CHAPTER XI. FOUCAULT OR EDDY CURRENTS. 86. While magnetic hysteresis or molecular friction is a magnetic phenomenon, eddy currents are rather an elec- trical phenomenon. When iron passes through a magnetic field, a loss of energy is caused by hysteresis, which loss, however, does not react magnetically upon the field. When cutting an electric conductor, the magnetic field induces a current therein. The M.M.F. of this current reacts upon and affects the magnetic field, more or less ; consequently, an alternating magnetic field cannot penetrate deeply into a solid conductor, but a kind of screening effect is produced, which makes solid masses of iron unsuitable for alternating fields, and necessitates the use of laminated iron or iron wire as the carrier of magnetic flux. Eddy currents are true electric currents, though flowing in minute circuits; and they follow all the laws of electric circuits. Their E.M.F. is proportional to the intensity of magneti- zation, (B, and to the frequency, N. Eddy currents are thus proportional to the magnetization, (B, the frequency, N, and to the electric conductivity, y, of the iron ; hence, can be expressed by The power consumed by eddy currents is proportional to their square, and inversely proportional to the electric con- ductivity, and can be expressed by W= 130 ALTERNATING-CURRENT PHENOMENA. or, since, ($>N is proportional to the induced E.M.F., E, in the equation it follows that, TJie loss of power by eddy currents is propor- tional to the square of the E.M.F., and proportional to tlie electric conductivity of the iron ; or, W=aE*y. Hence, that component of the effective conductance which is due to eddy currents, is that is, The equivalent conductance due to eddy currents in the iron is a constant of the magnetic circuit ; it is indepen- dent of ^M..^., frequency, etc., but proportional to the electric conductivity of the iron, y. 87. Eddy currents, like magnetic hysteresis, cause an advance of phase of the current by an angle of advance, ft ; but, unlike hysteresis, eddy currents in general do not dis- tort the current wave. The angle of advance of phase due to eddy currents is, sin/3 = £, where y = absolute admittance of the circuit, g = eddy current conductance. While the equivalent conductance, g, due to eddy cur- rents, is a constant of the circuit, and independent of E.M.F., frequency, etc., the loss of power by eddy currents is proportional to the square of the E.M.F. of self-induction, and therefore proportional to the square of the frequency and to the square of the magnetization. Only the energy component, g E, of eddy currents, is of interest, since the wattless component is identical with the wattless component of hysteresis, discussed in a preceding chapter. FOUCAULT OR EDDY CURRENTS. 131 88. To calculate the loss of power by eddy currents — Let V = volume of iron ; (B = maximum magnetic induction ; N= frequency; y = electric conductivity of iron ; £ = coefficient of eddy currents. The loss of energy per cm3, in ergs per cycle, is hence, the total loss of power by eddy currents is W = e y VN* (B2 10 - 7 watts, and the equivalent conductance due to eddy currents is o_ W _ IQey/ __ .507ey/ £> Tf"2 O 2 C^/2 C«2 * where : / = length of magnetic circuit, d S — section of magnetic circuit, n = number of turns of electric circuit. The coefficient of eddy currents, e, depends merely upon the shape of the constituent parts of the magnetic cir- cuit ; that is, whether of iron plates or wire, and the thickness of plates or the diameter of wire, etc. x i JC The two most important cases are : (a). Laminated iron. (b). Iron wire. 1 ' 1 89. (a). Laminated Iron. Let, in Fig. 79, i d = thickness of the iron plates ; (B = maximum magnetic induction ; JV = frequency ; y = electric conductivity of the iron. Fi 1.79. 132 ALTERNATING-CURRENT PHENOMENA. Then, if x is the distance of a zone, d x, from the center of the sheet, the conductance of a zone of thickness, */x, and of one cm length and width is y^x ; and the magnetic flux cut by this zone is (Bx. Hence, the E.M.F. induced in this zone is 8 E = V2 TrN($> x, in C.G.S. units. This E.M.F. produces the current : ///=SJ£y y x d x, in C.G.S. units, provided the thickness of the plate is negligible as compared with the length, in order that the current may be assumed as flowing parallel to the sheet, and in opposite directions on opposite sides of the sheet. The power consumed by the induced current in this zone, dx, is dP = §EdI= 2 7T2^2(B2 y x Vx, in C.G.S. units or ergs per second, and, consequently, the total power consumed in one cm2 of the sheet of thickness, d, is = C+* dP = 27rW2(B2y C ° in C.G.S. units; the power consumed per cm3 of iron is, therefore, . / = — = - — '- — , m C.G.S. units or erg-seconds, and the energy consumed per cycle and per cm3 of iron is N 6 The coefficient of eddy currents for laminated iron is, therefore, c = ^- = 1.645 d\ FOUCAULT OR EDDY CURRENTS. 