CHAPTER II MULTIPLE SQUIRREL-CAGE INDUCTION MOTOR 18. In an induction motor, a high-resistance low-reactance secondary is produced by the use of an external non-inductive resistance in the secondary, or in a motor with squirrel-cage secondary, by small bars of high-resistance material located clow* to the periphery of the rotor. Such a motor has a great slip of speed under load, therefore poor efficiency and poor speed regu- lation, but it has a high starting torque and torque at low and intermediate speed. With a low resistance fairly high-reactance secondary, the slip of speed under load is small, therefore effi- ciency and speed regulation good, but the starting torque arid torque at low and intermediate speeds is low, and the current in starting and at low speed is large. To combine good start- ing with good running characteristics, a non-inductive resistance is used in the secondary, which is cut out during acceleration. This, however, involves a complication, which is undesirable in many cases, such as in ship propulsion, etc. By arranging then two squirrel cages, one high-resistance low-reactance one, consisting of high-resistance bars clow* to the rotor surface, and one of low-resistance bars, located deeper in the armature iron, that is, inside of the first squirrel cage, and thus of higher reactance, a "double squirrel-cage induction motor" in derived, which to some extent combines the characteristics of the high- resistance and the low-resistance secondary. That is, at start- ing and low speed, the frequency of the magnetic flux in the arma- ture, and therefore the voltage induced in the secondary winding is high, and the high-resistance squirrel cage thus carries con- siderable current, gives good torque and torque efficiency, while the low-resistance squirrel cage is ineffective, due to its high reactance at the high armature frequency. At speeds near synchronism, the secondary frequency, being that of slip, is low, and the secondary induced voltage correspondingly low. The high-resistance squirrel cage thus carries little current and gives little torque. In the low-resistance squirrel cage, due to its low reactance at the low frequency of slip, in spite of the relatively 27 28 ELECTRICAL APPARATUS low induced e.m.f., considerable current is produced, which is effective in producing torque. Such double squirrel -cage induc- tion motor thus gives a torque curve, which to some extent is a superposition of the torque curve of the high-resistance and that of the low-resistance squirrel cage, has two maxima, one at low speed, Mid another near synchronism, therefore gives a fairly good torque and torque efficiency over the entire speed range from standstill to full speed, that is, combines the good features of both types. Where a very high starting torque requires locating the first torque maximum near standstill, and large size and high efficiency brings the second torque maximum very close to synchronism, the drop of torque between the two maxima may be considerable. This is still more the ease, when the motor is required to reverse at full speed and full power, that is, a very- high torque is required at full speed backward, or at or near slip s — 2. In this case, a triple squirrel cage may be used, that is, three squirrel cages inside of each other: the outermost, of high resistance and low reactance, gives maximum torque below standstill, at backward rotation; the second squirrel cage, of medium resistance and medium reactance, gives its maximum torque at moderate speed; and the innermost squirrel cage, of low resistance ami high reactance, gives its torque at full speed, near synchronism. Mechanically, the rotor iron may be slotted down to the inner- most squirrel cage, so as