Theory and Calculation of Electric Circuits — part 3 of 15

21. The arc stream is conducting only in the direction of its motion, but not in the reverse direction. Any body, which is . reached by the arc stream, is conductively connected with ‘it, if positive toward it, but is not in conductive connection, if negative or isolated, since, if this body is negative to the arc stream, an arc stream would have to issue from this body, to connect it con- ductively, and this would require energy to be expended on the body, before current flows to it. Thus, only if the arc stream is ; very hot, and the negative voltage of the body impinged by it very high, and the body small enough to be heated to high tem- . perature, an arc spot may form on it by heat energy. If, there- fore, a body touched by the arc stream is connected to an alternat- ing voltage, so that it is alternately positive and negative toward ; the arc stream, then conduction oecurs during the half-wave, when this body is positive, but no conduction during the negative

\ half-wave (except when the negative voltage is so high as to give . disruptive conduction), and the are thus rectifies the alternating voltage, that is, permits current to pass in one direction only. The arc thus is a unidirectional conductor, and as such extensively used for rectification of alternating voltages. Usually vacuum . ‘arcs are employed for this purpose, mainly the mercury arc, due : to its very great rectifying range of voltage.

Since the arc is a unidirectional conductor, it usually can not . exist with alternating currents of moderate voltage, as at the end

of every half-wave the arc extinguishes. To maintain an alterna- ting arc between two terminals, a voltage is required sufficiently high to restart the arc at every half-wave by jumping an elec- trostatic spark between the terminals through the hot residual vapor of the preceding half-wave. The temperature of this vapor : , is that of the boiling point of the electrode material. The voltage | required by the electrostatic spark, that is, by disruptive conduc- | tion, decreases with increase of temperature, for a 13-mm. gap | about as shown by curve I in Fig. 18. The voltage required to, | maintain an arc, that is, the direct-current voltage, increases with | increasing arc temperature, and therefore increasing radiation, |

etc., about as shown by curve II in Fig. 18. As seen, the curves I and II intersect at some very high temperature, and materials | as carbon, which have a boiling point above this temperature,

ELECTRIC CONDUCTION 33 require a lower voltage for restarting than for maintaining the arc, that is, the voltage required to maintain the arc restarts it : at every half-wave of alternating current, and such materials thus give a steady alternating arc. Even materials of a somewhat

; lower boiling point, in which the starting voltage is not much above the running voltage of the arc, maintain a steady alter- nating arc, as in starting the voltage consumed by the steadying resistance or reactance is available. Electrode materials of low

a Lp | FINA LZ Pa pf ty ptt tata tT BASE | SERENE | PET TTT PT ITN Tt | =) gage HELPS | || tf ee | fg | TONS TT te LETT] | at rar | DCB “Fat a}-— | PTT TT iy Et edt tt tt | H]AA = Pt tT ttt tt tt }-tfas PTT TT TT eden [| PTE p | d | fo | fo | xm | om | apo | mm | I . ia. 18. boiling point, however, can not maintain steady alternating arcs

. at moderate voltage. :

The range in Fig. 18, above the curve I, thus is that in which ; alternating arcs can exist; in the range between I and II, an alter- nating voltage can not maintain the arc, but unidirectional cur- rent is produced from an alternating voltage, if the are conductor is maintained by excitation of its negative terminals, as by an auxiliary arc. This, therefore, is the rectifying range of arc con- duction. Below curve II any conduction ceases, as the voltage is insufficient to’ maintain the conducting vapor stream.

Fig. 18 is only approximate. As ordinates are used the loga-

3

34 ELECTRIC CIRCUITS

rithm of the voltage, to give better proportions. The boiling points of some materials are approximately indicated on the curves,

It is essential for the electrical engineer to thoroughly under- stand the nature of the arc, not only because of its use as illumi-

nant, in arc lighting, but more still because accidental arcs are the foremost cause of instability and troubles from dangerous transients in electric circuits.

PRIN | TT TTT Ty [st PEAT TT TET tT TE Tt TA SNE NS Eee

TIN TRE TEE ETT TT PT REIN TE EE TT TTT he SENSE NEE Py TL KEEN EET TT TT Le PAT IN TT TSE TTT a PAT TINE Ty See TT PINE TT INE | TT PEL PN} f EB fae SRNR PT LENT ET ET Pr PCE ESS CCE PNET TE TET TP eee ee CCSSEE EEE CCC

. Pt tT TT TE TE TPT | EERE ER * PT tT ttt Tet tt tt ty Pte te ts mt ete ti Ls | Fig. 19.

22. The voltage consumed by an arc stream, é, at constant current, t, is approximately proportional to the arc length, l, or rather to the arc length plus a small quantity, , which probably 7! represents the cooling effect of the electrodes.

