A4 .,a er presented at tfle Eleventh General Meetizg
of the Amee rican Institute of Electrical Eng-in
eers, Philadelfhlia, May i8th, 1894, President
Hcnston in the Choir'
ON THE LAW OF HYSTERESIS (PART III.),
AND THE
THEORY OF FERRIC INDUCTANCES.
BY CHARLES PROTEUS STEINMETZ.
CHAPTER I.-COEFFICIENT OF MOLECUTLAR MAGNETIC FRICTION.
In two former papers, of January 19 and September 2T, 1892,
I have shown that the loss of energy by mnagnetic hysteresis, due to miolecular friction, can, with sufficient exactness, be expressed by the empirical formula-
:I = a B16 where H = loss of energy per cm3. and per cycle, in ergs,
B = amplitude of magnetic variation, coefficient of molecular friction,
the loss of energy by eddy currents can be expressed by h _1N B2,
where h = loss of energy per cm3. and per cycle, in ergs,
z coefficient of eddy currents.
Since then it has been shown by lMr. R. Arno. of Turiin, that
the loss of energy by static dielectric hysteresis, i.e., the loss of energy in a dielectric in an electro-static field can be expressed by the same formula:
H= aF where R = loss of energy per cycle,
F = electro-static field intensity or initensity of dielectric stress in the material,
a = coefficient of dielectric hysteresis. Here the exponent 2 was found approximately to = 1.6 at the low electro-static field intensities used. At the frequencies and electro-static field strengths met in
570

1894.]

S'EINYMETZ ON HYSTERESIS.

571

condensers used in alternate current circuits, I found the loss of
energy by dielectric hysteresis proportional to the square of the
field strength.

Watts
-24,000-

_

___ _

_

_ __ ___

-2-2-7000--

-2-0 000-

__

_.

_ ___. _

--1 470000-0--,C -1-2-,00-0-
40-,000
8TOGO0 -

_____---
-_ _

____. __ ___-_- ____

-000-

_

-4-,000 -- ___

-27-000---

Volts 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000
Bradley/ & PoatZes, Enar'>s, N. Y.
FIG. 1.
Other observations made afterwards agreed with this result. With regard to magnetic hysteresis, essentially new discoveries

572

STEINMETZ ON HYSTERESIS.

[May 18,

have not been mnade sinTce, and the explanation of this exponent

1.6 is still unknown.

In the calculation of the core losses in dynamo electrical ma-

chinery and in transformers, the law of hysteresis has found its

applicationa, and so far as it is not obscured by the superposition

of eddy currents has been fully confirmied by practical experi-

ence.

%

As anl instance is slhown in Fig. 1, the observed core loss of a

high voltage 500 E. w. altornate current generator for power

transmissioni. The curve is plotted with the core loss as abscisse

and the ter-minal volts as ordinates. The observed values are

marked by crosses, while the curve of 1.6 power is shown by

the drawni line.

The core loss is a very large and in alternators like the present

machine, eveni the largest part of the total loss of ener,gy in the

machine.

With regard to the numnerical values of the coefficient of

hysteresis, the observations up to the time of my last paper cover

the range,

97X j03=

Materials

From

Wrought iron,.....

Sheet iron and sheet steel ...(

2.00

Cast iron ..........

11I3

Soft cast steel and mitis metal ....... ..........

3.18

Hard cast steel .............................27.9

Welded steel Magnetite

.

................2........

.

.......

eI4.745.1
20.4

Nickel ..

2.2

Cobalt ..

..........

......

"I.9

To
5.48 T6.2 6.o 12 0
23.5
38.5

Average.
3.0 to 3.3 T3.0

While no new materials lhave been investigated in the meantimue, for some, especially sheet iron and slheet steel, the range of
observed value of i has been greatly extended, and, I am glad to state, mostly towards lower valuLe of -, that is, better iron.
While at the time of my former paper, the value of hysteresis
X 10' = 2.0, talen from Ewing's tests, was -unequaled, and the best material I could secure, a very soft Norway iron, gave
d X l03- 2.275, now quite frequently vaiues, considerably
better than Ewing's soft iron wire are found, as the following table shows, which gives the lowest and the highest values of hysteretic loss observed in sheet iron and sheet steel, intended
for electrical maehiniery.

1894.]

STEINMETZ ON HYSTERESIS.

573

The values are taken at random from the factory records of
the General Electric Company.
Values of X 10O.

LIowest. 1.24 1.33
1.35 1.58 1.59 1.59 1.66 1.66 1.68
1.70 1.71 1.76 1.80 1.82
1.88
1.90 1.93 1.94 1.94

Highest. 5.30 5.15 5.12 4.78 4.77 4.72 4.58 4.55 4.27

As seen, all the values of the first column refer to iron

superior in its quality eveni to the sample of Ewing ^q X 10 -- 2.0,
unequaled before.

The lowest valuie is ^ X 10' = 1.24, that is, 38 per cent. better

than Ewing's iron. A sample of this iron I have here. As you

see, it is very soft material. Its chemical analysis does not show

anything special. The chemical constitution of the next best

samnple j X 10 = 1.33 is almost exactly the same as the con-

stitution of samples C X 103 = 4.77 and ^ X 103 - 3.22, show-

ing quite conclusively that the chemical constitution has no

direct influenice upon the hysteretic loss'.

In consequence of this extenision of § towards lower values, the total range of C yet known in iron and steel is fromr

C X 101 = 1.24 in best sheet iron to q X 10( = 74.8 in glass-

hard steel, and a X 108 81.8 in manganese steel, giving a

ratio of 1 to 66.

With regard to the exponenit X in

H=a B
which I found to be approximnately = 1.6 over the whole range of magnetization, Ewing has investigated its variation, and found that it varies somnewhat at different magnetizationls, and that its variation corresponds to the shape of the magnetization curve, showing its three stages.'
1. J. A. Ewing, Philo8ophical Transaections of the Royal Society, London, Juine 15, 1893.

574

STE]NMETZ 0N HYSTERESIS.

[May 18,

Tests of the variation of the hysteretic loss per cvele as fune-
tion of the temperature have been published by Dr. W. Kunz',
for temnperatures from 20° and 800° Cent. They show that with
rising temperature, the hysteretic loss decreases very greatly,
and this decrease consists of two parts, one part, whieh disappears againi with the decrease of temiperature and is directly proportional to the increase of temperature, thus making the hysteretic loss a linear function of the temperature, anid another part, which has becomne permanent, anid seems to be due to a permanent ehange of the m-olecular structure produced by heating. This latter part is in soft iron, proportional to the temperature also, buit irregular in steel.
CHAPTER II.--MOLECULAR FRICTION AND MAGNETiC HYSTERESIS.
In an alternating magnetic circuit in iron and other magnetic material, energy is converted inito heat by molecular magnetic friction. The area of the hysteretic loop, with the AT. MI. F. as abscissse and the magnetization as ordinates, represents the energy expended by the M. I. F. during the cyclic ehange of
magnetization. If energy is neitlher consumed nor applied outside of the
magnetic circuit by any other souLrce, the area of the hysteretic loop, i. e., the energy consumed bv hysteresis, mneasures and represents the energy wasted by molecular magnetic friction.
In general, however, the energy expended by the M. M. F.the area of the hysteretic loop-needs not to be equal to the molecular friction. In the armature of the dynamno machine, it probably is not, but, while the hysteretic loop more or less collapses under the influence of mechanical vibrationi, the loss of energy by molecular friction remains the sa-me, hence is no longer measured by the area of the hysteretic loop.
Thus a sharp distinction is to be drawn between the phenomenon of 'magnetic hysteresis, which represents the expenditure of energy by the M. M. F., and the molecuilar friction.
In stationary alternating current apparatus, as ferric induc-
tances, hysteretic loss and inolecular magn-etic friction are generally idenrtical.
In revolving machinery, the discrepancy between molecular friction and magnetic hysteresis may become very large, and the magnetic loop may even he overturhred and represent, not expen-
1. eUtroteohni8che Zeitschrift, Arril 5th, 1894.

1894.]

STEINMETZ ON HYSTERESIS.

575

diture, but production of electrical energy from meebanical energy; or inversely, the magnetic loop may represent not only the electrical energy converted into heat by molecular friction, buLt also electrical energy converted into mechanical miotion.
Two such cases are shown in Figs. 2 and 3 and in Figs. 4 and Z In these cases the magnetic reluctance and thus the induetance of the circuit was variable. That is, the magnetic circuit was opened and closed by the revolution of a shuttle-shaped
armature.
The curve s represenits the inductan-ces of the mnagnetic circuit

_ E_

Bradley ~Poates, Enrs, N.Y.
FiG. 2.
as function of the position. The curve a, couLnter E. M. F. or, since the internal resistance is negligrible, the impressed E. M. F. and curve M -_ magnetismn. If the impressed -E. M. F., E iS a sine wave, the current c assumes a distorted wave shape, and
the produict of current anid E. M. F_, W -C E represents the
energy. As seen, in this case t-e total energy is not equal to -zero, i. e., the a. M. F. or self-induction E not wattless as usually supposed, but represe-nts production of electr'ical energy in the -first, conisumptlion in the second case. Thus, if the apparatus is driven by exterior power, it assumes the phase relation shown in

576

STEINMETZ ON HYSTERESIS.