to avoid the excessive reactance of a closed magnetic circuit, that is, have the magnetic leakage flux or self-inductive flux pass an air gap. 19. In the calculation of the standard induction motor, it is usual to start with the mutual magnetic flux, *, or rather with the voltage induced by this flux, the mutual inductive voltage E — e, as it is most convenient, with the mutual inductive voltage, c, as starting point, to pass to the secondary current by the self-inductive impedance, to the primary current and primary impressed voltage by the primary self-inductive impedance and exciting admittance. In the calculation of multiple squirrel-cage induction motors, it is preferable to introduce the true induced voltage, that is, the voltage induced by the resultant magnetic flux interlinked with the various circuits, which is the resultant of the mutual and the self-inductive magnetic flux of the respective circuit. This permits starting with the innermost squirrel cage, and INDUCTION MOTOR 29 gradually building up to the primary circuit. The advantage hereof is, that the current in every secondary circuit is in phase with the true induced voltage of this circuit, and is ix = — » where ri is the resistance of the circuit. As ei is the voltage induced by the resultant of the mutual magnetic flux coming from the primary winding, and the self-inductive flux corre- sponding to the i\X\ of the secondary, the reactance, Xif does not enter any more in the equation of the current, and Cj is the voltage due to the magnetic flux which passes beyond the cir- cuit in which e\ is induced. In the usual induction-motor theory, the mutual magnetic flux, , induces a voltage, E} which produces a current, and this current produces a self-inductive flux, 'j, giving rise to a counter e.m.f. of self-induction I\X\, which sub- tracts from E. However, the self -inductive flux, 'i, interlinks with the same conductors, with which the mutual flux, , inter- links, and the actual or resultant flux interlinkage thus is i = $ — 'i, and this produces the true induced voltage e\ = E — I\X\y from which the multiple squirrel-cage calculation starts.1 Double Squirrel-cage Induction Motor 20. Let, in a double squirrel-cage induction motor: $2 = true induced vpltage in inner squirrel cage, reduced to full frequency, It = current, and Zi = r2 + jx2 = self-inductive impedance at full frequency, reduced to the primary circuit. #i = true induced voltage in outer squirrel cage, reduced to full frequency, /i = current, and Z\ = t\ + jxi = self-inductive impedance at full frequency, reduced to primary circuit. jj? = voltage induced in secondary and primary circuits by mutual magnetic flux, #o = voltage impressed upon primary, /o = primary current, Z0 = r0 + jx0 = primary self -inductive impedance, and Yo = g — jb = primary exciting admittance. *8ee "Electric Circuits", Chapter XII. Reactance of Induction Apparatus. 30 ELECTRICAL APPARATUS The leakage reactance, Xj, of the inner squirrel cage is Hint due to the flux produced by the current in the inner squirrel cage, which passes between the two squirrel cages, and does not in- clude the reactance due to the flux resulting from the current, ft, which passes beyond the outer squirrel cage, as the latter is mutual reactance between the two squirrel cages, and thus meets the reactance, Si, It is then, at slip s: and: sEt •h + li+Yt (2) (3) E, = Et + jx, h- (4) E - E,+jx,Ut + h)- (5) E0 = E + Z„f„. (6) The leakage flux of the outer squirrel cage is produced by the m.tn.f. of the currents of both squirrel cages, /, + ft, and the reactance voltage of this squirrel cage, in (5), thus is jxt (ft + /»). As seen, the difference between E, and Et is the voltage in- duced by the flux which leaks