IV. Armature Current and Heating 88. The current in the armature conductors of a converter is the difference between the alternating-current input and the direct-current output. SYNCHRONOUS CONVERTERS 233 In Fig. 127, ai, a2 are two adjacent leads connected with the collector rings DI, D2 in an n-phase converter. The alternating e.m.f. between a\ and a2, and thus the power component of the alternating current in the armature section between a\ and a2, will reach a maximum when this section is midway between the brushes BI and Bz, as shown in Fig. 127. The direct current in every armature coil reverses at the mo- ment when the coil passes under brush BI or B2, and is thus a rec- tangular alternating current as shown in Fig. 128 as 7. At the moment when the power com- ponent of the alternating current is a maximum, an armature coil d midway between two adjacent alternating leads ai and a2 is midway between the brushes BI and B2} as in Fig. 127, and is thus in the middle of its rectan- gular continuous-current wave, and consequently in this coil the power component of the alternating current and the rectan- gular direct current are in phase with each other, but opposite, as FIG. 127. — Diagram for study of armature heating in synchronous converters. FIG. 128. — Direct current and alternating current in armature coil d, Fig. 127. FIG. 129. — Resultant current in coil d, Fig. 127. shown in Fig. 128 as 7i and /, and the actual current is their difference, as shown in Fig. 129. In successive armature coils the direct current reverses suc- cessively; that is, the rectangular currents in successive arma- 234 ELEMENTS OF ELECTRICAL ENGINEERING 7 FIG. 130. — Alternating current and direct current in coil between d and a\ or a* Fig. 127. FIG. 131. — Resultant of currents given in Fig. 130. FIG. 132. — Alternating current and direct current in coil between d and or a2, Fig. 127. FIG. 133. — Resultant of currents shown in 132. SYNCHRONOUS CONVERTERS 235 ture coils are successively displaced in phase from each other; and since the alternating current is the same in the whole section ai a2, and in phase with the rectangular current in the coil d, it becomes more and more out of phase with the rectangular current when passing from coil d toward ai or a2, as shown in Figs. 130 to 133, until the maximum phase displacement between alternating and rectangular current is reached at the alternating leads ai and a2, and is equal to -• li 89. Thus, if E = direct voltage, and I = direct current, in an armature coil displaced by angle T from the position d, mid- way between two adjacent leads of the n-phase converter, the direct current is ~ for the half period from 0 to ?r, and the alter- nating current is V2 I' sin (0 - r), where /' = n sin - n is the effective value of the alternating current. Thus, the actual current in this armature coil is io = \/2 /' sin (0 — T) - g _ / [4 sin (0-r) _ "2 nsin- ( n In a double-current generator, instead of the minus sign, a plus sign would connect the alternating and the direct current in the parenthesis. The effective value of the resultant converter current thus is: n sin - rnr sin - n n Since ~ is the current in the armature coil of a direct-current 236 ELEMENTS OF ELECTRICAL ENGINEERING generator of the same output, we have 7r = Jo' 2 o | i 16 COS*T 7 2 — 1 -1 n2 sin2 - n nir sin ^ 71 the ratio of the power loss in the armature coil resistance of the converter to that of the direct-current generator of the same output, and thus the ratio of coil heating. This ratio is a maximum at the position of the alternating leads, T = -, and is 7m = n* sin n It is a minimum for a coil midway between adjacent alter- nating leads, T = 0, and is = 8 ... i . 7T .IT n2 sin2 - mr sin - n n Integrating over T from 0 (coil d) to-, that is, over the whole phase or section 0,1 0,%, we have the ratio of the total power loss in the armature resistance of an n-phase converter to that of the same machine as direct- current generator at the same output, or the relative armature heating. Thus, to get the same loss in the armature conductors, and consequently the same heating of the armature, the current in the converter, and thus its output, can be increased in the pro- portion —7= over that of the direct-current generator. The calculation for the two-circuit single-phase converter is somewhat different, since in this in one-third of the armature the Pr loss is that of the direct-current output, and only in the 27f other two-thirds — or an arc -^ is there alternating current. o SYNCHRONOUS CONVERTERS 237 Thus in an armature coil displaced by angle r from the center of this latter section the resultant current is io = V2 /' sin (0 - r) - giving