book
Theory and Calculation of Alternating Current Phenomena (1900) — part 4 of 19
1 January 1900
let a reactance, x0 , be connected in series in a receiver cir- cuit of impedance
Z = r — jx, z = -/r2 -|- x'2.
IMPRESSED E.M.F. CONSTANT, E0=IOO IMPEDANCE OF RECEIVER CIRCUIT CONSTANT, Z - 1.0
LINE RESISTANCE CONSTANT n =.2
3 - -.4 T-5 ' '.6 T.7 r-8
Fig. 37. Variation of Voltage at Constant Series Resistance with Phase Relation of Receiver Circuit.
Then, the total impedance of the circuit is Z -jx0 = r—j(x +#e).
Er Er0
Fig. 38.
and the current is, /=
E Fig. 39.
Z-jx0 r—j(x + x0}' /hile the difference of potential at the receiver terminals
r—jx
62 ALTERNATING-CURRENT PHENOMENA.
Or, in absolute quantities : — Current,
/_ Eo EQ
•* ~
Vr* -f- (x + x0)'2 V 'z'1 + 2xx0 -- xa2 E.M.F. at receiver terminals,
r / r' + *« = J^
° V ra + (* + „) V** + 2*.r0 + *.a 5
difference of phase in receiver circuit,
x
tan <D = - ; r
difference of phase in supply circuit,
a.} If JT is small compared with r, that is, if the receiver circuit is non-inductive, / and E change very little for small values of x0 ; but if x is large, that is, if the receiver circuit is of large reactance, / and E change much with a change of x0.
b.} If x is negative, that is, if the receiver circuit con- tains condensers, synchronous motors, or other apparatus which produce leading currents — above a certain value of x the denominator in the expression of E, becomes < z, or E > E0 ; that is, the reactance, x0 , raises the potential.
c.) E = E0 , or the insertion of a series inductance, x0 , does, not affect the potential difference at the receiver ter-
minals, if
^z*--2xx0 + x02 = 2; or, x0 = — 2 x.
That is, if the reactance which is connected in series in the circuit is of opposite sign, but twice as large as the reactance of the receiver circuit, the voltage is not affected, but E = E0,I= E0/z. If x0 < — 2 x, it raises, if x0 > — Zv, it lowers, the voltage.
We see, then, that a reactance inserted in series in an alternating-current circuit will lower the voltage at the
RESISTANCE, INDUCTANCE, CAPACITY.
63
receiver terminals only when of the same sign as the reac- tance of the receiver circuit ; when of opposite sign, it will lower the voltage if larger, raise the voltage if less, than twice the numerical value of the reactance of the receiver circuit.
d.} If x = 0, that is, if the receiver circuit is non- inductive, the E.M.F. at receiver terminals is :
= (!-}- )•' expanded by the binomial theorem
= nx
Therefore, if x0 is small compared with r : —
That is, the percentage drop of potential by the insertion of reactance in series in a non-inductive circuit is, for small
Fig. 40.
values of reactance, independent of the sign, but propor- tional to the square of the reactance, or the same whether it be inductance or condensance reactance.
64
AL TERNA TING-CURRENT PHENOMENA.
- As an instance, in Fig. 41 the changes of current, /, and of E.M.F. at receiver terminals, E, at constant im- pressed E.M.F., E0, are shown for various conditions of a receiver circuit and amounts of reactance inserted in series.
Fig. 41 gives for various values of reactance, x0 (if posi- tive, inductance — if negative, condensance), the E.M.Fs., E, at receiver terminals, for constant impressed E.M.F.,
VOLTS E OR AMPERES I
100
IMPRESSED E.'M.F! CONSTANT, E IMPEDANCE OF RECEIVER CIRC.UI
I. r=l.o x=o
II. r=.6 X=H-,8
- r=.e i=-.8
=160 r CONS
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Fig. 41.
E0 = 100 volts, and the following conditions of receiver circuit •— z= 1 Qj r = 1>0> x= 0 (Curve j)
2=1.0, r= .6,^= .8(CurveII.) 2= 1.0, r= .6, AT= — .8 (Curve III.)
As seen, curve I is symmetrical, and with increasing x0 the voltage E remains first almost constant, and then drops off with increasing rapidity.
In the inductive circuit series inductance, or, in a con- denser circuit series condensance, causes the voltage to drop off very much faster than in a non-inductive circuit.
RESISTANCE, INDUCTANCE, CAPACITY.
