Theory and Calculation of Alternating Current Phenomena (1900) — part 17 of 19

282. We have thus, by Ohm's law and Kirchhoff 's law :

If *' E is the E.M.F. per circuit of the generator, be- tween the terminal i and the neutral point of the generator, or the star E.M.F.

/,- = the current issuing from the terminal i of the gen- erator, or the star current.

Zt = the impedance of the line connected to a terminal i of the generator, including generator impedance.

EL = the E.M.F. at the end of line connected to a ter- minal i of the generator.

458 ALTERNATING-CURRENT PHENOMENA.

Eik = the difference of potential between the ends of the lines i and k.

Iik = the current passing from line i to line k.

Zik = the impedance of the circuit between lines i and k.

Iio, Iioo . . . . = the current passing from line i to neu- tral points 0, 00, ....

Zio, Zioo . . . . = the impedance of the circuits between line i and neutral points 0, 00, ....

It is then :

Zio = Zoi, etc. 2.) Et =JE-ZiIi.

3.) Ei = Zi0fi0 = Zioofj00 = . . . .

4.) Eik = Et'- E{ = (t* - e') E - (Zklk - ZJ^).

5.) Eik = ZikIik.

7.) If the neutral point of the generator does not exist, as in ring connection, or is insulated from the other neutral points :

IE/,, =0;

n

5E/ioo = 0, etc. 1

Where 0, 00, etc., are the different neutral points which are insulated from each other.

If the neutral point of the generator and all the other neutral points are grounded or connected with each other, it is:

INTERLINKED POLYPHASE SYSTEMS. 459

If the neutral point of the generator and all other neu- tral points are grounded, the system is called a grounded system. If the neutral points are not grounded, the sys- tem is an insulated polyphase system, and an insulated polyphase system with equalizing return, if all the neutral points are connected with each other.

8.) The power of the polyphase system is —

P = ^f e1' E Ii cos $i at the generator

1

•f = "^i ^* Eik Iik cos <f>it in the receiving circuits.

4GO ALTERNATING-CURRENT PHENOMENA.

CHAPTER XXIX.

TRANSFORMATION OF POLYPHASE SYSTEMS.

283. In transforming a polyphase system into another polyphase system, it is obvious that the primary system must have the same flow of power as the secondary system, neglecting losses in transformation, and that consequently a balanced system will be transformed again in a balanced system, and an unbalanced system into an unbalanced sys- tem of the same balance factor, since the transformer is an apparatus not able to store energy, and thereby to change the nature of the flow of power. The energy stored as magnetism, amounts in a well-designed transformer only to a very small percentage of the total energy. This shows the futility of producing symmetrical balanced polyphase systems by transformation from the unbalanced single-phase system without additional apparatus able to store energy efficiently, as revolving machinery.

Since any E.M.F. can be resolved into, or produced by, two components of given directions, the E.M.Fs. of any polyphase system can be resolved into components or pro- duced from components of two given directions. This en- ables the transformation of any polyphase system into any other polyphase system of the same balance factor by two transformers only.

284. Let Elt E2, Ez . . . . be the E.M.Fs. of the primary system which shall be transformed into —

E{, £2', £s' . . . . the E.M.Fs. of the secondary system.

Choosing two magnetic fluxes, <£ and <£, of different

TRANSFORMATION OF POLYPHASE SYSTEMS, 461

phases, as magnetic circuits of the two transformers, which induce the E.M.Fs., e and ?, per turn, by the law of paral- lelogram the E.M.Fs., Elf E^, . . . . can be dissolved into two components, El and Elt E^ and Ez, .... of the phases* "e and J. Then, -

E!, £2, • • ' • are the counter E.M.Fs. which have to be- induced in the primary circuits of the first transformer;.

Ev E2, .... the counter E.M.F.'s which have to be in- duced in the primary circuits of the second transformer..

hence

EI 1 7, £2 1 J . . . . are the numbers of turns of the primary coils of the first transformer.

Analogously

EI /T £2 IT . . . . are the number of turns of the primary coils in the second transformer.

