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BELL SYSTEM TECHNICAL JOURNAL 



serious a problem as the high-frequency asymptote since it can be 

 placed as far from the band as we need by using large enough elements 

 in the power supply circuits. The superposition of a low-frequency 

 cutofif on the idealized loop gain and phase characteristics of a "low- 

 pass" circuit is illustrated by the broken lines in Fig. 20. 



fo 





Fig. 20- 



-Modification of loop characteristics to provide a lower cutoff in a broad- 

 band amplifier. 



If the band width is relatively narrow it is more efficient to use the 

 transformation in filter theory which relates a low-pass to a symmetrical 

 band-pass structure. The transformation is obtained by replacing 

 ///o in the low-pass case by (P - f 1/2/ f(f 2 -/i)), where /i and/2 are 

 the edges of the prescribed band. It substitutes resonant and anti- 

 resonant circuits tuned to the center of the band for the coils and con- 

 densers in the low-pass circuit. In particular each parasitic inductance 

 is tuned by the addition of a series condenser and each parasitic capac- 

 ity is tuned by a shunt coil. The parameters of the transformation 

 must, of course, be so chosen that the parasitic elements have the cor- 

 rect values for use in the new branches. . 



This leads to a simple but important result. If the inductance of a 

 series resonant circuit is fixed, the interval represented by /& — /„ in 

 Fig. 21, between the frequencies at which the absolute value of the 

 reactance reaches some prescribed limit Xo, is always constant and 

 equal to the frequency at which the untuned inductance would exhibit 

 the reactance Xo, whatever the tuning frequency may be. The same 

 relation holds for the capacity in an anti-resonant circuit. Thus the 

 frequency range over which the branches containing parasitic elements 

 exhibit comparable impedance variations is the same in the band-pass 

 structure and in the prototype low-pass structure. But since the 

 transformation does not afi^ect the relative impedance levels of the 

 various branches in the circuit, this result can be extended to the com- 

 plete yujS characteristic. We can therefore conclude that the feedback 

 which is obtainable in an amplifier of given general configuration and with 

 given parasitic elements and given margins depends only upon the breadth 



