Theory and Calculation of Electric Circuits — part 6 of 15

64. If a constant alternating potential, eo, is impressed upon an electromagnet, and the voltage consumed by the resistance, tr, can be neglected, the voltage consumed by the reactance, 2, is constant and is the terminal voltage, éo, thus the magnetic flux, ®, also is constant during the motion of the armature of the electromagnet. The current, 7, however, varies, and decreases from a maximum, 7), in the initial position, to a minimum, ¢2, in the end position of the armature, while the inductance increases from L; to Le.

The voltage induced in the electric circuit by the motion of the armature,

ef = n2@2 198 (30) dt , then is zero, and therefore also the electrical energy expended, w= 0.

That is, the electric circuit does no work, but the mechanical work of moving the armature is done by the stored magnetic energy.

The increase of the stored magnetic energy is

: w' = te"Le - 47D, (31) 2 and since the mechanical energy, in joules, is by (13), ~ wo = Flg 107 the equation of the law of conservation of energy, w=w' + wo (32) then becomes a2 — 4,2 o = Bila = tlh _e Lt 4. Fg 10-1, or 2 — 7,2 Fl = mee 107 gram-cm. (33) Since, from the equation of self-induction, in the initial posi- tion, €o = 2afliii (34) in the end position . €o = 2xfLiis (35) |

MAGNETISM 99 substituting (34) and (35) in (33), gives the equation of the constant-potential alternating electromagnet. .

_ Co(21 - i) 7 pm Fl = ~~ 4ahg 10’ gram-cm. (36) and _ colin == ta) gr 60 di ay F 4 afgl 10? = aafg dl 10 grams (37) or, in foot-pounds, ; ; Fy = 2586 ells — 2) eet (38) : _ 0.586 €0(21 - te) _ 0.586 @o di F= a EF Ib. (39) Substituting Q = et = volt-amperes, in equations (36) to . (39) of the constant-potential alternating electromagnet, and equations (22) to (25) of the constant-current alternating magnet, gives the same expression of mechanical work and pull: In metric system: = A2 107

• = anf 10 o (40) = A 7 — 1 _ ‘7 F= Tafgl 10 “Fafa dl 107 grams (41) . In foot-pounds: Fl = aceeae ft.-lb. "(49 e . _ 0.586 AQ _ 0.586 dQ F= a Fa Ib. (43) where AQ = difference in volt-amperes consumed by the magnet in the initial position, and in the end position of the armature. Both types of alternating-current magnet, then, give the same expression of efficiency, 1 = 28 (44) Qn where Q,, is the maximum volt-amperes consumed, corresponding to the end position in the constant-current magnet, to the initial position in the constant-potential magnet.

4. Short-circuit Stresses in Alternating-current Transformers _ 665. At short-circuit, no magnetic flux passes through the sec- ondary coils of the transformer, if we neglect the small voltage consumed by the ohmic resistance of the secondary coils. If

100 - ELECTRIC CIRCUITS the supply system is sufficiently large to maintain constant voltage at the primary terminals of the transformer even at short-circuit, full magnetic flux passes through the primary coils! In this case the total magnetic flux passes between primary coils and secondary coils, as self-inductive or leakage : flux. If then x = self-inductive or leakage reactance, ¢ = im- pressed e.m.f., i = = is the short-circuit current of the trans- former. Or, if as usual the reactance is given in per cent., that is, the 7x (where 7 = full-load current of the transformer) given in per cent. of e, the short-circuit current is equal to the full-load current divided by the percentage reactance. Thus a trans- former with 4 per cent. reactance would give a short-circuit cur- rent, at maintained supply voltage, of 25 times full-load current. To calculate the force, F, exerted by this magnetic leakage flux on the transformer coils (which is repulsion, since primary and secondary currents flow in opposite direction) we may assume, at constant short-circuit current, 49, the secondary coils moved against this force, F, and until their magnetic centers coincide with those of the primary coils; that is, by the distance, I, as shown diagrammatically in Fig. 45, thesection of a shell-type transformer. When brought to coincidence, no magnetic flux passes between primary and secondary coils, and during this motion, of length, l, the primary coils thus have cut the total magnetic flux, ®, of the transformer. . Hereby in the primary coils a voltage has been induced, , d® , c= ny 10-8 | where n = effective number of primary turns. : The work done or rather absorbed by this voltage, e’, at cur- rent, %, is , w= f e'igdt = niob 10-8 joules. (45) 1 If the terminal voltage drops at short-circuit on the transformer seconda- ries, the magnetic flux through the transformer primaries drops in the same proportion, and the mechanical forces in the transformer drop with the square of the primary terminal voltage, and with a great drop of the ter- minal voltage, as‘occurs for instance with large transformers at the end of a transmission line or long feeders, the mechanical forces may drop to a small fraction of the value, which they have on a system of practically un- limited power. : ; { |