In the same manner as the E.M.Fs. of the primary system have been resolved into components in phase with J and FJ the E.M.Fs. of the secondary system, E-^> E^, ....

are produced from components, E-f and E^, E£ and EJ, .... in phase with ~e and J, and give as numbers of second ary turns, —

£il / J, £2l /?»•••• in the first transformer ; EI 1 7, EZ / F, .... in the second transformer.

That means each of the two transformers m and m con- tains in general primary turns of each of the primary phases, and secondary turns of each of the secondary phases. Loading now the secondary polyphase system in any desired manner, corresponding to the secondary cur- rents, primary currents will flow in such a manner that the total flow of power in the primary polyphase system is the

4j^ ALTERNATING-CURRENT PHENOMENA.

same as the total flow of power in the secondary system, plus the loss of power in the transformers.

285. As an instance may be considered the transforma- tion of the symmetrical balanced three-phase system

E sin ft, E sin (ft — 120), E sin (ft — 240), in an unsymmetrical balanced quarter-phase system :

E' sin ft, E' sin (ft — 90). Let the magnetic flux of the two transformers be

(/> cos £ and </> cos (ft — 90).

Then the E.M.Fs. induced per turn in the transformers e sin ft and e sin (ft — 90) ;

hence, in the primary circuit the first phase, E sin ft, will give, in the first transformer, E/e primary turns; in the second transformer, 0 primary turns.

The second phase, E sin (ft — 120), will give, in the first transformer, — E / 2 e primary turns; in the second

E x ~/3

transformer, — — primary turns.

2 e

The third phase, E sin (ft — 240), will give, in the first transformer, — E /le primary turns; in the second trans- former, — primary turns.

2 e

In the secondary circuit the first phase E' sin ft will give in the first transformer: E' / e secondary turns; in the second transformer : 0 secondary turns.

The second phase : E' sin (ft — 90) will give in the first transformer : 0 secondary turns ; in the second transformer, E' I e secondary turns.

Or, if :

E = 5,000 E' = 100, e = 10.

TRANSFORMATION OF POLYPHASE SYSTEMS. 463

PRIMARY. 1st. 2d.

SECONDARY. 3d. 1st. 2d. Phase.

first transformer second transformer

• 500 0

• 250 - 250

4- 433 - 433

10 0

0 10 turns.

That means :

Any balanced polyphase system *.jm be transformed by two transformers only, without storage of energy, into any other balanced polyphase system.

286. Some of the more common methods of transfor- mation between polyphase systems are :

Fig. 799.

1. The delta -Y connection of transformers between three-phase systems, shown in Fig. 199. One side of the transformers is connected in delta, the other in Y. This arrangement becomes necessary for feeding four wires

rwi nnr V

Fig. 200.

three-phase secondary distributions. The Y connection of the secondary allows to bring out a neutral wire, while the delta connection of the primary maintains the balance be- tween the phases at unequal distribution of load.

464

ALTERNA TING-CURRENT PHENOMENA.

2. The L connection of transformers between three-phase systems, consisting in using two sides of the triangle only, as shown in Fig. 200. This arrangement has the disadvan- tage of transforming one phase by two transformers in series, hence is less efficient, and is liable to unbalance the system by the internal impedance of the transformers.

Fig. 201.

3. The main and teaser, or T connection of trans- formers between three-phase systems, as shown in Fig. 201.

V3 One of the two transformers is wound for -- times the

voltage of the other (the altitude of the equilateral triangle), and connected with one of its ends to the center of the

Fig. 202.

other transformer. From the point £ inside of the teaser transformer, a neutral wire can be brought out in this con- nection.

4. The monocyclic connection, transforming between three-phase and inverted three-phase or polyphase mono- cycle, by two transformers, the secondary of one being reversed regarding its primary, as shown in Fig. 202.

TRANSFORMATION OF POLYPHASE SYSTEMS. 465

5. The L connection for transformation between quar- ter-phase and three-phase as described in the instance, para- graph 257.

