Theory and Calculation of Transient Electric Phenomena and Oscillations — part 13 of 20

The free oscillation of a circuit which is closed upon itself isa ° full-wave oscillation, containing a fundamental wave of frequency 1 => ——) 60 . hie ©) and all the higher harmonics thereof, the even ones as well as the odd ones, f= nf,. (61) Substituting in (3), ¢" = ¢,' + je,” and C, = Cc,’ + je,” gives I = (c, + je,”’) cos Bl + (c,/’ — jc,’)sin Al and L (62) : B= VF {ley + icy”) cos Bl+ (e\” — fey) sin fi. Substituting the analytic expression, ce,’ + je,” = c,/ cos 2 xft +c,” sin 2 xft, etc., ‘ also 2 aft= 8, 63 2zft = nd, (63) and Al = ma = nt, oe 2 C where t= lL l, 0 that is, the length of the circuit, / = 1,, is represented by the angle t = 27, or a complete cycle, this gives

338 TRANSIENT PHENOMENA. I = (c, cos né + c,” sin n@) cos nt

• (¢,” cos n#—c,! sin n8) sin nt and . E= C { (c,’ cos n@ + ¢,’ sin n8) cos nO
• (c,” cos né— ¢,’ sin nf) sin nz}, or writing ce,’ = acosnr . ec,” =asinny c,’ = beos ny c,/’ = bsin nz gives 4 =acosn (6 — yr) cosnt — bsinn (6 — x) sinnt and L (66)

e= vz {bcos n (6 — x) cos nr — a sinn (6 — )sinnz}.

Thus in its most general form the full-wave oscillation gives the equations

t= > {a, cosn (9 — yn) cosnt — bz sinn (0 —7n) sin nt} 1 Le . . e=V- >" {b, cos n (8—yn) cos nt — a,sinn (0 — yn) sin nt}, C4 . (67) where 6 = 2xf,t, 1 f= LWLC’ (68) 27 T= 7” l, . and Qn, yn and bn, Yn are groups of four integration constants.

38. With a short circuit at the end of a transmission line, the | drop of potential along the line varies fairly gradually and | uniformly, and the instantaneous rupture of a short circuit — | as by a short-circuiting arc blowing itself out explosively — |

NATURAL PERIOD OF TRANSMISSION LINE 839 causes an oscillation in which the lower frequencies predominate, that is, a low-frequency high-power surge. A spark discharge : from the line, a sudden high voltage charge entering the line locally, as directly by a lightning stroke, or indirectly by induc- tion during a lightning discharge elsewhere, gives a distribution of potential which momentarily is very non-uniform, changes very abruptly along the line, and thus gives rise mainly to very high harmonics, but as a rule does not contain to any appre- ciable extent the lower frequencies; that is, it causes a high- frequency oscillation, more or less local in extent, and while of high voltage, of rather limited power, and therefore less destruc- tive than a low-frequency surge.

At the frequencies of the high-frequency oscillation neither capacity nor inductance of the transmission line is perfectly constant: the inductance varies with the frequency, by the

_ increasing screening effect or unequal current distribution in the conductor; the capacity increases by brush discharge over the insulator surface, by the increase of the effective conductor diameter due to corona effect, etc. The frequencies of the very high harmonics are therefore not definite but to some extent variable, and since they are close to each other they overlap; that is, at very high frequencies the transmission line has no definite frequency of oscillation, but can oscillate with any frequency.

A long-distance transmission line has a definite natural period of oscillation, of a relatively low fundamental frequency and its overtones, but can also oscillate with any frequency whatever, provided that this frequency is very high.

This is analogous to waves formed in a body of water of regular shape: large standing waves have a definite wave length, depending upon the dimensions of the body of water, but very short waves, ripples in the water, can have any wave length, and do not depend on the size of the body of water.

A further investigation of oscillations in conductors with distributed capacity, inductance, and resistance requires, how- ever, the consideration of the resistance, and so leads to the investigation of phenomena transient in space as well as in time, which are discussed in Section IV.

39. In the equations discussed in the preceding, of the free oscillations of a circuit containing uniformly distributed resist-

340 TRANSIENT PHENOMENA ance, inductance, capacity, and conductance, the energy losses in the circuit have been neglected, and voltage and current therefore appear alternating instead of oscillating. That is, these equations represent only the initial or maximum values of the phenomenon, but to represent it completely an exponential function of time enters as factor, which, as will be seen in Section IV, is of the form .

eM, (69) where u = ; G + *) may be called the “time constant”’ of the circuit.

While quarter-wave oscillations occasionally occur, and are of serious importance, the occurrence of half-wave oscillations and especially of full-wave oscillations of the character discussed before, that is, of a uniform circuit, is less frequent.