Plotting the arc voltage, e, as function of the current, i, at con- stant arc length, gives dropping volt-ampere characteristics, and the voltage increases with decreasing current the more, the longer

ELECTRIC CONDUCTION 35 the arc. Such characteristics are shown in Fig. 19 for the mag- : netite arcs of 0.3; 1.25; 2.5 and 3.75 cm. length. These curves can be represented with good approximation by the equation r ' cl+ 8 e=a+ Vi (4) This equation, which originally was derived empirically, can also be derived by theoretical reasoning: Assuming the amount of arc vapor, that is, the section of the : conducting vapor stream, as proportional to the current, and the heat produced at the positive terminal as proportional to the ; vapor stream and thus the current, the power consumed at the terminals is proportional to the current. As the power equals the current times the terminal drop of voltage, it follows that this terminal drop, a, is constant and independent of current or arc length—similar as the terminal drop at the electrodes in electro- lytic conduction is independent of the current. The power consumed in the are stream, pi = é;7, is given off from the surface of the stream, by radiation, conduction and con- vection of heat. The temperature of the arc stream is constant, as that of the boiling point of the electrode material. The power, therefore, is proportional to the surface of the arc stream, that is, proportional to the square root of its section, and therefore the square root of the current, and proportional to the arc length, i, plus a small quantity, 5, which corrects for the cooling effect of the electrodes. This gives Pi =ai=cvVi (+8) ” e(l + 8) a= Vi (5) as the voltage consumed in the arc stream. Since a represents the coefficient of power consumed in produc- ing the vapor stream and heating the positive terminal, and c the coefficient of power dissipated from the vapor stream, a and c are different for different mater-als, and in general higher for materials of higher boiling point and thus higher arc tempera- ture. c, however, depends greatly on the gas pressure in the space in which the arc occurs, and decreases with decreasing gas pressure. It is, approximately, when ! is given in centimeter at atmospheric pressure,

. . | { | | 36 ELECTRIC CIRCUITS | a= 13 volts for mercury, | = 16 volts for zinc and cadmium (approximately), | = 30 volts for magnetite, = 36 volts for carbon; ' c = 31 for magnetite, | . = 35 for carbon; 5 = 0.125 cm. for magnetite, ° | = 0.8 cm. for carbon. The least agreement with the equation (4) is shown by the car- bon arc. It agrees fairly well for arc lengths above 0.75 cm., but for shorter arc lengths, the observed voltage is lower than given by equation (4), and approaches for 1 = 0 the value e = 28 volts. _ It seems as if the terminal drop, a = 36 volts with carbon, con- sists of an actual terminal drop, a9 = 28 volts, and a terminal drop of a: = 8 volts, which resides in the space within a short distance from the terminals. { | Stability Curves of the Arc : 23. As the volt-ampere characteristics of the arc show a de- | crease of voltage with increase of current, over the entire range of current, the arc is unstable on constant voltage supplied to its | terminals, at every current. | Inserting in series to a magnetite arc of 1.8 cm. length, shown as curve I in Fig. 20, a constant resistance of r = 10 ohms, the vol- tage consumed by this resistance is proportional to the current, and thus given by the straight line II in Fig. 20. Adding this voltage IT to the arc-voltage curve I, gives the total voltage con- sumed by the arc and its series resistance, shown as curve III. | In curve III, the voltage decreases with increase of current, up to % = 2.9 amp. and the arc thus is unstable for currents below 2.9 amp. For currents larger than 2.9 amp. the voltage increases with increase of current, and the arc thus is stable. The point % = 2.9 amp. thus separates the unstable lower part of curve | III, from the stable upper part. With a larger series resistance, 7’ = 20 ohms, the stability range | is increased down to 1.7 amp., as seen from curve III, but higher voltages are required for the operation of the arc. With a smaller series resistance, r’’ = 5 ohms, the stability | range is reduced to currents above 4.8 amp., but lower voltages | are sufficient for the operation of the arc. | { | | 7 |

