RESISTANCES AND CAPACITANCES 



Now choose R, the geometric mean of i?i and R^, and define a such that 

 i?2 = aR and i?i = i?/a. Then 



out 



m 



-jaRX 



(R^-X')-jXR{2a + l'j 



Putting Z equal to 1/coC, we get the modulus of the transmission factor 

 ^out 



m 



(oCR 



a 



1 Y T nW^ 



This function is plotted in Graph 14, which may be used to design a filter 

 for a given band-width centred on a given frequency. At the centre frequency 

 the transmission factor rises from 1/3, at a = 1, asymptotically towards 1/2 

 as a approaches infinity. 



Rationalizing to find the phase shift produced by the filter 



f^out 



Fin 



(f) — tan 



, -°(f - f ) 



2a + - 

 a 



= tan" 



(oCR 



(oCR 



2 + 



a'' 



(Graph 75) 

 Wien bridge 



At coCR = I, the R-C band-pass filter dehvers a maximum output of 

 modulus Kout = {«/(2« + l/«)}f^in with zero phase shift. If we con- 

 nect across the input terminals a potential divider whose output is also 

 {al(2a + l/a)}Fin, then the potential difference between the terminals A-C 

 is zero at coCR — 1, and finite at other frequencies (Figure 3.49). The device 

 is called a Wien bridge, and appears more bridge-like if re-drawn as in 

 Figure 3.50. The Wien bridge is a null-transmission network. 



48 