ee MAGNETISM ; 101 . If L = leakage inductance of the transformer, at short-circuit, where the entire flux, ®, is leakage flux, we have @ = Hej93 (46) hence, substituted in (45) _ The stored magnetic energy at short-circuit is 2D, W1= > (48) and since at the end of the assumed motion through distance, I, the leakage flux has vanished by coincidence between primary and secondary coils, its stored magnetic energy also has vanished, _ and the change of stored magnetic energy therefore is , , to°L Hence, the mechanical work of the magnetic forces of the short- circuit current is , 2 wm=w-—w = we (50) It is, however, if F is the force, in grams, I, the distance between the magnetic centers of primary and secondary coils, w, = Flg 10-7 joules. Hence, , 2 Fl = — 107 gram-cm. (51) and — WL qr F = 2 gl 10’ grams (52) the mechanical force existing between primary and secondary coils of a transformer at the short-circuit current, to. Since at short-circuit, the total supply voltage, ¢, is consumed by the leakage inductance of the transformer, we have 9 = 2xfLig (53) hence, substituting (53) in (52), gives . _ Coto 10’ P= Taft SOS 3 ri grams (54)

102 ELECTRIC CIRCUITS Example.—Let, in a 25-cycle 1667-kw. transformer, the supply voltage, é& = 5200, the reactance = 4 per cent. The trans- former contains two primary coils between three secondary coils, and the distance between the magnetic centers of the adjacent coils or half coils is 12 cm., as shown diagrammatically in Fig. 45. What force is exerted on each coil face during short-circuit, in a system which is so large as to maintain constant terminal voltage?

At 5200 volts and 1667 kw., the full-load current is 320 amp. At 4 per cent. reactance the short-circuit current therefore, to = A = 8000 amp. Equation (54) then gives, for f = 25, lL = 12,

F=112X10%grams _ = 112 tons. :

This force is exerted between the four faces of the two primary coils, and the corresponding faces of the secondary coils, and on every coil face thus is exerted the force

F

4 = 28 tons This is the average force, and the force varies with double frequency, between 0 and 56 tons, and is thus a large force.

56. Substituting i) = - in (54), gives as the short-circuit force of an alternating-current transformer, at maintained terminal voltage, &o, the value

e210? 810 ec? F= dsfgla fla grams (55)

That is, the short-circuit stresses are inversely proportional to the Jeakage reactance of the transformer, and to the distance, l, between the coils.

In large transformers on systems of very large power, safety therefore requires the use of as high reactance as possible.

High reactance is produced by massing the coils of each cir-

cuit.

Let in a transformer

n = number of coil groups

MAGNETISM . 108 (where one coil is divided into two half coils, one at each end of the coil stack, as one secondary coil in Fig. 45, where n = 2) the . mechanical force per coil face then is, by (55), a= FL 60°10" _ 810 eo? Fo= 37 Bafgnle ~ Dfnle S°™5 (56) Let z = leakage reactance of transformer; l, = distance between coil surfaces; : l, = thickness of primary coil; l, = thickness of secondary coil.