6. The T connection of transformation between quarter- phase and three-phase, as shown in Fig. 203. The quar- ter-phase side of the transformers contains two equal and

Fig. 203.

independent (or interlinked) coils, the three-phase side two

Vs

coils with the ratio of turns 1 -=- — ^ connected in T.

7. The double delta connection of transformation from three-phase to six-phase, shown in Fig. 204. Three trans- formers, with two secondary coils each, are used, one set of

Fig 204.

secondary coils connected in delta, the other set in delta also, but with reversed terminals, so as to give a reversed E.M.F. triangle. These E.M.F.'s thus give topographically a six-cornered star.

466

AL TERN A TING-CURRENT PHENOMENA.

8. The double Y connection of transformation from three-phase to six-phase, shown in Fig. 205. It is analo- gous to (7), the delta connection merely being replaced by the Y connection. The neutrals of the two F's may be connected together and to an external neutral if desired.

9. The double T connection of transformation from

Fig. 205.

three-phase to six-phase, shown in Fig. 206. Two trans- formers are used with two secondary coils which are T con- nected, but one with reversed terminals. This method allows a secondary neutral also to be brought out.

287. Transformation with a change of the balance factor of the system is possible only by means of apparatus

\

\

•/

/

y

/ \

y

2' v '

Fig. 208.

able to store energy, since the difference of power between primary and secondary circuit has to be stored at the time when the secondary power is below the primary, and re- turned during the time when the primary power is below

TRANSPORMATION OF POLYPHASE SYSTEMS. 467

the secondary. The most efficient storing device of electric energy is mechanical momentum in revolving machinery. It has, however, the disadvantage of requiring attendance ; fairly efficient also are capacities and inductances, but, as a rule, have the disadvantage not to give constant potential.

468 ALTERNATING-CURRENT PHENOMENA.

CHAPTER XXX.

EFFICIENCY OF SYSTEMS.

288. In electric power transmission and distribution, wherever the place of consumption of the electric energy is distant from the place of production, the conductors which transfer the current are a sufficiently large item to require consideration, when deciding which system and •what potential is to be used.

In general, in transmitting a given amount of power at a given loss over a given distance, other things being equal, the amount of copper required in the conductors is inversely proportional to the square of the potential used. Since the total power transmitted is proportional to the product of current and E.M.F., at a given power, the current will vary inversely proportional to the E.M.F., and therefore, since the loss is proportional to the product of current- square and resistance, to give the same loss the resistance must vary inversely proportional to the square of the cur- rent, that is, proportional to the square of the E.M.F. ; and since the amount of copper is inversely proportional to the resistance, other things being equal, the amount of copper varies inversely proportional to the square of the E.M.F. used.

This holds for any system.

Therefore to compare the different systems, as two-wire single-phase, single-phase three-wire, three-phase and quar- ter-phase, equality of the potential must be assumed.

Some systems, however, as for instance, the Edison three-wire system, or the inverted three-phase system, have

EFFICIENCY OF SYSTEMS. 409

different potentials in the different circuits constituting the system, and thus the comparison can be made either —

1st. On the basis of equality of the maximum potential difference in the system ; or

2d. On the basis of the minimum potential difference in the system, or the potential difference per circuit or phase of the system.

In low potential circuits, as secondary networks, where the potential is not limited by the insulation strain, but by the potential of the apparatus connected into the system, as incandescent lamps, the proper basis of comparison is equality of the potential per branch of the system, or per phase.

On the other hand, in long distance transmissions where the potential is not restricted by any consideration of ap- paratus suitable for a certain maximum potential only, but where the limitation of potential depends upon the problem of insulating the conductors against disruptive discharge, the proper comparison is on the basis of equality of the maximum difference of potential in the system ; that is, •equal maximum dielectric strain on the insulation.

The same consideration holds in moderate potential power circuits, in considering the danger to life from live wires entering human habitations.

Thus the comparison of different systems of long-dis- tance transmission at high potential or power distribution for motors is to be made on the basis of equality of the maximum difference of potential existing in the system. The comparison of low potential distribution circuits for lighting on the basis of equality of the minimum difference of potential between any pair of wires connected to the receiving apparatus.

