CHAPTER IV. 

GRAPHIC REPRESENTATION. 

14. While alternating waves can be, and frequently are, 
represented graphically in rectangular coordinates, with the 
time as abscissae, and the instantaneous values of the wave 
as ordinates, the best insight with regard to the mutual 
relation of different alternate waves is given by their repre- 
sentation in polar coordinates, with the time as an angle or 
the amplitude, — one complete period being represented by 
one revolution, — and the instantaneous values as radii 
vectores. 


Fig. 8. 


Thus the two waves of Figs. 2 and 3 are represented in 
polar coordinates in Figs. 8 and 9 as closed characteristic 
curves, which, by their intersection with the radius vector, 
give the instantaneous value of the wave, corresponding to 
the time represented by the amplitude of the radius vector. 

These instantaneous values are positive if in the direction 
of the radius vector, and negative if in opposition. Hence 
the two half-waves in Fig. 2 are represented by the same 


20 


ALTERNA TING-CURRENT PHENOMENA. 


polar characteristic curve, which is traversed by the point of 
intersection of the radius vector twice per period, — once 
in the direction of the vector, giving the positive half-wave, 


Fig. 9. B, Fig. 10. 

and once in opposition to the vector, giving the negative 
half-wave. In Figs. 3 and 9, where the two half-waves are 
different, they give different polar characteristics. 

15. The sine wave, Fig. 1, is represented in polar 
coordinates by one circle, as shown in Fig. 10. The 
diameter of the characteristic curve of the sine wave, 
1= OC, represents the intensity of the wave ; and the am- 
plitude of the diameter, OC, /_& = AOC, is thefl/iase of the 
wave, which, therefore, is represented analytically by the 

function : — 

t = /cos (<£ — w), 

where </> = 2 IT / / T is the instantaneous value of the ampli- 
tude corresponding to the instantaneous value, 2, of the wave. 

The instantaneous values are cut out on the movable ra- 
dius vector by its intersection with the characteristic circle. 
Thus, for instance, at the amplitude AOBl = ^ = 2 ^ / T 
(Fig. 10), the instantaneous value is OB' ; at the amplitude 
AO£2 = <f>2 = 27T/2/ T, the instantaneous value is ~OJ3", and 
negative, since in opposition to the radius vector OBZ. 

The characteristic circle of the alternating sine wave is 
determined by the length of its diameter — the intensity 
of the wave ; and by the amplitude of the diameter — the 
phase of the wave. 


GRAPHIC REPRESENTATION. 21 

Hence, wherever the integral value of the wave is con- 
sidered alone, and not the instantaneous values, the charac- 
teristic circle may be omitted altogether, and the wave 
represented in intensity and in phase by the diameter of 
the characteristic circle. 

Thus, in polar coordinates, the alternate wave is repre- 
sented in intensity and phase by the length and direction of 
a vector, OC, Fig. 10, and its analytical expression would 
then be c = OC cos (<f> — w). 

Instead of the maximum value of the wave, the effective 
value, or square root of mean square, may be used as the 
vector, which is more convenient ; and the maximum value 
is then V2 times the vector OC, so that the instantaneous 
values, when taken from the diagram, have to be increased 
by the factor V2. 

Thus the wave, 

l> = £ cos 

= B cos (</> - fy 
is in Fig. 10# represented by 

T) 

vector OB = — , of phase 

A OB = G! ; and the wave, 
c= Ccos 


is in Fig. 10# represented by vector OC=—j=, of phase 
AOC= -£* 

The former is said to lag by angle ^, the latter to lead 
by angle £2, with regard to the zero position. 

The wave b lags by angle (o^ + £2) behind wave c, or c 
leads b by angle (wx + £2). 

16. To combine different sine waves, their graphical rep- 
resentations, or vectors, are combined by the parallelogram 
law. 

If, for instance, two sine waves, OB and OC (Fig. 11), 
are superposed, — as, for instance, two E.M.F's. acting in 
the same circuit, — their resultant wave is represented by 


22 


ALTERNATING-CURRENT PHENOMEA?A. 


OD, the diagonal of a parallelogram with OB and OC as 
sides. 

For at any time, /, represented by angle <f> = AOX, the 
instantaneous values of the three waves, OB, OC, OD, are 
their projections upon OX, and the sum of the projections 
of OB and OC is equal to the projection of OD ; that is, the 
instantaneous values of the wave OD are equal to the sum 
of the instantaneous values of waves OB and OC. 

From the foregoing considerations we have the con- 
clusions : 

The sine wave is represented graphically in polar coordi- 
nates by a vector, which by its length, OC, denotes the in- 


Fig. 11. 


tensity, and by its amplitude, AOC, the phase, of the sine 
wave. 

