Theory and Calculation of Alternating Current Phenomena (1900) — part 16 of 19

It follows herefrom that higher harmonics in the E.M.F. waves of generators and synchronous motors do not repre- sent a mere waste of current, but may contribute more or less to the output of the motor. Thus at 75 amperes total current, the percentage of increase of power due to the higher harmonic is equal to the increase of current, or in other words the higher harmonics of current do work with the same efficiency as the fundamental wave.

426 ALTERNATING-CURRENT PHENOMENA.

259. kth Instance: In a small three-phase induction motor, the constants per delta circuit are

Primary admittance Y= .002 + .03/

Self-inductive impedance ZQ = Zl = .6 — 2.4/

and a sine wave of E.M.F. e0 = 110 volts is impressed upon the motor.

The power output P, current input 7S, and power factor /, as function of the slip s are given in the first columns of the following table, calculated in the manner as described in the chapter on Induction Motors.

To improve the power factor of the motor and bring it to unity at an output of 500 watts, a condenser capacity is required giving 4.28 amperes leading current at 110 volts, that is, neglecting the energy loss in the condenser, capacity susceptance

In this case, let Is = current input into the motor per delta circuit at slip s, as given in the following table.

The total current supplied by the circuit with a sine wave of impressed E.M.F., is

/i = ls - 4.28/

energy current and heref rom the power factor = - ; — — , given in

total current the second columns of the table.

If the impressed E.M.F. is not a sine wave but a wave of the shape

E, = e, (lx + .12. - .235 - .134,)

to give the same output, the fundamental wave must be the same : e0 = 110 volts, when assuming the higher harmonics in the motor as wattless, that is

£0 = 110, + 13.2, - 25.3B - 14.7,

= *o + £<? where £0l = 13.2, - 25.3B - 14.7T

= component of impressed E.M.F. of higher frequency-

REPRESENTATION^ Of ALTERNATING WAVES. 427

The effective value is :

EQ = 114.5 volts.

The condenser admittance for the general alternating wave is

Yc= -.039«/;

Since the frequency of rotation of the motor is very small compared with the frequency of the higher harmonics, as total impedance of the motor for these higher harmonics can be assumed the stationary impedance, and by neglecting the resistance it is

Z1 = - njn (XQ + XJ

= - 4.8 njn

The exciting admittance of the motor, for these higher harmonics, is, by neglecting the conductance,

n and the higher harmonics of counter E.M.F.

Thus we have,

Current input in the condenser,

fc = E, Yc

= - 4.28/i - 1.54/3 + 4.93/5 + 4.02/7

High frequency component of motor impedance current,

|£ = .92/3 - 1.06y5 - .44/7 High frequency component of motor exciting current,

= .07/3 - -08/5 - .

428

AL TERN A TING-CURRENT PHEA'OAIENA.

thus, total high frequency component of motor current,

/o1 = |f + & y1

= .99y3 - 1.14,; - .47/7 and total current,

without condenser,

4 = 4 + 41

= Is + .99/3 - 1.14,; - .47/7

with condenser,

= 4 - 4.28,i - . and herefrom the power factor.

3.79,; + 3.55/7

T PER PHASE

In the following table and in Fig. 181 are given the values of current and power factor : —

