ELECTRIC WAVE FILTERS 269 



not be allowed for exactly in advance. Coil adjustment is limited by 

 another factor, as well; the adjustment of a single coil must be made in 

 units of one turn. For these reasons it is very difficult to adjust con- 

 densers to better than about ± 0.3 per cent, and coils to better than a 

 little over ± 1 per cent. 



The effect of the variation in element values may be manifested in 

 several ways. In a simple half-section low-pass filter, for example, 

 the cut-off frequency 



2WLC 



where L and C are the values of the series inductance and shunt 

 capacitance. A variation in L or C will produce a change in the cut- 

 off frequency proportional to the change in the element value. Chang- 

 ing the cut-off frequency is, of course, equivalent to shifting the whole 

 attenuation characteristic upward or downward in frequency. A change 

 in an element value also produces a change in the filter impedance. 

 Suppose that in a network composed of a number of series and shunt 

 arms, the impedance of one arm of the network, Z, is increased by an 

 amount AZ,^. The current through that branch is /,. The change 

 in impedance is approximately equivalent to introducing a voltage e in 

 the circuit. 



e = - I^l^Zq. 



If E is assumed to be the voltage in series with the terminal impedance 

 Zg at the sending end of the network, and /, is the input current, 



__ IgMq ^ E^ 



Ms ~ I, 

 from the reciprocity theorem. But 



AL AZ 



Is Z + Z, 



where Z is the input impedance of the network, and AZ is the change 

 in Z produced by the impedance change in the arm Z,. Furthermore, 



£ = (Z + Z,)/,. 

 Therefore 



^ ^' A// 



And 



AZ = AZ„ I ^-f 



