CHAPTER XIII. TRANSIENT TERM OF THE ROTATING FIELD. 106. The resultant of np equal m.m.fs. equally displaced from each other in space angle and in time-phase is constant in intensity, and revolves at constant synchronous velocity. When acting upon a magnetic circuit of constant reluctance in all directions, such a polyphase system of m.m.fs. produces a revolving magnetic flux, or a rotating field. (" Theory and Calculation of Alternating Current Phenomena," 4th edition, Chapter XXXIII, paragraph 368.) That is, if np equal mag- netizing coils are arranged under equal space angles of - np electrical degrees, and connected to a symmetrical np phase system, that is, to np equal e.m.fs. displaced in time-phase by 360 - degrees, the resultant m.m.f. of these np coils is a constant np and uniformly revolving m.m.f., of intensity SF0 = — &, where $ Zi is the maximum value (hence — — the effective value) of the \ V2 I m.m.f. of each coil. In starting, that is, when connecting such a system of mag- netizing coils to a polyphase system of e.m.fs., a transient term appears, as the resultant magnetic flux first has to rise to its constant value. This transient term of the rotating field is the resultant of the transient terms of the currents and therefore the m.m.fs. of the individual coils. 107. If, then, $ = nl = maximum value of m.m.f. of each coil, where n = number of turns, and / = maximum value of current, and r = space-phase angle of the coil, the instantaneous value of the m.m.f. of the coil, under permanent conditions, is f-frCOS^-T), (1) 191 192 TRANSIENT PHENOMENA and if the time 6 is counted from the moment of closing the circuit, the transient term is, by Chapter IV, /" = - gr£~ x 0 cos T} (2) where Z = r — jx. The complete value of m.m.f. of one coil is /i = /' + /" = & {cos(0 - r) -r-'cosr}. (3) In an np-phase system, successive e.m.fs. and therefore currents are displaced from each other by — of a period, or an angle — , np np and the m.m.f. of coil, i, thus is ( / 2n \ -r~6 I 2?r \) ff = $ < cos (0 — T — • — i — e x cos ( r H -- i]>. (4) ( \ n I \ n n p / 2 TT \ 2ji cos T H -- i) = 0; \ nv I The resultant of np such m.m.fs. acting together in the same direction would be (5) that is, the sum of the instantaneous values of the permanent terms as well as the transient terms of all the phases of a sym- metrical polyphase system equals zero. In the polyphase field, however, these m.m.fs. (4) do not act in the same direction, but in directions displaced from each other by a space angle — equal to the time angle of their phase np displacement. 108. The component of the m.m.f., fit acting in the direction (00 - T), thus is 27T .> // = ft cos (»„ - T - ~ i), (6) \ nn i TRANSIENT TERM OF THE ROTATING FIELD 193 and the sum of the components of all the np m.m.fs., in the direction (00 - r), that is, the component of the resultant m.m.f. of the polyphase field, in the direction (00 - T), is np f = X< fi I Transformed, this gives $ ( np / 4 TT \ "* / = ; ! 5/ cos (0 + 00 - 2 r - - i) + V» c ^ I i^ \ "^n / -Htf Hp I cos 0ft - s x Vt cos ( #n — 2 T — — r. "P 47r and as the sums containing - - i equal zero, we have np os(0-00) -cos00, (8) and for 6 = oo f that is as permanent term, this gives /0=^SFcos(0-00); (9) ft hence, a maximum, and equal to -~ &, that is, constant, for 00 = 0, that is, uniform synchronous rotation. That is, the resultant of a polyphase system of m.m.fs., in permanent con- dition, rotates at constant intensity and constant synchronous velocity. Before permanent condition is reached, however, the resultant m.m.f. in the direction #0 = 6, that is, in the direction of the synchronously rotating vector, in which in permanent condition 194 TRANSIENT PHENOMENA the m.m.f . is maximum and constant, is given during the transient period, from equation (8), by (10) that is, it is not constant but periodically varying. • As example is shown, in Fig. 48, the resultant m.m.f. /0 in the direction of the synchronously revolving vector, 00 = 6, for the 1600 fnoo 6 800 a 400 r \ I 08'( \ s~ N, fn = ' mBi _( -0. cos 6 [ j \ ^ X, 1 \ 1 \ 1 S ^ ^ — ^v s I \ 1 \ / s s ^ / ^.^ 1 ^ s 1 V y / -V 0 / 27T 37T 4 7T 5 7T c 7T 7 7T 87T 90 180 270 360 450 540 630 720 810 900 990 1060 1170 1260 1350 1440 Fig. 48. Transient term of polyphase magnetomotive force. constants np =3, or a three-phase system; SF = 667, and Z = r - jx = 0.32 - 4 j ; hence, /0 - 1000 (1 - £ — 1.080 cos d), with 6 as abscissas, showing the gradual oscillatory approach to constancy. 