book
Theory and Calculation of Alternating Current Phenomena (1900) — part 5 of 19
1 January 1900
ratio of potentials, in Curve II., Curve IV., and Curve VI.;
Curves I. and II. refer to a non-inductive receiver circuit ;
RESISTANCE OF TRANSMISSION LINES,
Curves III. and IV. refer to a receiver circuit of
constant susceptance b = .142
Curves V. and VI. refer to a receiver circuit of
constant susceptance b = — .142 ;
Curves VII. and VIII. refer to a non-inductive re- ceiver circuit and non-inductive line.
In Fig. 59, the output is shown as Curve I., and the ratio of potentials as Curve II., for the same line constants, fora constant conductance, ^- = .0592 ohms, and for variable susceptances, b, of the receiver circuit.
OUTPUT P /NO RATIO OF POTENTIAL a t SENDING END OF LINE OF IMPEDANCE. Z0
T RECEIV1 NG^ND =5.5 -3j
AT
CON
TAN
g= . 0592
1 OUTPUT II RATIO OF POTENTIALS —
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Fig. 59. Variation of Potential in Line at Various Loads.
3.) Maximum Efficiency.
- The output, for a given conductance, g, of a receiver circuit, is a maximum if b = — b0. This, however, is gen- erally not the condition of maximum efficiency.
94 ALTERNATING-CURRENT PHENOMENA.
The loss of energy in the line is constant if the current is constant ; the output of the generator for a given cur- rent and given generator E.M.F. is a^aximum if the cur- rent is in phase with the E.M.F. at the generator terminals. Hence the condition of maximum output at given loss, or of maximum efficiency, is —
tan £>0 = 0. The current is —
The current I0, is in phase with the E.M.F., E0, if its quadrature component — that is, the imaginary term — dis- appears, or
x + Xo = 0.
This, therefore, is the condition of maximum efficiency,
Hence, the condition of maximum efficiency is, that the reactance of the receiver circuit shall be equal, but of oppo- site sign, to the reactance of the line.
Substituting x = — x0, we have, ratio of E.M.Fs.,
power,
RESISTANCE OF TRANSMISSION LINES.
95
and depending upon the resistance only, and not upon the reactance.
This power is a maximum if g = g0, as shown before; hence, substituting g = g0, r = r0,
E 2
maximum power at maximum efficiency, Pm = —2— ,
at a ratio of potentials, am — — -2— ,
" ro
or the same result as in § 62.
.01 .03 • .03 .01 .05 .06 .07 .08
Fig. 60. Load Characteristic of Transmission Line.
In Fig. 60 are shown, for the constants — E0 = 1,000 volts, Z0 =2.5 — 6/; r0 = 2.5 ohms, x0 = 6 ohms, z0 = 6.5 ohms,
96 ALTERNATING-CURRENT PHENOMENA.
and with the variable conductances, g, of the receiver circuit as abscissae, the —
Output at maximum efficiency, (Curve I.) ;
Volts at receiving end of line, (Curve II.) ;
Efficiency = • , (Curve III.).
r + r0
4.) Control of Receiver Voltage by Shunted Snsceptance.
- By varying the susceptance of the receiver circuit, the potential at the receiver terminals is varied greatly. Therefore, since the susceptance of the receiver circuit can be varied at will, it is possible, at a constant generator E.M.F., to adjust the receiver susceptance so as to keep the potential constant at the receiver end of the line, or to vary it in any desired manner, and independently of the generator potential, within certain limits.
The ratio of E.M.Fs. is —
If at constant generator potential E0, the receiver potential E shall be constant,
a — constant ; hence,
#2' or, expanding,
which is the value of the susceptance, b, as a function of the receiver conductance, — that is, of the load, — which is required to yield constant potential, aE0, at the receiver circuit.
For increasing g, that is, for increasing load, a point is reached, where, in the expression —
b = -
RESISTANCE OF TRANSMISSION LINES.