65
Series inductance in a condenser circuit, and series con- densance in an inductive circuit, cause a rise of potential. This rise is a maximum for x0 = i .8, or, x0 = — x (the condition of resonance), and the E.M.F. reaches the value, E = 167 volts, or, E = E0z] r. This rise of potential by series reactance continues up to x0 = il.6, or, x0 = — %x,
Fig. 42.
where E = 100 volts again ; and for x0 > 1.6 the voltage drops again.
At x0 = ± -8, x = =f .8, the total impedance of the circuit is r — j (x -f x0} = r = .6, x + x0 = 0, and tan S>0 = 0 ; that is, the current and E.M.F. in the supply circuit are in phase with each other, or the circuit is in electrical resonance.
\
Fig. 43.
Since a synchronous motor in the condition of efficient working acts as a condensance, we get the remarkable result that, in synchronous motor circuits, choking coils, or reactive coils, can be used for raising the voltage.
In Figs. 42 to 44, the polar diagrams are shown for the conditions —
E0 = 100, x0 = .6, x = 0 . (Fig. 42) E = 85.7
x = + .8 (Fig. 43) E = 65.7
(Fig. 44) E = 158.1
66
ALTERNA TING-CURRENT PHENOMENA.
- In Fig. 45 the dependence of the potential, E, upon the difference of phase, oi, in the receiver circuit is shown for the constant impressed E.M.F., E0 = 100 ; for the con- stant receiver impedance, z = 1.0 (but of various phase differences to), and for various series reactances, as follows :
x0 = .2 (Curve I.)
x0 = .6 (Curve II.)
x0 = .8 (Curve III.)
xo = 1.0 (Curve IV.)
Xo = 1.6 (Curve V.)
x0 = 3.2 (Curve VI.)
Fig. 44.
Since z = 1.0, the current, /, in all these diagrams has the same value as E.
In Figs. 46 and 47, the same curves are plotted as in Fig. 45, but in Fig. 46 with the reactance, .*•, of the receiver circuit as abscissas ; and in Fig. 47 with the resistance, r, of the receiver circuit as abscissae.
As shown, the receiver voltage, E, is always lowest when x0 and x are of the same sign, and highest when they are of opposite sign.
The rise of voltage due to the balance of x0 and x is a maximum for x0= +1.0, x = — 1.0, and r = 0, where
RESISTANCE, INDUCTANCE, CAPACITY.
L Q. 4— PHASE D FFERENCE IN CONSUMER SIR UIT
l-90 80 70 bO 50 40 30 20 10 0 10 20 30 10 50 60 70 bO 90 OEUHE
fig. 45. Variation of Voltage at Constant Series Reactance with Phase Angle of Receiver Circuit.
Fig. 46. Variation of Voltage at Constant Series Reactance with Reactance of Receiver Circuit.
68
AL TERN A TING-CURRENT PHENOMENA.
E = oo ; that is, absolute resonance takes place. Obvi- ously, this condition cannot be completely reached in practice.
It is interesting to note, from Fig. 47, that the largest part of the drop of potential due to inductance, and rise to condensance — or conversely — takes place between r = 1.0 and r = .9 ; or, in other words, a circuit having a power
Volts E or Amperes I. 160 150 140 130 120 110 100 90 80 70
sfl
Fig. 47. Variation of Voltage at Constant Series Reactance with Resistance of Receiver Circuit.
factor cos & = .9, gives a drop several times larger than a non-inductive circuit, and hence must be considered as an inductive circuit.
3.) Impedance in series witJi a circuit. 48. By the use of reactance for controlling electric circuits, a certain amount of resistance is also introduced, due to the ohmic resistance of the conductor and the hys- teretic loss, which, as will be seen hereafter, can be repre- sented as an effective resistance.
