56 BELL SYSTEM TECHNICAL JOURNAL 



cal magnitudes of residual voltages in the isolated system of non- 

 triple frequencies. 



The residual voltage due to one of these non-triple frequencies, 

 which is three times the neutral voltage, is ^y' E'oi, according to 

 equation (6). Here y' is the fractional residual admittance and E'oi 

 may be taken as the induced voltage in the transformer for the fre- 

 quency in question. For a harmonic belonging to the triple series, 

 with neutral grounded through the reactor, the absolute value of the 



residual voltage is — ] — ^ (equation (8) ) m being the order of 

 w* — 1 



the harmonic and Eqi the induced voltage, if we assume the trans- 

 former bank provided with a low impedance path for triple harmonics, 

 and therefore neglect the difference between the induced and the 

 terminal voltages. The ratio of the triple-series residual voltage to 



the other is thus, in absolute value, 



(m^- l)|y£o'ir 



If we take the ninth as the harmonic of the triple series and assume 

 equal values of the induced voltages £01 and Fq'i it will be seen that 

 I y' I must be of the order of 0;01 if the residual voltage of the triple 

 harmonic series is to be as large as the other. This amount of un- 

 balance is somewhat larger than has been found at this frequency 

 (540 cycles/sec.) in an actual transposed line.^ If we consider the 

 higher harmonics of the triple series, | y' \ would have to be made 

 progressively smaller in order that the ratio might remain unity. 

 Thus, for the 21st harmonic, | y' \ would have to be of the order of 

 0.002. While, of course, | y' \ may be made as small as desired by 

 sufficiently close power circuit transpositions, it appears that in prac- 

 tical cases where transformer banks have delta windings, one may 

 expect the residual voltages of the triple series, introduced by changing 

 from an isolated system to one grounded through the reactor, to be 

 relatively unimportant except in the case of the third harmonic and 

 perhaps in that of the ninth. This statement would not be true if, 

 as with star-star transformers under some circumstances, no low 

 impedance path is provided for magnetizing currents of the 

 triple harmonic series. Such cases are not common in operating 

 practice. 



The method of estimating comparative effects here applied to the 

 case of triple harmonic residual voltages is not available for residual 

 currents. To take account of the latter in comparing the isolated 



• Inductive Interference between Power and Communication Circuits, California 

 Railroad Commission, Technical Report No. 51. 