When in a circuit, as a transmission line, a disturbance or oscillation occurs while this circuit is connected to other cir-

_ cuits — as the generating system and the receiving apparatus — as is usually the case, the disturbance generally penetrates into the circuits connected to the circuit in which the disturbance originated, that is, the entire system oscillates, and this oscilla- tion usually is a full-wave oscillation; that is, the oscillation of a circuit closed upon itself; occasionally a half-wave oscillation. For instance, if in a transmission system comprising generators, step-up transformers, high-potential lines, step-down trans-_ . formers, and load, a short circuit occurs in the line, the circuit comprising the load, the step-down transformers, and the lines from the step-down transformers to the short circuit is left closed upon itself without power supply, and its stored energy is, therefore, dissipated as a full-wave oscillation. Or, if in this system an excessive load, as the dropping out of step of a syn- chronous converter, causes the circuit to open at the generating station, the dissipation of the stored energy — in this case that | of the excessive current in the system — occurs as a full-wave oscillation, if the line cuts off from the ‘generating station on the low-tension side of the step-up transformers, and the oscillating circuit comprises the high-tension coils of the step-up trans- | ‘ formers, the transmission line, step-down transformers, and load. If the line disconnects from the generating system on the high-

| NATURAL PERIOD OF TRANSMISSION LINE 341 potential side of the step-up transformers, the oscillation is a half-wave oscillation, with the two ends of the oscillating circuit open.

Such oscillating circuits, however,— representing the most frequent and most important case of high-potential disturbances in transmission systems,—cannot be represented by the preced-

. ing equations since they are not circuits of uniformly distributed constants but complex circuits comprising several sections of

| different constants, and therefore of different ratios of energy

| consumption and energy storage, zand 3. During the free oscillation of such circuits an energy transfer takes place be- tween the different sections of the circuit, and energy flows from those sections in which the energy consumption is small com- —

pared with the energy storage, as transformer coils and highly inductive loads, to those sections in which the energy consump- . tion is large compared with the energy storage, as the more non-inductive parts of the system. This introduces into the equa- tions exponential functions of the distance as well as the time, and requires a study of the phenomenon as one transient in.

distance as well 4s in time. The investigation of the oscillation of a complex circuit, comprising sections of different constants, is treated in Section IV.

|

|

i

| ;

| ‘ CHAPTER IV. . DISTRIBUTED CAPACITY OF HIGH-POTENTIAL TRANSFORWYERS.

40. In the high-potential coils of transformers designed for very high voltages phenomena resulting from distributed capacity occur. ;

. In transformers for very high voltages — 100,000 volts and more, or even considerably less in small transformers — the high- potential coil contains a large number of turns, a great length of conductor, and therefore its electrostatic capacity is appreciable, and such a coil thus represents a circuit of distributed resistance, inductance, and capacity somewhat similar to a transmission line.

The same applies to reactive coils, etc., wound for very high voltages, and even in smaller reactive coils at very high frequency.

This capacity effect is more marked in smaller transformers, where the size of the iron core and therewith the voltage per turn is less, and therefore the number of turns greater than in very large transformers, and at the same time the exciting cur- rent and the full-load current are less; that is, the charging current of the conductor more comparable with the load current of the transformer or reactive coil.

However, even in large transformers and at moderately high voltages, capacity effects occur in transformers, if the frequency is sufficiently high, as is the case with the currents produced in overhead lines by lightning discharges, or by arcing grounds resulting from spark discharges between conductor and ground, or in starting or disconnecting the transformer. With such frequencies, of many thousand cycles, the internal capacity of the transformer becomes very marked in its effect on the dis- tribution of voltage and current, and may produce dangerous

; high-voltage points in the transformer.

The distributed capacity of the transformer, however, is differ-

ent from that of a transmission line. 342

HIGH-POTENTIAL TRANSFORMERS 843

In a transmission line the distributed capacity is shunted capacity, that is, can be represented diagrammatically by con- densers shunted across the circuit from line to line, or, what amounts to the same thing, from line to ground and from ground to return line, as shown diagrammatically in Fig. 88.

H HTT HTT ttt ito 7 Fig. 88. Distributed capacity of a transmission line.

The high-potential coil of the transformer also contains shunted capacity, or capacity from the conductor to ground, and so each coil element consumes a charging current proportional to its potential difference against ground. Assuming the circuit as insu- lated, and the middle of the transformer coil at ground potential, the charge consumed by unit length of the coil increases from zero at the center to a maximum at the ends. If one terminal of the circuit is grounded, the charge consumed by the coil ; increases from zero at the grounded terminal to a maximum at the ungrounded terminal.

