Substituting this in the equation of the counter E.M.F. of self-induction

we have

(R hence, the absolute admittance of the circuit is

(RIO8 = a&

E ~ 2 TT n*N ~ N '

108

where a = , a constant.

2 TT n

Therefore, the absolute admittance, y, of a circuit of neg- ligible resistance is proportional to the magnetic reluctance, (R, and inversely proportional to the frequency, N, and to the square of the number of turns, n.

82. In a circuit containing iron, the reluctance, (R, varies with the magnetization ; that is, with the E.M.F. Hence the admittance of such a circuit is not a constant, but is also variable.

In an ironclad electric circuit, — that is, a circuit whose magnetic field exists entirely within iron, such as the mag- netic circuit of a well-designed alternating-current trans-

EFFECTIVE RESISTANCE AND REACl^ANCE. 123

former, — (R is the reluctance of the iron circuit. Hence, if p. = permeability, since —

and g:A = jr/7=Zge = M.M.F.,

and <R, 10L

magnetic flux,

substituting this value in the equation of the admittance,

(R 108 Z 109 z

y= -z- nrv> we have 5— ;

where „ L W 127Z10'

TJierefore, in an ironclad circuit, the absolute admittance, y, is inversely proportional to the frequency, N, to the perme- ability, JJL, to the cross-section, S, and to the square of the number of turns, n ; and directly proportional to the length of the magnetic circuit, L.

The conductance is

=

and the admittance, y = - ;

yv/u.

hence, the angle of hysteretic advance is

or, substituting for A and z (p. 117), NA «Z1068

or, substituting J£

we have sin a = —

-4 '

1 24 AL TERN A TING-CURRENT PHENOMENA.

which is independent of frequency, number of turns, and shape and size of the magnetic and electric circuit.

Therefore, in an ironclad inductance, tJie angle of Jiysteretic advance, a, depends upon the magnetic constants, permeability and coefficient of hysteresis, and tipon the maximum magnetic induction, but is entirely independent of the frequency, of the shape and other conditions of the magnetic and electric circuit ; and, therefore, all ironclad 'magnetic circuits constructed of the same quality of iron and using the same magnetic density, give the same angle of Jiysteretic advance.

The angle of Jiysteretic advance, a, in a closed circuit transformer, depends tipon tJie quality of the iron, and upon the magnetic density only.

The sine of tJie angle of Jiysteretic advance equals 4 times the product of the permeability and coefficient of hysteresis, divided by the .4th power of tJie magnetic density.

83. If the magnetic circuit is not entirely ironclad, and the magnetic structure contains air-gaps, the total re- luctance is the sum of the iron reluctance and of the air reluctance, or

<R = (R { | <Rfl ;

hence the admittance is

TJierefore, in a circuit containing iron, the admittance ts the sum of the admittance due to the iron part of tJie circuit, yi = a&i/ N, and of the admittance due to the air part of the circuit, ya = a (&a / N, if the iron and the air are in series in the magnetic circuit.

The conductance, g, represents the loss of energy in the iron, and, since air has no magnetic hysteresis, is not changed by the introduction of an air-gap. Hence the angle of hysteretic advance of phase is

sm a = —

y

EFFECTIVE RESISTANCE AND REACTANCE. 125

and a maximum, gjyt, for the ironclad circuit, but decreases with increasing width of the air-gap. The introduction of the air-gap of reluctance, (R0, decreases sin a in the ratio,

<Rj

«* + <*« '

In the range of practical application, from (B = 2,000 to (B = 12,000, the permeability of iron varies between 900 and 2,000 approximately, while sin a in an ironclad circuit varies in this range from .51 to .69. In air, /t = 1.

If, consequently, one per cent of the length of the iron consists of an air-gap, the total reluctance only varies through the above range of densities in the proportion of 1^ to Ig^, or about 6 per cent, that is, remains practically constant ; while the angle of hysteretic advance varies from sin a = .035 to sin a = .064. Thus g is negligible compared with b, and b is practically equal to j.

Therefore, in an electric circuit containing iron, but forming an open magnetic circuit whose air-gap is not less than T^ the length of the iron, the susceptance is practi- cally constant and equal to the admittance, so long as saturation is not yet approached, or,

b = <Ra / N, or : x = N/ (Ra.

The angle of hysteretic advance is small, below 4°, and the hysteretic conductance is,

-= A

EAN* '

The current wave is practically a sine wave.

As an instance, in Fig. 71, Curve II., the current curve of a circuit is shown, containing an air-gap of only ^ of the length of the iron, giving a current wave much resem- bling the sine shape, with an hysteretic advance of 9°.

84. To determine the electric constants of a circuit containing iron, we shall proceed in the following way : Let —

E = counter E.M.F. of self-induction ;

126 ALTERNATING-CURRENT PHENOMENA.

then from the equation, E =

where,

N '= frequency,

n = number of turns,

we get the magnetism, <£, and by means of the magnetic cross section, S, the maximum magnetic induction : ($> = ® / S.