ELECTRIC CONDUCTION - 37 At the stability limit, 2, in curve III of Fig. 20, the resultant characteristic is horizontal, that is, the slope of the resistance curve II: r = a is equal but opposite to that of the arc charac- Pe Ett tT ET tT TE TE TT A HAE PIAL EET TE TT ET tT PTV TPT TT et TT Tad i ee PRIME TTT | etimy oT TT PANN TT eT TT Te BSNS ae CANECE TTT ert SENNA? eee PL INNES TT et Ta He) NSE Epa tip SRRENG eS ; PET TT PINT PTT Peet pt tT Tt eet ea ptt tT tet yt | ee oe Pt TT Taper tT eet tT Pt PT AT eet tt ptt ett rat tet Pt i eat ert tt anceps | wed Th es Et Fia. 20. teristic I: a. The resistance, r, required to give the stability limit at current, 7, thus is found by the condition r=— a (6) Substituting equation (4) into (6) gives = c(l + 8) "7 T= WE (7)

mT | | { 38 ELECTRIC CIRCUITS as the minimum resistance to produce stability, hence, . e(L+ 8) _ | i= aVi 0.5 €1 (8) | where e,; = arc stream voltage, and E=et+ri _ c(l + 8) a+ 1.5 — Vi (9) is the minimum voltage required by arc and series resistance, to just reach stability. (9) is plotted as curve IV in Fig. 20, and is called the stability

• curve of the arc. It is of the same form as the arc characteristic I, and derived therefrom by adding 50 per cent. of the voltage, é;, consumed by the arc stream.

The stability limit of an arc, on constant potential, thus lies at an excess of the supply voltage over the arc voltage e = a + é1, by 50 per cent. of the voltage, e:, consumed in the arc stream.

In general, to get reasonable steadiness and absence of drifting of current, a somewhat higher supply voltage and larger series resistance, than given by the stability curve IV, is desirable. .

24. The preceding applies only to those arcs in which the gas pressure an the space surrounding the arc, and thereby the arc . vapor pressure and temperature, are constant and independent , ; of the current, as is the case with arcs in air, at “atmospheric pressure.”

With arcs in which the vapor pressure and temperature vary with the current, as in vacuum arcs like the mercury arc, different considerations apply. Thus, in a mercury arc in a glass tube, if the current is sufficiently large to fill the entire tube, but not so large that condensation of the mercury vapor can not freely occur in a condensing chamber, the power dissipated by radiation, . etc., may be assumed as proportional to the length of the tube, and to the current

p= et = di thus, a=d (10) that is, the stream voltage of the tube, or voltage consumed by the arc stream (exclusive terminal drop) is independent of the | |

ELECTRIC CONDUCTION 39 current. Adding hereto the terminal drop, a, gives as the total voltage consumed by the mercury tube

e=a+el (11) for a mercury arc in a vacuum, it is approximately c= it (12) where d= diameter of the tube, since the diameter of the tube . is proportional to the surface and therefore to the radiation coefficient. Thus, - e= 13414 (13)

At high currents, the vapor pressure rises abnormally, due to incomplete condensation, and the voltage therefore rises, and Pitt tT ti ttt EE TT TT TT TT TT f { | VOLT- AMPERE CHARACTERISTIC OF VACUUM Pf] tt dl Pt ft L=40 CM, pn 2.2 cM. r { J tt

100 Pt tt APPROX, € ~"a19 —4.21 — 6.8 lf Fe pbb ‘AT Tt tt PINT TTP ye | eer ba POSSE ETE eee rte CCC SEES PTET ET Tete ey ey ye ey Te BREE HEHE EEE] PTT tt tT TT tT tT tt EE EE TE ET TT Pitt tty tit ty te tt Ee et BRR ER. eee Pri tet ate Pete te bt bb Fia. 21. at low currents the voltage rises again, due to the arc not filling the entire tube. Such a volt-ampere characteristic is given in Fig. 21. .

25. Herefrom then follows, that the voltage gradient in the mercury arc, for a tube diameter of 2 cm., is about 34 volts per centimeter or about one-twentieth of what it is in the Geissler tube, and the specific resistance of the stream, at 4 amp., is

40 ELECTRIC CIRCUITS about 0.2 ohms per cm.?, or of the magnitude of one one- thousandth of what it is in the Geissler tube. At higher currents, the mercury arc in a vacuum gives a rising volt-ampere characteristic. Nevertheless it is not stable on constant-potential supply, as the rising characteristic applies only to stationary conditions; the instantaneous characteristic is drop- ; ping. ‘That is, if the current is suddenly increased, the voltage drops, regardless of the current value, and then gradually, with the increasing temperature and vapor pressure, increases again, to the permanent value, a lower value or a higher value, which- ever may be given by the permanent volt-ampere characteristic. : In an are at atmospheric pressure, as the magnetite arc, the voltage gradient depends on the current, by equation (1), and at 4 amp. is about 15 to 18 volts per centimeter. The specific re- sistance of the arc stream is of the magnitude of 1 ohm per cm.®, and less with larger current arcs, thus of the same magnitude as | in vacuum arcs. Electronic Conduction 26. Conduction occurs at moderate voltages between terminals | in a partial vacuum as well as in a perfect vacuum, if the terminals are incandescent. If only one terminal is incandescent, the con- duction is unidirectional, that is, can occur only in that direction, ~ which makes the incandescent terminal the cathode, or negative. Such a vacuum tube then rectifies an alternating voltage and may be used as rectifier. Ifa perfect vacuum exists in the conducting | space between the electrodes of such a hot cathode tube, the con- | duction is considered as true electronic conduction. The voltage consumed by the tube is depending on the high temperature of the cathode, and is of the magnitude of arc voltages, hence very much lower than in the Geissler tube, and the current of the mag- | nitude of arc currents, hence much higher than in the Geissler tube. | 27. The complete volt-ampere characteristic of gas and vapor conduction thus would give a curve of the shape in Fig. 22. It | consists of three branches separated by ranges of instability or | discontinuity. The branch a, at very low current, electronic con- duction; the branch b, discontinuous or Geissler tube conduction; | and the branch c¢, arc conduction. The change from a to b oc- | curs suddenly and abruptly, accompanied by a big rise of current, | as soon as the disruptive voltage is reached. The change b to c | |