I. With sine wave of E.M.F., of 110 volts, and no condenser.

II. With sine wave of E.M.F , of 1 10 volts, and with condenser.

III. With distorted wave of E.M.F., of 114.6 volts, and no condenser.

IV. With distorted wave of E.M.F., of 114.5 volts, and with condenser.

REPRESENTATION OF ALTERNATING WAVES. 429

f 0 .01 .02 .035 .05 .07 .10 .13 .15

P

0 160 320 500 660 810 885 900 890

I, .24+ 3.10/ 1.73+ 3.16/ 3.32+ 3.47> 5.16+ 4.28/ 6.95+ 5.4/ 8.77+ 7.3; 10.1 + 9.85/ 10.45 + 11.45/ 10.75 + 12.9/

It 3.1 3.6 4.8 6.7 8.8 11.4 14.1 15.5 16.8

7.8 48 69 77 79 77 71.5 67.5 64

f

1.2 2.1 3.4 5.2 7.0 9.3 11.5 12.7 13.8

— •> P 20 84 97.2 100 98.7 94.5 87 82 78

i —

3.5 3.9 5.1 6.9 8.9 11.5 14.2 15.6 16.9

1 — \

6.6 43 64 72.5 76 73.5 68 64.5 61

/
I

5.2 5.5 6.1 7.2 8.6 10.6 12.6 13.7 14.7

— i

81 64

(18 7T 80 7T 73: 7Q/

The curves II. and IV. with condenser are plotted in dotted lines in Fig. 181. As seen, even with such a dis- torted wave the current input and power factor of the motor are not much changed if no condenser is used. When using a condenser in shunt to the motor, however, with such a wave of impressed E.M.F. the increase of the total current, due to higher frequency currents in the condenser, is greater than the decrease, due to the compensation of lagging cur- rents, and the power factor is actually lowered by the con- denser, over the total range of load up to overloads, and especially at light loads.

Where a compensator or transformer is used for feeding- the condenser, due to the internal self-induction of the com- pensator, the higher harmonics of current are still more accentuated, that is the power factor still more lowered.

In the preceding the energy loss in the condenser and compensator and that due to the higher harmonics of cur- rent in the motor has been neglected. The effect of this energy loss is a slight decrease of efficiency and correspond- ing increase of power factor. The power produced by the higher harmonics has also been neglected ; it may be posi- tive or negative, according to the index of the harmonic, and the winding of the motor primary. Thus for instance, the effect of the triple harmonic is negative in the quarter- phase motor, zero in the three-phase motor, etc., altogether,, however, the effect of these harmonics is very small.

430 ALTERNATING-CURRENT PHENOMENA.

CHAPTER XXV.

GENERAL POLYPHASE SYSTEMS.

260. A polyphase system is an alternating-current sys- tem in which several E.M.Fs. of the same frequency, but displaced in phase from each other, produce several currents of equal frequency, but displaced phases.

Thus any polyphase system can be considered as con- sisting of a number of single circuits, or branches of the polyphase system, which may be more or less interlinked with each other.

In general the investigation of a polyphase system is carried out by treating the single-phase branch circuits independently.

Thus all the discussions on generators, synchronous motors, induction motors, etc., in the preceding chapters, apply to single-phase systems as well as polyphase systems, in the latter case the total power being the sum of the powers of the individual or branch circuits.

If the polyphase system consists of n equal E.M.Fs. displaced from each other by 1 / n of a period, the system is called a symmetrical system, otherwise an unsymmetrical system.

Thus the three-phase system, consisting of three equal E.M.Fs. displaced by one-third of a period, is a symmetrical system. The quarter-phase system, consisting of two equal E.M.Fs. displaced by 90°, or one-quarter of a period, is an unsymmetrical system.

261. The flow of power in a single-phase system is pulsating ; that is, the watt curve of the circuit is a sine

GENERAL POLYPHASE SYSTEMS, 431

wave of double frequency, alternating between a maximum value and zero, or a negative maximum value. In a poly- phase system the watt curves of the different branches of the system are pulsating also. Their sum, however, or the total flow of power of the system, may be either constant or pulsating. In the first case, the system is called a balanced system, in the latter case an unbalanced system.

The three-phase system and the quarter-phase system, with equal load on the different branches, are balanced sys- tems ; with unequal distribution of load between the indi- vidual branches both systems become unbalanced systems.

Fig. 181.

Fig. 182.

The different branches of a polyphase system may be either independent from each other, that is, without any electrical interconnection, or they may be interlinked with each other. In the first case, the polyphase system is called an independent system, in the latter case an inter- linked system.

The three-phase system with star-connected or ring-con- nected generator, as shown diagrammatically in Figs. 181 and 182, is an interlinked system.

432

ALTERNATING-CURRENT PHENOMENA.

The four-phase system as derived by connecting four equidistant points of a continuous-current armature with four collector rings, as shown diagrammatically in Fig. 183,

Fig. 183.

is an interlinked system also. The four-wire quarter-phase system produced by a generator with two independent armature coils, or by two single-phase generators rigidly connected with each other in quadrature, is an independent system. As interlinked system, it is shown in Fig. 184, as star-connected four-phase system.

