CHAPTER I INTRODUCTION 1. In the practical applications of electrical energy, we meet with two different classes of phenomena, due respectively to the continuous current and to the alternating current. The continuous-current phenomena have been brought within the realm of exact analytical calculation by a few fundamental laws : c 1. Ohm's law: i = -, where r, the resistance, is a constant r of the circuit. 2. Joule's law: P = ^^r, where P is the power, or the rate at which energy is expended by the current, i, in the resistance, r. 3. The power equation: Po = ei, where Po is the power expended in the circuit of e.m.f., e, and current, i. 4. Kirchhoff's laws: (a) The sum of all the e.m.fs. in a closed circuit = 0, if the e.m.f. consumed by the resistance, ir, is also considered as a counter e.m.f., and all the e.m.fs. are taken in their proper direction. (b) The sum of all the currents directed toward a distributing point = 0. In alternating-current circuits, that is, in circuits in which the currents rapidly and periodically change their direction, these laws cease to hold. Energy is expended, not only in the con- ductor through its ohmic resistance, but also outside of it ; energy is stored up and returned, so that large currents may exist simultaneously with high e.m.fs., without representing any considerable amount of expended energy, but merely a surging fo and fro of energy; the ohmic resistance ceases to be the deter- 1 2 ALTERNATING-CURRENT PHENOMENA mining factor of current value; currents may divide into com- ponents, each of which is larger than the undivided current, etc. 2. In phice of the above-mentioned fundamental laws of continuous currents, we find in alternating-current circuits the following: Ohm's law assumes the form i = -, where z, the apparent resistance, or impedance, is no longer a constant of the circuit, but depends upon the frequency of the currents; and in circuits containing iron, etc., also upon the e.m.f. Impedance, z, is, in the system of absolute units, of the same dimension as resistance (that is, of the dimension lt~^ = velocity), and is expressed in ohms. It consists of two components, the resistance, r, and the reactance, x, or z — \/r" + X". The resistance, r, in circuits where energy is expended only in heating the conductor, is the same as the ohmic resistance of continuous-current circuits. In circuits, however, where energy is also expended outside of the conductor by magnetic hysteresis, mutual inductance, dielectric hysteresis, etc., r is larger than the true ohmic resistance of the conductor, since it refers to the total expenditure of energy. It may be called then the effective re- sistance. It may no longer be a constant of the circuit. The reactance, x, does not represent the expenditure of energy as does the effective resistance, r, but merelj^ the surging to and fro of energy. It is not a constant of the circuit, but depends upon the frequency, and frequently, as in circuits containing iron, or in electrolytic conductors, upon the e.m.f. also. Hence while the effective resistance, r, refers to the power or active component of e.m.f., or the e.m.f. in phase with the current, the re- actance, X, refers to the wattless or reactive component of e.m.f., or the e.m.f. in quadrature with the current. 3. The principal sources of reactance are electromagnetism and capacity. Electromagnetism An electric current, i, in a circuit produces a magnetic flux surrounding the conductor in lines of magnetic force (or more correctly, lines of magnetic induction), of closed, circular, or other form, which alternate with the alternations of the current, INTRODUCTION 3 and thereby generate an e.m.f. in the conductor. Since the magnetic flux is in phase with the current, and the generated e.m.f. 90°, or a quarter period, behind the flux, this e.m.f. of self-induction lags 90°, or a quarter period, behind the current; that is, is in quadrature therewith, and therefore wattless. If now $ = the magnetic flux produced by, and interlinked with, the current, i (where those lines of magnetic force which are interlinked ?i-fold, or pass around n turns of the conductor, are counted n times), the ratio, —, is denoted by L, and called the inductance of the circuit. It is numerically equal, in absolute units, to the interlinkages of