Between two adjacent coils, P and S in Fig. 45, the leakage flux density is uniform for the width 1) between the coil surfaces, | to too.

a a ito oF Fia. 45.

and then decreases toward the interior of the coils, over the dis- tance a respectively z, to zero at the coil centers. All the coil turns are interlinked with the leakage flux in the width, lk, but toward the interior of the coils, the number of turns interlinked with the leakage flux decreases, to zero at the coil center, and as the leakage flux density also decreases, proportional to the dis- tance from the coil center, to.zero in the coil center, the inter- linkages between leakage flux and coil turns decrease over the . space st respectively i proportional to the square of the distance from the coil center, thus giving a total interlinkage distance,

ql

2 du = 4,

f udu = F

where u is the distance from the coil center.

; 104 ELECTRIC CIRCUITS :

Thus the total interlinkages of the leakage flux with the coil turns are the same as that of a uniform leakage flux density over the width I) + i. + a, This gives the effective distance between coil centers, for the reactance calculation,

l= + ath (57)

Assuming now we regroup the transformer coils, so as to get m primary and m secondary coils, leaving, however, the same iron structure. ;

The leakage flux density between the coils is hereby changed in proportion to the changed number of ampere-turns per coil, that is, by the factor 2.

m

The effective distance between the coils, J, is changed by the

same factor —- m

The number of interlinkages between leakage flux and electric circuits, and thus the leakage reactance, 2, of the transformer, thus is changed by the factor

n 2 | (7)

That is, by regrouping the transformer winding within the same magnetic circuit and without changing the number of turns of the electric circuit, the leakage reactance, 2, changes inverse propor- tional to the square of the number of coil groups.

As by equation (56) the mechanical force is inverse propor-

2 tional to z, 1 and n, and xz changes proportional to (7) > 1 pro- portional to = » the mechanical force per coil thus changes proportional to . n? 2 m n? | C) XR a= Gi)

That is, regrouping the transformer winding in the same wind-

ing space changes the mechanical force inverse proportional to

MAGNETISM 105 the square of the coil groups, thus inverse proportional to the change of leakage reactance.

However, the distance ly between the coils is determined by in- . sulation and ventilation. Thus its decrease, when increasing the number of coil groups, would usually not be permissible, but more winding space would have to be provided by changing the mag- netic circuit, and inversely, with a reduction of the number of coil groups, the winding space, and with it the magnetic circuit, would be reduced.

Assuming, then, that at the change from n to m coil groups, the distance between the coils, lo, is left the same.

The effective leakage space then changes from

_ L+l L=Iybt+ —.” to . nht+ls pepe tbth itm 6 om 6 hth | lo + 6 and the leakage reactance thus changes from z to ,» nl, ge = — —2 ml hence the mechanical force per coil, from Fy, = & = oe 10", ° 2n Bafnglz to” Py = Fe 002 10" ° Im & xfngl’x’ la = Foe? if 2 = Fo (5) | lo + n 4, +0\2. = F{ —™o (58) L+h lo + 3

| 106 ELECTRIC CIRCUITS uth. . . | Thus, if —— is large compared with ly, | | r= (7)? | Fro = (=) Fo, | that is, the mechanical forces vary with the square of the number | of coil groups. It ath is small compared with ls, F,' = Fo ° that is, the mechanical forces are not changed by the change of the number of coil groups. In actual design, decreasing the number of coil groups usually materially decreases the mechanical forces, but materially less than proportional to the square of the number of coil groups. . 5. Repulsion between Conductor and Return Conductor 57. If i) is the current flowing in a circuit consisting of a con- ductor and the return conductor parallel thereto, and 1 the dis- tance between the conductors, the two conductors repel each other by the mechanical force exerted by the magnetic field of the circuit, on the current in the conductor. As this case corresponds to that considered in section 2, equa- tion (16) applies, that is,

