CHAPTER X. 

EFFECTIVE RESISTANCE AND REACTANCE. 

72. The resistance of an electric circuit is determined : — 

1.) By direct comparison with a known resistance (Wheat- 
stone bridge method, etc.). 

This method gives what may be called the true ohmic 
resistance of the circuit. 

2.) By the ratio : 

Volts consumed in circuit 

Amperes in circuit 

In an alternating-current circuit, this method gives, not 
the resistance of the circuit, but the impedance, 


3.) By the ratio : 

r__ Power consumed . 

(Current)2 

where, however, the "power" does not include the work 
done by the circuit, and the counter E.M.Fs. representing 
it, as, for instance, in the case of the counter E.M.F. of a 
motor. 

In alternating-current circuits, this value of resistance is 
the energy coefficient of the E.M.F., 

_ Energy component of E.M.F. 

Total current 

It is called the effective resistance of the circuit, since it 
represents the effect, or power, expended by the circuit. 
The energy coefficient of current, 

a._ Energy component of current 

Total E.M.F. 
is called the effective conductance of the circuit. 


EFFECTIVE RESISTANCE AND REACTANCE. 105 

In the same way, the value, 

_ Wattless component of E.M.F. 

Total current 
is the effective reactance, and 

, _ Wattless component of current 
TotafE.M.F. 

is the effective susceptance of the circuit. 

While the true ohmic resistance represents the expendi- 
ture of energy as heat inside of the electric conductor by a 
current of uniform density, the effective resistance repre- 
sents the total expenditure of energy. 

Since, in an alternating-current circuit in general, energy 
is expended not only in the conductor, but also outside of 
it, through hysteresis, secondary currents, etc., the effective 
resistance frequently differs from the true ohmic resistance 
in such way as to represent a larger expenditure of energy. 

In dealing with alternating-current circuits, it is necessary, 
therefore, to substitute everywhere the values "effective re- 
sistance," "effective reactance," "effective conductance," 
and " effective susceptance," to make the calculation appli- 
cable to general alternating-current circuits, such as induc- 
tances, containing iron, etc. 

While the true ohmic resistance is a constant of the 
circuit, depending only upon the temperature, but not upon 
the E.M.F., etc., the effective resistance and effective re- 
actance are, in general, not constants, but depend upon 
the E.M.F., current, etc. This dependence is the cause 
of most of the difficulties met in dealing analytically with 
alternating-current circuits containing iron. 

73. The foremost sources of energy loss in alternating- 
current circuits, outside of the true ohmic resistance loss, 
are as follows : 

1.) Molecular friction, as, 

a.) Magnetic hysteresis ; 
b.) Dielectric hysteresis. 


106 .ALTERNATING-CURRENT PHENOMENA. 

2.) Primary electric currents, as, 

a.} Leakage or escape of current through the insu- 
lation, brush discharge ; b.) Eddy currents in 
the conductor or unequal current distribution. 
3.) Secondary or induced currents, as, 

a.) Eddy or Foucault currents in surrounding mag- 
netic materials ; b.} Eddy or Foucault currents 
in surrounding conducting materials ; c.} Sec- 
ondary currents of mutual inductance in neigh- 
boring circuits. 

4.) Induced electric charges, electrostatic influence. 
While all these losses can be included in the terms effec- 
tive resistance, etc., only the magnetic hysteresis and the 
eddy currents in the iron will form the subject of what fol- 
lows, since they are the most frequent and important sources 
of energy loss. 

Magnetic Hysteresis. 

74. In an alternating-current circuit surrounded by iron 
or other magnetic material, energy is expended outside of 
the conductor in the iron, by a kind of molecular friction, 
which, when the energy is supplied electrically, appears as 
magnetic hysteresis, and is caused by the cyclic reversals of 
magnetic flux in the iron in the alternating magnetic field. 

To examine this phenomenon, first a circuit may be con- 
sidered, of very high inductance, but negligible true ohmic 
resistance ; that is, a circuit entirely surrounded by iron, as, 
for instance, the primary circuit of an alternating-current 
transformer with open secondary circuit. 

The wave of current produces in the iron an alternating 
magnetic flux which induces in the electric circuit an E.M.F., 
— the counter E.M.F. of self-induction. If the ohmic re- 
sistance is negligible, that is, practically no E.M.F. con- 
sumed by the resistance, all the impressed E.M.F. must be 
consumed by the counter E.M.F. of self-induction, that is, 
the counter E.M.F. equals the impressed E.M.F. ; hence, if 


EFFECTIVE RESISTANCE AND REACTANCE. 


