vol. 13, pp. 24-25; Encyclopedia of American Biography, 1974, pp. 1035-1036; P. L. Alger and C. D. Wagoner,
“Charles Proteus Steinmetz,” IEEE SPECTRUM, pp. 82-95, Apr. 1%5; Steinmetz the Philosopher, compiled by
Philip L. Algerand Ernest Caldecott (Schenectady, NY, 1%5); Ronald R. Kline,“Professionalismandthe Corporate Engineer:Charles P. Steinmetz and the American Institute of Electrical Engineers,” IEEE TRANS.
EDUCATION, vol. E-23, pp. 144-150, Aug. 1980; Ronald R. Kline “Charles P. Steinmetz and the Development of Engineering Science,” Ph.D. dissertation, University of Wisconsin-Madison, 1983; James E. Brittain, “C.P. Steinmetz and E. P. W. Alexanderson: Creative Engineering in a Corporate Setting,” Proc. /€€E, vol. 6 4 , pp. 1413-1 41 7,1976.

ON THE LAW OF HYSTERESIS.

BY CHAS. PROTEUS STEINMETZ.

In the number 137,of December 17th, 1890,-of theElectrical Engineer I published a short article under the title “Note on the Lawof Hysteresis,”where I showed that in a setof determinations of the lossof energydue to hysteresisby reversals of magnetism, for different magnetizations, made by Ewing, this loss of energy due to hysteresis can fairly well be expressed by the equation:
H =VB’.~,
where H istheenergy consumedbyhysteresisduringone magnetic cycle, in ergs per cubic centimetre,B the magnetiza-
tion in lines of magnetic force per squarecentm i etre,and q(’) a numerical coefficienti,n this case = .002.
Considering that even the simplelawofmagnetism-that is, the dependence of the magnetization B upon the magnet@
motive force F (for instance, in ampere turns per centimetre
length of themagneticcircuit) has until now defied all attempts of mathematicalformulation, it appeared astrange feature that the apparently much more intricate phenomenon of hysteresis, or rather of theconsumption of energy by hysteresis,shouldyieldtoanalyticalformulation in such a simple way, to be directly proportional to the 1.6th power of the magnetization. At the samtieme the coincidenceof Ewing’s tests with the curve of the 1.6th power was near enough to be considered as something more than a mere incident, but at least as a clue to a law of hysteresis, the moreas this law held not only for low and medium magnetizatiobnu,t even for very high saturation, withoutshowing any kink at that point where the magnetic characteristicgoes over the bend or “knee” and thereby entirely changes its shape, nor any marked tendency of deviation of the extremest observed values from the calculated m e .
Reprintedfromthe American lnstitute of ElecfricaEl ngineers Transactions, vol. 9, pp. 344,1892. Copyright 1892 by the American Institute of Electrical Engineers.
’If any quantity has a nght to be ded “magnetic resistance,” it is this
coefficient 7 ; for 7 is the c&cient of conversion of magnetic energy into
heat, while as “electric resistance” we define thecwfficent of conversion of electric energv into heat.
The term generaIiy denoted ‘‘magnetic resistance”-that is, the inverse value ofmagneticconductivity,docsnotdeserve this name at all, but is &re properly caned “reluctance.”

F
1.50 1.95 2.56 3.01 3.76 4.% 6.62 7.04 26.5 75.2

3,830 5,950
7,180 8,790 10,590 11,480 11.W 13,700 15,560

WLE I.

H:

1160 2190 2940 3990 5560 6160 6590 8690
10,040

CalC
375 1082 2190 2956 4080 5510 6260 6690 8310 10,190

iT

H - H:=
calc +3o5bs + 58
... - 16
-90
+so
- 100 - 100 + 380
- -150
Av: i98

-
s:
+ 8.5 + 5.0
... - .5 -23
+ .9
-1.7
- 1.5 + 4.4
-1.5
f2.6

In Fig. 1 and Table I, I give from the article referred to, the calculatedcurve of hystereticloss, as adrawnline,with Ewing’s tests marked as crosses, and in dotted line the curve of magnetomotiveforce F, c o r r v n - tothedifferent
magnetizations, as abscsisae.
In the table, I: F = the M.M.F.,in absolute units,
B = the magnetization,in lines of magnetic forceper square centimetre,
H = the observed values, and
ObS

STEINMETZ:ON THE LAW OF HYSTERESIS

197

H = the calculated valuesof hysteretic loss,in ergs per cubic centimetre,
H - H = thedifference between both, in ergs and in
calc obs percentages.

To study morecompletely this phenomenon of hysteresis

and of the energy consumption caused thereby, I endeavored

to make a number of determinations with different magnetic

circuits and at different magnetizations.

To be enabled to carry out these experiments, I am highly
obliged to Mr. Rudolph Eickemeyer, of Yonkers, N. Y.,who,

being greatly interestedin the laws of the magnetic circuit and

havingdoneconsiderablework himself in this branch of

msa,l electrical science, not only put the large facilities of his well-

known factory at my

but also guided the experiments

with his valuable advice. Apart of the instrumentsused in the

tests are of Mr. Eickemeyer's invention and coveredby his

patents.

To beabletodealnotonlywiththesmallamounts

of

energy which the reversal of magnetism in a tiny bit of iron

wire sends through the ballistic galvanometer, but to reduce

the determinations to readings of considerable power-values,

and where a much greater exactnesscan be reached, andat the

same time to determine the dependence of the hysteretic loss

of energy upon the velocity of the magnetic cycles, I decided

to use alternatingcurrents, at least as far as this could be

done, whereby the determination of the energy consumed by

hysteresis is r e d u d to a simultauwus wattmeter, voltmeter,

ammeter and speed reading.

At the same time this electro-dynamometer method has the

advantage that the magnetic cycle is completed in a steady,

continuous motion,while in the ballistic method the magnetic

cycle is completed by sudden changes in the magnetization,

which jumps from point to point,to enable the productionof

the induced current. This feature introduces an errorinto the

ballistic method, for if a magnetic cycle is gone through by

sudden changes, a larger amount of energy may be consumed

than if the magnetization varies steadily in harmonic vibra-

tion.

Suppose, around a magnetic circuit, an alternating current

of N complete periods per second is sent in n convolutions.

Let C=the effective strength of the current,

E =the effective E. bf. F. induced in the circuit by self-in-

duction, after subtracting theE. M. F.'S induced by the

self-induction of the instnunents,

W =the energy consumed in the circuit, after subtracting

the energy consumed by the electric resistance.

Then, I beingthelength and s thecross-section of the
magnetic circuit, allin centimetres, amperes,volts, watts, etc.,

Let B==themaximum magnetization in lines of magnetic

force per squareamtimetre,

H ==the loss of energy by hysteresis, in ergs per cycle and

cubic centimetre;it is

hence

w = L ~ N Hx 10-7
H = - LWrN X 10+7

the hysteretic loss of energy, and

hence

E = f i m B N n X lo-'

B=

E X
amNn

the maximum magnetism.

For higher frequencies, 80 to 200 periods per second, the

alternating currentwas derived from a1H. P. 50 volt Westing-

housedynamo. This was drivenbya 3 HP.. Eckemeyer

continuouscurrentmotor. Byvarying theexcitation of the

motor field and varying the E. M. F. supplied to the motor, the

speed and therefore the frequency of the alternating current

could be variedin wide limits.At the same time, supplied with

constant E. M. F. andlikealtlheEickemeyermotors

of

unusually small armature reaction, this electromotor kept al-

most absolutely constant speed under varying load, the more

as it never ran with full load.

For low frequencies, this bipolar continuous current motor

wasused as abipolaralternatingdynamo, as shown in a

patent of Mr. Stephen D. Field. On the continuous current

commutator two sliding rings weremountedandconnected

with opposite commutator bars. In the ordinary continuous

current brushes a continuous current was sent in, which set

themachine in motion as an electromotor,whilefromthe

slidmgringsby two separatebrushes,alternatingcurrents

were taken off. By varying the E. M. F. supplied to the motor,

the E. M. F. of thealternatingcurrent wasvaried,while a

variation of themotofrieldgavethevariations

of the

frequency. The curve of E. M. F. was very nearly a sine-wave,

the ratio of maximum E. M. F. to effective E. M. F. found

= 1.415,whilethesine-waverequires 1.414-ht is, essen-

tially the same.

To determinewhetherthechange of theshape of the

alternating currentby varyingload and varying excitationhad

any influenceupon the readmgs, the variationsof the alternat-

ing E. M.F. were produced:

1. By varying the excitation of the field of the Westmghouse

dynamo.

2. By running the Westmghouse dynamo fully excited feed-

ing the secondaries of a bank of converters, feedmg from

the fine wirecoils of these converters the fine wirceoils of

another bank of converters, and taking current off from

the secondariesof these converters, connected from onteo

six in series.

3. Bychangmg the E. M. F. bymeans of a Westinghouse

converter of variable ratio of transformation.

4.By loadmg the dynamo when small currents wereused for

the tests.

But afterhavingfound that allthesedifferentways of

varying the alternating E. M. F. gave no perceptible difference

whatever in the readmgs, I afterwards used the most conveni-

ent way to vary the excitationof the dynamo field and, where

higher E. M. F'S wereneeded,toincreasethe E. M. F. by an

interchangeable converterw, hich gave the ratios:1:1,2,3,4,5.

For thedetermination of thefrequency,adirect-reading

speed indicator (horizontalball governor, actinugpon a spring)

was used, which was carefully calibrated.

For theelectricreadings,instruments of theelectro-dy-

namometer type were used,zero-reading-that is, the movable

coil was carried back by the torsion of a steel spring to zero

position.

*This formula holds rigidly only for the sine-wave, but as shown in the follow& the currents &LI in the tests were at least very near sim-waves.
Besidegadeviationfromthcsiacshapewouldnotaltcrtherrsultsatall,
but only slightly change the coefficient q.

198

PROCEEDINGS OF THE IEEE. VOL. 72, NO. 2, FEBRUARY 1984

These instruments were specially built for alternating cur-

rents, with very low self-induction and low internal resistance,

using bifilar german silver wire as additional resistance.

In theammetertherange of readingswasfrom 3 to 40

amperes, the internal resistance = .011 o.

Thenormalinductance (that is, E. M. F. of self-induction

induced by one ampere alternating current, flowing through

the instrument with a frequency of 100 complete periods per

second): = ,045 o.

In thevoltmetertherange ofrea-was

from .5 volts

upwards, but to avoid the necessity e: corrections for self-in-

duction sufficient additional resistance was used to decrease

the correction under1 per cent., and then the lowest readings

were from 3 to 6 volts.

The internal resistance of thevoltmeter is = 2.5 w , its

normal inductance = 4.12 w .

In the wattmeter the resistanceof the coarse wire coil (fixed

coil) was = .026 o,is normal inductance = .073 u.

The internal resistanceof the fine wire coil was = .25 o,its

normal inductance = .33 a.

In most of the readings sufficient additional resistance was

used to make the correction for self-induction of the fine wire

coil neghgible. Only in a few readings where it exceeded 1 per

cent. it was taken in account.

For small currents an Eickemeyer ammeter was used, which,

while reading from .7 to 3 amperes, though built originally for

continuous currents, had alreadbyeen used by me for alternat-

ing currents and had been checked for its constancy of read-

ings several times, and always found to give

no perceptible

difference in its readingsfor continuous currentsand for

alternatingcurrentsupto over 200 complete.periods per

second, the highest frequency I could reach

Its internalresistance is = 1.1 o,its normalinductance

= 2.03 o.

Several sets of readings for different frequencies were taken

on an old Westinghouse voltmeterconverter.Thefinewire

coil and one of the 50 volt coils were left open.Into the other

coarse wire coil an alternating current was sent, in series to

ammeter and coarse wire coil of wattmeter, while the voltme-

ter and the fine wire coil of the wattmeter were connected in

shunt around the whole circuit.

Hence a correction had to be applied for the self-induction

of ammeter and coarse wire coil of the wattmeter and for the

resistance of thecircuitO. nly in veryfew

this correc-

tion amounted to somewhat more than 10 per cent. Generally

it was much smaller.

Theinstruments were calibratedseveral timesandtheir

constants found to remain constant.

The speed indicator was calibrated carefully andits correc-

tions added.

Each r e a consisted of an ammeter readmg, a voltmeter

reading, a wattmeter reading and aspeed readmg.

Before and after each set of readings the zero positions of

theinstruments weredetermined, and onlythose sets of

rea- readings used where the zero positionhad remained constant.

Before and after each set of alternating current

a

continuous currentwas sent into the circuit and faew readings

for different currents taken. Voltmeter and ammeter readmgs

combinedgavetheresistance of the circuit, andboth com-

bined with the wattmeter readmg gave a checkfor the instru-

ments,herebeing watts = volts X amperes. Only those sets

were used againwhere an entireagreementwasfound,and

with the alternating current firstreadings with small currents,

then with large currents, and then again with small currents

taken, so that I believe every possible care was exercised to

avoid any errors in the tests.

As beforesaid,thefirst sets of testsweremade on the

magnetic circuit of a small Westinghouse converter.

The constants of this converter, so far as they are of interest

here, are:

Mean length of magnetic circuit,21 cm.

