CHAPTER VI. TOPOGRAPHIC METHOD. 33. In the representation of alternating sine waves by vectors in a polar diagram, a certain ambiguity exists, in so far as one and the same quantity — an E.M.F., for in- stance — can be represented by two vectors of opposite direction, according as to whether the E.M.F. is considered as a part of the impressed E.M.F., or as a counter E.M.F. This is analogous to the distinction between action and reaction in mechanics. Further, it is obvious that if in the circuit of a gener- ator, G (Fig. 25), the current flowing from terminal A over resistance R to terminal B, is represented by a vector OI (Fig. 26), or by /= i -\-ji', the same current can be con- sidered as flowing in the opposite direction, from terminal B to terminal A in opposite phase, and therefore represented by a vector OI-± (Fig. 26), or by 7l = — i —ji'> Or, if the difference of potential from terminal B to terminal A is denoted by the E = e + je' , the difference of potential from A to B is El = — e — je' . 44 ALTERNA TING-CURRENT PHENOMENA. Hence, in dealing with alternating-current sine waves, it is necessary to consider them in their proper direction with regard to the circuit. Especially in more complicated circuits, as interlinked polyphase systems, careful attention has to be paid to this point. -*' Fig. 28. 34. Let, for instance, in Fig. 27, an interlinked three- phase system be represented diagrammatically, as consist- ing of three E.M.Fs., of equal intensity, differing in phase by one-third of a period. Let the E.M.Fs. in the direction Fig. 27 from the common connection O of the three branch circuits to the terminals A19 A2,AB, be represented by Elt E2, £3. Then the difference of potential from A2 to A± is £z — £lf since the two E.M.Fs., El and are connected in cir- cuit between the terminals A, and A*, in the direction, TOPOGRAPHIC METHOD. 45 Al — O — A2; that is, the one, Ez, in the direction OA2, from the common connection to terminal, the other, JS1, in the opposite direction, A^O, from the terminal to common connection, and represented by — El. Conversely, the dif- ference of potential from A1 to Az is El — Ez. It is then convenient to go still a step farther, and drop, in the diagrammatic representation, the vector line altogether ; that is, denote the sine wave by a point only,, the end of the corresponding vector. " Looking at this from a different point of view, it means that we choose one point of the system — for instance, the common connection O — as a zero point, or point of zero potential, and represent the potentials of all the other points of the circuit by points in the diagram, such that their dis- tances from the zero point gives the intensity ; their ampli- tude the phase of the difference of potential of the respective point with regard to the zero point ; and their distance and amplitude with regard to other points of the diagram, their difference of potential from these points in intensity and phase. Fig. 28. Thus, for example, in an interlinked three-phase system with three E.M.Fs. of equal intensity, and differing in phase by one-third of a period, we may choose the common con- nection of the star-connected generator as the zero point, and represent, in Fig. 28, one of the E.M.Fs., or the poten- 46 AL TERN A TING-CURRENT PHENOMEMA. tial at one of the three-phase terminals, by point Er The potentials at the two other terminals will then be given by the points Ez and E& which have the same distance from O as Ev and are equidistant from E± and from each other. The difference of potential between any pair of termi- nals — for instance E^ and E2 — is then the distance EZEV or E±EV according to the direction considered. 35. If now the three branches OEV ~OEZ and "OEW of the three-phase system are loaded equally by three currents equal in intensity and in difference of phase against their THUEE-PHA8E 8V8TEM 48° LAO BALANCED THREE-PHASE SYSTEM NON-INDUCTIVE LOAD E° Fig. 29. E.M.Fs., these currents are represented in Fig. 29 by the vectors 07^ = 072 = Ofs = I, lagging behind the E.M.Fs. by angles E.O^ = EZOIZ = EZOI& = Q. Let the three-phase circuit be supplied over a line of impedance Z± = r^ —jx\ from a generator of internal im- pedance Z0 = x0 -jx0. In phase OEV the E.M.F. consumed by resistance r^ is represented by the distance E^EJ = Irv in phase, that is parallel with current OIV The E.M.F. consumed by re- actance #! is represented by E^Ej' = Ixv 90° ahead of cur- TOPOGRAPHIC METHOD. 47 rent OIr The same applies to the other two phases, and it thus follows that to produce the E.M.F. triangle E^E^E^ at the terminals of the consumer's circuit, the E.M.F. tri- angle E^E^E? is required at the generator terminals. Repeating the same operation for the internal impedance of the generator we get E"E'" = Iroi and parallel to OIV E'"E° = Ixoy and 90° ahead of ~OTV and thus as triangle of (nominal) induced E.M.Fs. of the generator E°E£E°. In Fig. 29, the diagram is shown for 45° lag, in Fig. 30 for noninductive load, and in Fig. 31 for 45° lead of the currents with regard to their E.M.Fs. BALANCED THREE -PHASE SYSTEM 45° LEAD THREE-PHASE CIRCUIT 80°LA» TRANSMISSION LINE' WITH DISTRIBUTED CAPACITY, INDUCTANCB RESISTANCE AUD LEAKAQB •I, Fig. 31. Fig. 32. As seen, the induced generator E.M.F. and thus the generator excitation with lagging current must be higher, with leading current lower, than at non-inductive load, or conversely with the same generator excitation, that is the same induced generator E.M.F. triangle E°E£E°, the E.M.Fs. at the receiver's circuit, Ev Ez, E9 fall off more with lagging, less with leading current, than with non- inductive load. 36. As further instance may be considered the case of a single phase alternating current circuit supplied over a cable containing resistance and distributed capacity. 