CHAPTER VI 

MAGNETISM 

MECHANICAL FORCES 

1. General 

61. Mechanical forces appear wherever magnetic fields act on 
electric currents. The work done by all electric motors is the 
result of these forces. In electric generators, they oppose the 
driving power and thereby consume the power which finds its 
equivalent in the electric power output. The motions produced 
by the electromagnet are due to these forces. Between the 
primary and the secondary coils of the transformer, between 
conductor and return conductor of an electric circuit, etc., such 
mechanical forces appear. 

The electromagnet, and all electrodynamic machinery, are 
based on the use of these mechanical forces between electric 
conductors and magnetic fields. So also is that type of trans- 
former which transforms constant alternating voltage into con- 
stant alternating current. In most other cases, however, these 
mechanical forces are not used, and therefore are often neglected 
in the design of the apparatus, under the assumption that the 
construction used to withstand the ordinary mechanical strains 
to which the apparatus may be exposed, is sufficiently strong to 
withstand the magnetic mechanical forces. In the large appara- 
tus, operating in the modern, huge, electric generating systems, 
these mechanical forces due to magnetic fields may, however, 
especially imder abnormal, though not infrequently occurring, 
conditions of operation (as short-circuits), assume such formi- 
dable values, so far beyond the normal mechanical strains, as to re- 
quire consideration. Thus generators and large transformers on 
big generating systems have been torn to pieces by the magnetic 
mechanical forces of short-circuits, cables have been torn from 
ijieir supports, disconnecting switches blown open, etc. 

In the following, a general study of these forces will be given. 
This also gives a more rational and thereby more accurate de- 

91 


92 ELECTRIC CIRCUITS 

sign of the electromagnet, and permits the determination of what 
may be called the eflSciency of an electromagnet. 

Investigations and calculations dealing with one form of 
energy only, as electromagnetic energy, or mechanical energy, 
usually are relatively simple and can be carried out with very 
high accuracy. DiflSculties, however, arise when the calculation 
involves the relation between several different forms of energy, 
as electric energy and mechanical energy. While the elementary- 
relations between different forms of energy are relatively simple, 
the calculation involving a transformation from one form of 
energy to another, usually becomes so complex, that it either can 
not be carried out at all, or even only approximate calculation 
becomes rather laborious and at the same time gives only a low 
degree of accuracy. In most calculations involving the trans- 
formation between different forms of energy, it is therefore 
preferable not to consider the relations between the different 
forms of energy at all, but to use the law of conservation of energy 
to relate the different forms of energy, which are involved. 

Thus, when mechanical motions are produced by the action 
of a magnetic field on an electric circuit, energy is consumed 
in the electric circuit, by an induced e.m.f. At the same time, 
the stored magnetic energy of the system may change. By the 
law of conservation of energy, we have: 

Electric energy consumed by the induced e.m.f, = mechanical 
energy produced, + increase of the stored magnetic energy. (1) 
The consumed electric energy, and the stored magnetic energy, 
are easily calculated, as their calculation involves one form of 
energy only, and this calculation then gives the mechanical work 
done, = Fl, where F = mechanical force, and I = distance over 
which this force moves. 

Where mechanical work is not required, but merely the me- 
chanical forces, which exist, as where the system is supported 
against motions by the mechanical forces — as primary and 
secondary coils of a transformer, or cable and return cable of a 
circuit — the same method of calculation can be employed, by 
assuming some distance I of the motion (or dl) ; calculating the 
mechanical energy t^o =Fl by (1), and therefrom the mechanical 

. „ t^o „ dwo 

force as r = -^^ or F = -^r- 

I dl 

Since the induced e.m.f., which consumes (or produces) the 

electric energy, and also the stored magnetic energy, depend on 


MAGNETISM 93 

the current and the mductance of the electric circuit, and in 
alternating-current circuits the impressed voltage also depends 
on the inductance of the circuit, the inductance can frequently 
be expressed by supply voltage and current; and by substituting 
this in equation (1), the mechanical work of the magnetic forces 
can thus be expressed, in alternating-ciu'rent apparatus, by sup- 
ply voltage and current. 

