CHAPTER XIII FOUCAULT OR EDDY CURRENTS 105. While magnetic hysteresis due to molecular friction is a magnetic phenomenon, eddy currents are rather an electrical 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 produces 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 existing in minute circuits; and they follow all the laws of electric circuits. Their e.m.f. is proportional to the intensity of magnetization, B, and to the frequency, /. Eddy currents are thus proportional to the magnetization, B, the frequency, /, and to the electric conductivity, X, of the iron; hence, can be expressed by i = bXBf. The power consumed by eddy currents is proportional to their square, and inversely proportional to the electric conduc- tivity, and can be expressed by P = 62X52J2. or, since Bf is proportional to the generated e.m.f., E, in the equation E = V2Tr AnfB 10-^, it follows that, The loss of power by eddy currents is proportional to the square of the e.m.f., and proportional to the electric con- ductivity of the iron; or, P = aE^\. 136 FOUCAULT OR EDDY CURRENTS 137 Hence, that component of the effective conductance which is due to eddy currents is P . that is, The equivalent conductance due to eddy currents in the iron is a constant of the magnetic circuit; it is independent of e.m.f., frequency, etc., but proportional to the electric conductivity of the iron, X. Eddy currents, like magnetic hysteresis, cause an advance of phase of the current by an angle of advance, /3; but unhke hysteresis, eddy currents in general do not distort the current wave. The angle of advance of phase due to eddy currents is sin /3 = ^ » y where y = absolute admittance of the circuit, g = eddy current conductance. While the equivalent conductance, g, due to eddy currents, 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 power component, gE, of eddy currents, is of interest, since the wattless component is identical with the wattless com- ponent of hysteresis, discussed in the preceding chapter. 106. To calculate the loss of power by eddy currents. Let V = volume of iron; B = maximum magnetic induction; / = frequency; X = electric conductivity of iron; e = coefficient of eddy currents. The loss of energy per cubic centimeter, in ergs per cycle, is w = e\fB~; hence, the total loss of power by eddy currents is P = e\\T~B"~ 10-7 watts, and the equivalent conductance due to eddy currents is _ P _ 10 6X? _ 0.507 tXZ^ ^ ~ E^~ 2 ir'-An^ ~ An^ 138 ALTERNATING-CURRENT PHENOMENA •^du where I = length of magnetic circuit, A = 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 circuit; that is, whether of iron plates or wire, and the thickness of plates or the diameter of wire, etc. The two most important cases are: (a) Laminated iron. (b) Iron wire. 107. (a) Laminated Iron. Let, in Fig. 91, d = thickness of the iron plates; B = maximum magnetic induction; / = frequency; X = electric conductivity of the iron. Then, if u is the distance of a zone, du, from the center of the sheet, the conductance of a zone of thickness, du, and of one centimeter length and width is \du; and the magnetic flux cut by this zone is Bu. Hence, the e.m.f. induced in this zone is 8E = -s/^irfBu, in c.g.s. units. Fig. 91. This e.m.f. produces the current, dl = dE \du = \/2 TrfB udu, 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 parallel to the sheet, and in opposite directions on opposite sides of the sheet. The power consumed by the current in this zone, du, is dP = 8EdI = 2TrT~B-\u^du, in c.g.s. units or ergs per second, and, consequently, the total power consumed in one square centimeter of the sheet of thick- ness, d, is 8P X dP = 2x2/2B2X X ^ 2 6 u^du , in c.g.s. units; FOUCAULT OR EDDY CURRENTS 139 the power consumed per cubic centimeter of iron is, therefore, 7) = —- = -^ — , m c.g.s. units or erg-seconds, a 0 and the energy consumed per cycle and per cubic centimeter of iron is w = Y = a