CHAPTER VIII. 

<?IBCniTS CONTAININa RESISTANCX:, INDUCTANCX:, AND 

CAPACITY. 

42. Having, in the foregoing, reestablished Ohm*s law 

and Kirchhoff' s laws as being also the fundamental laws 

of alternating-current circuits, or, as expressed in their com- 

plexform. ^ _^ ' „_ 

E -= ZJ^ or, / = \Ey 

and S-f = in a closed circuit, 

5/ = at a distributing point, 

where J?, /, Z^ V, are the expressions of E.M.F*., current, 
impedance, and admittance in complex quantities, — these 
laws representing not only the intensity, but also the phase, 
of the alternating wave, — we can now — by application of 
these laws, and in the same manner as with continuous- 
current circuits, keeping in mind, however, that E, /, Z, V, 
are complex quantities — calculate alternating-current cir- 
cuits and networks of circuits containing resistance, induc- 
tance, and capacity in any combination, without meeting 
with greater difficulties than when dealing with continuous- 
current circuits. 

It is obviously not possible to discuss with any com- 
pleteness all the infinite varieties of combinations of resis- 
tance, inductance, and capacity which can be imagined, and 
which may exist, in a system or network of circuits ; there- 
fore only some of the more common combinations will here 
be considered. 

1.) R I sis fa nee in scries with a circuit, 

43. In a constant-potential system with impressed 
E.M.F., ^ 


§43] KESISTANCEy INDUCTANCE, CAPACITY, 69 

let the receiving circuit of impedance 


if = /• —jx^ z = V/"*"^ + xS 

be connected in series with a resistance, r^ . 
The total impedance of the circuit is then 

hence the current is 

/= ^o = ^o = ^oia+j'o+Jx) , 
Z + r^ r+ f\, —jx (/- + r^)^ + x^ * 

.and the E.M.F. of the receiving circuit, becomes 

r + to —jx \r + /-„)*-» + x^ 

•or, in absolute values we have the following : — 
Impressed E.M.F., 

current, 


V(r + r,y + x^ -y/z^ + 2 rr, + r,- 
E.M.F. at terminals of receiver circuit, 

£ s 


{r + r^f + X' V^^ + 2 rr, + r,^ 
difference of phase in receiver circuit. 


r 
tan 01 = - ; 

X 


difference of phase in supply circuit. 


tan ta^ = — ■ — ■ 


o 

X 


a.) If X is negligible with respect to r, as in a non-induc- 
tive receiving circuit. 


and the current and E.M.F. at receiver terminals decrease 
steadily with increasing r^ . 


Z' 


60 AL TERNA TIXG-CUKRENT PHENOMENA, [§ 44 

b.) If r is negligible compared with x^ as in a wattless 
receiver circuit, 


/ ^' , 


E — E ^ 


Vr.-' + A:'' 


V/o^ + a:^ 


or, for small values of r„ , 


1 


E-E.; 


that is, the current and E.M.F. at receiver terminals remain 
approximately constant for small values of r^ , and then de- 
crease with increasing rapidity. 

44. In the general equations, x appears in the expres- 
sions for / and E only as x^y so that / and E assume the 
same value when x is negative, as when x is positive ; or, in 
other words, series resistance acts upon a circuit with leading 
current, or in a condenser circuit, in the same way as upon a 
circuit with lagging current, or an inductive circuit. 

For a given impedance, s^ of the receiver circuit, the cur- 
rent /, and E.M.F., E^ are smaller, as r is larger; that is, 
the less the difference of phase in the receiver circuit. 

As an instance, in Fig. 37 are shown in dotted lines the 
current, /, and the E.M.F., E^ at the receiver circuit, for 
E^ = const. = 100 volts, -cr = 1 ohm, and — 

^.) r^ = .2 ohm (Curve I.) 
d.) r, = .8 ohm (Curve II.) 


with values of reactance, x= y/a^ — r^, for abscissae, from 
X = + 1.0 to ;r = + 1.0 ohm. 

As shown, / and E are smallest for ;r = 0, r = 1.0, 
or for the non-inductive receiver circuit, and largest for 
X = ± 1.0, r = 0, or for the wattless circuit, in which latter 
a series resistance causes but a very small drop of potential. 