109. The direction, 6Q = 0, is, however, not the direction in which the resultant m.m.f. in equation (8) is a maximum, but the maximum is given by df = 0, this gives hence, sin (0 - 60) + s x sin 00 = 0, cot 0n = COS 0 — £ sin 0 (11) (12) (13) that is, the resultant maximum m.m.f. of the polyphase system does not revolve synchronously, in the starting condition, but revolves with a varying velocity, alternately running ahead and TRANSIENT TERM OF THE ROTATING FIELD 195 dropping behind the position of uniform synchronous rotation, by equation (13), and only for 0 = oo, equation (12) becomes cot #0 = cot 6, or #0 = 6, that is, uniform synchronous rotation. The speed of rotation of the maximum m.m.f. is given from equation (12) by differentiation as dQ ^_0= £ dd ' dQ' where Q = sin (6 — 00) + £ x sin 00; _ - a cos {0 - 00) - -£ x sin 00 hence, S = -^— ~, (14) cos (0 - 00) - e~xfl cos 00 or approximately, 1 - £ x COS 6 (15) For 0 = oo, equation (14) becomes S = 1, or uniform syn- chronous rotation, but during the starting period the speed alternates between below and above synchronism. From (13) follows -- COS 0 — £ x and sin d sin 0 = (R where (16) \ (R = V os - e z + sn = -£ ' cos e (17) 196 TRANSIENT PHENOMENA 110. The maximum value of the resultant m.m.f., at time- phase 6, and thus of direction 6Q as given by equation (13) or (16), (17), is derived by substituting (16), (17) into (8), as: -2£ *os0 + e *, (18) hence is not constant, but pulsates periodically, with gradually decreasing amplitude of pulsation, around the mean value — $. 2i For 6 = 0, or at the moment of start, it is, by (13), -r-e cos 6 - £ * 0 ' sinfl Q hence, differentiating numerator and denominator, r -- —sin 6 + -s x X r cos 0 x tan00'=?; cot 007 = and that is, the position of maximum resultant m.m.f. starts from angle 00' ahead of the permanent position, where 00' is the time- phase angle of the electric magnetizing circuit. The initial value of the resultant m.m.f., for 0 = 0, is fm = 0, that is, the revolving m.m.f. starts from zero. Substituting (16) in (15) gives the speed as function of time (cos 6 -- sin 0) (19) 1 + s~*—2e~* cos# for 0 = 0 this gives the starting speed of the rotating field SQ = - , or, indefinite; TRANSIENT TERM OF THE ROTATING FIELD 197 hence, after differentiating numerator and denominator twice, this value becomes definite. So = ^; (20) that is, the rotating field starts at half speed. As illustration are shown, in Fig. 49, the maximum value of the resultant polyphase m.m.f., fm, and its displacement in +40 ' ° -40 ,400 Inten'sit^/ = 1000 /\ \f position |(0 ili + 8il mil 2 2 2 22 Fig. 49. Start of rotating field. position from that of uniform synchronous rotation, 00— 6, for the same constants as before, namely: np = 3; *F = 667, and Z = r - jx = 0.32 - 4 /; hence, fm = 1000 Vl - 2 £-C ' COS 0 + £-C •», with the time-phase angle ^ as abscissas, for the first three cycles. 111. As seen, the resultant maximum m.m.f. of the poly- phase system, under the assumed condition, starting at zero in the moment of closing the three-phase circuit, rises rapidly - within 60 time-degrees — to its normal value, overreaches and exceeds it by 78 per cent, then drops down again below normal, by 60 per cent, rises 47 per cent above normal, drops 37 per cent below normal, rises 28 per cent above normal, and thus by a series of oscillations approaches the normal value. The maximum value of the resultant m.m.f. starts in position 198 TRANSIENT PHENOMENA 85 time-degrees ahead, in the direction of rotation, but has in half a period dropped back to the normal position, that is, the position of uniform synchronous rotation, then drops still fur- ther back to the maximum of 40 deg., runs ahead to 34 deg., drops 23 deg. behind, etc. It is interesting to note that the transient term of the rotat- ing field, as given by equations (10), (13), (18), does not contain the phase angle, that is, does not depend upon the point of the wave, 0 = r, at which the circuit is closed, while in all preced- ing investigations the transient term depended upon the point of the wave at which the circuit was closed, and that this tran- sient term is oscillatory. In the preceding chapter, in circuits containing only resistance and inductance, the transient term has always been gradual or logarithmic, and oscillatory phenom- ena occurred only in the presence of capacity in addition to in- ductance. In the rotating field, or the polyphase m.m.f., we thus have a case where an oscillatory transient term occurs in a circuit containing only resistance and inductance but not capacity, and where this transient term is independent of the point of the wave at which the circuits were closed, that is, is always the same, regardless of the moment of start of the phe- nomenon. The transient term of the polyphase m.m.f. thus is independ- ent of the moment of start, and oscillatory in character, with an amplitude of oscillation depending only on the reactance factor, — , of the circuit.