In addition thereto, however, the transformer coil also con- tains a capacity between successive turns and between successive layers. Starting from one point of the conductor, after a certain

Cs Cs —- —- B & LI w CEUMOUOMOMOUoWuua O; LE MU EEHWHHHHEHHMH@007 Fig. 89. Distributed capacity of a high-potential transformer coil. length, the length of one turn, the conductor reapproaches the first point in the next adjacent turn. It again approaches the

| 344 TRANSIENT PHENOMENA first point at a different and greater distance in the next adjacent layer.

A transformer high-potential coil can be represented dia- grammatically as a conductor, Fig. 89. C, represents the capacity against ground, C, represents the capacity between adjacent turns, and C, the capacity between adjacent layers of the coil. |

The capacities C, and C, are not uniformly distributed but more or less irregularly, depending upon the number and arrange- ment of the transformer coils and the number and arrangement of turns in the coil. As approximation, however, the capacities . C, and C, can be assumed as uniformly distributed capacity between successive conductor elements. If / = length of con- ductor, they may be assumed as a capacity between / and / + dl, or as a capacity across the conductor element dl.

This approximation is permissible in investigating the general effect of the distributed capacity, but omits the effect of the irregular distribution of C, and C,, which leads to local oscilla- tions of higher frequencies, extending over sections of the circuit, and of lesser power. ;

41. Let then, in the high-potential coil of a high-voltage trans- former, e = the e.m.f. generated per unit length of conductor, as, for instance, per turn; Z = r — jx = the impedance per unit length; Y = g — jb = the capacity admittance against ground per unit length of conductor, and Y’ = pY= the capacity admittance, per unit length of conductor, between conductor elements distant from each other by unit length, as admittance between successive turns. Y’ is assumed to represent the total effective admittance representing the capacity between successive turns, successive layers, and successive coils, as represented by the condensers C’, and C, in Fig. 89.

The charging current of a conductor element di, due to the admittance Y’, is made up of the charging currents against the next following and that against the preceding conductor element.

Let J, = length of conductor; / = distance along conductor ; E = potential at point /, or conductor element di, and J = cur- rent in conductor element di; then

dE ta: ; dE = i dl = the potential difference between successive conductor elements or turns.

HIGH-POTENTIAL TRANSFORM ERS 345 dE ;

Y’ a dl = the charging current between one conductor ele- ment and the next conductor element or turn.

— yy’ te”) di = the charging current between one con- ductor element and the preceding conductor element or turn, hence,

CE .

YF dl = the charging current of one conductor element due . to capacity between adjacent conductors or turns.

If now the distance / is counted from the point of the con- ductor, which is at ground potential, YEdl = the charging cur- rent of one conductor element against ground, and

CE PE a -{reertBla~ rere tila is the total current consumed by a conductor element.

However, the e.m.f. consumed by impedance equals the e.m.f. consumed per conductor element; thus

dE, = ZI di. This gives the two differential equations: dl PE a YYE + PFS @ dE and e-a= ZI. (2) Differentiating (2) and substituting in (1) gives : CE ; eB} : Po ZY jE + Pops? transposing, . . , eh -# dP 1” (3) | P- ay PE °F ae ~ — F, (4)

346 TRANSIENT PHENOMENA | . ; 1 where a=: (5) P~ ZY ’ If Ww is small compared with p, we have, approximately, a=" © P and E = Acosal + Bsinal, (7) and since, for 1 = 0, E = 0, if the distance / is counted from the point of zero potential, we have E = Bsinal, (8) and the current is given by equation (2) as [= L e- di ;

• Zee dl ’ (9) substituting (8) in (9) gives [= 7 fe -aBeosal |. (10) ° Z .

42. If now J, = the current at the transformer terminals, lL = 1,, we have; from (10), ZI, =e — aB cos al, e —ZI,. and B= acos al, , (11) substituting in (8) and (10), . sin al E=(e- 41 ) Geos al, 1 cos al (12) and I = Fhe 21) 2S — ef for I, = 0, or open circuit of the transformer, this gives RB =e al . a cos al, (13) and I= £ (wet - 1)

• Z\cos al, , |

HIGH-POTENTIAL TRANSFORMERS 847 The e.m.f., #, thus is a maximum at the terminals, . _ E, = 5 tan al,, | the current a maximum at the zero point of potential, / = 0, where ef 1 | l= seal, ~ 3) (14) | . | |

| CHAPTER V. DISTRIBUTED SERIES CAPACITY.

43. The capacity of a transmission line, cable, or high-poten- i tial transformer coil is shunted capacity, that is, capacity from conductor to ground, or from conductor to return conductor, or shunting across a section of the conductor, as from turn to turn or layer to layer of a transformer coil.

In some circuits, in addition to this shunted capacity, dis- | tributed series capacity also exists, that is, the circuit is broken . at frequent and regular intervals by gaps filled with a dielectric |

‘ or insulator, as air, and the two faces of the conductor ends thus | constitute a condenser in series with the circuit. Where the | elements of the circuit are short enough so as to be represented, approximately, as conductor differentials, the circuit constitutes a circuit with distributed series capacity.

