Theory and Calculation of Alternating Current Phenomena (1900) — part 15 of 19

241. We find here a case of a circuit in which the power factor — that is, the ratio of watts to volt amperes — differs from unity without any displacement of phase ; that is, while current and E.M.F. are in phase with each other, but are distorted, the alternating wave cannot be replaced by an equivalent sine wave ; since the assumption of equivalent sine wave would introduce a phase displace- ment,

cos w =/

of an angle, w, whose sign is indefinite.

As an instance are shown, in Fig. 173 for the constants,

1= 12

r= 3

£ =.9

the resistance,

R = 3 {I + .9 cos 2 /3) ;

the current,

• = 17 sin /3 ;

tha potential difference,

e = 28 (sin ft + .82 sin 3 £). In this case the effective E.M.F. is £=25.5;

396 ALTERNATING-CURRENT PHENOMENA.

the apparent resistance,

the power,

the apparent power,

the power factor,

r0 = 2.13 ; P = 244 ;

El =307; / = .796.

Fig. 173. Periodically Varying Resistance.

As seen, with a sine wave of current the E.M.F. wave in an alternating arc will become double-peaked, and rise very abruptly near the zero values of current. Inversely, with a sine wave of E.M.F. the current wave in an alter- nating arc will become peaked, and very flat near the zero values of E.M.F.

242. In reality the distortion is of more complex nature ; since the pulsation of resistance in the arc does not follow

DISTORTION OF WAVE-SHAPE.

397

a simple sine law of double frequency, but varies much more abruptly near the zero value of current, making thereby the variation of E.M.F. near the zero value of current much more abruptly, or, inversely, the variation of current more flat.

A typical wave of potential difference, with a sine wave of current passing through the arc, is given in Fig. 174.*

1 13 13 1 15

ONE PAIR CARBONS

EG U LATE D BY HAND

A. C. dynamo e. m. f

•' " " current*.

" " " watts.

7 18 19 20 S

Fig. 174. Electric Arc.

243. The value of e, the amplitude of the resistance pulsation, largely depends upon the nature of the electrodes and the steadiness of the arc, and with soft carbons and a steady arc is small, and the power factor f of the arc near unity. With hard carbons and an unsteady arc, e rises greatly, higher harmonics appear in the pulsation of resis- tance, and the power factor f falls, being in extreme cases even as low as .6.

The conclusion to be drawn herefrom is, that photo- metric tests of alternating arcs are of little value, if, besides current and voltage, the power is not determined also by means of electro-dynamometers.

• From American Institute of Electrical Engineers, Transactions, 1890, p-

376. Tobey and Walbridge, on the Stanley Alternate Arc Dynamo.

398

A L TERN A TING-CURRENT PHENOMENA .

CHAPTER XXIII.

EFFECTS OF HIGHER HARMONICS.

244. To elucidate the variation in the shape of alternat- ing waves caused by various harmonics, in Figs. 175 and

Fig. 175. Effect of Triple Harmonic.

176 are shown the wave-forms produced by the superposi- tion of the triple and the quintuple harmonic upon the fundamental sine wave.

EFFECTS OF HIGHER HARMONICS. 399

In Fig. 175 is shown the fundamental sine wave and the complex waves produced by the superposition of a triple harmonic of 30 per cent the amplitude of the fundamental, under the relative phase displacements of 0°, 45°, 90°, 135°, and 180°, represented by the equations :

sin ft

sin ft — .3 sin 3 ft

sin ft- .3 sin (3/3-45°)

sin ft — .3 sin (3 ft — 90°)

s'm ft - .3 sin (3 ft - 135°)

sin ft — .3 sin (3/3 — 180°). •

As seen, the effect of the triple harmonic is in the first figure to flatten the zero values and point the maximum values of the wave, giving what is called a peaked wave. With increasing phase displacement of the triple harmonic, the flat zero rises and gradually changes to a second peak, giving ultimately a flat-top or even double-peaked wave with sharp zero. The intermediate positions represent what is called a saw-tooth wave.