RESISTANCE, INDUCTANCE, CAPACITY. 69
Hence the impedance of a reactive coil (choking coil) may be written thus : —
&Q = ro JXoi ZQ = V f0 -j- Xo ,
where r0 is in general small compared with x0. From this, if the impressed E.M.F. is
E0 = e0 +je0'> E0 = Ve02 + e0'2
and the impedance of the consumer circuit is
we get the current, /= ^- = -. —
and the E.M.F. at receiver terminals,
. . ° 7 \ 7 "° (r \ *-\ //„! „ \ '
•^I^o \r ~T ' o) J *- ~T •*<>/
Or, in absolute quantities, the current is,
~/(r -f- roy2 -|- (x -j- ^;0)2 V^2 + z02 + 2 (rr0
the E.M.F. at receiver terminals is,
E0z E0z
V(r + r0)'2 + (x + xoy V^2 + Z0* + 2 the difference of phase in receiver circuit is,
x
tan oi = - ; r
and the difference of phase in the supply circuit is,
- In this case, the maximum drop of potential will not take place for either x = 0, as for resistance in series, or for r = 0, as for reactance in series, but at an intermediate point. The drop of voltage is a maximum ; that is, E is a minimum if the denominator of E is a maximum ; or, since. zy z0, r0, x0 are constant, if rr0 + xx0 is a maximum, that is, since x = ~Vz2 — r2, if rr0 -f- x0 ~/z2 — r2 is a maximum.
70
AL TERN A TING CURRENT-PHEXOMENA.
A function, f = rr0 -+- x0 V^2 — r2 is a maximum when its differential coefficient equals zero. For, plotting f as curve with r as abscissae, at the point where f is a maxi- mum or a minimum, this curve is for a short distance horizontal, hence the tangens-function of its tangent equals zero. The tangens-function of the tangent of a curve, how- ever, is the ratio of the change of ordinates to the change of abscissae, or is the differential coefficient of the func- tion represented by the curve.
/
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Off. 48.
Thus we have : —
f = rr0 + *0 Vs2 — r2 = maximum or minimum, if
Differentiating, we get : —
RESISTANCE, INDUCTANCE, CAPACITY.
71
That is, the drop of potential is a maximum, if the re- actance factor, x I r, of the receiver circuit equals the reac- tance factor, *0/r0, of the series impedance.
Fig. 49.
''o Fig. 50.
- As an example, Fig. 48 shows the E.M.F., E, at the receiver terminals, at a constant impressed E.M.F., E0 = 100, a constant impedance of the receiver circuit, s = 1.0, and constant series impedances,
Z0= .S-/.4 (Curve I.)
Z0 = 1.2 — / 1.6 (Curve II.) as functions of the reactance, x, of the receiver circuit.
Fig. 51.
Figs. 49 to 51 give the polar diagram for E0 = 100, x = .95, x = 0, x = - .95, and Z0 = .3 -/ .4.
72 ALTERNATING-CURRENT PHENOMENA.
4.) Compensation for Lagging Currents by Shunted Condensance.
- We have seen in the latter paragraphs, that in a constant potential alternating-current system, the voltage at the terminals of a receiver circuit can be varied by the use of a variable reactance in series to the circuit, without loss of energy except the unavoidable loss due to the resistance and hysteresis of the reactance; and that, if the series reactance is very large compared with the resis- tance of the receiver circuit, the current in the receiver circuit becomes more or less independent of the resis- tance,— that is, of the power consumed in the receiver
Fig. 52.
circuit, which in this case approaches the conditions of a constant alternating-current circuit, whose current is.
/= — " . or approximately, / = — ° .
This potential control, however, causes the current taken from the mains to lag greatly behind the E.M.F., and thereby requires a much larger current than corresponds to the power consumed in the receiver circuit.
Since a condenser draws from the mains a leading cur- rent, a condenser shunted across such a circuit with lagging current will compensate for the lag, the leading and the lagging current combining to form a resultant current more or less in phase with the E.M.F., and therefore propor- tional to the power expended.
RESISTANCE, INDUCTANCE, CAPACITY. 73
In a circuit shown diagrammatically in Fig. 52, let the non-inductive receiver circuit of resistance, r, be connected in series with the inductance, x0 , and the whole shunted by a condenser of condensance, c, entailing but a negligible loss of energy.
Then, if E0 = impressed E.M.F.,—
the current in receiver circuit is,
the current in condenser circuit is,
and the total current is
— Jxo Jc
or, in absolute terms, I0
'•=VfeJ+fe-'/;
while the E.M.F. at receiver terminals is, r
- The main current, 70, is in phase with the impressed E.M.F., E0, or the lagging current is completely balanced, or supplied by, the condensance, if the imaginary term in the expression of I0 disappears ; that is, if
This gives, expanded :
Hence the capacity required to compensate for the lagging current produced by the insertion of inductance- in series to a non-inductive circuit depends upon the resis- tance and the inductance of the circuit. x0 being constant,
74 ALTERNATING-CURRENT PHENOMENA.
with increasing resistance, r, the condensance has to be increased, or the capacity decreased, to keep the balance.
r2 4- r2 Substituting c = ^/ " ,
we get, as the equations of the inductive circuit balanced by condensance : —
7 =
r — Jxo
and for the power expended in the receiver circuit : —
that is, the main current is proportional to the expenditure of power.