From (B, we get, by means of the magnetic characteristic of the iron, the M.M.F., = F ampere-turns per cm length, where

if OC = M.M.F. in C.G.S. units.

Hence, if Z, = length of iron circuit, JFj = Z, F = ampere-turns re-

quired in the iron ; if La = length of air circuit, CFa = — — - — = ampere-turns re-

quired in the air ;

hence, CF= JF, -)- $Fa = total ampere -turns, maximum value, and JF/ V2 = effective value. The exciting current is

and the absolute admittance,

If SF, is not negligible as compared with JFa, this admit- tance,^, is variable with the E.M.F., E.

If —

V = volume of iron,

rj = coefficient of hysteresis,

the loss of energy by hysteresis due to molecular magnetic friction is,

hence the hysteretic conductance is g = lV/£?, and vari- able with the E.M.F., E.

EFFECTIVE RESISTANCE AND REACTANCE. 127

The angle of hysteretic advance is, —

sin a=g/y;

the susceptance, b = Vj*2 — gz\

the effective resistance, r = g / y*\

and the reactance, x = b / y*.

85. As conclusions, we derive from this chapter the following : —

1.) In an alternating-current circuit surrounded by iron, the current produced by a sine wave of E.M.F. is not a true sine wave, but is distorted by hysteresis, and inversely, a sine wave of current requires waves of magnetism and E.M.F. differing from sine shape.

2.) This distortion is excessive only with a closed mag- netic circuit transferring no energy into a secondary circuit by mutual inductance.

3.) The distorted wave of current can be replaced by the equivalent sine wave — that is a sine wave of equal effec- tive intensity and equal power — and the superposed higher harmonic, consisting mainly of a term of triple frequency, may be neglected except in resonating circuits.

4.) Below saturation, the distorted curve of current and its equivalent sine wave have approximately the same max- imum value.

5.) The angle of hysteretic advance, — that is, the phase difference between the magnetic flux and equivalent sine wave of M.M.F., — is a maximum for the closed magnetic circuit, and depends there only upon the magnetic constants of the iron, upon the permeability, yu., the coefficient of hys- teresis, rj, and the maximum magnetic induction, as shown* in

the equation, 4

sin a = — f—i .

&'4

6.) The effect of hysteresis can be represented by an admittance, Y — g + j b, or an impedance, Z = r — j x.

7.) The hysteretic admittance, or impedance, varies with the magnetic induction; that is, with the E.M.F., etc.

128 ALTERNATING-CURRENT PHENOMENA.

8.) The hysteretic conductance, £•, is proportional to the coefficient of hysteresis, 17, and to the length of the magnetic- circuit, L, inversely proportional to the .4th power of the E.M.F., E, to the .6^h power of frequency, N, and of the cross-section of the magnetic circuit, S, and to the 1.6th power of the number of turns of the electric circuit, ;/, as expressed in the equation,

58 7 Z 103

9.) The absolute value of hysteretic admittance, —

is proportional to the magnetic reluctance : (R = (R, -f (Ra , and inversely proportional to the frequency, N, and to the square of the number of turns, n, as expressed in the

_(«. + «„) 10-

2-irNn*

10.) In an ironclad circuit, the absolute value of admit- tance is proportional to the length of the magnetic circuit, and inversely proportional to cross-section, S, frequency, Ny permeability, /*, and square of the number of turns, n, or 127 L 106

11.) In an open magnetic circuit, the conductance, gt is the same as in a closed magnetic circuit of the same iron part.

12.) In an open magnetic circuit, the admittance, yt is practically constant, if the length of the air-gap is at least TJC of the length of the magnetic circuit, and saturation be not approached.

13.) In a closed magnetic circuit, conductance, suscep- tance, and admittance can be assumed as constant through a limited range only.

14.) From the shape and the dimensions of the circuits, and the magnetic constants of the iron, all the electric con- stants, gy b,y; r, x, z, can be calculated.

FOUCAULT OR EDDY CURRENTS. 129

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</x = V2 TrN<$> 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 </2 10 -9.

Substituting for the conductivity of sheet iron the ap-

proximate value,

y = 105,

we get as the coefficient of eddy currents for laminated iron, 2-»= 1.645</210-9-

loss of energy per cm3 and cycle,

W= ey^Wfc2 = - //2y^(B210-9 = 1.645 </2y N<$? 10 ~9 ergs 6

= 1.645</27V~(B210-4ergs; or, W = c y NW 10 - 7 = 1.645 d* N <S? 10 - " joules ;

loss of power per cm3 at frequency, N,

p = NW '= cy^2«210-7 = 1.645 </W2(B2 10 ~n watts; total loss of power in volume, V,

p = vp = 1.645 ^/2^2(B210-n watts.

