CHAPTER VI. 

TOPOGRAPHIC METHOD. 

33. In the representation of alternating sine waves by 
vectors in a polar diagram, a certain ambiguity exists, in so 
far as one and the same quantity — an E.M.F., for in- 
stance — can be represented by two vectors of opposite 
•direction, according as to whether the E.M.F. is considered 
as a part of the impressed E.M.F., or as a counter E.M.F. 
This is analogous to the distinction between action and 
reaction in mechanics. 


Fig. 25. 


Further, it is obvious that if in the circuit of a gener- 
ator, G (Fig. 25), the current flowing from terminal A over 
resistance R to terminal B, is represented by a vector 0/ 
(Fig. 26), or by /= i +ji\ the same current can be con- 
sidered as flowing in the opposite direction, from terminal 
B to terminal A in opposite phase, and therefore represented 
by a vector 0/^ (Fig. 26), or by /j = — / —ji'' 

Or, if the difference of potential from terminal B to 
terminal A is denoted by the E = e +jyf the difference 
lOf potential from A to B is E^ = — e —je'. 


44 


AL TERN A TING-CURRENT PHENOMENA. 


[§34 


Hence, in dealing with alternating-current sine waves, 
it is necessary to consider thorn in their proper direction 
with regard to the circuit. Especially in more complicated 
circuits, as interlinked polyphase systems, careful attention 
has to be paid to this point. 


Fig. 26, 


34. Let, for instance, in Fig. 27, an interlinked three- 
phase system be represented diagrammatically, and consist- 
ing of three E.M.Fs., of equal intensity, differing in phase 
by one-third of a period. Let the RM.Fs. in the direction 


Fig. 27. 

from the common connection O of the three branch circuits 
to the terminals A^, A^f A^, be represented by E-^^ E^, E^. 
Then the difference of potential from A^ to A^ is E^ — E^y 
since the two E.M.Fs., E^ and E^^ are connected in cir- 
cuit between the terminals A-^ and A^y in the direction. 


§34] TOPOGRAPHIC METHOD. 45 

A^ — O — A^\ that is, the one, E^^ in the direction OA^^ 
from the common connection to terminal, the other, E^ , in 
the opposite direction, A-fi^ from the terminal to common 
connection, and represented by — -fi*!. Conversely, the dif- 
ference of potential from A<^ to A^ is E<^ — E^, 

It is then convenient to go still a step farther, and 
drop, in the diagrammatic representation, the vector line 
altogether ; that is, denote the sine wave by a point only, 
the end of the corresponding vector. 

Looking at this from a different point of view, it means 
that we choose one point of the system — for instance, the 
common connection O — as a zero point, or point of zero 
potential, and represent the potentials of all the other points 
of the circuit by points in the diagram, such that their dis- 
tances from the zero point gives the intensity ; their ampli- 
tude the phase of the difference of potential of the respective 
point with regard to the zero point ; and their distance and 
amplitude with regard to other points of the diagram, their 
difference of potential from these points in intensity and 
phase. 


Thus, for example, in an interlinked three-phase system 
with three E.M.Fs. of equal intensity, and differing in phase 
by one-third of a period, we may choose the common con- 
nection of the star-connected generator as the zero point, 
and represent, in Fig. 28, one of the E.M.Fs., or the poten- 


46 


ALTERNATING-CURRENT PHENOMENA, 


[§35 


tial at one of the three-phase terminals, by point E^ The 
potentials at the two other terminals will then be given by 
the points E^ and £'3, which have the same distance from 
^ as -ffj , and are equidistant from E^ and from each other. 


Fig, 29. 


The difference of potential between any pair of termi- 
nals — for instance E^ and E^ — is then the distance E^E^^ 
or E^E^y according to the direction considered. 


Fig, 30. 


35. If, now, in Fig. 29, a current, /j, in phase with 
E.M.F., E^, passes through a circuit, the counter E.M.F. 
of resistance, r, is E^ = /r, in opposition to /^ or E^^ 


135] 


TOPOGRAPHIC METHOD. 


47 


and hence represented in the diagram by point £",, and 
its combination with E^ by E(. The counter E,M.F. of 
reactance, x, is E^ = Ix, 90' behind the current /j, or 
E.M.F., E^, and therefore represented by point E^, and 
giving, by its combination with E^, the terminal potential 
of the generator E^, which, as seen, is less than the 
E.M.F., £■,. 

If all the three branches are loaded equally by three 
currents flowing into a non-inductive circuit, and thus in 


phase with the E.M,Fs, at the generator terminals (repre- 
sented in the diagram. Fig. 30, by the points E-^, E^, E^, 
equidistant from each other, and equidistant from the zero 
point, O), the counter E.M.Fs. of resistance, fr, are repre- 
sented by the distances EE', as EyE.^, etc., in phase with 
the currents, /; and the counter E.M.Fs. of reactance, /^, 
are represented by the distance, E'E" in quadrature with 
the current, thereby giving, at the generator E.M.Fs., the 
points £",", Ej", E^. 

