THEORY AND DESIGN OE WAVE-FILTERS 29 



numbered correspondingly with a subscript 2- We have then for 

 mid-series or mid-shunt sections: 



(a) IV=VIII+ IX, 



(44) 



etc. To verify these identities we need to consider the propagation 

 constants only since impedance equivalence is known to exist. This 

 is most easily accomplished in either the mid-series or mid-shunt 

 cases by using the formula for e~ r in (1) to show the sufficient rela- 

 tion for propagation constant equivalence, 



e -T = e-T' e -Y". (45) 



Here T represents the propagation constant of the section in the left- 

 hand member of (44) a, b, c, or d; V and Y" those of the correspond- 

 ing right-hand member sections. It can likewise be verified that 

 these identities hold even when dissipation is present if in both struc- 

 tures all inductances have the same time constants and if a similar 

 relation holds for all capacities. A comparison shows that the num- 

 bers of elements in the two structures corresponding to the left- and 

 right-hand members of (44) are, respectively, 8 and 10 in (a), 6 and 8 

 in (b), 7 and 9 in both (c) and (d), and 12 and 12 in (e). 



Equivalent impedance structures involving two inductances and 

 two capacities have already been mentioned in the discussion of the 

 " constant k " low-band-and-high pass wave-filter in Part I. These 

 also include equivalent three element structures. The formulae 

 which hold when a transformation is made from one structure to an 

 equivalent one follow directly from those for certain combinations 

 of two different general impedance components, as given inAppendix 

 III. Because of this generality of the components, equivalence exists 

 even when there is dissipation provided the inductances and capaci- 

 ties have time constants which are, respectively, the same in all. 

 Moreover, since the two structures are identical from an impedance 

 standpoint at all frequencies of the steady periodic state, they will 

 be identical similarly under any conditions of the transient state. 

 The method of deriving the formulae consists in first forming for 

 the two corresponding networks their general impedance expressions 

 which are found to have the same functional form in the two com- 



