Triple harmonics in transformers

Abstract

Although the prolbem of higher harmonics in the electrical circuits of transformers is a generation old and the solution has been given by a number of eminent engineers, still hardly a week passes in a department specializing on the manufacture of transformers that an instance does not appear of the lack of comprehension of the relations involved. If the problem were entirely confined to the realms of the manufacturing specialists, it might well be passed over and taken care of by local educational work. But the troubles which arise from time to time in the practise of transmission and distribution of electrical energy on polyphase circuits due entirely to the method of connection of the three phases of polyphase apparatus warrants a review of the subject in a simple form. What are the troubles which may arise in trails mission practise? The harmonics may, first set up potential strains in the transformer coils; second, raise the voltage of the line; third, burn out incandescent lamps; fourth, change the ratio of transformation of voltage under low-load conditions from its proper value as determined by the ratio of the number of turns on the primary coil to the number of turns on the secondary coil; fifth, produce a triple harmonic current in the neutral connection to ground; sixth, induce a distracting hum in telephone receivers connected to parallel telephone circuits; seventh, produce abnormally high voltages and large currents in odd places on the circuits due to a resonance with a natural frequency of the circuit; and so on. To get at the basic effect which causes these triple-harmonic troubles the magnetization current of a single-phase transformer may be considered. If the applied potential at the terminals of the transformer is the familiar smooth sine wave it is well-known that the magnetizing current is not a sine wave but is a smooth distorted wave. On the other hand, if a pure sine wave of current is forced through the primary there appears on both the primary and the secondary a smooth but distorted wave of potential. It must be one or the other. It often helps the understanding and avoids possible confusion to point out a resemblance which is actually a different phenomenon. Instead of finding by oscillographic tests that a current wave is smooth, it may be somewhat saw-toothed. This occurs when a generator supplies current to a condenser, such as an unloaded overhead line or electric cable. A generator usually has either twelve teeth on the armature per pair of poles (that is to say per cycle) or eighteen teeth. If there are twelve teeth, the nearest odd number for the necessarily odd harmonic is either eleven or thirteen. Likewise, if there are eighteen teeth there will be found either seventeen or nineteen saw-teeth or ripples on the main wave. These variations from a sine wave are known as teeth harmonics which entirely distinguish them from the distortions of the sine wave by the effect of variable permeability of the iron. The teeth harmonics, if they exist, are so numerous they can be counted on the oscillographic wave. The permeability harmonics being lower and nearer the first harmonic distort the general shape of the wave without making it visibly evident whether the third, fifth, or seventh is the cause. In passing — the even number of harmonics cannot exist continually in the generator wave of either current or voltage because an even number harmonic would make the positive half of the wave different from the negative half of the wave. The simplest proof, without mathematics, is to draw a sine wave, superpose on it a sine wave of twice the frequency, and combine the two to form a third wave. The distortion of the wave of magnetizing current away from a sine wave may be explained to the mathematically inclined by reference to Fourier's Theorem — a long involved trigonometric equation. It is more evident to put the analysis in words. The distorted current wave is apparently scrambled sine waves. Unscrambled, the elementary sine waves are found in perfect order. There is primarily the first harmonic or fundamental frequency (the generator frequency). This first harmonic is almost as large as the original wave. Since there can be no even harmonics, the next harmonic, somewhat smaller in amplitude, is the third — three times the frequency of the generator. Then comes the fifth harmonic, five times the frequency of the generator and very much smaller than the third. There is also a still smaller yet appreciable seventh harmonic, seven times the generator frequency. But all of this analysis is like describing an egg without saying anything about the bird that produced it. What caused all these sine waves of odd frequencies? It was not the eddy-current in the laminated iron. It was not the hysteresis loss in the iron. It did not come from any electrical effect in either the primary or the secondary electrical circuits. With this narrowing down of the cause, it may be definitely located in an intrinsic characteristic of magnetic iron. At different degrees of magnetizing force (ampere-turns) the molecular magnets add different degrees of magnetism to the iron core. If the distorted wave of magnetism is the egg, this permeability is the hen that produced it. It may also add to clarification to point out a discontinuity or missing link. Magnetization curves are usually given in coordinate graphs with current or ampere turns as the abscissas and magnetism as the ordinates. On the other hand, the magnetizing current of the transformer is usually given with an entirely different factor as abscissas, namely the time, and the ordinates as a near relative to magnetism, namely the voltage. The missing link between these two forms of graphic expression must be supplied. In the following treatment of this subject a coordination of the factors involved is obtained by starting with a fully analyzed wave of magnetization and synthetically constructing, step by step, the shape that the artificial permeability curve of the iron would have to assume to permit, first, the production of the first harmonic sine wave of magnetizing current from an applied sine wave of electromotive force and then the effect of the other harmonics. Fig. 5 shows the first harmonic (the larger wave), and Fig. 5A the corresponding artificial permeability curve, the straight line O M. Next the artificial permeability curve is synthetically constructed on the basis that the first harmonic and the third harmonic of Fig. 5 are combined. They produce the artificial permeability curve O A′ B P of Fig. 5A which is nearer the real permeability curve than a straight line O M. Next, the first harmonic and the fifth harmonic of Fig. 6 are combined to form the permeability curve O A′ B C′ F of Fig. 6A. It is readily seen that this curve does not resemble the well-known permeability curve which is shown in Curve 1 of Fig. 7. However, if in the next step the first, third, and fifth harmonics are combined and a permeability curve is constructed from the three of them, which is Curve 2 of Fig. 7, this curve approaches closely to the real permeability curve (Curve 1.) The difference between this last synthetic permeability curve and the real permeability curve is due to the fact that there is a seventh harmonic of current of small value necessary to correct the difference. Even still smaller values of higher harmonics may be necessary for still greater accuracy. The effects that may be obtained in Y connections, delta connections, and various combinations thereof are discussed in detail. An executive might well sum up the whole subject, without going into detail, by asking an engineer the simple question: Have you given a circuit for the circulation of the third harmonic? If not, can you be sure the third harmonic will not cause some trouble?