[May 18,

Fig. 2, arid yields electrical energy as a self-exciting alternate
current generator; if now the driving power is withdrawn it
drops into the phase relation shown in Fig. 4, and then continues
to revolve and to yield mechanical energy as a synchronous
motor.
The magnetic cycles or H-B curves, or rather for convenience,l the C-A curves, are shown in Figs. 3 and 5.
As seen in Fig. 5, the magnetic loop is greatly increased in
area and represents not only the energy consumed by molecular
magnetic friction, but also the energy converted into mechaniical
power, while the loop in Fig. 3 is overturned or negative, thus representing the electrical energy produced, minus loss by moleeular friction.

: X_

-~~~~~

FIG. 3.

This is the same apparatus, of which two hysteretic loops were shown in my last paper, an indicator-alternator of the "hhummning bird" type.
Thus magnetic hysteresis is not identical with molecular magnetic friction, but is one of the phenomrena caused by it.

CHAPTER III.-THEORY AND CALCULATION OF FERRJIC INDIUCTANCES.
In the discussion of inductive circuits, generally the assump-
tion is made, that the circuit contains no iron. Such non-ferric inductances are, however, of little interest, since inductances are almost always ironclad or ferric inductances,

1894.1

STEINMETZ ON HYSTERESIS.

5

With our present knowledge of the alternating magnetic cir-
cuit, the ferric inductances can now be treated analytically with the same exactness and almost the same siimplicity as non-ferrie
inductances.
Before entering into the discussion of ferric inductances, some ternms will be introduced, which are of great value in simplify-
ing the treatinent.
Referrilig back to the continuous current circuit, it is known that, if in a continu-ous current circuit a number of resistances)

__ __ __ _te __ __ __ _ _

_

A~~~~~~

____ _X_7 \

FIG. 4.

Bradley 'PoXates Engrs, N.Y.'

ri, r2, 93 . . . . are connected in series, their joint resistance, R, is. the sum of the individual resistances:
R= + r2 + r + *
If, however, a number of resistances, rI r 3..r. , are connected in parallel, or in multiple, their joint resistance, R, can-
not be expressed in a simple form, but is:

Hence, in the latter case, it is preferable, instead of the tern

578

STEINMIETZ ON HYSTERESIS.

[May 18,

4 resistance," to introduce its reciprocal, or inverse value, the
terim conduetanee" p = . Theen we get:
"If a number of conlductanices, pn P2, p3.. . are connected
in parallel, their joined conductance is the sum of the inidividual
conductances: p= P + P2 + p3 +
When usilng the term conductance, tlhe joined conductance of

t =XTtI+ /fI

_-M

FIG. 5.

Bradley & Poates, EBgr', N. Y.

a number of series connected conductances, Pl P2, p3 . .. becomes
a complicated expression
-P
Pt P2 P's Hence the use of the termn "resistance" is preferable in the case of series connection, the use of the reciprocal term. conductance," in parallel connection, and we have thus:
"The joined resistance of a number of series connected re-
si ts ces is eqtal to the sum of the individual resistances, the
Joined conductance of a number of parallel connected conductances is equal to the sum of the individual conductances."
In alternating current circuits, in place of the term "resist-

1894.]

STEINMETZ ON HYSTERESIS.

579

ance" we hiave the term "impedance,"' expressed in comnplex
quantities by the symbol: U r-J8
with its two components, the "resistacie" r and the "reactae s, in the formula of Ohm's law:
E= C U.'
The resistance, r, gives the coefficient of the E. M. F. in phase with the current, or tlhe energy component of E. M. F., Cr; the reactance, s, gives the coefficient of the E. M. F. in quadrature
with the current, or the wattless CoMponent of E. M. F., Cs, botl
combined give the total E. M. F.
CW= C Vr +s2
Thlis reactance, S, is positive as inductive reactance:
s _ 2 wr Nl, or negative as capacity reactance:

s

2 7r NK'

where,

N = frequency,

I = coefficient of self-induction, in h-enrys,

X = capacity, in farads.

Since F. M. F.'s are combined by adding their complex expres-

sions, we hlave:

"'The joinied impedance of a numiiber of series connected im-

pedances, is the sum of the individual impedances, when ex-

pressed in complex quantities."

In graphical representation, impedances have not to be added,

but combined in their proper phase, by the law of parallelogram,

like the 1.M. F.'S consumed by them.

The termn '4 impedance " becornes inconvenienlt, hiowever, when

dealinig with parallel connected circuits, or, in other words, when

several currents are produced by the same E. M. F., in cases where

Ohm's law is expressed in the form:

It is preferable then, to introduce tlhe reciprocal of "impe-
1." Complex Quantities and their use in Electrical Engineering,'" a paper read before Section A of the Initernational Electrical Congress at Chicago, 1893.

580

STElNAETZ ON HYSTERESIS.

[May 18,

dance," which may be called the "admittance" of the circuit:
F_1

As the reciprocal of the complex quantity
U = r -j8,
the admittanee is a complex quantity also:
Y p +H-J
consisting of the component, p, which represents the coefficient of current in phase with the E. M. F., or energy current, o E, in the equation of Ohm's law:
C = YE(p+j a) E,
and the component, CTwhich represents the coefficient of current
in quadrature with the E. M. F., or wattless component of current,
arE.
p may be called the " condcetance," a the "suseeptance" of the eirculit. Hence the conductance, p, is the energy component, the susceptance, ?, the wattless component of the admnittance
Yy +i n
anid the nLmerical value of admittanee is:
v=
the resistance, r, is the energy component, the reactance, 8, the wattless component of the impedance
U r --J 8r
and the numerical value of impedance is
u = t/r2 +e s8'l2.
As seen, the term " admittance " means dissolving the current into two components, in phase and in quadrature with the E. M. F,, or the energy current and the wattless current; while the term "' impedance" means dissolving the F. M. F. into twp coimponents, in phase and in qluadrature with the curreint, or the energy
E. M. F. and the wattless E. M. F. It must be understood, however, that the "conductance" is,
not the reciprocal of the resistance, but depends upon the resist-
ance as well as upon the reactance. Only when the reactance s - 0, or in continuous current circuits, is the conductance the reciprocal of resistance.
Again, only in circuits with zero resistance =- 0, is the sus-

1894.]

STEIN ETZ ON HYSTERESIS.

581

ceptance the reciprocal of reactance; otherwise the susceptance depends upon reactance and upon resistance.
From the definition of the admnittance:

Y =p +j a

.as the reciprocal of the impedance:

we get

U= r-j8
Y1

or
P +j - q = a1

or, multiplying on the right side numerator and denominator by

!(r +js):

hence, since

+j (r-j 8) (r +j 8)'

(r j s) (P +j 8) = r2 + 82 = 2:

r

8 . S

r+ S

Y/+8S2 u2 + u78

or,

P2

r+ - g

and inversely:

t_ P

..

-2+ 2 v2

C _C
S= 2 + ve2

By these equations, from resistanee and reactanee, the conduct-
ane and susceptance can be calculated, and inversely.
Multiplying the equations for p and r, we get:

Pr tl?2arvp 2,1

hence,

,2 2 (r2 + 82) (p2 + ?) = 1

and

1
U= -

1
;

:582

STEINMETZ 0N HYSTERESIS.

[May 18,

the absolute value of impedance,

iL

1

u 4 r2 + S2

the absolute value of admittance. The sign of " admittance " is always opposite to that of "im-
pedance," that means, if the cuirrent lags behind the E. M. F.,
the E. M. F. leads the current, and inversely, as obvious. Thus we can express Ohm's law in the two forms:

h' = U.

and have

C=E Y,

"The joined impedance of a number of series connected im-
pedlances is equal to the sum, of the individual impedances; the
joined admittance of a number ofparallel connected admittances
is eqlual to the sumn of the individual admittances, if expressed in complex quantities; in diagramm,natic representation, com-
bination by the parallelogram law takes the_place of addition of
the complex quantities."
The resistance of an electric circuit is determined:
1. By direct comparison with a known resistance (Wheatstone

bridge method, etc.). This method gives what may be called the truie ohinic resistance of the circuit.
2. By the ratio:
Volts consuLmed in circuit Amperes in circuit

In an alternating current circuit, this method gives not the re-
sistance, but the impedance

of the circuit.

u/= V'r2+ s2

3. By the ratio:

Power consumed - (E. M. .)2

(current)2

Power consumed'

where, however, the "'power " and the " E. M. F." do not inelude the work done by the circuit, and the counter E. M. F.'S representing it, as for instance, the counter E. N. F. of a motor.
In alternlating current circujits, this value of resistance is the
energy coefficient of the E. N. F., and is:

r Eniergy component of E. M. F.
Total current

1894.]

STEINMETZ ON HYSTERESIS.