-E

r

Fig. 184.

262. Thus, polyphase systems can be subdivided into : Symmetrical systems and unsymmetrical systems. Balanced systems and unbalanced systems. Interlinked systems and independent systems. The only polyphase systems which have found practical application are :

The three-phase system, consisting of three E.M.Fs. dis-

GENERAL POLYPHASE SYSTEMS. 433

placed by one-third of a period, used exclusively as inter- linked system.

The quarter-phase system, consisting of two E.M.Fs. in quadrature, and used with four wires, or with three wires, which may be either an interlinked system or an indepen- dent system.

The six-phase system, consisting of two three-phase sys- tems in opposition to each other, and derived by transforma- tion from a three-phase system, in the alternating supply circuit of large synchronous converters.

The inverted three-phase system, consisting of two E.M.F.'s displaced from each other by 60°, and derived from two phases of a three-phase system by transformation with two transformers, of which the secondary of one is reversed with regard to its primary (thus changing the phase difference from 120° to 180° - 120° = 60°), finds a limited application in low tension distribution.

434 ALTERNATING-CURRENT PHENOMENA.

CHAPTER XXVI.

SYMMETRICAL POLYPHASE SYSTEMS.

263. If all the E.M.Fs. of a polyphase system are equal in intensity, and differ from each other by the same angle of difference of phase, the system is called a symmetrical polyphase system.

Hence, a symmetrical w-phase system is a system of n E.M.Fs. of equal intensity, differing from each other in phase by 1 / n of a period :

*i = E sin (3 ; e2=£sm((3-^L',

en = E sin ( ft - L V* ~ - \

The next E.M.F. is again :

^ = E sin (ft — 2 TT) = E sin ft.

In the polar diagram the n E.M.Fs. of the symmetrical 0-phase system are represented by n equal vectors, follow- ing each other under equal angles.

Since in symbolic writing, rotation by l/« of a period, or angle 2ir/n, is represented by multiplication with :

the E.M.Fs. of the symmetrical polyphase system are:

SYMMETRICAL POLYPHASE SYSTEMS. 435

/ 9 T- ? -rr

E( cos — + / sin — = • '

n

„ f 2 (n — 1) TT . . . 2 (« — 1) ^ f cos — -i - L -- -j sm — ^ - ^

' V »

The next E.M.F. is again :

E ( cos 2 -n- +j sin 2 TT) = .£ e" = .£. Hence, it is

27T . • . 27T n/?

e = cos - - -f J sm - = V 1. ;z «

Or in other words :

In a symmetrical «-phase system any E.M.F. of the system is expressed by :

e'-Ej

where : e = -y/1.

264. Substituting now for n different values, we get the different symmetrical polyphase systems, represented by

*E\

, n/T 2 7T . . 2 7T

where, e = vl = cos -- -j sin — • .

n n

1.) « = 1 e = 1 c«'^ = .£, the ordinary single-phase system.

2.) « = 2 e = - 1 J £ = £ and - £.

Since — ^ is the return of E, n = 2 gives again the single-phase system.

3 -1-/V3

436 ALTERNATING-CURRENT PHENOMENA.

The three E.M.Fs. of the three-phase system are :

-i-yV3

Consequently the three-phase system is the lowest sym- metrical polyphase system.

4.) n = 4, c = cos — +/ sin — =/, £2 = — 1, e3 = - /. 4 4

The four E.M.Fs. of the four-phase system are:

*£ = £, J£, -E, -JE. They are in pairs opposite to each other : E and — E • j E and —JE.

Hence can be produced by two coils in quadrature with each other, analogous as the two-phase system, or ordinary alternating-current system, can be produced by one coil.

Thus the symmetrical quarter-phase system is a four- phase system.

Higher systems, than the quarter-phase or four-phase system, have not been very extensively used, and are thus of less practical interest. A symmetrical six-phase system, derived by transformation from a three-phase system, has found application in synchronous converters, as offering a higher output from these machines, and a symmetrical eight- phase system proposed for the same purpose.

