CHAPTER VI TOPOGRAPHIC METHOD 36. In the representation of alternating sine waves by vectors, a certain ambiguity exists, in so far as one and the same quantity — voltage, for instance — 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 voltage 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 generator, G (Fig. 25), the current in the direction from terminal A over re- sistance R to terminal B is represented by a vector, 01 (Fig. 26), or by 7 = z + ji' , the same current can be considered as being ' 7 ,,U— — L Fig. 25. Fig. 26. in the opposite direction, from terminal B to terminal A in op- posite phase, and therefore represented by a vector, OIi (Fig. 26), or by 7] = — 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'. 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 inter- linked polyphase systems, careful attention has to be paid to this point. 37. Let, for instance, in Fig. 27, an interlinked three-phase system be represented diagrammatically as consisting of three 39 40 ALTERNATING-CURRENT PHENOMENA voltages, of equal intensity, differing in phase by one-third of a period. Let the voltages in the direction from, the common con- nection, 0, of the three branch circuits to the terminals, Ai, A 2, Az, be represented by Ei, E^, E3. Then the difference of poten- tial from AitoAiisEi—Ei, since the two voltages, Ei and E2, are connected in circuit between the terminals, Ai and A2, in the direction Ai — 0 — A^; that is, the one, E2, in the direction, 0^1 2, from the common connection to terminal, the other, Ei, in the opposite direction, AiO, from the terminal to common connec- tion, and represented by — Ei. Conversely, the difference of potential from Ai to Az'is Ei — Ei. It is then convenient to go still a step farther, and drop the vector line altogether in the diagrammatic representation; that is, denote the sine wave by a point only, the end of the corre- sponding vector. Looking at this from a different point of view, it means that we choose one point of the system — for instance, the common O ^a § ^^^ El O E2 Fig. 27. Fig. 2S. connection, or neutral 0 — as a zero point, or point of zero poten- tial, and represent the potentials of all the other points of the circuit by points in the diagram, such that their distances from the zero point give the intensity, their amplitude 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. Thus, for exam.ple, in an interlinked three-phase system with three voltages of equal intensity, and differing in phase by one- third of a period, we may choose the common connection of the star-connected generator as the zero point, and represent, in Fig. 28, one of the voltages, or the potential at one of the three- TOPOGRAPHIC METHOD 41 phase terminals, by point Ei. The potentials at the two other terminals will then be given by the points E2 and £'3, which have the same distance from 0 as Ei, and are equidistant from Ei and from each other. The difference of potential between any pair of terminals, for instance, Ei and Ei, is then the distance EiEi, or E1E2, according to the direction considered, 38. If now the three branches, OEi, OE2 and OE3, of the three-phase system are loaded equally by three currents equal in intensity and in difference of phase against their voltages, BAtANCED THREE-PHASE SYSTEtif NON-INDUCTIVE LOAD Fig. 29. Fig. 30. these currents are represented in Fig. 29 by the vectors 01 1 = 01 2 = 01 3 = I, lagging behind the voltages by angles EiOIi = £20/2 = EsOh = d. Let the three-phase circuit be supplied over a line of impedance, Zi = ri -{- jxi, from a generator of internal impedance, Zo = ro + jxo. In phase OEi the voltage consumed by resistance ri is repre- sented by the distance, EiEi^ = Iri, in phase, that is, parallel with current OIi. The voltage consumed by reactance Xi is represented by Ei^Ei^^ = Ixi, 90° ahead of current OTu The same applies to the other two phases, and it thus follows that to produce the voltage triangle, E1E2E3, at the terminals of the consumer's circuit, the voltage triangle, Ei^^Ez^^Ea^^, is required at the generator terminals. 42 ALTERNATING-CURRENT PHENOMENA Repeating the same operation for the internal impedance of the generator, we get E^^E^^^ = /ro, and parallel to OTi, W^^'^ = Ixo, and 90° ahead of OIi, and thus as triangle of (nominal) gen- erated e.m.fs. of the generator, Ei^E2°Ez^. In Fig. 29 the diagram is shown for 45° lag, in Fig. 30 for non- inductive load, and in Fig. 31 for 45° lead of the currents with regard to their voltages. As seen, the generated e.m.f. and thus the generator excitation with lagging current must be higher, and with leading current lower, than at non-inductive load, or conversely with the same generator excitation, that is, the same internal generator e.m.f. SINGLE-PHASE CIRCUIT 60' LAG CABLE OF DISTRIBUTED CAPACIir AND RESISTANCE Fig. 31. Fig. 32. triangle, Ei^E^^Ez^, the voltages at the receiver's circuit, Ei, E2, Es, fall off more