An illustration of such a circuit is afforded by the so-called “multi-gap lightning arrester,’”’ as shown diagrammatically in Fig. 90, which consists of a large number of metal cylinders p, ¢ ..., With small spark gaps between the cylinders, connected between line L and ground G. This arrangement, Fig. 90, can be represented diagrammatically by Fig. 91. Each cylinder has | a capacity C, against ground, a capacity C against the adja- ! cent cylinder, a resistance 7,— usually very small,—and an inductance L.

If such a series of n equal spark gaps is connected across a constant supply voltage e,, each gap has a voltage e = “ . Tf, however, the supply voltage is alternating, the voltage does not divide uniformly between the gaps, but the potential difference . is the greater, that is, the potential gradient steeper the nearer the gap is to the line ZL, and this distribution of potential becomes | the more non-uniform the higher the frequency; that is, the . greater the charging current of the capacity of the cylinder | against ground. The charging currents against ground, of all

848

DISTRIBUTED SERIES CAPACITY 349 the cylinders from g to the ground G, Figs. 90 and 91, must pass the gap between the adjacent cylinders p and q; that is, the charging current of the condenser represented by two adjacent

7 Fig. 90. Multi-gap lightning arrester. cylinders p and q is the sum of all the charging currents from q to G; and as the potential difference between the two cylinders p and q is proportional to the charging current of the condenser L : rxe Py C,

Fig. 91. Equivalent circuit of a multi-gap lightning arrester. formed by these two cylinders, C’, this potential difference increases towards L, being, at each point proportional to the vector sum of all the charging currents, against ground, of all the cylinders between this point and ground.

The higher the frequency, the more non-uniform is the poten- tial gradient along the circuit and the lower is the total supply voltage required to bring the maximum potential gradient, near the line L, above the disruptive voltage, that is, to initiate the discharge. Thus such a multigap structure is discriminating regarding frequency ; that is, the discharge voltage with increas- .

350 TRANSIENT PHENOMENA

ing frequency, does’not remain constant, but decreases with increase of frequency, when the frequency becomes sufficiently high to give appreciable charging currents. Hence high fre- quency oscillations discharge over such a structure at lower voltage than machine frequencies. .

For a further discussion of the feature which makes such a multigap structure useful for lightning protection, see A. I. E. E. Transactions, 1906, pp. 431, 448, 1907, p. 425, etc. |

44, Such circuits with distributed series capacity are of great . interest in that it is probable that lightning flashes in the clouds are discharges in such circuits. From the distance traversed by lightning flashes in the clouds, their character, and the disruptive strength of air, it appears certain that no potential difference can exist in the clouds of such magnitude as to cause a disruptive discharge across a mile or more of space. It is probable that | as the result of condensation of moisture, and the lack of uni- . formity of such condensation, due to the gusty nature of air ! currents, a non-uniform distribution of potential is produced between the rain drops in the cloud; and when the potential gradient somewhere in space exceeds the disruptive value, an oscillatory discharge starts between the rain drops, and grad- | ually, in a number of successive discharges, traverses the cloud and equalizes the potential gradient. A study of circuits containing distributed series capacity thus leads to an under- standing of the phenomena occurring in the thunder cloud during . the lightning discharge.*

Only a general outline can be given in. the following.

45. Inacircuit containing distributed resistance, conductance, inductance, shunt, and series capacity, as the multigap lightning

arrester, Fig. 90, represented electrically as a circuit in Fig. 91, let r = the effective resistance per unit length of circuit, or per circuit element, that is, per arrester cylinder; g = the shunt . conductance per unit length, representing leakage, brush dis- charge, electrical radiation, etc.; L = the inductance per unit length of circuit: C = the series capacity per unit length of cir- cuit, or circuit element, that is, capacity between adjacent arrester cylinders, and C, = the shunt capacity per unit length of circuit, or circuit element, that is, capacity between arrester cylinder and

• See paper, ‘Lightning and Lightning Protection,’ N.E.L.A., 1907. Reprinted and enlarged in ‘ General Lectures on Electrical Engineering,” by Author. |

DISTRIBUTED SERIES CAPACITY 351 ground. If then f = the frequency of impressed -e.m.f., the , series impedance per unit length of circuit is Z=r-j(e—-2%,); (1) the shunt admittance per unit length of circuit is , Y= g- 7b, . (2) where xz =22fL, 1 t= 2 fC , (3) b = 22fC,; or the absolute values are 2= VP + eu | and (4) y ='VFFE If the distance along the circuit from line L towards ground . G is denoted by J, the potential difference between point / and ground by £, and the current at point J by /, the differential equations of the circuit are * ; dE oat ZI (5) and . dl , at YE. (6) Differentiating (5) and substituting (6) therein gives PE Equation (7) is integrated by : E=Ageo" + Ag", (8) where a=VYZ =a — jp, (9) a = V4{yz + gr — b (x — x} and (10) B= Vi{yz — gr +b (x — x,)}-

• Section III, Chapter II, paragraph 7.