the effective value I III 16 °W:= thus, the relative heating is //oV 11 16 ^ (I = \2/ with the minimum value at r = 0, it is •"• - T ~ = °-70' and with the maximum value at r = ^ it is o 11 8 =2-18; the average current heating in two-thirds of the armature is 11 48 TT •T dT = -3- - ^—/^ Sln 3 3 7T2 in the remaining third of the armature, Tz = 1, thus the average is 3 = 1.151, and therefore the rating is -4= = 0.93. Vr By substituting for n, in the general equations of current heat- ing and rating based thereon, numerical values, we get the following table: 238 ELEMENTS OF ELECTRICAL ENGINEERING d V •si 0> § 0) Type Direct-cur generator P jj-fl £•§ • a £ H a 1 1 8 n 2 2 3 4 6 12 1.00 0.45 0.70 0.225 0.20 0.19 0.187 7m • r 1.00 1.00 3.00 1.37 2.18 1.157 1.20 0.555 0.73 0.37 0.42 0.26 0.24 0.20 0.187 Rating (by mean arm. heating) 1.00 0.85 0.93 1.34 1.64 1.96 2.24 2.31 As seen, in the two-circuit single-phase converter the arma- ture heating is less, and more uniformly distributed, than in the single-circuit single-phase converter. 90. A very great gain is made in the output by changing from three-phase to six-phase, but relatively little by still further increasing the number of phases. In these values, the small power component of current supply- ing the losses in the converter has been neglected. These values apply only to the case where the alternating current is in phase with the supply voltage, that is, for unity power-factor of supply. If, however, the current lags, or leads, by the time angle 0, then the alternating current and direct current are not in opposition in the armature coil d midway between adjacent leads, Fig. 127, and the resultant current is a minimum and of the shape shown in Fig. 128, at a point of the armature winding displaced from mid position d by angle r = 0. At the leads the displacement between alternating cur- 7T • 7T rent and direct current then is not -, but - + 8 at the n n one, 6 at the other lead, and thus at the other side of the same n lead. The resultant current is thus increased at the one, de- creased at the other lead, and the heating changed accordingly. For instance, in a quarter-phase converter at zero phase dis- placement, the resultant current at the lead would be as shown in Fig. 134, - = 45 deg., while at 30 deg. lag the resultant currents in the two coils adjacent to the commutator lead are displaced SYNCHRONOUS CONVERTERS 239 respectively by- + & = 75 deg. and by - — d = 15 deg., and so of very different shape, as shown by Figs. 135 and 136, giving very different local heating. Phase displacement thus increases the heating at the one, decreases it at the other side of each commutator lead. Let again, I = direct current per commutator brush. The effective value of the alternating power current in the armature winding, or ring current, corresponding thereto, is n sn - n Let pi' = total power current, 'allowing for the losses of power in the converter; qlf = reactive current in the converter, assumed as positive when lagging, as negative when leading, and si' = total current, where s = Vp2 + tf2 is the ratio of total current to the load current, that is, power current corresponding to the direct-current output, and — = tan 6 is the time lag of the supply current; p is a quantity slightly larger than 1, by the losses in the converter, or slightly smaller than 1 in an inverted converter. The actual current in an armature coil displaced in position by angle r from the middle position d between the adjacent collector leads, then, is to = V2 If [p sin (0 - T) - q cos (0 - r) } - ^ 4 [p sin (0 — T) — q cos (0 — T)] — I 2 | n sin - n and, therefore, its effective value is = \l I * *vjo / I 8(p2 + g2) 16 (p COST + gsinr) n / 8 s2 16 s cos (T - 6) 2 11 + ~ rr^r • —^r n2 sin2 - TTH sin - n n 240 ELEMENTS OF ELECTRICAL ENGINEERING \ FIG. 134. — Quarter-phase converter unity power-factor, armature current at collector lead. \ \ v_ FIG. 135. — Quarter-phase converter phase displacement 30 ture current at collector lead. 7 FIG. 136. — Quarter-phase converter phase displacement 30 degrees, arma- ture current at collector lead. SYNCHRONOUS CONVERTERS 241 and herefrom the relative heating in an armature coil displaced by angle r from the middle between adjacent commutator leads: 8s2 16scos(r-0) n sn - irn sn — n n this gives at the leads, or for r = + — > 8s n2 sin2 — TTU sin — n n 16 s cos (— — 0) '8s2 \n I n2 sin2 - irn sin - n n Averaging from to H gives the mean current-heating of Ti ft the converter armature. r 1 + J+^ %* -? n2 sin2 — n2 sin - n n i i 8s2 16s cos 0 — i -f- 9 *7T 7T"2 n2 sin2 — n -t-i_ 8 (p2 + q* ) 16 P 2 2 - n 91. This gives for Three-phase, n = 3: 7T = 1 + 1.185 s2 - 1.955 s cos (T - 0), ym = 1 + 1.185 s2 - 1.955 s cos (60 ± 0), r = 1 + 1.185s2 - 1.620 p. Quarter-phase, n = 4: >YT = 1 + s2 - 1.795 s cos (r - 0), ym = 1 + s2 - 1.795 s cos (45 ± 0), T = 1 + s2 - 1.620 ». 