• i ig F= 29 a 1° grams, The inductance of two parallel conductors, at distance | from each other, and conductor diameter l, is, per centimeter length of conductor, . . L= (4 log a + 1) 10-° henrys (59) a Hence, differentiated, dL _ 4X 10-° dl l and, substituted in (16), — F= 50 gl grams (60) or substituting (12),

MAGNETISM 107 ‘a2 6 F= 20.4 4 10- grams (61) If 1 = 150 cm. (5 ft.) to = 200 amp. this gives .

F = 0.0054 grams per centimeter length of circuit, hence it is inappreciable.

If, however, the conductors are close together, and the current very large, as the momentary short-circuit current of a large alternator, the forces may become appreciable.

For example, a 2200-volt 4000-kw. quarter-phase alternator feeds through single conductor cables having a distance of 15 cm. (6 in.) fram each other. A short-circuit occurs in the cables, and the momentary short-circuit current is 12 times full-load current. What is the repulsion between the cables?

Full-load current is, per phase, 910 amp. Hence, short-circuit current, tp = 12 K 910 = 10,900 amp. ! = 15. Hence,

F = 160 grams per centimeter. Or multiplied by 30.5 454 F = 10.8 lb. per feet of cable. ' That is, pulsating between 0 and 21.6 Ib. per foot of cable. Hence sufficient to lift the cable from its supports and throw it aside.

In the same manner, similar problems, as the opening of dis- connecting switches under short-circuit, etc., can be investigated. 6. General Equations of Mechanical Forces in Magnetic Fields

58. In general, in an electromagnetic system in which mechan- ical motions occur, the inductance, L, is afunction of the position, l, during the motion. If the system contains magnetic material, in general the inductance, LZ, also is a function of the current, ¢, especially if saturation is reached in the magnetic material.

Let, then, Z = inductance, as function of the current, 7, and ; position, 1;

I, = inductance, as function of the current, 7, in the initial position 1 of the system;

Ly = inductance, as function of the current, 7, in the end position 2 of the system.

108 - ELECTRIC CIRCUITS If then 6 = magnetic flux, » = number of turns interlinked with the flux, the induced e.m.f. is 1742 19-8 ef =n 10 (62) . We have, however, n® = iL 108; hence, , _ ail) the power of this induced e.m.f. is Pp = te’ = a ae), and the energy ; 2 2 wf pat = {iaGit) 1 2 = f dL + f tLdi (64) 1 1 ‘ The stored magnetic energy in the initial position 1 is : 1 Wy, = f 4d(iL1) . (65) In the end position 2, | = { id(iLs) (66) : 0 and the mechanical work thus is, by the law of conservation of energy Wo = wW— Wet Wi 1 2 = f id(GQL) + f id(iL,) — f 4d (tL) (67) 1 0 0 and since the mechanical work is wo = Flg 10-7 (68) We have: 10’ | (?.,,. 1 oe Fl = -_ | f ad(iL) + f ey) - f iat | gram-cm. (69) 1 0 . ; |

MAGNETISM 109. If L is not a function of the current, 7, but only of the position, that is, if saturation is absent, ZL, and Zz are constant, and equa- tion (69) becomes, ‘7 742 — 42, Fl = * | f id(iL) + Mats | gram-cm. (70) 1 (a) If ¢ = constant, equation (70) becomes, . Fl = 10° t(Le —Iy) ~ 9 2 - (Constant-current electromagnet.) (b) If L = constant, equation (70) becomes, . e Fl = 0. That is, mechanical forces are exerted only where the in- ductance of the circuit changes with the mechanical motion which would be produced by these forces. (c) If iL = constant, equation (70) becomes, Fl= 10” tL(t, — #2) g 2 ; (Constant-potential electromagnet.) In the general case, the evaluation of equation (69) can usually be made graphically, from the two curves, which give the varia- tion of L; with 7 in the initial position, of Lz with 7 in the final position, and the curve giving the variation of L and 7 with the . motion from the initial to the final position. ; In alternating magnetic systems, these three curves can be determined experimentally by measuring the volts as function of the amperes, in the fixed initial and end position, and by measuring volts and amperes, as function of the intermediary ‘positions, that is, by strictly electrical measurement. As seen, however, the problem is not entirely determined by the two end positions, but the function by which 7 and L are . related to each other in the intermediate positions, must also be given. That is, in the general case, the mechanical work and thus the average mechanical force, are not determined by the end positions of the electromagnetic system. This again shows an analogy to thermodynamic relations. If then in case of a cyclic change, the variation from position . ;