352 TRANSIENT PHENOMENA Substituting (10) in (8) and eliminating the imaginary expo- nents by the substitution of trigonometric functions, E = Ag~@ (cos fl + j sin Bl) + Aye*™ (cos Bl — j sin fl). (11) 46. However, if n = the total length of circuit from line L to ground G, or total number of arrester cylinders between line and ground, for 1 = n, E=0, (12) and for / = 0, , E =e, = the impressed e.m.f. (13) Substituting (12) and (13) into (11) gives 0 = Aye~™ (cos Bn + j sin Bn) + A,s* (cos Bn — j sin fn) and . e, = A, + A,; hence, A. = ————__ 1" 1 = 67? (cos 2 Bn rt j sin 2 Bn)’ (14) A, =— A,e~** (cos 2 Bn + j sin 2 Bn), and the potential difference against ground is ib =e, e~ “(cos Al+j sin Bl) —e~*@"— [cos B (2 n—1) +] sin B(2 n—))] 1 — «?™ (cos 2 Bn + j sin 2 fn) (15) From equation (5), substituting (15) and (9), we have [ =- yz & e“(cos Bl+j sin Bl) +272 2"—” [cos 8 (2n—1) +jsin 2 (2n—D] 1 — <~**"(cos 2 Bn + j sin 2 Bn) , (16) Reduced to absolute terms this gives the potential difference against ground as . ' _ ge Fh 4 ge Fan) _ 2 -- 78" cog 2B (n — I) C= VT Fe ae cos dan YD

DISTRIBUTED SERIES CAPACITY 853 the current as yO fpn ee ett H—- Od DI OC DB (nT a v/s +e +2e¢ cos 2 8 (n — lt) += v: 1 +e~ 4" — 227? cos 2 Bn (18) and the potential gradient, or potential difference between adja- cent cylinders, is rie y fo te —O 42 6— cos 2 B (n—L) onae cay V Te 26 cos 2pm —* 19) For an infinite length of line, n = ©, that is, for a very large number of lightning arrester cylinders, where «~?™ is negligible, as in the case where the discharge passes from the line into the arrester without reaching the ground, equations (17), (18), (19) simplify to e= eg” , (20) | i=e / ; eet (21) | and d= en, VE 4; (22) that is, are simple exponential curves. Substituting (4) and (3) in (21) and (22) gives \ ; g 2 _ 4 Cy + (, 1) e’=eg tf ————"¥______ (93) C7§[1 — (2 xf)? CLF + (2 2fCr)’} ‘and t = 22fCe; (24) or, approximately, if r and g are negligible, we have C _ - ~al ~. roe C= ee Vo {1 — (22f)? CL} (25) and CC , = —al —____ 9? . t= 2afes “V 1 Qn CL (26) 47. Assume, as example, a lightning arrester having the fol- lowing constants: L = 2 x 10-* henry; C, = 10-" farads;

| 354 TRANSIENT PHENOMENA C =4 X'10-" farads;r = 1 ohm;g = 4 X 10-* mho;f = 10° = 100 million cycles per second; n = 300 cylinders, and e, = 30,000 volts; then from equation (3), z = 12.6 ohms, z, = 39.7 ohms, and, b = 62.8 x 10-* mhos; ~ from equation (1), Z =1 + 27.1j ohms; from equation (2), Y = (4 — 62.87) 10-* mho; from equation (4), z = 27.1 ohms and y = 62.9 x 10-* mho; from equation (10), a = 0.0021 and £ = 0.0412; from equation (17), e = 35,500 Ve 0? 4 0.08 etl — 0.568 cos (24.72 — 0.0824 1); from equation (18), dg = 54 Ve RTT 008 RTF 0.568 cos (24.72 — 0.0824 1), ; and from equation (19), ef = 2140 Ve? 4 0.08 eT 4 0.568 cos (24.72 — 0.0824 2).

Hence, at 1 = 0, e = 30,000 volts, « = 64.6 amperes, and é = 2560 volts; and at / = 300, e = 0, 7 = 57.5 amperes, and_ é = 2280 volts.

With voltages per gap varying from 2280 to 2560, 300 gaps would, by addition, give a total voltage of about 730,000, while the actual voltage is only about one-twenty-fourth thereof; that is, the sum of the voltages of many spark-gaps in series may be many times the resultant voltage, and a lightning flash may pass possibly for miles through clouds with a total potential of only a few hundred million. volts. In the above example the 300 cylinders include 7.86 complete wave-lengths of the discharge.

