188 BELL SYSTEM TECHNICAL JOURNAL 



that each structure is a purely reactive network with the exception of 

 the terminating resistance and finds the network elements in terms of 

 the short circuit reactance as measured from the resistance end of 

 the network. 



Use may be made of the following theorem : 



With any four-terminal reactive network the reactance measured at 

 terminals 3-4 with terminals 1-2 short-circuited is equal to the laftgent 

 of the phase shift between a voltage Eq applied to terminals 1-2 and the 

 resultant voltage Ei across a unit resistance connected to terminals 3-4. 



The open-circuit voltage across 3-4 due to Eo would be ± kEo, 

 where ^ is a real quantity, if the network contains only reactances. 

 By Thevenin's Theorem, then, 



_ ±kEo 

 ^'~ l-j-iX' 



where X is the reactance of the network from terminals 3-4. If (3 is 

 the phase shift between Eo and Ei, X = tan /5. 



Since this phase shift is given by (11) the short-circuit reactance is 

 known. At a value of X = i(l/Pi) or x = 1/Pi, the impedance of the 

 first shunt arm from the right of Fig. 5 is zero, so that the reactance 

 of the filter is simply the reactance of the arm aiX, which gives the 

 value of oi directly as 



ai = Pi(tan j8i)i, 



where (tan /3i)i denotes the value of tan ^i when x = 1/Pi. The 

 reactance of the network after subtracting oix is tan j8i — Pi(tan /3i)i.-v;. 

 At values of .v very close to 1/Pi this is the reactance of the first shunt 

 arm, or 



dx\ X 



J- (tan j8, - <hx) 



where, after differentiation, x = 1/Pi. Carrying through the differ- 

 entiation, 



2Pi2 



02 = 



(> + '-'^')'(f),-'" 



Similar formulas may be found for the rest of the elements. If Xm 

 denotes the reactance starting with the series arm OmX or with the 



