SIXTH LECTURE HIGHER HARMONICS OF THE GENERATOR WAVE mHE open circuit reactance of the transformer is the only reactance high enough to give resonance with the line capacity at fundamental frequency. All other reactances are too low for this. Since, however, the inductive reactance increases and the capacity reactance decreases proportionally to the frequency, the two reactances come nearer together for higher frequency; that is, for the higher harmonics of the generator wave, and for some of the higher harmonics of the generator wave resonance rise of voltage so may occur between the line capacity and the circuit inductance. The origin and existence of higher harmonics therefore bears investigation in transformers, transmlission lines and cable systems. ORIGIN OF HIGHER HARMONICS Higher harmonics may originate in synchronous machines, as generators, synchronous motors and converters, and in transformers. These two classes of higher harmonics are very different. The former have constant potential character; the latter, con- stant current character; their cure and prevention there- fore must be different, and the method of elimination of one may be very harmful with the other type of harmonics. For instance, the voltage produced by a constant current harmonic as coming from a transformer is eliminated by short circuit Short circuiting a generator harmonic, however, gives large 8o GENERAL LECTURES short circuit currents, due to the constant potential character, and is therefore dangerous. HIGHER HARMONICS OF SYNCHRONOUS MACHINES In synchronous machines, as alternating current genera- tors, the higher harmonics are : At No Load I St. The distribution of magnetism in the air gap depends on the shape of the field poles; it is not a sine wave; neither is the e. m. f . induced by it in an armature a sine wave. Since there are a number of conductors in series on the armature, the voltage wave is more evened out than that of a single conductor ; but still it is not a sine wave, that is, contains harmonics of which the third is the lowest. 2nd. The change of magnetic flux by the passage of open armature slots over the field pole produces harmonics of e. m. f . ; that is, when a large open armature slot stands in front of the field pole, the magnetic reluctance is high ; the magnetism is lower than when no slot is in front of the field pole ; that is, by the passage of the armature slots the field magnetism pul- sates, the more so the larger the slots and the fewer they are. If there are n slots per pole, this produces the two har- monics 2n — I and 2n + i. At Load 3rd. The armature reaction of a single-phase machine pulsates between zero at zero current and a maximum at maxi- mum current. The resultant armature reaction of a pol3rphase machine is constant, but locally there is a pulsation making as many cycles per pole as there are phases. HARMONICS OF GENERATOR WAVE 8i Since the field magnetism under load is due to the com- bination of field excitation and armature reaction, the pulsa- tion of armature reaction therefore causes a pulsation of field magnetism, and thereby higher harmonics of the e. m. f . wave. If m == number of phases, the higher harmonics : 2m — i and 2m + i are produced. 4th. The terminal voltage under load is the resultant of the induced e. m. f, and the e. m. f. consumed by the reactance of the armature circuit ; that is, the reactance produced by the magnetic flux produced by the armature current in the arma- ture iron. This armature reactance is not constant, but peri- odically varies, more or less, with double frequency; that is, when the armature coil is in front of the field pole its magnetic circuit is different than when it is between the field poles, and the reactance therefore is different. This pulsation of armature reactance produces the third harmonic, since it is of double frequency. The most common and prominent harmonic so is the third harmonic in a synchronous machine. These harmonics of synchronous machines are induced e. m. f's, that is, constant potential or approximately so. HIGHER HARMONICS OF TRANSFORMERS In a transformer the wave of e. m. f. depends on that of the magnetism and vice versa. That is, with a sine wave of e. m. f., the magnetism must also be a sine wave, and if the magnetism is not a sine wave, but contains higher harmonics, the e. m. f. is not a sine wave, but contains the harmonics induced by the harmonics of magnetism. The exciting current of the transformer depends on tht magnetism by the hysteresis cycle; if the magnetism is a sine wave, the exciting current therefore cannot be a sine wave, but 82 GENERAL LECTURES must contain higher harmonics- — mainly the third harmonic, which reaches 20 to 30% of the fundamental, or even more at saturation. Fig. 23. In a transformer, e. m. f. and exciting current therefore cannot both be sine waves, but a sine wave of e. m. f. requires an exciting current containing a third harmonic; and a sine wave of exciting current in a transformer or reactive coil thus produces a third harmonic of e. m. f. If therefore in a transformer the third harmonic is sup- pressed, and if this third harmonic should have been 20% of the fundamental, then its suppression produces a third har- monic of magnetism of 20% in the opposite direction. A third harmonic of magnetism, however, of 