CHAPTER VI. ALTERNATING MAGNETIC FLUX DISTRIBUTION. 48. As carrier of magnetic flux iron is used, as far as possible, since it has the highest permeability or magnetic conductivity. If the magnetic flux is alternating or otherwise changiiig rapidly, an e.m.f. is generated by the change of magnetic flux in the iron, and to avoid energy losses and demagnetization by the currents . produced by these e.m.fs. the iron has to be subdivided in the direction in which the currents would exist, that is, at right angles to the lines of magnetic force. Hence, alternating magnetic fields and magnetic structures desired to respond very quickly to changes of m.m.f. are built of thin wires or thin iron sheets, that is, are laminated. | Since the generated e.m.fs. are proportional to the frequency of the alternating magnetism, the laminations must be finer the higher the frequency. To fully utilize the magnetic permeability of the iron, it there- fore has to be laminated so as to give, at the impressed frequency, practically uniform magnetic induction throughout its section, that is, negligible secondary currents. This, however, is no longer the case, even with the thinnest possible laminations, at extremely high frequencies, as oscillating currents, lightning discharges, etc., and under these conditions the magnetic flux distribution in the iron is not uniform, but the magnetic flux density, ®, decreases rapidly, and lags in phase, with increasing depth below the surface of the lamination, so that ultimately hardly any magnetic flux exists in the inside of the laminations, but practically only a surface layer carries magnetic flux. The apparent permeability of the iron thus decreases at very high . frequency, and this has led to the opinion that at very high fre- quencies iron cannot follow a magnetic cycle. There is, however, no evidence of such a “viscous hysteresis,’’ but it is probable that iron follows magnetically even at the highest frequencies, ‘ traversing practically the same hysteresis cycle irrespective of 8655

356 TRANSIENT PHENOMENA the frequency, if the true m.m.f., that is, the resultant of the

: impressed m.m.f. and the m.m.f. of the secondary currents in the iron, is considered. Since with increasing frequency, at constant impressed m.m.f., the resultant m.m.f. decreases, due _ to the increase of the demagnetizing secondary currents, this simulates the effect of a viscous hysteresis.

Frequently also, for mechanical reasons, iron sheets of greater thickness than would give uniform flux density have to be used in an alternating field.

Since rapidly varying magnetic fields usually are alternating, and the subdivision of the iron is usually by lamination, it will be sufficient to consider as illustration of the method the dis- tribution of alternating magnetic flux in iron laminations.

49. Let Fig. 92 represent the section of a lamination. The alternating magnetic flux is assumed to pass in a direction perpendicular to the plane of the paper.

Let » = the magnetic permeability, 4 = the electric conductivity, J = the distance of a layer dl from the center line of the lamination, and 21, = the total thickness of the lamination. If then J = the current density in the layer di, and E = the e.m.f. per unit length generated in ‘>| the zone di by the alternating magnetic flux, we have ! I = 26. (i) The magnetic flux density @, at the surface 1 =1, of the lamination corresponds to the Fig. oe Alor impressed or external m.m.f. The density 8 fluxdistritation in the zone dl corresponds to the impressed in solidiron, 4 ™.m.f. plus the sum of all the m.m.fs. in the zones outside of dl, or from J to J,. The current in the zone di is Idl = Ed (2) and produces the m.m.f. . H = 0.4 xAE di, (3) which in turn would produce the magnetic flux density d® = 0.4 rap dl; (4)

ALTERNATING MAGNETIC FLUX DISTRIBUTION 3857 that is, the magnetic flux density @ at the two sides of the zone dl differs by the magnetic flux density d@ (equation (4)) pro- duced by the m.m.f. in zone dl, and this gives the differential equation between @, E, and 1,

• = 0.4 rlpE. (5)

The e.m.f. generated at distance / from the center of the lamination is due to the magnetic flux in the space from / to J,. Thus the e.m.fs. at the two sides of the zone dl differ from each other by the e.m.f. generated by the magnetic flux @dl in this zone.

Considering now @, £, and I as complex quantities, the e.m-f. dE, that is, the difference between the e.m.fs. at the two sides of the zone di, is in quadrature ahead of @dl, and thus denoted by

' dE = — j2xf@ 10- di, (6) where f = the frequency of alternating magnetism.