265. A characteristic feature of the symmetrical »- phase system is that under certain conditions it can pro- duce a M.M.F. of constant intensity.

If « equal magnetizing coils act upon a point under equal angular displacements in space, and are excited by the n E.M.Fs. of a symmetrical w-phase system, a M.M.F. of constant intensity is produced at this point, whose direction revolves synchronously with uniform velocity.

Let, n' =• number of turns of each magnetizing coil.

SYMMETRICAL POLYPHASE SYSTEMS. 437

E= effective value of impressed E.M.F. / = effective value of current.

Hence, & =n'f= effective M.M.F. of one of the magnetizing coils.

Then the instantaneous value of the M.M.F. of the coil acting in the direction 2 «•*'/» is :

The two rectangular space components of this M.M.F. are ;

and

Hence the M.M.F. of this coil can be expressed by the symbolic formula :

fi

n \ n

Thus the total or resultant M.M.F. of the n coils dis- placed under the n equal angles is :

or, expanded :

n

438 ALTERNATING-CURRENT PHENOMENA.

It is, however :

cos'2 — + / sin — cos — = £ ( 1 + cos — +/ sin —]

n n n V w w /

\ /

sin 2=1 cos ?Z£+ysin«2=£= ^Yl - cos i^'-ysin4^'

« » • « z y « «

_ ^ /I _ ,2A X

2(1-^

and, since:

5t<2< = 0,

it is, /= nn'f^ (-sin ft _ y cos ft),

or,

the symbolic expression of the M.M.F. produced by the « circuits of the symmetrical «-phase system, when exciting n equal magnetizing coils displaced in space under equal angles.

The absolute value of this M.M.F. is :

nn' I n"S n <5

V2 V2 2

Hence constant and equal w/V2 times the effective M.M.F. of each coil or «/2 times the maximum M.M.F. of each coil.

The phase of the resultant M.M.F. at the time repre- sented by the angle ft is :

tan w = — cot /8 ; hence w = /? — ^

That is, the M.M.F. produced by a symmetrical «-phase system revolves with constant intensity :

SYMMETRICAL POLYPHASE SYSTEMS. 439

F= — •

V25

and constant speed, in synchronism with the frequency of the system ; and, if the reluctance of the magnetic circuit is constant, the magnetism revolves with constant intensity and constant speed also, at the point acted upon symmetri- cally by the n M.M.Fs. of the w-phase system.

This is a characteristic feature of the symmetrical poly- phase system.

266. In the three-phase system, n = 3, F= 1.5 <5max where $max is the maximum M.M.F. of each of the magne- tizing coils.

In a symmetrical quarter-phase system, n = 4, F = 2 ^tnax, where $maje is the maximum M.M.F. of each of the four magnetizing coils, or, if only two coils are used, since the four-phase M.M.Fs. are opposite in phase by two, F = &max> where ^max is the maximum M.M.F. of each of the two magnetizing coils of the quarter-phase system.

While the quarter-phase system, consisting of two E.M.Fs. displaced by one-quarter of a period, is by its nature an unsymmetrical system, it shares a number of features — as, for instance, the ability of producing a constant result- ant M.M.F. — with the symmetrical system, and may be considered as one-half of a symmetrical four-phase system.

Such systems, consisting of one-half of a symmetrical system, are called hemisymmetrical systems.

440 ALTERNATING-CURRENT PHENOMENA.

CHAPTER XXVII.

BALANCED AND UNBALANCED POLYPHASE SYSTEMS.

267. If an alternating E.M.F. :

e = E V2 sin (3, produces a current :

• = 7V2sin (/? — a),

where u> is the angle of lag, the power is :

p = ei = 2 £Ssin ft sin (ft — S)

= £S(cos a — cos (2 £ — a)),

and the average value of power :

Substituting this, the instantaneous value of power is found as :

Hence the power, or the flow of energy, in an ordinary single-phase alternating-current circuit is fluctuating, and varies with twice the frequency of E.M.F. and current, unlike the power of a continuous-current circuit, which is

constant :

/-**

If the angle of lag £ = 0 it is :

p = P (1 — cos 2 0) ;

hence the flow of power varies between zero and 2 Pt where P is the average flow of energy or the effective power of the circuit.