In Fig. 176 are shown the fundamental sine wave and the complex waves produced by superposition of a quintuple harmonic of 20 per cent the amplitude of the fundamental, under the relative phase displacement of 0°, 45°, 90°, 135°, 180°, represented by the equations :

sin ft

sin ft — .2 sin 5 ft sin/3- .2 sin (5,8-45°) sin/3- .2 sin (5/3-90°) smft- .2 sin (5/3- 135°) sin/3- .2 sin (5/8- 180°).

The quintuple harmonic causes a flat -topped or even double-peaked wave with flat zero. With increasing phase displacement, the wave becomes of the type called saw- tooth wave also. The flat zero rises and becomes a third peak, while of the two former peaks, one rises, the other

400

AL TERN A TING- CURRENT PHENOMENA.

decreases, and the wave gradually changes to a triple- peaked wave with one main peak, and a sharp zero.

As seen, with the triple harmonic, flat-top or double- peak coincides with sharp zero, while the quintuple har- monic flat-top or double-peak coincides with flat zero.

Distortion of Wave Shapa by Quintuple Harmonfc Sin./S-.2sin.(5/?-S5j/

J

\J

Fig. 176. Effect of Quintuple Harmonic.

Sharp peak coincides with flat zero in the triple, with sharp zero in the quintuple harmonic. With the triple har- monic, the saw-tooth shape appearing in case of a phase difference between fundamental and harmonic is single, while with the quintuple harmonic it is double.

Thus in general, from simple inspection of the wave shape, the existence of these first harmonics can be discov- ered. Some characteristic shapes are shown in Fig. 177.

EFFECTS OF HIGHER HARMONICS.

401

Sin/?-.225 sinf3/?-180) , ""-.05 sin/5/3-180)

Sin./?- 15 sm.(3/?-180).

Sin./?-. 15' sin 3/?-.1Q sir (5/J-180)

f/jjr. 777. So/ne Characteristic Wave Shapes.

Flat top with flat zero :

sin /3 — .15 sin 3 /3 — .10 sin 5 0. Flat top with sharp zero :

sin 0 - .225 sin (3 /3 - 180°) - .05 sin (5 /3 - 180°). Double peak, with sharp zero :

sin (3 - .15 sin (30- 180°) - .10 sin 5 /?. Sharp peak with sharp zero :

sin {3 — .15 sin 3 0 — .10 sin (5 (3 — 180°).

245. Since the distortion of the wave-shape consists in the superposition of higher harmonics, that is, waves of higher frequency, the phenomena taking place in a circuit

402 ALTERNATING-CURRENT PHENOMENA.

supplied by such a wave will be the combined effect of the different waves.

Thus in a non-inductive circuit, the current and the potential difference across the different parts of the circuit are of the same shape as the impressed E.M.F. If self- induction is inserted in series to a non-inductive circuit, the self-induction consumes more E.M.F. of the higher harmon- ics, since the reactance is proportional to the frequency, and thus the current and the E.M.F. in the non-inductive part of the circuit shows the higher harmonics in a reduced amplitude. That is, self-induction in series to a non-induc- tive circuit reduces the higher harmonics or smooths out the wave to a closer resemblance with sine shape. In- versely, capacity in series to a non-inductive circuit con- sumes less E.M.F. at higher than at lower frequency, and thus makes the higher harmonics of current and of poten- tial difference in the non-inductive part of the circuit more pronounced — intensifies the harmonics.

Self-induction and capacity in series may cause an in- crease of voltage due to complete or partial resonance with higher harmonics, and a discrepancy between volt-amperes and watts, without corresponding phase displacement, as will be shown hereafter.

246. In long-distance transmission over lines of notice- able inductance and capacity, rise of voltage due to reso- nance may occur with higher harmonics, as waves of higher frequency, while the fundamental wave is usually of too low a frequency to cause resonance.

An approximate estimate of the possible rise by reso- nance with various harmonics can be obtained by the inves- tigation of a numerical instance. Let in a long-distance line, fed by step-up transformers at 60 cycles,

The resistance drop in the transformers at full load = 1%. The inductance voltage in the transformers at full load = 5%

with the fundamental wave. The resistance drop in the line at full load = 10%.

EFFECTS OF HIGHER HARMONICS. 403

The inductance voltage in the line at full load = 20% with the

fundamental wave. The capacity or charging current of the line = 20% of the full-

load current / at the frequency of the fundamental.

