Theory and Calculation of Electric Circuits — part 5 of 15

The next metal beyond nickel, in the periodic system of ele- ments, is copper, and this is non-magnetic, as far as known.

On the other side of iron, in the periodic system, is manganese.

This is very interesting in so far as it has never been observed in a strongly magnetic state, but many of the alloys of manganese are more or less strongly magnetic, and estimating from the satu- ration values of manganese alloys, the saturation value of man- ganese as pure metal should be about S = 30 x 10%. This would make it the most magnetic metal.

In favor of manganese as magnetic metal also is the unusual behavior of its alloys with iron: the alloys of nickel, and of cobalt with iron also show unusual characteristics, and this seems to be a characteristic of alloys between magnetic metals.

The best known magnetic manganese alloys are the Heusler alloys, of manganese with copper and aluminum, and the char- acteristics of three such alloys are given in the following table. The most magnetic shows about the same saturation value as magnetite, but higher saturation values, equal to those of nickel, have been observed. ;

A curious feature of some Heusler alloys is, that when slowly cooled from high temperatures, they are very little magnetic, and have low saturation values. The quicker they are cooled, the higher their permeability and their saturation value, and the best values have been reached by dropping the molten alloy into water, so suddenly chilling it.

In general, the Heusler alloys are especially sensitive to heat treatment, and some of them show the ageing in a most pro-

. 6

. 82 ELECTRIC CIRCUITS nounced degree, so that maintaining the alloy for a considerable time at moderate temperature, increases hardness and hysteresis loss more than tenfold.

. Magnetic alloys of manganese also are known with antimony, arsenic, phosphorus, bismuth, boron, with zinc and with tin, etc. Usually, the best results are given by alloys containing 20 to 30 per cent. of manganese. Little is known of these magnetic al- loys, except that they may be in a magnetic state, or in an unmagnetic stage. They are most conveniently produced by dissolving manganese metal in the superheated alloying metal, or in this metal with the addition of some powerful reducing metal, as sodium or aluminum, but the alloy is only sometimes magnetic, sometimes practically unmagnetic, and the conditions of the formation of the magnetic state are unknown.

Apparently, there also exists an intermediary oxide of mangan- ese, or & compound oxide of manganese with that of the other

’ metal, which is strongly magnetic. The black slag, appearing in the fusion of manganese with other metals such as antimony, zinc, tin, without flux, often is strongly magnetic, more so than the alloy itself.

A mixture of about 25 per cent. powdered manganese metal, and 75 per cent. powdered antimony metal, heated together to a moderate temperature—in a test-tube—gives a strongly mag- netic black powder, which can be used like iron filings, to show the lines of forces of the magnetic field, but has not further been investigated.

A considerable number of such magnetic manganese alloys have been investigated by Heusler and others, and their constants are given in the following table.

It is supposed that these magnetic manganese alloys are chem- ical compounds, similar as magnetite or magnetkies. Thus the copper-aluminum-manganese alloy of Heusler is a compound of 1 atom of aluminum with 3 atoms of copper or manganese: Al- (Mn or Cu)s, usually AIMnCus. Other magnetic manganese compounds then are:

With antimony........................ MnSb and Mn.8b With bismuth....................+.++. MnBi With arsenic......................+... MnAs With boron..............eeeeee+0e20+. MnB . With phosphorus...................... MnP With tin..........0c.. eee eeeeeeeeeeeee MngSn and MnSn {

MAGNETISM 83

Next adjacent to manganese in the periodic system of elements is chromium, Neither the metal, nor any of its alloys (except those with magnetic metals) have ever been observed in the mag- netic state. There is, however, an intermediary oxide of chro- mium, alleged to be CrsO¢ (a basic chromic chromate?) which is strongly magnetic. It forms, in black scales, in a narrow range of temperature, by passing CrO.Cl. with hydrogen through a heated tube. ,

A second strongly magnetic chromium oxide is Cr,O¢ (a basic chromic bichromate?). It is easily produced by rapidly heat- ing CrOs, but the product is not always the same. Their magnetic characteristics have never been investigated, and they are the only indication which would point to chromium having potentially magnetic qualities.

The metal next to chromium in the periodic system of elements, vanadium, is non-magnetic, as far as known.

. 50. On attached tables are given the magnetic constants of the better known magnetic materials, metals, alloys, mixtures and compounds:

The first tables give the saturation density, S, and the demag- netization temperature, that is, temperature at which the ma- terial ceases to be ferromagnetic, and its specific gravity.