This gives the second differential equation

a = — j 2nf@ 10-. (7) 60. Differentiating (5) in respect to /, and substituting (7) therein, gives a =— 0.8 jrfiu 10- @, (8) dt or, writing 2 = fa? = 0.4 fly 10-, (9) a = 0.4 Ap 10-8, (10) we have ; &B . GE 7 2126. - (11)

This differential equation is integrated by

@ = Ae-™; (12) this equation substituted in (11) gives v=— 2Qjc; (13)

358 TRANSIENT PHENOMENA. hence, . v= +(1—j)e (14) and @ = AyetOaey Aye i P, Since ® must have the same value for — / as for + 1, being | symmetrical at both sides of the center line of the lamination, A, =A, =A, , hence, B= Af{eta-Het + e~ A-Doky (15) or, substituting et J? — cos cl + j sincl (16) gives B® = A{(et*+ e-%) cosel — j (e**— e-) sincl}. (17) 51. Denoting the flux density in the center of the lamination, for 1 = 0, by @, from (17) we have B® = 24; hence, A =38, (18) and +el el +el _ .—cl , = 0} coset — j$ = sinalt, (19) . ‘. 2 2 Denoting the flux density at the outside of the lamination, for 1=1,, that is, the density produced by the external m.m.f., by @,, substituted in (19), we have . +cly —cly +cely _ .—cly 8, = 8, jee cos cl, — joo" ana,}, (20) and substituting (20) in (19), , (e+e? 4+ e~%) cos el — j (e+ — e~%) sin el @ = 6, 1 (eto + e~ oo) cos cl, — j (ete — e~ co) sin dl, (21) The mean or apparent value of the flux density, i.e., the average throughout the lamination, is . . 1 . Bn = tS 8 dl. (22) ’ ° Cy

ALTERNATING MAGNETIC FLUX DISTRIBUTION 359 Using equation (15) as the more convenient for integration gives Bm = — 4 p+a-na_ eS oty ty , (1 — 9) dl, A (et G-Sebe_ -—(1-J) ely _ A ett eh — e--Pehy (93) dl ~ D cl, and substituting herein (16), (18) and (20), gives 8, etl _ -— Che ete och . Bn a-)) a} 3 cos cl, —j 2 sin cl, os) — Be —e-%) cos cl, —j (e+ +e~%) sin cl, ~ (1-7) el, (e+ +6~%) cos cl, —j (e+ —e~-™) sin cl, The absolute values of the flux densities are derived as square root of the sum of the squares of real and imaginary terms in equations (19), (20), (21), and (24), as . B® = o Vettel 4 3c + 2 cos 2 cl, (25) ! " @= Sov/etioh + 2-2 + 2 cos 2dl,, (26) et2et 4 e-3eb + 2 cog el 8 = OV rr T+ D coe Del’ 7) | and Bn = — So _ et2elo 4 —¢—2ceh 2 cog 2 cl | 2 l,W2 ° (28) . _ 8: vee + e724 _ 2 cos 2 cl, | el, V2" et? 4 e204 2 cos 2 cl, 652. Where the thickness of lamination, 2 1,, or the frequency /, | is so great as to give cl, a value sufficiently high to make «%, or the reflected wave, negligible compared with the main wave ete the equations can be simplified by dropping «. In this | case the flux density, ®, is very small or practically nothing in the interior, and reaches appreciable values only near the surface. | It then is preferable to count the distance from the surface of the

360 TRANSIENT PHENOMENA lamination into the interior, that is, substitute the independent ‘ variable s=1,-1. (29) Dropping «~ and e~°* in equation (21) gives e* (cos cl — j sin cl) ® = (cos cl, — jsindl,) = @, 6°" {eos c (l,— 1) + jsine (l,— D}; hence, ® = B,« ~ (coscs + j sin cs); (30) or the absolute value is @® =, %, (31) and at the center of the lamination, 8, =, (coe el, + jsin ay (32) @, = Be, From equation (24) the mean value of flux density follows when dropping e~“ as negligible, thus: = a, 8m = G—pa’ (33) or the absolute value is 8 Gn = ——>: 34 " edlWV2 4) 63. As seen, the preceding equations of the distribution of alternating magnetic flux in a laminated conductor are of the same form as the equations of distribution of current and voltage in a transmission line, but more special in form, that is, the attenuation constant a and the wave length constant 8 have the same value, c. As result, the distribution of the alternating magnetic flux in the lamina depends upon one constant only, cl,. The wave length is given by . cl, = 27;

ALTERNATING MAGNETIC FLUX DISTRIBUTION 361

hence l, = 2a

c and by (9)

22

be = aVf 10,000 935 SO?