110 ELECTRIC CIRCUITS 1 to 2 is different from that from position 2 back to 1, such a

cyclic change produces or consumes energy.

1 1

w= f td(iL) + f id(iL) = f id(iL)

1 2 1 Such a case is the hysteresis cycle. The reaction machine (see Theory and Calculation of Electrical Apparatus) is based

on such cycle.

SECTION II CHAPTER VII SHAPING OF WAVES: GENERAL

59. In alternating-current engineering, the sine wave, as shown in Fig. 46, is usually aimed at as the standard. This is not due to ; any inherent merit of the sine wave.

For all those purposes, where the energy developed by the cur- rent in a resistance is the object, as for incandescent lighting, heating, etc., any wave form is equally satisfactory, as the energy of the wave depends only on its effective value, but not on its shape.

With regards to insulation stress, as in high-voltage systems, a flat-top wave of voltage and current, such as shown in Fig. 47, would be preferable, as it has a higher effective value, with the same maximum value and therefore with the same strain on the ; insulation, and therefore transmits more energy than the sine wave, Fig. 46.

Inversely, a peaked wave of voltage, such as Fig. 48, and such .

as the common saw-tooth wave of the unitooth alternator, is superior in transformers and similar devices, as it transforms the energy with less hysteresis loss. The peaked voltage wave, Fig. 48, gives a flat-topped wave of magnetism, Fig. 47, and thereby transforms the voltage with a lesser maximum magnetic flux, than a sine wave of the same effective value, that is, the same power. As the hysteresis loss depends on the maximum value of the mag- netic flux, the reduction of the maximum value of the magnetic flux, due to a peaked voltage wave, results in a lower hysteresis loss, and thus higher efficiency of transformation. This reduc- tion of loss may amount to as much as 15 to 25 per cent. of the total hysteresis loss, in extreme cases.

Inversely, a peaked voltage wave like Fig. 48 would be objec- tionable in high-voltage transmission apparatus, by giving an un- necessary high insulation strain, and « flat-top wave of voltage like Fig. 47, when impressed upon a transformer, would give a peaked wave of magnetism and thereby an increased hysteresis loss.

111

112 ‘ELECTRIC CIRCUITS — The advantage of the sine wave is, that it remains unchanged in shape under most conditions, while this is not the case with any other wave shape, and any other wave shape thus introduces the danger, that under certain conditions, or in certain parts of the - circuit, it may charige to a shape which is undesirable or even 7 \ 7 Fia. 46, 7 \ Tia. 47, a : ~ 7 Fia. 48, : \ / Fre. 49. Fias. 46 To 49. dangerous. Voltage,e, and current, i, are related to each other by proportionality, by differentiation and by integration, with re- sistance, r, inductance, L, and capacity, C, as factors, é=rt, : e=L a, . e=C f idt, | and as the differentials and integrals of sines are sines, as long as’ r, L and C are constant—which is mostly the case—sine waves of