ergs. The coefficient of eddy currents for laminated iron is, therefore, e = —^ ^ lM5d^ where X is expressed in c.g.s. units. Hence, if X is expressed in practical units or 10~^ c.g.s. units, , = "LR J^L^ = 1.645 dMO-^ D Substituting for the conductivity of sheet iron the approxi- mate value- X = 10^l we get as the coefficient of eddy currents for laminated iron, 6 = "^ d-nO-^ = 1.645 rf2 10-3; loss of energy per cubic centimeter and cycle, W = e\fB^ = Y d^\fB^ 10-3 = 1.645 d''\fB^ 10-^ ergs . = 1.645^2/^2 10-4 ergs; or, W = eXfBnO-^ = 1M5 d'-fBHO-^' joules. The loss of power per cubic centimeter at frequency, /, is p = fW = eXpSnO-'' = 1.645^2/2^210-" watts; the total loss of power in volume, V, is P = Vp = 1.645 VdT'BnO-'^ watts. As an example, d = 1 mm. - 0.1 cm.:/ = 100; B = 5,000; V = 1,000 c.c; 6 = 1,645 X 10-11; F = 4,110 ergs = 0.000411 joules; p = 0.0411 watts; P = 41.4 watts. ' In some of the modern silicon steels used for transformer iron, X reaches values as low as 2 X 10*, and even lower; and the eddy current losses are reduced in the same proportion (1915). 140 ALTERNATING-CURRENT PHENOMENA 108. (6) Iron Wire. Let, in Fig. 92, d = diameter of a piece of iron wire; then if u is the radius of a circular zone of thickness, du, and one cen- timeter in length, the conductance of this zone is ^ — , and the magnetic flux inclosed by the zone is SuV. Fig. 92. Hence, the e.m.f. generated in this zone is 8E = \/2Tr^fBu- in c.g.s. units, and the current produced thereby is dl = ^X V2w'fBu' 2tw = — ^r— \fBu du, in c.g.s. units. The power consumed in this zone is, therefore, dP = 8EdI = ir^\pB^uHu, in c.g.s. units; consequently, the total power consumed in one centimeter length of wire is 5P = j dW = 7r3 X/2B2 I uMu = TTT Xf-B'-d"^, m c.g.s. units. Since the volume of one centimeter length of wire is •^ = X' the power consumed in one cubic centimeter of iron is 8P w p = — = Ta XPB'^d'^, in c.g.s. units or erg-seconds, FOUCAULT OR EDDY CURRENTS 141 and the energy consumed per cycle and cubic centimeter of iron is W = ^ = ^ XJB^d^ ergs. Therefore, the coefficient of eddy currents for iron wire is e = fJrfS ^ 0.617 rf2; lb e = ^ rf2 10-9 = 0.617 d^ 10-». Id or, if X is expressed in practical units, or 10"^ c.g.s. units. Substituting X = 105, we get as the coefficient of eddy currents for iron wire, e = ^ rf2 10-9 = 0.617^2 10-9. lb The loss of energy per cubic centimeter of iron, and per cycle, becomes W = eX/52 = ^ (/2X/52 10-9 _ 0.617 d'~\fB'~ 10-^ = 0.617 rf2/B2 10-4 ergs, = eX/52 10-^ = 0.617 d'-JB^ 10-11 joules; loss of power per cubic centimeter at frequency, /, p = fW = eXN^B'^ 10-7 ^ 0.617 d'^N^B^ lO-^^ watts; total loss of power in volume, V, P = Vp = 0.617 7^2^-252 10-11 watts. As an example, d = 1 mm., = 0.1 cm.; / = 100; B^ = 5,000; V =1,000 cu. cm. Then, € = 0.617 X 10-11, W = 1,540 ergs = 0.000154 joules, p = 0.0154 watts, P = 1.54 watts, hence very much less than in sheet iron of equal thickness. 109. Comparison of sheet iron and iron wire. If di = thickness of lamination of sheet iron, and di = diameter of iron wire. 142 ALTERNATING-CURRENT PHENOMENA the eddy current coefficient of sheet iron being 61 = ^ di^ 10-«, and the eddy current coefficient of iron wire the loss of power is equal in both — other things being equal — if 61 = 62; that is, if dz"" = %di\ or da = 1.63-(/i. It follows that the diameter of iron wire can be 1.63 times or, roughly, 1^^ as large as the thickness of laminated iron, to give the same loss of power through eddy currents, as shown in Fig. 93. Fig. 93. 