The line capacity may approximately be represented by a condenser shunted across the middle of the line. The E.M.F. at the generator terminals E is assumed as main- tained constant.

The E.M.F. consumed by the resistance of the circuit from generator terminals to condenser is

Ir = .06 £, or, r = .06 -| .

The reactance E.M.F. between generator terminals and condenser is, for the fundamental frequency,

Ix = .15 £,

-IK E

or, x = .15 — ,

thus the reactance corresponding to the frequency (2/£ — 1) N of the higher harmonic is :

x(2k- 1) =.15(2£- 1) — . The capacity current at fundamental frequency is :

hence, at the frequency : (2 k — 1) N:

/ = .2(2£-l)/Z, if:

e' = E.M.F. of the (2 k — l)th harmonic at the condenser,

e = E.M.F. of the (2 k — l)th harmonic at the generator terminals.

The E.M.F. at the condenser is : —

e' = V*2 — iar2 + ix (2k — V) •

404 AL TERNA TING-CURRENT PHENOMENA.

hence, substituted :

' l — .059856 (2 k — I)2 + .0009 (2 k — I)4

the rise of voltage by inductance and capacity. Substituting :

k= 1 2 3 4 56

or, 2 £ - 1 = 1 3 5 7 9 11

it is, a = 1.03 1.36 3.76 2.18 .70 .38

That is, the fundamental will be increased at open circuit by 3 per cent, the triple harmonic by 36 per cent, the quintuple harmonic by 276 per cent, the septuple harmonic by 118 per cent, while the still higher harmonics are reduced.

The maximum possible rise will take place for :

= 0, or, 2,- 1 = 5.77

That is, at a frequency : N = 346, and a = 14.4.

That is, complete resonance will appear at a frequency between quintuple and septuple harmonic, and would raise the voltage at this particular frequency 14.4 fold.

If the voltage shall not exceed the impressed voltage by more than 100 per cent, even at coincidence of the maximum of the harmonic with the maximum of the fundamental,

the triple harmonic must be less than 70 per cent of the

fundamental, the quintuple harmonic must be less than 26.5 per cent of the

fundamental, the septuple harmonic must be less than 46 per cent of the

fundamental.

The voltage will not exceed twice the normal, even at a frequency of complete resonance with the higher har- monic, if none of the higher harmonics amounts to more

EFFECTS OF HIGHER HARMONICS. 405

than 7 per cent, of the fundamental. Herefrom it follows that the danger of resonance in high potential lines is in general greatly over-estimated, since the conditions assumed in this instance are rather more severe than found in prac- tice, the capacity current of the line very seldom reaching 20% of the main current.

247. The power developed by a complex harmonic wave in a non-inductive circuit is the sum of the powers of the individual harmonics. Thus if upon a sine wave of alter- nating E.M.F. higher harmonic waves are superposed, the effective E.M.F., and the power produced by this wave in a given circuit or with a given effective current, are increased. In consequence hereof alternators and synchronous motors of ironclad unitooth construction — that is, machines giving waves with pronounced higher harmonics — give with the same number of turns on the armature, and the same mag- netic flux per field pole at the same frequency, a higher output than machines built to produce sine waves.

248. This explains an apparent paradox :

If in the three-phase star-connected generator with the magnetic field constructed as shown diagrammatically in Fig. 162, the magnetic flux per pole = $, the number of turns in series per circuit = n, the frequency = N, the E.M.F. between any two collector rings is:

E= V2~7T^2;z<S>10-8.

since 2« armature turns simultaneously interlink with the magnetic flux 3>.

The E.M.F. per armature circuit is :

hence the E.M.F. between collector rings, as resultant of two E.M.Fs. e displaced by 60° from each other, is :

406 ALTERNATING-CURRENT PHENOMENA.

while the same E.M.F. was found by direct calculation from number of turns, magnetic flux, and frequency to be equal to 2e; that is the two values found for the same E.M.F. have the proportion V3 : 2 = 1 : 1.154.

Fig. 178. Three-phase Star-connected Alternator.