SHAPING OF WAVES 113 voltage produce sine waves of current and inversely, that is, the sine wave shape of the electrical quantities remains constant. A flat-topped current wave like Fig. 47, however, would by differentiation give a self-inductive voltage wave, which is peaked, ; like Fig. 48. A voltage wave like Fig. 48, which is more efficient in transformation, may by further distortion, as by intensifica- tion of the triple harmonic by line capacity, assume the shape, Fig. 50. Fig. 49, and the latter then would give, when impressed upon a transformer, a double-peaked wave of magnetism, Fig. 50, and such wave of magnetism gives a magnetic cycle with two small CCC EEE PT tT Tt | Leer ; PTT TET tee Vet PT itt tye tia” Tt PPE TZLT 7 Pit TT Vit tT Vy tt oe Pt tT tT tT AT TTA TTT YT Pitt itt Tt tT =) ee 4 PTT iA tere Et | | | eer tt PEt EET EET tE Tt tt Fie. 51. °° secondary loops at high density, as shown in Fig. 51, and an additional energy loss by hysteresis in these two secondary loops, ; which is considerable due to the high mean magnetic density, at which the secondary loop is traversed, so that in spite of the reduced maximum flux density, the hysteresis loss may be increased. : Therefore, in alternating-current engineering, the aim gener- 8 .

114 ELECTRIC CIRCUITS ‘ ally is to produce and use a wave which is a sine wave or nearly so.

60. In an alternating-current generator, synchronous or in- duction machine, commutating machine, etc., the wave of voltage induced in a single armature conductor or “face conductor”

; equals the wave of field flux distribution around the periphery of the magnet field, modified, however, by the reluctance pulsations ; of the magnetic circuit, where such exist. As the latter produce higher harmonics, they are in general objectionable and to be avoided as far as possible. .

By properly selecting the length of the pole arc and the length of the air-gap between field and armature, a sinusoidal field flux distribution and thereby a sine wave of voltage induced in the armature face conductor could be produced. In this direction, however, the designer is very greatly limited by economic con-

sideration: length of pole arc, gap length, etc., are determined within narrow limits by the requirement of the economic use of the material, questions of commutation, of pole-face losses, of field excitation, etc., so that as a rule the field flux distribution and with it the voltage induced in a face conductor differs materially from sine shape.

The voltage induced in a face conductor may contain even har- monics as well as odd harmonics, and often, as in most inductor alternators, a constant term.

The constant term cancels in all turn windings, as it is equal and opposite in the conductor and return conductor of each turn. Direct-current induction (continuous, or pulsating current) thus is possible only in half-turn windings, that is, windings in which each face conductor has a collector ring at either end, so-called unipolar machines (see “Theory and Calculation of Electrical Apparatus’’).

In every winding, which repeats at every pole or 180 electrical degrees, as is almost always the case, the even harmonics cancel, even if they existed in the face conductor. In any machine in which the flux distribution in successive poles is the same, and merely opposite in direction, that is, in which the poles aresymmet- rical, no even harmonics are induced, as the field flux distribution contains no even harmonics. Even harmonics would, however, exist in the voltage wave of a machine designed as shown diagram- matically in Fig. 52, as follows:

The south poles S have about one-third the width of the north

|

| | SHAPING OF WAVES 115 poles N, and the armature winding is a unitooth 50 per cent. pitch winding, shown as A in Fig. 52. Assuming sinusoidal field flux distribution in the air-gaps under the poles N and S of Fig. 52, curve I in Fig. 53 shows the field flux distribution and thus the voltage induced in a single-face con- ductor. Curve II shows the voltage wave in a 50 per cent. pitch ° turn and therewith that of the winding A. As seen, this contains & pronounced second harmonic in addition to the fundamental. | If, then, a second 50 per cent. pitch winding is located on the arma- ; { s N Kar N CNS —--~, } — (5 als ~~ a : N )\ SS N Ss . Fia. 52. ture, shown as B in Fig. 52, by connecting B and A in series with each other in such direction that the fundamentals cancel (that is, in opposition for the fundamental wave), we get voltage wave III of Fig. 53, which contains only the even harmonics, that is, is of double frequency. Connecting A and B in series so that the fundamentals add and the second harmonics cancel, gives the wave lV. If the machine is a three-phase Y-connected alterna- tor, with curve IV as the voltage per phase, or Y voltage, the delta or terminal voltage, derived by combination of two Y vol- tages under 60°, then is given by the curve V of Fig. 58. Fig. 54 shows the corresponding curves for the flux distribution of uni- form density under the pole and tapering off at the pole-corners, curve I, such as would approximately correspond to actual con-