V0.1 Auf G ) and the attenuation during one wave length, or decrease of intensity of magnetism, per wave length, is

e~?* = 0.0019, and per half-wave length is e~* = 0.043. At the depth ™ below the surface, the magnetic flux lags 90 degrees and has considerably decreased; at the depth it lags 180 degrees, that is, is opposite in direction to the flux at the surface of the lamination, but is very small, the intensity being less than 5 per cent of that at the surface, and at the depth J, the flux is again in phase with the surface flux, but its intensity is practically nil, less than 0.2 per cent of the surface intensity; that is, the penetration of alternating flux into the laminated iron is inappreciable at the depth of one wave length. By equations (33) and (34), the total magnetic flux per unit width of lamination is . ° 28 21 nm = —— ; im = De . 28, the absolute value is 21,8,, = —=; cv2 that is, the same as would be produced at uniform density in a thickness of lamination . L = 2 Pp a — D c or absolute value, v2 | mes

362 TRANSIENT PHENOMENA which means that the resultant alternating magnetism in the lamination lags 45 degrees, or one-eighth wave behind the im- pressed m.m.f., and is equal to a uniform magnetic density penetrating to a depth 1 . l,=-—=: 36 » = 5 (36) . I,, therefore, can be called the depth of penetration of the alternating magnetism into the solid iron.

Since the only constant entering into the equation is cl,, the distribution of alternating magnetism for all cases can be repre- sented as function of cl,.

If cl, is small, and therefore the density in the center of the lamination ®, comparable with the density ®, at the outside, the equations (19), (20), and (24) respectively (25), (26), and (28) are preferably used; if cl, is large, and the flux density B, in the center of the lamination is negligible, the equations (30) to (34) are preferably used.

54. As an example, let » = 1000 and 4 = 105; then a = 1.98, and for f = 60 cycles per second, c = aVf = 15.3; hence, the thickness of effective layer of penetration is

1 lL, = —= = 0.046 cm = 0.018 inches. » eV2 In Fig. 93 is shown, with cl as abscissas, the effective value of the magnetic flux, which from equation (25) is CT See TE B= > Ver Te 3l 4 2 cos 2 cl, and also the space-phase angle between ® and ®,, which from equation (19) is et! _ ed tan T= et gna ton cl. (37)

In Fig. 94 is shown, with cs as abscissas, the effective value

of the magnetic flux, which from equation (31) is B= Be %, |

ALTERNATING MAGNETIC FLUX DISTRIBUTION 363 and also the space-phase angle between ® and &,, which from equation (30) is

tan t, = tan cs. (38)

The thickness of the equivalent layer is marked in Fig. 94. ! VI TT TTT ttt ty tt tt TY PAT TT TTT TT bt TT Tt A PINT PT ee YT PAP ALT ae tp

ASAE ECE PCE

PIN ATT TTT Ta tt TE TE TAY.

| SENN RRR

| TTT AKT TTT TT TE tt PAA TT

| PT TINK TET TTT PA

PTT TIN We Ti T PY Tt PTT TT ATE TT TT Etta TT BRERA Aes PT TTT TIN TTT TT yt ttt et th, PT TTT TT NET TT Ae Tt Tt rT TT ttt | Nd TT felt tt 2.0 16 13 0.8 0.4 0.4 0.8 12 16 2.0

Fig. 98. Alternating magnetic flux distribution in solid iron.

As further illustrations are shown in Fig. 95 the absolute values of magnetic flux density ® throughout a layer of 14 mils thickness, that is, of 1, = 0.007 inches = 0.018 cm. thickness.

For 60 cycles, by Curve I, e= 15.3 cl, = 0.275 For 1000 cycles, by Curve II, c= 62.5 cl, = 1.125 | For 10,000 cycles, by Curve III, c = 198 cl, = 3.55 ‘ It is seen that the density in Curve I is perfectly uniform, while in Curve III practically no flux penetrates to the center. 55. The effective penetration of the alternating magnetism into the iron, or the thickness /, of surface layer which at con- stant induction ®, would give the same total magnetic flux as exists in the lamination, is . 1 | = THe’ (89) . or the absolute value is 1 | l, =——sz; cv2 |

364 TRANSIENT PHENOMENA a A AH eA ° AE ot IEE TT TT, PAL 4A ot WN ETE TT A+ PTY ETT aan of LATELY aan FONE EEL eof FEM sane? rLIN-EIYET TTT wl LEONE TAT TT TT, pare TN ZT TT E vee IX TT TT, TET ALN- FE ot IW TT NTT, rr IAC ENT . wt ZEEE 240 eee. se YLT test FTE LLL OT 04 0.8 12 16 2.0 2.4 28 32 ‘ Fig. 94. Alternating magnetic flux distribution in solid iron. TTTIT IIIT bead TTT TTT Ie NESS ANCE TET tT tT tT tt tt tt tT Pe / PY | RL LT tard 000 eyes | | EE | YT PRET TTT Tre Terre tA, PIN TT TTT TTT TT ET tT Tt PERT TT Ti Te tt PT AT PEIN GE ETE EEE TET ETT Ty eT , EEE A PTT TSNIT head adores LA? TT TT SERRE 1.0 0.8 0.6 4 0.2 0 02 0.4 0.6 as 1.0 Fig. 95, Alternating magnetic flux distribution in solid iron.