It is interesting to note that with some magnetic materials the demagnetization temperature is very close to, or within the range of, atmospheric temperature.

The second table gives more complete data of those materials, of which such data are available. It gives:

S = saturation density, or value of B — H for infinitely high.H;

a = coefficient of magnetic hardness;

o = coefficient of magnetic saturation.

. Where the reluctivity line shows a bend at some critical point, a and o are given for the lower range—which is the one indus- trially most useful—together with the range of field intensity, for which this value applies, and are given also for the highest range observed, together with the value of field intensity H, above which the latter values of « and o apply.

. n = coefficient of hysteresis, in the 1.6 power law.

B = coefficient of unsyfometrical cycle, for the two cases where this is known.

Demagnetization temperature, that is, temperature at which ferromagnetism ceases. :

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(beyond this, the linear law of reluctivity usually applies), for . . : : a number of magnetic materials of higher values of saturation os densities. . . : . . Fig. 43 gives, with twice the scale of ordinates, but the same wo. / ts a! . po €0 a ta fet cae 7 ; oa “ Supe , foe . ‘te 7 tee Loe : ‘ soe ‘ G 7 Diatizes ty GOOLE . Jos

85 low ‘of ials ne ater he ic, m: . . ISM me dt erist ET. 80 an ct - N of . hoe Hye AG istic Sciss: c oe M. ris ab tic _. ete of e oo ° ; ra ale agn: | : cha: e BC the m sas ic th f are 4 et 0 . ae i Hatae ™m ‘ niu et a . density inti onaaee _ con de wr the in a isse ate aaaes a i. ae i C18s: on . es, . aa HE 3 saurti inaies poate! ¢ eae eee —s f or aa feo He eH Hts Hest ie fey |. ca oO _ iveaal aoe HE 5 4 le eats ie ee . sea a LS qth i eceatt fi jo Lo | aa ; a 8 ee ao ES usu = i vo = i ee Baise aH iit rei $3 Huet HEE seeeee eee ae it HEH Lae enti if Hid oe eg. a a a ae i H ai = cai a oo aE i at vo dee ee fest = Bes ee Hin: _ HH ae HE HEH fees iF He a HH Hui ay ; oc Eee aaa if i ff fe | a ous HnE oo Hn ae a : sa fa ped ft i eat Bae hag StH aiff #4 fi a i a Hee sa RARE 4 ve ee a fi F Co og | He ag a ae #3: sash HE SSeSEE st ni HH Hii : a3 i IE SHH ie — = ee eeeoe Lo 4 a feet apa . ee a it ae fag ifs Hi ae ret Es sees ! tt # =o : ao pas ay rH Sie it Nits ane : ee ce 10 42, cs. Oe case peseteese : 2 ih ig. ye. _ vaniny . oe 8 0: pid istic [oe shee # ial ra r pee er 0 cte pe 43, hich cen chara ted i pie He ee patresit HHH posses . : a soft me heir deno eae ically w ft Le tic f a, t+ Oo 44 are a nenetic tial par nd le as em valu iti 42. 43 a 0g . ee th Ww e in ig. 42, O : ke ° for th ee th fF igs. y =ose : . b : u 16, wi that # in F itized = ials ity e 8¢ tics Dig H ater dens. th ris p to , m of in acte ul t values OW char: tha Vv: ell sh tic high e is not magnet “The dune 1 2