Mean cross-section of magnetic circuit,= 43.67 an.*

Hence volume of iron, = 917. a d .

Resistance of secondary coil, = .2 w .

Further sets of readings were taken on a magnetic circuit,

built up of very thin sheets of iron, alternately 8 in. x 1 in.

and 3 in. X 1 in., in rectangular shape,very carefully insulated

against eddy currents with layers of thin paper between the

sheets. On the two long sides two coils of each 50 turns,very

coarse wire (3 No. 10 in parallel), were wound and connected

in series, thereby givingn = 100 turns of an internal resistance

of .048 w .

Here the mean lengthof the magnetic circuitwas I = 41 c m .

The cross-section,s = 3.784

Thecircuitconsisted of 58 layers of sheet-iron of the
thicknesss = .02577 t3) and the width o = 2.579.

The whole volume of iron was = 155 c m . 3

The sheet-ironpieces were first freed from scales by dipping

into dilute sulphuric acid.

In one set of tests an open magnetic circuit was used, by

leaving the short end pieces (3 in.x 1 in.)off, and using two

pileseach of 66 pieces (8 in. X 1 in.) of thesame iron, the

same pieces as used in the former closed circuit tests.

In thesereadings,forthedetermination of thehysteretic

loss, onlyvoltmeterandwattmeter,but no ammeter,were

used,andtheconductivitycurvedeterminedseparatelyby

voltmeter and ammeter.

The calculation of the rea-was

done in the following

way:

After applying the corrections for self-induction of instru-

ments, resistance andspeed, the rea- were reduced to lines

of magnetic force per@are centimetre B and consumptionof

energy by hysteresis per magnetic cycleH , in ergs.

Then the resultswere plotted on cross-section paper and if

any valuewas foundto be verymuch out of thecurve

connecting the other values, it was stricken out as evidently

erroneous,not consideringit worth whileto determinewhether

it was a wrongreading of anyone of the instnunents or a

mistake in the calculation.

Then from the other values of B and H , under the supposi-

tion that H were proportional to anypower x of B:

H = qBX

this exponent x was determined. Thisvaluexwillbeseenalwaystobesonearto1.6that1.6
can be considered at least as first approximationto x. Then, under the assumption
x = 1.6
hence
H VB'.~

the coefficient q was calculated, and now the equation

H =vB'.~

STEINMETZ: ON T H E LAW OF HYSTERESIS

199

plotted in a curve, as given in the figures, and the observed

values of H drawn in and marked.

Fromthe c w e were taken thecalculatedvalues of H ,

correspondingtotheobservedvalues

of B, thedifference

Hdc - Bobs determined, and expressed in per cents. of Hdc.

These values are given in thetablesandshown in the

curves.

MAGNETIC CIRCUIT OF THE
WESTINGHOUSCEONVERTER. FIG.2; TABLE1S1.
MAGNEnC CRARACIERISIIC.

F. = M. M. F., in ampere turns per centimetre length of magnetic circuit.
B.= Magnetization, in lines of magnetic force per s q w e
centimetre.
TABLE II. (1)

F.
I:
8 9 10

B.

F.

1500
y.o~~ 0
6800
%a,

I I

::
i!

11.750

20

1i8m

25

13,600

30

14,100

35

14,350

40

B.
15.080 15570 15,630 15.880 16;4m 16.950 172m 17,780

F.

B.

18500

55

18,820

19,140

19,440

m

19,740

75

mom

80 85

2m,056w0

90

2o.m

Coefficient of hysteresis: q = BO2315
hence, theoretical curve: H = BO2315 B’.6

TABLE II. (4)

- Frequency: N = 137 complete periods per second:

B.

H.
ObS

H.
calc

cHa.k.

-

H.
obs.

%

4ooo 4670 5510 5760 5840 6690
E
12,430 13,7SO

1490 1818 2358
2482 2540 3285 3358 3374 8336
10.m

1410 1800 2350 2520 2580 3180 3290 3370 8610
10,1oO
av:

-80
- 18

-+

8 38

+40 - 105 - 68

+-2744

+loo

f 73.5

- 5.7 - 1.0 - .3 - 1.5
+ 1.6
- 3.3 - 2.1 - .1 + 3.6
+ 1.0 i-zb

Exponent of power, derived fromtests:
- X 1.5887 1.6
Coefficient of hysteresis: q = .002438
hence, theoretical curve.
H = .002438B’.6

WSTERESSI.
B.= Magnetization, in lines of magnetic force per square
centimetre. H . = Loss of energy by hysteresis, in ergs percycle,and
cubciecntimetre, = watt-second.
TABLE II. (2)
Freauencv: N.= 28 comlete Deriodsm second.

3510
-10,560 13,800 17,W

1178 6286
10286 15,357

110.180
15.600

E-1 ii 1 -106
+ 243

PV:

- Exponent of power, derived fromtests: X = 1.6111 1.6

TABLE II. ( 5 ) Frequency. N 5 205 complete periods per second.

I B. I E.

1790

376

1990

463

2380

585

2620

735

3060

893

3390

1054

3660

1297

3710

1288

4620

1822

mm 2024

4990

2034

5910

2693

6100

2844

6550

3039

7290

3673

4341

H.
- calc
400 460 610 720 920 1100 1240 1250
I1800
2070 2010 2620 27x1
4530 u60

%

-

+ 24

+ 6.0

-3
+ 35

+-5..77

+-2175
+46

+-2.1
+ 42..92

- 57

- 4.6

- 38

- 3.0

-22

- 1.2

+46

+ 2.2

- 24

- 1.2

-73

-2 8

-96

-+31.5.3

- .9

_- -+-

1.0 2.1 2.2

-- f 2 7

Coefficient of hysteresis: q = .002410
hence, theoretical curve:

Exponent of power, derived from tests:
- x = 1.6012 1.6
Coefficient of hysteresis:

H = .00241B’.6

q = .002434

TABLE! II.(3) Frequency: N = 36 complete periods persecond

hence, theoretical curve.
H = .002434B’.6

From these 4 sets of readings, we get the results:

7090 10,250
13,410 17,080 19,340

3333 5667
9694 14,417 16,111

3500 6310
9700 14,400 17,600

++ 614637 + 17
+ 1489

~-

Exponent of power, derived from tests:
- X = 1.6476 1.6

+ +

4.8 10.2

36

1. N = 28 4 m w :X =1.61117 =.002410

2.

5

”

1.6476 BO2315

+ +

.1 .1

+ 8.4

3. 137 10 ”

205 4.

18

”

1.5887 BO2438
1.6012 .m34

f.

Therefrom we derive the average, by giving to each value as

- weight the number of readmgs, whereit is based upon: X = 1.60513 1.6

200

PROCEEDINGS OF T H E IEEE, VOL. 72, NO. 2, FEBRUARY 1 9 6 4

= .0024164

Hence:

H -- .0024164B'.6
This curve is used for calculating the values given as H&, and is plotted in Fig. 2 in drawn line.
The observed values of H are drawn in Fig. 2: The magnetic characteristicis drawn in dotted lines.
From this curve of hysteretic loss

H = .0024164B'.6 we derive the values:

TABLE II. (6)

1 B. I H. 1 B.

H.

152 462 884
m 1400
2680 3430 4240
5130 6070
7070 8130

13,000 14,000 15,000 16,000 17,000 18.000 19,000 20,000 25,000
3c4000 35,000
40,000

9230
lop00
11,610
12Im
14,180
15.550 16,970 18,uy)
26,290 35,210 45,060 55,800

11,160

II.-MAGNE~c CIRCUIT BUILTUP OF WELL INSULATELDAYERSOF VERYTHIN SHEET-IROFNI.G.3; TABLE1S11.
MAGNElTC CHARACIERISTIC.

F = M. M. F. in ampere tums per centimetrelength of
magnetic circuit. B = magnetization in lines of magneticforce per square
centimetre.
TABLE III.(1)

F.

B.

F.

B.

F.

B.

2

1700

12

1137,7,50 2

45

3

4200

14

14260

50

17,wO

4

7400

16

14,600

55

lSf00

5

9200

18

14,900

60

18,650

6

10.400

20

15200

65

19,030

7

15,700

25

19,380

70

8 9

11,850 12,470

30 35

1169,2,70300

75

16,680

80

m,oso

10

13,070

40

17,050

85

m,m

90

20,750

HYsTmFSIs
B = magnetization in lines of magneticforcepersquare centimetre.

STEINMETZ: ON T H E LAW OF HYSTERESIS

201

H = loss of energy by hysteresis, in ergs per cycle and cubic centimetre, = 1 0 -w~att-seconds.

CLOSED MAGNETIC CIRCUIT.
Frequency: N = 85 complete periods per second.

4220 7690

10,470

7160

11,110

8370

14,030 12,600

14.890

13,730

' 17,190 17,040 17,940 17,570

TABLE III.(2)

H
cak.
3140
4700 7700 8464 12280 13,540 17,040
- 18,240 av :

H . - H.=
cak. obs

- 180 - 270 3420

+480

+540

+%

- 320
- 190

+ +

... 670

315 =

s

- 5.7

- 7.9

++170..02

- + -

1.1 2.6 1.4

...

x + 3.7

Exponent of power, derived from tests:
- X = 1.6041 1.6
Coefficient of hysteresis:
7 = .00285
hence, theoretical curve:

H = .900285B'.6
TABLE III. (3)
Frequency, N = 138 complete periods per second.

B.
5220 5750 6540 7070 8210 8520 9570 10,450 11,990 14,570 14,aaO 16,770 17,970 19,320

H.
obS.
3030 3620 4320 4830 5950 6090 7850 8780 11,060 15,840 16,160
20.350 20,620 23,180

H.
calC.
3015 3550 4355 4890 6160 6530 7840 9040 11,230 15,340 15,580 19260 21,440
- 24,120 av:

H. - H . =
calc. obs.

- 15 - 70 + 35
++26100
+440
- 10
++216700

+ 3.4
I i+21:;.5

- 1090

- 5.6

- + 820
+940
f 371

+ 3.9
+ 3.8

Exponent of power, derived from tests: x = 1.6044 = 1.6
Coefficient of hysteresis: q = .00335
hence theoretical curve:
H = .00335B'.6

TABLE III. (4)

I I I 1 Frequency,N = 205 completeperiods per second:

B.

H. H.

H -H.-

ob%

cak.

cak obs

6360 I340 10.030 10,860
1WO 14,600 14,700 15,750 16,700

4440 5380 9510
9980
13,700 17,390 17,830
19,700 21,990

4660 5780 9510
l0,6X,
12,940 17,160
17.343 19560 21500
aV:

+m
+400
. .. + 690 - I60
-230
- 490
*--364900 425

+ 4.8 + 6.9 ...
+6.5
- 5.9
-- 1.3 2.8
- 1.7
r-n3.2

Exponent of power, derived from tests: x = 1.697 = 1.6

Coefficient of hysteresis: I) = .00373
hence theoretical curve: H = .00373B1.6

OPEN MAGNETIC CIRCUIT.
- Twogapsof 4 cm. length.

I 2. z.;;= 1 ", TABLE III. (5)
Frequency, N = 138 complete periodsper second.

B.

H.

obS.

3150 3640 4690 5490 6270 10.250 11,Ooo 12280

1570

1560

2110

2020

2930

2950

-90
+ 20

4380

3780 3510 4690

++237100

10,290 10,450 -160

I11,810
14250

11,520
13a,v7:40

I
I

-+4..47
++76.26
-1.6

Exponent of power derived from tests:
- X = 1.6040 1.6
Coefficient of hysteresis: q = .00394
hence theoretical curve:
H = .00394B'.6 From these four sets of readings we get the results:

N = 85
138 205

CLOSED MAGNETIC CIRCUIT.

9 readings: x = 1.6041

14 I'

1.6044

9

'I

1.6970

7 = .OO285 .00335 .00373

OPEN MAGNETIC CIRCUIT.

N = 138 8 readings: x = 1.6040 q = .00393

Herefrom it seems that theconsumption of energyby

hysteresipsemr agneticcycle

increases with increashg

frequency-that is, with increasing velocity of the magnetic

change.

The three values of three coefficients of hysteresis for closed

circuit in their dependence upon the frequency N, can be

expressed by the empirical formula:

+ 7 = (0017 .000016N - .00000003 N2)

To compare the values of hysteretic loss for different fre-
quencies, in Fig. 3 the curve of hysteretic loss for N = 100
complete periodsper second is plotted, giving:

q100 = .003

hence

H = .003 B'.6
and the observed values of H are not cllrectly drawn in, but the observed valuesof H multiplied withthe factor:
'Iloo

I)&.

to compare the differentfrequencieswith each other. These valuesare plotted for:

202

PROCEEDINGS OF T H E IEEE, VOC. 72, NO. 2, FEBRUARY 1%

26.000,-

a

] N-8

5wit 138

h

the

m

a

r

k r+

closedmagcnirectuicit.

205

‘1

‘1

N = 138 with themark 0 ;Open magnetic circuit.

From this m e of hysteretic loss,

H = .003B1.’
we derive the values, for the frequency of N = 100 complete periods per second.

