TWO-TERMINAL BALANCING NETWORKS 281 



function as frequency varies from to oo . At frequency ^(X) = + jO 

 and at oo Z(X) = — ./O. The locus is a circle and crosses the real axis at R 

 and the frequency at which L and C are anti-resonant. The special cases 

 will be readily apparent and without further discussion attention will be 

 shifted to Fig. 1-c which is the generating function applied to obtain solu- 

 tion of the examples listed below, all of which are located in the fourth 

 quadrant. 



The impedance of this function (Fig. 1-c) at zero cycles is a real and has 

 the value R, and at infinite frequency its impedance is — jO. The locus 

 traced by this function in the fourth quadrant of the complex plane as / 

 varies from zero to infinity is a semicircle of radius R/2 whose center is at 

 R/2 on the axis of reals. Obviously, the impedance for any given fre- 

 quency depends only on C when R has been fixed. 



One of the most useful networks for voice frequency work is that in which 

 two such functions are added together but the second function is the special 

 case in which C — Q. We then have a network which consists of a resist- 

 ance i?i in series with the parallel combination Ri and Co, and is represented 

 by the semicircle just described but displaced to the right of the origin by 

 the distance R\. This form corresponds to a special case of the bilinear 

 transformation previously mentioned. 



As stated earlier a given impedance function can be obtained from a large 

 number of networks but when the impedance is to be simulated for a limited 

 frequency range, such as the voice band, the selection of the best network is 

 reduced to sorting through a relatively small range of networks to select 

 that one which is the best compromise for the given conditions. This then 

 is a restatement of the problem: To find the network having the minimum 

 number of circuit elements which will give the desired approximation to a speci- 

 fied impedance function. 



The other sections of Fig. 1 will be evident upon analysis. 



Method of Solution 



The first step to be followed in finding the solution to a given problem 

 is to plot in the complex plane the locus traced by the given impedance 

 function as the frequency varies over the range which is to be considered 

 and to mark the frequency at those impedances which are essential to the 

 problem. Having done this, the next step is to draw a semicircle with the 

 center on the real axis such that an arc of the semicircle approximates part 

 or all of the locus of the impedance function. In many cases this semi- 

 circle is a sufi&ciently good approximation but where it is not, it will be 

 necessary to add other functions. The examples given below are illustra- 

 tive of cases requiring three-, four- and five-element networks, 