This discrepancy is due to the existence of more pro- nounced higher harmonics in the wave e than in the wave E = e X V3, which have been neglected in the formula :

Hence it follows that, while the E.M.F. between two col- lector rings in the machine shown diagrammatically in Fig. 178 is only e x V3, by massing the same number of turns in one slot instead of in two slots, we get the E.M.F. 2 e or 15.4 per cent higher E.M.F., that is, larger output.

EFFECTS OF HIGHER HARMONICS. 407

It follows herefrom that the distorted E.M.F. wave of a unitooth alternator is produced by lesser magnetic flux per pole — that is, in general, at a lesser hysteretic loss in the armature or at higher efficiency — than the same effective E.M.F. would be produced with the same number of arma- ture turns if the magnetic disposition were such as to pro- duce a sine wave.

249. Inversely, if su<:h a distorted wave of E.M.F. is impressed upon a magnetic circuit, as, for instance, a trans- former, the wave of magnetism in the primary will repeat in shape the wave of magnetism interlinked with the arma- ture coils of the alternator, and consequently, with a lesser maximum magnetic flux, the same effective counter E.M.F. will be produced, that is, the same power converted in the transformer. Since the hysteretic loss in the transformer depends upon the maximum value of magnetism, it follows that the hysteretic loss in a transformer is less with a dis- torted wave of a unitooth alternator than with a sine wave.

Thus with the distorted waves of unitooth machines, generators, transformers, and synchronous motors — and induction motors in so far as they are transformers — operate more efficiently.

250. From another side the same problem can be approached.

If upon a transformer a sine wave of E.M.F. is im- pressed, the wave of magnetism will be a sine wave also. If now upon the sine wave of E.M.F. higher harmonics, as sine waves of triple, quintuple, etc., frequency are superposed in such a way that the corresponding higher harmonic sine waves of magnetism do not increase the maximum value of magnetism, or even lower it by a coincidence of their negative maxima with the positive maximum of the fundamental, — in this case all the power represented by these higher harmonics of E.M.F. will be

408 ALTERNATING-CURRENT PHENOMENA.

transformed without an increase of the hysteretic loss, or even with a decreased hysteretic loss.

Obviously, if the maximum of the higher harmonic wave of magnetism coincides with the maximum of the funda- mental, and thereby makes the wave of magnetism more pointed, the hysteretic loss will be increased more than in proportion to the increased power transformed, i.e., the efficiency of the transformer will be lowered.

That is : Some distorted waves of E.M.F. are transformed at a lesser, some at a larger, hysteretic loss than the sine wave, if the same effective E.M.F. is impressed upon the transformer.

The unitooth alternator wave and the first wave in Fig. 175 belong to the former class ; the waves derived from continuous-current machines, tapped at two equi-distant points of the armature, in general, to the latter class.

251. Regarding the loss of energy by Foucault or eddy currents, this loss is not affected by distortion of wave shape, since the E.M.F. of eddy currents, as induced E.M.F., is proportional to the secondary E.M.F. ; and thus at constant impressed primary E.M.F., the energy consumed by eddy currents bears a constant relation to the output of the secondary circuit, as obvious, since the division of power between the two secondary circuits — the eddy current circuit, and the useful or consumer cir- cuit — is unaffected by wave-shape or intensity of mag- netism.

252. In high potential lines, distorted waves whose maxima are very high above the effective values, as peaked waves, may be objectionable by increasing the strain on the insulation. It is, however, not settled yet beyond doubt whether the striking-distance of a rapidly alternat- ing potential depends upon the maximum value or upon

EFFECTS OF HIGHER HARMONICS. 409

some value between effective and maximum. Since dis- ruptive phenomena do not always take place immediately after application of the potential, but the time element plays ari important part, it is possible that insulation-strain and striking-distance is, in a certain range, dependent upon the effective potential, and thus independent of the wave-shape.

In this respect it is quite likely that different insulating materials show a different behavior, and homogeneous solid substances, as paraffin, depend in their disruptive strength upon the maximum value of the potential difference, while heterogeneous materials, as mica, laminated organic sub- stances, air, etc., that is substances in which the disruptive strength decreases with the time application of the potential difference, are less affected by very high peaks of E.M.F. of very short duration.