110. Demagnetizing, or screening effect of eddy currents. The formulas derived for the coefficient of eddy currents in laminated iron and in iron wire hold only when the eddy currents are small enough to neglect their magnetizing force. Other- wise the phenomenon becomes more complicated; the magnetic flux in the interior of the lamina, or the wire, is not in phase with the flux at the surface, but lags behind it. The magnetic flux at the surface is due to the impressed m.m.f., while the flux in the interior is due to the resultant of the impressed m.m.f. and to the m.m.f. of eddy currents; since the eddy currents lag 90 degrees behind the flux producing them, their resultant with the impressed m.m.f., and therefore the magnetism in the interior, is made lagging. Thus, progressing from the surface toward the interior, the magnetic flux gradually lags more and more in phase, and at the same time decreases in intensity. While the complete analytical solution of this phenomenon is beyond the FOUCAULT OR EDDY CURRENTS 143 scope of this book, a determination of the magnitude of this demagnetization, or screening effect, sufficient to determine whether it is neghgible, or whether the subdivision of the iron has to be increased to make it negligible, can be made by calcu- lating the maximum magnetizing effect, which cannot be exceeded by the eddys. Assuming the magnetic density as uniform over the whole cross-section, and therefore all the eddy currents in phase with each other, their total m.m.f. represents the maximum possible value, since by the phase difference and the lesser magnetic density in the center the resultant m.m.f. is reduced. In laminated iron of thickness d, the current in a zone of thick- ness du, at distance u from center of sheet, is dl = -\/2 irfEXu du units (c.g.s,) = \^ t/BXu du 10~^ a.mp.; hence the total current in the sheet is I = ' ^ rf/ - \/2 tt/BX 10-8 y udu V2 7r /BXd2 10-8 amp. Hence, the maximum possible demagnetizing ampere-turns, acting upon the center of the lamina, are i = —^-jBW 10-8 = 0.555/^X^2 10-8, = 0.555 fBXd" 10-8 ampere-turns per cm. Example: d = 0.1 cm.,/ - 100, B = 5000, X = 10^ or / = 2.775 ampere-turns per cm. 111. In iron wire of diameter d, the current in a tubular zone of du thickness and u radius is hence, the total current is ^d r^ ^d I = h dl ^^ TfB\ 10-8 h udx Jo 2 Jo V2 wfBXd^ 10-8 amp 144 ALTERNATING-CURRENT PHENOMENA Hence, the maximum possible demagnetizing ampere-turns, acting upon the center of the wire, are 7 = yAlf^x^o 10-8 ^ 0.2775/5X^2 10-8 = 0.2775 fBXd"^ 10~^ ampere-turns per cm. For example, if d = 0.1 cm.,/ = 100, B = 5000, X = 10^ then / = 1.338 ampere-turns per cm.; that is, half as much as in a lamina of the thickness d. For a more complete investigation of the screening effect of eddy currents in laminated iron, see Section III of "Theory and Calculation of Transient Electric Phenomena and Oscillations." 112. Besides the edd}^, or Foucault, currents proper, which exist as parasitic currents in the interior of the iron lamina or wire, under certain circumstances eddy currents also exist 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 produced outside of the mag- netic iron circuit proper, by the magnetic stray field cutting electric conductors in the neighborhood, especially when drawn toward them by iron masses behind, in electric 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. The power consumed by such large eddy currents frequently increases more than proportional to the square of the voltage, when approaching magnetic saturation, by the magnetic stray field reaching unlaminated conductors, and so, while negligible at normal voltage, this power may become large at over-normal voltage. Eddy Currents in Conductor, and Unequal Current Distribution 113. 