In this manner, it becomes possible, for instance, to express 
the mechanical work and thereby the pull of an alternating 
electromagnet, by simple expressions of voltage and current, or 
to give the mechanical strains occurring in a transformer under 
short-circuits, by an expression containing only the terminal 
voltage, the short-circuit current, and the distance between 
primary and secondary coils, without entering into the details 
of the construction of the apparatus. 

This general method, based on the law of conservation of 

energy, will be illustrated by some examples, and the general 

equations then given. 

2. The Constant-current Electromagnet 

62. Such magnets are most direct-current electromagnets, 
and also the series operating magnets of constant-current arc 
lamps on alternating-current circuits. 

Let io = current, which is constant dming the motion of 
the armatm'e of the electromagnet, from its initial position 1, 
to its final position 2,1 = the length of this motion, or the stroke 
of the electromagnet, in centimeters, and n = number of turns 
of the magnet winding. 

The magnetic flux ^, and the inductance 

L=-^10-8 (2) 

to 

of the magnet, vary during the motion of its armature, from a 
^^fiiiumum value, 

$, = Mil 108 (3) 

n 


^ the initial position, to a maximum value, 

^, = 12^ 108 (4) 

n 

^ the end position of the armature. 


94 ELECTRIC CIRCUITS 

Hereby an e.m.f. is induced in the magnet winding, 

e' = n_lO« = t.^ (5) 

This consumes the power 

p = toe' = io^ -^ (6) 

and thereby the energy 

ti; = I pdt = nio^(L2 — Li) (7) 

Assuming that the inductance, in any fixed position of the 
armature, does not vary with the current, that is, that magnetic 
saturation is absent,^ the stored magnetic energy is: 

In the initial position, 1, 

wi = — 2" (8) 

in the end position, 2, 

io^L2 ,n\ 

W2 = — ?r— (9) 


The increase of the stored magnetic energy, during the motioa 
of the armature, thus is 

w' = W2 — Wi = -^ (L2 — Li) (10 

The mechanical work done by the electromagnet thus i^^, 
by the law of conservation of energy. 

Wo = w — w' 
= —(L2 - Li) joules. (1 ^) 


If Z = length of stroke, in centimeters, F = average for< 
or pull of the magnet, in gram weight, the mechanical work: is 

Fl gram-cm. 
Since 

g = 981 cm.-sec. QJ-^) 

= acceleration of gravity, the mechanical work is, in absoliite 

units, 

Fig 

* If magnetic saturation is reached, the stored magnetic energy is taken 
from the magnetization curve, as the area between this curve and the 
vertical axis, as discussed before. 


MAGNETISM 95 

and since 1 joule = 10^ absolute units, the mechanical work is 

Wo = Fig 10+7 joules. (13) 

From (11) and (12) then follows, 

Fl = ]f^ (L2 - Li)10+7 gram-cm. (14) 

as the mechanical work of the electromagnet, and 

F = '^ ^^^-^ 10' grams (16) 

as the average force, or pull of the electromagnet, during its 
stroke I, 

Or, if we consider only a motion element dl, 

as the force, or pull of the electromagnet in any position L 

Reducing from gram-centimeters to foot-pounds, that is, giving 
the stroke I in feet, the pull F in pounds, we divide by 

454 X 30.5 = 13,850 
which gives, after substituting for g from (12) 

(14) : Fl = 3.68 tV(L2 - Li) ft.-lb (17) 

(15) : F = 3.68 zV ^' "^ ^' lb. (18) 

(16) : F = 3.68 tV ^ lb. (19) 

These equations apply to the direct-current electromagnet 
as well as to the alternating-current electromagnet. 

In the alternating-current electromagnet, if io is the effective 
value of the current, F is the effective or average value of the 
pull, and the pull or force of the electromagnet pulsates with 
double frequency between and 2F. 

63. In the alternating-current electromagnet usually the vol- 
tage consumed by the resistance of the winding, tV, can be 
neglected compared with the voltage consumed by the reactance 
of the winding, ioXy and the latter, therefore, is practically equal 
to the terminal voltage, e, of the electromagnet. We have then, 
by the general equation of self-induction, 

e = 27r fLio (20) 


96 ELECTRIC CIRCUITS 

where / = frequency, in cycles per second. 
From which follows, 

ij. = 2^ (21) 

and substituting (21) in equations (14) to (19), gives as the equa- 
tion of the mechanical workj and the pull of the alternating^current 
electromagnet. 