289. 1st. Comparison on the basis of equality of the minimum difference of potential, in low potential lighting circuits :

4TO ALTERNATING-CURRENT PHENOMENA.

In the single-phase alternating-current circuit, if e — E.M.F., i = current, r— resistance per line, the total power is = ei, the loss of power 2z'V.

Using, however, a three-wire system, the potential be- tween outside wires and neutral being given = e, the potential between the outside wires is == 2 e, that is, the dis- tribution takes place at twice the potential, or only -'• the copper is needed to transmit the same power at the same loss, if, as it is theoretically possible, the neutral wire has no cross-section. If therefore the neutral wire is made of the same cross-section with each of the outside wires, | of the copper of the two- wire system is needed ; if the neutral wire is £ the cross-section of each of the outside wires, T% of the copper is needed. Obviously, a single-phase five-wire system will be a system of distribution at the potential 4 e, and therefore require only TV °f the copper of the single- phase system in the outside wires ; and if each of the three neutral wires is of i the cross-section of the outside wires, /? = 10.93 per cent of the copper.

Coming now to the three-phase system with the poten- tial e between the lines as delta potential, if i = the current per line or Y current, the current from line to line or delta current = ^ / VB ; and since three branches are used, the total power is 3 e i\ / V3 == e z'x V3. Hence if the same power has to be transmitted by the three-phase system as with the single-phase system, the three-phase line current must be z'i = i / V3 where i — single-phase current, r = single-phase resistance per line, at equal power and loss; hence if 1\ = resistance of each of the three wires, the loss per wire is i? rt = iz rt /.3, and the total loss is z2 1, while in the single-phase system it is 2 t*r. Hence, to get the same loss, it must be : rv = 2 r, that is, each of the three three- phase lines has twice the resistance — that is, half the cop- per of each of the two single-phase lines ; or in other words, the three-phase system requires three-fourths of the copper of the single-phase system of the same potential.

EFFICIENCY OF SYSTEMS. 471

Introducing, however, a fourth or neutral wire into the three-phase system, and connecting the lamps between the neutral wire and the three outside wires — that is, in Y con- nection— the potential between the outside wires or delta potential will be = e X V3, since the Y potential = e, and the potential of the system is raised thereby from e to e V3 ; that is, only J as much copper is required in the out- side wires as before — that is \ as much copper as in the single-phase two-wire system. Making the neutral of the same cross-section as the outside wires, requires \ more copper, or \ = 33.3 per cent of the copper of the single- phase system ; making the neutral of half cross-section, requires \ more, or ^ = 29.17 per cent of the copper of the single-phase system. The system, however, now is a four-wire system.

The independent quarter-phase system with four wires is identical in efficiency to the two-wire single-phase sys- tem, since it is nothing but two independent single-phase systems in quadrature.

The four-wire quarter-phase system can be used as two independent Edison three-wire systems also, deriving there- from the same saving by doubling the potential between the outside wires, and has in this case the advantage, that by interlinkage, the same neutral wire can be used for both phases, and thus one of the neutral wires saved.

In this case the quarter-phase system with common neu- tral of full cross-section requires -fo = 31.25 per cent, the quarter-phase system with common neutral of one-half cross- section requires ^ = 28.125 per cent, of the copper of the two-wire single-phase system.

In this case, however, the system is a five-wire system, and as such far inferior to the five-wire single-phase system.

Coming now to the quarter-phase system with common return and potential e per branch, denoting the current in the outside wires by z'2, the current in the central wire is *a V2 ; and if the same current density is chosen for all

472 ALTERNATING-CURRENT PHENOMENA.

three wires, as the condition of maximum efficiency, and the resistance of each outside wire denoted by rz, the re- sistance of the central wire = r2/V2, and the loss of power per outside wire is z'22 r2 , in the central wire 2 z'22 r2 / V2 = z'22 r2 V2 ; hence the total loss of power is 2 z'22 r2 + z'22 r2 V2 = z'22 r2 (2 -f V2). The power transmitted per branch is z'2 ^, hence the total power 2 z'2 e. To transmit the same power as by a single-phase system of power, e z, it must

be z2 = z'/2; hence the loss, *2;a(2 + ^ . Since this loss shall be the same as the loss 2z'2r in the single- phase system, it must be 2 r = - — — r2 , or r2 = ~ . .