107 


the impressed E.M.F. is a sine wave, the counter E.M.F., 
and, therefore, the magnetic flux which induces the counter 
E.M.F. must follow a sine wave also. The alternating wave 
of current is not a sine wave in this case, but is distorted 
by hysteresis. It is possible, however, to plot the current 
wave in this case from the hysteretic cycle of magnetic flux. 
From the number of turns, n, of the electric circuit, 
the effective counter E.M.F., E, and the frequency, N, 
of the current, the maximum magnetic flux, <j>, is found 
by the formula : 


hence, 


E 108 


A maximum flux, <£, and magnetic cross-section, S, give 
the maximum magnetic induction, (B = $ / 6". 

If the magnetic induction varies periodically between 
+ (B and — (B, the M.M.F. varies between the correspond- 
ing values -f ff and — JF, and describes a looped curve, the 
cycle of hysteresis. 

If the ordinates are given in lines of magnetic force, the 
abscissae in tens of ampere-turns, then the area of the loop 
equals the energy consumed by hysteresis in ergs per cycle. 

From the hysteretic loop the instantaneous value of 
M.M.F. is found, corresponding to an instantaneous value 
of magnetic flux, that is, of induced E.M.F. ; and from the 
M.M.F., JF, in ampere-turns per unit length of magnetic cir- 
cuit, the length, /, of the magnetic circuit, and the number of 
turns, «, of the electric circuit, are found the instantaneous 
values of current, i, corresponding to a M.M.F., JF; that is, 
magnetic induction (B, and thus induced E.M.F. e, as : 


75. In Fig. 65, four magnetic cycles are plotted, with 
maximum values of magnetic inductions, (B = 2,000, 6,000, 
10,000, and 16,000, and corresponding maximum M.M.Fs., 


108 


AL TERNA TING-CURRENT PHENOMENA. 


SF = 1.8, 2.8, 4.3, 20.0. They show the well-known hys- 
teretic loop, which becomes pointed when magnetic satu- 
ration is approached. 

These magnetic cycles correspond to average good sheet 
iron or sheet steel, having a hysteretic coefficient, 77 = .0033, 
and are given with ampere-turns per cm as abscissae, and 
kilo-lines of magnetic force as ordinates. 


a 


M 


«</. 65. Hysteretic Cycle of Sheet Iron. 

In Figs. 66, 67, 68, and 69, the curve of magnetic in- 
duction as derived from the induced E.M.F. is a sine wave. 
For the different values of magnetic induction of this sine 
curve, the corresponding values of M.M.F., hence of current, 
are taken from Fig. 65, and plotted, giving thus the exciting 
current required to produce the sine wave of magnetism ; 
that is, the wave of current which a sine wave of impressed 
E.M.F. will send through the circuit. 


EFFECTIVE RESISTANCE AND REACTANCE. 109 

As shown in Figs. 66, 67, 68, and 69, these waves of 
alternating current are not sine waves, but are distorted by 
the superposition of higher harmonics, and are complex 
harmonic waves. They reach their maximum value at the 
same time with the maximum of magnetism, that is, 90° 


1=2000 


1.6 


N 


^ 


\ 


(Bfeooo 


T2.8 


3 =2.S 


M\ 


\\ 


Figs. 66 and 67. Distortion of Current Waue by Hysteresis. 

ahead of the maximum induced E.M.F., and hence about 
90° behind the maximum impressed E.M.F., but pass the 
zero line considerably ahead of the zero value of magnet- 
ism, or 42°, 52°, 50°, and 41 °, respectively. 

The general character of these current waves is, that the 
maximum point of the wave coincides in time with the max- 


110 


ALTERNA TING-CURRENT PHENOMENA. 


imum point of the sine wave of magnetism ; but the current 
wave is bulged out greatly at the rising, and hollowed in at 
the decreasing, side. With increasing magnetization, the 
maximum of the current wave becomes more pointed, as 
shown by the curve of Fig. 68, for (B = 10,000 ; and at still 


(B- 


10000 


4. 


