184 BELL S YS TEM TECH NIC A L JOURNA L 



values of oj and negative real values of p correspond to positive imaginary 

 values of oj. The axes of Fig. 3 have been labelled accordingly. 



It is at this point that the electrostatic analogy begins to come into play. 

 Assume that an infinite wire filament, positively charged throughout its 

 length, is run through the zero /> = — ^„ + ;/a)„ perpendicular to the plane 

 of the paper and that a unit positive charge is placed at an arbitrary point, 

 CO, along the real frequency axis. The component of the force normal to 

 the o) axis exerted on the unit charge may be written in the form 



1 



(3) 



Jn 



J , (w — Oin) 



When distances in equation (3) are identified with frequencies in equation 

 (2), the two expressions are identical. A similar argument applies to the 

 other zero, and also to the two poles provided that the filaments passing 

 through the poles have charges of the opposite polarity. Thus we may say 

 that the network of Fig. 2 will have a delay proportional to that component 

 of the electric field intensity which is normal to the co axis, when a positive 

 filament passes through each zero and a negative filament through each 

 pole. Fig. 4 indicates the character of the delay as a function of frequency. 

 Parenthetically we may note that the component of the field intensity 

 parallel to the w axis is proportional to the derivative of the loss. Since this 

 component is zero, the loss will be constant at all frequencies. In the case 

 of the reactance networks with which we are dealing here, the loss is zero. 



Although the usefulness of the electrostatic analogy lies principally in 

 its application to more complex networks, several conclusions may be 

 drawn from Fig. 4. The right-hand zero and pole, because of their sym- 

 metrical spacing and opposite charges, make equal contributions to the total 

 delay. The same statement holds true for the left-hand zero and pole 

 combination. As the zeros and poles approach the real-frequency axis, 

 the delay peaks become sharper and higher because of the increased local 

 field intensity. The figure also shows that the slope of the delay curve is 

 zero at zero frequency and that, unless lOn is large compared to kn , the delay 

 at zero frequency is of appreciable magnitude. These isolated facts will 

 be exploited later in considering more complex networks. 



Assume, now, a tandem series of sections of the type shown in Fig. 2, 

 in which the zeros and poles are so selected that they are evenly spaced at 

 intervals, a, along straight lines parallel to the real-frequency axis as shown 

 in Fig. 5. It was pointed out by H. W. Bode^ that the resulting field in- 

 tensity may be approximated by distributing the total of the discrete 

 charges on the plates of an equivalent condenser passing through the zeros 

 and poles and extending a distance of a/2 beyond the extreme zeros and 