ELECTRIC CONDUCTION ‘ 41 occurs suddenly and abruptly, by the formation of a cathode spot, anywhere in a wide range of current, and is accompanied by a sudden drop of voltage. To show the entire range, as abscisse are used ~/7 and as ordinates +/e. . 1-4} ete _] APPROXIMATE VOLT AMPERE eon CER" eiesenee a ITE IN | tT TT EE te | lel] | Ne TE TT EP EP EE LH ET TaN EE EE PTT | TE eT eT a HTT TE EE TEP EEE ET EE PHT TTT aE TE TE TT TT Lotte? uti? Pet Tt ied PT tT Tt ti PE ET TS SERRE REEREe Pt tT tT tT det PE EE ET PTT TLE EE EEE ET TT pt tad | AT Tt | Pt} tT tT tT | Peete Pt Tt Ete ey ty ty pe Salle fellll | biel | | dT dda | ’ Waa, 22. Review 28. The various classes of conduction: metallic conduction, electrolytic conduction, pyroelectric conduction, insulation, gas - vapor and electronic conduction, are only characteristic types, but numerous intermediaries exist, and transitions from one type to another by change of electrical conditions, of temperature, etc. As regards to the magnitude of the specific resistance or resist- ivity, the different types of conductors are characterized about as follows:

42 ELECTRIC CIRCUITS The resistivity of metallic conductors is measured in microhm- centimeters. | The resistivity of electrolytic conductors is measured in ohm- | centimeters. Theresistivity of insulators is measured in megohm-centimeters and millions of megohm-centimeters. The resistivity of typical pyroelectric conductors is of the mag- : nitude of that of electrolytes, ohm-centimeters, but extends from this down toward the resistivities of metallic conductors, and up toward that of insulators. The resistivity of gas and vapor conduction is of the magnitude of electrolytic conduction: arc.conduction of the magnitude of lower resistance electrolytes, Geissler tube conduction and corona. conduction of the magnitude of higher-resistance electrolytes. Electronic conduction at atmospheric temperature is of the _ magnitude of that of insulators; with incandescent terminals, it reaches the magnitude of electrolytic conduction. _ While the resistivities of pyroelectric conductors extend over the entire range, from those of metals to those of insulators, ‘ typical are those pyroelectric conductors having a resistivity of electrolytic conductors. In those with lower resistivity, the drop of the volt-ampere characteristic decreases and the insta- bility characteristic becomes less pronounced; in those of higher -| resistivity, the negative slope becomes steeper, the instability in- | , creases, and streak conduction or finally disruptive conduction . appears. The streak conduction, described on the pyroelectric conductor, probably is the same phenomenon as the disruptive conduction or breakdown of insulators. Just as streak conduc- | tion appears most under sudden application of voltage, but less under gradual voltage rise and thus gradual heating, so insulators of high disruptive strength, when of low resistivity by absorbed moisture, etc., may stand indefinitely voltages applied intermit- tently—so as to allow time for temperature equalization—while quickly breaking down under very much lower sustained voltage. | |

CHAPTER III MAGNETISM Reluctivity

29. Considering magnetism as the phenomena of a “magnetic circuit,” the foremost differences between the characteristics of the magnetic circuit and the electric circuit are:

(a) The maintenance of an electric circuit requires the ex- penditure of energy, while the maintenance of a magnetic circuit does not require the expenditure of energy, though the starting of a magnetic circuit requires energy. A magnetic circuit, there- fore, can remain “remanent” or “permanent.”

(b) All materials are fairly good carriers of magnetic flux, . and the range of magnetic permeabilities is, therefore, narrow, from 1 to a few thousands, while the range of electric conductivi- ties covers a range of 1 to 10%. The magnetic circuit thus is analogous to an uninsulated electric circuit immersed in a fairly good conductor, as salt water: the current or flux can not be carried to any distance, or constrained in a “conductor,” but divides, “leaks” or “strays.”