48 ALTERNATING-CURRENT PHENOMENA. Let in Fig. 33 the potential midway between the two terminals be assumed as zero point 0. The two terminal voltages at the receiver circuit are then represented by the points E and El equidistant from 0 and opposite each other, and the two currents issuing from the terminals are rep- resented by the points / and I1, equidistant from 0 and opposite each other, and under angle & with E and El respectively. Considering first an element of the line or cable next to the receiver circuit. In this an E.M.F. EEl is consumed by the resistance of the line element, in phase with the current OI, and proportional thereto, and a current //x con- sumed by the capacity, as charging current of the line element, 90° ahead in phase of the E.M.F. OE and propor- tional thereto, so that at the generator end of this cable element current and E.M.F. are OI^ and OEl respectively. Passing now to the next cable element we have again an E.M.F. E1EZ proportional to and in phase with the current OI^ and a current IJZ proportional to and 90° ahead of the E.M.F. OEV and thus passing from element to element along the cable to the generator, we get curves of E.M.Fs. e and e1, and curves of currents i and il, which can be called the topographical circuit characteristics, and which corre- spond to each other, point for point, until the generator terminal voltages OE0 and OE0l and the generator currents OI0 and OIJ are reached. Again, adding 'E~Er' = I0r0 and parallel OI0 and E"E° = I0x0 and 90° ahead of ~OIM gives the (nominal) induced E.M.F. of the generator OE°, where Z0 = r0 — jx0 = inter- nal impedance of the generator. In Fig. 33 is shown the circuit characteristics for 60° lag, of a cable containing only resistance and capacity. Obviously by graphical construction the circuit character- istics appear more or less as broken lines, due to the neces- sity of using finite line elements, while in reality when calculated by the differential method they are smooth curves. TOPOGRAPHIC METHOD. 49 37. As further instance may be considered a three-phase circuit supplied over a long distance transmission line of distributed capacity, self-induction, resistance, and leakage. Let, in Fig. 38, O£v ~OEy ~OEZ = three-phase E.M.Fs. at receiver circuit, equidistant from each other and = E. Let OIV Oly Of3 = three-phase currents in the receiver circuit equidistant from each other and = /, and making with E the phase angle <3. Considering again as in § 35 the transmission line ele- ment by element, we have in every element an E.M.F. consumed by the resistance in phase with the current n^ proportional thereto, and an E.M.F. E^, Ef con- sumed by the reactance of the line element, 90° ahead of the current OIV and proportional thereto. In the same line element we have a current IJ^ in phase with the E.M.F. OEV and proportional thereto, representing the loss of energy current by leakage, dielectric hysteresis, etc., and a current ^V/', 90° ahead of the E.M.F. OEV and proportional thereto, the charging current of the line ele- ment as condenser, and in this manner passing along the line, element by element, we ultimately reach the generator terminal voltages E°, E°, Es°, and generator currents //, /2°, 78°, over the topographical characteristics of E.M.F. ev ev es, and of current iv z'2, z'3, as shown in Fig. 33. The circuit characteristics of current i and of E.M.F. e 50 ALTERNATING-CURRENT PHENOMENA. correspond to each other, point for point, the one giving the current and the other the E.M.F. in the line element. TRANSMISSION WITH DISTRIBUTED CAPACITY, INDUCTANCE RESISTANCE AND LEAKAGE 90° LAO Fig. 34. Only the circuit characteristics of the first phase are shown as ^ and z'r As seen, passing from the receiving end towards the generator end of the line, potential and TRANSMISSION LINE WITH DISTRIBUTED CAPACITY, INDUCTANCE RESISTANCE AND LEAKAGE Fig. 35. current alternately rise and fall, while their phase angle changes periodically between lag and lead. TOPOGRAPHIC METHOD. 51 37. a. More markedly this is shown in Fig. 34, the topo- graphic circuit characteristic of one of the lines with 90° lag in the receiver circuit. Corresponding points of the two characteristics e and i are marked by corresponding figures 0 to 16, representing equidistant points of the line. The values of E.M.F., current and their difference of phase are plotted in Fig. 35 in rectangular co-ordinates with the distance as abscissae, counting from the receiving circuit towards the generator. As seen from Fig. 35, E.M.F. and current periodically but alternately rise and fall, a maximum of one approximately coinciding with a minimum of the other and with a point of zero phase displacement. The phase angle between current and E.M.F. changes from 90° lag to 72° lead, 44° lag, 34° lead, etc., gradually decreasing in the amplitude of its variation. 52 ALTERNATING-CURRENT PHENOMENA.