For r = 0 we have c = x0, or the condition of balance.
Complete balance of the lagging component of current by shunted capacity thus requires that the condensance, <:, be varied with the resistance, r; that is, with the varying load on the receiver circuit.
In Fig. 53 are shown, for a constant impressed E.M.F., E0 = 1000 volts, and a constant series reactance, x0 = 100 ohms, values for the balanced circuit of,
current in receiver circuit (Curve I.), current in condenser circuit (Curve II.), current in main circuit (Curve III.),
E.M.F. at receiver terminals (Curve IV.),
with the resistance, r, of the receiver circuit as abscissae.
RESISTANCE, INDUCTANCE, CAPACITY.
75
IMPRESSED E.M.F. CONSTANT, E0 = IOOO VOLTS. SERIES REACTANCE CONSTANT, X0= IOO OHMS. VARIABLE RESISTANCE IN RECEIVER CIRCUIT. BALANCED BY VARYING THE SHUNTED CONDENSANCE,
I. CURRENT IN RECEIVER CIRCUIT.
II. CURRENT IN CONDENSER CIRCUIT.
III. CURRENT IN MAIN CIRCUIT. JV. E.M.F. AT RECEIVER CIRCUIT.
100 /
r. OF RECEIVER
CIRCUIT OHMS
10 20 30 40 50 60 70 80 90 100 110 120 130 HO 150 160 170 180 190 200
Fig. 53. Compensation of Lagging Currents in Receiving Circuit by Variable Shunted Condensance.
- If, however, the condensance is left unchanged, c = x0 at the no-load value, the circuit is balanced for r = 0, but will be overbalanced for r > 0, and the main current will become leading.
We get in this case : —
r-jx
The difference of phase in the main circuit is, —
tan u>0 = ,
«0
which is = 0.
76
ALTERNA TING-CURRENT PHENOMENA.
when r = 0 or at no load, and increases with increasing resistance, as the lead of the current. At the same time, the current in the receiver circuit, 7, is approximately con- stant for small values of r, and then gradually decreases.
IMPRESSED E.M.F. CONSTANT, EO—IOOO VOLTS.
SERIES REACTANCE CONSTANT, Xt, -<OO OHMS. SHUNTED CONDENSANCE CONSTANT, C= IOO OH VARIABLE RESISTANCE. IN RECEIVER CIRCUIT- •(.CURRENT IN RECEIVER CIRCUIT. II. CURRENT IN CONDENSER C RCUIT. III. CURRENT IN MA N CIRCUIT. IV.E.M.F. AT RECEIVER CIRCUIT.
MS.
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RESISTANCE r— OF RECEIVER CIRCUIT, OHMS.
2
MINI
JO 20 80 40 50 60 70 80 90 100 110 120 ' 130 140 150 100 170 1
JO 190 200 OHMS
Fig. 54.
In Fig. 54 are shown the values of /, 71} 70, 7f, in Curves
I., II., III., IV., similarly as in Fig. 50, for E0 = 1000 volts,
c = x = 100 ohms, and r as abscissas.
5.) Constant Potential — Constant Current Transformation.
54. In a constant potential circuit containing a large
and constant reactance, x0, and a varying resistance, r, the
current is approximately constant, and only gradually drops
off with increasing resistance, r, — that is, with increasing
load, — but the current lags greatly behind the E.M.F. This
lagging current in the receiver circuit can be supplied by a
shunted condensance. Leaving, however, the condensance
constant, c = x0, so as to balance the lagging current at no
RESISTANCE, INDUCTANCE, CAPACITY. .
77
load, that is, at r = 0, it will overbalance with increasing
load, that is, with increasing r, and thus the main current
will become leading, while the receiver current decreases
if the impressed E.M.F., E0, is kept constant. Hence, to
keep the current in the receiver circuit entirely constant, the
impressed E.M.F., E0, has to be increased with increasing
resistance, r; that is, with increasing lead of the main cur-
rent. Since, as explained before, in a circuit with leading
current, a series inductance raises the potential, to maintain
the current in the receiver circuit constant under all loads,,
an inductance, x^ , inserted in the main circuit, as shown ia
the diagram, Fig. 55, can be used for raising the potential
E0, with increasing load.
Fig. 55.