In the metric system: 

PI = ^ \ . ^ gram-cm. (22) 

„ io(e2 — ei) 10^ _Jo^ de ^ ,^. 

^ — 4irp 1^-^^^^"^ (23) 

In foot-pounds: 

Fl = 0-586 io{e, - e^) ^^ _j^^ ^24) 

„ 0.586 io(e2 — ei) 0.586zo de ,, .^_x 

^ Ji = —f—ll^^- ^^^ 

Example. — In a 60-cycle alternating-current lamp magnet, 
the stroke is 3 cm., the voltage, consumed at the constant alter- 
nating current of 3 amp. is 8 volts in the initial position, 17 
volts in the end position. What is the average pull of the 
magnet? 


I 


= 


3 cm. 


ei 


=: 


8 


€2 


= 


17 


f 


= 


60 


io 


= 


3 


hence, by (23), 

F = 122 grams (= 0.27 lb.) 

The work done by an electromagnet, and thus its pull, depended, 
by equation (22), on the current io and the difference in voltfuge 
between the initial and the end position of the armature, 62 — ^1 5 
that is, depend upon the diJGference in the volt-amperes con- 
sumed by the electromagnet at the beginning and at the end of fclie 
stroke. With a given maximum volt-amperes, io62, avaiIjLl>l^ 
for the electromagnet, the maximmn work would thus be doxie, 
that is, the greatest pull produced, if the volt-amperes at "fche 
beginning of the stroke were zero, that is, ei = 0, and tbe 
theoretical maximum output of the magnet thus would be 

FJ = i^ (26) 

4 7r/g 


MAGNETISM 97 

and the ratio of the actual output, to the theoretically maximum 
output, or the efficiency of the electromagnet, thus is, by (22) 
and (26) > 

^^F^e^ (27) 

or, using the more general equation (14), which also appUes to 
the direct-current electromagnet, 

r, = ^^^ (28) 

The efficiency of the electromagnet, therefore, is the dif- 
ference between maximum and minimum voltage, divided by the 
maximum voltage; or the dijGference between maximum and 
minimum volt-ampere consumption, divided by the maximmn 
volt-ampere consumption; or the diiEference between maximum 
and minimum inductance, divided by the maximum inductance. 
As seen, this expression of efficiency is of the same form as 
that of the thermodynamic engine, 

T2 

Prom (26) it also follows, that the maximum work which can 

be derived from a given expenditure of volt-amperes, io«2, is 

linoited. For 1062 = 1, that is, for 1 volt-amp. the maximum 

work, which could be derived from an alternating electro- 

oi^et, is, from (26), 

FJ = j^ = -J- gram-cm. (29) 

That is, a 60-cycle electromagnet can never give more than 
^3.5 gram-cm., and a 25-cycle electromagnet never more than 
32.4 gram-cm. pull per volt-ampere suppUed to its terminals. 

Or inversely, for an average pull of 1 gram over a distance of 

cm., a minimum of y^-r volt-amp. is required at 60 cycles, 

^^d a minimum of ^^T volt-amp. at 25 cycles. 

Or, reduced to poimds and inches: 

^or an average pull of 1 lb. over a distance of 1 in., at least 
^^ volt-amp. are required at 60 cycles, and at least 36 volt- 
^*^p. at 25 cycles. 

This gives a criterion by which to judge the success of the 
^^sign of electromagnets. 


98 ELECTRIC CIRCUITS 

3. The Constant-potential Alternating Electromagnet 

64. If a constant alternating potential, eo, is impressed upon an 
electromagnet, and the voltage consumed by the resistance, 
ir, can be neglected, the voltage consumed by the reactance, x, is 
constant and is the terminal voltage, eo, thus the magnetic flux, 
$, also is constant during the motion of the armature of the 
electromagnet. The current, i, however, varies, and decreases 
from a maximiun, ^l, in the initial position, to a minimum, 12, in 
the end position of the armature, while the inductance increases 
from Li to L2. 

The voltage induced in the electric circuit by the motion of 
the armature, 

e' = n^ 10« (30) 

at 

then is zero, and therefore also the electrical energy expended, 

w = 0. 

That is, the electric circuit does no work, but the mechanical 
work of moving the armature is done by the stored magnetic 
energy. 