583

It is called the " equivalent resistanc" of the circuit, and the energy coefficient of current:
- Energy comnponent of current Total E, M. F.
is called tlle " equivalent conductance" of the circuit. In the same way the valie:
8 = WattIess component of E. M. F. Total current
is the "equivalent reactance," and
Wattless comnponent of current Total E. M. F,
is the "equivalent 8suceptance" of the circuit. While the true ohmic resistance represents the expenditnre of
energy as heat, inside of the electric conductor, by a current of uniform deensity, the " equivalent resistance " represents the total expenditure of energy.
Since in an alternating current circuit in general, energy is expended nlot only in the conductor, but also outside thereof, by hysteresis, secondary currents, etc., the equivalent resistance frequently differs from the true ohmic resistanee, in such way as to represent a larger expendituire of energy.
In dealing with alternating current circuits, it is necessary, therefore, to substitute everywhere the values " equivalent resistance," "equivalent reactance," "equivalent conductance," "equivalent susceptance," to iiiake the calculation applicable to genaeral alternating current circuits, as ferric inductance, etc.
While the true ohmic resistance is a conistant of the circuit, depending upon the temuperature only, but not upon the E. M. F.e etc., the "'equivalent resistance"- and "equivalent reactanee" is in general not a constant, but depends upon the E. M. F., cur-
relt, etc.
This depenidence is the cause of most of the difficulties mret in dealing analytically with alternating cuLrrent circuits containing
iron. The foremost sources of energy loss in alternating current cir-
cuits, oLutside of the true ohmic resistance loss, are: 1. Molecular friction, as: (a) magnietic hysteresis; (b) dielectric hysteresis.

M84

STEINMETZ OJV HYSTERESIS.

[May 18,

2. Primary electric currents, as: (a) leakage or escape of cuLrrent through the insulation, brush discharge; (b) eddy-eurrents in the conductor, or unequal current distribution.
3. Secondary or induced currents, as:
(a) eddy or Foucault currents in surrounding miagnetic materials;
(b) eddy or Foucault currents in surroundino, conducting materials;
(e) secondary currents of mutual inductanlce in neighboring circuits.
4. Induced electric charges, electro-static influence. While all these losses can be included in the terms "1 equivalent -resistance," etc., only the magnetic hysteresis and the eddy-currents in the iron will form the object of the present paper.

I.-Alfaynetic IJsteresi.S.

To examinle this phenomenon, first a cireuit of very high inductanee, but negligible true ohmic resistance may be considered, that is, a circuit entirely surrounded by iron ; for iiistance, the primary circuit of an alternating current transformer with open
secondary circuit.
The wave of current produces in the iron an alternating mag-
netic flux, which induces in the electric eireuit all . M. F., the
,counter E. M. F. of self-induction. If the ohmic resistance is
negligible, the counter E. M. F. equals the impressed E. M. F., hence,
if the impressed . M. F. is a sine-wave, the counter E. M. F., and therefore the magnetism which induces the counter F. M. F. must
be sine-waves also. The alternating wave of current is not a sine-wave in this case, but is distorted by hysteresis. It is possible, however, to plot the current wave in this case from the hystereticeycle of magnetization.
From the number of turns n of the electric circuit, the effective couniiter E. M. F. L and the frequenley X of the current, the maximum magnetic flux M1 is found by the formula:

hence:

E= 4/2NiXt X10;

M4/2E7t1fl0NV

1894.]

STEINYMETZ ON HYSTERESIS.

585

Maximum flux X1 anid magnetic cross-section S give the maximumn magnetic induction B 31
If the miagnetic induction varies periodically between + B
and - B, the m. M. F. varies between the corresponding values + Fand -F and describes a looped curve, the cycle of hys-
teresis.
If the ordinates are given in lines of nmagnetic force, the ab-

. r16,000
4-___ 14 00

04,000

_-E__

m--12-0-

-- -- W -L -- - - -

/--2 00
,0

--

-

B. __ ____ fC 2_10

IL /,040 +0 4- +10

.__
____ ____ __ :L 14000 -

444--108 4_20
.1-

Bradley 4 Poates, Bgr'e, N. Y.
FIG. 6.
Scissoe in tens of ampere-turns, the area of the loop equals the energy consumed by hysteresis, in ergs per cycle.
From the h-ysteretic loop is found the instantaneous value of M. M. F. corresponding to an instantaneous value of magnetic flux, that is of induced E. M. F., and from the m. M. F., F, in amperetuLrns per unit lenigth of magnetic circuit, the length I of the magnetic circuit, and the number of turns n of the electric circuit, are found the iiistantaneons values of current c correspond-
ing to a M. M. F. F, that is a magnetic induction B anld thus in-
duiced E. M. F. e, as:
n

586

STEINMETZ ON HYSTERESIS.

[May 18,

In Fig. 6 four magnetic cycles are plotted, with the maximumlh
values of magnetic inlductions: B = 2,000, 6,000, 10,000 and
16,000, and the corresponding maximuM M. M. F.'S: F= 1.8, 2.8,, 4.3, 20.0. They show the well-known h-ysteretic loop, which be-
conies pointed when magnetic saturation is approached.
These magnetic cycles correspond to average good sheet iron
or sheet steel of hysteretic coefficient: .0033, aind are given

0F

B 2000

__ _____

6a 00

i1=7i9 .8 \ X F 2.8
_1-W__ _.1 i. Ct~~-02.bl

Bradley Poates, Engr'8, N.Y.
with ampere-turns per cmi. as abscissoe and kilolinies of mnagnetic~
force as ordinates.
In Figs. 7, 8, 9 and 10 the mnagnetism, or rather the magnetic, induction, as derived from the i-nduced 'E. M. F.1 is assumed as, sine-curve. For the ditfer-ent values of magnetic inductioni of' this sine-curve, the corresponding values of m. m. F.~hence of~ c-urrent, are taken from Fig. 6, a-nd plotted, givi-ng thius the excit'ing currenit required to produce the si-ne-wave of miagnetism;

1894.]

STEINf2TETZ ON HYSTERESIS.

587

that is, the wave of current, which a sine-wave of impressed
E. M. F. will send through the circuit. As seen fromn Figs. 4 to 10, these waves of alternating current
F are not sine-waves, but are distorted by the superposition of higlher harmonies, that is, are complex harmonic waves. They
reach their maxinmum value at the same tiimne with the maximum of magnetism, that is, 900 ahead of the naximum induced E. M. F.,

hence about 90° behind the maximum impressed E. M. F., but; pass the zero line considerably ahead of the zero valule of magnetismu: 42, .52, 50 and 41 degrees respectively.
The general character of these curtrent waves is, that the maaximum point of thle wave coincides inL timne with the maximumn point of the sine-wave of mlagnetism, but the current wave is bulged out greatly at the risinlg; hollowed in at the decreasing side. With increasing mnagnetization, thle maxsimum of the current

IX~ ~--388

STEINMETZ ON HYSTERESIS.

[May 18,

wave becomes more pointed, as the curve of Fig. 9, for B = 10,000 shows, and at still higher saturation a peak is formed at the max-
imum point. as in the curve of Fig. 10, for B - 16,000. This
is the case, when the cuarve of magnetization reaches within the
range of magnetic saturation, since in the proximity of saturation

--FiFiq. 11 - 1-\7 _ _

-_ 1<

~ /~
.~~t~l~~4 T W</X<1-- t-4-I ;-X.' I.

_r rT I _
Bradley Poates, Engrls, NJ.
the current near the nmaximum point of magnetization has to rise
abnormally, to cause a small increase of magnetization only. The distortion of the wave of magnetizing current is so large
as shown here, onlv in an iron closed magnetic circuit expending energy by hysteresis ornly, as in the ironclad transformer at open

1894.]

STEINMETZ ON HYSTERESIS.

M8

secondary circuit. As soon as the circuit expends energy in any
other way, as in resistance, or by mutual inductance, or if an airgap is introduced in the magnetic circuit, the distortion of thecurrent wave rapidly decreases and practically disappears, and
the current beconles more sinuLsoidal. That is, while the distort-
ing component rem-ains the same, the sinusoidal component of
current greatlv increases, and obscures the distortion. For in-
stance, in Figs. 11 and 12 two waves are shown, corresponding in mnagnetization to the curve of Fig. 8, as the worst distorted. The curve in Fig. 11 is the current wave of a transformer at 1
load. At higher load the distortion is still correspondingly less.
The curve of Fig. 12 is the exciting current of a magnetic cir-
cuit, containing an air-gap, whose length equals .1 the lenigth
of the magnetic circuit. These two curves are drawn in 3 the
size of the curve in Fig. S. As seen, both curves are practically
sine-waves.
The distorted wave of current can be dissolved in two com-
ponents: a true sine-wave of equal efective intensity and equal power wit1l the distorted wave, called the "equivalent sine-wave," and a wattless hAiher harmonic, consistinig chiefly of a term of
trip]e frequiency. In Figs. 7 to 12 are shown, in drawn linaes, the equ:ivalent sine-
waves, and the wattless complex higher harmonics, whlich together formri the distorted current wave. The equivalent sinle-wave of
M. M. F., or of ecurrent, in Figs. 7 to 10, leads the magnetism by
34, 44, 38 and 15.5 degrees respectively. In Figs. 11 and 12 the
equivalent sine-wave almost coincides with the distorted curve,
and leads the magnetism by only 90, It is interesting to note, that even in the oreatly distorted
curves of Figs. 7 to 9 the maximum valne of the equivalent:
sine-wave is nearly the same as the maximuin value of the original distorted wave of M. m. ia., as long as magnetic saturationi is not approached, being 1.8, 2.9 and 4.2 respectively, agaim st l.8, 2.8 and 4.3 as inaximnuin values of the distorted curve. Since by the
definition the effective valuLe of the equivalent sine-wave is the same as that of the distorted wave, this meanis, that the distorted wave of exciting current shares with the sine-wave the feature, that
the maximn-um value and the effective value h-iave the ratio: 4/2 + 1. Hence, below saturation, the inaxim-num value of tlhe distorted
curve can be calculated from the effective value-wlmieh is given by the reading of an electro-dynamometer-by the same ratio as

:<590

STEIN1XETZ ON HYSTERESIS.