In general, as conclusions may be derived that the im- portance of a proper wave-shape is generally greatly over- rated, but that in certain cases sine waves are desirable, in other cases certain distorted waves are preferable.

410 ALTERNATING-CURRENT PHENOMENA.

CHAPTER XXIV.

SYMBOLIC REPRESENTATION OF GENERAL ALTERNATING WAVES.

253. The  vector  representation, 
     

A = a1 +y<zu = a (cos a --j sin d) of the alternating wave,

A — a0 cos (<£ — a)

applies to the sine wave only.

The general alternating wave, however, contains an in- finite series of terms, of odd frequencies,

A = Al cos (<£ — #1) 4- Az cos (3 <£ — #3) + A& cos (5 <£ — #5) -f

thus cannot be directly represented by one complex vector quantity.

The replacement of the general wave by its equivalent sine wave, as before discussed, that is a sine wave of equal effective intensity and equal power, while sufficiently accu- rate in many cases, completely fails in other cases, espe- cially in circuits containing capacity, or in circuits containing periodically (and in synchronism with the wave) varying resistance or reactance (as alternating arcs, reaction ma- chines, synchronous induction motors, oversaturated mag- netic circuits, etc.).

Since, however, the individual harmonics of the general alternating wave are independent of each other, that is, all products of different harmonics vanish, each term can be represented by a complex symbol, and the equations of the general wave then are the resultants of those of the indi- vidual harmonics.

REPRESENTATION OF ALTERNATING WAVES. 411

This can be represented symbolically by combining in one formula symbolic representations of different frequen- cies, thus,

00

A = £.»-i (a* +jn */)

i where,

and the index of the/M merely denotes that the/s of differ-

entindices n, while algebraically identical, physically rep-

resent different frequencies, and thus cannot be combined.

The general wave of E.M.F. is thus represented by,

the general wave of current by,

if,

is the impedance of the fundamental harmonic, where

xm is that part of the reactance which is proportional to

the frequency (inductance, etc.).

x0 is that part of the reactance which is independent of

the frequency (mutual induction, synchronous motion, etc.). xc is that part of the reactance which is inversely pro-

portional to the frequency (capacity, etc.).

The impedance for the nth harmonic is,

r —Jnn xm

This term can be considered as the general symbolic expression of the impedance of a circuit of general wave shape.

412 ALTERNATING-CURRENT PHENOMENA.

Ohm's law, in symbolic expression, assumes for the general alternating wave the form,

/-Jo,

E = IZ or,

Z = £or,

Z = r -n

The symbols of multiplication and division of the terms E, /, ^f, thus represent not algebraic operation, but multi- plication and division of corresponding terms of E, T, Z, that is, terms of the same index «, or, in algebraic multipli- cation and division of the series E, /, all compound terms, that is terms containing two different w's, vanish.

254. The  effective  value  of  the  general  wave  : 
     

a = AI cos (<£ — «,) + As cos (3 <£ — a8) +^5 cos (5 <f> — #6) +. .

is the square root of the sum of mean squares of individual harmonics,

A= V i { A? + A82 + A? + . . . |

Since, as discussed above, the compound terms, of two different indices «, vanish, the absolute value of the general alternating wave,

REPRESENTATION OF ALTERNATING WAVES. 413

is thus,

A

which offers an easy means of reduction from symbolic to absolute values.

Thus, the absolute value of the E.M.F.

s,

the absolute value of the current,

is,

255. The double frequency power (torque, etc.) equa- tion of the general alternating wave has the same symbolic expression as with the sine wave :

= Pl +JPJ

1

where,

41-4 ALTERNATING-CURRENT PHENOMENA.

The jn enters under the summation sign of the " watt- less power " 1$, so that the wattless powers of the different harmonics cannot be algebraically added.

i Thus,

The total " true power" of a general alternating current circuit is the algebraic sum of the powers of the individual harmonics.

The total "wattless power" of a general alternating current circuit is not the algebraic, but the absolute sum of the wattless powers of the individual harmonics.

Thus, regarding the wattless power as a whole, in the general alternating circuit no distinction can be made be- tween lead and lag, since some harmonics may be leading, others lagging.