TABLE III. (6)

II1740 2490 3330 4260

4M 780oao0oQo0o

9oM)

6360

10.000

7530

11,000
w@Jo

8790 10.080

B.
13,000 14,000 15,000 16,000 17,000 18,000 19,000
m2so;oaO,o
30,000 35,000 40,000

H.
11,460
15,990 17,610 19590 21,060 22.830 32340 43,680 55,950
a92m

Especiallynoteworthy is thelastset ofreadings, an open magneticcircuit, in so far as it provesthefallacyofthe general opinion that the hysteretic lossof energy in the iron is smaller in the open magnetic circuit than in the closed circuit.
For the coefficient of hysteresis observed on open magnetic circuit
q = m393
is even greater than that for closed magnetic circuit,
q = .00335
But this discrepancy is easily explained by the fact that in theclosedmagneticcircuitthemagnetization is nearly uniformthroughoutthewhole iron. But in the open magnetic circuit the magnetic field intensity differs considerably from point to point, bemg a maximum in the middle of the mag-
netizing coils,a minimumat the ends of the iron sheets.Now,
the values ofB given in the table,are the average vahaesof the
magnetization, and the values H,the average valuesof hyster-
etic loss. But the average valuoef the 1.6th powersof different
quantitiesB is larger than the 1.6th powerof the average value of B.

STEINMETZ: ON THE LAW OF HYSTERESIS

203

For instance, in acubic cm. of iron magnetized to B = 12,000 is H = 10,080; in a cubic cm. of iron magnetized to B = 6OOO is H = 3330; hence of these 2 cubic centimetres the
- average magnetization is B = 9o00,and the averageH 6,705 ergs
but to B = 9o00 c o r r q n h H = 6360 ergs; that is, about 5 per cent. less, and the difference becomes still greater, if the values B differ still more.
Talung this into account,it seems that the loss of energy due to hysteresis depends onlyupon the intensityof magnetization, and perhaps upon the frequencyb,ut is independent of open or c l o d magnetic circuit, as is to be expected.
III.-FIG. 4. TABLEISV.
A third setof determinationsof the hystereticloss of energy is given in the following:
Again a magnetic circuit was built upof 17 layers of a soft kind of sheet-iron, each layer consisting of two pieces of 20 cm.length, 2.54 cm. width, and two piecesof7.6 cm.length
and 2.54 cm.width, of the thicknessS =i .M86cm., that is, of considerably greater thickness than in the former set of tests.
Here evident proof of the inductionof eddy-aments in the iron was found. EspeQally perceptible was a decrease in the watts consumedby the iron, when a larger M. M. F. ofhigh frequency was leftacting upon the ironT. his decrease mustbe attributed to the increaseof the electricresistance of the iron, caused by its increasing temperature.
To eliminate this source of error as far as possible, before eachset of tests an alternating ament ofhigh frequency ( N = 200) andconsiderablestrengthwassent through the magnetizing coils and left on for ten to fifteen minutes, and thenfirstreadingswith lowmagnetization,thenwithhigh, and thenagainwithlowmagnetizationweretakenB. ut, nevertheless, as was to be expected, in these tests the observed values agreedless with each other thanin the former readings.
The method of determination, the apparatus, etc., were the same as in the second set of tests, only that ammeter, voltmeter, and wattmeter were used at the same time. In calculating these tests, the law of the 1.6th power was assumed as true, and the loss of energy in the iron expressed by the equation,
+ H = vB'.~ tNB2
where
Hl = T B ' . ~
is the true hystereticloss per cycle and ad., which is indepen-
dent of the frequency, and
Hz = cNB2
is the lossofenergyby eddy-currents per cyclewhich is proportional to the frequency N.
From this expression
+ H = H I Hz
the coefficients 7 and c were calculated and the agreement or dsagreement of thex coefficients7 and c allow now to check the correctness or incorrectness of the law of the 1.6th power.
These tests gave the following results:

Fig. 4.

MAGNmC CHARACIWSTICS.

1 1 F = M. M. F., in ampere turns percentimetrelength
magnetic circuit. B = magnetization, in lines of magneticforceper
centimetre.
TABLE TV. (1)

of

B.

B. F. B.

1.5

2,700

11,700

18

15,450

4,350 7,100

lZ,ZIIO 20 15,800

12,700

25

16,400

4

8,850

10

13,100

30 16,800

5

10.000

12

13,900 35

17,200

6

10,800

14

14,500 40 17,500

16

15,000

WsrmEslS.
B = magnetization, in lines of magnetic force per
-centimetre. H loss of energy by hysteresis,in ergs per cycle and a d .
(= joules) = Hl + Hz
Hl = v B ' . ~ = lossof energy by hysteresis proper, in ergs
per cyclaend ad. (= joules).

204

PROCEEDINGS OF THE IEEE, VOL. 72, NO. 2, FEBRUARY 1 9 8 4

H2cNB2= loss of energy by eddy-currents,in ergs per cycle and ad.( = joules).

TABLE IV.(2)

Frequency, N = 78.

q = .00331

c = .751 X

B.
4171 58x) 9520 13,160 14,320 16,050

HI
2,060 3340 7,740 12,w
17280

HZ
1,080
12,720 15,900

H.(4) H.

calc

obs.

3,140 3,060 5,660 2,152,6040 13,340 51,630,4040 23,670 120,47,15040 1247,,86800 26,460 33,180 33,180

av:

A
+ 80 + 20
- 100 - 870
-1140
.._
( T::;}

=%

+ +

2.6 .3

- .8

-3.7

+4.0

*1.9(+.4)

B.
4880 6780 7720 10,200 12,080

TABLE IV.(3)

Frequency, N = 140.

q = .00331

c = .730 X

HI
4.490 5,530 8,640

H 2 H. (4) H.

A

calc.

obs.

2,720 2,650 9,760
6,830 11,940

5,280 5,297,4020 12,600 201400

- 5,36++0

80 340

2102,,53+8600124800

16,700 33,840

11,300 19,860

29,100 1573,2,000

2583,,07-+00100170000

=%

+1.5

+ 3.4

- 1.9

+ .9

+

4.0 1.3

(T:::} av:

+2.2(+.2)

TABLE N.(4)

Frequency, N = 207

q = .00336

c = .757 x 1 0 - 6

1 1 1 1 1 1 -670 1 1 I 2710 4720
7540

- I
1,030
5;320

- I calc. I obs. I

I

1.290

2,340 2,32-0 20

- .8

Tf 6.480 6.430- 35.0910 2.510- .8
9;970 15390 l 5 & 0

12380 112,7603,80803,5080.5000

13200 1340,043,402.40,6000

+800

+1.8

L

I

I

I

I

.

I

J

Therefrom we get the results:

N = 78, 6 readmgs, q = .00331 c = .751 x

140, 6 If

.00331 .730 X

207, 5 "

.00336

.757 x 1 0 - 6

Thevaluesfoundfor q are so nearlyalike that we can consider them as constant, and take theirmean value
q = .00333
as the coefficient of hysteresis. Eventhevaluesfoundfor c are not much different from
eachother,not more than was to be expectedfromthe unavoidable differences in the temperature of the iron, which because of thehighelectrictemperaturecoefficient of iron makes E rather variable.
Taking the average of E, we derive
c = .746 x
and as formula of iron loss,
+ H = .00333B'.6 .I46 X 10-6NB2

4 H d is calculated by using for 7 the mean value 7 = ,00333, but for c the individual values, corresponding to the particular set ofobservations.

In Fig. 4 are drawn the four curves,

1.True hysteretic loss, H = .00333 B'.6

2. Iron loss for N = 78

+ .00333 B'.6 .oooO5856 B2

3. ' I

'I

140

.0001022 B2

4. If

If

209

.0oO1567 B2

The observed valuesare plotted by crosses, +

IV.-FIGS.5 AND 6;TABLEVS AND VI.

Two other sets of determinations of the hysteretic lossof

energy, for the frequency 170 complete periods per second,

weremade on two laminated horse shoe magnetsw, ith

laminated keeperor armature.

The method of observation and of calculation was the same

as in III., and the same precautions weretaken.

The dimensions of the horse shoe magnets were:

Mean length of magnetic circuit: 38 cm.

" cross-section: 70

" volumeof iron: 2660 ~ m . ~

I' distance of keeper from magnet, in the first case:

, .15 cm.

' I distance of keeper from magnet, in the second case:

.08cm.

eachmagnet consisting of 300 sheets well insulated iron, of

the thickness .0405 cm.

In the first set of readmgs, considerable eddy-currents were

found; in the second set, only asmall amount of eddies.

The magnetic conductivityof the iron was not determined,

becausethereluctance of themagneticcircuitmainly con-

sisted of that of the air gap between magnet and keeper.

The results were,

B =magnetization, in hes per

Hobs.= observed loss of energy in the iron,in ergs per cycle

anfodr

N = 170.

Hl =true hysteretic loss of energy.

H2 =loss of energy by eddy-currents.
+ HdC.=whole calculated loss of energy, = Hl H2

TABLE V.

Frequency, N = 170.

7-= .-w5

E = 1.16 X

670
1020 1100 1200 1310 1490 1930 2600

- Hl - H 2 51 23 68 34
108 59 132 78 150 90 178 111
210 138 293 208 345 234 392 290 436 343 539 445 820 742 1310 1280

- - H. H.
calC. ObS.

74

70

102 102

166 166

210 219

240 234

289 300

348 333

501 524

579 549

682 695

779 795

984 985

1562 1547

2590 2670

-
- av:

Therefore we get the formula for theloss in the iron,
+ H = .0045B'.6 1.16N X 10-6B2

STEINMETZ: ON THE LAW OF HYSTERESIS

205

zation up to high saturation, while the tests in IV. cover the

rangefrom 85 to 2600 lines per

that is, frommedium

down to very low magnetization.

The law is found exactly the same,

H = vB'.~+ ENB'

and herewith proved for the full range from 85 lines per m 2 up to 19,340 lines, a ratio from 1 + 230.

In Fig. 5 are shown

1. Thecurve of true hysteretic loss,
Hl= .0045B'.6
2. Thecurveof the wholeloss in the iron,
+ H = HI H 2 with the observed values marked by crosses +

TABLE VI.

Frequency, N = 170

-1 = .00421

I B.
- H1

I 2;.

85 5.2 .3 5.5 I82 17.3 1.3 18.6 211 22.0 1.7 23.7
560* 105 11 116
670 140 15 155 685 145 16 161 775 176 21 197 800 186 22 208 loo0 265 35 300 lorn 2% 41 337 1130 322 47 369 1250 319 56 435 1380 445 69 514 2200 940 170 1110 2420

-c = .2083 X

H.
- obs.
5.6 16.9 23.5 122 146 157 202 200 300 353 386 430 514 1130 1268

H. - H.
calc. obs.
- .1 + 1.7 + .2
-6 +9 +4 -5 +8 ...
- 16 - 17
is
- 26 - 20 + 30

+ 38

- av :

*-90 10

-

=%

-

- 1.8

+ 10.0

+ +

.9 5.0 6.1

+ 2.6

- +

2.4 4.0

... - 4.0 - 4.3
+ 1.2 - 4.1 - 1.8
- + 2.4
+ 21.2

- 24.6
+- 34

Therefore we get the formula for the lossin the iron,
H = .00421B1.6+ .2083 X 10-6NB2
In Fig. 6 are shown, 1. The curve of true hysteretic loss,

Hl= .00421B'.6

2. The curve of the whole loss in the iron,

H HI + H 2

+ with the observed values marked by c r o w

Especially interesting are these two sets of readmgs in so far

as they cover quite a different range of magnetization as the

tests in I. to rn.
In I. to m. the tests cover the range from 1790 to 19,340

lines ofmagnetic forceper

that is, for mediummagneti-

This seems not to agree with Ewing's theory of the molecular magnets. According to this theory, for very small magnetizationthehysteresisshould be expected to dmppear, or almost disappear, and thceycle be reversible. Thenfor medium magnetization, where the chains of molecular magnets break up and rearrange, hysteresis should increase very rapidly, and slowly again for saturation. Nothing of this is the case, but hysteresis seems to follow the same law over the whole range of magnetization, and is certainly not zero for even such a low magnetization as 85 lines per
MAGNETOMETER TESTS.
The method used in the foregoing has the great advantage that
1.It allowsthetaking of agreaternumber of readings, over a wide range of magnetization, in a short time, bmy erseimultaneouisnstrumenrteadingsa,nd thereby reduces the probable error by increasing the number of observations.
2. It allowsthe use of electro-dynamometers,as themost reliable electric measuring instruments.
3. It dealswithlargeramounts ofenergy, countingby wattsor even hundreds of watts,wherebyamuch greater accuracy can be reached than by the ballistic galvanometer.
4. It measures the hysteresisundertheinfluence of an harmonically, andnot suddenly varyingM.bi. F., that is underthe same conditions, where it becomes of importance for practical engineering.
But it has the great disadvantagtehat it canbe used only for testing sheet-iron or other thoroughly laminated iron, where
eddies are either inappreciableor can be calculated also. For

M6

PROCEEDINGS OF THE IEEE, VOL. 72,N O . 2, FEBRUARY 1984

testing solidiron and steel pieces,this method cannotbe used, because of thetremendousamount of eddieswhichwould flow in a solid piece of iron.
To determine.thehystereticloss ofenergy in steeland cast-iron the Eickemeyer differential magnetometer was used. Complete description of this instrument and its use is to be found in the Electrical Engineer, March 25th. 1891,wherefrom is taken apart of the following description. In Fig. 7 is shown this instrument, which I shall be gladto show in our factory to a n m y who is interested in it. In Figs. 8 and 9 are diagrams of its action.
Theprinciple of this instrument resemblessomewhatthe principle of the well-known differential galvanometer, applied to the magnetic circuit. In Fig. 8, suppose F1 and F2 were two E. M. F.’S connected in series; forinstance, two cellsof a battery, x and y the two resistances which we want to compare. Either resistance x and y is shunted respectively by a conductor a and b of equalresistance,whichinfluencesa galvanometer needle G in opposite directions but with equal strength.
Thenthezeroposition of theneedle G shows that the electric currentc,, flowing in u, is equal to the currenctb in b. But let the currentin x be c,, and iny, cy; then we must have
+ + ca cy = cb cx

0
Fre. 8

becausethecurrents c, and cy arethe two branches of the same integral currentas c b and c,
Therefore, if c, = cb, then

c, = cy

But if c, = cb, and a = b, the difference of potential at the

R e .9

ends of a (or,what is thesame thing, y ) is equal to the

difference of potentialattheends of b or x and,therefore,

or “number of lines ofmagneticforce;”insteadof“elec-

the current in x and y , and the potential differences being thetromotiveforce”or

“potential difference,”say “magneterne

same, it follows that x = y .

tive foinracsnetde;”ad

of “elercetsrisctanscaey,”

“reluc-

That is, this method of connection allows us to compare an tance,” and we have the principle of this instrument.

unknown resistance x with a standard resistancey .