» lo \ / ; 86 ELECTRIC CIRCUITS by numbers, and these numbers refer to the materials given in the table of ‘Magnetic Constants” under the same numbers. With regards to the magnetic data, it must be realized, however, ! . that the numerical values, especially of the less-investigated | : materials, are to some extent uncertain, due to the great diffi- | culty of exact magnetic measurements. SEEEEEE ST HEEER REET cate E ARERR RETOGEERRECEEE ESTEE EUOCRET EG ETERGCER EEE ERECT (EERO REET ESE ERUREE REE Ce eee een ee ee eee ee Hee eee ee ee ee Bbipesis: Sttetesel gy 5 <Sbbi Sees seedan bee cesaged tetsecseet pucssceses casetscees sestectsas tug W tne! toesesee se ti i rest igaEEHEE HSHGEY, 328228 ateanetay asses cee Fc EEo SEES SEES oot ee Ena ROE Eee PRE HE ee HH sHitigittsstismrttiitisetistisstss ited tee teed etess | 3353 SoSo8S eos! | sSsSosoe: Beets aces. titties 583: + Lint 33 a il aL ers Se te E He ee Hay Fssee ests stir bustes tessenasts coisssaaet tagersetas suazeesegs tescseaay. scetsestad teestsasnd testasteas Stisseee Sh35 Pcoatised tteestase! Hest: . rr us ae | ft eS ce gee eee duties | SR de GR a eM | Hg EE fT a | ee ae eee Fe | ee ay Hetiefitt atts REE Een ateRRe HE fedtai anes rcnttee teeta tatanee atte Hain teenie ae He Estibrtititirisiimettstttscteettisttiapeereretteattiittissreietisetitiststi tia icsasl Siebasedastisecesteatesestsssartsaeesees tessetared sepesses! . A : oe ee . | Cee : ei pera sre trien eiadiaeenp aid THR a ere Sao efi | ; ya. 44, The saturation density, S, which is the most constant and | most definite and permanent magnetic quantity, can be measured either directly, by measuring B in such very high fields, —H = 10,000 and over—that B—H does not further increase, or in- directly, by observing the B, H curve up to moderately high fields, therefrom derive the reluctivity curve: p, H, and from the straight-line law of the latter curve determine o and therewith B. Digitized by Google |

MAGNETISM 87 Tasie I.—Saruration Densiry . S=(B—H)p-o Satura- | Demag- 8 . tion netization| Specific Material Authority | density, | tempera- | gravity 8 ture, °C. x 10+3 . Iron: Most probable value...........0..eeeeeeeees 1916 21.0-21.5 760 7.70 Best standard sheet steel, annealed.........| 1915-16 20.70 ° Average standard sheet steel, annealed......| 1915-16 20.20 Pure iron.......... 0s see eeneeeeeeereeeee+| Wedekind | 21.10 765 Swedish wrought iron.....................| Ewing 21.25 Tron, 99.88 per cent................-+.-+--| Hatfield 21.15 Tron... . cece cece eee eee eee eeeee eee} Gumlich 21.60? Electrolytic iron................+.+++++++-| Gumlich 21.70? Commercial steel.................0-++++++| Williams 22.00? Vacuum-melted electrolytic iron............| Williams 22.60? Pure iron. ........c cc cece ee ee eee eeeeeeee+| DuBois 23.207 756 7.86 Average sheet iron,..............--.ese0ee 1892 20.10 Average medium silicon steel, annealed, 2.5 PCr CEN... ... eee eee cece ence cece s| 1915-16 19.25 Average soft steel castings...........-..-.- 1915-16 20.20 Bpecia tungsten steel...............-- 200 -[eseeeeeees| 20.30 agnet steel. ....... eee eee eee ee eee tlecese cece | 18.50 Average cast iron..............5..00++2-++| 1915-16 15.00 FesCo, cobalt iron.............+2+++,-+++++-| Williams 22.50? 520 Fe:Co, cobalt iron, vacuum-melted and forged.| Williams 25.80? FesCo, cobalt-iron, probable value, about.....|..........| 23.50 Scale of silicon steel........... 00. eee le cence eee 9-10 Iron amalgam, 11 per cent...............0005 1892 0.90 Magnetite, FesOs............. 2.0 e eee e eee 1802 4.70 |.........] 5.10 : Magnetite, FesOi.............222+-+0+-+++-| DuBois 5.46-5.91| 536-589 Magnetkies, FerSs or FesSe........-.--+++-+- DuBois 0.88 |.........]| 4.60 Cu e204, copper ferrite........-- 02-020 -01 ++ ns Ce 280 CoFes04, cobalt ferrite.......... 00: cee cece elec eee eee cfereeeeees| 280-290 Cobalt: Probable value.........0cceeeeeeeeee ee ereen 1916 12-14 Cast cobalt......ccccc ese c cece e eee e eee eeleseeceeeee | 21,10 Cobalt..... cece cece cece cece cere ee ttettessececee.| 12.10 Cobalt..... 0. cece eee e eee e reese eeeeeeess| Wedekind} 13.30 Cobalt, 1.66 per cent. Fe.........-.....-..] Ewing 16.45? Cobalt...... cc cece e cece eee eeeeeeeeees| DuBois 17.20? 1075 8.70 Cobalt, pure........... cece eee eee ses | Stifler 17.85? Cobalt, vacuum-melted.............-...--| Williams 18.85? Nickel: Probable value. sosssscccaseccrssersrertert: 1916 87. ickel, 99 per cent. pure..........0--- eee le eeee sees : Nickel wire, soft........ccceeeeeeereeere-| 1892 5.88 Nickel, cast........ccccesscc cece cere eee fes enc eeeee 6.52 Nickel... ......ccccccececeseeseeeeeess++| DuBois 7.27? | 340-376 | 8.93 Nickel... . 0. ccc cece cece cece cece eet tle ae eeeeeee 8.17? Monel metal...........ccc cee e cece eet ete licesaneeee 2.22 Binel metal......... cece cece eee e eee leneeeeeees 0.26 . Manganese, Heusler alloys: AiMnCus, soft, high permeability. ....-.++-+-[.ss0se000+ 4.67 AlMnCuz, hard, high permeability...........-).......... 3.92 AlMnCuz, soft, low permeability..........-.-)00....00 0s 1.56 AlMnCus, highest values.............0-00eeelic cece eee 7.00 310 Manganese-antimony, MnSb..............0--hocleeeeees 7.00 | 310-330 Manganese-antimony, MnoSb..........000- selec cece eens 3.85 j Manganese-boron ee ee 3.10 Manganese-phosphorus MmP..... cee cee eli cece eens 0.70 18-26 Manganese-bismuth, MoB o.oo ccc cece cece s[occccesces[eeeeee ees] 360-380 Manganese-arsenio, MnA8..........0s-000--- veccaeeeccfeseceeees| 40-50 Manganeee-tin, Mn«Sn Chromium: CriOv, chromic bichromate CrsOs, chromic chromate nn