Hence the control of a circuit by series resistance de- 
pends upon the difference of phase in the circuit. 

For r^ = .8 and x = 0,x = + .8, and x = — .8, the polar 
diagrams are shown in Figs. 37, 38, 39. 


1451 RESISTANCE, INDUCTANCE, CAPACITY. 


2.) Reactance in series with a circuit. 
4p. In a constant potential system of impressed E.M.F., 


let a reactance, x^ , be connected in series in a receiver cir- 
cuit of impedance 

Z=r—jx, s = Vr*^fx*. 


>u 


fM 


aat 


E. 


CO 


sr 


NT 


E. 


100 


" 


■'^ 


^ 


1 


r. 


2 


_ 


' 


c 


" 


■: 


_^ 


-J 


s 


— 


1 


— ' 


' 


e 


s" 


rT, 


ION 


3E« 


OM 


,, 


'■ 


r. 


„ 


.J..I. 


, 


„ 


.1 .= 


.3 


, 


■ J 


., 


,, 


... -„| 


FIf. 37. Varlatlan of Vottaii at Oomtant 8*r/w ffn/ttonc* w/M Mow 8«(atlM ef 


Then, the total impedance of the circuit is 

Z + x^ = r-j(x+x.). 


m. 


fig. 38. 
and the current is, 


while the difference of potential at the receiver terminals 


62 AL TERN A TING-CURRENT PHENOMENA, [§ 46 

Or, in absolute quantities : — 
Current, 


V/"-* + {x + x,f y/'z '' + L> XX, + x,^ 
E.M.F. at receiver terminals, 

I 

V r'+i,x + x,y v^-* + 2xx, + x^'' 

difference of phase in receiver circuit, 

r 
tan 01 = - ; 

X 

difference of phase in supply circuit, 

r 
tan o)^ = — I . 

a.) If X is small compared with r, that is, if the receiver 
circuit is non-inductive, / and £ change very little for small 
values of x^ ; but if x is large, that is, if the receiver circuit 
is of large reactance, / and £ change much with a change 

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 < ;?, or 
E > E^\ that is, the reactance, x^ , raises the potential. 

c.) E = E^f or the insertion of a series inductance, ;r^, 
does not affect the potential difference at the receiver ter- 
minals, if 

V^* -f 2 XX, + xj' = z ; 

or, x,= — 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 = E^y I^E^j z. If x^ > — 2,t-, it raises, if ;r < — 2 jr, 
it lowers, the voltage. 

We see, then, that a reactance inserted in series in 
an alternating-current circuit will lower the voltage at the 


«45] 


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 ;r = 0, that is, if the receiver circuit is non- 
inductive, the E.M.F*. at receiver terminals is : 

Ei^r Ea 


E== 


V^ + 


' v/i+(^J 


= (1 -j- x)~i expanded by the binomial theorem is 

(1 + ;c)» = 1 + «x + " (^^ - ^^ X* + . . .y 

Therefore, if x^ is small compared with r : — 

1/*, ^» 


2v r 


E.-E 


) 


-E _ v-t-.y 


That is, the percentage drop of potential by the insertion 
of reactance in series in a non-inductive circuit is, for small 


Fiq. 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. 


tS46 


46. As an instance, in Fig. 41 the changes of current, 
/, and of E.M.F. at receiver tenninals, E, at constant im- 
pressed E.M.F., Eg, are shown for various conditions of a 
receiver circuit and amounts of reactance inserted in series. 

Fig. 41 gives for various values of reactance, x^ (if posi- 
tive, inductance — if negative, condensance), the E.M.Fs,, 
E, at receiver terminals, for constant impressed E.IVI.F., 


i,.,.....6,.k,i,.i.ri..,..u,^ "YlVTTi I I I I 


ii-;"."!"!'. /"^ ' ' ^^ 


f 3 "^ ?: 5 


J. L > ' ^ i„ 


t ^\^Jl 3 


I us/ 


T yr 


/- ^-^ AK'"-^ 


»- J^' ^/-j CS^ 


f" y^ ,^ J^ ' ^^N 


K ^'^ y' ^^ _, ^^ ^^ 


°« -<^ -'' >-^ J ^ 


!«•' ---'■"'' 