The apparent power, or total volt-amperes, of the circuit is,

The power factor of the circuit is,

The term "inductance factor," however, has no mean- ing any more, since the wattless powers of the different harmonics are not directly comparable.

The quantity,

,...._ ... wattless power

has no physical significance, and is not =

total apparent power

REPRESENTATION OF ALTERNATING WAVES. 4] >

The term, /#.

El

= 2/n~17

where,

consists of a series of inductance factors qn of the individual harmonics.

As a rule, if <f = 2^-1 ^n2,

for the general alternating wave, that is q differs from

fo=vr^72

The complex quantity,

Q El ~ El

1

takes in the circuit of the general alternating wave the same position as power factor and inductance factor with the sine wave.

p

17= -~ may be called the " circuit factor "

It consists of a real term /, the power factor, and a series of imaginary terms jn qn, the inductance factors of the individual harmonics.

416 ALTERNATING-CURRENT PHENOMENA.

The absolute value of the circuit factor :

as a rule, is < 1.

256. Some  applications  of  this  symbolism  will  explain 
     

its mechanism and its usefulness more fully.

\st Instance : Let the E.M.F.,

be impressed upon a circuit of the impedance,

7 • ( *CN

Z = *•—./„ \nxm --

that is, containing resistance r, inductive reactance xm and capacity reactance xc in series.

Let

e? = 720 ef = 540

V = 283 4" = - 283

e£ = - 104 *6" = 138

or,

^ = 900 tan e^ = .75

*, = 400 tan o)3 = - 1

^5 = 173 tan w5 = - 1.33

It is thus in symbolic expression,

Zj = 10 + 80/; *! = 80.6

Z3 = 10 zz = 10

ZB = 10 - 32/; 25 = 33.5

and, E.M.F.,

^ = (720 + 540/0 + (283 - 283y;) + (- 104 + 138/5)

or absolute,

E = 1000

REPRESENTATION OF ALTERNATING WAVES. 417

and current,

_ £ _ 720 + 540/t 283 - 283/8 - 104 + 138./; Z~~ 10 + 80/i " 10 10-32y5

= (7.76 - 8.04/i) + (28.3 - 28.3/8) + (- 4.86 - 1.73 A)

or, absolute,

7=41.85

of which is of fundamental frequency, ll = 11.15 " " " " triple " I3 = 40

« « « quintuple " I5 = 5.17

The total apparent power of the circuit is,

Q = £7=41,850 The true power of the circuit is :

/» = [7i 7]1 = 1240 + 16,000 + 270

= 17,510 the wattless power,

j PJ =/ [7i 7]J = 10,000^ - 850/6 thus, the total power,

P= 17,510 + 10,000/; - 850y5

That is, the wattless power of the first harmonic is leading, that of the third harmonic zero, and that of the fifth harmonic lagging.

17,510 = I2 r, as obvious. The circuit factor is,

• Q El

= .418 + .239 j\ - .0203/5

or, absolute,

u = V.4182+ .2392 + .02032 = .482

The power factor is,

p = .418

418 ALTERNATING-CURRENT PHENOMENA.

The inductance factor of the first harmonic is : ql = .239, that of the third harmonic ft = 0, and of the fifth harmonic ft = - -0203.

Considering the waves as replaced by their equivalent sine waves, from the sine wave formula,

f + qf = 1 the inductance factor would be,

ft = -914 and the phase angle,

tan a, = ^= '-^=2.8 « = 65.4°

p .41o

giving apparently a very great phase displacement, while in reality, of the 41.85 amperes total current, 40 amperes (the current of the third harmonic) are in phase with their E.M.F.

We thus have here a case of a circuit with complex har- monic waves which cannot be represented by their equiva- lent sine waves. The relative magnitudes of the different harmonics in the wave of current and of E.M.F. differ essentially, and the circuit has simultaneously a very low power factor and a very low inductance factor; that is, a low power factor exists without corresponding phase displace- ment, the circuit factor being less than one-half.

Such circuits, for instance, are those including alternat- ing arcs, reaction machines, synchronous induction motors, reactances with over-saturated magnetic circuit, high poten- tial lines in which the maximum difference of potential ex- ceeds the voltage at which brush discharges begin, polariza- tion cells, and in general electrolytic conductors above the dissociation voltage of the electrolyte, etc. Such circuits cannot correctly, and in many cases not even approxi- mately, be treated by the theory of the equivalent sine waves, but require the symbolism of the complex harmonic wave.