the circuit with the magnetic flux produced by unit current, and is, in the system of abso- lute units, of the dimension of length. Instead of the inductance, L, sometimes its ratio with the ohmic resistance, r, is used, and is called the time-constant of the circuit, r If a conductor surrounds with n turns a magnetic circuit of reluctance, (R, the current, i, in the conductor represents the m.m.f. of ni ampere-turns, and hence produces a magnetic flux m of — lines of magnetic force, surrounding each n turns of the n 1/ conductor, and thereby giving ^ = ~^ interlinkages between Oi. the magnetic and electric circuits. Hence the inductance is _ $ _ n^ ~ i ~ (R ' The fundamental law of electromagnetic induction is, that the e.m.f. generated in a conductor by a magnetic field is pro- portional to the rate of cutting of the conductor through the magnetic field. Hence, if i is the current and L is the inductance of a cir- cuit, the magnetic flux interlinked with a circuit of current, i, is Li, and 4/L* is consequently the average rate of cutting; that is, the number of lines of force cut by the conductor per second, where / = frequency, or number of complete periods (double reversals) of the current per second, i = maximum value of current. Since the maximum rate of cutting bears to the average rate the same ratio as the quadrant to the radius of a circle (a sinu- 4 ALTERNATING-CURRENT PHENOMENA soidal variation supposed), that is, the ratio ^ -^ 1, the maxi- mum rate of cutting is 2 7r/, and, consequently, the maximum vahic of c.m.f. generated in a circuit of maximum current value, i, and inductance, L, is e = 2TrfLi. Since the maximum values of sine waves are proportional (by factor V^) to the effective values (square root of mean squares), if i = effective value of alternating current, e = 2irfLi is the g effective value of e.m.f. of self-induction, and the ratio, -. — 2 tt/L, is the inductive reactance, Xm = 2 7r/L. Thus, if r = resistance, Xm = reactance, z = impedance, the e.m.f. consumed by resistance is ei = ir; the e.m.f. consumed by reactance is 62 = iXm', and, since both e.m.fs. are in quadrature to each other, the total e.m.f. is e = Ver + 62^ = i Vr^ + x„,^ = iz; that is, the impedance, z, takes in alternating-current circuits the place of the resistance, r, in continuous-current circuits. Capacity 4. If upon a condenser of capacity C an e.m.f., e, is impressed, the condenser receives the electrostatic charge, Ce. If the e.m.f., e, alternates with the frequency, /, the average rate of charge and discharge is 4 /, and 2 irf the maximum rate of charge and discharge, sinusoidal waves supposed; hence, i = 2 irfCe, the current to the condenser, which is in quadrature to the e.m.f. and leading. It is then _e 1__ ^' ~ i ~ 2 7r/C' the " condensive reactance.^' Polarization in electrolytic conductors acts to a certain extent like capacity. The condensive reactance is inversely proportional to the frequency and represents the leading out-of-phase wave; the inductive reactance is directly proportional to the frequency, and represents the lagging out-of-phase wave. Hence both are INTRODUCTION 5 of opposite sign with regard to each other, and the total react- ance of the circuit is their difference, x = Xm — Xc. The total resistance of a circuit is equal to the sum of all the resistances connected in series; the total reactance of a circuit is equal to the algebraic sum of all the reactances connected in series; the total impedance of a circuit, however, is not equal to the sum of all the individual impedances, but in general less, and is the resultant of the total resistance and the total reactance. Hence it is not permissible directly to add impedances, as it is with resistances or reactances. A further discussion of these quantities will be found in the later chapters. 