— ^ , , 


11 


U tit i% U ii t'l >a 11 It u a i'p .i .j j t J -J .' , J Lo Lj-A 


£■„ = 100 volts, and the foilowing conditions of receiver 

■^'^^^'^^^ ,.,.0.,= 1.0,..= 0(CurveI.) 
e = 1.0, r= .6,:.:= .8(CurveII.) 
s=1.0, r= .6. *=-.8(Curve III.) 
As seen, curve / is symmetrical, and with increasing x^ 
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 
o£f very much faster than in a non-inductive circuit. 


846] 


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 jr^ = i .8, or, ;r^ = — x (the 
condition of resonance), and the E.M.F. reaches the value, 
E = 167 volts, or, E = E^zj r. This rise of potential by 
series reactance continues up to ;r^ = ± 1.6, or, ;r = — 2;r, 


Rg, 42, 

where J? = 100 volts again ; and for x^> 1.6 the voltage 
drops again. 

At ;r^ = ± .8, ;r = ^ .8, the total impedance of the circuit 
is r - y (.r + ;r^) = r = .6, x + x^ = 0, and tan w^ = ; 
that is, the current and E.M.F. in the supply circuit are 
in phase with each other, or the circuit is in electrical 
resofiafice. 


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 — 

£, = 100, ^. = .6, ^ = (Fig. 42) £ = 85.7 

a: = + .4 (Fig. 43) £ = 73.7 

x= -A (Fig. 44) ^ = 106.6 


66 


ALTERNA TIXG-CURRENT PHENOMENA. 


[§47 


47. In Fig. 45 the dependence of the potential, Ey upon 
the difference of phase, w, in the receiver circuit is shown 
for the constant impressed E.M.F*., E^ = 100 ; for the con- 
stant receiver impedance, -s: = 1.0 (but of various phase 
differences w), and for various series reactances, as follows : 


jr«= :i 


•^0 = 


-^0 = 

X^ = 


2 

.6 

.8 

1.0 

1.6 

3.2 


(Curve I.) 
(Curve II.) 
(Curve III.) 
(Curve IV.) 
(Curve V.) 
(Curve VI.) 


Fig, 44, 


Since s = 1.0, the current, /, in all these diagrams has 
the same value as E, 

In Figs. 40 and 47, the same curves are plotted as in 
Fig. 45, but in ¥\g. 46 with the reactance, x, of the receiver 
circuit as abscissae ; and in Fig. 47 with the resistance, r, of 
the receiver circuit as abscissae. 

As shown, the receiver voltage, E, is always lowest when 
x^ and X are of the same sign, and highest when they are 
of opposite sign. 

The rise of voltage due to the balance of x^ and x is a 
maximum for x^ = +10, x = — 1.0, and r = 0, where 


S47] J!ESISTA/fCE, IXDUCTAXCE, CAPACITY. 61 


( 


] 


i /i^ 


lu 


£HEa5£ 


fn 


rn 


"il 


hln 


m-^ 


'"i 


^_ 


i^# 


t 


^ii 


'I' 


I.O 


II 


I. 


.8 


VI, 


3.3 


/ 


//I 


' 


/' 


/ 


— 


.^^^/i '7 


J, 


,,-r-' 


'y1^/ 1 / 


i 


J- 


— 


"T^. ; _i/ 


<y/i 


/ 


i 


L--\^ 


/i 


/ 


i- 


b^ 


t23 


— - 


r^ 


y 


I 


Jl 


^ 


3 


" r L' 


^ 


4-^ 


f_ 


J 


, 


— 


- 


1 


- 


- 


4^ 


= — 


- 


p 


- 


- 


- 


- 


— 


— 1~ 


"" 


^ 


' 


ffcr 


~" 


" 


1 


~ 


~ 


~- 


~ 


- 


r 


^-^y^^ 


1 


_ 


I'TTTT 


_ 


'.A^ 


u KtaOaxBa w/U Mom itflgte of 


■^' T^ T 


X ^E 


?i"S^ 


iii,j...e vi,|t.-s.-a 


? J 


ft- t 


j'c^'^'2- 


^J-:.-^/ / 


i, . ^ — 1 — 1 — 


\^Jy^ J 


ZX-~^'~ J^ 


"---'C/ / 


„ j.^**— -L-^ ^.._-4^ 


-^ y 


-""iTj^ -"^"^ "^ ^^ 


_-— __jj^ii- __ 


---' 


LjiisiS, — — 


__ ^ 


' r"^ jj^_ 


— Li— - 


."-I tis 


c=|s.yai«p.J ' 


r. ., -s .. -5 L, -, ..Lw. 