[May 18

with a true sine-wave, and the mlagnetic characteristic can thus
be determined by means of alternating currents, by the electrodynamometer method, witlh su-fficient exactness.
In. Fig. 13 is shown the truie magnetic characteristic of a sample of average good sheet iron, as found by the metthod of slow

14

'~~~~~~~~_ 1si5

_ _ __ li

tt_

Ii__

/
iiii-__

tr4-1X

;

<

- I__

-

X

i

3~~~~~~~~~~~~~~~~~~~~~~~rde r, PoaesEnql,N

12 X1 __

1
__ -

-reversals by the magnetomueter, and for coi-nparisoir in dotted
-lines the saiine el-aracteristic, as determined by alternating cur.-
rents, by the electro-dynai-nometer, with amnpere-t-urris per cm. as ordinates, and magnetic inductio-ns as abscissoe. As seen, the
-two c,urves practically coin2ide9 up1B 0to,000= ,114,000 .

1894.]

STEINMIETZ ON HYSTERESIS.

591

For higher saturations, the curves rapidly diverge, and the
electro-dynamnometer curve shows comparatively small M. M. F.Ss producing apparently very high magnetizations.
The sane Fig. 13 gives the curve of hysteretic loss, in ergs per cm.3 and cycle, as ordinates, and magnetic inductions as abscisse.
So far as current strength and energy consumption is coneerned, the distorted wave can be replaced by the equivalent sine-wave, and the higher harimonies nleglected.
All the measurements of alternating currents, with the only exception of instantaneous readings, yield the equivalent sine-wave only, but snppress the higher harmonic, since all mneasuring instruments give either the mean square of the cuirrent wave, or the mean product of inistantaneous values of current and E. M. F. which are by definition the same in the equivalent sine-wave as
in the distorted wave.
Hence, in all practical applications, it is permissible to neglect the higher harmonic altogether, and replace the distorted wave by its equivalent sine-wave, keeping iu minid, however, the existence
of a higher harmonic as a possible disturbing factor, wicth may become noticeable in those very infrequLent cases, where the frequency of the higher harinonic is near the frequency of resonance of the circuit.
The equivalent sine-wave of exciting current leads the sine-
wave of magnetismn by an angle a, which is called the " angle of
Aysteretic advance ofphase." Hence the current lags behind the E. M. F. by 90 - a, and the power is, therefore:
P = CLecos (90°- a) = CEsin a.
Thus the execiting current C consists of an energy comuponent: C sin a, -which is called the " hy.3teretic energy current," and a wattless component: C cos a, which is called the "mnagnetizing
eurrent." Or inversely, the E. M. F. consists of an energy component:: E sin a, the "hysteretic envergy E. M. F." anid a wattless
component: EGcos a, the E. m. F. of self-rnductton." Denoting the absolute value of the impedance of the eircuit
by u-where u is determined by the magnetic characteristic of
the iron, and the shape of the magnletic and electric circuit-the
impedance is represented, in phase and intensity, by the synmbolic,
expression:
U r-j s = u sin a-j u Cos a,

592

STEINMETZ ON HYSTERESIS.

LMay 18.1

and the adimittance by:

C

+ os a = v sin a + ICcos a.

The quantities: u, r, s and v, p, a are not constanits, however,
in this case, as in the circuit without iron, but depend upon the
intensity of magnetization, B, that is, upon the E. M. F.
This dependence comnplicates the in-vestigation of circuits con-
taining iron.
In a eirenit entirely enelosed by iron, a is quite considerable, from 30 to 50 degrees for values below saturation. Hence even
with negligible true ohmic resistance no great lag can be pro-
dueed in ironclaCl alternating current irculits. As I have proved, the loss of energy by hysteresis due to
molecular friction is with sufficient exactness proportional to the
l.6th power of magnetic induction, B. Hence, it can be expressed by the formula:
15 ~B16, where
AI= loss of energy per cycle, in ergs or (c. G. S.) units (- 10- Joules) per cm.,
B = maximnum magnetic induction, in lines of force per cm.',
and, = the ccoefficient of hysteresis." At the frequency, N, in the volume, VT, the loss of power is by this formnula:
P = N FB-B' 10-7 watts,

- N V ( 1)0-7 watts,

where 8 is the cross-section of the total magnetic flux, Xl.
The maximum magnletic flux, Mf, depends upon the counter
E. M. F. of self-ind-uction, E, by the equation:
E = V/2 7r N n M 1O-8,
or,
iL= - E10

where n = number of turns of the electric circuit.
Substituting this in the value of the'power, P, and cancelling,

we get:

1.6 V 1058

E16 V 108

= ,N6 28 i-6X 81.6 n'6- 58 N1.6 81.6 n,6

1894.]

STEINMETZ ON HYSTERESIS.

593

or
E1.6
P= a Nf where: a =

V10ti
<-

8

y3103
St.6 fl65

or, substituLting

.0033:

a 191.4 5 l 6'

or, substitutinig

V = S 1l, where I = length of mnagnetic circuit:

L 01. _ 58 V L103 1914

/ 28 r16 S 6 A1.6 - S6 16

5.6 n1.6

and

58 E16L103 191A4El6 I

NAT6 5.6 n16 AT=5.6 n1.6

As seen, thle hysteretic loss is proportional to the 1.6th power

of the E. M. F., inverse proportional to the 1.6th power of the

number of turnls, and, inverse proportional to the .6th power of

frequency, and of cross-section.

If o = equivalent conductance, the energy coinponent of cur-

rent is C' = Ep, and the energy consumed in conductance p is:

Since, however,

P = C E= ' p. p= CC EF:11-.66

it is:
or,
a
P-5j17. 1]t94

a

p16
N,6

E2p

sAE' 582A1rV.6LS1.Z603nn61.46 L1119.1.4EEF4 j1.l6 5.6 n1.6

That is:
"TChe ewivvalent conductctnce due to magnetic hysteresis, is pro-
portional to the coejfieient of hysteresis, i, and to the tengtht of
the ?maqnetic ctrcut, I, and inverse proportional to the .4th
power of the E. 3. F -E, to the .6th power of the frequency, -Y, and opf the cross-seetion of the magnetic cirauit, 5, and to the
1.6th power of the number of turns, n."
Hence, the equivalent hysteretic conductance increases with decreasing E. M. F., and decreases with increasing E. M. F.; it varies, however, much slower than the E. M. F., SO that, if the hysteretic conductance represents only a part of the total energy consumnp-

Z94

STEINMETZ ON HYSTERESIS.

[May 18,

tion,it can within a limited range of variation, as for instance,
un constant potential transformers, without serious error be as-

.surmed as constant. If:
P = inagnietic reluctance of a circuit,
= maximum M. M. F., C = effective current, hence

f' 4 = maxim-um current, it is the magnetic flux:

_

F

_
=

C 4/ 2

'Substitulting this in the equation of the counter E. M. F. of ,self-induction:
E= 4/2X n IO8,
it is:
2_ n' X C1O-8

hence, the absolute admittance of the circuit:

a
0n=v2 + ar2

Pi0,

-11

2-f=

Where

b = () is a constant.
2 r n2
Thus:
T"he absolute admittan ce v, of a circuit of neyligible resistce is propwotional to the magnetic reluetance, P, and inverse
proporftional to the frequency, N, and to the squatre of the numn-
ber of turns, na." In a circuit containing iron, the reluctanice, P, varies witlh the
magnetization, that is, with the E. M1. F. HSence, the admittance of such a circuit is not a. constant, but is; variable also.
In an ironclad electric circuit, that is, a circuit whose magnetic field exists entirely within iron, as the magnetic circuit of a well-
designed alternating current transformer, P, is the reluctance of
the iron circuit. Hence, if y permleability,
since,
Jz)-

;iand
ff = L F 4wr JL M=. . F., M - S B _u S H magnetism,

1894.]

ST'EINMETZ ON HYSTERES18.

