IRREGULARITIES IN WIRE TRANSMISSION CIRCUITS 543 



lion (phase or envelope), which (at the higher frequencies) is inversely 

 proportional to the square root of the product of the capacitance by the 

 inductance. Consequently the portion of the fractional deviation in 

 capacitance which is due to geometrical deviations correlates with an 

 equal and opposite fractional deviation in inductance. Since in prac- 

 tice the contribution from the geometrical deviation is apt to be 

 dominating, that due to the variation in dielectric constant will be 

 neglected and the above correlation assumed as 100 per cent. 



The standard deviation of the capacitance of the successive ele- 

 mentary lengths, as a fraction of the average capacitance, will be 

 designated as 8. 



The secondary constant of the line most affected by these irregulari- 

 ties is the sending end (or similarly receiving end) impedance. If we 

 consider a large ensemble of lines of infinite length of similar manufac- 

 ture (and equal average characteristics and 8) but in which the indi- 

 vidual irregularities are uncorrelated, then the sending end impedances 

 of these lines, measured at a given frequency, also form an ensemble. 

 The standard deviation of the real parts in this latter is ^AKr^, and 

 that of the imaginary parts ^AKi^. 



In general, the departure in the impedance of one individual line 

 from the average will vary with frequency; and perhaps over a moder- 

 ate frequency range a sizeable sample can be collected which is fairly 

 typical of the ensemble of the departures at a fixed frequency in the 

 interval. If this is the case, and if at the same time the average im- 

 pedance varies smoothly and slowly with frequency, and the standard 

 deviation of the ensemble of departures also varies smoothly and 

 slowly with frequency, then the standard deviation of the sample of 

 departures over the moderate frequency interval is substantially equal 

 to that of the ensemble of departures at a fixed frequency in this inter- 

 val. (It is clear that this disregards exceptional lines in the ensemble, 

 characterized by regularity in the array of their capacitance deviations, 

 for which these conditions do not hold.) Under the circumstances 

 where this observation is valid it makes it possible to correlate measure- 

 ments on a single line, provided it is not too exceptional, with theory 

 deduced for an ensemble. 



The irregularities in the transmission line will also affect its attenua- 

 tion. If again we consider an ensemble of lines and measure the at- 

 tenuation of each at a given frequency these attenuations will also 

 form an ensemble. 



It will be found in this case, as will be demonstrated further below, 

 that the average attenuation is a little higher than that of a single 

 completely smooth line having throughout its length a characteristic 