^ 4(L foriatroB o/ yoltagt at Conianl Serl. 


AL TKKA'A TIXG-Cl'KKENT PlIEXOMKA 


(§48 


^ = 00; that is, absolute resonance takes place. Obvi- 
ously, this condition cannot be completely reached in 
practice. 

It is interesting to note, from Fig. 4T, 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 


1 


1 


//' 


ir 


■TSf 


'n'l 


m 


BT( 


^3^ 


, 


.w 


1. 


F--r 


— 


.0 


"t 


h 


/ 


Ill, 


•■'^ 


■8 


■— 


..» 


1 


/ 


// 


/ 


//! J, 


/ 


^ 


— 1 


— 


/JM- 


/ 


1 


> 


/ 


' 


ki 


;i. 


i 


- 


/ 


1 


y. 


' 1/ 


— i — -Tv"'jl^Ia.-^3--^r^ 


/ 


1 


«- 


r.U 


^ 


__, 


_ 


_ 


_ 


-' 


1 


- 


T ■ 


r 


L» 


c 


RR 


NT 


' 


. .I> 


. 


., 


,■ 


, 


, 


1 


np. n. VarbUha of Voltag* at Comtanl Strin Rtactatiet aHh Hmtlrtanem s/ 


factor of cos S> = -ft, gives a drop several times larger than 
a non-inductive circuit, and hence must be considered as a 
highly inductive circuit. 

3.) Impcilatue in sirirs with a circuit. 
48. Hy 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 hcrenftcr, can be repre- 
sented as an effective resistance. 


S 40] RESISTANCE, INDUCTANCE, CAPACITY. 69 

Hence the impedance of a reactive coil (choking coil) 
may be written thus : — 

where r^ is in general small compared with x^ , 
From this, if the impressed E.M.F. is 


and the impedance of the consumer circuit is 


we get the current, / — ^ "*,^ = ^ 

^ Z + Z, (r + r,)-y(^ + ;c,) 

and the E.M.F. at receiver terminals, 

?__ - /r r — jx 


E^E.-^^—^E 


Z + Z. "(r + rJ-Xx + jr.) 

Or, in absolute quantities, 
the current is, 


V(r + r,)2 + (^ + ;i:„)*^ V^* + ^o' + 2 (rr, + .v;r,) ' 
the KM.F. at receiver terminals is, 


V(r + r^ + (x+ x^y ^z' + zJ^ + 2 (rr, + xx;) 
the difference of phase in receiver circuit is, 

r 


tan <o = 

X 


and the difference of phase in the supply circuit is, 


tan <o„ = 


X +jr 


49. In this case, the maximum drop of potential will not 
take place for either ^ = 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 /: is a maximum ; or, 
since ^, ^^, r^^ x^ are constant if rr^ -\- xx^ is a maximum, 
since x = y/ z^ — r^ if rr^ + ^o ^^^ "~ ^'^ ^s a maximum. 


70 


AL TEA'A'A TIXG CUKREXT-FilENQMEA'A. 


[§4& 


a function, / = rr„ -j- .r„ V^ — r'^ is a maximum when 
its differential coefTicient equals zero. For, plotting f as 
curve with r as abscissa;, at the point where / is a maxi- 
mum or a mhiimum, this curve is for a short distance 
horizontal, hence the tangens-fiinction of its tangent equals 
zero. The tangens-f unction of the tangent of a curve, how- 
ever, is the ratio of the change of ordinates to the change 
of abscissje, or is the differential coefficient of the func- 
tion represented by the curve. 


o. J 


- t 


I J 


^^^ 


,^ _^ia:^:=:t 


"— Tr -/ 


m — r 


3— 


Fig. 48. 