595

it is: 10 I
4 w ,u .'

and, substituting this valule in the equation of the admittance:

P 108

10

d

wlhere: Thus:

2 n2nf - S27

SN A' TNy'

d 109 8 7 =n2

127 106 nS

"in an ironclad &rculit, the absolut6e admittance, vis inverse proportional to the frequency. X, to the permeablity, a, the
cross-section, 5, and square of the number of turns, n, and
directly proportional to the length of the magnetic circuit, 7."
The conductance is:

a
7;.6 e'4

the admittance:
d
V Np.'; hence, the angle of lhysteretic advanee:

sin a = v - adENs
or, substituting for a and d:
N4t10 sin a = _Y 4 L05 8 8 w7r'2n^22 4P' 2-8 w6 56 n'6 I 101
- A 4 nn 4 S.4 7.4 22.2 -E4 1032
or, substituting: E= 2-w NYnSBO '8:
sin a 4 Bp'r4

hence, independent of frequency, number of turns, shape and size of mragnetic and electric eircuit.
Thus:
nIn an ironclad inductance, the angle of hysteretic advance, a, depends upon the maynetic constan,ts.: permeability and coefficient of hysteresis, andtpon the maeimum magnetic indutction, but is entirely independent of the frequency, of the shape and other7 coniditions of the mnagnetic and electr c circutit, and,

596

STEILNETZ ON HYSTER EIT S [May 18,

therefore, all the ironclad magnetic circuits constructed of the
same quality of iron, and using the same magnetic density, give the same angle of hysteretic advanee."
"The angle of hysteretic advance, a, in a closed circuit transformer, depends upon the quality of the iron, and the mnagnletic density only."
"The sine of the angle of hysteretic advaance equals four times the product of permeability and coefficient of hysteresis, divided by the .4th power of the magnetic density:
sin a-4 _4BuA '
If the magnetic circuit is not entirely ironclad, but the magnetic structure contains air-gaps, the total reluctance is the sum of the iron reluctance and the air reluctance:
p = Pi + i'a hence, tlhe admittance is:

P + 62 bP (P V
or: "In a circuit containiing iron, the admnittance is the sum of the
adiimittance due to the iron part of the circuit:
Vi X Pi,
and the admittance due to the air part of the circuit:

QJa ~--Vpal

if the iron and tlle air are in series in the magnetic circuit." The conductance, o, represents the loss of energy in the iron,
and, sinice air has no imagnetic hysteresis, is not changed by the introductionl of an air-gap.
Hence, the angle of hysteretic advance of plhase is:

P= sin a

Pv

-Xvi +,o Va -

p
VI P1 + f a

anid is a inaximnunm =°I for the ironcelad circuit, but decreases

with increasing width of the air-gap. The introduiction of the

air-gap of reluctIannkceer,nPpaa,ctdeiccraeasPesi si=n a in the ratio

Pi
2,

Ina the range of practical applicationi, from B = 2,000 to

-1894]

STEINMETZ ON HYSTERESIS.

597

B 12,000, the permeability of the iron varies between 900 and 2,000 approximately, while sin a in an ironclad circuit varies in this range fromn .51 to .69. In air, v- 1.
If, consequently, onie per cent. of the length of the iron is replaced by an air-gap, the total reluctance varies only in the pro-
portion of 1i9 to 1- 0 or by abolut six per cent.; that is, is practically constan-t, while the angle of hysteretic advance varies
from sill a .035 to sin a - .064. Thus p is already negligible comnpared witl a, and a practically equal to v.
Hence: "cIn an electric eirciiit containinig iron, but forming an open
magnetic circuit whose air-gap is not less than t-. the length of
the iron, the susceptance is practically constant and equal to the admittance, as long as saturation is not yet approached, and it is:
r PN,Pb, or:s= pN
The angle of hysteretic advance is small, below 4', and the
hysteretic conduetance is
a

At a sine-wave of impressed E. M. F., the current wave is practically a sine-wave."
To determine the electric constants of a circuit containing
iron, we slhall proceed in the followinlg way: Let E = conLlter E. M. F. of self-induction;
theni from the equation.

where:

E= 2wn NXM1O08,

N = frequency, f = number of turns, we get the magnetism, M, and by means of the magnetic cross-
section, S, the maximum magnetic induction:

From B we get, by means of the magnetic ehara@teristic of the iron, the M. M. F., ]F in ampere-turnis per cm. length, where
10
47
Hi = M. M. F. in (C. G. s.) units.

598

STEINMETZ ON HYSTERESIS.

[May 18,

Hence, if

=i length of iron circuit, F -= l F= ampere-turns requtired
in the iron,

'a

=

length

of

air

circuit,

Fa

10

'a B
47w

ampere-turns re-

quired in the air,

hence,

F = F1 ± Fa = total ampere-turns, maximum value, and

-Ff_ = effective value.

The exciting current is: F

and the absolute admittance:
v- Vfo2+a2=E.
If -FA is not negligible against Fa, this adinittance, v, is vari-
able witli the E. M. F., E. If:
VT volume of iron,
= coefficient of hysteresis, the loss of energy by hysteresis due to molecular magnetic friction is:

hence the lhvsteretic conductance:

0-

and is variable with the E. M. F., E. The angle of hysteretic advance is:
Sill at ,1
the susceptance:
T = 24/%_ 02,
the equivalent resistance:
=

the reactance:

q S _2

1894.]

STEINMETZ ON HYSTERESIS,

59%

As conclusions we derive from this chapter: 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.
2. This distortioni is excessive only with a closed magnetic circuit tranisferring 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 effective intensity
and equal power, and the superposed higher harmonic, consisting nainly of a term of triple frequency, can be neglected except in resonating circuits.
4. Below satur-ation, the distorted curve of current and its equivalent sine-wave have approximately the same maximum valuLe.
5. The angle of hysteretic advance, that is, the phase difference between magnetism and equivalent sine-wave of M. M. F. is a,
maximum for the closed magnetic circuit, and depends then only
upon the magnetic constants of the iron: the permeability ,a and the coefficient of hysteresis r, and upon the maximum magnetie induction, by the equation:
sin a B-
6. The effect of hysteresis can be represented by an admnit-
tance: Y +j a, or an impedance: U r -j s.
7. The hysteretic admittance, or impedance, varies with the
'magnetic induction, that is, with the E. M. F.? etc.
8. The hysteretic conductance p is proportional to the coeffici-
ent of hysteresis C and to the length of the magnetic circuit 1,
inverse proportional to the .4th power of the E. . F., L to the .6th power of frequeney N and of cross-section of the magnetie circuit S, and to the 1.6th power of the number of turns of the
electric circuit n thus expressed by the equation:
s 58 L,103
EL4 jV.6 AS16 qib
9. Tlle absolute value of hysteretic admittancee v 4/ pt + a2 ig: proportional to the magnetic reluctanee: P = Pi + Pa, and inverse proportional to tlle frequency N and to the square of the;
number of turns n hence expressed by the equation:
- (Pi + Pa) 108
V wN 2

600

STEINMILETZ ON HYSTERESIS.

[May 18,

10. In an ironclad circuit, the absolute value of admrlittance is proportional to the length of the miiagnetic circuit, and inverse proportional to cross-section 8, frequeney N, permiieability u, and
square of the number of turns n:
127
11 06
11. In an open magnetic eircuit, the conductance 0 is the same
as in a closed magnetic circuit of the same ironl part. 12. In an open magnetic eircuit, the admittance v is practically
constant, if the length of the air-gap is at least T& of the leingth of the magnetic circuit, and saturation is not approached.
13. In a closed magnetic circuit, coniductance, suseeptance and
admittance can be assumed as constant in a limited range only.
14. From the shape and the dimensions of the cireuits, and the magnetic constants of the iron, all the electric conistanits: o, vTv;
v8s, can be calculated.

IT.-Foucault or Eddy-Ourrents.
While magnetic hysteresis or molecular frictioni is a irmagnetic phenomienon, eddy-currents are rather an electrical phenomenon. Wheni passing through the iron, the magnetic field causes a loss of energy by hysteresis, which, however, does not react rrmagnetically upon the field. When impinging upon an electric con-
ductor, the magnetic field induces a current therein. The M. M. F. of this current reacts uponi anid affects the magnetic field more or less, and thus an alternating magnetic field cannot pen-
etrate deeply into a solid conductor, but a kind of screening effect is produced which makes solid miiasses of iron unsuitable for alternating fields, and necessitates the use of lamrinated iron, or
iron wire, as the carrier of magnetism.
The eddy-currernts are truie electric currenits, thouigh flowing in
minute circuits, and follow all the laws of electric circuits.
Their E. M. F. is proportional to the intensity of mnagnietization B, and to the frequency N.
Thus the eddy-currents are proportional to the magnetization B, the frequency 1N and the electric conductivity r of the iron, hence can be expressed by:
c - r B N.
The power consumed by the eddy-currents is proportional to

1894.]

STEINMETZ ON HYSTERESIS.

601

their square, and inversely proportional to the electric conduetiv-
ity, hence can be expressed by:
w= po r B2 N2,
or, since B A is proportional to the iniduced E. M. F., ]E by the equation:
E- 2 7 nn1NB0B8.
"The loss of power by eddy-currents is proportional to Ihe
'square of the E. M. F.,andProportional to 'the electric conductivity
of the iron: W - aa 2 "
hIeince that component of the effective conductaniee, whieh is due to eddy-currents, is:
TIF a
-that is:
"T he equivalent conductance due to eddy-eurrents bin the iron is a constant of the nagnetic circuit, independent of E. M. F., frequency, etc., but proportional to the electric conductivity qf
the iron rEddy-currents cause anr advance of phase of the current also,
like magnetic hysteresis, by aT angle of advance, /9, but unlike hysteresis, eddy-currents in general do not distort the current
wave.
The angle of advance of phase due to eddy-currents is:
sin j9 - _
v
where v absolute admittance of the circuit, eddy-current conductance.
While the equivalent conductance, p d'ae to eddy-currents, is a constanit of the circuit, independent of E. M. F., frequiency, etc., the loss of power by eddy-cutrrents is proportional to the square of the E. ME. F., of self-induction, hence proportional to the square of frequency and the square of magnetization.
Of eddy-currents, only the ener-gy cotponent, p E, is of interest, since the wattless componenit is idenitical with the wattless Component of hysteresis, discussed before.
The calclliation of the losses of power by eddy-cuirrents is the -following:
Let Y = volumne of iron, B = imaximum magnetic induction,

602

STEINMETZ ON HYSTERESIS

[May 18,

x - frequiency, r =electric conductivity of iron.,
=C coefficient of eddy-currents.
The loss of energy per cm.3 in ergs per cycle, is

A = ty 1NVB2,

hence, the total loss of power by eddy-currents is:

w = r V N2 B2 10- watts,

and the equivalent conductance due to eddy-currents:

_o- _VF - B2

-

10O I 2278,n2

_

.50O7
$X2

I

where:

I -length of magnetic circuit,

Se section of magnetic circuit,

An number of turns of electric circuit.