BALANCED POLYPHASE SYSTEMS. 441

If the current lags or leads the E.M.F. by angle £ the power varies between

and

cos u>

that is, becomes negative for a certain part of each half- wave. That is, for a time during each half-wave, energy flows back into the generator, while during the other part of the half-wave the generator sends out energy, and the difference between both is the effective power of the circuit. If £ = 90°, it is :

O rt ,

" p >

that is, the effective power : P = 0, and the energy flows to and fro between generator and receiving circuit.

Under any circumstances, however, the flow of energy in the single-phase system is fluctuating at least between zero and a maximum value, frequently even reversing.

268. If in a polyphase system

*D ez> *s> • • • • = instantaneous values of E.M.F. ; h) *2, t'a, • • • • = instantaneous values of current pro- duced thereby ;

the total flow of power in the system is :

p = glt\ -f <?2/2 -j- e,j, -f . . . . The average flow of power is :

P = £i /i cos £>! -(- E<i /2 cos w2 -f- . . . .

The polyphase system is called a balanced system, if the flow of energy :

/ = e\i\ + <V2 + W, +.'.'.;..

is constant, and it is called an unbalanced system if the flow of energy varies periodically, as in the single-phase sys- tem ; and the ratio of the minimum value to the maximum value of power is called the balance factor of the system.

442 ALTERNATING-CURRENT PHENOMENA.

Hence in a single-phase system on non-inductive circuit, that is, at no-phase displacement, the balance factor is zero ; and it is negative in a single-phase system with lagging or leading current, and becomes = — 1, if the phase displace- ment is 90° — that is, the circuit is wattless.

269. Obviously, in a polyphase system the balance of the system is a function of the distribution of load between the different branch circuits.

A balanced system in particular is called a polyphase system, whose flow of Energy is constant, if all the circuits are loaded equally with a load of the same character, that is, the same phase displacement.

270. All the symmetrical systems from the three-phase system upward are balanced systems. Many unsymmetrical systems are balanced systems also.

1.) Three-phase system : Let

^ = E V2 sin ft, and t\ = I V2 sin (ft — w) ;

ez = E V2 sin (ft - 120), /2 = / V2 sin (0 - « - 120) ;

ez = E V2 sin (ft - 240), /3 = / V2 sin (ft - & - 240) ;

be the E.M.Fs. of a three-phase system, and the currents produced thereby.

Then the total flow of power is :

/ = 2 .57 (sin {3 sin (ft — fi) + sin ((3 — 120) sin (ft — & — 120)

• sin (ft — 240) sin ($ — <* — 240)) = 3 .£7 cos w = T5, or constant.

Hence the symmetrical three-phase system is a balanced system.

2.) Quarter-phase system :

Let £l = £^2s\nft, t\ = I /2 sin (ft - 5) ;

e2 = E V2 cos ft, 4 = 7 V2 cos (ft - £) ;

BALANCED POLYPHASE SYSTEMS. 443

be the E.M.Fs. of the quarter-phase system, and the cur- rents produced thereby.

This is an unsymmetrical" system, but the instantaneous flow of power is :

/ = 2 £I(sm J3 sin (/? — 5) + cos ft cos (0 — £>)) = 2 £Scos w = P, or constant.

Hence the quarter-phase system is an unsymmetrical bal- anced system.

3.) The symmetrical «-phase system, with equal load and equal phase displacement in all n branches, is a bal- anced system. For, let :

e( = E V2 sin ( ft - — "\ = E.M.F. ; V » /

/ 2 IT A

*',- = 7V2 sin O — S — = current

V » V

the instantaneous flow of power is :

l V « 7 \ »

EI \ yr cos a -57-035^2 /?-£- —

or p = n E I cos w = T7, or constant.

271. An unbalanced polyphase system is the so-called inverted three-phase system,* derived from two branches of a three-phase system by transformation by means of two transformers, whose secondaries are connected in opposite direction with respect to their primaries. Such a system takes an intermediate position between the Edison three- wire system and the three-phase system. It shares with the latter the polyphase feature, and with the Edison three-

• Also called "polyphase monocyclic system," since the E.M.F. triangle is similar to that usual in the single-phase monocyclic system.