Sine waves are combined or resolved graphically, in polar 
coordinates, by the law of parallelogram or tJie polygon of 
sine waves. 

Kirchhoff's laws now assume, for alternating sine waves, 
the form : — 

a.) The resultant of all the E.M.Fs. in a closed circuit, 
as found by the parallelogram of sine waves, is zero if 
the counter E.M.Fs. of resistance and of reactance are 
included. 

b.} The resultant of all the currents flowing towards a 


GRAPHIC REPRESENTATION. 


23 


distributing point, as found by the parallelogram of sine 
waves, is zero. 

The energy equation expressed graphically is as follows : 
The power of an alternating-current circuit is repre- 
sented in polar coordinates by the product of the current , 
/, into the projection of the E.M.F., E, upon the current, or 
by the E.M.F., E, into the projection of the current, /, upon 
the E.M.F., or by IE cos 


17. Suppose, as an instance, that over a line having the 
resistance, r, and the reactance, x = ZirNL, — where N = 
frequency and L = inductance, — a current of / amperes 
be sent into a non-inductive circuit at an E.M.F. of E 


Fig. 12. 

volts. What will be the E.M.F. required at the generator 
end of the line ? 

In the polar diagram, Fig. 12, let the phase of the cur- 
rent be assumed as the initial or zero line, Of. Since the 
receiving circuit is non-inductive, the current is in phase 
with its E.M.F. Hence the E.M.F., E, at the end of the 
line, impressed upon the receiving circuit, is represented by 
a vector, OE. To overcome the resistance, r, of the line, 
an E.M.F., Ir, is required in phase with the current, repre- 
sented by OEr in the diagram. The self-inductance of the 
line induces an E.M.F. which is proportional to the current 
/ and reactance x, and lags a quarter of a period, or 90°, 
behind the current. To overcome this counter E.M.F. 


24 


ALTERNA TING-CURRENT PHENOMENA. 


of self-induction, an E.M.F. of the value Ix is required, 
in phase 90° ahead of the current, hence represented by 
vector OEX. Thus resistance consumes E.M.F. in phase, 
and reactance an E.M.F. 90° ahead of the current. The 
E.M.F. of the generator, E0, has to give the three E.M.Fs., 
E, Ery and Ex, hence it is determined as their resultant. 
Combining by the parallelogram law, OEr and OEX, give 
OEZ, the E.M.F. required to overcome the impedance of 
the line, and similarly OEZ and OE give OE0, the E.M.F. 
required at the generator side of the line, to yield the 
E.M.F. E at the receiving end of the line. Algebraically, 
we get from Fig. 12 — 


or, E = VX2 — (/*)2 - Jr. 

In this instance we have considered the E.M.F. con- 
sumed by the resistance (in phase with the current) and 
the E.M.F. consumed by the reactance (90° ahead of the 
current) as parts, or components, of the impressed E.M.F., 
E0, and have derived E0 by combining Er, Ex, and E. 


E'. 


E? 0 


Fig. 13. 


18. We may, however, introduce the effect of the induc- 
tance directly as an E.M.F., Ex , the counter E.M.F. of 
self-induction = Ix, and lagging 90° behind the current ; and 
the E.M.F. consumed by the resistance as a counter E.M.F., 
Ef = Ir, but in opposition to the current, as is done in Fig. 
13 ; and combine the three E.M.Fs. E0, EJ, Ex , to form a 
resultant E.M.F., E, which is left at the end of the line- 


GRAPHIC REPRESENTA TION. 


25 


Ef and £a! combine to form Eg) the counter E.M.F. of 
impedance ; and since Eg and E0 must combine to form 
E, E0 is found as the side of a parallelogram, OE0EEg) 
whose other side, O£z', and diagonal, OE, are given. 

Or we may say (Fig. 14), that to overcome the counter 
E.M.F. of impedance, OEZ, of the line, the component, OEZ, 
of the impressed E.M.F. is required which, with the other 
component OE, must give the impressed E.M.F., OE0. 

As shown, we can represent the E.M.Fs. produced in a 
circuit in two ways — either as counter E.M.Fs., which com- 
bine with the impressed E.M.F., or as parts, or components, 


E.V o 


Fig. 14. 

of the impressed E.M.F., in the latter case being of opposite 
phase. According to the nature of the problem, either the 
one or the other way may be preferable. 