The coefficienit of eddy currents, a, depends merely upon theA

shape of the constituent parts of the magnetic eireuit, that is,

whether iron plates or wire, and thiekness of, plates or diameter

of wire, etc.

The two most importanit cases are:

(a), laminiated iron,

(b) iron wire.

a. Laminated Iron.
Let, in Fig. 14,
d = thickness of the iron plates,
B = maximum. magnetic induction, N = frequieney,
= electric conductivity of the iron.
Then, if x is the distance of a zone, d ?, frorn. the center of the sheet, the conductance of a zone of thickness, d x, anid one cm. length and width is, d x; and the magnetic flux ecut bythis zone is, B x. Hence, the E. M. F. iniduced in this zone is:

8E = 7-2NW Bx (c. G. S.) UDitS.
This E. M. F. produces the current:
d C = a E;, dx 4/2 N7Af B r x d x (C. G. S.) Units,
if the thickness of the plate is negligible compared with the length, so that the current can be assumed as flowinig parallel to the sheet, in the one direction at the one, in the other direction at the other side.

1894.]

STEINYIETZ ON HYSTERESIS.

603

The power consumed by-the induced current in thi8 zone,

d xis:

d W=6Ed C=2w2NI2B' yX d X (C.G.S.) units or erg seconds,

and, consequenXtly, the total power consumed in one cm.2 of the

sheet of thickness, d:

d

+~~~~~~~~~~d

wTf. 2d4v 22 2 _d

I xd1/C _d

N22 B' r (C. G. S.) units
6

henee) the power consumed per cm.3 of iron:

8 TF _ w2 N2 B2 r d2 (C. G. s.) units or erg seconds,

and the energy consumed per cycle and per cm.3 of iron;
A w - 2rd2 NB2 ergs

Thus, the coefficient of eddy-eurrents for laminated iron is: 72 6d 2 = =1.d645 (j2

where r is expressed in. (C. G. s.) units. Hence, if r is expressed in practical units, or mho-centimetres, it is:

S, __2dd2-2 _10-9 -= 1.645 d 2 10-. 6

Substituting for the conductivity of sheet iron the approxi-

mate value:

we get:

10r )

Coefficient of eddy-currents for laminated iron:

c-d2 10- = 1.645 d 10-9. 6 Loss of energy per cm.' and cycle:

h

re

TNB2=

72
6

IA d2 r B2 10--=1.645

d2 r NB2 10-9 ergs

1.645 d2 NB2 10- ergs; or,
h = r XNB' 10-' 1.645 d2 NB2 10-"1 joules.

Loss of power per em.3 at frequency N:

604

STEIINMETZ ON HYST"ERESIS.

[May 18,

w N A = r XNI2 B2 10-7 1.645 dQ2 N2 B21 0-11 watts, and, total loss of power, in volume V:
W Vw 1.645 V d2IXI B2 10-11 watts.
Instance: d =1 mm. .1cm. N= 100. B - 5,000. V= 1,000 cm.3
= 1,645 X 10-",t
h = 4110 ergs = .000411 joules, w = .0411 watts, W 41.1 watts.

II

FIG. 14.

FIG. 15.

b. Jron Wire.-Let, in Fig. 15 d diameter of wire; then, if x is the radius of a circular zone of thickness, dx, and
one cm. length, the conductance of this zone is d and the

magnetic flux enclosed by the zone is B x2 7. Hence, the E. M. F. induced in this zone is:
JE 42r2NXBx2(c. G. S.) units,

1894.]

STEINAETZ ON HYSTERESIS,

605

and the current produced thereby:
dC 22rzx:SX 4/2 7r2 -TB x
= V 27 rNB X d X (C. G. S.)units,
2
hence, the power consumed in this zone:
d W = 8 Ed C w73 rN2 B' X2 d X (C. G. s.) units, and, consequently, the total power consumed in one cm. length
of wire:
C2d
W fdo Wo 7rN2B2fx2d x

: N ]3B24 ' (C. G. s.) units.

Since the volunme of o-ne cm. length of wire is: d2 w
4,
power consumed in one cm.' of iron is:

w=W - 176 r N2 B2 d2 (a. G. s.) units or erg seconds,

and the energy consumed per cycle and cm.' of iron:

2
N- 16

r NIVB2

ergs.

Thus, the coefficient of eddy-currents for iron wire is:

£- d 2 =.617 d2

or if r is expressed in practical units or mlho centimetres - 10-O
absolute units:

Substituting: we get:

_T d2 1O- .617 d2 iO-n.
16
r 101,

Coefficient of eddy-eurrents for iron wire:

s= d2 10-
16,

617 d2 109.

Loss of ener-gy per cm.' of iron, and per cycle:

h-e rN1VB2 _ 71T6 d2rJNB2l O .61T d2 r N B2 to-9

606

STEINMETZ ON HYSTER4'SIB.:

[May 18,

- .61 d2 NIV 1)-4 ergs,

=sr NvB2 1t-7- .67 de -ATB2 1)-0 joules.

Loss of powver per cm.13, at frequiency N:

w V-Ah

N2 B2 10-= .617 d2 N2 B2 10-" watts,

and, total loss of power, in volume lY:

W Vw = .61NVdI 2 B2 10-"1 watts.

instance:
d 1 mm.he = 1 cmi. N 100. B 5000. V = 1000 cmA .617 X 10-1,
h 1540 ergs .000154 joules,
w .0154 watts, W = 15.4 watts, hence very mueh less than in
sheet ironl of equal thicknless.

Comparison of sheet iron and iron wire. If
d, thickness of lamination of sheet iron, and
d= diameter of ironl wire, it is: ,coefficient of eddies in sheet iron:

El= 6 d12 10-9;
coefficient of eddies in iron wire:
2
- d22 1(i-9

The loss of power is equal in both-other tlings being equalifs1 = s2,that is:
d22 Q d

or,
That is:

d2 = 1.63 di.

The dianeter of iron wire can be 1.63 tiunes, or roughly l-

as large as the thickness of laminated iron, to give the same loss

of energy by eddy-currents.

ALTERNATING C-URRENT TRANSFORMER.

The relative proportionis of wire and lamina are shown in
Fig. 16. The samie formulas obviously apply to the eddy-currenits in
masses of ally other material, substituting for r the proper valuie.

1894.]

STEINM ETZ ON HYSTERESIS.

607

As an instance of the calculation of ferric inductances, the general equations of the alternate current transformer iiay be given.
Let:
Y , + j co hysteretic admittaniee of primary coil, CDo = -o j o = impedance of primary coil,
U1 - -, rJ S1 - imnpedance of secondary coil,
where the inductances, s8 and s,, refer to the flow of trie self-in-
duction, that is, that magnetism, which surroutnds one of the
transformer coils onily, but not the other.
Let a = - ratio of tuirns of primiiary and of secondary
nl
'Coil.

FIG. 16.

Then, denoting th-e terminal voltage of primary anid of

,secondary coil by FE0 and FE, and the E. M. F.'S indu-ced in these

coils by the magnietic flux surroun-dinig them by ElV and F,', it is:

E0' a FEl.

Denotinig the total. admittance of the seconidaryv cir-cuLit-ini-

eluding the internal impedance of the secondary coil-by:

Y/ P' + a

the secondary cuirrent is:

, YE,El,

consistinig of the eniergyv compo-ne-nt, p, E,', a-nd the wattless

component, or1 El'.

Hereto corresponds the compoinent of primary current, by the

ratio of turns:

oC -_ El,'

la

a

608

STEINMETZ -ON HYSTERESIS.

[Mlay 18,

The primary exciting currenit (current at open secondary circuit) is
Coo = Yo0Eo'
Yo a E11,
henice, the total primiary current:

cO=C+ coo _ I ( Y1 + a2 y), a
and, the ratio of primary and of secondary current:

C, a (+ Y, The terminial voltage of the secondary coil is:
E1 = E1- U C1 = E1 (1- U1 Y1)
The terminal voltage oL the primary coil is
Eo =Eko + Uo Co aE,' + I 0 (Yl+a Y0)

a E1 1 + O0 = 2 )

hence the ratio of primary and of secondary terminal voltage:

1+U~Y0

EO
El

_

a

1

+

ta4o

YO + ul Y

a2

That is, if, at the primary impressed E. M. F., Eo, the secondary

circuit is closed by the adimittance Y1, it is:

IRatio of transformnation of E. M. F.'S:

+ 1 + UO Yo

,

F, a

Itatio of transformation of currents:

C +( a Yo

where these ratios aie complex quantities of the form:
p (Cos + j SillC),
thlls denoting the numerical value of the ratio of transformation
by the vector_p, and tlhe plhase diflerence between primarv and
seconidary eircuit by angle c(.