2 -}- V 2

° 4- V^ Therefore each of the outside wires must be — — times

o

as large as each single-phase wire, the central wire V2 times larger ; hence the copper required for the quarter- phase system with common return bears to the copper required for the single-phase system the relation :

2 (2 + V2) (2 + V5) V2 . 9 3 + 2V2

^~ T ~T~~

per cent of the copper of the single-phase system.

Hence the quarter-phase system with common return saves 2 per cent more copper than the three-phase system, but is inferior to the single-phase three-wire system.

The inverted three-phase system, consisting of two E.M.Fs. e at 60° displacement, and three equal currents /8 in the three lines of equal resistance r3, gives the out- put 2^z'3, that is, compared with the single-phase system, /8 = z'/2. The loss in the three lines is 3 z'32 r3 = | z2 rs. Hence, to give the same loss 2 z'2 r as the single-phase sys- tem, it must be rs = f r, that is, each of the three wires must have f of the copper cross-section of the wire in the two-wire single-phase system ; or in other words, the in- verted three-phase system requires ^ of the copper of the two-wire single-phase system.

EFFICIENCY OF SYSTEMS.

473

We get thus the result,

If a given power has to be transmitted at a given loss, and a given minimum potential, as for instance 110 volts for lighting, the amount of copper necessary is :

2 WIRES : Single-phase system, 100.0

3 WIRES : Edison three-wire single-phase sys-

tem, neutral full section, 37.5 Edison three-wire single-phase sys- tem, neutral half-section, 31.25 Inverted three-phase system, 56.25 Quarter-phase system with common

return, 72.9

Three-phase system, 75.0

4 WIRES : Three-phase, with neutral wire full

section, 33.3

Three-phase, with neutral wire half- section, 29.17 Independent quarter-phase system, 100.0

5 WIRES : Edison five-wire, single-phase system,

full neutral, 15.625

Edison five-wire, single-phase system,

half-neutral, 10.93

Four-wire, quarter-phase, with com- mon neutral full section, 31.25 Four-wire, quarter-phase, with com- mon neutral half-section, 28.125

We see herefrom, that in distribution for lighting — that is, with the same minimum potential, and with the same number of wires — the single-phase system is superior to any polyphase system.

The continuous-current system is equivalent in this' comparison to the single-phase alternating-current system of the same effective potential, since the comparison is made on the basis of effective potential, and the power depends upon the effective potential also.

474 AL TERNA TING-CURRENT PHENOMENA.

290. Comparison on the Basis of Equality of the Maximum Difference of Potential in the System, in Long- Distance Transmission, Power Distribution, etc.

Wherever the potential is so high as to bring the ques- tion of the strain on the insulation into consideration, or in other cases, to approach the danger limit to life, the proper comparison of different systems is on the basis of equality of maximum potential in the system.

Hence in this case, since the maximum potential is fixed, nothing is gained by three- or five-wire Edison sys- tems. Thus, such systems do not come into consideration.

The comparison of the three-phase system with the single-phase system remains the same, since the three- phase system has the same maximum as minimum poten- tial ; that is :

The three-phase system requires three-fourths of the copper of the single-phase system to transmit the same power at the same loss over the same distance.

The four-wire quarter-phase system requires the same amount of copper as the single-phase system, since it con- sists of two single-phase systems.

In a quarter-phase system with common return, the potential between the outside wire is V2 times the poten- tial per branch, hence to get the same maximum strain on the insulation — that is, the same potential e between the outside wires as -in the single-phase system — the potential per branch will be ej V2, hence the current z'4 = t/ V2, if i equals the current of the single-phase system of equal power, and t\ V2 = i will be the current in the central wire.