(c) In the electric circuit, current and e.m.f. are proportional, in most cases; that is, the resistance is constant, and the circuit therefore can be calculated theoretically. In the magnetic circuit, in the materials of high permeability, which are the most important carriers of the magnetic flux, the relation between flux, m.m.f. and energy is merely empirical, the “reluctance” or mag- netic resistance is not constant, but varies with the flux density, the previous history, etc. In the absence of rational laws, most of the magnetic calculations thus have to be made by taking numerical values from curves or tables.

The only rational law of magnetic relation, which has not been disproven, is Fréhlich’s (1882):

“The premeability is proportional to the magnetizability”

n= a(S — B) (1) where B is the magnetic flux density, S the saturation density, 43

44 ELECTRIC CIRCUITS and S — B therefore the magnetizability, that is, the still avail- able increase of flux density, over that existing. . From (1) follows, by substituting, B B= (2) and rearranging, H B= a+ocH (3) where

c= ‘ = saturation coefficient, that is, the reciprocal of the

saturation value, S, of flux density, B, and -~t_¢% as’ for B = O, equation (1) gives . : 1 1 wo = aS = —; enn (4) that is, a is the reciprocal of the magnetic permeability at zero flux density.

A very convenient form of this law has been found by Kennelly (1893) by introducing the reciprocal of the permeability, as reluctivity p,

-1_H

P= = B’ in the form, which can be derived from (3) by transposition. p=a+oH (5) , As a dominates the reluctivity at lower magnetizing forces, and thereby the initial rate of rise of the magnetization curve, which is characteristic of the “magnetic hardness” of the material,

it is called the coefficient of magnetic hardness.

30. When investigating flux densities, B, at very high field intensities, H, it was found that B does not reach a finite satura tion value, but increases indefinitely; that, however, reaches a finite saturation value S, which with iron usually is not , far from 20 kilolines per cm.*, and that therefore Fréhlich’s and Kennelly’s laws apply not to B, but to Bo. The latter, then,

MAGNETISM 45

is usually called the metallic magnetic density or ferromagnetic

density.

Bo may be considered as the magnetic flux carried by the mole-

cules of the iron or other magnetic material, in addition to the CULL ERREEEELEEE] Case eeeE ec fiat ttt tt tt | | ed, pti tt | lg err |

A ere ptf tp Pieter it tt tT tT tt,

Vr ti tt tit tT ey er : me | (CCC ee Ho |} el df Titi tit t tt tt tt

fy iA TT EEE EE TE ET

Hi Yi tt ET tet tt tt et, ptAtT TT tT ttt | ir { ,

Piet} | PE tt ea PE TT tt | feet eee SESRnE576>-2an0 SERRE a 22eees 4 NERV a> Za0eeReee | Pecle= 41 ty teat | i tt | ree ft Za eeerreerers

Fig. 23.

space flux, H, or flux carried by space independent of the material .

in space.

The best evidence seems to corroborate, that with the excep-

tion of very low field intensities (where the customary magneti-

zation curve usually has an inward bend, which will be discussed

later) in perfectly pure magnetic materials, iron, nickel, cobalt,

46 ELECTRIC CIRCUITS etc., the linear law of reluctivity (5) and (3) is rigidly obeyed by the metallic induction Bo. In the more or less impure commercial materials, however, the , : p — Hrelation, while a straight line, often has one, and occasion- ally two points, where its slope, and thus the values of a and change.

Fig. 23 shows an average magnetization curve, of good standard iron, with field intensity, H, as abscisse, and magnetic induction, B, as ordinates. The total induction is shown jn drawn lines, the : metallic induction in dotted lines. The ordinates are given in . kilolines per cm.?, the abscisse in units for B,, in tens for B2, and in hundreds for B3.

The reluctivity curves, for the three scales of abscisse, are plotted as 1, p2, os, in tenths of milli-units, in milli-units and in

; tens of milli-units.

Below H = 3, pis not a straight line, but curved, due to the in- ward bend of the magnetization curve, B, in this range. The straight-line law is reached at the point c;, at H = 3, and the re-

_ luetivity is then expressed by the linear law pi = 0.102 + 0.059 H (7) for 3<H < 18, giving an apparent saturation value, ‘ Si = 16,950. At H = 18, a bend occurs in the reluctivity line, marked by point ce, and above this point the reluctivity follows the equation p2 = 0.18 + 0.0548 H (8) . for 18 <H < 80, giving an apparent saturation value S2 = 18,250.

