6-A] GENERAL STUDY. 2I 7 



APPENDIX II. 

 TRIPLE HARMONIC IN DELTA AND STAR CONNECTION. 



41. In a circuit supplying current to a transformer, induction 

 motor or similar apparatus with iron, hysteresis in the iron intro- 

 duces* in the exciting current odd harmonics of 3, 5, 7, 9, etc., times 

 the fundamental frequency. 



In a 3-phase system, if three transformers have their primaries 

 either star- or delta-connected, the currents in the three transformers 

 will have a phase difference of one third of the fundamental period. 

 The third harmonic due to hysteresis will accordingly have the same 

 phase in each of the three transformers. This will be seen by 

 sketching curves for the fundamental and third harmonic, and shift- 

 ing the curves to left or right one third of the fundamental period, 

 which is one full period for the third harmonic. In a 3-phase system 

 all harmonics divisible by 3, as the 9th, I5th, etc., will likewise have 

 the same phase in each transformer. 



For a 5- or 7~phase system, the harmonics thus appearing would 

 be 5, 15, 25 and 7, 21, 35, etc., respectively. 



In an even phase (single- or 2-phase) system, even harmonics only 

 could appear; but no even harmonics are produced by hysteresis. 



These facts can be shown by curves taken by the method of instan- 

 taneous contact or the oscillograph. A set of such curves has been 

 published and discussed by E. J. Berg (Electrical Energy, p. 154). 



42. Thirdf Harmonic in Delta-connection. If the transformer 

 primaries are delta-connected, the harmonics due to hysteresis for 

 the three transformers are in phase and form a current which circu- 

 lates around the delta but does not appear in the line. The delta 

 current ID may accordingly be 5 or 10 per cent, more than the line 

 current I, divided by V3- If H is the current (third and higher har- 

 monics) caused by hysteresis, we have:}: ID = V (/ -f- V3) 2 + H 2 . 



* Compare " The Effect of Iron in Distorting Alternating Current Wave 

 Form," A. I. E. E., September, 1906, and its discussion by Steinmetz. 



fThe third harmonic is mentioned, being most important; it will be 

 understood that the ninth, fifteenth, etc., are included when only the third 

 is mentioned. 



$ ( 42a). If A and B are currents or voltages of any two frequencies, 

 the total effective value is ^A* + B 2 . This is easily shown experimentally 