& 


NX 


\L 


. 16000 


20 


\ 


G 


13 


\ 


F/SfS. 88 and 69. Distortion of Current Waue by Hysteresis. 

higher saturation a peak is formed at the maximum point, 
as in the curve of Fig. 69, for (B = 16,000. This is the case 
when the curve of magnetization reaches within the range of 
magnetic saturation, since in the proximity of saturation the 
current near the maximum point of magnetization has to 
rise abnormally to cause even a small increase of magneti- 
zation. The four curves, Figs. 66, 67, 68, and 69, are not 
drawn to the same scale. The maximum values of M.M.F., 


EFFECTIVE RESISTANCE A.\D REACTANCE- 111 

corresponding to the maximum values of magnetic induction, 
(B = 2,000, 6,000, 10,000, and 16,000 lines of force per cm2, 
'arc & = 1.8, 2.8, 4.3, and 20.0 ampere-turns per cm. In 
the different diagrams these are represented in the ratio of 
8 : 6 : 4 : 1, in order to bring the current curves to approxi- 
mately the same height. The M.M.F., in C.G.S. units, is 
J#r=47r/103r = 1.257 IF. 

76. The distortion of the wave of magnetizing current 
is as large as shown here only in an iron-closed magnetic 
circuit expending energy by hysteresis only, as in an iron- 
clad transformer on Open secondary circuit. As soon as the 
circuit expends energy in any other way, as in resistance, or 
by mutual inductance, or if an air-gap is introduced in the 
magnetic circuit, the distortion of the current wave rapidly 
decreases and practically disappears, and the current becomes 
more sinusoidal. That is, while the distorting component 
remains the same, the sinusoidal component of the current 
greatly increases, and obscures the distortion. For example, 
in Figs. 70 and 71, two waves are shown, corresponding in 
magnetization to ^the curve of Fig. 67, as the one most 
distorted. The curve in Fig. 70 is the current wave of a 
transformer at TV load. At higher loads the distortion is 
correspondingly still less, except where the magnetic flux of 
self-induction, that is, flux passing between primary and sec- 
ondary, and increasing proportionally to the load, is so large 
as to reach saturation, in which .case a distortion appears 
again and increases with increasing load. The curve of Fig. 
71 is the exciting current of a magnetic circuit containing 
an air-gap whose length equals ?^ the length of the magnetic 
circuit. These two curves are drawn to £ the size of the curve 
in Fig. 67. As shown, both curves are practically sine waves. 
The sine curves of magnetic flux are shown dotted as <£. 

77. The distorted wave of current can be resolved into 
two components : A true sine wave of equal effective intensity 
nnd equal power to the distorted wave, called the equivalent 


112 


ALTERNATING-CURRENT PHENOMENA. 


sine wave, and a wattless JiigJier harmonic, consisting chiefly 
of a term of triple frequency. 

In Figs. 66 to 71 are shown, as /, the equivalent sine' 


\ 


\ 


v 


\ 


Figs. 70 and 71. Distortion of Current Wave by Hysteresis. 

waves and as i, the difference between the equivalent sine 
wave and the real distorted wave, which consists of wattless 
complex higher harmonics. The equivalent sine wave of 
M.M.F. or of current, in Figs. 66 to 69, leads the magnet- 


EFFECTIVE RESISTANCE AND REACTANCE. 113 

ism by 34°, 44°, 38°, and 15°. 5, respectively. In Fig. 71 
the equivalent sine wave almost coincides with the distorted 
curve, and leads the magnetism by only 9°. 

It is interesting to note, that even in the greatly dis- 
torted curves of Figs. 66 to 68, the maximum value of the 
equivalent sine wave is nearly the same as the maximum 
value of the original distorted wave of M.M.F., so long as 
magnetic saturation is not approached, being 1.8, 2.9, and 
4.2, respectively, against 1.8, 2.8, and 4.3, the maximum 
values of the distorted curve. Since, by the definition, the 
effective value of the equivalent sine wave is the same as 
that of the distorted wave, it follows, that this distorted 
wave of exciting current shares with the sine wave the 
feature, that the maximum value and the effective value 
have the ratio of V2 -f- 1. Hence, below saturation, the 
maximum value of the distorted curve can be calculated 
from the effective value — which is given by the reading 
of an electro-dynamometer — by using the same ratio that 
applies to a true sine wave, and the magnetic characteris- 
tic can thus be determined by means of alternating cur- 
rents, with sufficient exactness, by the electro-dynamometer 
method, in the range below saturation. 