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 FOUCAULT OR EDDY CURRENTS 145 particular case, where the shape of the conductor and the dis- tribution of the magnetic field are known. Only in the case where the magnetic field is produced by the current in the conductor can a general solution 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 con- ductor generate e.m.fs. in their interior only. Thus the counter e.m.f. of self-induction is largest at the axis of the conductor, and least at its surface; consequently, the current density at the sur- face will be larger than at the axis, or, in extreme cases, the cur- rent may not penetrate at all to the center, or a reversed current may exist there. Hence it follows that only the exterior part of the conductor may be used for the conduction of electricity, thereby causing an increase of the ohmic resistance due to unequal current distribution. The general discussion of this problem, as applicable to the distribution of alternating current in very large conductors, as the iron rails of the return circuit of alternating-current rail- ways, is given in Section III of "Theory and Calculation of Tran- sient Electric Phenomena and Oscillations." In practice, this phenomenon is observed mainly with very high frequency currents, as lightning discharges, wireless tele- graph and lightning arrester circuits; in power-distribution cir- cuits it has to be avoided by either keeping the frequency suffi- ciently low or having a shape of conductor such that unequal current-distribution does not take place, as by using a tubular or a flat conductor, or several conductors in parallel. 114. It will, therefore, here 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 in phase be- hind the current at the surface, due to the increased effect of self-induction. This time-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 10 146 ALTERNATING-CURRENT PHENOMENA than it is in realit3^ Hence this assumption brings us on the safe side, and at the same time greatly simplifies the calculation; however, it is permissible only where the current density is still fairly uniform. Let Fig. 94 represent a cross-section of a conductor of radius, R, and a uniform current density, I where / = total current in conductor. The magnetic reluctance of a tubular zone of unit length and thickness du, of radius u, is 2w7r (H« = du Fig. 94. The current inclosed by this zone is /„ = iu^ir, and therefore, the m.m.f. acting upon this zone is Fu = 0.4 tIu = 0.4 7rHu\ and the magnetic flux in this zone is F d^ = (Ru 0.2 TTtu du. Hence, the total- magnetic flux inside the conductor is Jo 10 Jo 10 10 From this we get, as the excess of counter e.mi. at the axis of the conductor over that at the surface, AE = \/2 7r/$ 10^8 ^ -v/2x// 10~9, per unit length, = V2 Try? 7^2 10-9; and the reactivity, or specific reactance at the center of the con- A J^ ductor, becomes k = —r- = \/2 w^fR^ 10~^. Let p = resistivity, or specific resistance, of the material of the conductor. We have then, ^ _ V2ir^fRnO-\ P P and P the ratio of current densities at center and at periphery. FOUCAULT OR EDDY CURRENTS 147 For example, if, in copper, p = 1.7 X 10^^, and the percentage decrease of current density at center shall not exceed 5 per cent., that is, P ^ Vk" + p' = 0.95 -- 1, we have k = 0.51 X 10-«; hence 0.51 X 10-« = V2 7r^fR''10 -9 > or fR^ = 36.3; hence, when / = 125 100 60 25 R = 0.541 0.605 0.781 1.21 cm, D = 2R = 1.08 1.21 1.56 2.42 cm Hence, even at a frequency of 125 cycles, the effect of unequal current distribution is still negligible at one centimeter diameter of the conductor. Conductors of this size are, however, excluded from use at this frequency by the external self-induction, which is several times larger than the resistance. We thus see that un- equal 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 conductor of iron of permeability Ai, then d^ = — ^ ; and thus ultimately, k = \/2 tr'^ftxR'^ 10~^, and k /^- JixRnO-' n^u V ■ . r • • . — = v2 TT- . thus, tor instance, tor iron wire at p = P P 10 X 10~^, fjL — 500, it is, permitting 5 per cent, difference be- tween center and outside of wire, k = 3.2 X 10"*^, and fR^ = 0.46; hence, when / =' 125 100 60 25 R= 0.061 0.068 0.088 0.136 cm.; thus the effect is noticeable even with relatively small iron wire. Mutual Induction 115. When an alternating magnetic field of force includes a secondary electric conductor, it generates therein an e.m.f. which produces a current, and thereby consumes energy if the circuit of the secondary conductor is closed. 