Its magnetic circuit consists of two pieces of best Norway

Now, instead of “electric current,” say “magnetic current” iron, Ln shaped, shown in theillustration of thecomplete

STEINMETZ: ON THE LAW OF HYSTERESIS

207

instrument, Fig. 7,and in the diagramFig. 9, at F~ and F2. The

middle portion is surrounded by a magnetizing coil c. There-

fore if coil c is traversed by an electric current, the frontpart

s1 of the left iron piece becomes south, and the back part n

north polarity.Thefront part of therightironpiece n be-

comes north, and the back part south;and the lines of

magnetic force travel in the front from the right to the left,

from n 2 to s,; in the back the opposite way, from the left to

the right, or from nl to sz, either through the air or, when n 2

and sl,or nl and s2, are connected by a piecoef magnetizable

metal, through this and through the air.

In the middle of the coil c stands a small soft iron needle

with an aluminium indicator,which plays over a scale K, and

is held in a vertical position by the lines of magnetic force of

the coil c itself, deflected to the left by the lines of magnetic

force traversing thefront part of the instrument fromn z to sl,

deflected to the right by the lines traversing the back fromn,

to s2. This needle shows byits zero position that the magnetic

flowthroughthe air in front from n 2 to s1 hasthesame

strength as themagneticflowinthebackfrom

n1 to s2

through the air.

Nowwe put apiece of soft iron x on thefront of the

instrument. A large numberof lines go throughx , less through

the air from n 2 to s,, but all theselinesgofrom nl to s2

throughthe air at theback part of themagnetometer,the

front part and back part of the instrumentbeing connected in

series in the magnetic circuit. Therefore the needisledeflected

to the right by the magnetic flow in the back of the instru-

ment.

Now we put another piece of iron,y, on the backpart of the

instrument. Then equilibriumwould be restoredas soon as the

samenumber of lines of magneticforcegothrough x , as

through y, because then also thesamenumber of lines go

through air in the front as in the back. As will be noted, the

air here takes the place of the resistancesa and b, influencing

the galvanometer needle G, as in the diagram,Fig. 8.

The operationof the instrument is exceedingly simple andis

as follows: Into the coil c an electric current is sent which is

measured by the ammeter A, and regulated by the resistance-

switch R Then the needle which before hadno fixed position,

points to zero.

Now the magneticstandard,consistingof a cylindrical piece

ofNorway iron of 4 cross-sectionand 20 cm. length is

laid against the back of the instrument, with both ends fitted

into holes in large blocks of Norway iron, A,, A , , which are

laid against the poles S,N of the magnetometer, so that the

transient resistance from pole-faceto iron is eliminated.

The sample of iron that we wish to examine is turned off to

exactly the same size, 4 cm.’ cross-section and 20 cm. length,

and fitted into blocks A,A2 in front of themagnetometer.

Then so many fractional standard-piecesof Norway iron are

added in front, that theneedleof the instrument points to

zero. This means that the 4 cm.’ Norway iron in the back,

cany underthesamedifference of magneticpotential,the

samemagnetism as the 4 of the examined sampleplus
the x cm.’ of fractional standard, added in the front. Hence,

4 cm.’ of the examined sample are equal in magnetic Conduc-

tivityto (4 - x ) cm.’ of Norwayiron,andthemagnetic

conductivity of this sample is (4 - x)/4 x 100 per cent. of

that of Norway iron,for that differenceof magnetic potential,

viz., magnetization, that corresponds to themagnetometer

current.

To get absolute values, the instrument has been calibrated

in the followingway: In the front and in the back the

magnetic circuit of the instrument has been closed by 4 cm.’

Norway iron. Then another piece of iron, and of any desired

size, has been addedin the front.Tbis piece, y, carrying some

magnetism also, equilibrium was disturbed. Then through a

coil of exactly 110turns,surrounding this piece y, an electric

current i was sentandregulated so that equilibrium was

restored. In this case no magnetism passed through y , or in

other words, the M. FM..of the currenti 110 i ampere turns,is

equal to thedifferences of magneticpotentialbetweenthe

pole-faces of the instrument. In this way, for any strength of

current in the maincoil C of the magnetometer, the difference

of magnetic potential produced therebeytween the pole-faces

of the instrument, was determined and plotted in a curve, for

convenience in ampere turnsper cm. length.

Now,theNorwayironstandardwascompared

on the

magnetometer with sheet-iron, of which, from tests with low

frequency alternating currents, the magnetization correspond-

ing to any M. M. F. wasknown, andtherefromderivedthe

magneticcharacteristic of theNorway iron standard, and

plotted in a curve also.

In the way explained before, theiron sample that was to be

determined, was balanced by themagnetometerbyNorway

iron, thereby giving its magnetic conductivity.in per cent. of

that of the Norway iron standard, the magnetometer current

read, from the curves taking theM.M.F. corresponding thereto

-denotedwithF-andthemagnetization

of theNorway

iron, corresponding to this m M. F., F, andfromthede-

terminedpercentage of conductivity of theexamined sam-

ple, the magnetization B of this sample correspondingto the

M. M. F. F.

With this instrument a numberof magnetic cyclesof differ-

ent samples of steel and cast-iron were determined.

First, a powerful alternating current was sent through the

magnetometer and around all the iron pieces used, to destroy

any trace of permanent or remanentmagnetism.

Then the examined sample was laid against the front, the

standard against the back of the magnetometer, balanced, and

a larger number of magnetic cycles completed between given

limits, for instance, +95 and -95 ampere turns M. M. F. per

cm. length. Then readings were taken from maximum m m F.
+95 down to zero, and again up to the maximum -95, down

over zero and up to +95, thereby completing a whole mag-

neticcycle, andthen of a secondmagneticcycle, a few

readings were taken as check for the first one.

In this way for different M. M. F.’s the curve of hysteresis

was found, and by measuring its area the loss by hysteresis

determined.

The further calculation was done in a somewhat different

way. Generally the number of cycles was not large enough to

determine conveniently the exponent by analytical methcds.

Therefore the law of the 1.6 M. power:

H =vB’.~

wasassumed as true,andforeach cyclefrom the known values of H and B determined the co-efficientq.
If fordifferent cycles the valuesof q agreed, this would provetheassumption,the correctness of thelaw of1.6th power, while a dtsagreement would disprove it.
In the following fora number of samplesthemagnetic cycles are given:
F=M.M.F.,inampereturnspercm.length. B, and Bd = theintensity of magnetization, in kilolines,
corresponding to M. M. F. F,for the rising and the decreasing
branch of the magnetic curve. The area of the looped curve, representing the loosfsenergy

208

PROCEEDINGS OF THE IEEE, VOL. 7 2 , NO.2, FEBRUARY 1%

by hysteresis is derivedby adding the values of B,, and subtractingtherefrom the s u m of thevalues B,,Bd and B,, being given from5 to 5 ampere turns,or .5 absolute units, the difference of the sums of Bd - B, just gives thelossby hysteresis, in ergs per cycle.
CAST-STEEAL,NNEALEDAND HARDENED. FIG.10; TABLVEII.
Of one kind of steel, two test pieces were cast, at the same casting, turned off to standard size and, by comparing them in the magnetometer, foundto be exactly alike.
Then the one piece was hardened, the other left annealed. Magnetometer tests gave the following magnetic cycles:

Herefrom as coefficient of hysteresis, was found

T = .024941.025121.02490I.0079971.007%2,

- - 7 = .024987 a- 1 = .007980

.025

.0080

Hence, when anneded, the hysteretic loss is

when hardened

H = .008 B'.6

H = .025B'.6 and calculated by means of these formulas, we derive

H = 48,40707,501001,50304,73405,100
calc.

- r

-F. Br Bd

0

f 5.0

5 -4.4 + 5.6

10 -3.1 6.1

15 6-.25.7

20 0 6.9

25 63.9 7.3

30 5.5 1.6

35 6.1 8.0

40 7.1 8.3

45

8.5

50 (44.5.)

55

60

65

10

15

80

a5

90

9s

100

105

- 110

H-
- Ibz

r TABLEVII.
Md.

Br Bd
* 1.0
-6.4 + 1.5
-5.6 1.9 -4.4 8.2 -1.9 8.6 +1.9 9.0
4.2 9.3 6.2 9.6 7.6 9.9 8.1 10.2 9.6 10.5 10.4 10.8 10.9 11.1
11.4 (64.5.)
77,800

Br Bd

Br Bd

f 1.8

f 6.6

-7.3 + 8.2 - 1.4 + 10.7

-6.8 8.6 C3.4 11.9

-5.6 8.9 128..54

-2.3 9.2 1120.8.9

+.4 9.5 12.2 13.1

2.5 9.8 13.0 13.4

4.2 10.1 13.5 13.7

5.8 10.4 13.9 14.0

1.2 10.1 14.1

8.4 11.0
9.6 11.2 10.4 11.5 10.9 11.8 11.4 12.0
11.9 12.3 12.2 12.5 12.5 12.7 12.98
13.0 13.1
13.2 13.3 13.4 13.4

(44.5.)

13.5 (108.0.)

101,100

Br Bd

f -2.6

8.6
+ 11.3

t3.7 12.3

8.4 12.7

10.8 13.0

12.0 13.3

12.1 13.6

13.2 13.9

13.5 14.2

13.8 14.5

14.1 14.1

14.4 15.0

14.7 15.2

15.0 15.4

15.3 15.6

15.6 15.8

15.8 16.0

16.0 16.1

16.2 16.3

16.4 16.5

16.6

(101,O.)

45,000

STEINMETZ: ON THE LAW OF HYSTERESIS

209

and
H - H = +lo0 -300 +400 -70 +lo0
calc. obs.
= per cent. of
H + .2 - .4 + .4 - .2 + .2
calC.
In Fig. 10 are drawn some of the magnetic curves for both samples.
It is especially interesting to note that though the chemical constitution of both samples is exactly the same, their magnetic behavior is entirely different,so that the magnetic properties of iron seem to be determined much more byitsphysical than its chem'cd constitution.
ANOTHER SAMPLEOF CAST-STEEOLF Low MAGNETIC CONDUCTMTY. FIG. 11.

TABLE VIII.

Br Bd

Br bd

Br Bd

10

* 2.5
+ .6 4.1

k 2.8
- 1.9 + 3.6 - .4 4.3

f -2.1

3+.13.9

- 4.6

f -2.7

3+.44.2

-1.3 4.8

15

2.7 4.6

+2.7 4.9

+2.2 5.2

+2.3 5.4

3.9 5.1

4.0 5.5

4.2 5.8

5.93.8

5.64.7

4.9 6.0

5.1 6.2

4.8 6.4

6.0

5.5

30

5.6 6.4

5.7 6.6

5.5 6.7

35

6.63.2

6.2 6.7

6.1 6.9

7.16.0

40

6.38

6.6 7.0

7.26.6

7.46.5

45

(37.0)

7.0 7.3

7.57.0

7.7 .0

50

7.4 7.5

7.78.4

7.79.4

55

7.64

8.17.8

8.27.8

60

(52.0)

8.48.1

8.58.1

65

8.68.4

8.8 .4

70

8.7 8.8

8.7 9.0

75

9.928..095

80

(75.0)

9.59.3

85

9.5 9.6

90
I 95

9.8 9.8

14,600

19,900

4=

,0119

,0122

- Average, TJ = .001195 .012

Herefrom,

H =
calc
H-H
calc. obs.
= per cent. of
H
calc.

