DESIGN OF REACTIVE EQUALIZERS 727 





> for CO > 0. This condi- 



network of the tvpe described is that 



tion will be insured if the polynomial, 1 -|- Bicj + • • • + /i„w ", has no 

 negative real X- roots of odd multiplicity.^^ In addition to the sufficiency 



of eq. (7), if the -^ — = fV4 derived from "" in the usual manner 



Ao //(A) 1 Ro 



is to be the transfer impedance of a lossless network terminated in resistance, 

 it is necessary that g{\) be either even or odd and that li(\) be a Hurwitz 

 polynomial.'- In this problem g(X) = 1 is surely even since all zeros of 



-^ — occur at inhnity; and the method of forming — ^ — always insures 



that h{\) = m -\- n, wliere m is the even part and ;/. is the odd part of /;(X), 

 is a Hurwitz polynomial. Thus, the fulfillment of the sufficient condition that 

 there be no negative real X^ roots of odd multiplicity of B(co') is the assurance 

 that the JB's of eq. (7) will always produce a physical network of the con- 

 figuration of Fig. 9. 



Once the conditions for physical realizability have been fulfilled, and a 



ZM 

 R, 



.... . , / m 



elements are easily calculated from a partial traction expansion of Z22 = — 



has been found in the final stages of a particular design, the network 



according to the following relation: 



Zi2(X) _ ZuO^)/Ro _ g(X) _ g(X)/n 



Ro 1 + S22(X)/i?c ni+n I + m/n' 



(9) 



where zU\) = ^^ and zUX) 



n n 



The previous discussion of the special problems of input and output 



coupling circuit design has been based, broadly, on (1) a consideration of 



the terminating or load impedance, (2) a consideration of the shape of the 



transfer characteristic, and (3) a consideration of the conditions for physical 



realizability. A major problem in the design is the choice of an appro.ximat- 



ing function which satisfactorily matches the stated transfer characteristic 



over the useful frequency band and, at the same time, sharply changes slope 



near the cut-off frequency so that it approximates zero outside the useful 



band in a prescribed manner. When the transfer characteristic is a constant 



over the useful frequency band, e.g., the impedance matching and low-pass 



filter cases, techniques which employ TchebychefT [iolynomials as the ap- 



11 Ref . 4. 



'2 A Hurwitz polynomial is defined as a polynomial in X which has the property that the 



quotient of its even and odd parts, <p{\) = — , yields a reactance function. 