At H = 80, another bend occurs in the reluctivity line, marked by point cz, and above this point, up to saturation, the reluctivity follows the equation

ps = 0.70 + 0.0477 H (9) for H> 80 giving the true saturation value, S = 20,960.

MAGNETISM 47 Point c; is frequently absent. ‘ Fig. 24 gives once more the magnetization curve (metallic in- duction) as B, and gives as dotted curves B,, B; and B; the mag- netization curves calculated from the three linear reluctivity equa- tions (7), (8), (9). As seen, neither of the equations represents | bh bh bw hh ® & wo sto wo wo we |B ; Pitt ttt yet tT dy HERRERO : COCO eee CCC ee TEE} : LAL ptt tT A | Yel | | | pt ty fie | /L | [abet | tt, it FV | AR eT Tt tT | HA TA et HiiTA Vi | tt tt ed PP YUL TT TT | eet | PTAA LTT | | cert tt tl HE iA | | tet tt TT TET Ut, evi | | leet | tt tT tT tL I LY P| ett tt ET Et TATA LETT TET TTT Tt, WALT PtP tt te tt tt, WY TTT tt ety tt te te } , 3 _ B 9 10 p18 14 4 Fia. 24,

B even approximately over the entire range, but each represents

it very accurately within its range. The first, equation (7), prob-

ably covers practically the entire industrially important range.

87. As these critical points cz and cs do not seem to exist in per- fectly pure materials, and as the change of direction of the re-

48 ELECTRIC CIRCUITS luctivity line is in general the greater, the more impure the mate- rial, the cause seems to be lack of homogeneity of the material; that is, the presence, either on the surface as scale, or in the body, . as inglomerate, of materials of different magnetic characteristics: magnetite, cementite, silicide. Such materials have a much greater hardness, that is, higher value of a, and thereby would give the observed effect. At low field intensities, H, the harder material carries practically no flux, and all the flux is carried by the soft material. The flux density therefore rises rapidly, giving low a, but tends toward an apparent low saturation value, as , . the flux-carrying material fills only part of the space. At higher field intensities, the harder material begins to carry flux, and while in the softer material the flux increases less, the increase of flux in the harder material gives a greater increase of total flux density and a greater saturation value, but also a greater hard- ness, as the resultant of both materials. Thus, if the magnetic material is a conglomerate of fraction p of soft material of reluctivity p: (ferrite) and g = 1 — p of hard material of reluctivity, p, (cementite, silicide, magnetite), pr = a, toi
pe = ae + oof (10) | at low values of H, the part p of the section carries flux by p,, the . part q carries flux by p:, but as ps is very high compared with 1, | the latter flux is negligible, and it is re A Ma 11 ie + > | (11) | At high values of H, the flux goes through both materials, more or less in series, and it thusis __ , | p” = poi + gor = (par + gaz) + (por + go2)H (12) if we assume the same saturation value, o, for both materials, and | neglect a: compared with az, it is pe” =qa,+oH (13) Substituting, as instance, (7) and (9) into (11) and (13) | respectively, gives “* — 0,102, i p . Z = 0.059, ° |

MAGNETISM 49 qas = 0.70, . o = 0.0477, hence, . p = 0.80: p: = 0.082 + 0.0477 H, g= 0.20: pr= 3.5 + 0.0477 H.

However, the saturation coefficients, ¢, of the two materials probably are usually not equal.

The deviation of the reluctivity equation from a straight line, by the change of slope at the critical points, czand cs, thus probably is only apparent, and is the outward appearance of a change of the flux carrier in an unhomogeneous material, that is, the result of a second and magnetically harder material beginning to carry flux.

Such bends in the reluctivity line have been artificially produced by Mr. John D. Ball in combining by superposition two different materials, which separately gave straight-line, p, curves, while combined they gave a curve showing the characteristic bend.

Very impure materials, like cast iron, may give throughout a curved reluctivity line.

32. For very low values of field intensity, H < 3, however, the straight-line law of reluctivity apparently fails, and the mag- netization curve in Fig. 23 has an inward bend, which gives rise of p with decreasing H.

This curve is taken by ballistic galvanometer, by the step-by- step method, that is, H is increased in successive steps, and the increase of B observed by the throw of the galvanometer needle.

; It thus is a “rising magnetization curve.”

The first part of this curve is in Fig. 25 reproduced, as B,, in twice the abscisse and half the ordinates, so as to give it an average slope of 45°, as with this slope curve shapes such as the inward bend of Bi below H = 2, are best shown (“Engineering Mathematics,” p. 286).

Suppose now, at some point, By) = 13.15, we stop the increase of H, and decrease again, down to 0. We do not return on the same magnetization curve, B:, but on another curve, B’,, the “decreasing magnetic characteristic,” and at H = 0, we are not back to B = 0, but a residual or remanent flux is left, in Fig. 25: . R =7.A.