H = .012B'.6
14,62019,52025,140 +20 -380 +140
+.l -1.9 +.6

30,020 +420
+1.4

With regard to hysteresis, this kind of cast-steel is 50 per cent. worse than the aunealed cast-steelNo. 1,but still twice
as good as the hardened sample.But, magnetically,it is poor
- -that is, oflow conductivity, giving for 40 ampere turns
M. M. F. per centimetre length only 6600 lines of magnetic
forcepersquarecentimetre,whilethe annealed steelgives
- 14,OOO-that is, more than twice as many, and even the
- hardened steelgives more, 8OOO.

SOFTMACHINE STEELF. IG.12.

TABLE E.

I. III. 11.

F

Br Bd

Br Bd

Br Bd

0 5

+i8.3
-5.7 10.2

10

+l.2 11.6

15

18.017.72700 14.210.913.411.0

18.218.02755 14.812.413.812.6

30

13.5 14.2

35

40

14.8 18.8

45

(39.0)

H-
obs.
'I=

~,400
.w

5 -7.5

9+.611.2

-2.0 12.4

+7.2 127.16.43.5

1153..33 1114145.4.5.2.70 164.07 1165.4.3

50

15.9 16.8

55 17.016.4

60

16.9 17.4

65

17.3 17.7

80 18.418.3 85 18.718.6 90
(90.0)
64.ooo
,00928

Average, TJ = .00936 hence

H-

calc

A=

-400

64,600
+600

= 2 1.0per cent.

21 0

PROCEEDINGS OF THE I E E E , VOL. 72, NO. 2, FEBRUARY 1984

CAST-IRONFI.G.13.

TABLE X.

I

_+ 2.5

* 3.5

- F
50

55

10 3.9- .6
15 4+.4 .9

-1.7
5-.2 .2

4.7 60 65 8.2 8.9

20

2.6 4.9 +1.6 5.7 70 8.6 9.2

25 6.1 3.83.0 5.4

75 9.0 9.4

30 6.5 4.64.0 5.8

80 9.4 9.7

35

5.24.9 6.1

6.8 85 9.1 9.9

40

5.8 6.47.2 5.5

90 10.0 10.1

6.3

45 7.6 6.1 6.6

95

50

6.8

(95.0)

H =
&IS.
7-

(50.0) 22,300ergs

I 42,000ergs ,01589 .01647

Average, 7 = .01616

H-
CalC.
H- H=
Calc. obs.
= per cent.,

22,000
- 300 - 1.5

42,800
+800
+ 1.9

MAGNETIC IRONORE.FIG.14; TABLEXI.
In the following are given the magneticcurves of a piece of magnetic iron ore, apparently pureFe3 04,of the dimensions,
1 in.x 1in.x 2 3 in.

TABLE XI.

MAGNETIC CHARACTERISTIC.
F = M. M. F., in ampere turnsper centimetrelength of
magnetic circuit. B = magnetization, in lines of magnetic force per
square centimetre.

F

F

B

F

B

10 20 30 40 50 60 1

750

70

1510

80

2000

90

2320

100

2354670 0

110

2760

120

I

23973700

140

3309830 0

160

3220

180

3350

200

220

3580

240

1

4070 4200
4310 4400

IF
0 10 20 30 40
3410

& I TABLEXII.
CYCLIC MAGNETIZATION. AdI&IF

I 1 *m

37346041f0310020

0 + 1520 -200 + 1660 140

+1200 1920 +lo00 2020 150

1800

2280 160

2160 223m0

2117%50 2520 170

3440

3530

3280 3410 3530 3640

33783200

33820
3910
4050 4120 4190 4250 4320 4Mo

3900 3980
4110 4170 4230 4280 4uO 4370

(240.)

H = 9,340 ergs 13,780 ergs
obS.

STEINMETZ:O N THE LAW OF HYSTERESIS

21 1

t) = .02.00429041

Average, t) = .02045

C w e of hysteresis, H = .02045 B'.6

H = 9,320er1g3s,810-
calc.

H - H = -2Oergs +30ergs

calc. obs.

+ = - .2per cent.

.2 per cent.

As seen, thecoefficient of hysteresis of magnetic iron ore,

t) = .020, ranges between that of cast-iron, t) = .016,and of

hardened steel, 7 = ,025.

The magnetic conductivity is approximately 20 per cent. of

that of wrought-iron.

In Fig. 15 is given a comparison of the hysteretic curves of

Hardened steel,

h e a e ld steel,

cast-iroq

Magnetic iron ore,

inthesamesize.

This figureshowsweltlhe

three characteristicforms of

21 2

PROCEEDINGS OF THE IEEE, VOL. 72, NO. 2, FEBRUARY 1984

Flu. 15.

TABLE XIII.

H in ergspercycleand
I

H = vB’.~[+ d B in lines of magnetic

R21
Zmel per

I

F i n ampere turnsper - cmI.

Material.

Hysteretic Coefficient.

Magnetizationat the M.M. F.

Residual MagnetismR./Coercitive ForceC

- t C

9

F=lO =40 -90 F o r F - 4 0 =90 F o r F = 4 0 =90 F o r F = 4 0 =90

very soft iron wire @wing) ...................
WestlngbwEe converter,s h e e t - i r o n ........... Very thin sheet-iron, standard ................. n i c k sbcet-iron.. ............................... Sheet-iron ....................................... Sheet-iron ....................................... Soft annealed cart-steel ........................ Soft machinc steel............................... Cast-steel of low mapetic conductivity ...... CaJt-iron ......................................... Hard& cast-steel ............................ Magnetic iron ore...............................

,0020 ,0024
.00”
.00333t 00421$
.w509
.0080
.oow ,0120 ,0162 ,0250 ,02045

12800 14400 13100 13100
...
251
m
1600 1200
750

14700 17800 17100 17m
... ...
14OOO 14800 6400
6100 8ooo 2320

16600 20800 20700
...
... ... 1683200 18800
9800 10100
12903 3220

5100 8300 3300
4m

(1.9) (1.5)1

(1.8)

(2.0)

(2.3)

(2.8)

(2.5)

(3.1)

(3.2)

(3.9)

(3.4)

(4.2)

6.0

7.0.00114,00133

91M1).o1

9.1

.0.00100038*5***

2600

9.1

11.6.00104,00132

21304000

10.4

15.2 .,0000110566..

752030.5

19.0

,00107,00132

900

10.0

Average .......0.0108,00132

.XI204

*, For N = 100.
+,€ = .746 x 10-6. *,c = .2083 X *, c = 1.16 X

II, This, and thefollowingvalues of this c o l u m n are derived as averageof risinganddecreasing characteristic, because at F = 10 the magnetism is still very unstable.
’,Computed by means of the averagevalues of f = .00132 and = .00108. L
**, Left out by takingtheaverage of 3 .
C

branch of themagnetic

chuyrvsteesr:etic
1. The hardened steel m e , of high coercitive force,has the bend or “knee” on the negutiue side, so that for zero M. M. F. the ‘‘remanent” magnetismis still in the satura-

tion part of the curve-that is, in stable equilibrium; therefore permanently magnetizable. 2. The soft iron curve, with the bend on the positiue side, so that for zero M.M.F. the “remanent” magnetism, though
still very htgh, is already below the range of saturation,

STEINMETZ:ON T H E LAW OF HYSTERESIS

213

TABLE X I V .
W = yH',6
Win watts per cubic inch an1d00complete periods per second. H in lirles of magnetic force per square inch.

7

Very soft iron wire @wing) ...................166 X IO-''

Watlnghousc convener, sheet-iron ........... 201

2.90

2V.4e9ry thin shee2t-.i1r1on .............1...7..4........... 249

2.77 'Thickk t -2i.r34o n . . .............1...9..4.............. 277

sheet-iron ...........................6..6..9........ 4.69350 4.07
sheet-iron ....................................... 374 7.17

8.88 7.73 6.63Soft a5n.6n0ealed4c.6a4st-sle3d.75.......2..9..3.......2...1.9.. 663
9.06 S7o.7f8t mac6h.5in7e ste5.e4l5......4....4.0.......3...4..4......2..5..7 778 Cast-steelof low magxtic conductivity ...... 994 Cast-iron........................................ I346

Hardened cast-steel .......................... .2077

H - 10,000 20,000 30,000 40,05000,06000,07000,000 1.66

3.50

2.96

2.45

1.918.15 1.55

5.02

4.36

3.75

3.17 1.23 2.612.66

5.62 4.39 3.28

13.140.99.471.650.944.44

14.6

11.7 39.8 9.18 27.8 6.85 24.2

8 0 , 0 0 0 90,010000,000
1.40 1.16 2.13.412.01 1.70

110,OOO 120,010500,000
13.9.148 2.23 269 3.84

2.12

'6.95 8.39

20.7

17.5

'9.94 11.6 25.187.165.7

' 10.4 13.3

' ~ 14.9 19.0

H = 25,000; alternate current transformer, American styl(ehighfrequency).

H = 35,000; "

"

European " low

"

Only the values smaller than .25 W , can be of practical use; in those larger than 10 the iron gets at least red hot if in larger quantities.

on the branchof unstable equilibre. Therefore the rema-

nent magnetism is very unstable and easily destroyed,

the more as the mcitive force is very small.

3. The cast-ironc w e , which has no marked knee at all, but

a steady curvature of low remanent magnetization, but

with regard to coercitive force ranging between1and 2.

The curve of the magneticiron ore showsall the characteris-

tics of a cast-iron curve.

Havingderived,now,alargernumber

of values of the

hysteretic coefficient q for different kinds of iron and other

material, we shall put them together for comparison in Table
xIn.

It is remarkable, in these results, that for several samplesof

each set the quotientq / C gives almost exactly the same value,

while other valuesdisagree therefrom. Fromthis average value

of q/C are calculated the values of the coercitive force C of

sheet-iron, given in the brackets.

For convenience, in the following table are given the values

W of consumption of energy in watts per cubic inch, for100

completeperiods(magneticcycles)persecond,andforthe

magnetization of H lines of force per square inch, giving as

coefficient of hysteresis the value 7 = 8.3 x 10-6q

In Table XIV., I havegiven anumber of experimental

values of the consumption of energy by hysteresis and believe

to have shown that this consumption of energy can fairly well

be expressed by the empirical formula,

H = qBX

where the exponent x is equal, or at least very nearly, to 1.6,

and the coefficient q a constant of the material, which ranges

from .002 up to .025 and more, and may possibly have aslight

dependence upon the velocity wherewith the magnetic cycle is

performed, as thesecondsetof

alternate-ament readings

seems to indicate.

In the following table, I give the valuesof thehysteretic

resistance TJ for some iron samples, subjected to a magnetic

cycle between F = +190 and -190 ampere tums per centi-

metre, calculated from Hopkinson's tests' by the assumption

of the lawof hysteresis.

q =the coefficientof hysteresis. B =the maximum magnetization in lines of magnetic force per
square centimetre.
5From"Mender ftu Electrotechniker," by Uppenborn, &rlin, Germany.

R =the remanent magnetization in lines of magnetic force per square centimetre.

TABLEX V .

Material.

coaditiom

- 1 B R

wrought-iron ............ SoftBcJscmrstcel ...... Soft Wittworth steel .....

Annealed ........... ,00202 18,250 7250

.OM62 1 8 7,~860

,00257

7,080 19,840

.00598 18,740 9,840

.a9 " .32 ''

.00786 16,120 10,740 .00954 18,800 11,040

.89 "
Siliconsteel ................ 3.44 "

.01844 16,120 8,740

,00937

11,070 15,150

3.44 "

,00784 14,700 8,150

3.44 It
Manganevsteel ........... 4.73
8.74
12.36 I'

,01282 14,700 8,080

.05%3
...

4,620 747

220 ...

... 310 ...

4.73

,04146 10,580 5,850

8.74 '*

.08184 1,985 540

4.73

I 8.74
Chrome-steel ............. .62
1.2 .62

''
" " "

1.2

"

"

"

,06706
...

4,770 733

2,.1.6.0

,01179 15,780 9,320

,01851 14,680 7,570

,00897 14,850 7370

,01638 13,230 6,490

.62 " " Oil-hard. ,03958 13,960 8,600

1.2

"

"

"

,04442 12,870 7,890

Tungsf.=sf=l ............

,01516 15,720 10,140 ,01435 16,Mo 11,010

,04776 14,480 8,640

2.35

very hard

Grey cast-iron............ 3.45 p. c. C.1.;7 p. c. Me
.. White cast-iron........... 2.04 C.: .34 " " .. . . . . . . . . . . . .

,05778 12,130 6,820

,01826 9,150 3,160

,01616 9,340 5,550

- ... 385

77

These valuesof the hysteretic resistancveary from .002up to

082,41 times the first value.

But especially markedis, that q depends much lessupon the

chemical constitutionof the iron sample, than upon its physi-

cal condition, annealing decreasing, and hardening increasing

the hysteresis very considerably.

So far as the chemical constitution is concerned, the purer

the iron the lower is its hysteresis, while any kind of foreign

matter increases the hysteresis. Especiallymanganeseincreases

the hysteretic loss enormously, much less wolfram and chre

mium, least silicon and carbon. Comected with the increak

of hysteresis is always a decrease in magnetic conductivity.

I wish to adda few remarks on twoallegedphenomena

connected with hysteresis, which have been talked about con-

siderablyw, ithoutyetbeingmadeclear;thedecreaseof

hysteresisfor open magneticcircuit,andthedecrease

of

hysteresis of a transformerwith increasing load.