Let —
be the impressed E.M.F. of the generator, or of the mains,
and let the condensance be xc = x0\ then — •
Current in receiver circuit,
r —jx0
current in condenser circuit,
T
/I = —
X0
Hence, the total current in main line is
r— x x
78 A L TERN A TING-CURRENT PHENOMENA.
and the E.M.F. at receiver terminals,
r —JXo
E.M.F. at condenser terminals,
E.M.F. consumed in main line,
hence, the E.M.F. at generator is
and conversely the E.M.F. at condenser terminals,
current in receiver circuit,
7
r —jx0 r (x0 — xj —jx? '
This value of / contains the resistance, r, only as a fac-
tor to the difference, x0 — x^\ hence, if the reactance, ;r2 ,
is chosen = x0 , r cancels altogether, and we find that if
#2 = *0, the current in the receiver circuit is constant,
/-/A,
X0
and is independent of the resistance, r ; that is, of the load.
Thus, by substituting xz = x0, we have,
Impressed E.M.F. at generator,
E<i = <?2 + Je*'i Ez = V^22 + ^2' 2 = constant ;
current in receiver circuit,
/ =j%L, 7 = ^? = constant;
x0 xa
E.M.F. at receiver circuit,
E = Ir=jE-^-, E ~ ^^, or proportional to load r;
'
RESISTANCE, INDUCTANCE, CAPACITY. 79
E.M.F. at condenser terminals,
E* 1 +/ - , £0= ^2 V 1 + - , hence > E, •
.V
current in condenser circuit,
main current,
r
° *.(*.+./>) '
( proportional to the load,
T JZI<L f 1 , . , . ,
/o = — V ' J r» anC^ ln Pnase Wlt"
° X° ( E.M.F., Ez .
The power of the receiver circuit is,
the power of the main circuit,
f0Ez = 2 r , hence the same.
*02
55. This arrangement is entirely reversible ; that is,
if Ez = constant, / = constant ; and
if I0 = constant, E = constant.
In the latter case we have, by expressing all the quanti-
ties by 70 : —
Current in main line,
I0 = constant;
E.M.F. at receiver circuit,
E = I0x9 = constant ;
current in receiver circuit,
/ =f0 — , proportional to the load -;
current in condenser circuit,
80 AL TERNA TING-CURRENT PHENOMENA.
E.M.F. at condenser terminals,
Impressed E.M.F. at generator terminals,
x 2 1
£2 = —I0 , or proportional to the load - .
From the above we have the following deduction :
Connecting two reactances of equal value, x0, in series
to a non-inductive receiver circuit of variable resistance, r,
and shunting across the circuit from midway between the
inductances by a capacity of condensance, xc = x0, trans-
forms a constant potential main circuit into a constant cur-
rent receiver circuit, and, inversely, transforms a constant
current main circuit into a constant potential receiver cir-
cuit. This combination of inductance and capacity acts as
a transformer, and converts from constant potential to con-
stant current and inversely, without introducing a displace-
ment of phase between current and E.M.F.
It is interesting to note here that a short circuit in the
receiver circuit acts like a break in the supply circuit, and a
break in the receiver circuit acts like a short circuit in the
supply circuit.
As an instance, in Fig. 56 are plotted the numerical
values of a transformation from constant potential of 1,000
volts to constant current of 10 amperes.
Since E^ = 1,000, 7=10, we have : x0 = 100 ; hence
the constants of the circuit are : —
E* = 1000 volts ;
7 = 10 amperes ;
E — 10 r, plotted as Curve I., with the resistances, r, as abscissa;;
E0 = 1000 1/1 + I — Y plotted as Curve II. ;
»' V 100 y
7t = 10 i/1 + ( -£-Y, plotted as Curve III.-
V ^-^^ J
70 = .1 r, plotted as Curve IV.
RESISTANCE, INDUCTANCE, CAPACITY.
81
56. In practice, the power consumed in the main circuit
will be larger than the power delivered to the receiver cir-
cuit, due to the unavoidable losses of power in the induc-
tances and condensances.
u
13
12
11
10
9
j«
|.7
6
6
1
3
2
1
—
CURRENT IN RECEIVER CIRCUIT CONSTANT,
IMPR£SSED E.M, F.CONSTANT, E8=IOOO VOL
2 REACTANCES OFOTo =IOO OHMS EACH, SH
THE CONDENSANCE, ZC = IOO OHMS.