Where the magnetic circuit contains an air-gap, as the field circuits of electrical machinery, the decreasing magnetic charac- teristic, B’:, is very much nearer to the increasing one, B;, than in

4

on 7 , | 50 ELECTRIC CIRCUITS the closed magnetic circuit, Fig. 25, and practically coincides for higher values of H. . There appears no theoretical reason why the rising character- | istic, B,, should be selected as the representative magnetization curve, and not the decreasing characteristic, B’:, except the inci- . dent, that B; passes through zero. In many engineering applica- _ tions, for instance, the calculation of the regulation of a generator, that is, the decrease of voltage under increase of load, it is ob- viously the decreasing characteristic, B’,, which is determining. | Suppose we continue B’; into negative values of H, to the point | Ai, at H = —1.5, B = —4, and then again reverse, we get a ris- ; | ing magnetization curve, B’, which passes H = 0 at a negative remanent magnetism. Suppose we stop at point A:, at H = | | —1.12, B = —1.0: therising magnetization curve B’” then passes H = 0 at a positive remanent magnetism. There must thus be | a point, Ao, between A, and As, such that the rising magnetiza- tion curve, B’, starting from Ao, passes through the zero point | | H= 0, B = 0, and thereby runs into the curve, B:. The rising magnetization curve, or standard magnetic charac- | teristic determined by the step-by-step method, B,, thus is noth- | | ing but the rising branch of an unsymmetrical hysteresis cycle, traversed between such limits +B) and —Ao, that the rising . branch of the hysteresis cycle passes through the zero point. 33. The characteristic shape of a hysteresis cycle is that it is a | loop, pointed at either end and thereby having an inflexion point . about the middle of either branch. In the unsymmetrical loop +B, —Apo of Fig. 25, the zero point is fairly close to one extreme, Ao, and the inflexion point, characteristic of the hysteresis loop, thus lies between 0 and Bo, that is, on that part of the rising branch, which is used as the “magnetic characteristic,” B,, and thereby produces the inward bend in the magnetization curve at low fields, which has always been so puzzling. : If, however, we would stop the increase of H at B’y, we would get the decreasing magnetization curve, B’’;, and still other curves for other starting points of the decreasing characteristic. Thus, the relation between magnetic flux density, B, and mag- metic field intensity, H, is not definite, but any point between the various rising and decreasing characteristics B’’, Bi, B’”’, BY, B’,, and for some distance outside thereof, is a possible B-H relation. B, has the characteristic that it passes through the zero point. But it is not the only characteristic which does this:

. MAGNETISM 51 if we traverse the hysteresis cycle between the unsymmetrical : limits +Ao and —Bo, as shown in Fig. 26, its decreasing branch

. __ By passes through the zero point, that is, has the same feature as B;. It is interesting to note, that Bs does not show an inward bend, and the reluctivity curve of Bs, given as ps in Fig. 28, apparently is a straight line. .

Magnetic characteristics are frequently determined by the method of reversals, by reversing the field intensity, H, and ob- serving the voltage induced thereby by ballistic galvanometer, Te Tet Tt tet tl el PEE tE EET ETE TTT ETE Tt TTT TT ht TE | PT ce] . pete || pad tit tH COCR CCC CCE PT TT Tye verte tt itt tt Se eean J | eld | ft | feet tk ta | PV it et pa ee EL eH Shee aA yf Ree ee AEE 4} fy Pt tt | ler ee tt tet tT tt Pete TTT tte tt ET Ta a

Figs. 25 anp 26. or using an alternating current for field excitation, and observing the induced alternating voltage, preferably by oscillograph to eliminate wave-shape error.

This “alternating magnetic characteristic” is the one which is of consequence in the design of alternating-current apparatus. It differs from the “rising magnetic characteristic,” B, by giving lower values of B, for the same H, materially so at low values of H. It shows the inward bend at low fields still more pronounced than B, does. It is shown as curve B; in Fig. 27, and its reluctivity

52 ELECTRIC CIRCUITS . line given as p,in Fig. 28. At higher values of H: from H = 3 up- ward, B, and B; both coincide with the curve, Bo, representing the straight-line reluctivity law. HE te Pt tT TTT Tt er peer Ty eer HE oe eee P| ttt | Ve Lar HEH Apa IF SERRE (oft e o ttt Pi TTT TAR AAA PT TT | Vie ee BRRESG SARA REE PL i tT VIA T yt Tt E E op Agee EE | VY pA] Vit} tt tt tt tt EVAR eee AGRE REED COREE | ; pit | pt er PE PE TE ET Tt tT ett tt tt tt tt PE EE EL Fria. 27. REECE EEE REE EES NER RRR VIAL ITT TTT Tt tT ya te tt ee Ave tT tT Pt TT Et TT ert hee} CNY eee PA | fel ert tT TT TT TE EE TT BSS =e eee Lert | TTT TTT TT wee TT PTE ETE TT ET eT Fig. 28. . The alternating characteristic, B, is not a branch of any hystere- sis cycle. It is reproducible and independent of the previous — history of the magnetic circuit, except perhaps at extremely low values of H, and in view of its engineering importance as repre-

MAGNETISM 53 senting the conditions in the alternating magnetic field, it would appear the most representative magnetic characteristic, and is commonly used as such.

