CHAPTER XII. 

MAGNETIC SATURATION AND HYSTERESIS IN ALTERNAT- 
ING-CURRENT CIRCUITS. 

99. If an alternating e.m.f. is impressed upon a circuit con- 
taining resistance and inductance, the current and thereby the 
magnetic flux produced by the current immediately assume 
their final or permanent values only in case the circuit is closed 
at that point of the e.m:f. wave at which the permanent current 
is zero. Closing the circuit at any other point of the e.m.f. wave 
produces a transient term of current and of magnetic flux. So 
for instance, if the circuit is closed when the current i should 
have its negative maximum value - 70, and therefore the 
magnetic flux and the magnetic flux density also be at their 
negative maximum value - ^>0 and - (B0 — that is, in an 
inductive circuit, near the zero value of the decreasing e.m.f. 
wave — during the first half wave of e.m.f. the magnetic flux, 
which generates the counter e.m.f., should vary from — 4>0 to 
+ <I>0, or by 2 4>0; hence, starting with 0, to generate the same 
counter e.m.f., it must rise to + 2 <l>0, that is, twice its permanent 
value, and so the current i also rises, at constant inductance L, 
from zero to twice its maximum permanent value, 2 70. Since 
the e.m.f. consumed by the current during the variation from 
0 to 2 70 is greater than during the normal variation from — 70 
to + 70, less .e.m.f. is to be generated by the change of magnetic 
flux, that is, the magnetic flux does not quite rise to 2 4>0, but 
remains below this value the more, the higher the resistance of 
the circuit. During the next half wave the e.m.f. has reversed, 
but the current is still mostly in the previous direction, and the 
generated e.m.f. thus must give the resistance drop, that is, the 
total variation of magnetic flux must be greater than 2 4>0, 
the more, the higher the resistance. That is, starting at a value 
somewhat below 2 4>0, it decreases below zero, and reaches a 
negative value. During the third half wave the magnetic flux, 
starting not at zero as in the first half wave, but at a negative 

179 


180 TRANSIENT PHENOMENA 

value, thus reaches a lower positive maximum, and thus grad- 
ually, at a rate depending upon the resistance of the circuit, the 
waves of magnetic flux 4>, and thereby current i, approach their 
final permanent or symmetrical cycles. 

100. In the preceding, the assumption has been made that 
the magnetic flux, <£, or the flux density, (B, is proportional to 
the current, or in other words, that the inductance, L, is con- 
stant. If the magnetic circuit interlinked with the electric 
circuit contains iron, and especially if it is an iron-clad or closed 
magnetic circuit, as that of a transformer, the current is not 
proportional to the magnetic flux or magnetic flux density, but 
increases for high values of flux density more than proportional, 
that is, the flux density in the iron reaches a finite limiting value. 
In the case illustrated above, the current corresponding to 
double the normal maximum magnetic flux, 3>0, or flux density, 
®0, may be many times greater than twice the normal maximum 
current, 70. For instance, if the maximum permanent current 
is 70 = 4.5 amperes, the maximum permanent flux density, 
(B0 = 10,000, and the circuit closed, as above, at that point of 
the e.m.f. wave where the flux density should have its negative 
maximum, — &0 = — 10,000, but the actual flux density is 0, 
during the first half wave of e.m.f., the flux density, when 
neglecting the resistance of the electric circuit, should rise from 
0 to 2 (B0 = 20,000, and at this high value of saturation the 
corresponding current maximum would be, by the magnetic 
cycle, Fig. 43, 200 amperes, that is, not twice but 44.5 times 
the normal value. With such excessive values of current, the 
e.m.f. consumed by resistance would be in general considerable, 
and the e.m.f. consumed by inductance, and therefore the 
variation of magnetic flux density, considerably decreased, that 
is, the maximum magnetic flux density would not rise to 20,000, 
but remain considerably below this value. The maximum 
current, however, would be still very much greater than twice 
the normal maximum. That is, in an iron-clad circuit, in 
starting, the transient term of current may rise to values very 
much higher than in air magnetic circuits. While in the latter 
it is limited to twice the normal value, in the iron-clad circuit, 
if the magnetic flux density reaches into the range of magnetic 
saturation, very much higher values of transient current are 
found. Due to the far greater effect of the resistance with such 


MAGNETIC SATURATION AND HYSTERESIS 181 

excessive values of current, the transient term of current during 
the first half waves decreases at a more rapid rate ; due to the 
lack of proportionality between current and magnetic flux 
density, the transient term does not follow the exponential law 
any more. 