With regard to the first, as shown, actual tests do not show

a smaller valueof hysteresis foropen than for closedmagnetic

circuit.

21 4

PROCEEDINGS OF THE IEEE, VOL. 72, NO. 2, FEBRUARY 1964

And it can not be understood how that could be.

For consider an iron moleculeof the magnetic circuit ex-

posed to the harmonically varying M. M. F. and performing a

magnetic cycle. Evidently it can make no difference for this

iron molecule, whether some trillion of molecules distant the

magneticcircuitendsin air, or is closedentirely in iron,

supposing that the M. M. F. and the magnetism, and therefore

also the magnetic reluctivity, are thseame in both cases.

Neither can it make any difference whether the M. M. F. is

causedonlybyonesine-wave of electriccurrent, or is the

resultant of several M. M. F.’s,as in the loaded transformer. It

is the same as with theelectriccurrent,wheretheenergy

converted into heat in each molecule of the conductor does

not depend either, whether the material of the conductor on

some other point changes, or whether one or more E. M. F.’S

are acting upon the circuit.

Hence, until absolutely exact and undoubtable determina-

tions of the hysteretic loss for fully loaded transformers araet

hand, the assumptionof a decrease of hysteresis with increas-

ing load must be rejected.

That an apparent decreasewithincreasingloadhas been

observedseveraltimesmay

be conceded,forbesidesthe

exceedingly great liability to errors in these tests, where the

hysteretic loss comes out as the small difference of two large

values, primary energy and secondary energy, and therefore is

verymuchaffectedbytheslightest error in anyone of the

components, it mustbeunderstoodthatthemainpossible

errors in the determinations on fully loaded transformers all

point this way. Neglect of secondary self-induction, decrease

of magnetization with increasing load, slowing down of the

dynamealternator, etc., all cause an apparent decrease in the

hysteretic loss for increasing load. At least in one set of tests,

those made by Prof. Ryan, at Cornell University, on a small

Westinghouse converter, I was able to show in my “Elemen-

tary GeometricalTheory of theAlternateCurrentTrans-

former”6thattheobserveddecrease of thehystereticloss

disappears by reducingthedifferentreadingstothesame

magnetization and the samefreq~ency.~

If, indeed, the shape of the waveof M. MF..varies, then a

certaindifference in thevalue of thehysteretic loss can be

imagined.Compare it with amechanical or elasticcycle.A

moving pendulum, or an oscillating spring, for instance, con-

tinuouslyconvertspotentialenergy into kineticenergyand

back; in each oscillation consuming, that is, converting into

heat, a part of the energy by internal and external friction.

Now, if this motion of spring or pendulumis truly harmonic,

less energy is converted into heat than if the motion varies

abruptly, is jerking, etc. S o , in a magnetic cycle, between the

same limits of magnetizationthehystereticlossmight

be

smallest, when the cycle is entirely harmonid, but mght be

larger if the M.M.F.varies abruptly; forinstance, when caused

by an intermittent current.

Now, in atransformer with open secondary the M. M. F.

acting upon the iron is that of the primary current, and this

current is rigidly determined in its shape by the E. M. F. of the

d ~ and th~e E. M.oF. ofSelf-indUctiOn.But in a loaded

transformer the secondary current is proportional to the

changes of the magnetism, therefore increases very consider-

ably in the moment of a sudden changeof magnetism. Hence,

w - w 6 D e c 1891. Electrid Engineer, New Yoh

7~ latest tests of

prov~that,in a

traaJfm the

lossbyhystacJisisnotsmaIkrthanforopensccondaycircrdt

if a sudden and abrupt change in the primary current OCCUTS, just as suddenly the secondary current increases in the oppe site direction, and therebymakes a sudden changeof resulting M. M. F. andmagnetismimpossible, so that the fully loaded transformer compares with the elastic spring which oscillates freely, while the open-circuited transformer compares with a spring,wherethemotion is determinedbyarigidly-acting outside force.
Hence, if the shape of thealternatingprimarycurrent differs considerably from the sine law, a certain decrease of
the hysteretic lossfor increasing loadcan be expected, though certainly not such an enormous decreaseas some former tests
seemed to point out.These tests must undoubtedlyhave given erroneous results, perhaps causedbytheneglectofthesecondary self-induction, which, even if very small and causing onlyaslighterror in thesecondaryenergy,mustcause an enormouserrorinthehystereticloss,thesmalldifference between the two large values-primary and secondary energy.
That an electremagnet without keeper loses its magnetism quickerthanamagnetwithkeeper, or a closedmagnetized iron ring, is a phenomenon, which hasnothing whatever to do with this loss of energy by hysteresis, but is merely due to the demagnetizingforce of theremanentmagnetism. For the remanent magnetism in an open magnetic circuit causes between itspolesacertaindifference of magneticpotential, whichinthemomentof breaking the electric circuit acts as demagnetizingM. M. F., and, if the coercitive force is small,as in wrought-iron or annealed steeal,lmost entirely destroys the remanent magnetism, while in aniron of large coercitive force it affectsthepermanentmagnetismverylittle. In theclosed magnetic circuit the remanent magnetism causes no or very littledifference of magneticpotential,andtherefore no destruction of the remanent magnetism by its own demagnetizing M.M.F. takes place. But with the hysteretic loss of energy this phenomenon has nothingto do.
To combine the results, whaIt believe to have proved isthat loss of energy in iron caused by reversals of magnetism can be expressed by the analytical formula:
+ H = v B ’ . ~ r N B 2 .

where

q =the cuefficient of hysteresis,

e -the cuefficient of eddy currents,

N =the frequency of the altemations of magnetism,

qB’.6 =the loss of energy by hysteresis propero, r by

molecularfriction, and

eN, B2 =the loss of energy by eddy currents,per magnetic

cycle andperproportional

to thefrequency N.

- n
Pa 1 w
42 85 115 142 164 185 205 223 2u) 258 275 292 308 324 339 353 366

TABLEX V I .

B 81.6
9ooo 2122 9500 2313 10.000 2.511 10,500 2.716 llp00 2.925 11,500 3.141 12.000 3.363 12,500 3.589 13,000 3.821 13,500 4.060 14,000 4.303 14Joo 4.580 15,000 4.807 15,500 5.062 16,000 5.329 16,500 5.598

pcr 1 w
378 389 400 41 42 43 44 45 46 47 48 49 50 51 53 54

m per 100
55 56 57 58 59 59 60 63 66 70 73 76 82 89 96 103

STEINMETZ: O N THE LAW OF HYSTERESIS

215

For convenience, I give in Table XVI, the values of the 1.6th powerof the numbers, from 500 to 50,OOO with the parts proportional, or the increaseof B’.6 for 100 lines of magnetic force.
YonLca N.Y..Dmmber 7th 1891.

DISCUSSION.*

THECmuwm:-Gentlemen, thepoet has informed us

that “better Gfty years of EuropethanacycleofCathay.”

What he would have done had he met a cycle of magnetism,

we can but conjecture. “he Institute has therefore good rea-

son,I conceive, to congratulateitself that one of its members

does not shrink from such a conflict. I am sure I shall but

express the sentiments ofeverymember present, when I say

that we are much obligedto Mr. Steinmea for his very elegant

and exhaustive treatment of a subject whose title, to say the

least,hasamostunpromisinganduninterestingsound-a

subject d e d q with the causes of those indspositions of iron

to change its magnetic conditionwhich in our old telegraphic

days wewerewont to s u m up by theunscientificterm of

“ residual magnetism.”

Beforecallingforgeneral discussion, I wouldliketo ask

Mr. Steinmetz whether, in his experiments and tests, he had

determined whether or not there was any real foundation in

factforthedistinction whichProfessor Ewing has drawn

between the molecular friction, which hecalls “static hystere-

sis,’’ andtherealtime-lag,whichhedenominated

“viscous

hysteresis.”

MR. STEINMEIZ:--I. reallaym not yet prepared to answer

the question whether viscous or time hysteresis exists or not.

My tests in only one setof determinationsgave me an increase

of hysteretic loss with increasing frequency, which seems to

point to the existence of a viscous hysteresis. For if a viscous

hysteresis exists,it would show byan apparent increaseof the

coefficient of hysteresis, with increasing frequency. But most

of the tests do not show this, but give the same coefficient of

hysteresis for different hequencies.

At any rate, if there exists such a time-hysteresis-which I

shall try to find out-it follows the law of the 1.6th power

also.

But I think, only at much higher frequencies than those I

have used in my tests, can we hope to meet with viscous

hysteresis. I hope to be able at a future meetingto give more

detailed informationon this and some other phenomena con-

nected with the magnetic hysteresis.

THE Cmmwm-Gentlemen, thesubject is beforeyou.

While a fewof us were in the parlor, prior to the reading of

the paper, I heard Mr. Steinmek condolmgwithhimself in

relation to the weather and expressing the hope

that there

would still be a very considerable discussion.It is therefore to

be hoped that any of us who may feel able to grapple with

such a subject will not hesitate to do so.

MR. CHAWSS. BRADLEY:-I do not feel able to discuss

this paper, but I know it will prove very valuable to us. Our

work of late has been upon transformers.I am connected with

the Fort Wayne Electric Company, whose transformers now

use about 2,000 lines of force to the square centimetre, andwe

have been trying to increasethelines oforceW. e

en-

counteredthe veryphenomenatreated in this paper,and

*Discussionby Messrs. Bradley, Kenuelly, Lochvood and Pupin.

therefore it is very interesting to me, and I think that we ought

to congratulateourselves upon havinga member who can

tacklesuchasubject. It is veryseldom that in America,
anything of this kind is taken up. We see it very often in

Europe, but ourcommercialage will hardly permit us to

devote our time to such experiments and carry them out as
they should be.

MR. JOSEPH WETZLER:-Agentlemanwho is present but

who is not a member, has asked me to inquire of the author

whetherhemadeany experiments on mitis iron and, if so,

what his results were.

MR. STEINMETZ:-m~ e r made any experiments with re-

gard to hysteresis, on mitis iron-only on different kinds of

cast-iron.

MR. A. E. KENNELLyz-Mr. Pr&dent and gena-

I

think that we have to congratulate ourselvesupon a magnetic

and physical treat in the paper that we have just listened to.

Mr.Steinmekhasbeen,Ithink,thefirsttopointoutthis

remarkablelawofhysteresis-thevariation

of theenergy

consumed per cycle, with the total flux per square centimetre

that passes through it. I think that it is perhaps preferable to

expresstheexponent in theequation as a vulgar fraction

instead of as a decimal-not that it alters the factsin any way,

but merely because it gives us a little more hope of being able

to understand whatthe equation means,if not now, at least let

us say in thefuture. If, instead of writing the energy-

~ rste.inmetz calls it H,as q ~ 1 . 6w, e write it q ~ fit, gives us

some hopeof being able to transform that in a simpleher,

which will give us thefundamental law concerned I think

there is very little doubt that the law Mr. Steinmek giva is

the true one. It is, first of aU, as he showed us some time ago,

in accordance with the values observed by Professor Ewing,

and so far as my own knowledge goesI am able to corroborate

it, for I have observed the same law in the case of one sample

of wrought-irontakenbyaballisticmethod,

and another

sample of wrought-irontakenbywattmetermethcd,both
givingthe 9 power,although I do not knowwhat the exact

value of the coefficient q was in those particular instances.It
is very puzzling to understand what that peculiar fraction 9

means. It is rather too high andunwieldyafraction to be

understood at a glance. But whatever its inner meaning may

be, its outward and visible indications are clear enough, be-

cause if you double the flux density in a piece of iron you will

trebletheenergywhich

is consumed init per cycle, by

hysteresis, independentof the energythat is umsumed in it by
eddy currents. Of course,if you have any m e which starts

from the zero point andrises up in that way, and if you take

arbitrary distances like thisin the formof u, u2, ua3n,d so on,

then if you want to find out whether that curve follows any

such law as

Y - bX”

you have only got to mark off the ordinates corresponding to

those abscissae, and to see if with the powers ofu along Xyou

have a constant ratio from one to another in the ordinates. If

you do, thatratio will be un. In this case,if u is 2, un is almost

exactly 3. For the 1.6thpower of 2 is 3.03, which means that if

yoduoubltehe

maximum magnetization in a piece of

wrought-iron, youwill have 3.03 times the hysteresis loss, and

this is a simple way of stating the resultswhich Mr. Steinmek

has pointed out.

MR. STEINMETZ:p-~ointed out by Mr. Kennelly, thislaw

of hysteresis gives a very simple numericalmeaning. It means

21 6

P R O C E E D I N G S O FTHE IEEE, VOL. 72, NO. 2, FEBRUARY 1%

that by doubling the magnetization you approximately treble

thehystereticloss and quadrupletheeddy loss. So if you

make but two tests therefrom, you can find out the amountof

energyconsumedbyeddiesandtheamountconsumedby

hysteresis for any magnetization.

And, in general, youwill see at once whether theratio of the

iron lossfordoubledmagnetization is nearer to three, or

rather 3.031, or to four, that is, whether hysteresis or eddies

consume more energyin the iron.