As an example, the E.M.F. consumed by the resistance 
is Ir, and in phase with the current ; the counter E.M.F. 
of resistance is in opposition to the current. ' The E.M.F. 
consumed by the reactance is Ix, and 90° ahead of the cur- 
rent, while the counter E.M.F. of reactance is 90° behind 
the current ; so that, if, in Fig. 15, OI, is the current, — 

OEr = E.M.F. consumed by resistance, 
OEr' = counter E.M.F. of resistance, 
OEX = E.M.F. consumed by inductance, 
OEX' = counter E.M.F. of inductance, 
OEZ = E.M.F. consumed by impedance, 
OEt ' = counter E.M.F. of impedance. 


26 ALTERNATING-CURRENT PHENOMENA. 

Obviously, these counter E.M.Fs. are different from, for 
instance, the counter E.M.F. of a synchronous motor, in so 
far as they have no independent existence, but exist only 
through, and as long as, the current flows. In this respect 
they are analogous to the opposing force of friction in 
mechanics. 

if. 


\f 
—X« 


Fig. 15. 


19. Coming back to the equation found for the E.M.F. 
at the generator end of the line, — 


we find, as the drop of potential in the line 


A E = E — E = V£ />'2 /*2 — E. 


This is different from, and less than, the E.M.F. of 
impedance — 

Hence it is wrong to calculate the drop of potential in a 
circuit by multiplying the current by the impedance ; and the 
drop of potential in the line depends, with a given current 
fed over the line into a non-inductive circuit, not only upon 
the constants of the line, r and *, but also upon the E.M.F., 
E, at end of line, as can readily be seen from the diagrams. 

20. If the receiver circuit is inductive, that is, if the 
current, /, lags behind the E.M.F., E, by an angle w, and 
we choose again as the zero line, the current OI (Fig. 16), 
the E.M.F., OE is ahead of the current by angle £. The 


GRAPHIC REPRESENTA TION. 


27 


E.M.F. consumed by the resistance, Ir, is in phase with the 
current, and represented by OEr; the E.M.F. consumed 
by the reactance, Ix, is 90° ahead of the current, and re- 
presented by OEX. Combining OE, OEr, and OEX, we 
get OE0, the E.M.F. required at the generator end of the 
line. Comparing Fig. 16 with Fig. 13, we see that in 
the former OE0 is larger ; or conversely, if E0 is the same, 
E will be less with an inductive load. In other words, 
the drop of potential in an inductive line is greater, if the 
receiving circuit is inductive, than if it is non-inductive. 
From Fig. 16, — 

E0 = V(^ cos w + Ir)2 -f- (E sin w + Ix)z. 


Fig. 18. 

If, however, the current in the receiving circuit is 
leading, as -is the case when feeding condensers or syn- 
chronous motors whose counter E.M.F. is larger than the 
impressed E.M.F., then the E.M.F. will be represented, in 
Fig. 17, by a vector, OE, lagging behind the current, Of, 
by the angle of lead £'; and in this case we get, by 
combining OE with OEr, in phase with the current, and 
OEX, 90° ahead of the current, the generator E.M.F., OE~0, 
which in this case is not only less than in Fig. 16 and in 
Fig. 13, but may be even less than E ; that is, the poten- 
tial rises in the line. In other words, in a circuit with 
leading current, the self-induction of the line raises the 
potential, so that the drop of potential is less than with 


28 


AL TERN A TING- CURRENT PHENOMENA. 


a non-inductive load, or may even be negative, and the 
voltage at the generator lower than at the other end of 
the line. 

These diagrams, Figs. 13 to 17, can be considered polar 
diagrams of an alternating-current generator of an E.M.F., 
E0> a resistance E.M.F., Er = fr, a reactance E.M.F., 
Ex = fx, and a difference of potential, E, at the alternator 
terminals; and we see, in this case, that with an inductive 
load the potential difference at the alternator terminals will 
be lower than with a non-inductive load, and that with a 
non-inductive load it will be lower than when feeding into 


'E. 


Fig. 17. 

a circuit with leading current, as, for instance, a synchro- 
nous motor circuit under the circumstances stated above. 

21. As a further example, we may consider the dia- 
gram of an alternating-current transformer, feeding through 
its secondary circuit an inductive load. 

For simplicity, we may neglect here the magnetic 
hysteresis, the effect of which will be fully treated in a 
separate chapter on this subject. 

Let the time be counted from the moment when the 
magnetic flux is zero. The phase of the flux, that is, the 
amplitude of its maximum value, is 90° in this case, and, 
consequently, the phase of the induced E.M.F., is 180°, 


GRAPHIC REPRESEiVTA TIOiV. 