between the two squirrel cages, in the path of the reactance, x?, or the reactance voltage, xtft', the difference between E and E, is the voltage induced by the rotor flux leaking outside of the outer squirrel cage. This has the m.m.f. f i + fi, and the reactance X\, thus is the reactance voltage xi (fi + /a). The difference between E <, and E is the voltage consumed by the primary impedance: Zafn- (4) and (5) are the voltages reduced to full frequency; the actual voltages are s times as high, but since all three terms in these equations are induced voltages, the s cancels. 21. From the equations (1) to (6) follows: f.-f(i+if) -*l(»-^)+*S+; - ft (»J + JOi), (7) (8) INDUCTION MOTOR 31 where: (10) Oi = 1 ! /Xi Xi XjV I as = * (- H h — 1 I \ri ra r*/ ; thus the exciting current : = Ei (g - >&) (ai + jot) -R(*i+jW, (11) where: 6i = <*\9 + Oa6'v and the total primary current is (3) : '•-*£ + £(1+i?)+*-+*} (13) = -Fl (Ci + JCi), where: (12) ci = - H 1- 6i Ct = — — + bt (M) rjr, and the primary impressed voltage (6) : Eo = Ei{ai + jat + (r0 + jx0) (c, + jc,) } = & (* + jdi), (15) where: di = a\ + roCi — XoC» d* = a, + r^ + XoCi hence, absolute: (16) C = -=*=• (17) to = e,\/ci* + c,*. (18) 22. The torque of the two squirrel cages is given by the product of current and induced voltage in phase with it, as: Dt = /£,, /,/' (19) «622 _ sei2 <(>+>?)■ «°> 32 ELECTRICAL APPARATUS hence, the total torque: D = D2 + Dh (21) and the power output: P = (1 - s) Z>. (22) (Herefrom subtracts the friction loss, to give the net power output.) The power input is: Po = /#o, Io/' = e22(cidi + c2d2), (23) and the volt-ampere input: Q = eoio. P Herefrom then follows the power-factor -gr * the torque effi- ciency y, , the apparent torque efficiency 7^-, the power efficiency P P jj- and the apparent power efficiency 7^ 23. As illustrations arc shown, in Figs. 14 and 15, the speed curves and the load curves of a double squirrel-cage induction motor, of the constants: Co = 110 volts; Zo = 0.1 + 0.3j; Z, = 0.5 + 0.2 j; Z2 = 0.08 + 0.4 j ; }ro = 0.01 - 0.1 j; the speed curves for the range from s = 0 to s = 2, that is, from synchronism to backward rotation at synchronous speed. The total torque as well as the two individual torques are shown on the speed curve. These curves are derived by calculating, for the values of s: s = 0, 0.01, 0.02, 0.05, 0.1, 0.15, 0.2, 0.3, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0, INDUCTION MOTOR Mill i ".".' ! '■ : DOUBLE SQUIRREL CAGE INDUCTION MOTOR SPEED CURVES ? ■\J ■ri \l .4JS AA 140 a _1 T* m. D?" U Ds in .,„ -£i -8 -.7 -.6 -.6-. 0 .1 .2 .3 A .6 .8 .7 .8 .9 1.0 s o/ douhle squirrel-cage induct iu DOUBLE SQUIRREL CAQE INDUCTION MOTOR LOAD CURVES << 1 i ••'■ i" y~ — ^\J\ /o 'ii , h . A, u J. . n : ■■ r g j ■■■ i o — Fig. 15.— Load c s of double squirrel -cage induction motoi 34 ELECTRICAL APPARATUS the values: . S2X\Xi a,\ = 1 > rir2 lX\ X\ . £2\ a% = *( )» \r\ r% T\i b\ = aig + a26, bi = a^g — a\b, ci = V + V + bu c2 = h btt rirt d\ = a\ + r0Ci — XoCj, da = a2 + roC2 + XqCi, . *o* e2* = and: di2 + = Z>! + Z>2, P = (1 - s) D, Po = e*1 (cidx + c2d2), Q = eot'oi P D P D Po Po'Po'Q'Q' Q* Triple Squirrel-cage Induction Motor 24. Let: <*> = flux, E = voltage, / = current, and Z = r + jx = self- inductive impedance, at full frequency and reduced to primary circuit, and let the quantities of the innermost squirrel cage be denoted by index 3, those of the middle squirrel cage by 2, of the outer squirrel cage by 1, of the primary circuit by 0, and the mutual inductive quantities without index. Also let: Yo = g — jb = primary exciting admittance. It is then, at slip s: current in the innermost squirrel cage: INDUCTION MOTOR 35 current in the middle squirrel cage: /* — zr> current in the outer