with lagging, and less with leading current, than with non-inductive load. 39. As a further example may be considered the case of a single-phase alternating-current circuit supplied over a cable containing resistance and distributed capacity. Let, in Fig. 32, the potential midway between the two ter- minals be assumed as zero point 0. The two terminal voltages at the receiver circuit are then represented by the points E and E^, equidistant from 0 and opposite each other, and the two cur- rents at the terminals are represented by the points I and P, equidistant from 0 and opposite each other, and under angle d with E and E^ respectively. Considering first an element of the line or cable next to the receiver circuit. In thi^ voltage, EE^, is consumed by the re- sistance of the line element, in phase with the current, 01, and proportional thereto, and a current. Hi, consumed by the TOPOGRAPHIC METHOD 43 capacity, as charging current of the hne element, 90*' ahead in phase of the voltage, OE, and proportional thereto, so that at the generator end of this cable element current and voltage are 01 1 and OEi respectively. Passing now to the next cable element we have again_ajyoltage, EiEo, proportional to and in phase with the current, Oli, and a current, /1/2, proportional to and 90° ahead of the voltage, 0E\, and thus passing from element to element along the cable to the generator, we get curves of voltages, e and e^ ^-^id curves of cur- rents, i and i^, which can be called the topographical circuit characteristics, and which correspond to each other, point for point, until the generator terminal voltages, OEo and OE^'^, and the generator currents, OIq and 0/oS are reached. Again, adding E^E^^ = I^r^ and parallel to OTi and E^^E"^ = loXo and 90° ahead of O/o, gives the (nominal) generated e.m.f. of the generator OE^, where Zo = Tq + jxo = internal impedance of the generator. In Fig. 32 is shown the circuit characteristics for 60° lag of a cable containing only resistance and capacity. Obviously by graphical construction the circuit characteristics appear more or less as broken lines, due to the necessity of using finite line elements, while in reality they are smooth curves when calculated by the differential method, as explained in Section III of "Theory and Calculation of Transient Electric Phenomena and Oscillations." 40, As further example may be considered a three-phase cir- cuit supplied over a long-distance transmission line of distrib- uted capacitj^, self-induction, resistance, and leakage. Let, in Fig. 33, OEi, OE2, OEz = three-phase voltages at re- ceiver circuit, equidistant from each other and = E. Let Oh, Oh, Oh = three-phase currents in the receiver cir- cuit equidistant from each other and = 7, and making with E the phase angle, 6. Considering again as in §3 the transmission line, element by element, we have in every element a voltage, EiEi^, consumed by the resistance in phase with the current. Oh, and proportional thereto, and a voltage, Ei^, iJi", consumed by the reactance of the Hne element, 90° ahead of the current, Oh, and proportional thereto. In the same Hne element we have a current, hh^, in phase with the voltage, OEi, and proportional thereto, representing 44 ALTERNATING-CURRENT PHENOMENA the loss of current by leakage, dielectric hysteresis, etc., and a current, /i^ /i^\ 90° ahead of the voltage, 0E-[, and proportional thereto, the charging current of the line element as condenser; and in this manner passing along the line, element by element, we ultimately reach the generator terminal voltages, Ei^, E^^, £'3", THREE PHASE CIRCUIT eO°LAG TRANSMISSION LINE WITH DISTRIBUTED CAPACITY, INDUCTANCE RESISTANCE AND LEAKAGE Fig. 33. 16 I TRANSMISSION WITH DISTRIBUTED 15 t CAPACITY, INDUCTANCE RESISTANCE AND LEAKAGE 90" LAG Fig. 34. and generator currents, /i", 72°, I^, over the topographical char- acteristics of voltage, ei, e^, 63, and of current, ii, U, is, as shown in Fig. 33. The circuit characteristics of current, ?', and of voltage, e, cor- respond to each other, point for point, the one giving the current and the other the voltage in the line element. Only the circuit characteristics of the first phase are shown, TOPOGRAPHIC METHOD 45 as Ci and ^l. As seen, passing from the receiving end toward the generator end of the hne, potential and current alternately rise and fall, while their phase angle changes periodically be- tween lag and lead. 41. More markedly this is shown in Fig. 34, the topographic 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 equi- distant points of the Hne. The values of voltage, current and TRANSMISSION LINE WITH DISTRIBUTED CAPACITY, INDUCTANCE RESISTANCE AND LEAKAGE Fig. 35. their difference of phase are plotted in Fig. 35 in rectangular coordinates with the distance as abscissas, counting from the receiving circuit toward the generator. As seen from Fig. 35, voltage 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.