1894.]

DISCUSSION.

609

DISCUSSION. DR. BEDELL:-Mr. President, I would like to comment on the remarks of Mr. Steinmetz on the idea of the equivalent sinewave.' The distorted natnire of actual current waves has been particularly emphasized in the valuable paper to which we have listened this morning. Although we know that this distortion exists, we still find it convenient to make what we call the "1 sine assumption." Now this sine assuimption does ilot mean as commonly supposed, that we consider that the current is actually harmonic. When- we assume a harmonic current we simply assumle a harmonic current to which the actRal cuirrent is equivalent. This has, I think, been already pointed out by Mr. Steiunietz as well as by Dr. Crehore and myself.2 The sine assumption with this meaninig has proved very useful in combining experimental and theor etical results, and is not open to the criticism which is often given, that we do not have perfect sine cuirrents unider ordiniary eircumstances. I would like to question Mr. Steinz-etz in regard to one other point; that is in regard to the hysteresis loss in the revolving
armature as compared to the Ihysteresis loss in the transformner,
and I would like to ask how he applies his law to the two cases. MR. STEINMETZ :-With regard to the loss of energy by mag-
netic friction in a rotary mnagnetic field, as for instanee in the
revolving armature of a bipolar smooth core dyniamo, I found
no essential difference with the loss in an alternating field. But I found that occasionally the observed core loss in the armature of a machine is not the molecular magnetic friction only, but superimposed upon it are eddy-current losses in the iron, the shields, etc., and in the coniductors, which losses are proportional to the square of the magnetization. Thus, the observed core loss sometiimes rises with a power higher than 1.6, sometimes
nearly approaching the square. But by lam inatitig the iron very carefully, designing the mechanical construction so as to expose no solid mnetal to the alternating field, and shapinig the conduetors
so as to exclude eddy currents, 1 always got curves very nearly
proportional to the 1.6 power, like the one I show here for a variation of voltage up to 9,000 volts, that is, up to very high
magnetic densities (about B = 19,000). There vou see the curve
of 1.6 power in drawn line, very closely representing the ob-
served core losses. The points marked by crosses are the observed values of the power consuimed by the generator less the
friction of the belt. So I thinlk the law holds for generators just
the same, and therefore I believe the law applies not to the hysteresis loss, but to the loss by molecular magnetic friction, since in the generators we probably have no hysteresis. I took
1. TRANSACTIONS, vol. xi, p. 46. 2. Geometrical Proof of the Three-ammeter Method of Measuring Power. PAysical Review, vol. 1, No. 1, p. 61.

610

STEINMETZ ON HYSTERESIS.

[May 18,

pains once to find out if there is a lag of the magnetism belhind the resultant magnetizing force in a generator, which wouild distort the wave of electromotive force, but I did not find anything of the kind. I found no hysteretic lag. Thus the total loss of energy, which as you see here in this case is many kilowatts, is supplied directly by the mlechanical power, in which way I am not able to say, but it is not in the form of a hysteretic loop, at least nrot a hysteretic loop of noticeable size.
PROF. ANTHONY:-I would like to ask one question simply to see whether I have properly understood Mr. Steinmetz. I understand hi'm to mean when lhe speaks of equivalent sinecurves the several component sine-cutrves into which the distorted curve could be resolved.
MR. STrEINMETZ :-No, I meanit a true sine-wave of current of
the same frequency as the fundamental, the same effective intenlsity as the total distorted wave, and shifted against the equivalent sine-wave of electromotive force by such an angle that its power in watts equals that of the distorted wave. I can say that the eqivalent sine wave is not identical with the fundamental sine-wave, except in the case where the sum total of higher harmonics is wattless, because the equivalenrt sine-wave includes the energy of the higher harmonics also, and thus the remnainder, or the difference between distorted wave and equiva-
lent sine-wave, genierally includes a component of the same fre-
queney as the fundamenital. MR. KENNELLY :-This paper seems to me to be valuable,
first for its bearing upon the subject of hysteresis and its nature, and, secondly, upon the practical determination of inductances or of equivalent inductances in coils containing iron, such as transformers. The main point, it seems to me, ca-n be stated in a very few words. Whe-n the current is no longer a sinusoidal wave, if it becomes distorted by the action of iron in the circuit, it is a complicated wave such as shown at F in the Figs. 7 and 8, etc. But the ammeter or dynamiiometer which is used to measuLre
that distorted current will show some effective current strength
which might be attributable to a pure sinusoidal current. It would show a currenit strength in amperes which would be rep-
resen1ted by the curve c, so that the real current F, whose shape can only be determined by a long series of experimnents, has an equivalent representation in the dynamom-eter such as would be
produced by a current of the pure sine shape of c. But if you do not carry the magnetization too highs the amplitude of the pure sine-wave a, such as the dynamometer, would lead you to suppose exists, and the amplitu-ide of tlhe actual distorted wave F are equal. This, if true, is an important and valuable proposition, because it gives you the maximum number of ampere-turns on the magnetic circuit, the niaximunii cyclic magneto-motive force. But it is pointed out that when you get beyond 10 kilogausses in your iron, you will no longer have this relation main-

1894.]

DISCUSS] ON.

611.

taimed. That is inl agreement with the observations in Dr. Pu pin's valuable paper read this morning, where it is shown that the harmonics of his primary currents remained proportional to the current strength if he did not go up too far in flux intensity, and that is bearing directly on this paper. If you do not go beyond 10 kilogausses you will probably have those two wave crests on the same line.
DR. BEDELL :-There is one point to which a little further attention might be given, and that is in regard to the lag of the current behind the electromotive force when the current and electromotive force are not harmonic. Those who have had occasion to yake a study of currents which are not strictly harmonic and desire to find the phase relationis, have doubtless met this question. The phase difference between the maximum values and zero valuLes or any other values of the current and electromotive force are not the same. The use of the equivalent sine function is the solution of this question. We assume an eqluivalent electromotive force which is harmonic and has the same mean square value as the electromotive force which is not harmonic, and we do the same with the current. We then set these two with such an angle of lag between them that the power is the same. Now we can get our power from other measurements and by these measurements of the power, the cnrrent and the electromotive force, we thuis have a measure of the angle of lag in degrees, which cannot be otherwise obtained when the currents are far from being harmonic. In other words, we say the power
is W = E Icos 0. By measuring W, E and I, we may find a avaue for the angle 0, whether the current is harmonic or not.
MR. STEINMETZ:-1 would like to point out one thing here, not to allow a misconception to arise. This dissolving of the distorted wave into an equivalent sine-wave, and a wattless remiainider is not identical with the dissolving of it by Fourier'& theorem into a series of sine-waves, becaLLse the equivalent sinewave c is not the fundamental component of the total wave, but the wattless remainder of apparently triple frequency, shown here, may contain a term of simple frequency.
To fix a definition of this equivalent sine-wave, it is " a sinewave of equal effective intensity and equal power witlh the true wave." If you take a wave of electromotive force, for instance, and a wave of current, then thle higher harmonies may, but need not, be powerless. This is especially the case if you have the cuirrent distorted by hysteresis.
DR. PuPiN:--1 might say a word or two on this paper of Mr. Steininetz, a very interesting paper indeed. In the first place
in studying these harmonies in the course of last year I had,
especially, Prof. Rowland's paper of 1892 to guide me, in which a radically different view was taken from that of Prof. Fleming. Comparing these two views with my own work, it seemed to miie that they could be reconeiled to a certain extent in this way.

612

STEIN El Z ON HYSTERESIS.

[May 18,

The hysteresis loop reminds us of two things: In the first place, of the loss of energy, and, in the second place, of the variation
of permeability. Now Dr. Fleming ascribed the generation of harmonics to the action of hysteresis in general, not saying exactly what he mean-t by it. Hysteresis is a very broad term and may be made to miean a great many things. Prof. Rowland specified his view and ascribed the presence of harimionies to the
Variation of permeability. Both views, therefore, refer to the
bysteresis loop for an explanationi of the distortion of the current wave. In the course of a discussion' at a meeting of this INSTITUTE, I suggested that the distortion of alternating current waves
could be very well studied by studying, with the aid of the hysteresis loop, the process of magnetization and demagnetization during each cycle. Mr. Steinmetz's mnethlod is exactly the
method to whiclh I referred at that time. I am sorry that Mr.
SteinTmetz has not explained the details of the miethod of hiis ilvestigation and the data obtained by it, which enabled hin to plot the harmonics of various frequiencies fromn the hysteretic
loop. Another point that I would like to mention refers to what Mr.
Steinmetz calls " molecular friction." The distinction between mnolecular frietion and hysteresis does not seemii quite clear from Mr. Steinrnetz's paper. I have expressed my opinion on several occasions in the course of this and last year, that there are certain
bphenomena going on during each complete cycle of magnetization
of iron which cannot very well be explained by Foucanlt current and hysteresis as commonly understood, but which phenomena seem to point out clearly the existence of additional passive resistances. Possibly Mr. Steinmetz means the same thing when he speaks of molecnlar friction. There is certainly a very marked difference betweena the action of iron when it forms a closed magnetic cireuit and when it does not formr such a circuit especially in its damiping action upon a resonating current. Again, certain kinds of iron may have a large hysteretic constant, but only a small damping constant, etc. These differences appear at all magnetizations, even at miagnetizations due to telephonic currents, and are especially marked at higher frequencies. There is a certain magnetic sluggishness in every piece of iron, and it is my opinion that this sluggishness is not measured by the hysteretic action as ordinarily understood, nor by Foucault current losses. Now what this sluogishlness is, it is difficult to tell. The invention of a new name like "mmolecular friction" certainly does not advance our knowledge one bit. It may retard it if the new name should lead us to believe that further inquiry into the matter will lead to nothing more than mere commonplace molecular friction.
MR. STEINMETZ:-I think Dr. Puipin is mistaken in his state-
1. See discussion of Dr. Bell's paper, " Practical Properties of Polyphase Apparatus." TRANSACTIONS, V01. Xi, P. 46.

