A METHOD OF IMPEDANCE CORRECTION 807 



and admittance relation at a single junction, can be developed. Using 

 the previous notation, the impedance Z„ will be in the required form if 

 Z„+] is in that form, and not otherwise. Instead of assuming that the 

 impedance at one junction and the admittance at an adjacent junction 

 can be fitted into the formula, therefore, it is sufficient to assume that 

 both the impedance and admittance at a single junction satisfy the 

 formula in order to show that the impedance and admittance at the 

 next succeeding junction satisfy this formula also. 



Application of General Analysis to Filter Impedance 



Correction 



The reciprocal property of the impedances at the terminals of a 

 reactive network indicates two possible methods of applying a ladder 

 network of the sort we have been describing to the correction of wave 

 filter impedances. We can either terminate the network by the filter 

 impedance and adjust its parameters to match the line impedance, or 

 we can consider that the load impedance of the network is a constant 

 pure resistance, representing the line impedance, and attempt to pro- 

 duce a match at the filter terminals. These two methods of procedure 

 lead to distinct results, since in one case the reactance or susceptance 

 correcting branch adjoins the line, while in the other it adjoins the 

 filter. Both are, however, admissible under the general mathematical 

 specifications we have set up for the load impedance of the resistance 

 or conductance controlling network and both lead to reasonably 

 satisfactory impedance correction. 



The fact that a constant pure resistance is an admissible load im- 

 pedance for the ladder network is easily established by inspection. 

 The rational function Gi{x)IG2{_x), representing the imaginary com- 

 ponent, reduces to zero, of course, while the rational function 

 Fi{x)/F2{x), whose square root represents the real component becomes 

 a constant. A filter image impedance within transmission bands is 

 similarly a pure resistance. As a function of frequency it may be 

 defined as the geometric mean of the open and short-circuit impedances. 

 An open or short-circuit filter, whatever its configuration is, however, 

 simply a network of pure reactances. The open and short-circuit 

 impedances are therefore rational functions of frequency and the 

 image impedance they define falls within the scope of the mathematical 

 specification we have set up for the load impedance of the correcting 

 network. 



Terminating Networks of the First Type 



While both of these methods of approaching the problem lead to 

 satisfactory impedance correction, other considerations to be discussed 