78. In Fig. 72 is shown the true magnetic character- 
istic of a sample of good average sheet iron, as found by 
the method of slow reversals with the magnetometer ; for 
comparison there is shown in dotted lines the same char- 
acteristic, as determined with alternating currents by the 
electro-dynamometer, with ampere-turns per cm as ordi- 
nates, and magnetic inductions as abscissas. As repre- 
sented, the two curves practically coincide up to a value of 
& = 13,000 ; that is, up to the highest inductions practicable 
in alternating-current apparatus. For higher saturations, 
the curves rapidly diverge, and the electro-dynamometer 
curve shows comparatively small M.M.Fs. producing appar- 
ently very high magnetizations. 


114 


AL TERN A TING-CUR RE KT PHENOMENA. 


The same Fig. 72 gives the curve of hysteretic loss, in 
ergs per cm3 and cycle, as ordinates, and magnetic induc- 
tions as abscissae. 


TT 

\ 

/ 

/ 

/ / 
/ 

18 

/ 

/ 
/ 

17 

r 

1 
1 

/I 

' 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

' 

/ 

/ 

1 

/ 

/ 

I. 

' 

/ 

// 

/ 

/ 

1 

/ 

I 

/ 

/ 

/ 

/' 

/ 

^ 

^ 

/ 

^" 

^ 

^^ 

^ 

; j^ 

* 

x 

2=EE 

x^^ 

£=1,000 2,000 3,000 1.0CO 5,000 6,000 7,000 8,000 9,000 10,000 11,000 12,000 13,0<W14,000 15, 
Fig. 72. Magnetization and Hysteresis Curve. 

woie.ooo iv.ow 

The electro-dynamometer method of determining the 
magnetic characteristic is preferable for use with alter- 
nating-current apparatus, since it is not affected by the 
phenomenon of magnetic "creeping," which, especially at 


EFFECTIVE RESISTANCE AND REACTANCE. 115 

low densities, may in the magnetometer tests bring the mag- 
netism very much higher, or the M.M.F. lower, than found 
in practice in alternating-current apparatus. 

So far as current strength" and energy consumption are 
concerned, the distorted wave can be replaced by the equi- 
valent sine wave, and the higher harmonics neglected. 

All the measurements of alternating currents, with the 
single exception of instantaneous readings, yield the equiv- 
alent sine wave only, and suppress the higher harmonic ; 
since all measuring instruments give either the mean square 
of the current wave, or the mean product of instantaneous 
values of current and E.M.F., which, by definition, are the 
same in the equivalent sine wave as in the distorted wave. 

Hence, in all practical applications, it is permissible to 
neglect the higher harmonic altogether, and replace the dis- 
torted wave by its equivalent sine wave, keeping in mind, 
however, the existence of a higher harmonic as a possible 
disturbing factor which may become noticeable in those cases 
where the frequency of the higher harmonic is near the fre- 
quency of resonance of the circuit, that is, in circuits con- 
taining capacity besides the inductance. 

79. The equivalent sine wave of exciting current leads 
the sine wave of magnetism by an angle a, which is called 
the angle of Jiysteretic advance of phase. Hence the cur- 
rent lags behind the E.M.F by ^ 90° — a, and the power 


is therefore, p=f£ cog (9QO _ a) = /E sin a 

Thus the exciting current, 7, consists of an energy compo- 
nent, / sin a, called the Jiysteretic or magnetic energy current, 
and a wattless component, / cos a, which is called the mag- 
netizing current. Or, conversely, the E.M.F. consists of an 
energy component, E sin a, the Jiysteretic energy E.M.F., 
and a wattless component, E cos a, the E.M.F. of self- 
induction. 

Denoting the absolute value of the impedance of the 


116 A L TERNA TING-CURRENT PHENOMENA . 

circuit, E 1 1, by s, — where s is determined by the mag- 
netic characteristic of the iron, and the shape of the 
magnetic and electric circuits, — the impedance is repre- 
sented, in phase and intensity, by the symbolic expression, 

Z = r — jx = z sin a — jz cos a ; 
and the admittance by, 

Y = g + j b = - sin a -j- j - cos a = y sin a -f- jy cos a. 

z z 

The quantities, z, r, x, and y, g, b, are, however, not 
constants as in the case of the circuit without iron, but 
depend upon the intensity of magnetization, (B, — that is, 
upon the E.M.F. This dependence complicates the investi- 
gation of circuits containing iron. 

In a circuit entirely inclosed by iron, a is quite consider- 
able, ranging from 30° to 50° for values below saturation. 
Hence, even with negligible true ohmic resistance, no great 
lag can be produced in ironclad alternating-current circuits. 

