Theory and Calculation of Alternating Current Phenomena (1900) — part 2 of 19

In alternating-current circuits, the power equation con- tains a third term, which, in sine waves, is the cosine of the difference of phase between E.M.F. and current : —

P0 = ei cos <£.

Consequently, even if e and i are both large, P0 may be very small, if cos <f> is small, that is, <f> near 90°.

Kirchhoff's laws become meaningless in their original form, since these laws consider the E.M.Fs. and currents as directional quantities, counted positive in the one, nega- tive in the opposite direction, while the alternating current has no definite direction of its own.

6. The alternating waves may have widely different shapes ; some of the more frequent ones are shown in a later chapter.

The simplest form, however, is the sine wave, shown in Fig. 1, or, at least, a wave very near sine shape, which may be represented analytically by : —

/ = / sin ^ (/ - 4) = /sin 2 TT yV (/ - 4) ;

INTRO D UC TION.

where / is the maximum value of the wave, or its ampli- tude ; T is the time of one complete cyclic repetition, or the period of the wave, or N = 1 / T is the frequency or number of complete periods per second ; and t\ is the time, where the wave is zero, or the epoch of the wave, generally called the pliasc*

Obviously, "phase" or "epoch" attains a practical meaning only when several waves of different phases are considered, as "difference of phase." When dealing with one wave only, we may count the time from the moment

T\

rS

Fig. 1. Sine Wave,

where the wave is zero, or from the moment of its maxi- mum, and then represent it by : —

« = / sin 2 TT Nt ; or, / = /cos 2 TT Nt.

Since it is a univalent function of time, that is, can at a given instant have one value only, by Fourier's theorem, any alternating wave, no matter what its shape may be, can be represented by a series of sine functions of different frequencies and different phases, in the form : —

/ = 7i sin 2 irN(t — A) + 72 sin 4 TrJV(t - /2)

• 73 sin

• " Epoch " is the time where a periodic function reaches a certain value, for instance, zero; and "phase" is the angular position, with respect to a datum position, of a periodic function at a given time. Both are in alternate- current phenomena only different ways of expressing the same thing.

8

ALTERNA TING-CURRENT PHENOMENA.

where fv 72, 73, . . . are the maximum values of the differ- ent components of the wave, fv fv /3 . . . the times, where the respective components pass the zero value.

The first term, 7X sin lir N (t — tj, is called the fun- damental wave, or the first harmonic; the further terms are called the higher harmonics, or "overtones," in analogy to the overtones of sound waves. In sin 2 mr N (t — /„) is the «th harmonic.

By resolving the sine functions of the time differences, / — fp t — /2 . . . , we reduce the general expression of the wave to the form :

Al sin 2 TrNt + A* sin 4 vNt + Az sin G TT Nt + . . . 1cos27rA?-f^2cos47rA?-f ^8cos67ry\7+ . . .

F/g. 2. Wave without Even Harmonics.

The two half-waves of each period, the positive wave and the negative wave (counting in a definite direction in the circuit), are almost always identical. Hence the even higher harmonics, which cause a difference in the shape of the two half -waves, disappear, and only the odd harmonics exist, except in very special cases.

Hence the general alternating-current wave is expressed

ty : i = 7i sin 2 TT N(t — A) + 7, sin 6 TT N (t — /3)

• 75 sin 10 TT A^(/ — /5) + ... or,

/ = ^ sin 2 TT A7 + Az sin 6 TT A7 + A& sin 10 w A? + . . . cos 2 TT Nt + ^8 cos 6 TrNt + ^5 cos 10 vNt + . . .

INTR OD UC TION.

9

Such a wave is shown in Fig. 2, while Fig. 3 shows a wave whose half-waves are different. Figs. 2 and 3 repre- sent the secondary currents of a Ruhmkorff coil, whose secondary coil is closed by a high external resistance : Fig. 3 is the coil operated in the usual way, by make and break of the primary battery current ; Fig. 2 is the coil fed with reversed currents by a commutator from a battery.

7. Self-inductance, or electro-magnetic momentum, which is always present in alternating-current circuits, — to a large extent in generators, transformers, etc., — tends to

Fig. 3. Wave with Even Harmonics.

suppress the higher harmonics of a complex harmonic wave more than the fundamental harmonic, since the self-induc- tive reactance is proportional to the frequency, and is thus greater with the higher harmonics, and thereby causes a general tendency towards simple sine shape, which has the effect, that, in general, the alternating currents in our light and power circuits are sufficiently near sine waves to make the assumption of sine shape permissible.

Hence, in the calculation of alternating-current phev nomena, we can safely assume the alternating wave as a sine wave, without making any serious error ; and it will be

10 AL TERN A TING-CURRENT PHENOMENA.

sufficient to keep the distortion from sine shape in mind as a possible disturbing factor, which generally, however, is in practice negligible — perhaps with the only exception of low-resistance circuits containing large magnetic reactance, and large condensance in series with each other, so as to produce resonance effects of these higher harmonics.

INSTANTANEOUS AND INTEGRAL VALUES.

