CHAPTER IX. THE USE OF SINE CURVES IN ALTERNATING-CURRENT PROBLEMS. EFFECT OF HIGHER HARMONICS. 50. The representation of alternating currents and E.M.F.s by sine, or simple periodic, curves is frequently objected to on the ground that they do not accurately represent the actual variations of the current or E.M.F., as the case may be, and consequently cannot lead to accurate results. Let us examine carefully the value of this objection, and ascertain whether we may expect to obtain true results when we assume that alternating currents and E.M.F.s may be expressed as sine functions of the time. It will readily be granted that all alternating currents and E.M.F.s are periodic; that is, that they are all of such a nature that there is a certain time, T, called the periodic time, in which their values go through a complete cycle of changes, and that in each succeeding time T this cycle is repeated. It is quite correct, then, to represent any alternating current or E.M.F. whatever by some periodic function of the time. Now, by a theorem due to Fourier, any periodic function of the time of frequency n may be represented by an expression of the form a1 sin (pt) + a2 sin (2pt — 02) + as sin (3pt - 03) + ... etc. 2πn, and a1, a2, etc., are the amplitudes, and 01, 02, etc., where The frequency of the first term is n, and those of the other terms are respectively 2n, 3n, etc. These terms are called the first, second, etc., harmonics of the first term, which is itself called the fundamental term. Fig. 23 shows a curve C compounded of the fundamental and the first harmonic; in Fig. 24 the curve C is compounded of the fundamental and the second harmonic; in Fig. 25 the curve C is compounded of the fundamental and the third harmonic; in Fig. 26 the curve C is compounded of the fundamental and the first and second harmonics; in Fig. 27 the first and third harmonics are present; in Fig. 28 the second and third harmonics are present; whilst in Fig. 29 the curve C is compounded of the fundamental and the first, second, and third harmonics. In each case A is the fundamental curve, az, as, a4, the first, second, and third harmonics respectively, and C the resultant curve. The first, third, etc., harmonics are called the even harmonics, because their periodic times are even submultiples of that of the fundamental. The second, fourth, etc., harmonics are, for like reason, called the odd harmonics. The reader should practise drawing curves compounded of the fundamental and harmonics, as they are very instructive. 51. It is a matter of experience that the even harmonics are generally absent from the curves representing alternating currents and E.M.F.s, so that we may legitimately represent them by expressions of the form a1 sin (pt - 01) + a3 sin (3pt — 03) + a5 sin (5pt — 05) + . . . etc. It may be that, in a transformer or other induction machine, either the current or E.M.F. follows the simple sine law. The presence of iron, however, causes one of them to deviate from the sine law. If a sine potential difference is applied between the terminals of the primary of a transformer, the primary-current wave will be compounded of a sine function and some of its higher odd harmonics, and will consequently be distorted. If a sine current is produced, the applied potential difference is distorted. 52. Root Mean Square Value of a Curve compounded of a Fundamental Sine Curve and its Higher Odd Harmonics. that is, the mean square value of y is the sum of a series of terms, such as together with the sum of a series of terms, such as |