As an example,

d = 1 mm = .1 cm ; N= 100 ; OS = 5000; V = 1000 cm8. e = 1,645 X 10-"; ^F= 4110 ergs

= .000411 joules; / = .0411 watts; P = 41.1 watts.

90. (6): Iron Wire.

Let, in Fig. 80, d = diameter of a piece of iron wire ; then if x is the radius of a circular zone of thickness, d x, and one cm in length, the conductance of this pig. so.

134 ALTERNATING-CURRENT PHENOMENA.

zone is, y^/x/2 TT x, and the magnetic flux inclosed by the zone is (B x2 *.

Hence, the E.M.F. induced in this zone is :

8£ = V2 7r2^(B x2, in C.G.S. units, and the current produced thereby is,

, in C.G.S. units.

The power consumed in this zone is, therefore,

dP= §EdI = 7T8 y N'2 (B2 x3 d x, in C.G.S. units

consequently, the total power consumed in one cm length of wire is

8 P = f~ dW = 7T3 y N'1 ®2 f * xa dx

= ^-y^2&V4, in C.G.S. units. Since the volume of one cm length of wire is

/ ,*?, - 'I

the power consumed in one cm3 of iron is

x P 2

P = -^- = ^ y ^2(BV2, in C.G.S. units or erg-seconds,

and the energy consumed per cycle and cm3 of iron is

ergs.

Therefore, the coefficient of eddy currents for iron wire is c = ^^2 = .617 </2;

or, if y is expressed in practical units, or 10 ~9 C.G.S. units,

c = -^ 10

FOUCAULT OR EDDY CURRENTS. 135

Substituting ^ = ^

we get as the coefficient of eddy currents for iron wire,

e= — ^210~9 = .617 </210-9. 16

The loss of energy per cm3 of iron, and per cycle becomes

= .617 d*N®? 10~4 ergs,

loss of power per cm3, at frequency, N,

p = Nh = ey^2(B210-7 = .617 d 2 N*<$? 10 -" watts; total loss of power in volume, V,

P= Vp = .617 FVJV'&'IO-11 watts. As an example, d = 1 mm, = .1 cm ; N= 100 ; «2 = 5,000 ; V= 1000 cm8.

e = .617 X 10-11, W= 1540 ergs = .000154 joules, p = .0154 watts, P = 15.4 watts,

hence very much less than in sheet iron of equal thickness.

91. Comparison of sheet iron and iron wire.

If

//! = thickness of lamination of sheet iron, and dz = diameter of iron wire,

the eddy-coefficient of sheet iron being T* j 2 10-9

• T?

and the eddy coefficient of iron wire

136 AL TERNA TING-CURRENT PHENOMENA.

the loss of power is equal in both — other things being equal — if ex = e2 ; that is, if,

= !</!», or 4 = 1.63 ^.

o

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 energy through eddy currents, as shown in Fig. 81.

Fig. 81.

92. Demagnetizing, or screening effect of eddy currents.

The formulas derived for the coefficient of eddy cur- rents in laminated iron and in iron wire, hold only when the eddy currents are small enough to neglect their mag- netizing force. Otherwise the phenomenon becomes more complicated; the magnetic flux in the interior of the lam- ina, or the wire, is not in phase with the flux at the sur- face, but lags behind it. The magnetic flux at the surface is due to the impressed M.M.F., while the flux in the inte- rior 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° behind the flux producing them, their resultant with the impressed M.M.F., and therefore the magnetism in the

FOUCAULT OR EDDY CUKREN7*S. 137

interior, is made lagging. Thus, progressing from the sur- face towards 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 scope of this book, a determina- tion of the magnitude of this demagnetization, or screening effect, sufficient to determine whether it is negligible, or whether the subdivision of the iron has to be increased to make it negligible, can be made by calculating the maxi- mum 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 thickness, dx at distance x from center of sheet, is :

dl = -rrN&jxdx units (C.G.S.)

= V2 TT N&jxdx 10 - 8 amperes ; hence the total current in sheet is

/=

amperes.

Hence, the maximum possible demagnetizing ampere-turns acting upon the center of the lamina, are

A/9

• 8 = .555 N&jd* 10 - 8

8 = .555 ./V(B</210~3 ampere-turns per cm

Example : d = .1 cm, N= 100, (B = 5,000, or / = 2.775 ampere-turns per cm.

138 ALTERNATING-CURRENT PHENOMENA.

93. In iron wire of diameter d, the current in a tubular zone of dx thickness and x radius is

dl= — TT JV&j'x dxlO-* amperes; hence, the total current is

I = f$4I~?2. vN&j 10-« f* xdx

Jo " Jo

A/9

~ * amperes. 16

Hence, the maximum possible demagnetizing ampere-turns, acting upon the center of the wire, are

10 -

16

= .2775 N(S> 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,