11

CHAPTER II

INSTANTANEOUS VALUES AND INTEGRAL VALUES.

8. IN a periodically varying function, as an alternating current, we have to distinguish between the instantaneous value, which varies constantly as function of the time, and the integral value, which characterizes the wave as a whole.

As such integral value, almost exclusively the effective

Fig. 4. Alternating Wave.

value is used, that is, the square root of the mean squares ; and wherever the intensity of an electric wave is mentioned without further reference, the effective value is understood.

The maximum value of the wave is of practical interest only in few cases, and may, besides, be different for the two half-waves, as in Fig. 3.

As arithmetic mean, or average value, of a wave as in Figs. 4 and 5, the arithmetical average of all the instan- taneous values during one complete period is understood.

This arithmetic mean is either = 0, as in Fig. 4, or it differs from 0, as. in Fig. 5. In the first case, the wave is called an alternating wave, in the latter a pttlsating wave.

12

ALTERNA TING-CURRENT PHENOMENA.

Thus, an alternating wave is a wave whose positive values give the same sum total as the negative values ; that is, whose two half-waves have in rectangular coordinates the same area, as shown in Fig. 4.

A pulsating wave is a wave in which one of the half- waves preponderates, as in Fig. 5.

By electromagnetic induction, pulsating waves are pro- duced only by commutating and unipolar machines (or by the superposition of alternating upon direct currents, etc.).

All inductive apparatus without commutation give ex- clusively alternating waves, because, no matter what con-

Fig. 5. Pulsating Wave.

ditions may exist in the circuit, any line of magnetic force, which during a complete period is cut by the circuit, and thereby induces an E.M.F., must during the same period be cut again in the opposite direction, and thereby induce the same total amount of E.M.F. (Obviously, this does not apply to circuits consisting of different parts movable with regard to each other, as in unipolar machines.)

In the following we shall almost exclusively consider the alternating wave, that is the wave whose true arithmetic mean value = 0.

Frequently, by mean value of an alternating wave, the average of one half-wave only is denoted, or rather the

INSTANTANEOUS AND INTEGRAL VALUES.

13

average of all instantaneous values without regard to their sign. This mean value is of no practical importance, and is, besides, in many cases indefinite.

9. In a sine wave, the relation of the mean to the maxi- mum value is found in the following way : —

Fig. 8.

Let, in Fig. 6, AOB represent a quadrant of a circle with radius 1.

Then, while the angle <£ traverses the arc -n- / 2 from A to B, the sine varies from 0 to OB = 1. Hence the average variation of the sine bears to that of the corresponding arc the ratio 1 -j- 7r/2, or 2 / TT •+- 1. The maximum variation of the sine takes place about its zero value, where the sine is equal to the arc. Hence the maximum variation of the sine is equal to the variation of the corresponding arc, and consequently the maximum variation of the sine bears to its average variation the same ratio as the average variation of the arc to that of the sine ; that is, 1 -f- 2 / 77-, and since the variations of a sine-function are sinusoidal also, we have,

o

Mean value of sine wave -r- maximum value = • — • -f- 1

7T

= .63663.

The quantities, "current," "E.M.F.," "magnetism," etc., are in reality mathematical fictions only, as the components

14 AL TERNA TING-CURRENT PHENOMENA.

of the entities, "energy," "power," etc. ; that is, they have no independent existence, but appear only as squares or products.

Consequently, the only integral value of an alternating wave which is of practical importance, as directly connected with the mechanical system of units, is that value which represents the same power or effect as the periodical wave. This is called the effective value. Its square is equal to the mean square of the periodic function, that is : —

TJie effective value of an alternating wave, or tJie value representing the same effect as the periodically varying wave, is the square root of the mean square.

In a sine wave, its relation to the maximum value is found in the following way :

Fig. 7.

Let, in Fig. 7, AOB represent a quadrant of a circle with radius 1.

Then, since the sines of any angle </> and its complemen- tary angle, 90°— <£, fulfill the condition, —

sin2 $ + sin2 (90 — <£) = 1,

the sines in the quadrant, AOB, can be grouped into pairs, so that the sum of the squares of any pair = 1 ; or, in other words, the mean square of the sine =1/2, and the square root of the mean square, or the effective value of the sine, = 1/V2. That is:

INSTANTANEOUS AND INTEGRAL VALUES.

15

The effective value of a sine function bears to its mum value the ratio, — 1

V2

Hence, we have for the sine curve the following rela- tions :

1 = .70711.

MAX.

EFF.

ARITH. MEAN.

Half Period.

Whole Period.

1

1

V2

2

7T

0

1

.7071

.63663

0

1.4142

1

.90034

0

1.5708

1.1107

1

0

10. Coming now to the general alternating wave,

/ = Ai sin 27r Nt + Az sin 4-n- Nt + A3 sin GTT Nt + . . .

• BI cos 2-n-Nt + B* cos ±TrNt + £s cos GTT Nt + . .

we find, by squaring this expression and canceling all the products which give 0 as mean square, the effective value, —

1= V* W

The mean value does not give a simple expression, and is of no general interest.