148 ALTERNATING-CURRENT PHENOMENA Particular cases of such secondary currents are the eddy or Foucault currents previously discussed. Another important case is the generation of secondary e.m.fs. in neighboring circuits; that is, the interference of circuits run- ning parallel with each other. In general, it is preferable to consider this phenomenon of mutual induction as not merely producing a power component and a wattless component of e.m.f. in the primary conductor, but to consider explicitly both the secondary 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 primary energy, as in the discussion of the disturbance caused by one circuit upon a parallel circuit, may the effect on the primary circuit be con- sidered analogously as in the chapter on eddy currents by the introduction of a power component, representing the loss of power, and a wattless component, representing the decrease of self-induction. Let X = 2 7r/L = reactance of main circuit; that is, L = total num- ber of interlinkages with the main conductor, of the lines of magnetic force produced by unit current in that conductor; Xi — 2 tt/Li = reactance of secondary circuit; that is, Li = total number of interlinkages with the secondary conductor, of the lines of magnetic force produced by unit current in that con- ductor; Xm = 2x/Li = mutual inductive reactance of the circuits; that is, L„i = total number of interlinkages with the secondary conductor, of the lines of magnetic force produced by unit cur- rent in the main conductor, or total number of interlinkages with the main conductor of the lines of magnetic force produced by unit current in the secondary conductor. Obviously: x^^ ^ xxi.^ 1 As self-inductance L, Li, the total flux surrounding the conductor is here meant. Usually in the discussion of inductive apparatus, especiallj^ of trans- formers, as the self-inductance of circuit is denoted that part of the mag- netic flux which surrounds one circuit but not the other circuit; and as mutual inductance flux which passes between both circuits. Hence, the total self-inductance, L, is in this case equal to the sum of the self-induc- tance, Li, and mutual inductance, L,„. The object of this distinction is to separate the wattless part, Li, of the FOUCAULT OR EDDY CURRENTS 149 Let Ti = resistance of secondary circuit. Then the imped- ance of secondary circuit is Zi = ri-{- jxi, zi = V^i^ + xi^; e.m.f. generated in the secondary circuit, Ei = — jxml, where I = primary current. Hence, the secondary current is J El ~ JXm J zi ri -i-jxi . ' and the e.m.f. generated in the primary circuit by the secondary current, /i, is E = - jx„,Ii = ^ . . I; or, expanded, Ti -\r JXi ■ 77, J Xm"^ 1 j^ jXm Xl I J Hence, the e.m.f. consumed thereby, „, f Xm~ri JXm ^1 I 7 / I • NT X ^ri — effective resistance of mutual inductance; Tl^ + Xi^ 2 — X X\ X = „ r — ^ = effective reactance of mutual inductance. ri^ + Xl- The susceptanceof mutual inductance is negative, or of opposite sign from the reactance of self-inductance. Or, Mutual inductance consumes energij and decreases the self-in- ductance. For the calculation of the mutual inductance between circuits Lm, see "Theoretical Elements of Electrical Engineering," 4th Ed. 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, L, 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 trans- forming; while the flux surrounding both circuits is called the mutual induc- tance, or useful magnetic flux. With this denotation, in transformers the mutual inductance, Lm, is usually very much greater than the self-inductance, L', and L/, while, if the self-inductance, L and Li, represent the total flux, their product is larger than the square of the mutual inductance, Lm', or LLi > L„2; (L' + L,„) (L/ -|- Lm) > Lm^