133 where y is expressed in C.G.S. units. Hence, if y is ex- pressed in practical units or 10 ~9 C.G.S. units, c = 7rVn°'- = 1.645 d* 10 - 8 ampere-turns per cm. For example, if d= .1 cm, N = 100, « = 5,000, then /= 1,338 ampere-turns per cm; that is, half as much as in a lamina of the thickness d. 94. Besides the eddy, or Foucault, currents proper, which flow as parasitic circuits in the interior of the iron lamina or wire, under certain circumstances eddy currents also flow in larger orbits from lamina to lamina through the whole magnetic structure. Obviously a calculation of these eddy currents is possible only in a particular structure. They are mostly surface currents, due to short circuits existing between the laminae at the surface of the magnetic structure. Furthermore, eddy currents are induced outside of the magnetic iron circuit proper, by the magnetic stray field cutting electric conductors in the neighborhood, especially when drawn towards them by iron masses behind, in elec- tric conductors passing through the iron of an alternating field, etc. All these phenomena can be calculated only in particular cases, and are of less interest, since they can and should be avoided. FOUCAULT OR EDDY CURRENTS. 139 Eddy Currents in Conductor, and Unequal Current Distribution. 95. If the electric conductor has a considerable size, the alternating magnetic field, in cutting the conductor, may set up differences of potential between the different parts thereof, thus giving rise to local or eddy currents in the copper. This phenomenon can obviously be studied only with reference to a particular case, where the shape of the conductor and the distribution of the magnetic field are known. Only in the case where the magnetic field is produced by the current flowing in the conductor can a general solu- tion be given. The alternating current in the conductor produces a magnetic field, not only outside of the conductor, but inside of it also ; and the lines of magnetic force which close themselves inside of the conductor induce E.M.Fs. in their interior only. Thus the counter E.M.F. of self- inductance is largest at the axis of the conductor, and least at its surface ; consequently, the current density at the surface will be larger than at the axis, or, in extreme cases, the current may not penetrate at all to the center, or a reversed current flow there. Hence it follows that only the exterior part of the conductor may be used for the conduc- tion of the current, thereby causing an increase of the ohmic resistance due to unequal current distribution. The general solution of this problem for round conduc- tors leads to complicated equations, and can be found else- where. In practice, this phenomenon is observed only with very high frequency currents, as lightning discharges ; in power distribution circuits it has to be avoided by either keeping the frequency sufficiently low, or having a shape of con- ductor such that unequal current distribution does not take place, as by using a tubular or a flat conductor, or several conductors in parallel. 140 ALTERNATING-CURRENT PHENOMENA. 96. It will, therefore, be sufficient to determine the largest size of round conductor, or the highest frequency, where this phenomenon is still negligible. In the interior of the conductor, the current density is not only less than at the surface, but the current lags behind the current at the surface, due to the increased effect of self-inductance. This lag of the current causes the magnetic fluxes in the conductor to be out of phase with each other, making their resultant less than their sum, while the lesser current density in the center reduces the total flux inside of the conductor. Thus, by assuming, as a basis for calculation, a uniform current density and no difference of phase between the currents in the different layers of the conductor, the unequal distribution is found larger than it is in reality. Hence this assumption brings us on the safe side, and at the same time simplifies the calculation greatly. Let Fig. 82 represent a cross-section of a conductor of radius R, and a uniform current density, where / = total current in conductor. Fig. 82. The magnetic reluctance of a tubular zone of unit length and thickness dxt of radius x, is FOUCAULT OR EDDY CURRENTS. 141 The current inclosed by this zone is Ix = zW, and there fore, the M.M.F. acting upon this zone is $x = 47r Ix/ 10 = 4 **«»/ 10, and the magnetic flux in this zone is d$> = $x I G(x = 2 Trixdx / 10. Hence, the total magnetic flux inside the conductor is , 27T . CR . TTiR* I From this we get, as the excess of counter E.M.F. at the axis of the conductor over that at the surface — &E = V27r^0> 10 ~8 = V27r7W10 -9, per unit length, and the reactivity, or specific reactance at the center of the conductor, becomes k = &E / i = V2 i^NR* 10 ~9. Let p = resistivity, or specific resistance, of the material of the conductor. We have then, k/p = V^TrW^lO-9/?; and p/ VFT7, the ratio of current densities at center and at periphery. For example, if, in copper, p = 1.7xlO— 6, and the percentage decrease of current density at center shall not exceed 5 per cent, that is — P -H VF+72 = .95 - 1, we have, £ = .51xlO-«; hence .51 x 10-6= V^TrW^lO-9 or N2? = 36.6 ; hence, when N= 125 100 60 25 £ = .541 .605 .781 1.21 cm. D = 1R= 1.08 1.21 1.56 2.42cm. Hence, even at a frequency of 125 cycles, the effect of unequal current distribution is still negligible at one cm diameter of the conductor. Conductors of this size are, however, excluded from use at this frequency by the exter- nal self-induction, which is several times larger than the. 142 ALTERNATING-CURRENT PHENOMENA. resistance. We thus see that unequal current distribution is usually negligible in practice. The above calculation was made under the assumption that the conductor consists of unmagnetic material. If this is not the case, but the con- ductor of iron of permeability p., then ; d$ = pffx / (&x and thus ultimately ; k = V2 wW/^10 ~" and ; k / P = V2 ** NpR* 10— '//»• Thus, for instance, for iron wire at /> = 10xlO-6, ft = 500 it is, permitting 5% difference between center and outside of wire; k = 3.2 X 10 ~6 and NR* = .46, hence when, N = 125 100 60 25 X = .061 .068 .088 .136 cm. thus the effect is noticeable even with relatively small iron wire. Mutual Inductance. 97. When an alternating magnetic field of force includes a secondary electric conductor, it induces therein an E.M.F. which produces a current, and thereby consumes energy if the circuit of the secondary conductor is closed. A particular case of such induced secondary currents are the eddy or Foucault currents previously discussed. Another important case is the induction of secondary E.M.Fs. in neighboring circuits ; that is, the interference of circuits running parallel with each other. In general, it is preferable to consider this phenomenon of mutual inductance as not merely producing an energy component and a wattless component of E.M.F. in the primary conductor, but to consider explicitly both the sec- ondary and the primary circuit, as will be done in the chapter on the alternating-current transformer. Only in cases where the energy transferred into the secondary circuit constitutes a small part of the total pri- mary energy, as in the discussion of the disturbance caused by one circuit upon a parallel circuit, may the effect on the primary circuit be considered analogously as in the chapter •on eddy currents, by the introduction of an energy com- FOUCAULT OR EDDY CURRENTS. 143 ponent, representing the loss of power, and a wattless component, representing the decrease of self-inductance. Let — x = 2 TT N L = reactance of main circuit ; that is, L = total number of interlinkages with the main conductor, of the lines of magnetic force produced by unit current in that conductor ; .#! = 2-jrNL1 = reactance of secondary circuit ; that is, Ll = total number of interlinkages with the secondary conductor, of the lines of magnetic force produced by unit current in that conductor ; xm = 2 TT N Lm = mutual inductance of circuits ; that is, Lm = total number of interlinkages with the secondary conductor, of the lines of magnetic force produced by unit current in the main conductor, or total number of inter- linkages with the main conductor of the lines of magnetic force produced by unit current in the secondary conductor. Obviously : xm* < xx^* * As coefficient of self-inductance L, L^, the total flux surrounding the conductor is here meant. Usually in the discussion of inductive apparatus, especially of trans- formers, that part of the magnetic flux is derroted self-inductance of the one circuit which surrounds this circuit, but not the other circuit ; that is, which passes between both circuits. Hence, the total self-inductance, L, is in this ease equal to the sum of the self-inductance, Z,j, and the mutual inductance, Lm. The object of this distinction is to separate the wattless part, Z1? of the total self-inductance, L, from that part, Lm, which represents the transfer of E.M.F. into the secondary circuit, since the action of these two components is essentially different. Thus, in alternating-current transformers it is customary — and will be done later in this book — to denote as the self-inductance, Z, of each circuit only that part of the magnetic flux produced by the circuit which passes between both circuits, and thus acts in " choking " only, but not in transform- ing; while the flux surrounding both circuits is called mutual inductance, or useful magnetic flux. With' this denotation, in transformers the mutual inductance, Lm, is usu- ally very much greater than the self-inductances, //, and Z/, while, if the self-inductances, Z and Zj , represent the total flux, their product is larger than the square of the mutual inductance, Lm ; or 144 ALTERNATING— CURRENT PHENOMENA. Let rx = resistance of secondary circuit. Then the im- pedance of secondary circuit is ^i = rv — /*! , zl = V/v + xi2 ; E.M.F. induced in the secondary circuit, £± = jxmf, where / = primary current. Hence, the secondary current is and the E.M.F. induced in the primary circuit by the secon- dary current, 7l is or, expanded, Y zr j~. 