101. In an iron-clad magnetic circuit, the current is not only 
not proportional to the magnetic flux density, but the same 
magnetic flux density can be produced by different currents, or 
with the same current the flux density can have very different 
values, depending on the point of the hysteresis cycle. Therefore 
the magnetic flux density for zero current may equal zero, or, on 
the decreasing branch of the hysteresis cycle, Fig. 43, may be 
+ 7600, or, on the increasing branch, — 7600. Thus, when 
closing the electric circuit energizing an iron-clad magnetic 
circuit, as a transformer, at the moment of zero current, the 
magnetic flux density may not be zero, but may still have a high 
value, as remanent magnetism. For instance, closing the 
circuit at the point of the e.m.f. wave where the permanent 
wave of magnetic flux density would have its negative maximum 
value, — OJ0 = -- 10,000, the actual density at this moment may 
be <$>r = + 7600, the remanent magnetism of the cycle. During 
the first half wave of impressed e.m.f. the variation of flux 
density by 2 (B0, as required to generate the counter e.m.f., when 
neglecting the resistance, would bring the positive maximum of 
flux density up to (Br + 2 (B0 = 27,600, requiring 1880 amperes 
maximum current, or 420 times the normal current. Obviously, 
no such rise could occur, since the resistance of the circuit would 
consume a considerable part of the e.m.f., and so lower the flux 
density by reducing the e.m.f. consumed by inductance. 

It is obvious, however, that excessive values of transient 
current may occur in transformers and other iron-clad magnetic 
circuits. 

102. When disconnecting a transformer, its current becomes 
zero, that is, the magnetic flux density is left at the value of the 
remanent magnetism ± (Br, and during the period of rest more 
or less decreases spontaneously towards zero. Hence, in con- 
necting a transformer into circuit its flux density may be any- 
where between + ($>r and - (Br. The maximum magnetic flux 
density during the first half cycle of impressed e.m.f. therefore is 
produced if the circuit is closed at the moment where the per- 


182 TRANSIENT PHENOMENA 

manent value of the flux density should be a maximum, ± (B0, 
and the actual density in this moment is the remanent magnetism 
in opposite direction, T (Br, and the maximum value of 
density which could occur then is ± ((Br + 2 (B0). If therefore 
the maximum magnetic flux density <B0 in the transformer is 
such that <Br + 2 (B0 is still below saturation, the transient term 
of current cannot reach abnormal values. At & = 16,000, the 
flux density is about at the bend of the saturation curve, and 
the current still moderate. Estimating (Br = 0.75 (B0 as approx- 
imate value, (Br + 2 (B0 = 16,000 thus gives ffi = 5800, or 
37,500 lines of magnetic flux per square inch. 

In 125-cycle transformers, (B0 is below 5800 or not much above, 
for reasons of heating, and this phenomenon of excessive tran- 
sient currents in starting thus does not appear. At 60 cycles, 
(B0 is usually above this value, and under unfavorable conditions 
considerable transient current may be observed. However, 
for (Br = 0, the limit is (B0 = 8000, or 51,600 lines per square 
inch; and since in 60-cycle transformers the flux density rarely 
exceeds this value to a great extent, and in starting the remanent 
magnetism is rarely very high, this phenomenon of an excessive 
transient current is not very marked. At 25 cycles, however, 
higher densities are used and -the transient starting current 
may then reach formidable values. 

103. Since the relation between the current, i, and the mag- 
netic flux density, (B, is empirically given by the magnetic cycle 
of the material, and cannot be expressed with sufficient accuracy 
by a mathematical equation, the problem of determining the 
transient starting current of a transformer cannot be solved in 
general, but must be investigated in the individual case by 
constructing the curves of current and magnetic flux density. 

Let the normal magnetic cycle .of a transformer be represented 
by the dotted curve in Figs. 43 and 44; the characteristic points 
are: the maximum values, ± OS0 = ± 10,000; the remanent 
values, ± (Br = ± 7600, and the maximum exciting current, 
im = ± 4.5 amp. 

