THE ELECTRIC H'AVE-FILTER 31 



ladder network having the same propagation constant and the same 

 iterative impedance at mid-series or mid-shunt. 



Lattice Network P'ormulas Fig. 14 



The impedances of a single section between terminals 1 and 2, with 

 the far end of the section 3 and 4 either short-circuited or open, are 

 readily seen to be 



Zo = 



2Z,Z, ^ _ 1/1 ./ , o7\- (10) 



\z, + 2Z, 



,Z. =2(2^' + 2^0' 



Since vZoZ 00 and 'V Zq/Zco are the iterative impedance and the 

 hyperbolic tangent of the propagation constant for any symmetrical 

 artificial line, we have the following analytically equivalent formulas 

 for the lattice network where 7 = v Z1/Z2, and k = V Z1Z2 as for 

 the ladder type. 



Lattice Formulas 

 (V = 2 tanh-i ^ ^1|-; = 2 tanh-1^7, (11) 



K = VZi Z2 = k. (12) 



Zi = 2K tanh^ r, (13) 



2 



2" --"^2 



Zo = ^i^coth^r. (14) 



1 + T 7^ 



r = i2 tan-1;^. =icos-i ^ (I5) 



' 1 1 2 



1 1 : 1 ^ 



2 T 1 + 4 '>'" 



= 2 tanh-i? = 2 sinh-i -=== = cosh-i 



2 i i , 1 



^1-57^ 1-tr 



1 + - 7 



= log ^-, (15a) 



1-27 