The increase of the stored magnetic energy is 

w' = ''"^' - ''"^' (31) 

and since the mechanical energy, in joules, is by (13), 

w^ = Fig 10^ 
the equation of the law of conservation of energy, 

w = w' + Wo (32) 


then becomes 


= '''^' Z '''^' + Fig 10-^ 


or 


Fl = '''^' ;r '''^' 10^ gram-cm. (33) 

Since, from the equation of self-induction, in the initial posi- 
tion, 

Co = 2 wfLiii (34) 

in the end position 

eo = 2 TfL2i2 (35) 


MAGNETISM 99 

substituting (34) and (35) in (33), gives the equation of the 
constant-potential alternating electromagnet. 

Fl •= ^"^V 7 ''^ 10^ gram-cm. (36) 

and 

F ^ £o(n_lli2) 10, ^ eo d» 107 grams (37) 

or, in foot-pounds, 

Fl = 0-586 «°(^'» - ^•») ft..ib. (38) 

„ 0.586 60(21 — 2*2) 0.586 Co di ,, , ^ 

F jj^ J—Jl^^- (39) 

Substituting Q = ei = volt-amperes, in equations (36) to 
(39) of the constant-potential alternating electromagnet, and 
equations (22) to (25) of the constant-current alternating magnet, 
gives the same expression of mechanical work and pull: 

In metric system: 

Fl = '^%- W gram-cm. (40) 

In foot-pounds: 

fZ = 5:^ft.-lb. (42) 

F = 0:^ = 0^ fib. (43) 

where AQ — difference in volt-amperes consumed by the magnet 
in the initial position, and in the end position of the armature. 

Both types of alternating-current magnet, then, give the same 
expression of eflSciency, 

-5? *«> 


where Q^ is the maximum volt-amperes consumed, corresponding 
to the end position in the constant-current magnet, to the initial 
position in the constant-potential magnet. 

4. Short-circuit Stresses in Alternating-current Transformers 

66. At short-circuit, no magnetic flux passes through the sec- 
ondary coils of the transformer, if we neglect the small voltage 
oonsiuned by the ohmic resistance of the secondary coils. If 


100 ELECTRIC CIRCUITS 

the supply system is sufficiently large to maintain constant 
voltage at the primary terminals of the transformer even at 
short-circuit, full magnetic flux passes through the primary 
coils. ^ In this case the total magnetic flux passes between 
primary coils and secondary coils, as self-inductive or leakage 
flux. If then x = self-inductive or leakage reactance, eo = im- 
pressed e.m.f., io = — is the short-circuit current of the trans- 
former. Or, if as usual the reactance is given in per cent., that 
is, the ix (where i = full-load current of the transformer) given 
in per cent, of e, the short-circuit current is equal to the full-load 
current divided by the percentage reactance. Thus a trans- 
former with 4 per cent, reactance would give a short-circuit cur- 
rent, at maintained supply voltage, of 25 times full-load current. 

To calculate the force, F, exerted by this magnetic leakage 
flux on the transformer coils (which is repulsion, since primary 
and secondary currents flow in opposite direction) we may assume, 
at constant short-circuit current, io, the secondary coils moved 
against this force, F, and until their magnetic centers coincide 
with those of the primary coils; that is, by the distance, I, as shown 
diagrammatically in Fig. 45, the section of a shell-type transformer. 
When brought to coincidence, no magnetic flux passes between 
primary and secondary coils, and during this motion, of length, Z, 
the primary coils thus have cut the total magnetic flux, €», of the 
transformer. 

Hereby in the primary coils a voltage has been induced, 

at 

where n = effective number of primary turns. 

The work done or rather absorbed by this voltage, e', at 
rent, io, is 


= I e'iodt = 


w = I e%dt = nio^ 10~® joules. {41 


* If the terminal voltage drops at short-circuit on the transformer secon 
ries, the magnetic flux through the transformer primaries drops in the 
proportion, and the mechanical forces in the transformer drop with 
square of the primary terminal voltage, and with a great drop of the 
minal voltage, as occurs for instance with large transformers at the en 
a transmission line or long feeders, the mechanical forces may drop 
small fraction of the value, which they have on a system of practically 
limited power. 


