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 

\ 

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. ; 

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 — 

/ 

\ 

/ 

\ 

/ 

\ 

/ 

\ 

/ 

\\ 

/ 

\\ 

/ 

I 

/ 

/ 

Ns 

f 

\ 

1 

/ 

\ 

\ 

/ 

\\ 

/ 

5 

/ 

\\ 

/ 

'/ 

\ 

\ 

/ 

7 

\° 

/ 

/ 

\ 

\ 

/ 

P 

\ 

* 

\ 

X 

-<, 

~^_ 

^ 

^« 

' — -. 

SUSCEfA 

°T' 

iECE 

IVE 

R C 

KCU 

IT 

-.3 -.2 -.1 0 +.1 +.2 +.3 +.4 

Fig. 59. Variation of Potential in Line at Various Loads. 

3.) Maximum Efficiency. 

65. 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. 

66. 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. 


67. 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. 

68. 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. 

69. 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 

0= I 

~ 

300 

• ' . 

DLTS 
1000 

uoo 

too 

roo 

GOO 
M) 
400 
300 
200 
100 
0 

~""~- 

\: 

~~~~-. 

-^. 

— 

— 

*-•, 

~— ^. 

* 

V 

nr 

\ 

x 

//' 

"^ 

-^ 

//s 

\ 

\ 

"*x- 

^^ 

\ 

x 

2 

S 

/^ 

A 

1 

, 

s 

^- 

^ 

^T 

) 

^S 

^~ 

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•*^=: 

— 

— 

— 

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^ 

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•*fc 

x 

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— - 

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—- 

, — • 

01 

r?v 

T IN 

RE 

iEIV 

x c 

RC 

IT, 

<ILO 

.VAT 

TS 

30 W 50 CO 70 80 

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. 


\ 

s 

\ 

\ 

VOLT 

^ 

\ 

\ 

\ 

\ 

1SOO 
1700 
1COO 
1500 
-1400 

\ 

\ 

\\ 

\ 

\ 

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 

\ 

IN 

\ 

IVE 

1200 

1100 
1000 

l\ 

> 

s 1 

GOO 
800 
700 
COO 

* 

"-^ 

•^ 

SEC 

JFF 

C1EN_ 

*-. 

-* 

fa 

"N 

^ 

^ 

/^ 

5 

L 

•\ 

// 

tV 

i 

/ 

^ 

\ 

8 

/f 

// 

300 
200 
100 

A 

^ 

' ., 

/\v 

^ 

tc)P 

^> 

4 

/ 

^ — 

*.ovJS 

^"\ 
PUT 

PUT 

K.W 

0 ' 

i) i 

it 

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.