97
the term under the root becomes imaginary, and it thus becomes impossible to maintain a constant potential, aE0. Therefore, the maximum output which can be transmitted at potential aE0, is given by the expression —
hence b = — o0 ,
and g = — g0 --
the susceptance of receiver circuit, the conductance of receiver circuit;
°- —f» the output.
- If a = 1, that is, if the voltage at the receiver cir- cuit equals the generator potential —
P=E*(ty00'-g0). If a = 1 when g = 0, b = 0
when g > 0, b < 0 ; if a > 1 when g = 0, or g > 0, b < 0,
that is, condensance; if a < 1 when g = 0, b > 0,
when g = - #, + /f — ^ - <V, ^ = 0 ; when^> -g0 + V/f — ^ - V, * < 0,
or, in other words, if a < 1, the phase difference in the main line must change from lag to lead with increasing load.
- The value of a giving the maximum possible output in a receiver circuit, is determined by dP / da = 0 ;
expanding : 2 a (yJL - g\ _ f!f' = 0 ;
\a J a
hence, y0 = 2ag0,
yo 1 Zo
" = = =
98 ALTERNATING-CURRENT PHENOMENA.
the maximum output is determined by —
S == So i = So I
and is, P = —2- .
4 r
From : a = ^ = -^- ,
the line reactance, x0, can be found, which delivers a maximum output into the receiver circuit at the ratio of potentials, a, and z0 = 2 r0 a,
for a == 1,
If, therefore, the line impedance equals 2# times the line resistance, the maximum output, P = E* j ± r0, is trans- mitted into the receiver circuit at the ratio of potentials, a.
If z0 = 2 r0, or x0 = r0 V3, the maximum output, P = £02/4:r0, can be supplied to the receiver circuit, without change of potential at the receiver terminals.
Obviously, in an analogous manner, the law of variation of the susceptance of the receiver circuit can be found which is required to increase the receiver voltage proportionally to the load ; or, still more generally, — to cause any desired variation of the potential at the receiver circuit indepen- dently of any variation of the generator potential, as, for in- stance, to keep the potential of a receiver circuit constant, even if the generator potential fluctuates widely.
- In Figs. 61, 62, and 63, are shown, with the output, P = E* g a2, as abscissae, and a constant impressed E.M.F., E0 = 1,000 volts, and a constant line impedance, Z0 = 2.5 — 6/, or, r0 = 2.5 ohms, x0 = 6 ohms, z = 6.5 ohms, the following values :
RATIO'OF RECEIVER VOLTAGE TO SENDER VOLTAGE: d =I.O
LINE IMPEDANCE: Z0= a. 5— 6;
ENERGY CURRENT CONSTANT GENERATOR
TOTAL CURRENT
CURRENT IN NON-INDUCTIVE RECEIVER CIRCUIT WITHOUT COMPENSATION
OUTPUT] IN RECEIVER CIPJCUIT, KILOWJATT 50 60 70 80
Fig. 61. Variation of Voltage Transmission Lines.
• .
RATIO OF RECEIVER VOLTAGE TO SENDER VOLTAGE: LINE MPEDANCE:Z_ = 2.5.— 6J . ENERGY CURRENT CONSTANT GENRATOR POT II. REACTIVE CURRENT III. TOTAL CURRENT IV. POTENTIAL IN NON-INDUCTIVE CIRCUIT WITHOUT C
~|Tt-MJJ MINI
a =.7 :NTIAL E
OMPENS
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Fig. 62. Variation of Voltaqe Transmission Lines.
100
AL TERNA TING-CURRENT PHENOMENA.
RATIO OF RECEIVER VOLTAGE TO SEN DER VOLTAGE: a =1.3
INE IMPEDANCE: Z0=2.5.— ej"
CONSTANT GENERATOR POTENTIAL E0=IOOOl
I. ENERGY CURRENT
II. "REACTIVE CURRENT
III. TOTAL CURRENT
IV. POTENTIAL IN NON-INDUCTIVE RECEIVER CIRCUIT WITHOUT COMPENSATION
OUTPUT N RECEIVER C RCUIT, KILOWATTS
30 10 80 60 70 80 90
Fig. 63. Variation of Voltage Transn\jssion Lines.