Hence, if r± = resistance per outside wire, r± / V2 = resistance of central wire, and the total loss in the sys- tem is :

, (2 + V2) =

EFFICIENCY OF SYSTEMS. 475

Since in the single-phase system, the loss = 2 i 2 r, it is :

2 + ~v/2 That is, each of the outside wires has to contain — — - -

4 times as much copper as each of the single-phase wires.

2 x V2 /- The central wires have to contain - - V 2 times as

^ (^ -4- ~v/2^ much copper ; hence the total system contains

2 +V2 — T - V2 times as much copper as each of the single-

3 + 2 ~/2

phase wires ; that is, - — times the copper of the

4

single-phase system. Or, in other words, A quarter-phase system with common return requires

3 + 2 A/2

— == 1.457 times as much copper as a single-phase

system of the same maximum potential, same power, and same loss.

Since the comparison is made on the basis of equal maximum potential, and the maximum potential of alter- nating system is A/2 times that of a continuous-current circuit of equal effective potential, the alternating circuit of effective potential e compares with the continuous-cur- rent circuit of potential e A/2, which latter requires only half the copper of the alternating system.

This comparison of the alternating with the continuous- current system is not proper however, since the continuous- current potential introduces, besides the electrostatic strain, an electrolytic strain on the dielectric which does not exist in the alternating system, and thus makes the action of the continuous-current potential on the insulation more severe than that of an equal alternating potential. Besides, self- induction having no effect on a steady current, continuous current circuits as a rule have a self-induction far in excess

476 ALTERNATING-CURRENT PHENOMENA.

of any alternating circuit. During changes of current, as make and break, and changes of load, especially rapid changes, there are consequently induced in these circuits E.M.F.'s far exceeding their normal potentials. At the voltages which came under consideration, the continuous current is excluded to begin with.

Thus we get :

If a given power is to be transmitted at a given loss, and a given maximum difference of potential in the system, that is, with the same strain on the insulation, the amount of copper required is :

2 WIRES : Single-phase system, 100.0

[Continuous-current system, 50.0]

3 WIRES : Three-phase system, 75.0

Quarter-phase system, with common return, 145.7

4 WIRES : Independent Quarter-phase system, 100.0

Hence the quarter-phase system with common return is practically excluded from long-distance transmission.

291 . In a different way the same comparative results between single-phase, three-phase, and quarter-phase sys- tems can be derived by resolving the systems into their single-phase branches.

The three-phase system of E.M.F. e between the lines can be considered as consisting of three single-phase cir- cuits of E.M.F. ^/V3, and no return. The single-phase system of E.M.F. e between lines as consisting of two single-phase circuits of E.M.F. <?/2 and no return. Thus, the relative amount of copper in the two systems being inversely proportional to the square of E.M.F., bears the relation ( V3 / e)2 : (2 / ef = 3 : 4 ; that is, the three-phase system requires 75 per cent of the copper of the single- phase system.

The quarter-phase system with four equal wires requires the same copper as the single-phase system, since it consists

EFFICIENCY OF SYSTEMS. 477

of two single-phase circuits. Replacing two of the four quarter-phase wires by one wire of the same cross-section as each of the wires replaced thereby, the current in this wire is V2 times as large as in the other wires, hence, the loss twice as large — that is, the same as in the two wires replaced by this common wire, or the total loss is not changed — while 25 per cent of the copper is saved, and the system requires only 75 per cent of the copper of the single-phase system, but produces V2 times as high a potential between the outside wires. Hence, to give the same maximum potential, the E.M.Fs. of the system have to be reduced by V2, that is, the amount of copper doubled, and thus the quarter-phase system with common return of the same cross-section as the outside wires requires 150 per cent of the copper of the single-phase system. In this case, however, the current density in the middle wire is higher, thus the copper not used most economical, and transferring a part of the copper from the outside wire to the middle wire, to bring all three wires to the same current density, reduces the loss, and thereby reduces the amount of copper at a given loss, to 145.7 per cent of that of a single-phase system.

