COAXIAL AND BALANCED TRANSMISSION LINES 287 



range of impedance for a single frequency. For this case 



tan— |- 

 cosh d = cos — - / 1 + 



V ^ I C0I2 



tan — 



V 



Ai = Zo, a/— tan — ^tan— ^ -\ (27) 



\ V V 



co/i 

 tan — 



(p2 = 1 -I . 



tan — 



V 



The pass band of the filter lies between the values 



and hence between these frequencies the structure will act as a trans- 

 former. The ratio of the transformer, however, varies with frequency, 

 and hence two given impedances can only be matched at one frequency. 

 By adjusting the values of h and I2 it is possible to transform between 

 any two resistances at a given frequency. 



For a number of purposes it is desirable to transform a wide band 

 of frequencies between two constant impedances. This requires a 

 transformer with a constant transformation ratio over the whole band 

 of frequencies. As can be seen from equation (26) this can be accom- 

 plished with the structure considered above if we let h = h = I. For 

 this case 



\ TT col 



cosh = ^ 1 -i-f^cos" ; 



Kr = Zo,^ I ^ -,; (29) 



1-l^cot^^^ 



The mid-band frequency occurs when cosh d = cos (col/v) = 0. Hence 

 / is I wave-length at the mid-band frequency. At the mid-band fre- 

 quency the impedance of the transformer is 



KiQ = Zoyip. (30) 



This is the impedance that the transformer should be connected to since 