squirrel cage: *l — 7~> (2) (3) primary current: /o = U + I* + U + Fo#. (4) The voltages are related by: & = #» + J*./., (5) tf i = #* + jx, (It + /•), (6) # = #i + J'x, (/, + /, + /,), (7) #o - # + Zo/o, (8) where x3 is the reactance due to the flux leakage between the third and the second squirrel cage; x% the reactance of the leak- age flux between second and first squirrel cage; X\ the reactance of the first squirrel cage and x0 that of the primary circuit, that is, Xt + xo corresponds to the total leakage flux between primary and outer most squirrel cage. #8, fit and #i are the true induced voltages in the three squirrel cages, $ the mutual inductive voltage between primary and secondary, and $o the primary impressed voltage. 26. From equations (1) to (8) then follows: ^l-^{l+J- + -(l+J-j+J- (9) (10) where: = #3 (ai + ja2)i (ID - 82XzXz d\ = 1 — - — r»r3 (x% . «*=*(- + X2 , xi\ r3 r3/ (12) 3 /, = - #, (O, + jttj), (13) 36 ELECTRICAL APPARATUS .SXi E = Es\ai+ jai + r:' («. + jo*) + 3 ^ (l + 3 ?) + 3 ? = ^3(6i+j62), (14) where : ri r2 r8 / J hi = aj — SXifl2 S #1X3 ri r2r3 fi r2 r8 thus the exciting current: /oo = * ov = #3 (&i + j6«) (g - jfc) = #3(Cj +JC2), where: C\ = feigr + 626, c2 = 62g — 616, and the total primary current, by (4) : /o = #3 8(ai+ja2)+^(l+;Sf3)+J+Cl+jc, r2 \ T3 / r8 where : di = ai H 1 h Ci ri r2 r3 , s s2xs . a2 = - a2 + ■ — t f2 n r2r3 Zo/o = #3 (di + jrf2) (r0 + ix0) = #3(/, +#2), where : /j = r0di — Xo^2 /2 = f(K*2 T" Xo«l thus, the primary impressed voltage, by (8) : where: #o = #3 (6i + jbt + /i + jf«) = ^3 (j/i + jg2), (/j = 6i + /i (Jz = b2 + /2 (15) (16) (17) (18) (19) (20) (21) (22) (23) INDUCTION MOTOR 37 hence, absolute: Vgi- + gS U = ez Vrfi8 + _dS, (25) e* =e3yJl+ ***?> (26) e, = e3 Va,2 + a**- (27) 26. The torque of the innermost squirrel cage thus is : D, = *?-; (28) that of the middle squirrel cage : z>2 = *ea2; (29) r2 and that of the outer squirrel cage: 0, = s- '*; (30) the total torque of the triple squirrel-cage motor thus is: D = D, + D2 + D3, (31) and the power: P = (1 - s) Z>, (32) the power input is : PQ = /#o, /o/' = £ 3.0 m -SB — - N ^ S, „. 60 JL , r I'l . D, -Hf«» Fit . i . — Speed curves of triple si]iiirre]-(.':igi> induction mo \ \ TRIPLE sguiRREL OAQE INDUCTION MOTOR LOAD CURVES * \ "n s _ ,-UXL H«. \ ^ ^ N / / •" / ^> _L - /o /" A V • % ' T / i. i 1 1 ! J. Fio. 17. — Load curves of triple squirrel-cape induction motor. INDUCTION MOTOR 39 the speed curves are shown from « = 0 to « = 2, and on them, the individual torques of the three squirrel cages are shown in addition to the total torque. These numerical values are derived by calculating, for the values of *: s = 0, 0.01, 0.02, 0.05, 0.1, 0.15, 0.20, 0.30, 0.40, 0.60, 0.80, 1.0, 1.2, 1.4, 1.6, 1.8, 20, the values: ai - 1 - 8*XiXz rtfz /Xt . Xi . Xi\ 8Xidi 82X\Xt 6l = Ol - n r2r9 hx\d\ , sxi 8X\ . 8XiG\ . SXi . 8X\ bt = «i H h H i r\ r2 rs c\ — big + 626, ci = btg + bib, dx = -- + - + - + c, r\ r2 rz j *a2 *2x« £ n + - + ci} TiTz rz /1 = rodi — Xodj, /2 = rodt + Xodi, 9\ = bi + /1, gi = bt + ft, e8* = «3, ove and the motor torque below full syn- chronism. Thus, while a concatenated couple of induction motors has two operative motor speeds, half synchronism and full synchronism, the latter is uneconomical, as the second motor holds back, and in the second or full synchronism speed range, it is more economical to cut out the second motor altogether, by short-circuiting the secondary terminals of the first motor. With resistance in the secondary of the second motor, the maximum torque point of the second motor above half syn- chronism is shifted to higher speeds, nearer to full synchronism, and thus the speed between half and full synchronism, at which the concatenated couple loses