1894.]

DISCUSSIO .

ment with regard to the name hysteresis. The word has a well-

defined nmeaning. It was introduced merely to denote the lag of

the imagnetism behind the magnetonmotive force, as the derivation

of the word signifies, which lag eauses the magnietisim as function

of an alternatinig m. M. F. to describe a closed curve, the "loop of

hysteresis."

Afterward it was shown by Warburg and Ewing that the area

of the hysteretic loop represents energy, and represents the energy

expended by the magnetomotive force during the cycle of mag-

netism, and from this, the erroneous conclusion has been drawin

that this hysteretic energy is the energy lost in the iron by molec-

ular magnetic friction, that is, by changing the magnetic state

of the iron. That is what I want to make clear-that this con-

clusion is wrong; that this energy expeTnded by the magneto-

mtiotive force is not necessarily the energy wasted in the iron.

The energy represented by the hysteretic loop or a part of it i ay-

be converted into mechlanical motion, or the energy lost inl molec-

ular magnetic friction may be supplied by meehanical energy,

and the hysteretic loop may collapse, oir may expand considerably,

so that between the area of the hysteretic loop and the loss of

energy in the iron there is no direct relation. I have explained

this quite fully and slhowni by tests in my second paper on hlys-

teresis.1 Since, however, it seems to have escaped attention,,

probably advisable

due to the length of to discuss it again more

aforesaid paper, I fully in my present

thtoughgt
paper.

it

Now with regard to the changes of permeability and to hyster-

esis as producers of higlher harmonics, ttie statement that hysteresis

produices higher lharmonics, is quite correct. It produces higher

harmonies, but change of permeability does the same, or rather,

hysteresis is nothing but a change of permeability. Take this

case I sihow lhere on pages 575-7, Figs. 2 and 4. There you have the

loop of hysteresis produced by the variable permeability. What

Prof. Pupin means in his statement that hysteresis does not pro-

duce higher harmonics is probably that molecular magnetic fric-

tion does not necessarily cauise higher harmonics, and with that I

agree; higher harmiionies of current appear onlv when the molec-

ular magnetic friction causes a variation of permeability in the

form of hysteresis. But beside this, there are undoubtedly still

other causes, which produce higher harmonies, whiclh are neithei-

change of permeability nor hysteresis.

Of any sluggishness displayed by the iron in changing its

magnetic state, I have never found any trace wlhiel could not be ex-

plained as the effect of the hysteretic loop, and thius do not believe

that aniy such sluggishness or viscous hysteresis exists at ordinary

frequencies of a few hnndred cycles.

The difference in the action of a closed circuit transformer and

an open circuit transformer is fully explained by the fact that the

open circuit transformner is at open secondary eircuit highly in-

1. TRANSACTIONS, 1892, vol. ix, cha1pter v, p. 711.

614

STEINMETZ ON HYSTERESIS.

[May 18,

ductive; that is, the current passing through it is almost all idle or wattless current, having a small energy component only. In the closed circuit transformtier the magnetizing cuirretit is so small that the exciting cutrrent is largely energy current-hysteretic energy current-the angle of lag being even at open secondary circuit only from 40 to 60 degrees. This explains that no reson-
anice can be produieed by a closed circuit transformer, since re-
sonance presupposes a highly inductive circuit, which the transforrner is not.
Can anyone inform me when the relation between the distortion of the alternating current wave and the hysteretic loop was first stated by Fleming?
DR. PUPIN:-It is in the second volume of his book. Mu. STEINMETZ :-If you go back, for instance, in our TRANS-
AC rIONS to Prof. Ryan's paper', I think it came out in 1889, he plotted the hysteretic loop from the wave shape of the current,
thereby making use of the feature, that the distortioni of the cur-
renit wave is due to the hysteresis, and that the hysteretic loop
can be reproduced from the distortior. What I did here was merely to reverse the process. Buit this has probably also been done before that.} Thus I did not need to give a very explicit description. But I think the credit of having first shown this rela-
tion between distortion and hy)7steresis is due to Prof. Ryan.
DP. PuPIN:--I do not tllink that Prof. Ryan employed the hysteretic loop for plotting the various harmonics. If I remember correctly, the curves of current and electromotive force were plotted by sliding contact, and then the harmonies were deternined by the ordinary method of harmonic analysis.
AIR. STEINMETZ:-I think hie did it directly froni the slhape of the wave of the current, iot from the watt curve, if I am not mistaken. I really do not remember exactly.
DR. PUPIN:-Perhaps Dr. Bedell can tell us? DR. BEDELL:--I think that the relation between hysteresis and the shape of the current curve was first brought out by Professor Ryan and described by hiim in his paper2 oni transformer, before
this INSTITUTE in I889. In conjunction withl Professor Merritt, he constructed a hysteresis loop from. the cnrves of- current and
electromotive force taken by the method of inistantaneous con-
tact. From these curves for current and electromotive force, they did construct a watt curve, as Dr. Pupin states, but they made no nse of thiis in determining tihe hysteresis loop, obtaining the latter directly froin the instatntaneous curves. That this
relation between the cuirrenit curve anid the hysteresis loop existed
had been pointed out a little before this tine by Dr. Hopkinson,' who showed the relation by ineans of a gi'aphical construction
1. TRANSACTIONS, vol. vii. p. 1. -2. Jbid; 3. Hopkinson: " Induction Coils or Transforrners." Proceedings of the Royal
Society, Feb. 17, 1887. Also given on p. 184 of his re-printed papers.

1894.]

DISCUSSION.

615

involving three dimensions whichl was based upon somne results obtained analytically fromi fundamental differential equations. As far as f am aware, however, it has not been until receintly that Dr. Hopkinson has made any investigations in this direction. In a paper' published a year ago or so, he described an extended investigation in which lhysteresis loops were obtaiined for different frequencies from curves taken by the miethod of instantaneous contact. This is, I think, the most complete investigation uipon this line of work which has thus far been published; but it differs fron the work of Ryan and Merritt only in its greater comnpleteness.
In a paper2 published about a year before the work done by Professors Ryan and Merritt, Dr. Sumpner showed a very pretty graphical construction for obtaining the cutrrent curve when we are given the electromotive force and a curve showing the relation betweenI the current and the time-constant of the circuit. This is at least of considerable theoretical interest; but he could
have carried it further. Furtlhermnore, if I remember rightly, he
did not take a different time-constant curve for his ascending and
descending values. He did not accomplish by his rmietlhod, hlow-
ever, that which was done by Ryan and MJerritt, viz., the construction of a hysteresis loop from the culrrenit curve.
Dielectric hysteresis, as well as magnetic, affects the shape of the current curve. I have already had the pleasure of calling the attention of the INSTITUTE to this relation, and of describing a method for deteriniininz the hysteresis loop for a condenser. Such a loop is given in the TRANSACTIONS' for last year.
Each one of the papers I have referred to has contributed
something of value to the question at hand, and due credit should
be given to each of the several writers; but I think that to Professors Ryan and Merritt inust be given the credit for the practical development of the subject. The harmonic analysis of these curves accordinLg to Folurier's theorem was worked out by them and is given by Dr. Fleming in the second volute4 of his work on transformers. The fundamlenital together with the third and fifth harmonics were found to closely represent the actual distorted wave.
In conclusion I would say that I consider all this work of par-
ticular significance, combining, as it does, observed phenomena and nathematical analysis. Theoretical deductions are always
based upon certain premises, and in many cases these premises
have consisted of artificial coniditions. The conelusionis are rigor-
ously true under the assumed conditions, but the conditions are unobtainable. We are acquiriing greater ability in making our
1. Drs. J. and B. Hopkinson : London Electrician, Sept. 9, 1892. Also: "Gray's Absolute Measuremiients in Electricity and Magnetism," vol. ii, p. 752.
2. Sumpner: Plhilosophical Jlagazimc, Junie, 1888, p. 468. 3. TRANSACTIONS, Vol. X, P. 525. 4. "Alternate Current Transformer," vol. ii, p. 452.

616

STEINJiETZ IOV HYSTERESIS.

[May 18,

conditions accord with facts. It has often happened tllat our conclusions are only true in case hysteresis be absent and the current is a true sine-wave. But this need not be: we may
make quantitative assumptions as to the hysteresis present, and may assume the presence of such hlarmonics in addition to the
fiundamental wave as- occasion demands; predetermination becomes possible, and ouir work becomes definite and exact.