MAGNETISM 101 

If L = leakage inductance of the transformer, at short-circuit, 
where the entire flux, $, is leakage flux, we have 

<l>=:^108 (46) 

n 

hence, substituted in (45) 

w = io^L (47) 

The stored magnetic energy at short-circuit is 

wi = -^ (48) 

and since at the end of the assumed motion through distance, Z, 
the leakage flux has vanished by coincidence between primary 
and secondary coils, its stored magnetic energy also has vanished, 
and the change of stored magnetic energy therefore is 

w = Wi = —^ (49) 

Hence, the mechanical work of the magnetic forces of the short- 
circuit current is 

Wo = w — w' = - y- (50) 

It is, however, if F is the force, in grams, I, the distance between 
the magnetic centers of primary and secondary coils, 

wi = Fig 10~^ joules. 
Hence, 

Fl = y^ 10^ gram-cm. (51) 

if 

and 

P = JJl 10^ grams (52) 

the mechanical force existing between primary and secondary 
coils of a transformer at the short-circuit current, io. 

Since at short-circuit, the total supply voltage, eo, is consumed 
by the leakage inductance of the transformer, we have 

eo = 2 TfLio (53) 

hence, substituting (53) in (52), gives 

' — :ij — grams (54) 


102 ELECTRIC CIRCUITS 

Example. — ^Let, in a 25-cycle 1667-kw. transformer, the supply 
voltage, Co = 6200, the reactance = 4 per cent. The trans- 
former contains two primary coils between three secondary coils, 
and the distance between the magnetic centers of the adjacent 
coils or half coils is 12 cm., as shown diagrammatically in Fig. 45. 
What force is exerted on each coil face dm*ing short-circuit, in a 
system which is so large as to maintain constant terminal voltage? 

At 5200 volts and 1667 kw., the full-load current is 320 amp. 
At 4 per cent, reactance the short-circuit ciurent therefore, 

320 
z'o = TT?^ = 8000 amp. Equation (54) then gives, for / = 25, 

I = 12, 

F = 112 X 10« grams 
= 112 tons. 

This force is exerted between the four faces of the two primary 
coils, and the corresponding faces of the secondary coils, and 
on every coil face thus is exerted the force 

F 

-: = 28 tons 

4 

This is the average force, and the force varies with double 
frequency, between and 56 tons, and is thus a large force. 

56. Substituting to = — in (54), gives as the short-circuit force 

of an alternating-current transformer, at maintained terminal 
voltage, Co, the value 

„ eo" 10^ 810 eo« ,„. 

That is, the short-circuit stresses are inversely proportional 
to the leakage reactance of the transformer, and to the distance, 
Z, between the coils. 

In large transformers on systems of very large power, safety 
therefore requires the use of as high reactance as possible. 

High reactance is produced by massing the coils of each cir- 
cuit. 

Let in a transformer 

n = number of coil groups 


MAGNETISM 


103 


F 


(where one coil is divided into two half coils, one at each end of 
the coil stack, as one secondary coil in Fig. 45, where n = 2) the 
mechanical force per coil face then is, by (55), 

eo^ W 810 eo^ ,-^, 

2n ~ S^^gnTx = 27^ ^'^"^^ ^^^^ 

Let X = leakage reactance of transformer; 
lo = distance between coil surfaces; 
li = thickness of primary coil; 
h = thickness of secondary coil. 

Between two adjacent coils, P and S in Fig. 45, the leakage flux 
density is uniform for the width lo between the coil surfaces, 


Fig. 45. 
and then decreases toward the interior of the coils, over the dis- 
tance K respectively ^, to zero at the coil centers. All the coil 

turns are interlinked with the leakage flux in the width, lo, but 
toward the interior of the coils, the number of turns interlinked 
with the leakage flux decreases, to zero at the coil center, and as 
the leakage flux density also decreases, proportional to the dis- 
tance from the coil center, to zero in the coil center, the inter- 
linkages between leakage flux and coil turns decrease over the 

space ^ respectively ^, proportional to the square of the distance 

from the coil center, thus giving a total interlinkage distance, 


X 


u^du = —1 
6 


where u is the distance from the coil center. 