444 ALTERNATING-CURRENT PHENOMENA.

wire system the feature that the potential difference be- tween the outside wires is higher than between middle wire and outside wire.

By such a pair of transformers the two primary E.M.Fs. of 120° displacement of phase are transformed into two secondary E.M.Fs. differing from each other by 60°. Thus in the secondary circuit the difference of potential between the outside wires is V3 times the difference of potential between middle wire and outside wire. At equal load on the two branches, the three currents are equal, and differ from each other by 120°, that is, have the same relative proportion as in a three-phase system. If the load on one branch is maintained constant, while the load of the other branch is reduced from equality with that in the first branch down to zero, the current in the middle wire first decreases, reaches a minimum value of 87 per cent of its original value, and then increases again, reaching at no load the same value as at full load.

The balance factor of the inverted three-phase system on non-inductive load is .333.

272. In Figs. 185 to 192 are shown the E.M.Fs. as e and currents as i in drawn lines, and the power as / in dotted lines, for :

Fig. 185. Single-phase System on Non-inductive Load.

Balance Factor, 0.

BALANCED POLYPHASE SYSTEMS. 445

Fig. 186. Single-phase System on Inductiue Load of 60° Lag.

Balance Factor, - .333.

Fig. 187. Quarter-phase System on Non-inductiui Load.

Balance Factor, + 1.

Fig. 183. Quarter-phase System on Inductiue Lozd of 60° Lag. Balance Factor, + 1.

446 ALTERNATING-CURRENT PHENOMENA.

Fig. 189. Three-phase System on Non-induct'we Load.

Balance Factor, + 1.

Fig. 190. Three-phase System on Inductive Load of 60° Lag.

Balance Factor, + 1.

Fig. 191. Inverted Three-phase System on Non-inductive Load.

Balance Factor, + .333

BALANCED POLYPHASE SYSTEMS.

447

Fig. 174. Inverted Three-phase System on

Inductive Load of 60° Lag.

Balance Factor, 0.

273. The flow of power in an alternating-current system is a most important and characteristic feature of the system, and by its nature the systems may be classified into :

Monocyclic systems, or systems with a balance factor zero or negative.

Polycyclic systems, with a positive balance factor.

Balance factor — 1 corresponds to a wattless circuit, balance factor zero to a non-inductive single-phase circuit, balance factor + 1 to a balanced polyphase system.

274. In polar coordinates, the flow of power of an alternating-current system is represented by using the in- stantaneous flow of power as radius vector, with the angle ($ corresponding to the time as amplitude, one complete period being represented by one revolution.

In this way the power of an alternating-current system is represented by a closed symmetrical curve, having the zero point as quadruple point. In the monocyclic systems the zero point is quadruple nodal point ; in the polycyclic system quadruple isolated point.

Thus these curves are sextics. «

448 ALTERNATING-CURRENT PHENOMENA.

Since the flow of power in any single-phase branch of the alternating-current system can be represented by a sine wave of double frequency :

the total flow of power of the system as derived by the addition of the powers of the branch circuits can be rep- resented in the form :

/ = />(! + « sin (2 £- a.))

This is a wave of double frequency also, with c as ampli- tude of fluctuation of power.

This is the equation of the power characteristics of the system in polar coordinates.

275. To derive the equation in rectangular coordinates we introduce a substitution which revolves the system of coordinates by an angle o>o/2, so as to make the symmetry axes of the power characteristic the coordinate axes.

hence, sin (2 ft - S>0) = 2 sin ^ - ^ ) cos (/? - ^ j = substituted,

^M' + ^j.

or, expanded :

— P2 (x2 + /* + 2 e A:^)2 = 0,

the sextic equation of the power characteristic. Introducing :

a = (! + «)/'= maximum value of power, b = (1 — c) P'= minimum value of power;

BALANCED POLYPHASE SYSTEMS. 449

it is **?>

a + b hence, substituted, and expanded :