At very high values of flux density an appreciable part of the 
total magnetic flux $ may be carried through space, outside of 
the iron, depending on the construction of the transformer. 
The most convenient way of dealing with such a case is to 
resolve the magnetic flux density, (B, in the iron into the " metallic 


MAGNETIC SATURATION AND HYSTERESIS 


183 


600— WIOCO- -1400- -1800- -2200 


Fig. 43. Magnetic cycle of a transformer starting with low stray field. 


Fig. 44. Magnetic cycle of a transformer starting with high stray field. 


184 


TRANSIENT PHENOMENA 


flux density," ($>' = & - X, which reaches a finite limiting 
value, and the density in space, oe. The total magnetic flux 
then consists of the flux carried by the molecules of the iron, 
$>' = A'(B', where A' is the section of the iron circuit, and the 
space flux, $" = A"3C, where A" is the total section interlinked 
with the electric circuit, including iron as well as other space. 


If then A" = &A/, that is, the total space inside of the coil is 
k times the space filled by the iron, we have 

$ - A' (&' + te), 

or the total magnetic flux even in a case where considerable 
stray field exists, that is, magnetic flux can pass also outside of 


^m 


-55 
500 — 

-45 
400- 


-25 

200- 
-15 

100- 
-5 
-0 
—5 


100 200 300 400 500 600 700 800 900 1000 
Degrees 


Fig. 45. Starting current of a transformer. Low stray field. 

the iron, can be calculated by considering only the iron section 
as carrying magnetic flux, but using as curve of magnetic flux 
density not the usual curve, 

(B = &' + 3C, 
but a curve derived therefrom, 

(B = &' + AflC, 

where k = ratio of total section to iron section. 

This, for instance, is the usual method of calculating the 
m.m.f. consumed in the armature teeth of commutating machines 
at very high saturations. 


MAGNETIC SATURATION AND HYSTERESIS 


185 


In investigating the transient transformer • starting current, 
the magnetic density curve thus is corrected for the stray field. 

Figs. 43 and 45 correspond to k = 3, or a total effective air 
section equal to three times the iron section, that is, (fc = (&' + 

3oe. 

Figs. 44 and 46 correspond to k = 25, or a section of stray 
field equal to 25 times the iron section, that is, & = (&' + 25 JC. 


22 


,16 


12 


-2 


L 


-CO 


45 


250-25 
200-20 


150-15 
100-10 


50-5 


-5 


100 200 300 400 500 600 700 

Degrees 

Fig. 46. Starting current of a transformer. High stray field. 

104. At very high values of current the resistance consumes 
a considerable voltage, and thus reduces the e.m.f. generated 
by the magnetic flux, and thereby the maximum magnetic flux 
and transient current. The resistance, which comes into con- 
sideration here, is the total resistance of the transformer primary 
circuit plus leads and supply lines, back to the point where the 
voltage is kept constant, as generator, busbars, or supply main. 

Assuming then at full load of im = 50 amperes effective in the 
transformer, a resistance drop of 8 per cent, or the voltage con- 
sumed by the resistance, as er = 0.08 of the impressed e.m.f. 

Let now the remanent magnetic flux density be &r = + 7600, 
and the circuit be closed at the moment 0 = 0, where the flux 


186 TRANSIENT PHENOMENA 

density should be <& = — <B0 = — 10,000; then the impressed 
e.m.f. is given by 

e = - E sin 0 = E ^ (cos 0). (1) 

au 

It is, however, 

(KB 

e = A — + Ci, (2) 

where A and C are constants; that is, the impressed e.m.f., e, is 
consumed by the self-inductance, or the e.m.f. generated by the 

changing magnetic density, which is proportional to — , and by 

the voltage consumed by the resistance, which is proportional 
to the current i. 
Combining (1) and (2) gives 


p 

However, at full load, we have -^— = effective impressed 

v2 

e.m.f. and im = 50 amperes = effective current; hence 
Cim = 50C = e.m.f. consumed by resistance, 
and since this equals er = 0.08 of impressed e.m.f., 


V2 
0.08 


or 


E imV2 50 
From (3) follows 


: = 0.00113. (4) 


d<S>=^dcosd -^dO (5) 

A A 


and 

It IT 

- | f*d cos 0 - '% f*idd; 

A J n 4»^n 


A 


MAGNETIC SATURATION AND HYSTERESIS 187 

hence, for i = 0, or negligible resistance drop, that is, permanent 
condition, 

».-'|'-io,poa (6) 

