ELECTRIC WAVE-FILTERS 303 



c. The series and shunt image impedances of a pair are inverse nehvorks 



of impedance product E?. 



Such results would also follow from (a) above together with the 

 consideration of mid-point terminations. They are verified by com- 

 parison of formulas (10) and (12) which give 



Wu:W-ik = Wi,(m)W2,(?n) = Wi>c(m, m')W2k(m, m') 



= Wik(m, m', m")Wik{m, m' , m") = • • • = R-. 



d. Both image impedances of either MM'-type, or of either one of a 



higher order pair, may be adjusted dependently without changing 



its transfer constant; the ratio of the two image impedances is 



fixed when the transfer constant is fixed. 



This can be seen from the fact that the transfer constant depends upon 



the parameters only in their product, g, and from the formulas for two 



consecutive impedances in (10) or (12). 



1.7 M-Type Wave- Filters 



These are the wave-filters of the first order in each sequence and 

 contain one arbitrary parameter, m. Although they are quite well- 

 known, it is necessary to include them here for the sake of continuity 

 and because of the fact that they are to be used later. 



The series .l/-type has the formulas 



Zik(m) = mzik, 



,, . 1 — m- 1 



cosh r.(;;0 = 1 + ^^'iUk + iV,] 



(13) 



1-f (1 -m''){Uu + iVk) 



and 



Wik = i?Vl+ Uk + iVk, 



^ i^[l+(l-^n-)(^. + .-F.)] 



Vl+ Uk + iVk 

 In the shunt .l/-type 



zik"{m) = — -^ , 



mzik 4 m 



1 — m'' 



.V(m)=^s,„ (14) 



cosh Tk{m) = same as in (13), 



i?Vl+ Uk-hiVk 



W,k{m) = 



[1+ (1 - w2)(f/, + iF,)J' 