Energy component of current, gE, (Curve I.) ;
Reactive, or wattless component of current, bE, (Curve II.) ;
Total current, yE, (Curve III.) ;
for the following conditions :
a = 1.0 (Fig. 61) ; a = .7 (Fig. 62) ; a = 1.3 (Fig. 63).
For the non-inductive receiver circuit (in dotted lines),
the curve of E.M.F., E, and of the current, I = gE, are
added in the three diagrams for comparison, as Curves IV.
and V.
As shown, the output can be increased greatly, and the
potential at the same time maintained constant, by the judi-
cious use of shunted reactance, so that a much larger out-
put can be transmitted over the line at no drop, or even at
a rise, of potential.
RESISTANCE OF TRANSMISSION LINES.
101
5.) Maximum Rise of Potential at Receiver Circuit.
70. Since, under certain circumstances, the potential at
the receiver circuit may be higher than at the generator,
it is of interest to determine what is the maximum value of
potential, E, that can be produced at the receiver circuit
with a given generator potential, E0 .
The condition is that
a = maxmum or — = mnmum :
a2
that is,
substituting,
r0g +
(*0g -
and expanding, we get,
dg = °; gss~"£''
— a value which is impossible, since neither r0 nor g can be
negative. The next possible value is g — 0, — a wattless
circuit.
Substituting this value, we get,
and by substituting, in
,
b + b0 = 0 ;
that is, the sum of the susceptances = 0, or the condition
of resonance is present.
Substituting,
*=-*-£,
we have
102 AL TERNA TING-CURRENT PHENOMENA.
The current in this case is,
or the same as if the line resistance were short-circuited
without any inductance.
This is the condition of perfect resonance, with current
and E.M.F. in phase.
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CONSTANT IMPRESSED E. M. F. Eo^lOOO
" LINE IMPEDANCE Z0=2.5- €
1 MAXIMUM OUTPUT BY COMPENSATION
II MAXIMUM EFFICIENCY BY COMPENSATIC
III NON-INDUCTiVE RECEIVER C RCU T
IV NON-INDUCTIVE LINE AND NON-INDUCT
RECEIVER CIRCUIT
If
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Fig. 64. Efficiency and Output of Transmission Line.
71. As summary to this chapter, in Fig. 64 are plotted,
for a constant generator E.M.F., E0 = 1000 volts, and a
line impedance, Z0 = 2.5 — 6/, or, r0 = 2.5 ohms, x0 = 6
ohms, z0 = 6.5 ohms ; and with the receiver output as
RESISTANCE OF TRANSMISSION LINES. 103
abscissae and the receiver voltages as ordinates, curves
representing —
the condition of maximum output, (Curve I.) ;
the condition of maximum efficiency, (Curve II.) ;
the condition b = 0, or a non-inductive receiver cir-
cuit, (Curve III.) ;
the condition b = 0, b0 = 0, or a non-inductive line and non-
inductive receiver circuit.
In conclusion, it may be remarked here that of the
sources of susceptance, or reactance,
a choking coil or reactive coil corresponds to an inductance ;
a condenser corresponds to a condensance ;
a polarization cell corresponds to a condensance ;
a synchronizing alternator (motor or generator) corresponds to
an inductance or a condensance, at will;
an induction motor or generator corresponds to an inductance.
The choking coil and the polarization cell are specially
suited for series reactance, and the condenser and syn-
chronizer for shunted susceptance.
104 ALTERNATING-CURRENT PHENOMENA.
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
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T2.8
3 =2.S
M\
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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
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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
Provenance
- Shelf
- Reference library
- Author
- Charles Proteus Steinmetz (with Ernst J. Berg)
- Rights
- Published in 1900, before 1929, and therefore in the public domain in the United States.
- Collected By
- StanBot reference library