16 ALTERNATING-CURRENT PHENOMENA,

CHAPTER III.

LAW OF ELECTRO-MAGNETIC INDUCTION.

11. If an electric conductor moves relatively to a mag- netic field, an E.M.F. is induced in the conductor which is proportional to the intensity of the magnetic field, to the length of the conductor, and to the speed of its motion perpendicular to the magnetic field and the direction of the conductor ; or, in other words, proportional to the number of lines of magnetic force cut per second by the conductor.

As a practical unit of E.M.F., the volt is defined as the E.M.F. induced in a conductor, which cuts 108 = 100,000,000 lines of magnetic force per second.

If the conductor is closed upon itself, the induced E.M.F. produces a current.

A closed conductor may be called a turn or a convolution. In such a turn, the number of lines of magnetic force cut per second is the increase or decrease of the number of lines inclosed by the turn, or n times as large with n turns.

Hence the E.M.F. in volts induced in n turns, or con- volutions, is n times the increase or decrease, per second, of the flux inclosed by the turns, times 10~8.

If the change of the flux inclosed by the turn, or by n turns, does not take place uniformly, the product of the number of turns, times change of flux per second, gives the average E.M.F.

If the magnetic flux, 4>, alternates relatively to a number of turns, n — that is, when the turns either revolve through the flux, or the flux passes in and out of the turns, the total flux is cut four times during each complete period or cycle, twice passing into, and twice out of, the turns.

LAW OF ELECTRO-MAGNETIC INDUCTION. 17

Hence, if N= number of complete cycles per second, or the frequency of the flux 3>, the average E.M.F. induced in n turns is,

£&vg, = 4 « 3> N 10 ~ 8 volts.

This is the fundamental equation of electrical engineer- ing, and applies to .continuous-current, as well as to alter- nating-current, apparatus.

12. In continuous-current machines and in many alter- nators, the turns revolve through a constant magnetic field ; in other alternators and in induction motors, the mag- netic field revolves ; in transformers, the field alternates with respect to the stationary turns.

Thus, in the continuous-current machine, if n = num- ber of turns in series from brush to brush, <I> = flux inclosed per turn, and N = frequency, the E.M.F. induced in the machine is E = 4«4>7V10~8 volts, independent of the num- ber of poles, of series or multiple connection of the arma- ture, whether of the ring, drum, or other type.

In an alternator or transformer, if n is the number of turns in series, $ the maximum flux inclosed per turn, and JV the frequency, this formula gives,

£avg = 4 « 4> JVW ~ 8 volts. Since the maximum E.M.F. is given by, —

•^maz. = £ ^avg

we have

^"max. = 27r»<S>7V710-8VOltS.

And since the effective E.M.F. is given by, —

we have

£es. =

= 4.44 n 4>^10- 8 volts,

which is the fundamental formula of alternating-current induction by sine waves.

18 AL TERN A TING-CURRENT PHENOMENA,

13. If, in a circuit of n turns, the magnetic flux, <t>, inclosed by the circuit is produced by the current flowing in the circuit, the ratio —

flux X number of turns X 10~8

current .

is called the inductance, L, of the circuit, in henrys.

The product of the number of turns, n, into the maxi- mum flux, <S>, produced by a current of / amperes effective, or / V2 amperes maximum, is therefore —

n® =Z/V2 108; and consequently the effective E.M.F. of self-inductance is:

E = V2

=' 2 TT NLI volts.

The product, x = 2 vNL, is of the dimension of resistance, and is called the reactance of the circuit ; and the E.M.F. of self-inductance of the circuit, or the reactance voltage, is

E = Ix,

and lags 90° behind the current, since the current is in phase with the magnetic flux produced by the current, and the E.M.F. lags 90° behind the magnetic flux. The E.M.F. lags 90° behind the magnetic flux, as it is propor- tional to the change in flux ; thus it is zero when the mag- netism is at its maximum value, and a maximum when the flux passes through zero, where it changes quickest.

GRAPHIC REPRESENTA TION,

19

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.

CHAPTER V.

SYMBOLIC METHOD.

23. The graphical method of representing alternating, current phenomena by polar coordinates of time affords the best means for deriving a clear insight into the mutual rela- tion of the different alternating sine waves entering into the problem. For numerical calculation, however, the graphical method is generally not well suited, owing to the widely different magnitudes of the alternating sine waves repre- sented in the same diagram, which make an exact diagram- matic determination impossible. For instance, in the trans- former diagrams (cf. Figs. 18-20), the different magnitudes will have numerical values in practice, somewhat like El — 100 volts, and 1-^ = 75 amperes, for a non-inductive secon- dary load, as of incandescent lamps. Thus the only reac- tance of the secondary circuit is that of the secondary coil, or, x-^ = .08 ohms, giving a lag of ^ = 3.6°. We have also,

n^ = 30 turns.

n0 = 300 turns.

CFi = 2250 ampere-turns.

y = 100 ampere-turns.

Er = 10 volts.

JSX = 60 volts.

E{ = 1000 volts.