80. The loss of energy by hysteresis due to molecular 
friction is, with sufficient exactness, proportional to the 
1.6th power of magnetic induction <&. Hence it can be ex- 
pressed by the formula : 


where — 

IV a = loss of energy per cycle, in ergs or (C.G.S.) units (= 10~7 
Joules) per cm8, 

(ft = maximum magnetic induction, in lines of force per cm2, and 

77 = the coefficient of hysteresis. 

This I found to vary in iron from .00124 to .0055. As a 
fair mean, .0033 * can be accepted for good average annealed 
sheet iron or sheet steel. In gray cast iron, 17 averages 
.013 ; it varies from .0032 to .028 in cast steel, according 
to the chemical or physical constitution ; and reaches values 
as high as .08 in hardened steel (tungsten and manganese 

* At present, with the improvements in the production and selection of sheet steel far 
alternating apparatus, .0025 can be considered a fair average in selected material (1899). 


EFFECTIVE RESISTANCE AND REACTANCE. 117 

steel). Soft nickel and cobalt have about the same co- 
efficient of hysteresis as gray cast iron ; in magnetite I 
found rj = .023. 

In the curves of Fig. 62 to 69, r, = .0033. 

At the frequency, N, the loss of power in the volume, V, 
is, by this formula, — 

P=-t]N F&1-6 10 - ' watts 


where S is the cross-section of the total magnetic flux, <£. 

The maximum magnetic flux, <E>, depends upon the 
counter E.M.F. of self-induction, 

E = V2 -IT Nn 4> 10 - 8, 


V2 TT Nn 

where n = number of turns of the electric circuit. 

Substituting this in the value of the power, P, and 
canceling, we get, — 

E1-' FIO 5-8 E™ F108 


no5-8 Ka no3 
» where ^ = ^ o.R i.« oi.fi ..,.. = 58 -n 


T/- 

or, substituting •>; = .0033, we have ^4 = 191.4 —^ — — ; 

o ' /? * 

or, substituting F= SL, where L = length of magnetic circuit, 
•n L 10 5-8 58 » Z 103 Z 

— — 


and 103 191.4 E 


In Figs. 73, 74, and 75, is shown a curve of hysteretic 
loss, with the loss of power as ordinates, and 

in curve 73, with the E.M.F., E, as abscissae, for L = 6, 
S = 20, N= 100, and n = 100 ; 


118 


AL TERNA TING-CURRENT PHENOMENA. 


RELATION 

BE 

TW = 

EN 

EA 

NDP 

F 

OR 

_— 

5,8 

= 20 

N = 

10 

r5 

= 1 

oo 

/ 

/ 

/ 

K 

/ 

o 

/ 

^/ 

Q. 

x 

^ 

x 

X 

^ 

x 

x 

x 

x 

^ 

X* 

^ 

. • 


^ 

E.IV 

l.F. 

Fig. 73. Hysteresis Loss as Function of £. M. F. 


BETW 
OR LT6. S=20, ^ 


= 100.E= 


SO 100 160 200 250 300 

Fig. 74. Hysteresis Loss as Function of Number of Turns. 


EFFECTIVE RESISTANCE AND REACTANCE. 


119 


II I I II I 


RELATION BETWEEN N AND P 
FOR 8=20, L=6, 71 = 100. E = 100. 


Fig. 75. Hysteresis Loss as Function of Cycles. 

in curve 74, with the number of turns as abscissae, for 
Z = 6, S = 20, JV= 100, and E = 100 ; 

in curve 75, with the frequency, JV, or the cross-section, S, 
as abscissae, for L = 6, n = 100, and E = 100. 

As shown, the hysteretic loss is proportional to the 1.6th 
power of the E.M.F., inversely proportional to the 1.6th 
power of the number of turns, and inversely proportional to 
the .6th power of frequency, and of cross-section. 

81. If g = effective conductance, the energy compo- 
nent of a current is / = Eg, and the energy consumed in 
a conductance, g, is P = IE = Ezg. 

Since, however : 

P = A , we have A = E2 g ; 

or 

A 58r)L 10s 


191.4 


From this we have the following deduction : 


120 


ALTERNA TING-CURRENT PHENOMENA. 


The effective conductance due to magnetic hysteresis is 
proportional to the coefficient of hysteresis, rj, and to the length 
of the magnetic circuit, L, and inversely proportional to the 
Jj!h power of the E.M.F., to the .6th power of the frequency, 
N, and of the cross-section of tlie magnetic circuit, S, and to 
tlie 1.6th power of the number of turns, n. 