Multiplying (4) and (6) gives 

? = ^7= = H-3, (7) 


and substituting (6) and (7) in (5) gives 
d& = (Bdcos0 - i 


= 10,000 d cos 0 - 11.3 idB. (8) 

Changing now from differential to difference, that is, replacing, 
as approximation, d by A, gives 

eJB 

A(B = (BnA cos d -i 7^—^-^d 

lmV2 

= 10,000 A cos 0 - 11.3 t'A0. (9) 

Assuming now 

A0 = 10° = 0.175 (10) 

gives for the increment of magnetic flux density during 10° 
change of angle the value 

ACS - 10,000 A cos 0 - 2 i (11) 

and & = <B' + A(B 

= &' + 10,000 A cos d - 2 i. (12) 

From equation (12) the instantaneous values of magnetic 
flux density ®, and therefrom, by the magnetic cycles, Figs. 43 
and 44, respectively, the values of current i are calculated, by 
starting, for 0 = 0, with the remanent density CB' = (Br = 7600, 
adding thereto the change of cosine, 10,000 A cos 0, which gives 
a value (Bt = &' + 10,000 A cos 0, taking the corresponding 
value of i from the hysteresis cycle, Figs. 43 and 44, subtracting 
2 i from (Bt, and then correcting i for the value corresponding to 
<B = (Bj - 2 i. 

The quantity 2 i is appreciable only during the range of the 
curve where i is very large. 


188 


TRANSIENT PHENOMENA 


105. The following table is given to illustrate the beginning 
of the calculation of the curve for low stray field. 

STARTING CURRENT OF A TRANSFORMER. 


0° 

cos 6 

A(& = 

10 A cos 0 

(&- 

(B + A(B, 

tm 

PI- 

2 1 X 10~3 

(B 

o 

+ 1 00 

7 6 

o 

10 

0 98 

+ 0.2 

7.8 

1 0 

20 
30 

0.94 
0.87 

0.4 
0.7 

8.2 
8.9 

1.9 
2 9 


40 

0.77 

1.0 

9.9 

3 8 

50 

0 64 

1.3 

11 2 

5 2 

60 

0.50 

1.4 

12.6 

7.3 

70 
80 
90 
100 
110 
120 
130 
140 
150 
160 
170 
180 
190 
200 
210 
220 
230 
240 
250 
260 
270 

0.34 
+ 0.17 
0 
-0.17 
0.34 
0.50 
0.64 
0.77 
0.87 
0.94 
0.98 
1.00 
0.98 
0.94 
0.87 
0.77 
0.64 
0.50 
0.34 
-0.17 
0 

1.6 

.7 
.7 
.7 
.7 
.6 
.4 
.3 
.0 
0.7 
0.4 
+ 0.2 
-0.2 
0.4 
0.7 
1.0 
.3 
.4 
.6 
.7 
.7 

14.2 
15.9 
17.6 
19.2 
20.6 
21.75 
22.6 
23.0 
23.0 
22.7 
22.2 
21.7 
21.0 
20.2 
19.15 . 
17.9 
16.45 
14.85 
13.2 
11.5 
9.8 

12.0 
27 
70 
138 
220 
270 
450 
510 
510 
440 
350 
250 
200 
180 
130 
76 
40 
14 
7 
3.2 
-1.0 

0.1 
0.3 
0.45 
0.55 
0.9 
1.0 
1.0 
0.9 
0.7 
0.5 
0.4 
0.35 
0.25 
0.15 
0.2 
0.05 

"i-r.'s" 

18.9 
20.15 
21.2 
21.7 
22.0 
22.0 
21.8 
21.5 
21.2 
20.6 
19.85 
18.9 
17.75 
16.25 
14.8 