(*»+/)» - {a (x + j)2 + b (x -X>T> = 0

the equation of the power characteristic, with the main power axes a and b, and the balance factor: b I a. It is thus :

Single-phase non-inductive circuit : / = /> (1 + sin 2 <£), b = 0, a = 2P

Single-phase circuit, 60° lag : / = P (1 + 2 sin 2 <£),

i*.~+"

Single-phase circuit, 90° lag :/ = ^ /sin 2 <£, b = — E I,

a = + El

2/, &/a= -1. Three-phase non-inductive circuit : p = P, ^ = 1, a =

x^+y* — P2 = 0: circle. & / a = + 1. Three-phase circuit, 60° lag : / = P, 6 = 1, a = 1

a? +/- 7>a = 0 : circle. £/«= + !. Quarter-phase non-inductive circuit :p = P,b = ]-) a =

x* | y» _ ^2 = o . circlei ^ / ^ = | i.

Quarter-phase circuit, 60° lag : p = P, b = 1, tf = 1

450 ALTERNATING-CURRENT PHENOMENA.

Inverted three-phase non-inductive circuit :

Inverted three-phase circuit 60° lag :/ = f (1 -- sin 2 <£), b = 0, a = 2 P

(y? + /)3 _ />2 (• x | yy = 0< fila = Qf

a and <5 are called the main power axes of the alternating- current system, and the ratio b [a is the balance factor of the system.

Figs. 193 and 104. Power Characteristic of Single-phase System, at 60° and 0° Lag.

276. As seen, the flow of power of an alternating-cur- rent system is completely characterized by its two main power axes a and b.

The power characteristics in polar coordinates, corre-

BALANCED POLYPHASE SYSTEM.

451

spending to the Figs. 185, 186, 191, and 192 are shown in Figs. 193, 194, 195, and 196.

Figs. 195 and 196. Power Characteristic of Inverted Three-phase System, at 0° and 60° Lag.

The balanced quarter-phase and three-phase systems give as polar characteristics concentric circles.

452 ALTERNATING-CURRENT PHENOMENA.

CHAPTER XXVIII.

INTERLINKED POLYPHASE SYSTEMS.

277. In a polyphase system the different circuits of displaced phases, which constitute the system, may either be entirely separate and without electrical connection with each other, or they may be connected with each other electrically, so that a part of the electrical conductors are in common to the different phases, and in this case the system is called an interlinked polyphase system.

Thus, for instance, the quarter-phase system will be called an independent system if the two E.M.Fs. in quadra- ture with each other are produced by two entirely separate coils of the same, or different but rigidly connected, arma- tures, and are connected to four wires which energize inde- pendent circuits in motors or other receiving devices. If the quarter-phase system is derived by connecting four equidistant points of a closed-circuit drum or ring-wound armature to the four collector rings, the system is an inter- linked quarter-phase system.

Similarly in a three-phase system. Since each of the three currents which differ from each other by one-third of a period is equal to the resultant of the other two cur- rents, it can be considered as the return circuit of the other two currents, and an interlinked three-phase system thus consists of three wires conveying currents differing by one- third of a period from each other, so that each of the three currents is a common return of the other two, and inversely.

278. In an interlinked polyphase system two ways exist of connecting apparatus into the system.

INTERLINKED POLYPHASE SYSTEMS.

453

1st. The star connection, represented diagrammatically in Fig. 197. In this connection the n circuits excited by currents differing from each other by 1 / n of a period, are connected with their one end together into a neutral point or common connection, which may either be grounded or connected with other corresponding neutral points, or insu- lated.

In a three-phase system this connection is usually called a Y connection, from a similarity of its diagrammatical rep- resentation with the letter Y, as shown in Fig. 181.

2d. The ring connection, represented diagrammatically in Fig. 198, where the n circuits of the apparatus are con- nected with each other in closed circuit, and the corners or points of connection of adjacent circuits connected to the n lines of the polyphase system. In a three-phase system this connection is called the delta connection, from the similarity of its diagrammatic representation with the Greek letter Delta, as shown in Fig. 182.