5. In Joule's law, P = i^r, r is not the true ohmic resistance, but the "effective resistance;" that is, the ratio of the power component of e.m.f. to the current. Since in alternating-cur- rent circuits, in addition to the energy expended iii the ohmic re- sistance of the conductor, energy is expended, partly outside, partly inside of the conductor, by magnetic hysteresis, mutual induction, dielectric hysteresis, etc., the effective resistance, r, is in general larger than the true resistance of the conductor, sometimes many time larger, as in transformers at open sec- ondary circuit, and is no longer a constant of the circuit. It is more fully discussed in Chapter VIII. In alternating-current circuits the power equation contains a third term, which, in sine waves, is the cosine of the angle of the difference of phase between e.m.f. and current: Po = ei cos d. Consequently, even if e and i are both large, Po may be very small, if cos d is small, that is, 6 near 90°. Kirchhoff's laws become meaningless in their original form, since these laws consider the e.m.fs. and currents as directional quantities, counted positive in the one, negative in the opposite direction, while the alternating current has no definite direction of its own. 6. The alternating waves may have widely different shapes; some of the more frequent ones are shown in a later chapter. The simplest form, however, is the sine wave, shown in Fig. 1, or, at least, a wave very near sine shape, which may be repre- sented analytically by 2ir 2 = / sin -— {t — ti) = / sin 2Trf{t — ti), 10 6 ALTERNATING-CURRENT PHENOMENA where / is the maximum value of the wave, or its amplitude; to is the time of one complete cyclic repetition, or the period of the wave, or / = 7 is the frequency or number of complete to periods per second; and ^1 is the time, where the wave is zero, or the epoch of the wave, generally called the phaseA — 1 r / 1 \ ■ / \, / I \ / \ 1 1 1 \ / \ / 0 1— \ wJ 1 f «=\ / \ \, y \ s y \ ^ .. Fig. 1. — Sine wave. Obviously, "phase" or "epoch" attains a practical meaning only when several waves of different phases are considered, as "difference of phase." When dealing with one wave only, we may count the time from the moment when the wave is zero, or from the moment of its maximum, representing it respec- tively by i = 1 sin 2 -wjl, and i = I cos 2 tt/^. Since it is^ univalent function of time, that is, can at a given instant have one value onlj^, by Fourier's theorem, any alter- nating wave, no matter what its shape may be, can be represented by a series of sine functions of different frequencies and different phases, in the form i = /1 sin 2-KJ{t — ti) + h sin 4 7r/(i — ^2) + 73 sin 6 irfit - /a) + . . . where h., 1 2, h, . . . are the maximum values of the different components of the wave, ti, t-i, tz . . . the times, where the respective components pass the zero value. ^ "Epoch" is the time where a periodic function reaches a certain value, for instance, zero; and "phase" is the angular position, with respect to a datum position, of a periodic function at a given time. Both are in alter- nate-current phenomena only different ways of expressing the same thing. INTRODUCTION 7 The first term, 7i sin 2Trf{t — t^), is called the fundamental wave, or the first harmonic; the further terms are called the higher harmonics, or "overtones," in analogy to the overtones of sound waves. /„ sin 2mrf{t — tn) is the n^^ harmonic. By resolving the sine functions of the time differences, 2! — ti, t — ti . . ., we reduce the general expression of the wave to the form: i = Ai sin 2 tt/^ + A2 sin 4 x// + A3 sin 6 tt/^ + . . . + Bi cos 2 irft + B2 cos 4 Tft + ^3 cos Qirft + . . . The two half-waves of each period, the positive wave and the negative wave (counting in a definite direction in the circuit), are usually identical, because, for reasons inherent in their construc- tion, practically all alternating-current machines generate e.m.fs. in which the negative half-wave is identical with the positive. Hence the even higher harmonics, which cause a difference in the shape of the two half-waves, disappear, and only the odd harmonics exist, except in very special cases. Hence the general alternating-current wave is expressed by: i = 7] sin 2 Trf(t — ti) + 1 3 sin 6 Trf(t — ts) -\- hsin lOirfit -U) -{- . . . or, i = A\ sin 2Tvft + A3 sin 6x/f + ^5 sin lOvr/^ + . . . + Ex cos 2 irjt + Bz cos 6 irft + B5 COS 10 tt/^ + . . . I I I I I I I I I I I ) i Fig. 2. — Wave without even harmonics. Such a wave is shown in Fig. 2, while Fig. 3 shows a wave whose half-waves are different. Figs. 2 and 3 represent the sec- ondary currents of a Ruhmkorff coil, whose secondary coil is closed by a high external resistance; Fig. 3 is the coil operated in the usual way, by make and break of the primary