its generator torque and again becomes motor, is shifted closer to full synchronism, and the motor torque in the second speed range, below full synchronism, is greatly reduced or even disappears. That is, with high resist- ance in the secondary of the second motor, the concatenated couple becomes generator or brake at half synchronism, and remains so at all higher speeds, merely loses its braking torque when approaching full synchronism, ami regaining it again beyond full synchronism. The speed torque curves of the concatenated couple, shown in Fig. 18, with low-resistance armature, and in Fig. 19, with high resistance in the armature or secondary of the second motor, illustrate this. 30. The numerical calculation of a couple of concatenated induction motors (rigidly connected together on the same shaft or the equivalent) can be carried out as follows: Let: = number of pairs of poles of the first motor, = tiiiuiljer of pairs of poles of the second motor, CONCATENATION 43 ; a = — = ratio of poles, (1) m f = supply frequency. Full synchronous speed of the first motor then is: Se = £ (2) of the second motor: 5'. - £ (3) At slip 9 and thus speed ratio (1 — s) of the first motor, its speed is: S- (1-«)S0- U-«)£ (4) and the frequency of its secondary circuit, and thus the frequency of the primary circuit of the second motor: */; synchronous speed of the second motor at this frequency is: sS'o = s *,; n the speed of the second motor, however, is the same as that of the first motor, S, hence, the slip of speed of the second motor below its synchronous speed, is: .Z_(1_.)/.(«,.L=i)/f n n \n n / and the slip of frequency thus is: s' = 8 (1 + a) - a. (5) This slip of the second motor, «', becomes zero, that is, the couple reaches the synchronism of concatenation, for: « = ^ (6) 44 ELECTRICAL APPARATUS The speed in this case is: So0 = (1 - so) I (7) n(l+«) 31. If: a = 1, that is, two equal motors, as for instance two four-polar motors n = W = 4, it is: while at full synchronism : If: it is: so = 0.5, i • So0 = / 2 71 4 sm: So = n i a — s 2, n — 4, n' = 8, So = 2 r So0 = / 3n f that is, corresponding to a twelve-polar motor. While: if: it is: So = n i a = 0.5, n = 8, n' = 4, So 5L o = 1 3' / f 1.5n 12 CONCATENATION 45 that is, corresponding to a twelve-polar motor again. That is, as regards to the speed of the concatenated couple, it is immaterial in which order the two motors are concatenated. 32. It is then, in a concatenated motor couple of pole ratio: ri a = - > n if: * = slip of first motor below full synchronism. The primary circuit of the first motor is of full frequency. The secondary circuit of the first motor is of frequency s. The primary circuit of the second motor is of frequency «. The secondary circuit of the second motor is of frequency s' = * (1 + a) — a. Synchronism of concatenation is reached at: a 1 + a Let thus: eo = voltage impressed of first motor primary; Yo = g — jb -= exciting admittance of first motor; Y'o = g* — jb' = exciting admittance of second motor; Zo = ro + jxo = self-inductive impedance of first motor primary; Z'o = r'o + jx'o = self-inductive impedance of second motor primary; Z\ = T\ + jxi = self-inductive impedance of first motor second- ary; Z\ = r'i + jx\ = self-inductive impedance of second motor secondary. Assuming all these quantities reduced to the same number of turns per circuit, and to full frequency, as usual. If: e = counter e.m.f . generated in the second motor by its mutual magnetic flux, reduced to full frequency. It is then: secondary current of second motor: r/ _ *'* [« (1 + a) - a] e 1 * "" PT+ fix>\ - ?;+j\MV+ a) - a] x\ = e(fll " Ja'^ (8) 46 ELECTRICAL APPARATUS where: Ol = a» = r'x [s (1 + o) - a] m x', [«.