20%, induces a third harmonic of e. m. f . of 3 x 20 = 60% ; the e. m. f. being proportional to magnetism and frequency. HARMONICS OF GENERATOR WAVE 83 The third harmonic of exciting current is positive at the maximum of magnetism, and the third harmonic of magnetism is negative at the maximum, hence is zero and rising at the zero of the magnetism ; and at this moment the e. m. f . induced by the third harmonic and by the fundamental therefore are both maxima and in the same direction, that is, add. The suppres- sion of the third harmonic of exciting current thus produces a very high third harmonic of e. m. f., which greatly increases the maximum e. m. f. ; that is, the e. m. f. wave is very low for a large part of the cycle and then rises to a very high peak, as shown by Fig. 23 ; and the maximum e. m. f . may exceed that of a sine wave by 50% and more, thus giving high insulation stress and the possibility of resonance voltages. EFFECTS OF HIGHER HARMONICS In a three-phase system the three phases are 120° apart, and their third harmonics are 3 x 120° = 360° apart, that is, in phase with each, and for the third harmonic the three-phase system therefore is a single-phase system. In a balanced three-phase system, the third harmonics can not exist in the voltages between the lines and in the line currents, if there is no return over the neuitral. The three voltages between lines, from i to 2, 2 to 3, and 3 to i, must add up to zero; but since the third harmonics would be in phase with each other, they would not add up to zero, therefore they cannot exist. The three currents, if there is no return over the neutral or the ground, must add up to zero; and since their third harmonics must be in phase with each other, they must be absent. In a balanced three-phase system, third harmonics can exist only in the voltage from line to neutral or Y voltage, in the current from line to line or delta current, and in the 84 GENERAL LECTURES line current only if there is a neutral return or ground return to the generator neutral or transformer neutral. In a three-phase generator, if the e. m. f . of one phase con- tains a third harmonic, as is usually the case, then by connect- ing the three phases in delta connection, the third harmonics of the generator e. m. f.'s are short circuited and so produce a triple frequency current circulating in the generator delta. This triple frequency circulating current can be measured by connecting an ammeter in one corner of the generator delta, and the sum of voltages of the three third harmonics can be measured by putting a voltmeter in a corner of the generator delta. This local current in the generator winding is the triple frequency voltage divided by the generator impedance (the stationary impedance, at triple frequency, but not the syn- chronous impedance, since the latter includes armature reac- tion). In generators of low impedance or close regulation, as turbine alternators, this local current may be far more than full load current ; delta connection of generator windings there- fore is unsafe. As a result, generator windings are almost always connected in Y, Even with delta connection of gener- ator windings no triple frequency appears at the terminals, since its voltage disappears by short circuit. If the generator winding is connected in Y, the triple frequency voltages from terminal to neutral are in phase with each other; that is, in a three-phase Y connected generator, a single-phase voltage of triple frequency exists between the neutral and all three terminals, and the neutral therefore is not a true neutral. Between the lines no triple frequency voltage exists, since from terminal to neutral and from neutral to the other terminal the two third harmonics are in opposition and so neutralize. i ' I I ^ HARMONICS OF GENERATOR WAVE 85 This third harmonic between generator neutral and line must be kept in mind, since when large it may produce danger- ous voltages by resonance with the line capacity. When the generator neutral is grounded, the potential difference from line to ground is not line voltage divided by V3, that is, the true Y voltage of the system ; but superimposed upon it is this single-phase triple frequency voltage; and the voltage from line to ground, especially its maximum, may be greatly increased, thus increasing the insulation strain. For this single-phase voltage all three lines go together, and so may cause static induction on other circuits, as telephone lines. A circuit of this single-phase triple frequency voltage then exists frcmi the generator neutral over the inductance of all three generator circuits in multiple, and over the capacity of all three lines to ground, back to the generator neutral ; that is, we have capacity and inductance in series in a circuit of the triple har- monic, and if capacity and inductance are high enough, we may get a dangerous voltage rise. In this case of grounded generator neutral, if the neutral of the Y connected step-down transformers is grounded also, and the low tension side of these transformers connected in Y, the third harmonic of the generator has no path; the cur- rent produced by it would have to return over the open circuit reactance of the step-down transformer, and is limited there- by to a negligible value. If, however, the secondaries of the