478 ALTERNATING-CURRENT PHENOMENA.

CHAPTER XXXI.

THREE-PHASE SYSTEM.

292. With equal load of the same phase displacement in all three branches, the symmetrical three-phase system offers no special features over those of three equally loaded single-phase systems, and can be treated as such ; since the mutual reactions between the three phases balance at equal distribution of load, that is, since each phase is acted upon by the preceding phase in an equal but opposite manner as by the following phase.

With unequal distribution of load between the different branches, the voltages and phase differences become more or less unequal. These unbalancing effects are obviously maxi- mum, if some of the phases are fully loaded, others unloaded,

Let:

E — E.M.F. between branches 1 and 2 of a three-phaser. Then:

« E = E.M.F. between 2 and 3, (*£= E.M.F. between 3 and 1,

where, e= ^1= ~ -

Let

ZD Z2, Zs = impedances of the lines issuing from genera-

tor terminals 1, 2, 3, and Yl} Y2, Ys = admittances of the consumer circuits con-

nected between lines 2 and 3, 3 and 1, 1 and 2. Jf then,

ID It, /8, are the currents issuing from the generator termi- nals into the lines, it is,

/I + /2 + /3 = 0. (1)

THREE-PHASE SYSTEM. 479

If //, 72', 7/ = currents flowing through the admittances Y1, F2, F3, from 2 to 3, 3 to 1, 1 to 2, it is,

/! = /,'-/,', or, /1 + /2'_/3' = Ol

,->/-/.', or, /2 + /3'-7/ = o[ (2)

3 = //->/, or, /3 + >1/-// = OJ

These three equations (2) added, give (1) as dependent equation.

At the ends of the lines 1, 2, 3, it is :

(3) Il + ztIt) •

the differences of potential, and

ti

(4)

the currents in the receiver circuits.

These nine equations (2), (3), (4), determine the nine quantities : flt 72, /3, //, 7a', 73', ^', Ti^ £&•

Equations (4) substituted in (2) give :

(5)

These equations (5) substituted in (3), and transposed, give,

since £l = c E

Ez = £ E \ as E.M.Fs. at the generator terminals.

480 AL TERNA TING-CURRENT PHENOMENA.

as three linear equations with the three quantities 2T/,

Substituting the abbreviations :

a I \7 7 I I/" 7 \ I/" 7 ~\7 7 i

~T * 1^2 ~T *1^3)> -tZ^S) •*8^'2 I

7 V 7 /1_1_V7_1_V7N>/

^zt y 2-^D — V*1 ~r -^s^i T *»^V /

A

c, F2Z3, F3Z2

a, - (1 + ^^3 +

, Y,Zlt -(1 + F3Z1+F3Z2)

• (1 + Y,Z2 + FiZ,), c, F3Z2 F.Z3, c2, YtZ, Y.Z,, 1, - (1 + F3ZX + F3Z2)

(i + ^iz. + yiz,), F2z3, £

A = / FIZS, - (i

FaZ2, F2ZX,

it is:

D

72 = i

__ F2Z>2-

hence,

(8)

(9)

(10)

(11)

THREE-PHASE SYSTEM.

293. SPECIAL CASES.

A. Balanced System

Y, = F2 = F8 = F Z, = Z2 = Z3 = Z.

Substituting this in (6), and transposing :

481

c E

£s = £

EI =

3FZ

1 + 3FZ

1 + 3YZ EY

1 + 3KZJ

3FZ

3FZ

3 YZ

(12)

The equations of the symmetrical balanced three-phase system.

B. One circuit loaded, two unloaded:

F! = F2 = 0, F8 = F Zj = Z2 = Z3 = Z.

Substituted in equations (6) :

= ( unloaded branches. E — E3'(l + 2 FZ) = 0, loaded branch.

hence : r./

,

2KZ

2FZ

1 + 2 FZ

unloaded ;

loaded ;

all three

KM.F.'s

unequal, and (13) of unequal phase angles.

482

AL TERNA TING-CURRENT PHENOMENA.