2 xm^ JXm 2 _i_ r 2 r2 i JT T^ ^i "l " •* 2 Hence, the E.M.F. consumed thereby effective resistance of mutual inductance ; ^ = effective reactance of mutual inductance. The susceptance of mutual inductance is negative, or of opposite sign from the reactance of self-inductance. Or, Mutual inductance consumes energy and decreases the self- inductance. Dielectric and Electrostatic Phenomena. 98. While magnetic hysteresis and eddy currents can be considered as the energy component of inductance, con- densance has an energy component also, namely, dielectric hysteresis. In an alternating magnetic field, energy is con- sumed in hysteresis due to molecular friction, and similarly, energy is also consumed in an alternating electrostatic field in the dielectric medium, in what is called electrostatic or dielectric hysteresis. FOUCAULT OR EDDY CURRENTS. 145 While the laws of the loss of energy by magnetic hys- teresis are fairly well understood, and the magnitude of the effect known, the phenomenon of dielectric hysteresis is still almost entirely unknown as concerns its laws and the magnitude of the effect. It is quite probable that the loss of power in the dielec- tric in an alternating electrostatic field consists of two dis- tinctly different components, of which the one is directly proportional to the frequency, — analogous to magnetic hysteresis, and thus a constant loss of energy per cycle, independent of the frequency ; while the other component is proportional to the square of the frequency, — analogous to the loss of power by eddy currents in the iron, and thus a loss of energy per cycle proportional to the frequency. The existence of a loss of power in the dielectric, pro- portional to the square of the frequency, I observed some time ago in paraffined paper in a high electrostatic field and at high frequency, by the electro-dynamometer method, and other observers under similar conditions have found the same result. Arno of Turin found at low frequencies and low field strength in a larger number of dielectrics, a loss of energy per cycle independent of the frequency, but proportional to the 1.6th power of the field strength, — that is, following the same law as the magnetic hysteresis, ^ = ^(B'-6. This loss, probably true dielectric static hysteresis, was observed under conditions such that a loss proportional to the square of density and frequency must be small, while at high densities and frequencies, as in condensers, the true dielectric hysteresis may be entirely obscured by a viscous loss, represented by W^ = e7V(B2. 99. If the loss of power by electrostatic hysteresis is proportional to the square of the frequency and of the field intensity, — as it probably nearly is under the working con- 146 AL TERNA TING-CURRENT PHENOMENA. ditions of alternating-current condensers, — then it is pro- portional to the square of the E.M.F., that is, the effective conductance, g, due to dielectric hysteresis is a constant ; and, since the condenser susceptance, — b= b', is a constant also, — unlike the magnetic inductance, — the ratio of con- ductance and susceptance, that is, the angle of difference of phase due to dielectric hysteresis, is a constant. This I found proved by experiment. This would mean that the dielectric hysteretic admittance of a condenser, Y=g+jb=g-jb', where : g = hysteretic conductance, b' = hysteretic suscep- tance ; and the dielectric hysteretic impedance of a con- denser, „ . . . Z = r — jx — r +jxc, where : r = hysteretic resistance, xc — hysteretic condens- ance ; and the angle of dielectric hysteretic lag, tan a = b' / g = xc / r, are constants of the circuit, independent of E.M.F. and frequency. The E.M.F. is obviously inversely propor- tional to the frequency. The true static dielectric hysteresis, observed by Arno as proportional to the 1.6th power of the density, will enter the admittance and the impedance as a term variable and dependent upon E.M.F. and frequency, in the same manner as discussed in the chapter on magnetic hysteresis. To the magnetic hysteresis corresponds, in the electro- static field, the static component of dielectric hysteresis, following, probably, the same law of 1.6th power. To the eddy currents in the iron corresponds, in the electrostatic field, the viscous component of dielectric hys- teresis, following the square law. As a rule however, these hysteresis losses in the alter- nating electrostatic field of a condenser are very much smaller than the losses in an alternating magnetic field, so that while the latter exert a very marked effect on the de- sign of apparatus, representing frequently the largest of all the losses of energy, the dielectric losses are so small as to be very difficult to observe. FOUCAULT OR EDDY CURRENTS. 147 To the phenomenon of mutual inductance corresponds, in the electrostatic field, the electrostatic induction, or in- fluence. 