Thus we have : — 

/ = rr,-\-x, Vs* — r* = maximum or minimum, if 

Differentiating, we get : — 


or, expanded, - 


§60] 


RESISTANCE, INDUCTANCE, CAPACITY. 


71 


That is, the drop of potential is a maximum, if the re* 
actance factor, x jvy of the receiver circuit equals the reac- 
tance factor, x^l r^y of the series impedance. 


»r 1,. 


_I 


,. 1'' 


4,E 


AE. 


'.. 


\ 


^ 


^ 


1 


«• 


n\f. 40. 


Fig. 60. 


E. 


50. As an example, Fig. 48 shows the E.M.F., E, 
at the receiver terminals, at a constant impressed E.M.F., 
E^ = 100, a constant impedance of ,the receiver circuit, 
s = 1.0, and constant series impedances, 

Z = .3-y.4 (Curve I.) 

Z, = 1.2 — y 1.6 (Curve II.) 

as functions of the reactance, x, of the receiver circuit. 


F/g. 61. 


Figs. 49 to 51 give the polar diagram for E^ = 100,. 
X = .9, X = Oy X = — .9, and Z^ = .3 —j .4. 


72 


ALTERNATING-CURRENT PHENOMENA. 


[§61 


4.) Campettsation for Lagging Currents by Shunted 
Condensance, 

51. 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 


X* 


Fig, 62. 


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 IC.M.F., and therefore propor- 
tional to the power expended. 


S52] RESISTANCE, INDUCTANCE, CAPACITY, 7S 

In a circuit shown diagrammatically in Fig. 52, let the 
non-inductive receiver circuit of resistance, r, be connected 
in series with the inductance, x^ , and the whole shunted by 
a condenser of condensance, f, entailing but a negligible loss 
of energy. 

Then, if E^ = impressed E.M.F., — 

the current in receiver circuit is, 

T ^o T ^9 . 

* . I '' ,— ■— 


the current in condenser circuit is, 

7 -^0 T ^o 

JC c 

and the total current is 

/. = / + A = E. |-J-^ + ^ 


= £o 


\r^+x.'^\r^+x,' Or 


or, in absolute terms, fo^ E^K^ { — -^ ■ ) + ( ■ 

^ \^+xjy V 


^ 1 V; 


r' + x,^ c 
while the E.M.F. at receiver terminals is, 

E^ Ir = E. "^ . , E= ^"^ 


52. The main current, /^, is. in phase with the impressed 
E.M.F., E^y or the lagging current is completely balanced, 
or supplied by, the condensance, if the imaginary term in 
the expression of /^ disappears ; that is, if 


This gives, expanded : c = 


X 


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, x^ being constant. 


74 ALTERNA TING-CURRENT PHENOMENA, [§ 62 

with increasing resistance, r, the condensance has to be 
increased, or the capacity decreased, to keep the balance. 

Substituting c = — lj_5. ^ 

we get, as the equations of the inductive circuit balanced 
by condensance : — 

r^ -Y Xo 

'" f' + xy 

r-JXo' V>' + V 

and for the power expended in the receiver circuit : — 


/= 


Eo 


Vr« + X* 


/l = 


■^'oXq 


/« = 


P-or . 


^ + V 


E = 


E„r 


that is, the main current is proportional to the expenditure 
of power. 

For r = we have x = x^, or the condition of balance. 

Complete balance of the lagging component of current 
by shunted capacity thus requires that the condensance, c, 
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., 
E^ = 1000 volts, and a constant series reactance, x^ = 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. 


(63] SES/STAWCE. INDUCTAXCE, CAPACITY. 75 


'e 


M 


t^^ E.;..r: 00 


■IST 


'"'.£ 


-1 


OO'VO^TB 


- 


- 


- 


- 


ll'. CUIIHENT IN CONOenSES OIIOu'lT, 
IV.' e!ii.F. «T HEOEl'vEROIROljiT. 


«|M 


^ 


V 


IV 


^ 


— 


" 


s 


. 