REPRESENTATION OF ALTERNATING WAVES. 419

257. 2d instance: A condenser of capacity C0 = 20 m.f. is connected into the circuit of a 60-cycle alternator giving a wave of the form,

e = E (cos <£ - .10 cos 3 <£ - .08 cos 5 <f> + .06 cos 7 <£) or, in symbolic expression,

£ = e(!1- .10, - .085 + .067) The synchronous impedance of the alternator is, ZQ = r0 —jnnx0 = .3 — 5 njn

What is the apparent capacity C of the condenser (as cal- culated from its terminal volts and amperes) when connected directly with the alternator terminals, and when connected thereto through various amounts of resistance and induc- tive reactance.

The capacity reactance of the condenser is, 106

or, in symbolic expression,

Let

Z^ =.r — jn nv = impedance inserted in series with the condenser.

The total impedance of the circuit is then,

n The current in the circuit is,

(.3 + r) - j (x - 132) (.3 + r) -j3 (3 x - 29)

^8 ^6 -j

(.3 + r) -j, (5x- 1.4) (.3 + r) -j(7x + 16.1)J

420 ALTERNATING-CURRENT PHENOMENA.

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

; Jn V

4.4 js

(.3 + r) -A (x - 132) (.3 + r) - jz (3 * - 29)

__ 2.iiy5 1.13;; -i

(.3 + r) -j6 (5x- 1.4) ^ (.3 + r) -/7 (7 x + 16.1) J thus the apparent capacity reactance of the condenser is,

and the apparent capacity,

106

^.) ^r = 0 : Resistance r in series with the condenser. Reduced to absolute values, it is,

1 .01 .0064 .0036

17424 19.4

(.8+r)a+ 17424 (.3 +r)2 + 841 (.3 + r)2 + 1.96 (.3 -f r)2 +2

(£.) r = 0 : Inductive reactance x in series with the condenser. Reduced to absolute values, it is,

1 .01 .0064 __ .0036

— 1.42 "*".

1.4)2 .09+(7;r-f 16.

— 132)2 .

From —g are derived the values of apparent capacity,

c=

and plotted in Fig. 179 for values of r and x respectively varying from 0 to 22 ohms.

As seen, with neither additional resistance nor reactance in series to the condenser, the apparent capacity with this generator wave is 84 m.f., or 4.2 times the true capacity,

REPRESENTATION OF ALTERNATING WAVES. 421

and gradually decreases with increasing series resistance, to C= 27.5 m.f. = 1.375 times the true capacity at r= 13.2 ohms, or TV the true capacity reactance, with r = 132 ohms, or with an additional resistance equal to the capacity reac- tance, C = 20.5 m.f. or only 2.5% in excess of the true capacity C0, and at r = oo , C = 20,3 m.f. or 1.5% in excess of the true capacity.

With reactances, but no additional resistance r in series, the apparent capacity C rises from 4.2 times the true capacity at x = 0, to a maximum of 5,03 times the true capacity, or C= 100.6 m.f. at x = .28, the condition of res- onance of the fifth harmonic, then decreases to a minimum of 27 m.f., or 35 % in excess of the true capacity, rises again to 60.2 m.f., or 3.01 times the true capacity at x = 9.67, the condition of resonance with the third harmonic, and finally decreases, reaching 20 m.f., or the true capacity at x = 132, or an inductive reactance equal to the capacity reactance, then increases again to 20.2 m.f. at x = oo .

This rise and fall of the apparent capacity is within cer- tain limits independent of the magnitude of the higher harmonics of the generator wave of E.M.F., but merely de- pends upon their presence. That is, with such a reactance connected in series as to cause resonance with one of the higher harmonics, the increase of apparent capacity is ap- proximately the same, whatever the value of the harmonic, whether it equals 25% of the fundamental or less than 5%, provided the resistance in the circuit is negligible. The only effect of the amplitude of the higher harmonic is that when it is small, a lower resistance makes itself felt by re- ducing the increase of apparent capacity below the value it would have were the amplitude greater.