step-down trans- formers are connected in delta, so that the third harmonic can circulate in the secondary delta, the third harmonic can flow through the transformer primary by inducing an opposite cur- rent in the secondary; in this case the step-down trans- former short circuits the third harmonic of the generator. Grounding the primary neutral of step-down transformers 86 GENERAL LECTURES with grounded generator neutral therefore is permissible only if the transformer secondaries are also connected in Y. With delta connected transformer secondaries, however, it is not safe to ground the generator neutral and transformer neutral ; since this produces a triple frequency current in generator, line and transformer ; and even if the generator reactance is so high that the generator is not harmed by this current, it may burn out at the transformer, and probably will do so if the trans- former is small compared with the generator. This therefore is a case where delta connection of the transformer secondaries does not eliminate the trouble from the third harmonic, but makes it worse. The triple frequency voltage from line to ground would be eliminated by short circuiting it in this manner, by Y delta connection of step-down transformer with grounded generator and transformer neutral, and static induction on other circuits so would disappear; but we get magnetic induction from the three triple frequency single-phase currents which now flow over the lines to the ground. If the generator neutral is not gfrounded, it is safe to ground transformer neutrals. With ungrounded generator neutral, a triple frequency voltage can be measured by volt- meter, which then appears between generator neutral and ground; this voltage under unfavorable conditions, may give insulation strains in the generator by resonance rise; in the circuit from generator neutral over triple frequency voltage, generator inductance, capacity from line to ground and capac- ity from ground to generator winding in series. In this case the capacity is much lower and the power therefore much less, that is, less danger exists. When running two or more three-phase generators in parallel, with grounded neutrals : HARMONICS OF GENERATOR WAVE 87 a. If the generators have different third harmonics, these harmonics are short circuited from neutral over generator to the other generator and back to neutral; a triple frequency current thus flows between the generators, that is, the current between the generators can never be made to disappear. That is, for the third harmonic, the two generators are two single-phase machines of different voltage, having the neutral as one terminal and the three three-phase terminals as the other single-phase terminal. b. With two identical generators running in multiple, if the excitation is identically the same, no current flows between the grounded neutrals. If the excitation of the two generators is different, one is over-excited the other is under-excited (that is, one carries leading, the other lagging current) then a triple frequency current flows between the neutrals of identical generators. Since in parallel operation the terminal voltages are in phase, if by difference of excitation the two terminal voltages have a different lag behind the induced e. m. f /s, the third harmonics, which lag three times as much as the funda- mentals, cannot be in phase in the two machines; and thus triple frequency current flows between the machines. In machines of very low reactance as turbo-alternators, even small differences in excitation of identical machines with grounded neutral may thus cause very large neutral currents. In parallel operation of three-phase machines with grounded neutral, machines of different wave shapes frequently cannot be run together at all without excessive neutral currents, and the ground has to be taken off of one of the machine typts. Even with identical machines, such care has to be taken in keeping the same excitation that it is frequently undesirable to ground all the neutrals, but only the neutral of one machine is grounded and the other machine neutrals are left isolated. In 88 GENERAL LECTURES this case, provisions must be made to ground the neutral of some other machine, if the first one is out of service. The best way is, when grounding generator neutrals, to ground through a separate resistance for every generator and to choose this resistance so high as to limit the neutral current, but still low enough so that in case of a ground on one phase, enough current flows over the neutral to open the circuit breaker of the grounded phase. The use of a resistance in the generator neutral is very desirable also, since it eliminates the danger of a high frequency oscillation between line and ground through the generator reactance in the path of the third harmonic, by damping the oscillation in the resistance. For this reason, the resistance should be non-inductive. To ground the gener- ator neutral through a reactance is very dangerous since it intensifies the danger of a resonance voltage rise. In grounding the generator neutral, special care is neces- sary to get perfect contact, since an arc or loose contact would generate a high frequency in the circuit of the third harmonic and so may lead to a higher frequency oscillation between line and ground.