battery 8 ALTERNATING-CURRENT PHENOMENA current; Fig. 2 is the coil fed with reversed currents by a com- mutator from a battery, 7. Inductive reactance, or electromagnetic momentum, which is always present in alternating-current circuits — to a large ex- tent in generators, transformers, etc. — tends to suppress the higher harmonics of a complex harmonic wave more than the Fig. 3. — Wave with even harmonics. fundamental harmonic, since the inductive reactance is pro- portional to the frequency, and is thus greater with the higher harmonics, and thereby causes a general tendency toward simple sine shape, which has the effect that, in general, the alternating currents in our light and power circuits are sufficiently near sine waves to make the assumption of sine shape permissible. Hence, in the calculation of alternating-current phenomena, we can safely assume the alternating wave as a sine wave, with- out making any serious error; and it will be sufficient to keep the distortion from sine shape in mind as a possible disturbing factor, which, however, is in practice generally negligible — except in the case of low-resistance circuits containing large inductive reactance and large condensive reactance in series with each other, so as to produce resonance effects of these higher harmonics, and also under certain conditions of long-distance power transmission and high-potential distribution. 8. Experimentally, the impedance, effective resistance, induc- tance, capacity, etc., of a circuit or a part of a circuit are con- veniently determined by impressing a sine wave of alternating e.m.f. upon the circuit and measuring with alternating-current INTRODUCTION 9 ammeter, voltmeter and wattmeter the current, ^, in the circuit, the potential difference, e, across the circuit, and the power, p, consumed in the circuit. Then, e The impedance, z = -.; The phase angle, cos 6 = —.: ^ ° ei P The effective resistance, r = — . i- From these equations, The reactance, x = s/z'^ — r-. If the reactance is inductive, the inductance is If the reactance is condensive, the capacity or its equivalent is ^ 2 7rfx wherein/ = the frequency of the impressed e.m.f. If the react- ance is the resultant of inductive and condensive reactances connected in series, it is L and C can be found by measuring the reactance at two different frequencies, /i and fo, as follows; Xr =27r/iL-2^. then, J ^ X^fl — X2/2 ■ 2 7r(/,^-/2^)' and c = 27r/i/o(a:i/2- 0:2/1) A moderate deviation of the wave of alternating impressed e.m.f. from sine shape does not cause any serious error as long as the circuit contains no capacity. In the presence of capacity, however, even a very slight dis- tortion of wave shape may cause an error of some hundred per cent. 10 ALTERNATING-CURRENT PHENOMENA To measure capacity and condensiye reactance by ordinary alternating currents it is, therefore, advisable to insert in series with the condensive reactance a non-inductive resistance or induc- tive reactance which is larger than the condensive reactance, or to use a source of alternating current, in which the higher har- monics are suppressed, as the ^-connection of Constant Potential — Constant-current Transformation, paragraph 64. In iron-clad inductive reactances, or reactances containing iron in the magnetic circuit, the reactance varies with the magnetic induction in the iron, and thereby with the current and the im- pressed e.m.f. Therefore the impressed e.m.f. or the magnetic induction must be given, to which the ohmic reactance refers, or preferably a curve is plotted from test (or calculation), giving the ohmic reactance, or, as usually done, the impressed e.m.f. as function of the current. Such a curve is called an excitation curve or impedance curve, and has the general character of the magnetic characteristic. The same also applies to electrolytic reactances, etc. The calculation of an inductive reactance is accomplished by calculating the magnetic circuit, that is, determining the ampere- turns m.m.f. required to send the magnetic flux through the magnetic reluctance. In the air part of the magnetic circuit, unit permeabihty (or, referred to ampere-turns as m.m.f., reluc- tivity 4—) is used; for the iron part, the ampere-turns are taken from the curve of the magnetic characteristic, as discussed in the following.