242 ELEMENTS OF ELECTRICAL ENGINEERING Six-phase, n = 6: 7r = 1 + 0.889 s2 - 1.695 s cos (r - 0), 7m = 1 + 0.889 s2 - 1.695 s cos (30 + 0), r = 1 + 0.889s2 - 1.62 p, oo -phase, n = co : TT = 7m = r = 1 + 0.810 s2 - 1.62 s cos 0 = 1 + 0.810s2 - 1.62 p. Choosing p = 1.04, that is, assuming 4 per cent, loss in friction and windage, core loss and field excitation — the z'2r loss of the armature is not included in p, as it is represented by a drop of direct-current voltage below that corresponding to the alternat- ing voltage, and not by an increase of the alternating current over that corresponding to the direct current — we get, for dif- ferent phase angles from 0 = 0 deg. to 0 = 60 deg., the values given below: 0=0 10 20 30 40 50 60 s = -^ =1.04 1.0561.1081.20 1.36 1.62 2.08 cos 0 q = s sin 0 = react' cur> =. 0 0.184 0.379 0.60 0.876 1.24 1.80 power cur. tan0 =0 0.176 0.364 0.577 0.839 1.192 1.732 Three-phase: _ i , 1.62 2 08 2.70 3. 65 5. 19 8.16 '*£ = *: 26 1.00 0 80 0.68 0. 70 0. 99 2.06 r = 0. 60 0.64 0 77 1.02 1. 51 2. 43 4.45 Quarter-phase : 7m = 1.02 1 39 1.88 2. 64 3. '87 6.30 7ft 7m' = • 4 \J 0.55 0 43 0.38 0. 42 0. 73 1.71 r = - - 0. 40 0.43 0 54 0.75 1. 16 1. 94 3.64 Six-phase : 7m = \ o 44 0.62 0 88 1.27 1. 86 2. 85 4.85 7m' - J 1 0.31 0 24 0.25 0. 38 0. 75 1.79 r = 0. 28 0.31 0 41 0.60 0. 97 1. 65 3.17 oo -phase: -v — 'v ' — r Jm jm ~ •*• = 0 20 0.22 0 32 0.49 0. 82 1. 45 2.82 92. The values are shown graphically in Figs. 137 and 138, SYNCHRONOUS CONVERTERS 243 reactive current , . with tan 6 = - TT~ as abscissas, and 7 as ordinates energy current in Fig. 137, T as ordinates in Fig. 138. As seen, with increasing phase displacement, irrespectively whether lag or lead, the average as well as the maximum arma- ture heating very greatly increases. This shows the necessity of keeping the power-factor near unity at full load and overload, and when applied to phase control of the voltage by converter, means that the shunt field of the converter should be adjusted so as to give a considerable lagging current afho load, so that the ~7 7 I 7 GENERATOR HEA FIG. 137. — Maximum 72r heating in converter armature coil expressed in per cent, of direct-current generator 72r heating. current comes into phase with the voltage at about full load. It therefore is very objectionable in this case to adjust the con- verter for minimum current at no load, as occasionally done by ignorant engineers, since such wrong adjustment would give con- siderable leading current at load, and therewith unnecessary armature heating. It must be considered, however, that above values are referred to the direct-current output, and with increase of phase angle the alternating-current input, at the same output, increases, 244 ELEMENTS OF ELECTRICAL ENGINEERING and the heating increases with the square of the current. Thus at 60 deg. lag or lead, the power-factor is 0.5, and the alternating- current input thus twice as great as at unity power-factor, corre- sponding to four times the heating. It is interesting therefore to refer the armature heating to the alternating-current input, that is, compare the heating of the converter with that of a synchronous motor of the same alternating-current input. This is given by r r 1 1 — ~S PER CENT 60 / z / 320 / / 300/ / /, / / 260^ / / / ^ / / / / V / / / 200 <% f // / / / / 180 A 7 3\ &< / / 160 / ^ '* y / 140 / i / / r & 130 D REG" CU iREf T / / / / // £\ 100 CEf ERA! OR HEAT ING, /^ / s / /" " 80 . ' ^ « X x 60 -—• ^ ^ x^ •^ 40 , ' ^ ^^ F EAC1 nvF CUR RFN- 1 PF CF^ T ?n 1 0 2 D 3 ) 4 OF 0 5 FUUL UOAD POWER CU.RRE 0 6,0 If) 80 90 100 1 NT 0 1 0 1 0 1 0 1 o FIG. 138. — Average Izr heating in converter armature expressed in per cent. of direct-current generator 72r heating. and, for p = 1.04; gives the following values: 0= 0 ^ 10 20 30 40 50 60 tan (9 = 0 0.176 0.364 0.577 0.839 1.192 14.32 Three-phase: Quarter-phase : 0.5550.57 0.63 0.71 0.82 0.93 1.03 0.37 0.385 0.44 0.52 0.63 0.74 0.84 SYNCHRONOUS CONVERTERS 245 Six-phase: Ti = oo -phase: 0.26 0.28 0.335 0.42 0.52 0.63 0.73 0.185 0.197 0.26 0.34 0.44 0.55 0.65 It is seen that, compared with the total alternating-current input, the armature heating increases much less with increasing phase displacement, and is almost always much lower than the heating of the same machine at the same input and phase angle, when running a synchronous motor, as shown in Fig. 139. FIG. 139. — Average I*r heating in converter armature expressed in per cent, of synchronous motor 72r heating at the same power-factor.