant attenuation, whereas all those much below \\CR are reduced. The 

 frequency \\CR, which marks the transition between the two regimes, is 

 called the 'turn-over frequency'; it is also sometimes, rather optimistically, 

 called the 'cut-oflf frequency'. In general we shall call a filter any network 

 whose transmission factor is frequency dependent. 



Use of log frequency and linear dB scales — The use of logarithmic scales 

 for frequency and transmission factor (the latter is, of course, the same as 

 saying the scale is linear in dB's) is practically universal in plotting the per- 

 formance of all kinds of filter, for by this means the filter properties can be 

 most clearly exhibited. In addition, the use of these scales allows the per- 

 formance of many types of filter to be sketched freehand. For example, in 

 the case of the R-C high-pass filter, the transmission characteristic may be 

 sketched as follows: 



Mark a point, ^, at cu = \jCR, insertion loss = zero. 



From A, draw a dotted straight fine horizontally to the right. 



35 