(13)

(13)

C. Two circuits loaded, one tinloaded.

F! = F2 = F, F8 = 0, Zt = Z2 = Z3 = Z.

Substituting this in equations (6), it is :

e E — E{ (1 + 2 FZ) + .£/ FZ = 0) £E — El (1 + 2 FZ) + E{ FZ = 0 J

E — £s' + (,£,' + ^2') FZ = 0 unloaded branch, or, since :

E — Ez''— EZ'Y2 :'= 0,

E1 = ?

\ + FZ

thus:

1 + 4 FZ + 3 F2Z2

1 + 4 FZ + 3 F2Z2 E

I+'FZ

loaded branches.

unloaded branch.

(14)

As seen, with unsymmetrical distribution of load, all three branches become more or less unequal, and the phase displacement between them unequal also.

QUARTER-PHASE SYSTEM. 483

CHAPTER XXXII.

QUARTER-PHASE SYSTEM.

294. In a three-wire quarter-phase system, or quarter- phase system with common return wire of both phases, let the two outside terminals and wires be denoted by 1 and 2> the middle wire or common return by 0.

It is then :

EI = E = E.M.F. between 0 and 1 in the generator. Ez=jE = E.M.F. between 0 and 2 in the generator.

Let:

./i and 72 = currents in 1 and in 2, 70 = current in 0,

Z-L and Zz = impedances of lines 1 and 2, Z0 = impedance of line 0.

Yl and Y2 = admittances of circuits 0 to 1, and 0 to 2, // and //= currents in circuits 0 to 1, and 0 to 2, Eia.-ndE2'= potential differences at circuit 0 to 1, and 0 to 2.

it is then, 7, -f 78 + 70 = 0 ) «v

or, I0 =-(/; + 72) j

that is, 70 is common return of 7: and 72.

Further, we have,

El =JE - 72 Z0 + 0Z0 =jE - 72 (Z2 + Z0) - A

and

A = K, E{

(3)

484 AL TERNA TING-CURRENT PHENOMENA.

Substituting (3) in (2) ; and expanding : *•/ - *• _ l + F2Z2 + F2Z0(l-y) _

'. (4)

• 2 • /1_l_VX_l_V7'W'l_l_V7_l_V5^ V V '7 2

*- i * 1^0 "T" *)~ i * i **• T * i^ij — *i *J ^o

Hence, the two E.M.Fs. at the end of the line are un- equal in magnitude, and not in quadrature any more.

295. SPECIAL CASES :

A. Balanced System.

Z0 = Z / V2 ; F, = F2 = F

Substituting these values in (4), gives :

i + 1 + V2-yrz

' 1 + V2 (1 + V2) FZ + (1 + V2) F2Z;

_ E 1 + (1.707 - .707/) FZ • 1 + 3.414 FZ + 2.414 F2Z2

(5)

V2 ~J • 1 + V2 (1 + V2) FZ + (1 + V2) F2Z2

_ . ^ 1 + (1.707 + .707.;) FZ ' 1 + 3.414 FZ + 2.414 F2Z2

Hence, the balanced quarter-phase system with common return is unbalanced with regard to voltage and phase rela- tion, or in other words, even if in a quarter-phase system with common return both branches or phases are loaded equally, with a load of the same phase displacement, nevertheless the system becomes unbalanced, and the two E.M.Fs. at the end of the line are neither equal in magnitude, nor in quadrature with each other.

QUARTER-PHASE SYSTEM. B. One branch loaded, one unloaded.

485

a.) b.)

Substituting these values in (4), gives :

i + V2 — y

b.}

l + FZ

a.) £1 = E

V2

1 + V2 V2 j

2.414 +

1.414 YZ

*+'*f

= /^l4-1.707FZ

1+^1^

• 1 + 1.707 FZ

-t I ^/O

1 + F2

V2

,

FZ

1 +

V2

2.414 +

1.414 FZ

(6)

486 AL TERNA TING-CURRENT PHENOMENA.

These two E.M.Fs. are unequal, and not in quadrature with each other.