29 


since the induced E.M.F. lags 90° behind the inducing 
flux. Thus the secondary induced E.M.F., JE1, will be 
represented by a vector, O£l} in Fig. 18, at the phase 
180°. The secondary current, flf lags behind the E.M.F., 
Elt by an angle a>1} which is determined by the resistance 
and inductance of the secondary circuit ; that is, by the 
load in the secondary circuit, and is represented in the dia- 
gram by the vector, OFl} of phase 180 + Gj. 


Fig. 18. 


Instead of the secondary current, flt we plot, however, 


the secondary M.M.F., 


where n1 is the number 
This. 


of secondary turns, and $l is given in ampere-turns. 
makes us independent of the ratio of transformation. 

From the secondary induced E.M.F., Ely we get the flux» 
3>, required to induce this E.M.F., from the equation — 


where — 

£i = secondary induced E.M.F. , in effective volts, 
JV = frequency, in cycles per second, 
;/1 = number of secondary turns, 
3> = maximum value of magnetic flux, in webers. 
The derivation of this equation has been given in a 
preceding chapter. 

This magnetic flux, 4>, is represented by a vector, O<b, at 
the phase 90°, and to induce it an M.M.F., ff is required, 


30 ALTERNATING-CURRENT PHENOMENA. 

which is determined by the magnetic characteristic of the 
iron, and the section and length of the magnetic circuit of 
the transformer ; it is in phase with the flux $, and repre- 
sented by the vector OF, in effective ampere-turns. 

The effect of hysteresis, neglected at present, is to shift 
OF ahead of O®, by an angle a, the angle of hysteretic 
lead. (See Chapter on Hysteresis.) 

This M.M.F., O7, is the resultant of the secondary M.M.F., 
JFlf and the primary M.M.F., SF0; or graphically, OF is the 
diagonal of a parallelogram with OFl and OF0 as sides. OF1 
and OF being known, we find OF0, the primary ampere- 
turns, and therefrom, and the number of primary turns, n0, 
the primary current, I0 = &0/ n0, which corresponds to the 
secondary current, 71. 

To overcome the resistance, r0, of the primary coil, an 
E.M.F., Er = f0r0, is required, in phase with the current, 
J0, and represented by the vector, OEr. 

To overcome the reactance, x0 = 2 •*• n0 L0 , of the pri- 
mary coil, an E.M.F. Ex = I0x0 is required, 90° ahead of 
the current f0, and represented by vector, OEX. 

The resultant magnetic flux, 4>, which in the secondary 
coil induces the E.M.F., EI} induces in the primary coil an 
E.M.F. proportional to E± by the ratio of turns n0/ nl} and 
in phase with El , or, — 

77 f "o zr 
£, *m—2£lf 

»1 

•which is represented by the vector OE%'. To overcome this 
counter E.M.F., Et't a primary E.M.F., Et, is required, equal 
but opposite to Et', and represented by the vector, OE,. 

The primary impressed E.M.F., E0, must thus consist of 
the three components, OEit OEr, and OEX, and is, there- 
fore, their resultant OE0, while the difference of phase in 
the primary circuit is found to be <30 = E0OF0. 

22. Thus, in Figs 18 to 20, the diagram of a trans- 
former is drawn for the same secondary E.M.F., Ev sec- 


GRAPHIC REPRESENTA TION. 


31 


ondary current, 7L and therefore secondary M.M.F., &v but 
with different conditions of secondary displacement : — 

In Fig. 18, the secondary current, /i , lags 60° behind the sec- 
ondary E.M.F., EI. 

In Fig. 19, the secondary current, 71} is in phase with the 
secondary E.M.F., El. 

In Fig. 20, the secondary current, 7: , leads by 60° the second- 
ary E.M.F., £lf 


These diagrams show that lag in the secondary circuit in- 
creases and lead decreases, the primary current and primary 
E.M.F. required to produce in the secondary circuit the 
same E.M.F. and current ; or conversely, at a given primary 


Fig. 20. 

impressed E.M.F., E0, the secondary E.M.F., E^ will be 
smaller with an inductive, and larger with a condenser 
(leading current) load, than with a non-inductive load. 

At the same time we see that a difference of phase 
existing in the secondary circuit of a transformer reappears 


32 AL TERNA TING-CURRENT PHENOMENA. 

in the primary circuit, somewhat decreased if leading, and 
slightly increased if lagging. Later we shall see that 
hysteresis reduces the displacement in the primary circuit, 
so that, with an excessive lag in the secondary circuit, the 
lag in the primary circuit may be less than in the secondary. 
A conclusion from the foregoing is that the transformer 
is not suitable for producing currents of displaced phase ; 
since primary and secondary current are, except at very 
light loads, very nearly in phase, or rather, in opposition, 
to each other. 


SYMBOLIC METHOD.