(1 + a) - a]* to m - r',« + *V (« (1 + a) - a)*; exciting current of second motor: /'00 - eY' = e (g' - jb'), (9) (10) (ID hence, primary current of second motor, and also secondary current of first motor: where: /os/i = /] + /' 00 = e (bi - j6a)> bi = ax + g', bt = a* + 6', (12) (13) the impedance of the circuit comprising the primary of the second, and the secondary of the first motor, is: Z = Z/ + ZV - (n + r'0) + js (*, + x'0), (14) hence, the counter e.m.f., or induced voltage in the -secondary of the first motor, of frequency is: s$i = se + IiZ, hence, reduced to full frequency : where: C - 1 + *!-« + — = e (ci + jc2), ri + r-6i + (x1 + x/0)6a (15) 8 c% = (xj + x'o) 6i - ri + r'0 8 bt (16) 33. The primary exciting current of the first motor is: loo = $]Y = e (di - jd2), where: di = C\Q + c»6 dt = C!& — Cjj/ (17) (18) CONCATENATION 47 thus, the total primary current of the first motor, or supply current: /o = /i + /oo -«(fi-ifi), (19) where: /i = 61 + di ft - 62 + d2 (20; and the primary impressed voltage of the first motor, or supply voltage: $0 = -Fi + £0/0 -«(0i+J0t), (21) where: and, absolute: thus: 0i = Ci + ro/i + X0/2 i 02 = Ci + X0/1 — r0/2> (22) eo - e VST+'fifi*. (23) e = 77-^Y Y (24) Vgi2 + 02* Substituting now this value of e in the preceding, gives the values of the currents and voltages in the different circuits. 34. It thus is, supply current : to - e VP~+'f22 = e0 Jfll *-h]\ power input: Po = /#o, /V' = e2 (fig 1 - /202) 012 + 022 volt-ampere input: Q = Wo, and herefrom power-factor, etc. The torque of the second motor is : r « /«,/,/' The torque of the first motor is : 7\ = /#„ /o/' = C2 (C1/1 - C2/2), 48 ELECTRICAL APPARATUS hence, the total torque of the concatenated couple: T = f + Ti - e= (oj + d/. - c/i), and herefrom the power output: J3 - (1 -%) T, thus the torque and power efficiencies and apparent efficiencies, etc. 35. As instances are calculated, and shown in Fig. 18, the speed | MINI „„, £ , .. * ^£>f ^4 i L jjdro_ ..„' 1 , ■■ 1/ * k u" E » « M T r ^ ,» \ 1 s ,(. \ g »>.<*■ ~c » Wl Fig tor a = I anc I n at thL niL 18.— Sp iue cur l,oftr Y Zo Zl iB. 18 a the su ig. 19 a 'd eouj second Tw loat ning, a (Wit torque curves of concalenHted couple with low resist secondary. es of the concatenated couple of two equal mot econstants:co = IlOvolts. - Y' - 0.01 - 0.1 j; - Z', - 0.1+ 0.3 j; = Z', = 0.1 + 0.3.). ao shows, separately, the torque of the second mc j pi jr current. jowa the Bpeed torque curves of the - e oofii !e with an additional resistance r = 0.5 inserted iry of the second motor. curves of the same motor, Fig. 18, for concaten id also separately 1 he load curves of either mc ors: tor, ate- nto ted tor, CONCA TEN A TION 49 are given on page 358 of "Theoretical Elements of Electrical Engineering." 36. It is possible in concatenation of two motors of different number of poles, to use one and the same magnetic structure for both motors. Suppose the stator is wound with an n-polar primary, receiving the supply voltage, and at the same time with an n' polar short-circuited secondary winding. The rotor is wound with an n-polar winding as secondary to the n-polar primary winding, but this n-polar secondary winding is not short-circuited, but connected to the terminals of a second &« -"■' ..„ *■ .„- o.i] s i ; \ N\ _j <£ .