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90 ELECTRIC CIRCUITS Such extremely high fields, as to reach complete magnetic ° saturation, are produced only between the conical pole faces of a very powerful large electromagnet. The area of the field then is very small, and it is difficult to get perfect uniformity of the field. The tendency is to underestimate the field, and this gives too high values of S. Thus, in the following table those values of S, which appear questionable for this case, have been marked by the interrogation sign.

The indirect method, from the straight-line reluctivity curve, gives more accurate values of S, as S is derived from a complete curve branch, and this method thus is preferable. However, the value derived in this manner is based on the assumption that there is no further critical point in the reluctivity curve beyond the observed range. This is correct with iron, as the best tests by the direct method check. With cobalt, there may be a critical point in the reluctivity curve beyond the observed range, as there are several observations by the direct method, which give very much higher, though erratic, values of saturation, S.

The value of the magnetic hardness, a, also is difficult to de- termine for very soft materials, especially where the method of . observation requires correction for joints, etc., and the extremely high values of permeability—over 15,000—therefore appear questionable.

CHAPTER VI MAGNETISM MECHANICAL FORCES

• 1. General

61. Mechanical forces appear wherever magnetic fields act on electric currents. The work done by all electric motors is the result of these forces. In electric generators, they oppose the driving power and thereby consume the power which finds its ; equivalent in the electric power output. The motions produced by the electromagnet are due to these forces. Between the primary and the secondary coils of the transformer, between conductor and return conductor of an electric circuit, etc., such . mechanical forces appear. The electromagnet, and all electrodynamic machinery, are . based on the use of these mechanical forces between electric conductors and magnetic fields. So also is that type of trans- former which transforms constant alternating voltage into con- stant alternating current. In most other cases, however, these mechanical forces are not used, and therefore are often neglected in the design of the apparatus, under the assumption that the construction used to withstand the ordinary mechanical strains to which the apparatus may be exposed, is sufficiently strong to withstand the magnetic mechanical forces. In the large appara- ; tus, operating in the modern, huge, electric generating systems, these mechanical forces due to magnetic fields may, however, especially under abnormal, though not infrequently occurring, conditions of operation (as short-circuits), assume such formi- dable values, so far beyond the normal mechanical strains, as to re- quire consideration. Thus generators and large transformers on big generating systems have been torn to pieces by the magnetic mechanical forces of short-circuits, cables have been torn from their supports, disconnecting switches blown open, ete. In the following, a general study of these forces will be given. This also gives a more rational and thereby more accurate de- 91

92 ELECTRIC CIRCUITS

sign of the electromagnet, and permits the determination of what |

may be called the efficiency of an electromagnet. | Investigations and calculations dealing with one form of .

energy only, as electromagnetic energy, or mechanical energy, ;

usually are relatively simple and can be carried out with very

high accuracy. Difficulties, however, arise when the calculation |

; involves the relation between several different forms of energy, | as electric energy and mechanical energy. While the elementary | relations between different forms of energy are relatively simple, | the calculation involving a transformation from one form of ; energy to another, usually becomes so complex, that it either can not be carried out at all, or even only approximate calculation becomes rather laborious and at the same time gives only a low

degree of accuracy. In most calculations involving the trans- | formation between different forms of energy, it is therefore preferable not to consider the relations between the different forms of energy at all, but to use the law of conservation of energy to relate the different forms of energy, which are involved.