VARIABLE RES STANCE IN RECEIVER CIRCUI
1 E.M.F. AT RECEIVER C RCUIT.
1 II E.M-F. AT CONDENSER CIRCUIT.
Ill CURRENT IN CONDENSER CIRCUIT.
IV CURRENT IN MAIN LINE
V CURRENT IN MAIN LINE INCLUDING tC
VI EFFICIENCY OF TRANSFORMATION,
1^10 AMPERES 1
rs. '~
UNTED IN THEIR MID:
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HM8
F/3. 50. Constant-Potential — Constant-Current Transformation.
Let —
ri = 2 ohms = effective resistance of condensance ;
r0 = 3 ohms = effective resistance of each of the inductances.
We then have : —
Power consumed in condensance, I* r± = 200 + .02 r2 ;
power consumed by first inductance, 72 r0 = 300 ;
power consumed by second inductance, /02r0 = .03 r*.
Hence, the total loss of energy is 500 + -05 r2 ;
output of system, /2 r = 100 r
input, 500 + 100 r -\
effidenCy' 500 + 1W M
It follows that the main current, f0, increases slightly
by the amount necessary to supply the losses of energy
in the apparatus.
82 ALTERNATING-CURRENT PHENOMENA.
This curve of current, I0, including losses in transforma-
tion, is shown in dotted lines as Curve V. in Fig. 56 ; and
the efficiency is shown in broken line, as Curve VI. As
shown, the efficiency is practically constant within a wide
range.
RESISTANCE OF TRANSMISSION LINES.
CHAPTER IX.
RESISTANCE AND REACTANCE OF TRANSMISSION LINES.
57. In alternating-current circuits, E.M.F. is consumed
in the feeders of distributing networks, and in the lines of
long-distance transmissions, not only by the resistance, but
also by the reactance, of the line. The E.M.F. consumed by
the resistance is in phase, while the E.M.F. consumed by the
reactance is in quadrature, with the current. Hence their
influence upon the E.M.F. at the receiver circuit depends
upon the difference of phase between the current and the
E.M.F. in that circuit. As discussed before, the drop of
potential due to the resistance is a maximum when the
receiver current is in phase, a minimum when it is in
quadrature, with the E.M.F. The change of potential due
to line reactance is small if the current is in phase with
the E.M.F., while a drop of potential is produced with a
lagging, and a rise of potential with a leading, current in
the receiver circuit.
Thus the change of potential due to a line of given re-
sistance and inductance depends upon the phase difference
in the receiver circuit, and can be varied and controlled
by varying this phase difference ; that is, by varying the
admittance, Y = g -f jb, of the receiver circuit.
The conductance, gy of the receiver circuit depends upon
the consumption of power, — that is, upon the load on the
circuit, — and thus cannot be varied for the purpose of reg-
ulation. Its susceptance, b, however, can be changed by
shunting the circuit with a reactance, and will be increased
by a shunted inductance, and decreased by a shunted con-
densance. Hence, for the purpose of investigation, the
84 ALTERNATING-CURRENT PHENOMENA.
receiver circuit can be assumed to consist of two branches,
a conductance, g, — the non-inductive part of the circuit, —
shunted by a susceptance, b, which can be varied without
expenditure of energy. The two components of current
can thus be considered separately, the energy component as
determined by the load on the circuit, and the wattless
component, which can be varied for the purpose of regu-
lation.
Obviously, in the same way, the E.M.F. at the receiver
circuit may be considered as consisting of two components,
the energy component, in phase with the current, and
the wattless component, in quadrature with the current.
This will correspond to the case of a reactance connected
in series to the non-inductive part of the circuit. Since the
effect of either resolution into components is the same so
far as the line is concerned, we need not make any assump-
tion as to whether the wattless part of the receiver circuit
is in shunt, or in series, to the energy part.
Let—
Z0 = r0 —,jx0 = impedance of the line ;
z0 = Vr02 + ^2;
Y = g -\-jb = admittance of receiver circuit;
y = VFTT2;
E0 = e0 -f /<?</ = impressed E.M.F. at generator end of line ;
E0 =
E = e +/<?' = E.lVf.F. at receiver end of line ;
E =
I0 = i0 -\-jio = current in the line ;
I0 = Vtf + 4".
The simplest condition is the non-inductive circuit.
1.) Non-inductive Receiver Circuit Sripplied over an
Inductive Line.
58. In this case, the admittance of the receiver circuit
is Y = g, since b = 0.