It has, -however, the disadvantage that it represents an un- stable condition.

Thus in Fig. 27, an alternating field H = 1 gives an alternating flux density, B, = 2.6. If, however, this field strength H = 1 is left on the magnetic circuit, the flux does-not remain at B, = 2.6, but gradually creeps up to higher values, especially in the presence of mechanical vibrations or slight pulsations of the magnetizing current. To a lesser extent, the same occurs with the values of curve, B,, to a greater extent with B;. At very low densities, this creepage due to instability of the B-H relation may amount to hundreds of per cent. and continue to an appreciable extent for minutes, and with magnetically hard materials for many years. Thus steel structures in the terrestrial magnetic field show immediately after erection only a small part of the magnetization, which they finally assume, after many years.

Thus the alternating characteristic, B:, however important in electrical engineering, can, due to its instability, not be considered as representing the true physical relation between B and H any ~ 6 more than the branches of hysteresis cycles B; and B:.

34. Correctly, the relation between B and H thus can not be expressed by a curve, but by an area.

Supposea hysteresis cycle is performed between infinite values of field intensity: H = + «, that is, practically, between very high values such as are given for instance by the isthmus method of magnetic testing (where values of H of over 40,000 have been reached. Very much lower values probably give practically the same curve). This gives a magnetic cycle shown in Fig. 5 as B’, B”. Any point, H, B, within the area of this loop between B’ and B” of Fig. 27 then represents a possible condition of the magnetic circuit, and can be reached by starting from any other point, Ho, Bo, such as the zero point, by gradual change of H.

Thus, for instance, from point Po, the points P;, Ps, Ps, etc., are reached on the curves shown in the dotted lines in Fig. 27.

• As seen from Fig. 27, a given value of field intensity, such as H = 1, may give any value of flux density between B = —4.6 and B = +13.6, and a given value of flux density, such as B = 10, may result from any value of field intensity, between H = —0.25toH = +34

54 ELECTRIC CIRCUITS The different values of B, corresponding to the same value of H in the magnetic area, Fig. 27, are not equally stable, but the val- ues near the limits B’ and B” are very unstable, and become more stable toward the interior of the area. Thus, the relation of point P,, Fig. 27: H = 2, B = 13, would rapidly change, by the flux density decreasing, to Po, slower to P; and then still slower, while from point Ps the flux density would gradually creep up. . If thus follows, that somewhere between the extremes B’ and | B’, which are most unstable, there must be a value of B, which is stable, that is, represents the stationary and permanent relation between B and H, and toward this stable value, Bo, all other val- . ues would gradually approach. This, then, would give the true magnetic characteristic: the stable physical relation between B . and H. | At higher field intensities, beyond the first critical point, c:, this stable condition is rapidly reached, and therefore is given by all the methods of determining magnetic characteristics. Hence, the curves B,, Bs, Bo coincide there, and the linear law of re- luctivity applies. Below c:, however, the range of possible, B, | “values is so large, and the final approach to the stable value so . slow, as to make it difficult of determination. 35. For H = 0, the magnetic range is from —Ry = —11.2 to +o = 11.2; the permanent value is zero. The method of reach- ing the permanent value, whatever may be the remanent mag- | netism, is well known; it is by “demagnetizing’’ that is, placing | the material into a powerful alternating field, a demagnetizing coil, and gradually reducing this field to zero. .That is, describ- } ing a large number of cycles with gradually decreasing amplitude. The same can be applied to any other point of the magnetiza- | tion curve. Thus for H = 1, to reach permanent condition, an alternating m.m-f. is superimposed upon H = 1, and gradually | , decreased to zero, and during these successive cycles of decreas- ing amplitude, with H = 1, as mean value, the flux density gradu- | ally approaches its permanent or stable value. (The only re- quirement is, that the initial alternating field must be higher than | any unidirectional field to which the magnetic circuit had been | exposed.) | This seems to be the value given by curve Bo, that is, by the | straight-line law of reluctivity. In other words, it is probable | that:

. Frohlich’s equation, or Kennelly’s linear law of reluctivity | | | I