It thus follows that the true capacity of a condenser cannot even approximately be determined by measuring volts and amperes if there are any higher harmonics present in the generator wave, except by inserting a very large re- sistance or reactance in series to the condenser.

422

ALTERNATING-CURRENT PHENOMENA.

258. §d  instance :    An  alternating  current  generator 
     

of the wave,

E. = 2000 [lt + .12, - .23B - .13,]

and of synchronous impedance,

Z0 = .3-5*/; feeds over a line of impedance,

C4PJ

CITV

Co =

= 20

mf i

CM

CL'IT

OF

r,E\

HAT

R

1

8

= EI O-J--I.L— .ya-t-uc/ OF

Zo^S-S), n WITH RESIS

fASC

DANCE

k r(I)

!

c

R RE

ACT

NCE

*^

I) 1

SE

!ES

C:

£

100

/\

0

90

J

i

^ft

I

k

5

rn

I

\

\

i

H

^

/

\

.w

\

\

/

X

10

REE

X

STAC

ii

^=^~

CE r

=

;="

^

^

=

REA(

— - TAN!1

X

•-

^S

•^

• ,

;

—

=

^=

_»

=3<F

10

I ;

,

i !'o !

1

2 1

1

1

1

1

• 1 
  

-t 1

r, 2,

1

0

a synchronous motor of the wave,

EI = 2250 [(cos oj +/i sin «) + .24 (cos 3 w -(-y's sin 3 o>)] and of synchronous impedance,

Z2 = .3 - C «/;

The total impedance of the system is then, Z = ZQ + Zl + Z2 = 2.6-15«/n

REPRESENTATION OF ALTERNATING WAVES. 423

thus the current,

_ 2000 - 2250 cos o> - 2250/\ sin o> 240 - 540 cos 3a> - 540/; sin 3a> 2.6 - 15/i 2.6 - 45y8

460 260

~~ 2.6 - 75 j\ 2.6 - 105 jj

= «

where,

aj1 = 22.5 - 25.2 cos co + 146 sin a>

ag1 = .306 - .69 cos 3 to + 11.9 sin 3

a,1 = - .213

«7i = - .061

V1 = 130 - 146 cos w - 25.2 sin a>

^8« = 5.3 - 11.9 cos 3 o> - .69 sin 3 o>

a* = - 6.12

a7u = - 2.48

or, absolute,

1st harmonic,

3d harmonic,

5th harmonic,

a6 = 6.12 7th harmonic,

«7 = 2.48

/= V while the total current of higher harmonics is,

424 ALTERNATING-CURRENT PHENOMENA.

The true input of the synchronous motor is,

= ( 2250 a£ cos o> + 2250 a? sin o> ) + ( 540 a? cos 3o> + 540 asn sin 3o>)

= /V + /'s1 ^ = 2250 (a? cos <o + af sin o>)

. 780. Synchronous Motor,

REPRESENTATION OF ALTERNATING WAVES. 425

is the power of the fundamental wave,

P£ = 540 (a,,1 cos 3 w + as11 sin 3 o>)

the power of the third harmonic.

The 5th and 7th harmonics do not give any power, since they are not contained in the synchronous motor wave. Substituting now different numerical values for u> the phase angle between generator E.M.F. and synchronous motor counter E.M.F., corresponding values of the currents / 70, and the powers P\ P*, /Y are derived. These are plotted in Fig. 180 with the total current /as abcissae. To each value of the total current / correspond two values of the total power P\ a positive value plotted as Curve I. — synchronous motor — and a negative value plotted as Curve II. — alternating current generator — . Curve III. gives the total current of higher frequency I0, Curve IV., the difference between the total current and the current of fundamental frequency, / — alt in percentage of the total current /, and V the power of the third harmonic, Pj, in percentage of the total power P1.

Curves III., IV. and V. correspond to the positive or synchronous motor part of the power curve P\ As seen, the increase of current due to the higher harmonics is small, and entirely disappears at about 180 amperes. The power of the third harmonic is positive, that is, adds to the work of the synchronous motor up to about 140 amperes, or near the maximum output of the motor, and then becomes negative.