««. , • « * » 3 «!l 0 i •. n'-polar winding, also located on the rotor. This latter thus receives the secondary current from the n-polar winding and acts as n'-polar primary to the short-circuited stator winding as secondary. This gives an n-polar motor concatenated to an n'-polar, and the magnetic structure simultaneously carries an n-polar and an n'-polar magnetic field. With this arrangement of "internal concatenation," it is essential to choose the number of poles, n and n', so that the two rotating fields do not interfere with each other, that is, the n'-polar field does not induce in the n-polar winding, nor the n-polar field in the n'-polar winding. This is the case if the one field has twice as many poles as the other, for instance a four-polar and an eight -polar field, If such a fractional-pitch winding is used, that the coil pitch is suited for an n-polar as well as an n'-polar winding, then the same winding can be used for both sets of poles. In the stator, the e qui potential points of a 2 p-polar winding are points of opposite polarity of a p-polar winding, and thus, by connecting together the equipotential points of a 2 p-polar primary winding, 50 ELECTRICAL APPARATUS this winding becomes at the same time a n-polar short-circuited winding. On the rotor, in some slots, the secondary current of the n-polar and the primary current of the n'-polar winding flow in the 3ame direction, in other Blots flow in opposite direction, thus neutralize in the latter, and the turns can be omitted in concatenation — but would be put in for use of the structure as single motor of n, or of »' poles, where such is desired. Thus, on the rotor one single winding also is sufficient, and this arrange- ment of internal concatenation with single stator and single rotor winding thus is more efficient than the use of two separate motors, and gives somewhat better constants, as the self-induclive im- pedance of the rotor is less, due to the omission of one-third. of the turns in which the currents neutralize (Hunt motor). The disadvantage of this arrangement of interna) concatenation with single stator and rotor winding is the limitation of the avail- able speeds, as it is adapted only to 4 -r- 8 + 12 poles and multiples thereof, thus to speed ratios of I + % + \i, the last being the concatenated speed. Such internally concatenated motors may be used advantage- ously sometime as constant -speed motors, that is, always run- ning in concatenation, for very slow-speed motors of very large number of poles. 37. Theoretically, any numl>er of motors may be concatenated. It is rarely economical, however, to go beyond two motors in concatenation, as with the increasing number of motors, the constants of the concatenated system rapidly become poorer. If: Y% - 9 ~ A Zo = r0+ jxa, Zi « Tx + 3*u are the constants of a motor, and we denote: Z = Z„ + Z, = (r0 + ri) + j (xn + x,) = r + jx then the characteristic constant of this motor — which char- acterizes its performance — is : (J = yz; if now two such motors are concatenated, the exciting admittance of the concatenated couple is (approximately): 1" = 2 >\ CONCATENATION 51 as the first motor carries the exciting current of the second motor. The total self-inductive impedance of the couple is that of both motors in series: Z' = 2 Z; thus the characteristic constant of the concatenated couple is: #' = y'z' = 40, that is, four times as high as in a single motor; in other words, the performance characteristics, as power-factor, etc., are very much inferior to those of a single motor. With three motors in concatenation, the constants of the system of three motors are: Y" = 3 7, Z" = 3 Z, thus the characteristic constant : 0" = y"z" = 9yz = 9 0, or nine times higher than in a single motor. In other words, the characteristic constant increases with the square of the number of motors in concatenation, and thus concatenation of more than two motors would be permissible only with motors of very good constants. The calculation of a concatenated system of three or more motors is carried out in the same manner as that of two motors, by starting with the secondary circuit of the last motor, and building up toward the primary circuit of the first motor.