I would like to add a few remarks regarding the results of

the tests given in the paper.This law of hysteresisis of interest

from another pointof view:

We all know,now, that energy is always the same and

indestructible, and merely changes its form and appearance,

so that a certain quantity of any kindof energy converted into

any other kind of energy always gives an exactly determined

amount of the other formof energy, which we call the law of

conservation of energy.

But this lawof conservation of energy needs a certain

restrictionor,rather, addition, because everyconversionof

one formof energy into another is not possible, but only those

where the value of a certain integral, called by Clausius the

"entropy," is positive or more correctly, is not negative, though

the case, that the integralof entropy equals zero, hardly exists

in nature otherwise but as mathematical fiction, or, in plain

FJ1PljSh. only those conversions whereby the sum of the latent

heat of the universe increases.

Accordmg to this law of entropy, if the complete conversion

of one form of energy into another is possible, the opposite

conversion is notcompletelypossible. Or if we converta

certain amount of one formof energy into another form of

energy,and this back again into thefirstform ofenergy,

whichwe call a cyclicconversionofenergy-we do not get

back the origrnal amount of energy, but less, and a part of the

energy has been lost;that means, converted into and dis-

sipated as heat.

Therefore no completecyclicconversionofenergyexists,

but byanysuchcycletheamountofavailableenergy

has

decreased by that fraction that was converted into heat.

Now, these cyclic conversionsof energy are of great impor-

tance in nature.

For instance, a moving pendulum, an osciuating spring, a

discharging condenser completes cyclic processeIsn. the mov-

ing pendulum, continuously kineticmechauical energy is con-
verted into potential decal energy, when it moves from

the vertical position into its greatest elongation, while when

movingfromelongation into verticalposition its potential

energy is reconverted into kinetic e n e r g y , thereby completing a

cycle, so that in vertical position all the energy is kinetic, in

elongation all the energy potential.

In the same way, in the oscillating spring, a cycle is per-

formed between potentialenergy of elasticity and kinetic

energy of motion, in the discharging condenser between elec-

trostatic and electrodynamic energy, and that the pendulum

and the spring come to rest, and the condenser discharges, is

due to the continuous lossof energy by dissipation as heat,

caused by the law of entropy.

Now, in none of these cyclic conversionsof energy, so far as

I know, was the law known,which determinesand analy-hcally

formulatesthe lossof energy by conversion into heat. The

electromagneticcycle is thefirstone where in the lawof

hysteresis, this law of dissipation of energy by heat, finds an

analytical formulation.

In the alternating electromagnetism wehave such a cyclic conversion of energy from electric into magnetic energy and back. Magnetism represents a certain amount of stored up or potential energy determined by the integral
1FdB

Now, as long as the magnetism increases, electric energy is

transferred from the electric curreanntd converted into poten-

tial magnetic energy. While the magnetism decreases, potential

magnetic energy is reumverted into electric energy, and ap-

pears in the electric CirCUit aS E. M. F.

But the full amount ofenergy is not givenback to the

electric circuit, but less. Less by that amount that has been

converted into heat by hysteresis.

Hence the lawof hysteresisis the dependenceof the integral

of entropy in the electromagnetic cycle, upon the intensity of

magnetization, and thereforeof interest.

DR M. 1. Pwnc-I agree fully with Mr. Steinmetz's last

remarks that no process in nature is perfectly reversible and

that the phenomenon of magnetic hysteresis is only a special

case of theirreversibility of ~ ~proceasses. Ilt is onlya

special case of the general law which was first announced by

the late Professor Clausius, thelaw namely that the entropyof

the universe is tending toward a maximum, that is, that there

is a certainfunctionof the propertiesof matter of the universe

which increases as the amount of heat energy increases in the

universe. Now, as in every process thereis a certain amounotf

energy convertedinto heat, the amountof heat in the universe

is continually increasing. "herefore the entropy is continually

increasing and thereforesteadilyapproaching its maximum.

Professor Rankine made aguess as to how many years would

elapse before the whole energy of the universe will be con-

verted into heat, when there will be no life, no natural phe-

nomena excepting heat vibrations.It is very far off yet.

Closely connected with this magnetic hysteresis is, I think

the so called electrestatic hysteresis. Of course experimental

researches in this field have not been carried on far enough

yet, to enable us to speak with any definiteness, but stin it is

beyond all doubt that if youpolarizeadielectricandde-

polarize it again, a certain amount of heat is developed. I

think oneof the obstacles to the commercial introduction of

the condenser,is its getting hot.Now some thinkit gets hot on

account of the convection currents which are passing between

the plates of the condenser by means of the air currents and

the dust that is in the air; but if you use paraffine so that it

will preventthoseconvectioncurrents,eventhenyou

will

0bseri.e heat developed in the paraffine which must be attri-

butedtothe same cause whichdevelopsheatwhen iron is

magnetized and demagnetized; that is hysteresis. Polarization

and depolarization of paraffine, and in fact any other dielec-

tric, is not a perfectly reversible process.

Allow me now to commentupon a few points broughtup in

Mr. Steinmetz's paper. I always believed thoroughlyin Profes-

sor Ewing's views with regard to the following experimentally

well supported assumption, namely that in very low magneti-

zationsthe actof magnetizing and demagnetizinigs practically

reversible,and that when a high point of saturation,say

24,000 or 25,000 lines per square centimetre is reached, that

after that the loss dueto hysteresis doesnot increase. I do not

see why it should increase,because after that the iron does not

receiveanystrongermagnetization.Theadditional lines of

forceafterpassingthesaturationpointaredue

to thein-

STEINMETZ: ON THE LAW OF HYSTERESIS

21 7

creased magnetizationof the air itself, and that magnetization

is practically re~ersible.I~see that Mr. Steinmetz has found

out anincrease,independent of thedegree of saturation.

There is a discrepancy, and I am inclined to side with Profes-

sorEwing, until I am convincedby Mr. Steinmetzthathis

method of measurement and observationcouldnot be ob-

jected to in any particular whatever. Unfortunately

Mr. Steinmetz has not discussed his method so that one can

examine it critically. He has given the general idea, theinstru-

ments employed, etc.,but there is no discussion of the theory

of themethod,and also of theprobablepercentageof his

errors of observation. I am sure that Mr. Steinmetz will do

that at some future time. It would be very interesting andvery

important indeed to knowwhetherthatdisagreement is in

favor of Mr. Steinmetz orof Professor Ewing.

There is on page 49 adiscussion of thevariation of the

hysteresislosswiththeload." In that discussion Mr. Stein-

metz says as long as the secondary currentis open, the form of

the wave of the primary currentmay not be a sine curve;but

that whenthesecondarycurrent is started, the waveof the

magneto-motiveforce is forced into theshape of thesine

curve on account of thereaction of thesecondarycurrent.

Now I would beg to disagree withMr. Steinmetz; I think it is

just theopposite. It doesnotmakeanydifferencewhatthe

electromotiveforceis, as long as thereisaverylargeself-

induction in the circuit,-as there certainly is in the primary

circuit as long as thesecondary is open,the waveof the

primary circuit is independent of the waveof the impressed

electromotive force and is practically a sine wave.Butwhen

the secondary circuit is closed, then the impressed electromo-

tive force, being assisted by the electromotive forces

in the

secondary circuit, asserts itself and gives the primary current

its own shape,andthestrongerthesecondarycurrent,the

largerassistancetheprimaryimpressedelectromotiveforce

getsfrom it. Thesecondarycurrent aids theprimaryim-

pressed E. M. F. to assert itself and force the primary current

into its shape, that is, the shapeof the impressed E. M. F. That

can be proved very easily both from theoretical and practical

standpoints. So that I do not see the force of Mr. Steinmetz's

argument.

MR. SnINMETz:-Themethod used in my tests was the

well-known electredynamometer method, as explained in the

paper, withsomeslightmodifications to insurethegreatest

possible exactnessin the results.

With regard to the difference between open circuited and

fullyloadedtransformers,I think ProfessorPupin misun-

derstood me. I didnotsaythatthe waveof theprimary

crurenr in the transformer under full I d resembles the sine

wavemore than with open circuit, for that wouldhave been

wrong. WhatI said was that thewave of the magnetism and of

the resulting M. M. F. in the transformer underfull load resem-

blesmore the sine wave than it does in the open circuited

transformer.

Suppose theimpressed E. M. F. at the terminals of the transformer differs from the sineshape, differs even consider-

ably. Then the primary current, which at open circuit repre-

sents the resulting M. M. F., will differ much less from the sine

Shape than the impressed E. M. F.,

Smoothed out and

rounded off to a very great extent by the heavy self-induction

of the open circuittransformer. For in themoment of any

9AE Kenwlly,on "Magnetic ReluaancC" W s c n o ~ svo,l viii.No.
m.

lli'kw%cnO~

vol ix, p. 49 (thisissuc p. 215).

sudden rise of the impressed E. M. F., already a small rise of

theprimarycurrentand,therefore, of themagnetism, will

induce sufficient counter E. M. F. to make a rapid increase of

the primary current impossible.

Hence, in the open circuited transformer, the waveof the

magnetism will resemble the sine wave more thanthe wave of

impressed E. M. F. But, nevertheless, it must differ from the

sine wave if theimpressed E. M. F. differsfromsineshape.

For, as before said, the resulting or current producingE. M. F.

and, therefore, the current, is rigidly determined by the small

difference of impressed and induced E. M. F., and the induced

E. M. F. must therefore have a shape very similar to the im-

pressed E. M. F., hence differing from sine shape the more the

impressed E. M. F. differs therefrom.

Now, the induced E. M. F. is the diffexential quotient of the

magnetism.Hence, if themagnetism is asinewave its dif-

ferential quotient, the induced E. M. F., has to be a sine wave

alsoand, on theother hand, themoretheinduced E. M. F.

differs fromsine shape, the more its integralfunction,the

magnetism, is forcedtodiffer.Indeed,themagnetismmay

apparently differ, in its absolute value,lessfromsinusoidal

form than the impressedE. M.F., for it is not the instantaneous

values of the magnetism which are directly influenced by the

shape of impressed E. M. F., but thegreatersteepness or

flatness of the curve of magnetism which is directly caused by

theimpressed E. M. F. But it is just this difference in the

velocity of change, that is, in the quickness of rise or decrease

of the magnetism, and not the magnetism itself which would

have to account for an increased loss by hysteresis. Heintcies,

really not the difference of the curve of magnetism, from sine

shape, but that of thecurve of inducedand,therefore of

impressed E. M.F.,which may possibly cause an increase in the

loss by hysteresis.

Qute different in the transformer at full load. Indeed, its

apparent self-inductionis essentially decreased and the primary

current will thereforeresemble the shape of theimpressed

E. M. F., and differ from the sinusoidal form, much morethan

for open circuit.

But at fullloadthe waveofmagnetism and of resulting

M. MF..is much more independent of that of primary current

and primary E. M. F. It is caused by the combined action of

the instautanmusvalues of primary andof secondary current,

and thesecondarycurrent, again, is inducedbythemag-

netism.Hencetheresult

will be, if asudden changeof

impressed E. M. F. occurs and produces a sudden change of

primary current,just as suddenly as the opposite changeof the

secondary currents will take place,so that the resultanMt. FM..

of both combined currents will not change perceptibly, but

practically independent of tither current, will alternate freely

in sinusoidal waves, in spite of anydifference in the wave

shape of primary and secondary current from the sine law.

And, indeed, a glance over the curves of instantaneous

values of theelectricquantities in the transformer, as they

have been determined, for instance+by ProfessorRyan, at

Cornell University, and communicated to this Institute some

time ago:' shows a considerable discrepancy at open circuit

betweentheprimarycurrentandthesinewave,whileinthe

loaded tranSfOlTtler the secondary E. M. P. and, therefore, the

magnetism,almost universally resemblessine shape.

With regard to Ewing's theory of the molecular magnets, I

do not say that I disbelieve in it, wither that I believe in it. At

thefirst view, this theory did not seem to agree with the

21 8

P R O C E E D I N G S O F T H EIEEE, VOL. 72, NO. 2, FEBRUARY 1'3%

results ofmy tests, as I said in my paper, but I did not take

the time to think it over more completely whether this theory

could be made to agree with the tests; my aim was to gather

facts, being convincedthatbased upon alargenumber of

facts, a theorywill be found in duetime to explain them. [ S e e

appendix, p. 221.1

DR hrpIN:-hbgnetiC force is certainly a resultant of the

primary and secondary currents. As long as the sewndary is

open, the primary current wiU be a sine wave, practically. It

does not make any difference what the impressed electromo-

tive force is otfhe alternator,and therefore the magneto-motive

force will be a sine wave and the magnetic inductionwill vary

liksaeine

wave. If you dose thseecondarcyircuit,

the self-inductionin the primaryis reduced, and therefore the

backelectromotiveforce in theprimary is smallerandthe

impressed electromotive forcebegins to assert itself more and

moreandgives to theprimarycurrent its ownshape.‘The

shape of thesecondarycurrent, as long as thesecondary‘s

resistance is very large and stheecondary current is small-that

too is practically a sine wave, the primary current being also

practically a sine wave, the resultant of the two-that is, the

magneto-motive force-must also be a sine wave. But now, if

you diminish the resistance in the secondary circuit, that is,

increase the load, then thsehape of the primary currenbt egm

to correspondtothe shape of theimpressedelectromotive

force,and also theshape of secondarycurrentbeginsto

correspond to theimpressedelectromotiveforce, and the

resultant of the two, the magnetizing current, must also begin

tocorrespondmoreandmore

in shape to theimpressed

electromotive force-that is, the magnetemotive force begins

tocorrespondtothe shape of\,the impressedelectromotive

force. The same is trueof the magnetic induction.We are not

to forget that the secondary current does not depend on the

rate of change of the primary current only. The relation is a

little more complicated. ‘Thereis a differencein phase between

theprimaryand secondary, varyinganywhere between 90

degreesand180degrees.Whenthedifference

in phase is

nearly180degrees, that is, at full load,thentheprimary

current and the secondary current correspond to each other

almostexactlyinshape,andhavethesame

shape as the

impressed electromotive force.