Hence, the effective hysteretic conductance increases 
with decreasing E.M.F., and decreases with increasing 


RELATION 
FOR L=6, 

BE- 

PWEEN 0AND E 
00. S = 20,?l = 1O 

V 

\ 

\ 

\ 

^ 

\ 

> 

^. 

.^^ 

__9 

a 

1 -, 

- — -. 

— ^ 

—— . 

. 

• , 

E 

Ftg. 76. Hysteresis Conductance as Function of E.M.F. 

E.M.F. ; it varies, however, much slower than the E.M.F., 
so that, if the hysteretic conductance represents only a part 
of the total energy consumption, it can, within a limited 
range of variation — as, for instance, in constant potential 
transformers — be assumed as constant without serious 
error. 

In Figs. 76, 77, and 78, the hysteretic conductance, g, is 
plotted, for L = 6, E = 100, N= 100, 5 = 20 and n = 100, 
respectively, with the conductance, g, as ordinates, and with 


EFFECTIVE RESISTANCE AND REACTANCE. 


1-21 


RELATION BETWEEN Q AND N 
FOR L-6, E = IOO. S = 20, n=IOO 


Fig. 77. Hysteresis Conductance as Function of Cycles, 


• 

R 

LAI 

,0, 

BE 

WE 

EN 

,AS 

D(/ 

FOP 

L= 

6,E 

= 1( 

50, 

00 

,8= 

2a 

\ 

b 

V 

a 

\ 

\ 

s 

\ 

X. 

E 

- 

T 

-NL 

\. 

M~B~ 

:RO 

• — , 

F T 


r= 

200 250 300 350 

Fig. 78. Hysteresis Conductance as Function of Number of Turns. 


122 ALTERNATING-CURRENT PHENOMENA. 

E as abscissae in Curve 76. 
.A^ as abscissas in Curve 77. 
n as abscissas in Curve 78. 

As shown, a variation in the E.M.F. of 50 per cent 
causes a variation in g of only 14 per cent, while a varia- 
tion in N or 6" by 50 per cent causes a variation in g of 21 
per cent. 

If (R = magnetic reluctance of a circuit, £FA = maximum 
M.M.F., I — effective current, since /V2 = maximum cur- 
rent, the magnetic flux, 


(R (R 

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


we have 


(R 
hence, the absolute admittance of the circuit is 

(RIO8 = a& 

E ~ 2 TT n*N ~ N ' 

108 

where a = , a constant. 

2 TT n 

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

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

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


EFFECTIVE RESISTANCE AND REACl^ANCE. 123 

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


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


and <R, 10L 


magnetic flux, 


substituting this value in the equation of the admittance, 

(R 108 Z 109 z 

y= -z- nrv> we have 5— ; 


where „ L W 127Z10' 


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


The conductance is 


= 

and the admittance, y = - ; 

yv/u. 

hence, the angle of hysteretic advance is 


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


or, substituting 
J£ 

we have sin a = — 

-4 ' 


1 24 AL TERN A TING-CURRENT PHENOMENA. 

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

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

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

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

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

<R = (R { _|_ <Rfl ; 

hence the admittance is 


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

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


sm a = — 

y 


EFFECTIVE RESISTANCE AND REACTANCE. 125 

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

<Rj 

«* + <*« ' 

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

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

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

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

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

-= A 

EAN* ' 

The current wave is practically a sine wave. 

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

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

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


126 ALTERNATING-CURRENT PHENOMENA. 

then from the equation, 
E = 


where, 

N '= frequency, 

n = number of turns, 


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

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


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

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

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

quired in the air ; 

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


and the absolute admittance, 


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

If — 

V = volume of iron, 

rj = coefficient of hysteresis, 

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


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


EFFECTIVE RESISTANCE AND REACTANCE. 127 

The angle of hysteretic advance is, — 

sin a=g/y; 

the susceptance, b = Vj*2 — gz\ 

the effective resistance, r = g / y*\ 

and the reactance, x = b / y*. 

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

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

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

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

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

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

the equation, 4 

sin a = — f—i . 

&'4 

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

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


128 ALTERNATING-CURRENT PHENOMENA. 

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

58 7 Z 103 


9.) The absolute value of hysteretic admittance, — 


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


> _(«. + «„) 10- 

2-irNn* 

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


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

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

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

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


FOUCAULT OR EDDY CURRENTS. 129