IRREGULARITIES IN WIRE TRANSMISSION CIRCUITS 545 



the same standard deviation also holds for the envelope delay depar- 

 tures over a range of frequencies on one line. 



Let Fig. 1 represent a line of the type we have been discussing. 

 The successive t/'s represent the reflection coefficients between succes- 

 sive elementary pieces of line. As mentioned before, to avoid correla- 

 tion, each 77 is broken up as shown into two h's, representing reflections 

 between the elementary pieces and infinitesimal lengths of average 

 line. 



The main signal transmission will flow as shown by the arrow a in 

 Fig. 1. In addition there will be single reflections as shown by the 

 arrow h. Following the assumptions we have set up, this really con- 

 sists of two reflections from infinitesimally separated points. Further 

 there will be double reflections, that is reflections of reflections, as 

 shown by c. Here again each reflection point, according to our assump- 

 tions, consists of two infinitesimally separated ones. There will be a 

 variety of double reflections according to the number of elementary 

 lengths between reflection points. Finally there will be triple, quad- 

 ruple and higher order reflections which are not shown. The wave 

 amplitude after reflection is cut down by the reflection coefficient. 

 Consequently, even though there are more of them, the total of any 

 given higher order reflections can always be made smaller than that of 

 lower order reflections by a small enough reflection coefficient. We 

 will here study only small reflection coefficients and therefore neglect 

 all reflections of higher order than needed to give a finite result. For 

 effects on the impedance this means neglect of all but. first-order reflec- 

 tions. For the other effects studied it means neglect of all but first- and 

 second-order reflections. 



The reflection coefficient between two successive impedances (one 

 being K), is, approximately 



h = AKJilK). (2) 



Following our earlier assumptions, namely that the principal cause of 

 impedance departures lies in geometrical irregularities, and that these 

 may be expressed in terms of capacitance departures, 



^ = ^, or h = ^, or Vp=6/2. (3) 



K C IC 



Consequently the reflection coefficients are real, namely, they intro- 

 duce no phase shifts other than or tt in the reflections. 



The irregularities in sending-end impedance have been computed in 

 Appendix I from the single reflections of the type h in Fig. 1, The 



