THEORY AND DESIGN OF WAVE-FILTERS 37 



networks of impedance product D 2 . This is readily shown, for here 

 we have 





(z'M')z'>+(zlz' 2 )z',. _ 

 z'i+z'1 



Similarly z, and z { in parallel, and z> and z> in series are another pair 

 of impedance product D 2 . 



The simplest pair of inverse networks in the case of reactances is 

 an inductance and a capacity. If in an elementary application of the 

 above relations the element L\ corresponds to z b C 2 to z 2 , C { to z : , 

 and L> to z, , where then 



j r j n 



~i = % = D\ (13) 



it follows that L x and C x in series or in parallel, and L" and C{ in 

 parallel or in series, respectively, are inverse networks of impedance 

 product D 2 . By successive applications of these relations we may con- 

 struct any given reactance and its inverse network. 



It should be mentioned that these inverse network relations are 

 even more general than has been considered above, for an elemental 

 pair of inverse networks, besides an inductance and a capacity, is 

 two resistances. 



Phase Constant 



To show that the phase constant increases with frequency through- 

 out each transmitting band of a wave-filter, we may proceed as fol- 

 lows, basing the proof primarily upon the fact that the slopes with 

 frequency of non-dissipative reactances are positive. Consider a 

 mid-series section of the ladder type Z\, z 2 . The impedances as meas- 

 ured across one pair of terminals when the other pair is open or short- 

 circuited are, respectively, 



Z = \Zi-\-Z<i, 



and Z J = Azi + 



Zi-f-2z 2 ' 



whose derivatives with respect to frequency may be written 



dZ . dZ s . 



-=j = i s, and -vx = i r , 

 df df 



where 5 2 and t 2 represent essentially positive quantities in accordance 

 with the above underlying fact. 