. Thus, when mechanical motions are produced by the action of a magnetic field on an electric circuit, energy is consumed | in the electric circuit, by an induced e.m.f. At the same time, the stored magnetic energy of the system may change. By the | law of conservation of energy, we have:

Electric energy consumed by the induced emf. = mechanical energy produced, + increase of the stored magnetic energy. (1) | The consumed electric energy, and the stored magnetic energy, are easily calculated, as their calculation involves one form of , energy only, and this calculation then gives the mechanical work | done, = Fl, where F = mechanical force, and 1 = distance over | which this force moves. Where mechanical work is not required, but merely the me- chanical forces, which exist, as where the system is supported against motions by the mechanical forces—as primary and secondary coils of a transformer, or cable and return cable of a _ ¢ircuit—the same method of calculation can be employed, by assuming some distance | of the motion (or dl); calculating the mechanical energy wo =Fl by (1), and therefrom the mechanical Wo dwo . force as F = T orF = a Since the induced e.m.f., which consumes (or produces) the electric energy, and also the stored magnetic energy, depend on | i

MAGNETISM 93 the current and the inductance of the electric circuit, and in alternating-current circuits the impressed voltage also depends on the inductance of the circuit, the inductance can frequently be expressed by supply voltage and current; and by substituting this in equation (1), the mechanical work of the magnetic forces can thus be expressed, in alternating-current apparatus, by sup- . ply voltage and current.

In this manner, it becomes possible, for instance, to express the mechanical work and thereby the pull of an alternating electromagnet, by simple expressions of voltage and current, or _to give the mechanical strains occurring in a transformer under short-circuits, by an expression containing only the terminal voltage, the short-circuit current, and the distance between

primary and secondary coils, without entering into the details of the construction of the apparatus.

This general method, based on the law of conservation of energy, will be illustrated by some examples, and the general equations then given. .

‘2. The Constant-current Electromagnet

52. Such magnets are most direct-current electromagnets, and also the series operating magnets of constant-current arc lamps on alternating-current circuits.

Let i) = current, which is constant during the motion of the armature of the electromagnet, from its initial position 1,

to its final position 2,1 = the length of this motion, or the stroke of the electromagnet, in centimeters, and n = number of turns of the magnet winding. _ The magnetic flux ®, and the inductance L=%* jo (2) to of the magnet, vary during the motion of its armature, from a minimum value, ; &, = 12h4 19s (3) n in the initial position, to a maximum value, 7 oy = 2 1s a in the end position of the armature.

| 94 ELECTRIC CIRCUITS Hereby an e.m.f. is induced in the magnet winding, d® dL | ‘= n— 8 = 4, — c= ne 10-8 = ty di (5) | This consumes the power | oe .,4aL | p = toe’ = io? a (8) . and thereby the energy 1 w= [pdt = niga — Ls) —@ 2 Assuming that the inductance, in any fixed position of the armature, does not vary with the current, that is, that magnetic saturation is absent,' the stored magnetic energy is: In the initial position, 1, tol W1= “7 (8) in the end position, 2, 2 Ww. = is (9) The increase of the stored magnetic energy, during the motion of the armature, thus is -] to! = wy — wy = (Le — In) (10) The mechanical work done by the electromagnet thus is, by the law of conservation of energy, W=w- w' , 2 = > (Le _ 11) joules. (11) If 1 = length of stroke, in centimeters, F = average force, or pull of the magnet, in gram weight, the mechanical work is Fl gram-cm. Since g = 981 cm.-sec. (12) = acceleration of gravity, the mechanical work is, in absolute _ units, Flg 1 If magnetic saturation is reached, the stored magnetic energy is taken from the magnetization curve, as the area between this curve and the vertical axis, as discussed before.

MAGNETISM 95 and since 1 joule = 10’ absolute units, the mechanical work is . wo = Flg 10+? joules. (13) From (11) and (12) then follows, ye Fl = 3 (Ly — L1)10** gram-cm. (14) as the mechanical work of the electromagnet, and i? Le — 11 ,,,

F 29 — 107 grams (15) as the average force, or pull of the electromagnet, during its stroke 1. .

