A METHOD OF IMPEDANCE CORRECTION 803 



load impedances are those whose real components can be written as 

 the square roots of rational functions^ of x and whose imaginary com- 

 ponents are rational functions of x. We can make this conclusion 

 plausible by direct inspection. It is obvious that the general nature 

 of the mathematical expression for the impedance of the network can- 

 not change radically as we add successive branches. When we add a 

 series branch, however, the reactance is increased by a,.r, while the 

 resistance is not altered. The functional form of the impedance then 

 will be unchanged if the reactance was originally an algebraic function 

 of X. But, since we must add shunt as well as series arms to the net- 

 work the functional forms must be symmetrical whether taken on an 

 impedance or admittance basis. By analogy, therefore, the suscept- 

 ance also must be a rational algebraic function. The susceptance B 

 is expressed in terms of R and X, the resistance and reactance, by 

 B = XI{R? + X2), but X (and therefore X"^) has already been fixed as a 

 rational algebraic function and R^ must have a similar form if the 

 whole susceptance expression is to be such a function. This conclu- 

 sion, since it applies equally at any part of the network, must, of course, 

 be valid for the load impedance also. 



This argument is sufficient to indicate what sort of a load impedance 

 might have the property for which we are looking — that of allowing 

 the change in resistance or conductance produced by the insertion of 

 the ladder network to be expressible as a simple polynomial. In order 

 to show definitely that this type of load impedance will have that 

 property it is simplest to begin by finding out whether the relation 

 holds when the network consists of a single branch. In accordance 

 with the previous discussion, the load impedance will be taken as 



\F2(x) G2{x) 



' Fiix) 



where Fi{x), F2{x), Gi{x), and G-^ix) are polynomials in x. Upon 

 multiplying and dividing the resistance expression by ■^F2{x)C{x), 

 where C{x) is a new polynomial so chosen that when the product 

 F2{x)C{x) is divided by G^{x) the quotient is a polynomial, the load 

 impedance is transformed into 



^lFl{x)F2{x)C^{x) . Gijx) _ F{x) . Gi{x) 



Fiix) C{x) "^ * GaCx) ~ Fiix) C{x) "^ * GgCx) * 



F{x) is a new symbol, written for ^|Fl{x)F2{x)C^{x), and, as we shall 



^ Including as special cases real components which are simply rational functions, 

 without the square root. 



