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. 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 it in the receiving circuits. 4GO ALTERNATING-CURRENT PHENOMENA.