280 

+ 0.17 

.7 

8.1 

- .4 

290 

0.34 

.7 

6.4 

-1.3 

300 

0.50 

.6 

4.8 

— 1 9 

310 

0.64 

.4 

3.4 

-2 2 

320 
330 

0.77 
0.87 

.3 
1.0 

2.1 
1.1 

-2.5 
-2 6 


340 

0.94 

0.7 

.4 

-2.75 

350 

0.98 

0.4 

0 

— 2 8 

360 

+ 1.00 

-0.2 

- .2 

— 2 8 

370 

0.98 

+ 0.2 

0 

-1 3 

380 
390 
400 

0.94 
0.87 
0.77 

0.4 
0.7 
.0 

+ 0.4 
1.1 
2 1 

-0.5 
+ 0.3 
1 0 


410 
420 

0.64 
0.50 

.3 
.4 

3.5 
4.9 

1.7 
2 2 

430 
440 
450 

0.34 
+ 0.17 
0 

.6 

.7 
.7 

6.5 

8.2 
9.9 

2.7 
3.3 
4 3 

460 
470 
480 

-0.17 
-0.34 
-0.50 

.7 
.7 
.6 

11.6 
13.3 
14.9 

6.2 
9.5 
16.5 

MAGNETIC SATURATION AND HYSTERESIS 


189 


The first column gives angle 0, 

The second column gives cos d, 

The third column gives A(Bt = 10 A cos 0, in kilolines per 
sq. cm., 

The fourth column gives 0^ = (B + Afl^, 

The fifth column gives i, 

The sixth column gives D1 = 2 i X'lCT3, and 

The seventh column gives (B = (Bt — Dv 

i in the fifth column being chosen, by trial, so as to corre- 
spond, on the hysteresis cycles, not to <&v but to (B = (Bt — Dr 

These values are recorded as magnetic cycles on Figs. 43 and 
44, and as waves of flux density, current, etc., in Figs. 45 and 46. 

The maximum values of successive half waves are : 


A. Low Stray Field. 
fc= 3 

B. High Stray Field. 
fc= 25 

9 

(B 

im 

e 

(B 

*m 

0 
145° 
360° 
530° 
720° 
900° 
1080° 
1260° 
1440° 
1620° 

7.6 
22.0 
- .2 
18.6 
-2.2 
18.0 
-2.8 
17.4 
-3.1 
16.9 

0 
510 
-2.8 
120 
-2.9 
92 
— 30 

0 

160° 
360° 
530° 
720° 

7.6 
24.6 
+ 2.0 
20.7 
- .4 

0 
230 
-2.4 
117 
-2.7 

66 
-3.0 
50 

Permanent 

±10.0 

±4.5 

±10.0 

±4.5 

As seen, the maximum value of current during the first cycle, 
510, is more than one hundred times the final value 4.5, and more 
than 7 times the maximum value of the full-load current, 50\/2 
= 70.7 amperes, and the transient current falls below full-load 
current only in the fourth cycle. That is, the excessive value of 
transient current in an ironclad circuit lasts for a considerable 
number of cycles. 

In the presence of iron in the magnetic field of electric circuits, 
transient terms of current may thus occur which are very large 
compared with the transient terms in ironless reactors, which do 
not follow the exponential curve, can usually not be calculated 


190 


TRANSIENT PHENOMENA 


by general equations, but require numerical investigation by 
the use of the magnetic cycles of the iron. 

These transient terms lead to excessive current values only if 
the normal magnetic flux density exceeds half the saturation 
value of the iron, and so are most noticeable in 25-cycle circuits. 


A !\ l\ 


YYVV 


Fig. 47. Starting current of a 25-cycle transformer. 

As illustration is shown, in Fig. 47, an oscillogram of the 
starting current of a 25-cycle transformer having a resistance 
in the supply circuit somewhat smaller than in the above 
instance, thus causing a still longer duration of the transient 
term of excessive current. 

These starting transients of the ironclad inductance at high 
density are unsymmetrical waves, that is> successive half waves 
have different shapes, and when resolved into a trigonometric 
series, would give even harmonics as well as the odd harmonics. 

Thus the first wave of Fig. 45 can, when neglecting the tran- 
sient factor, be represented by the series: 

i = + 108.3 - 183.8 cos (0 + 28.0°) 

+ 112.4 cos 2 (0 + 29.8°) - 53.1 cos 3 (6 + 33.3°) 
+ 27.2 cos 4 (0 + 39.1°) - 18.4 cos 5 (0 + 38.1°) 
+ 13.6 cos 6 (0 + 33.4°) - 8.1 cos 7 (0 + 32.7°) 
or, substituting: 6 = ft + 150°, gives: 

e = E sin (ft + 150°) 

i = 108.3 + 183.8 cos (/? - 2.0°) + 112.4 cos 2 (p - 0.2°) 
+ 53.1 cos 3 0? + 3.3°) + 27.2 cos 4 (/? + 9.1°) 
+ 18.4 cos 5 (/? + 8.1°) + 13.6 cos 6 (p + 3.4°) 
+ 8.1 cos 7 (/? + 2.7°).