100. The alternating electrostatic field of force of an electric circuit induces, in conductors within the field of force, electrostatic charges by what is called electrostatic influence. These charges are proportional to the field strength ; that is, to the E.M.F. in the main circuit. If a flow of current is produced by the induced charges, energy is consumed proportional to the square of the charge ; that is, to the square of the E.M.F. These induced charges, reacting upon the main conduc- tor, influence therein charges of equal but opposite phase, and hence lagging behind the main E.M.F. by the angle of lag between induced charge and inducing field. They require the expenditure of a charging current in the main conductor in quadrature with the induced charge thereon ; that is, nearly in quadrature with the E.M.F., and hence consisting of an energy component in phase with the E.M.F. — representing the power consumed by electrostatic influence — and a wattless component, which increases the capacity of the conductor, or, in other words, reduces its capacity reactance, or condensance. Thus, the electrostatic influence introduces an effective conductance, g, and an effective susceptance, b, — of the same sign with condenser susceptance, — into the equations of the electric circuit. While theoretically g and b should be constants of the circuit, frequently they are very far from such, due to disruptive phenomena beginning to appear at high electro- static stresses. Even the capacity condensance changes at very high potentials ; escape of electricity into the air and over the surfaces of the supporting insulators by brush discharge or electrostatic glow takes place. As far as this electrostatic 148 ALTERNATING-CURRENT PHENOMENA corona reaches, the space is in electric connection with the conductor, and thus the capacity of the circuit is deter- mined, not by the surface of the metallic conductor, but by the exterior surface of the electrostatic glow surround- ing the conductor. This means that with increasing po- tential, the capacity increases as soon as the electrostatic corona appears ; hence, the condensance decreases, and at the same time an energy component appears, representing the loss of power in the corona. This phenomenon thus shows some analogy with the de- crease of magnetic inductance due to saturation. At moderate potentials, the condensance due to capacity can be considered as a constant, consisting of a wattless component, the condensance proper, and an energy com- ponent, the dielectric hysteresis. The condensance of a polarization cell, however, begins to decrease at very low potentials, as soon as the counter E.M.F. of chemical dissociation is approached. The condensance of a synchronizing alternator is of the nature of a variable quantity ; that is, the effective reactance changes gradually, according to the relation of impressed and of counter E.M.F., from inductance over zero to condensance. Besides the phenomena discussed in the foregoing as terms of the energy components and the wattless compo- nents of current and of E.M.F., the electric leakage is to be considered as a further energy component ; that is, the direct escape of current from conductor to return con- ductor through the surrounding medium, due to imperfect insulating qualities. This leakage current represents an effective conductance, g, theoretically independent of the E.M.F., but in reality frequently increasing greatly with the E.M.F., owing to the decrease of the insulating strength of the medium upon approaching the limits of its disruptive strength. • FOUCAULT OR EDDY CURRENTS. 149 101. In the foregoing, the phenomena causing loss of energy in an alternating-current circuit have been dis- cussed ; and it has been shown that the mutual relation between current and E.M.F. can be expressed by two of the four constants : Energy component of E.M.F., in phase with current, and = current X effective resistance, or r ; wattless component of E.M.F., in quadrature with current, and = current 'X effective reactance, or x • energy component of current, in phase with E.M.F., and = E.M.F. X effective conductance, or g ; wattless component of current, in quadrature with E.M.F., and = E.M.F. X effective susceptance, or b. In many cases the exact calculation of the quantities, r, x, g, b, is not possible in the present state of the art. In general, r, x, g, b, are not constants of the circuit, but depend — besides upon the frequency — more or less upon E.M.F., current, etc. Thus, in each particular case it be- comes necessary to discuss the variation of r, x, g, b, or to determine whether, and through what range, they can be assumed as constant. In what follows, the quantities r, x, g, b, will always be considered as the coefficients of the energy and wattless components of current and E.M.F., — that is, as the effec- tive quantities, — so that the results are directly applicable to the general electric circuit containing iron and dielectric losses. Introducing now, in Chapters VII. to IX., instead of " ohmic resistance," the term " effective resistance," etc., as discussed in the preceding chapter, the results apply also — within the range discussed in the preceding chapter — to circuits containing iron and other materials producing energy losses outside of the electric conductor. 150 ALTERNATING-CURRENT PHENOMENA.