■ 


^ 


^ 


1 


y 


1, 


•^ 


■^ 


^ 


- 


'^ 


~ 


"— 


' 


-- 


1 


J~ 


^ 


., 


/ 


-- 


/ 


■~ 


/ 


HL 


,s.^. 


-■' 


Ul 


't 


„.,. 


IHl 


'i' 


UK 


Fig. S3. CompmaatlBii of lagging CurrtnU fa Sttleliit CIrailt bg Varlatli StHmtit 
Condantana. 

53. If. however, the condcnsance is left unchanged, 
c = j-„ at the no-load value, so that if the circuit is balanced 
for r = 0, it will be overbalanced for r > 0, and the main 
current will become leading. 

We get in this case : — 


/„=/+/,- 


Xoi'.+jr) 


, A=- 


£=/r= 


E^t 


E = 


E„r 


The di£ference of phase in the main circuit is, - 
tan So = - — = 


76 


ALTERNATING-CVRKENT PHENOMENA. 


[554 


when r = 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, /, is approximately con- 
stant for small values of r, and then gradually decreases. 


Mf 


IS 


EO 


r. 


COH 


■HT 


E. 


-10. 


X> 


DL 


B, 


BMUNTID COMOENaUHCE COBSTIBT. C- 100 OMy* 
»*HI«BLEI<EBiaT*«CElN BECEIV EH CIRCUIT. 

IILOUnniNT IN HIIN eiHCUIT. 
IV.t.M.f . iT RECEIVER CIRCUIT. 


HKI 


,". 


, 


— 


■^ 


_ 


•m, 


~- 


^ 


"" 


I 


^ 


IV 


^ 


■^ 


'i. 


-- 


_ 


_ 


/. 


■nn 


/ 


/ 


REBISTANCir— OF RECEIVER CIRCUIT, OKMS. 


\A 


J— 


— 


-UUiUA14A.-iir 


rr 


Iri 


_ 


In Fig. 54 are shown the values of /, /j, /„,£', in Curves 
I., II., III., IV.. similarly as in Fig. 50, for E^ = 1000 volts, 
c x= X, = 100 ohms, and r as abscissas. 


5.) CoMstant Potential — Constant Current Transformation. 

54. In a con.stant potential circuit containing a large 
and constant reactance, .\\ , and a varying resistance, r, the 
current is appro.ximatcly 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 conden.sance. Leaving, however, the condensance 
constant, c = x,, so as to balance the lagging current at no 


§64] 


KES/STANCE, 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., E^, is kept constant. Hence, to 
keep the current in the receiver circuit entirely constant, the 
impressed E.M.F., E^^ 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^y inserted in the main circuit, as shown in 
the diagram. Fig. 55, can be used for raising the potential, 
E^y with increasing load. 


Fig. 55. 


Let — 


^a = ^2 + jWy Eo = V^s^ + e^\ 


be the impressed E.M.F. of the generator, or of the mains, 
and let the condensance be x^=^ x^\ then — 
Current in receiver circuit, 


r—JXo 
current in condenser circuit, 

Ji — . 

Hence, the total current in main line is 


/. = /+/. =^.1—^7- -^ 


\r- 


JX^ x^ 


£,r 


Jx.(r—jx,)' 


/ 


^ 


78 AL TERN A TING-CURRENT PHENOMENA. [ § 64 

and the E.M.F. at receiver terminals, 

E.M.F. at condenser terminals, 
E.M.F. consumed in main line, 

hence, the E.M.F. at generator is 


E^^E. + E'^E. \ 1- 


r 


2 


\ Xo(r—jx,) ) ' 


or =E ^(^o-'^ i)--/V) . 

and the E.M.F. at condenser terminals, 

F — ^2X o(r —j\ ,x) . 

current in receiver circuit, 


2 


r —JXo r(x^ — Ao) —jx^ 

This value of / contains the resistance, r, only as a fac- 
tor to the difference, x^ — x^\ hence, if the reactance, x^f 
is chosen = x^y r cancels altogether, and we find that if 
x^ = x^y the current in the receiver circuit is constant, 

and is independent of the resistance, r ; that is, of the load. 