104 ELECTRIC CIRCUITS 

Thus the total interlinkages of the leakage flux with the coil 
turns are the same as that of a uniform leakage flux density over 

the width U + ^ + c * This gives the effective distance between 

coil centers, for the reactance calculation, 

l = lo + ^-^ (57) 

Assuming now we regroup the transformer coils, so as to get 
m primary and m secondary coils, leaving, however, the same iron 
structure. 

The leakage flux density between the coils is hereby changed in 
proportion to the changed number of ampere-turns per coil, that 

is, by the factor — 

The effective distance between the coils, Z, is changed by the 

same factor — • 

m 

The number of interlinkages between leakage flux and electric 
circuits, and thus the leakage reactance, x, of the transformer, 
thus is changed by the factor 


(-) 

\m/ 


That is, by regrouping the transformer winding within the same 
magnetic circuit and without changing the number of turns of the 
electric circuit, the leakage reactance, Xy changes inverse propor- 
tional to the square of the number of coil groups. 

As by equation (56) the mechanical force is inverse propor- 

(7l\ ^ 
— ) > I pro- 
portional to —f the mechanical force per coil thus changes 
proportional to 

\m/ m n \m/ 

That is, regrouping the transformer winding in the same wind- 
ing space changes the mechanical force inverse proportional to 


MAGNETISM 105 

the square of the coil groups, thus inverse proportional to the 
change of leakage reactance. 

However, the distance lo between the coils is determined by in- 
sulation and ventilation. Thus its decrease, when increasing the 
number of coil groups, would usually not be permissible, but more 
winding space would have to be provided by changing the mag- 
netic circuit, and inversely, with a reduction of the number of 
coil groups, the winding space, and with it the magnetic circuit, 
would be reduced. 

Assmning, then, that at the change from n to m coil groups, 
the distance between the coils, Zo, is left the same. 

The effective leakage space then changes from 


l = lo + 


li + l 


2 


6 
to 


■f 


V = I A- — ^^ "^ ^^ = Z ° ^ ^ 


m 6 1 . h + h 

Zo + — g— 

and the leakage reactance thus changes from 

X 


I 


to 


, n I , 

X = — ^Xi 

m I 


hence the mechanical force per coil, from 

2 n 8 vfnglx 
to 


^0 = 


= Fo 


2m 8 irfngVx' 
nlx 


mVx' 
Z\2 


= ^. ^) 


= ^ol "I ,^ I (58) 


h,+ 


h + h 


106 ELECTRIC CIRCUITS 

Thus, if -^-^ — is large compared with Zo, 

2 


n..©V., 


that is, the mechanical forces vary with the square of the number 
of coil groups. 

If — ^ — is small compared with Zo, 

Fo' = Fo 

that is, the mechanical forces are not changed by the change of 
the number of coil groups. 

In actual design, decreasing the number of coil groups usually 
materially decreases the mechanical forces, but materially less 
than proportional to the square of the number of coil groups. 

5. Repulsion between Conductor and Return Conductor 

67. If to is the current flowing in a circuit consisting of a con- 
ductor and the return conductor parallel thereto, and I the dis- 
tance between the conductors, the two conductors repel each 
other by the mechanical force exerted by the magnetic field of 
the circuit, on the current in the conductor. 

As this case corresponds to that considered in section 2, equa- 
tion (16) applies, that is. 

The inductance of two parallel conductors, at distance I froxn 
each other, and conductor diameter h is, per centimeter length of 
conductor, 

2J 

la 

Hence, differentiated, 

dL ^ 4 X 10-» 
dl I 

and, substituted in (16), 


= (4 log y- + m) 10-» henrys (5»^ 


10=" 


^ = m ^""^ ^^^^^ 


or substituting (12), 


MAGNETISM 107 

„ 20.4tV10-« .^,. 

F = -J grains (61) 

If I = 150 cm. (5 ft.) 

io = 200 amp. 
this gives 

F = 0.0054 grams per centimeter length of circuit, hence it is 
inappreciable. 

If, however, the conductors are close together, and the current 
very large, as the momentary short-circuit current of a large 
alternator, the forces may become appreciable. 

For example, a 2200-volt 4000-kw. quarter-phase alternator 
feeds through single conductor cables having a distance of 15 cm. 
(6 in.) from each other. A short-circuit occurs in the cables, and 
the momentary short-circuit current is 12 times full-load current. 
What is the repulsion between the cables? 