RESISTANCE OF TRANSMISSION LINES. 85
We have then —
current, 70 = Eg;
impressed E.M.F., E0 = E + Z0 70 = E (1 + Z.g).
Hence —
E.M.F. at receiver circuit,
= \^Z0g~ \-\-gr.-jgxJ
current, 70 = JA|_ = ^ .
Hence, in absolute values —
E.M.F. at receiver circuit, E
current, 70 :
The ratio of E.M.Fs. at receiver circuit and at genera-
tor, or supply circuit, is —
and the power delivered in the non-inductive receiver cir-
cuit, or
output, P = I0 E =
As a function of g, and with a given Eot r0, and x0, this
power is a maximum, if —
that is —
-l+^-V^+^^^O;
hence —
conductance of receiver circuit for maximum output,
Vr02 + V ^o
Resistance of receiver circuit, rm = — = z0 ;
86 AL TERNA TING-CURRENT PHENOMENA.
and, substituting this in P —
Maximum output, Pm = 2 = — g —
and —
ratio of E.M.F. at receiver and at generator end of line,
am = -=r =
efficiency,
That is, the output which can be transmitted over an
inductive line of resistance, r0 , and reactance, x0 , — that is,
of impedance, z0 , — into a non-inductive receiver circuit, is
a maximum, if the resistance of the receiver circuit equals
the impedance of the line, r = z0) and is —
The output is transmitted at the efficiency of
and with a ratio of E.M.Fs. of
1
59. We see from this, that the maximum output which
can be delivered over an inductive line is less than the
output delivered over a non-inductive line of the same
resistance — that is, which can be delivered by continuous
currents with the same generator potential.
In Fig. 57 are shown, for the constants
E0 = 1000 volts,
Zg = 2.5 — 6/ ; that is, r, = 2.5 ohms, x0 — 6 ohms, z0 = 6.5 ohms,
with the current I0 as abscissae, the values —
RESISTANCE OF TRANSMISSION LINES.
87
E.M.F. at Receiver Circuit, E, (Curve I.) ;
Output of Transmission, P, (Curve II.) ;
Efficiency of Transmission, (Curve III.).
The same quantities, E and P, for a non-inductive line of
resistance, r0 = 2.5 ohms, x0 = 0, are shown in Curves IV.,
V., and VI.
SUPFUED'OVER INDUCTIVE LINE OF IMPEDAN
AND OVER NON-INDUCTIVE LII^E OF RESISTAr.
T0 = 2.5
CURVE 1. E. M. F. AT RECEIVER CIRCUIT, INDUCTIVE LI
3E
CE
SE
UK
100
90
80
70
CO
50
40
30
40
10
^^x-
---1
.
ii V. 11 ii ii ii NON-INDUCTIVE »
^
x'.
0
t
VI
"
"
sos
NDL
CTIV
/
•4
"?"
z
5
-S-
o
/
cr:
fc
/
5
0
/
o
^
/
|
co
/
jjj
0
/,
/*
*~^
IIMl'
m
+*
"^
!5^-
//
/
<>
,„,
m
^^
^
^
^^^
**as.
\
gpj
JQJ
/
^^
^^
<^
^~,
f^
\
B3
TOO
/
\
>>
/r
5
-~^.
jj^
300
^
Xs-
x
S
x
\
.-»i ) 1
"~ — .
no
/
\
\
\
40j
wo
/
s
x\
ai-r
.300
/
s
\\
L'O'
L'OO
/
\y
n&
100
1
cu
^RE
NT
N L
!NE
AMF
ERE
s
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10 20 30 40 50 60 70 80
Fig. 57. Non-inductive Receiver Circuit Supplied Over Inductive Line.
2.) Maximum Power Supplied over an Inductive Line.
60. If the receiver circuit contains the susceptance, b,
in addition to the conductance, g, its admittance can be
written thus : —
Then —
current,
Impressed E.M.F.,
/„ = E Y;
E0 = E + I0Z0 == E (1 + KZ0).
88 AL TERNA TING-CURRENT PHENOMENA.
Hence —
E.M.F. at receiver terminals,
1 + FZ0 (1 + r.g + x.S) - J (x.g - r.6)'
current,
or, in absolute values —
E.M.F. at receiver circuit,
V(l + r.f + x,bf + (x.g - r.
current,
= E J _ jr2 + ^2 _ .
° V (i + rog + Xoby + (Xog - r0t>y'
ratio of E.M.Fs. at receiver circuit and at generator circuit,
E 1
and the output in the receiver circuit is,
P=E*g= E?o?g.