Thus, by substituting, we have, x^= x^y- 
Impressed E.M.F. at generator, 


-£"2 = ^2 + y^a'j -^2 = V^^a'* + <'a' * = constant ; 

current in receiver circuit, 

/ = y — ^ , / =z -^ = constant ; 

•*o Xq 

E.M.F. at receiver circuit, 

E =/r=y — ^, E = — ^, or proportional to load r; 


§55] I^ES/S7\4.VCK, IXDUCTANCK, CAPACITY. 79 

E.M.F. at condenser terminals, 

= E^ U +j ^\ ^,= E^\J\ + f^\\ hence > £^ ; 
current in condenser circuit, 

main current, 


/r ( proportional to the load, 

= — ^ » /, = — ~- , < /-, and in phase with 

■^^ "^^ ( E.M.F., E^ . 

The power of the receiver circuit is. 


3 


IE ="'^ 
the power of the main circuit. 


•^0 


E ^ f 
I^E^ = ^^ , hence the same. 

X 

55. This arrangement is entirely reversible ; that is, 

if E^ = constant, I = constant ; and 
if To = constant, E — constant. 

In the latter case we have, by expressing all the quanti- 
ties by /„ : — 
Current in main line, 

/„ = constant; 
E.M.F. at receiver circuit, 

E = /^jr^ =5 constant ; 
current in receiver circuit, 

X 1 

/ =/,— , proportional to the load -; 

r r 

current in condenser circuit, 


80 ALTERNATING-CURRENT PHENOMENA, [§65 

E.M.F. at condenser terminals, 


E.^x.I.\J\+h^\ 


Impressed E.M.F. at generator terminals, 

x^ 1 

E^ = -2-7^ , or proportional to the load - . 

r r • 

From the above we have the following deduction : 

Connecting two reactances of equal value, x^, 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, x^ = •*'o» 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 F^ig. 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, / = 10, we have : x^ = 100 ; hence 
the constants of the circuit are : — 

£^ = 1000 volts ; 

/ = 10 amperes; 

£ = 1 Or, plotted as Curve I., with the resistances, r, as abscissae; 

E, = 1000 1/ 1 + ( -I- ]' plotted as Curve II. ; 


\ = lOy 1 + (t^Y, plotted as Curve III.- 


/o = .1 r, plotted as Curve IV. 


I6ei 


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. 


cj 


's;. 


EC 


1" 


hoIbc 


IT 


oAtJ 


"T 


HP 


«E- 


! HEMTuictB Hit -ro<5 difilia 'ii.au, shunteo in their 


i-i^i^i-li^iiiSi^ 


LOSS 


■•-■ 


_^ 


■■ t 


ill 


— - 


r^ 


, 


1 


VI 


,'- 


-" 


( 


— ' 


-^ 


r' 


; 


™ 


/ 


-1 


,.- 


^ 


. 


>i. 


•- 


^ 


x 


i™ 


M 


.- 


•; 


lEB 


NC 


- 


EO 


VE 


10 


tc 


(T, 


3 


n 


t 


_ 


L 


n 


_ 


r" 


_ 


BT 


^ 


vJ 


ns 


flj. 5a Comfoirt-AMnltoi — eDBrtant-CBfMn* r™iMf«r«rtloji. 

Let — 

ri = 2 ohms = effective resistance of condensance ; 

r, = 3 ohms = effective resistance of each of the inductances. 

We then have : — 

Power consumed in condensance, /i' r, = 200 + ,02 r* ; 
power consumed by first inductance, /* r, = 300 ; 
power consumed by second inductance, /," r, = .03 ^. 
Hence, the total loss of energy is 500 -(- .05 H ; 
output of system, /' r = 100 r 

input, 500 + 100 /• + .05 r'; 

a ■ 100 r 

^^"'^'^^y- 500 + 100r + .05H- 

It follows that the main current, /„, increases slightly 
by the amount necessary to supply the losses of energy 
in the apparatus. 


82 AL TERN A TING-CURRENT PHENOMENA. [ § 66 

This curve of current, /^, including losses in transforma- 
tion, is shown in dotted lines as Curve V. in Fig. 50 ; and 
the efficiency is shown in broken line, as Curve VI. As 
shown, the efficiency is practically constant within a \\iclc 
range. 


§67] RESISTANCE OE TRANSMISSION LINES. 83