Full-load current is, per phase, 910 amp. Hence, short-circuit 
current, io = 12 X 910 = 10,900 amp. I = 15. Hence, 

F = 160 grams per centimeter. 
Or multiplied by j^ 

F = 10.8 lb. per feet of cable. 

That is, pulsating between and 21.6 lb. per foot of cable. 
Hence sufficient to lift the cable from its supports an(J throw it 
aside. 

In the same manner, similar problems, as the opening of dis- 
connecting switches under short-circuit, etc., can be investigated. 

^- General Equations of Mechanical Forces in Magnetic Fields 

68. In general, in an electromagnetic system in which mechan- 
^cal motions occur, the inductance, L, is a function of the position, Z, 
P^^ing the motion. If the system contains magnetic material, 
^^ general the inductance, L, also is a function of the current, i, 
^^Pecially if saturation is reached in the magnetic material. 

I-»et, then, L = inductance, as function of the current, i, and 
I^sition, I; 

iri = inductance, as function of the current, t, in the initial 
J^sition 1 of the system; 

1^2 = inductance, as function of the current, i, in the end 
I^sition 2 of the system. 


108 ELECTRIC CIRCUITS 

If then $ = magnetic flux, n = number of turns interlinked 
with the flux, the induced e.m.f . is 

e' = n ^ 10"» (62) 

at 

We have, however, 

n$ = iL 10®; 

hence, 

e' = ^ (63) 

the power of this induced e.m.f. is 

. , . d(iL) 
p = %e = t \^ > 

at 

and the energy 


w 


/2 /»2 

pdt = j id(iL) 


= f i^dL + r iLdi (64) 

The stored magnetic energy in the initial position 1 is 

w\ = \ id(iLi) (65 

In the end position 2, 

W2 =ridiiL2) (6^) 

and the mechanical work thus is, by the law of conservation of 
energy 

Wo = W — W2+ Wi 

= f id(iL) + f idiiLi) - I id(iL,) (6y) 

Ji Jo Jo 

and since the mechanical work is 

Wo = Fig 10-" («^) 

We have: 

Fl= — \ ridiiL) + f id(iLi) - f id(iLi) gram-«m. (i^*^) 


MAGNETISM 109 

If L is not a function of the current, i, but only of the position, 
that Ls, if saturation is absent, Li and L2 are constant, and equa- 
tion (69) becomes, 


Fi = '4 \£ ml) + '''''' -'''''' 


gram-cm. (70) 


9 \Ji " ' 2 

(a) If i = constant, equation (70) becomes, 

pj^ ^ 10_^ iKL2 -U) 
9 2 

(Constant-current electromagnet.) 
(6) If L = constant, equation (70) becomes, 

Fl = 0. 

That is, mechanical forces are exerted only where the in- 
ductance of the circuit changes with the mechanical motion 
which would be produced by these forces. 

(c) If iL = constant, equation (70) becomes, 

g 2 

(Constant-potential electromagnet.) 

In the general case, the evaluation of equation (69) can usually 
^ made graphically, from the two curves, which give the varia- 
tion of Li with i in the initial position, of L2 with i in the final 
'^ition, and the curve giving the variation ofL and i with the 
'Motion from the initial to the final position. 

^n alternating magnetic systems, these three curves can be 

determined experimentally by measuring the volts as function 

^^ the amperes, in the fixed initial and end position, and by 

^^^uring volts and amperes, as function of the intermediary 

^^^itions, that is, by strictly electrical measurement. 

As seen, however, the problem is not entirely determined by 
'^^ two end positions, but the function by which i and L are 
^*^ted to each other in the intermediate positions, must also be 
^^^n. That is, in the general case, the mechanical work and 
'^Us the average mechanical force, are not determined by the 
'^^ positions of the electromagnetic system. This again shows 
^ analogy to thermodynamic relations. 

If then in case of a cyclic change, the variation from position 


110 ELECTRIC CIRCUITS 

1 to 2 is different from that from position 2 back to 1, such a 
cyclic change produces or consumes energy. 


w 


= I id(iL) + I id(iL) = f id(iL) 

Jl J2 Jl 


Such a case is the hysteresis cycle. The reaction machine 
(see Theory and Calculation of Electrical Apparatus) is based 
on such cycle. 


SECTION II