Or, if we consider only a motion element dl,

a 10 AL os F= 3g dl 10+? grams (16) as the force, or pull of the electromagnet in any position 1. Reducing from gram-centimeters to foot-pounds, that is, giving the stroke ! in feet, the pull F in pounds, we divide by 454 X 30.5 = 13,850 which gives, after substituting for g from (12) (14): FU = 3.68 io?(L_ — Ly) ft.-Ib (17) (15): F = 3.68%¢ 27" pp, (18) (16): F = 3.6846¢ 2b. (19)

These equations apply to the direct-current electromagnet as well as to the alternating-current electromagnet.

In the alternating-current electromagnet, if to is the effective value of the current, F is the effective or average value of the 7 pull, and the pull or force of the electromagnet pulsates with double frequency between 0 and 2F.

53. In the alternating-current electromagnet usually the vol- tage consumed by the resistance of the winding, tor, can be neglected compared with the voltage consumed by the reactance of the winding, oz, and the latter, therefore, is practically equal to the terminal voltage, e, of the electromagnet. We have then, by the general equation of self-induction,

e = 2x fLlio (20)

• 96 ELECTRIC CIRCUITS ’ where f = frequency, in cycles per second. From which follows, . . e ; toL = Qaf (21) and substituting (21) in equations (14) to (19), gives as the equa- tion of the mechanical work, and the pull of the alternating-current electromagnet. In the metric system: . , _— ‘7 . Fl = foes UG gram-om, (22) , ° to(es _ é1) 107 to de . P= —"Gafgl ~ Fafg aio erems (23) In foot-pounds: FI = 9:586 tole — es) “aes —*) ft -1b, (24) _ 0.586 to(ee _ €1) = 0.58670 de F a Ib. (25) . Example—tIn a 60-cycle alternating-current lamp magnet, the stroke is 3 cm., the voltage, consumed at the constant alter- nating current of 3 amp. is 8 volts in the initial position, 17 volts in the end position. What is the average pull of the magnet? UL =3 cm. . a= 8 ee = 17 : f = 60 to = 3 hence, by (23),
• F = 122 grams (= 0.27 Ib.) ‘ . _ The work done by an electromagnet, and thus its pull, depend, by equation (22), on the current i) and the difference in voltage between the initial and the end position of the armature, e; — 1; . that is, depend upon the difference in the volt-amperes con- sumed by the electromagnet at the beginning and at the end of the stroke. With a given maximum volt-amperes, ioé¢2, available for the electromagnet, the maximum work would thus be done, that is, the greatest pull produced, if the volt-amperes at the beginning of the stroke were zero, that is, e: = 0, and the theoretical maximum output of the magnet thus would be 49€2107 F,l = “4af9 (26) (

. | MAGNETISM 97 and the ratio of the actual output, to the theoretically maximum output, or the efficiency of the electromagnet, thus is, by (22) and (26), F _ 62 — & . 7 Fa = es (27) or, using the more general equation (14), which also applies to the direct-current electromagnet, _in-h 7= Ls (28)

The efficiency of the electromagnet, therefore, is the dif- ference between maximum and minimum voltage, divided by the maximum voltage; or the difference between maximum and minimum volt-ampere consumption, divided by the maximum volt-ampere consumption; or the difference between maximum and minimum inductance, divided by the maximum inductance.

As seen, this expression of efficiency is of the same form as that of the thermodynamic engine,

T,— 71 T:

From (26) it also follows, that the maximum work which can be derived from a given expenditure of volt-amperes, i¢2, is limited. For ioe, = 1, that is, for 1 volt-amp. the maximum work, which could be derived from an alternating electro- _ magnet, is, from (26),

107 _ 810 F,/ = ——- = — gram-cm. 29 . anfg f ® @)

That is, a 60-cycle electromagnet can never give more than 13.5 gram-cm., and a 25-cycle electromagnet never more than 32.4 gram-cm. pull per volt-ampere supplied to its terminals.

Or inversely, for an average pull of 1 gram over a distance of 1 cm., a minimum of a volt-amp. is required at 60 cycles, . and a minimum of =A volt-amp. at 25 cycles.

Or, reduced to pounds and inches:

For an average pull of 1 lb. over a distance of 1 in., at least 86 volt-amp. are required at 60 cycles, and at least 36 volt- amp. at 25 cycles.

This gives a criterion by which to judge the success of the design of-electromagnets.

’ 98 ELECTRIC CIRCUITS 3. The Constant-potential Alternating Electromagnet