61. a.) Dependence of the output upon the susceptance of
the receiver circuit.
At a given conductance, g, of the receiver circuit, its
output, P = E?a?g, is a maximum, if a2 is a maximum ; that
is, when —
/=!=(! + r.g + x.Vf + (x.g - r0b?
is a minimum.
The condition necessary is —
or, expanding, ,., ,N , ,N A
5'. *. (1 + rog + jf0^) - r0 (Xog - r0b} = 0.
Hence —
Susceptance of receiver circuit,
t=~^^)=~^= ~b°'
or b + b0 = 0,
RESISTANCE OF TRANSMISSION LINES. 89
that is, if the sum of the susceptances of line and of receiver
circuit equals zero.
Substituting this value, we get —
ratio of E.M.Fs. at maximum output,
E0 z0 (g
maximum output,
Pl = -
current,
E0Y E0 (g
E0(g-jb0}
og - x0b.} -J(r0b0
Io = E° V (1 + rog - Xob0? + (r0b0 + Xog)*>
and, expanding,
r = *
'
phase difference in receiver circuit,
tan « = * = - A .
^ A"
phase difference in generator circuit,
62. b.} Dependence of the output upon the conductance
of the receiver circuit.
At a given susceptance, ^, of the receiver circuit, its
output, P — Eo<?g, is a maximum, if —
dP dl\\
-r = 0, or — I - I = 0,
dg d^P]
)* + (Xog -
90 ALTERNATING-CURRENT PHENOMENA.
that is, expanding, —
C1 + r0g -f x0 b}2 + (Xog — r0by — 2g(r0 + r*g -f x*g) = 0 ;
or, expanding, —
Substituting this value in the equation for a, page 88,
we get -
ratio of E.M.Fs.,
power
As a function of the susceptance, b, this power becomes
a maximum for dP^j db = 0, that is, according to § 61, if —
*'--*„.
Substituting this value, we get —
£= — bt> g = So* y = y<n hence: Y= g-\- jb= g0 — jb0\
x = - x0 , r = r0 , z = z0, Z = r — Jx = r0 + jx0 ;
substituting this value, we get —
ratio of E.M.Fs., m .
power, ^m = i-2- ;
that is, the same as with a continuous-current circuit ; or,
in other words, the inductance of the line and of the receiver
circuit can be perfectly balanced in its effect upon the
output.
63. As a summary, we thus have :
The output delivered over an inductive line of impe-
RESISTANCE OF TRANSMISSION LINES. 91
dance, Z0 = r0 —jx0 , into a non-inductive receiver circuit, is
a maximum for the resistance, r = z0, or conductance, g =
y0 , of the receiver circuit, or —
2 (r. +
at the ratio of potentials,
With a receiver circuit of constant susceptance, b, the out-
put, as a function of the conductance, g, is a maximum for
the conductance, —
and is
EO ' y?
= 2(^+Vo)'
at the ratio of potentials,
With a receiver circuit of constant conductance, g, the
output, as a function of the susceptance, b, is a maximum
for the susceptance, b = — b0, and is
P=
tffe+JJ?'
at the ratio of potentials,
1
7o (£• + go) '
The maximum output which can be delivered over an in-
ductive line, as a function of the admittance or impedance
of the receiver circuit, takes place when Z = r0 -\-jx0, or
y=jTo~J6o> that is, when the resistance or conductance
of receiver circuit and line are equal, the reactance or sus-
ceptance of the receiver circuit and line, are equal but of
opposite sign, and is, P = E? / 4 r0 , or independent of the
reactances, but equal to the output of a continuous-current
92
AL TERN A TING-CURRENT PHENOMENA.
circuit of equal line resistance. The ratio of potentials is, in
this case, a = zo j 2 roi while in a continuous-current circuit
it is equal to £. The efficiency is equal to 50 per cent.
.03 .01 .05 .08 ,07 .08 .09 .10 .11 .12 .13 .14 J5 J6 33
Fig. 58. Variation of the Potential in Line at Different Loads.
64. As an instance, in Fig. 58 are shown, for the
constants —
E0 = 1000 volts, and Z0 = 2.5 — 6/; that is, for
r0 = 2.5 ohms, x0 = Gohms, z0 = 6.5 ohms,
and with the variable conductances as abscissae, the values
of the —
output, in Curve I., Curve III., and Curve V. ;
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