MR. STEINMETZ:-I can not yet quite agree with Dr. Pupin.

‘Iheresultant Of two M. M.F.’S Of

s h a p e , but different

phase, need not have the same shape, but can have an entirely

differentform. So forinstancetheresultant otfwovery

ragged-looking waves can be acompletesinewave.Let us

come down to numerical values. Take for instance a loo0 volt

alternator, feeding into the primarycoil of a transformer. ‘The

internal resistance of the primary coil is 20 o. The current

flowing through theprimary, at open secondary circuit,a

small fraction of an ampere. Hence, what I call the “resulting

E. M. F.,” that is the E. M. F. which sends the current through

the resistance, is only a few volts.

But this “resulting E. M. F.,” is the difference of this instan-

taneous values of primaryimpressed, and primaryinduced

E. M. F. Thedifference is only a fewvolts,the primary im-

pressed E. M. F. = loo0 volts,hencetheprimaryinduced

E. M. F. must be almost like the impressed E. M. F., and must

differ fromsine-shape,therefore, if theimpressed E. M. F.

differs;and if thedifferentialquotient of magnetism,the

induced E. M. F., is non-sinusoidal, the curve of magnetism is

non-sinusoidal also.

In the transformerat full load the current and therefore the

differencebetweeninducedandimpressed E. M. F. is much

greater, the induced E. M. F. is therefore much more indepen-

dent of the impressed E. M. F., the more, the greater the load

is, hence the curve of magnetism alternating freerthan at open

circuit, and thereforemore approximating the harmonic vibra-

tion of the sine-wave.

DR.PUPIN:-It does not by anymeans follow that at every

moment the difference between theimpressed E. M. F. and the

back E. M. F. is smallwhenaveragevalueof thecurrent is

small, and that is the pointin your argument. And evenif it is

I do not see how that can prove that the shape of the current
and the impressed E. M. F. are the same.

MR. S ~ ~ : - Whavee seen that the effective value of

the current, and therefore the effectivoer average value of the

difference of primary impressed and primary induced E. M. F.

must be small. This indeed does not prove that some of the

instantaneous values of this difference may not be consider-

able. But first, this couldbeonlythe case withveryfew

values,because, if for anygreatlength of time thecurrent

were considerable, this would show in the average or effective

value,themore, as this is the averageof thesquares of

instantaneous values.

On the other hand, to make the current considerable only

for a moment, while immediately before and after it is small,

eithertheinduced E. M. F. mustsuddenlydecreaseenor-

mously, and the next moment increjauset as suddenly-which

is impossible, because it is the differential quotient ofmag-

netism-or the primaryE. M. F. had to rise and decrease again

verysuddenlya, ndsuchasudden

rise, andimmediately

afterwards decreaseof primary impressed E. M. F., not only is

an electredynamic alternator unable to produce, but no elec-

tric circuit would permit a current ofsuch enormously large

value andshort duration topass. Hencewe canfrom thesmall

value of effectiveprimarycurrent,conclude that also its

instantaneous values without exception must be small.

DR. F”IN:-I do not suppose that a wave which is not a

sine, must neceSSarily be a wave that goes up and down with

suddenvariations. I think that everygoodcommercialma-

chine is constructedin such a way that the electromotive force

is a perfectly smooth curve. There may be small comers, but

even those comers are very nicely rounded. Generally speak-

ing it is a sign of good construction of the machine when the

impressedelectromotiveforce is asmoothcurve-certainly

not a curve that haskinks in it. Kinks in the currentcurve are

producedbyaharmonicallyvaryingresistance. It would be

almost impossible to construct a machine so badly as to give

kinks in the electromotive force curve. ‘Ihe current may run

smoothly,but still be veryfar from a sine waveA. sine wave is

not theonlysmoothlyrunningwave.Therearemanyother

waves that arenice and smooth. The only possibilitoyf having

such a currentas Mr. Steinmetz described, would be simply to

introduce into the circuit a harmonically variable resistance.

An arc light circuit represents a harmonically variable resis-

tance, and introduces those complications, the kinks. An arc

lightmachineviolatesmost of the well establishedrules in

dynamoconstruction,but it does the workof the arc light

circuit admirably, and it does it because it encourages kinks

and other irregularities in the currenwt ave.

I v l ~STEINMETZ-Ientirely agree withProfessorPupin,

that there is really nowadays almost no possibility of getting

suchsharppointed wavesof alternating E. M. F. that a dif-

ference of the hysteretic loss between open circuit and closed

circuit couldbe expected. And I did not believe myselfin this

cause of the discrepancyof former testson transformers under

full load and with open secondary circuit.I made this remark

STEINMETZ: ON THE LAW OF HYSTERESIS

21 9

only to be absolutely just, and not entirely to reject as err* circuitanE.M.~.,E~isconsumed,whi~lagsonoquartcrofa

neous, determinations made by others buatt least to point out phase, or 90 degrees, behind the current, and is proportional

a cause which might produce, though not at all likely, a slight to the currentC , with a coefficient ofproportionality I , which

difference between the values found under full load and with I callthe Inductance of the circuit:

open circuit. Indeed, all our modern alternatorsproduce wavesvery

E2 s= IC

much resembhg sine curves, andtheonlyway to getfrom

This inductance, I , is of equal dimension with the resistance

them such rapidly changing E. M. F.’s is, as Dr. pupin pointed R,hencemeasuredinohmsalso.

out, the introductionof variable resistances, as arc lamps, into

This inductance, I , is proportional also to the frequency of

the circuit.

the alternatingcurrent. Hence, if I call the inductance for 100

But some of the older types of alternators, as, for instance, complete periods per second the N d inductunceI,, for any

the Klimenko alternator at the Vienna exhibition,1882,’2 gave other frequencyN the inductance is simply

evidently sharp pointed E. M. F.’S, as 1 found by drawing the

curve of instantaneous values of E. M. F. of an alternator of a

similar type, where induction was produced by making and
breaking the magnetic circuitA.s you see,this is a verysimilar case to that referred to by Dr. Pupin, only that in this case a
variablemagneticreluctanceandnotavariableelectrical

Now, the “normal inductance”is a constant of the circuit just as well as the“resistance” or the “coefficient of selfinduction,” and only dependsupon the latter by the equation,

resistance was introduced into the circuit. MR. KENNEuY:-It is unfair, perhaps, when we have such

z, = 200rrL

a good paper, to offercriticisms upon it, but when it is as likely as this is to become classical Ithink that in self defense

resistance. onlythat“inductance” is measuredin ohms also, therefore
most easily combined with the

we ought to try to keep it as freefromallimperfections as

Thecombination of theresistance-which determine the

possible. I am taking the liberty of makmg a criticism on one E. M. F. of equal phase with the current-with the inductance,

term Mr. Steinmetz has used. He has spoken of the normal which determines the E. M. F. lagging onequarter phase be-

inductance of the coil of his ammeter as so many ohms, and I hind the current,is the “impedance,”or “apparentresistance.”

would suggest that it would be preferable to employ the word

Hence,

impedance, instead of inductance, because an inductance is a henryandanimpedanceisanohm,andIthinkitisapityto

+ Impedance = {(Resistan~e)~ (Inductance)2

confuse the two ideas. MR. STEINMETZ:-I didnot use theterminductance as
synonymous with coefficient of self-induction, where it would be expressed in henrys,but I used inductance in the verysense that Mr. Kennelly means with impehnce.

Thequotient of inductance and resistance is theangleof difference of phase between current and impressed E. M. F.

tancp

=

Inductance Resistance

I intentionally used the term inductance, following a pro-

You see, it is easyto make a person understand that he has

position which I read once, I do not remember where,

but in an alternatingcurrentcircuit two kinds of resistancesa

which seemed to meso highly commendable, that I should like “resistance” which consumes energyand an “inductance”

to see it introduced in practical engineering.

which does not consume energy, and make him calculate the

Indeed,the“coefficient of self-induction”givesallthe

apparent resistance or “impedance” as the hypothenuse of a

information needed for determining the electric phenomena in right-angled triangle, with resistance and inductanascecatheti;

inductivecircuits.Buteverybcdy will concedethat it is a while the coefficientof self-induction will frighten the “practi-

tedious, cumbersomeworkf,romthe“coefficientof

self- calman” still for quite awhile.

induction” to- calculate, for instance, the instrument correc-

On the other hand, “inductance” is more convenient than

tions for a whole set of tests made with somewhat differing “coefficient of self-induction,” because expressed in the same

frequencies. Besides, I think it will be some time before the

dimensions as resistance, in ohms.

“practical electrician” will handle the “coefficient of self-in-

I used the term “normal inductance,” because in reducing

duction” just as easily as he now does ohms and amperes.

the rea- I found it much more convenient than the use of

Let us considersomewhatcloserthephenomena

in an the “coefficient of self-induction,” and therefore recommend

inductive circuit. If a sine waveof alternating current flows its use.

through an inductive circuit, a certain E. kf. F. is consumed by
Opposing E. M. F.’S.

MR. WETzLER:-Before moving to adjourn, I would like to
move a vote of thanks to Mr. Steinmetz for his admirable and

First, by the electric resistance of the circuit, an E. M. F. E, interesting paper this evening.

is consumed, which is proportional to the current C, with a

THE Chmww:--Gentlemen, it is with fee@ of peculiar

coefficientof proportionality, R , which is called the true or gratification that I put this motion. I was very glad indeed to

ohmic resistance,or, in short,the Resirtance of the circuit.

hear Mr. Bradley,in his initiatory remarks speak of the

This E. M. F. is of equal (but opposite) phase with the cur- markedexcellences of the paper wehave heard read, and I

rent C:

was pleased also to hear him remark upon the rarity of such

E1 = RC

papers in America Mr. Bradley, I think,did our sister societies of Europe more thanjustice, becauseit is in but few of the

Then by the action of the changing magnetic field of the

societiesoverthere, and I am speaking of Englishspeaking

countries of course, that we find such papers as this-leaving

‘2Aremarkabkfeaturewasthatitconsumcd4w.~.whcnmnning~~ outthePhysicalSociety and thatotherin which the most

M load, but almost 6 R P. when runningM y excited but without taking m t off, that is, without load.

distinguished member of our own profession now presides so

220

PROCEEDINGS OF THE IEEE, VOL. 72, NO. 2, FEBRUARY 1 9 6 4

ably (I mean the Royal Society), thereis none in which papers of this character are of high frequency.
[A vote of thanks was carried and the meeting adjourned.]
-
APPENDIX

[COMMUNICATED BY MR SEXNMXZ AFTER ADJOURNMENT.]

Having had time in thelast fewdays to considermore

deeply the relation of this law of hysteresis to Ewing's theory

of magnetism, I found that this law of hysteresis agrees very

nicelywith Ewing's theory, giving just thephenomena this

theory leadsus to expect.

According to Ewing's 'Iheory, for very low M. M. F.'s, forces

too small to affecthechains of molecularmagnets,the

magnetic cycle should be almost reversible, that is, the hys-

teresis very small or almost nil.

For medium M. M. F.'s, that is M. M. F.'S largeenough to

break up the chainsof molecular magnets, the magnetic cycles

mustbecomemarkedlyirreversiblea, ndthehysteresis

as

function of the M. M. F., must rapidly increase.

For high M. M. F.'s, where the chains of molecular magnets

are mostly broken by the superior outsidMe. M. F., the hyster-

eticloss, as function of the M. M. F., should be expected to

increase slower again and always slower.

This is exactly thecase,when the hysteretic lossf,ollows the law of the 1.6th of the magnetizm'on B, as shown best by the
affixed m e Fig. 16.13 In Fig.16the dorred m e gives themagnetization B, in
lines of magnetic forceper cm.*,as function of theM. FM..F, in ampere turnsper cm.
?he drawn m e gives the hysteretic loss, in ergs per and cycle, calculated by the equation:
H = .003507B1.6")
but not plotted, as in the former curves, with the magnetiza-
tions B as abscisste,but with theM. M. F.'s:F as ab&, that
is in the form:
H = f( F).
- As seen,the hysteresis H for low M. M. F.'s,F = 0 1, is - verylow andalmost nil, increasesveryrapidly for medium
M. M. F., F = 2 5, andthenincreasesslower again and always slower, just as Ewing's theory leads us to expect.
Yonkers, N.Y.,February 7th,1892.
curve corresponds to a set of tests n o t contained in the paper, being made after its completion. I chose this particular set of tests, because it covers a larger range of magnetization than any set of tests given in the
Paper.

STEINMETZ: ON THE LAW OF HYSTERESIS

221