Thus, the triangle of generator E.M.Fs. E^'E^E^, pro- 
•duces, with equal load on the three branches and non- 


48 


ALTERXA TING-CURRENT PHENOMENA, 


[§3e 


inductive circuit, the equilateral triangle, E^E^E^^ of ter- 
minal potentials. 

If the load is inductive, and the currents, /, lag behind 
the terminal voltages, E^ by, say, 40°, we get the diagram 
shown in Fig. 31, which explains itself, and shows that the 
drop of potential in the generator is larger on an inductive 
load than on a non-inductive load. 

Conversely, if the currents lead the terminal E.M.Fs. 
by, say, 40°, as shown in Fig. 32, the drop of potential in 
the generator is less, or a rise may even take place. 


Flq. 32. 


36. If, however, only one branch of the three-phase 
circuit is loaded, as, for instance, E{E^^ the E.M.F. pro- 
ducing the current (Fig. 33), is E^E^ ; and, if the current 
lags 20°, it has the direction 01^ where 01 forms with 
E^E^ the angle «> — 20° ; that is, the current in E^ is 01^, 
and the current in E^ is Ot^ , the return current of 01^, 

Hence the potential at the first terminal is E^, as de- 
rived by combining with E{ the resistance E.M.F., E{E^^ 
in phase, and the reactance, E.M.F., E^E^, in quadrature, 
with the current; and in the same way, the E.M.F. at the 


536] TOPOGRAPHIC METHOD. 49 

second terminal is E^^ derived by the combination of E^ 
with E^E^ in phase, and E^E^ in quadrature, with the 
current. 

Hence the three terminal potentials are now, E^y E^, 
E^y and the differences of potential between the terminals 
of the generators are the sides of the triangle, E^E^E^\ 
or, in other words, the equilateral triangle of E.M.F., 
E{E^E^y produces at the generator terminals the triangle 
of voltages, E^E^E^^ whose three sides are unequal ; one, 
E^E^t or the loaded branch, being less than E^E^^ or the 
two unloaded branches. That is, the one has decreased, 
the other has increased, and the system has become un- 
balanced. 


Dufc- 


ii 


Fig. 33. 

In the same manner, if two branches, E^E^^ and 
Ei^E^f are loaded, and the third, E^E^, is unloaded, and 
the currents lag 20°, we find the current /g in E{ to be 
20° behind E{E^\ the current /^ in E^^ 20° behind E^^E^^, 
and the current I^ in E^y the common return of /^ and 
/g; by combining again the generator E.M.Fs., E^^ with 
the resistance E.M.Fs., E^E\ in phase, and the reactance 
E.M.Fs., E' E, in quadrature, with the respective currents, 
we get the terminal potentials, E. We thus see that the 
E.M.F. triangle, E^E^E^, is, by loading two branches, 
changed to an unbalanced triangle of terminal voltages, 
E^E^E^y as shown in Fig. 34. 


60 


ALTERNATING-CURRENT PHENOMENA. 


[§3T 


If all the three branches of the three-phase system are 
loaded equally, we see, from Fig. 31, that the system 
remains balanced. 

37. As another instance, we may consider the unbal- 
ancing of a two-phase system with a common return. 

If, in a two-phase system, we choose the potential of the 
common return at the generator terminal as zero, the poten- 
tials of the two outside terminals of the generator are repre- 


sented by -f*!** and E^^ at right angles to each other, and 
equidistant from (?, as shown in Fig. 35. 

Let, now, both branches be loaded equally by currents 
lagging 40**. Then, the currents in E^ and E^ are repre- 
sented by /j and /j, and their common return current by 
/g. If, now, these currents are sent over lines containing 
resistance and reactance, we get the potentials at the end 
of the line by combining the generator potentials E^^ E^, 
and Oy with the resistance E.M.Fs., E{E^, E^E^ and 
OE^, in phase with the currents, and with the reactance 
E.M.Fs., EyE^, E^E^y E^E^, in quadrature with the cur- 
rents ; and thereby derive as the potentials at the end of the 
line the points E^^ E^^ Eg , which form neither an isosceles 


§37] 


TOPOGRAPHIC METHOD, 


61 


nor a rectangular triangle. That is, the two-phase system 
with a common return becomes, even at equal distribution 
of load, unbalanced in intensity and in phase. 

These instances will be sufficient to explain the general 
method of topographic representation of alternating sine 
waves. 

It is obvious now, since the potential of every point of 


Fig. 35. 


the circuit is represented by a point in the topographic 
diagram, that the whole circuit will be represented by a 
closed figure, which may be called the topographic circuit 
characteristic. 

Such a characteristic is, for instance, OE^E-IE^E^E^E^^ 
in Figs. 31 to 34, etc. ; further instances are shown in the 
following chapters, as curved characteristics in the chapter 
on distributed capacity, etc. 


62 AL TERNA TJNG-CURRENT PHENOMENA. ^ [§ 38