In consequence hereof we distinguish between star- connected and ring-connected generators, motors, etc., or

454 ALTERNATING-CURRENT PHENOMENA.

Fig. 198.

in three-phase systems Y- connected and delta-connected apparatus.

279. Obviously, the polyphase system as a whole does not differ, whether star connection or ring connection is used in the generators or other apparatus ; and the trans- mission line of a symmetrical «-phase system always con- sists of n wires carrying current of equal strength, when balanced, differing from each other in phase by l/« of a period. Since the line wires radiate from the n terminals of the generator, the lines can be considered as being in star connection.

The circuits of all the apparatus, generators, motors, etc., can either be connected in star connection, that is, between one line and a neutral point, or in ring connection, that is, between two adjacent lines.

In general some of the apparatus will be arranged in star connection, some in ring connection, as the occasion may require.

INTERLINKED POLYPHASE SYSTEMS. 455

280. In the same way as we speak of star connection and ring connection of the circuits of the apparatus, the term star potential and ring potential, star current and ring current, etc., are used, whereby as star potential or in a three-phase circuit Y potential, the potential difference be- tween one of the lines and the neutral point, that is, a point having the same difference of potential against all the lines, is understood ; that is, the potential as measured by a volt- meter connected into star or Y connection. By ring or delta potential is understood the difference of potential between adjacent lines, as measured by a voltmeter con- nected between adjacent lines, in -ring or delta connec- tion.

In the same way the star or Y current is the current flowing from one line to a neutral point ; the ring or delta current, the current flowing from one line to the other.

The current in the transmission line is always the star or Y current, and the potential difference between the line wires, the ring or delta potential.

Since the star potential and the ring potential differ from each other, apparatus requiring different voltages can be connected into the same polyphase mains, by using either star or ring connection.

281. If in a generator with star-connected circuits, the E.M.F. per circuit = E, and the common connection or neutral point is denoted by zero, the potentials of the n terminals are :

or in general : t* JS,

at the z'th terminal, where :

• = 0, 1, 2 ....»- 1, e = cos — +j sin — = -/l.

456 ALTERNATING-CURRENT PHENOMENA.

Hence the E.M.F. in the circuit from the zth to the £* terminal is :

Eki = ** E — ^E = (c* — e') E.

The E.M.F. between adjacent terminals i and i + 1 is :

(e.+i -J)E = e* (e - 1) E.

In a generator with ring-connected circuits, the E.M.F.

per circuit :

cl E

is the ring E.M.F., and takes the place of

while the E.M.F. between terminal and neutral point, or the star E.M.F., is :

Hence in a star-connected generator with the E.M.F. E per circuit, it is :

Star E.M.F., IE.

RingE-M.F., c'Xc-1)^.

E.M.F. between terminal / and terminal k, (c* — e') E.

In a ring-connected generator with the E.M.F. E per circuit, it is :

Star E.M.F., — ^— E. e — 1 '

Ring E.M.F., C E.

E.M.F. between terminals * and k, e ~ e* E.

£ — 1 '

In a star-connected apparatus, the E.M.F. and the cur- rent per circuit have to be the star E.M.F. and the star current. In a ring-connected apparatus the E.M.F. and current per circuit have to be the ring E.M.F. and ring current.

In the generator of a symmetrical polyphase system, if : c'' E are the E.M.Fs. between the n terminals and the neutral point, or star E.M.Fs.,

INTERLINKED POLYPHASE SYSTEMS. 457

If = the currents issuing from terminal i over a line of the impedance Z{ (including generator impedance in star connection), we have :

Potential at end of line i :

Difference of potential between terminals k and i :

where /,. is the star current of the system, Zt the star im- pedance.

The ring potential at the end of the line between ter- minals i and k is Eik, and it is :

Eile = — Eti.

If now Iik denotes the current passing from terminal i to terminal k, and Zik impedance of the circuit between ter- minal i and terminal k, where :

fit = ~ /,, Zt = Zti,

it is Eik = ZitIik.

If Iio denotes the current passing from terminal i to a ground or neutral point, and Zio is the impedance of this circuit between terminal i and neutral point, it is :

Eio = €*£- ZiSi = Ziolio.