116. ELECTRIC CIRCUITS

• ditions. As seen, curve III as well as V are approximately sine waves, but the one of twice the frequency of the other. Thus, such a machine, by reversing connections between the two wind- ings A and B, could be made to give two frequencies, one double the other, or as synchronous motor could run at two speeds, one one-half the other. 1 L__S
  . I, Fia. 53.

. 61. Distribution of the winding over an arc of the periphery of | the armature eliminates or reduces the higher harmonics, so that the terminal voltage wave of an alternator with distributed wind- ing is less distorted, or more nearly sine-shaped, than that of a single turn of the same winding (or that of a unitooth alternator). The voltage waves of successive turns are slightly out of phase | with each other, and the more rapid variations due to higher har- monics thus are smoothed out. In two armature turns different

SHAPING OF WAVES 117 in position on the armature circumference by 5 electrical degrees (“electrical degrees”? means counting the pitch of two poles as 360°), the fundamental waves are 5 degrees out of phase, the third harmonics 36 degrees, the fifth harmonics 55 degrees, and so on, and their resultants thus get less and less, and becomes zero for

that harmonic n, where nd = 180°.

/
. Fie. 54. If e = e, sin @ + e; sin 3 (®—ay) + €, sin 5 (@—as)

• e;sin 7 (@—a7) +... (1) is the voltage wave of a single turn, and the armature winding of m turns covers an arc of w electrical degrees on the armature periphery (per phase), the coefficients of the harmonics of the resultant voltage wave are

118 ELECTRIC CIRCUITS E, = Me, avg. COS (2) ° _ ne , 2
or, since | nw : + 2 | avg. cos = 2 sin B- ~ nw 2 _ ne 2 : E, = 2™ e, sin ™ (3)

"nw" 2 | and : | in @ gi + % gi 30 sin 3($ — )

@ @1 Sl 9 sine 3 Sn -5 St as

• %sin 52 sin 5(¢ —a)+... \ (4) Thus, in a three-phase winding like that of the three-phase synchronous converter, in which each phase covers an arc of 120° = 2, it is 5 = 3 hence, E= anv { e.sin ¢ — & sin 5(¢ — as)

+%sin7(6-a)-+.... } © that is, the third harmonic and all its multiples, the ninth, fif- teenth, ete., cancel, all other harmonics are greatly reduced, the . more, the higher their order.

In a three-phase Y-connected winding, in which each phase _ covers 60° = 5 of the periphery, as commonly used in induction and synchronous machines, it is 3 = a hence, E= Sm { e,sin ¢ + * essin 3(¢ — a3) + 5 essin 5(¢ — as)

• er sin 7(¢@ — a7) — es sin 9(¢ — ap)
• dan sin 116 — an) + J. ysin 13(@ — au) +—... } (6) 11 13 |

SHAPING OF WAVES 119

Here the third harmonics do not cancel, but are especially large. Thus in a Y-connected three-phase machine of the usual 60° winding, the Y voltage may contain pronounced third harmonics, which, however, cancel in the delta voltage.

Thus with the distributed armature winding, which is now al- most exclusively used, the wave-shape distortion due to the non- sinusoidal distribution of the field flux is greatly reduced, that is, the higher harmonics in the voltage wave decreased, the more so, the higher their order, and very high harmonics, such as the seven- : teenth, thirty-fifth, etc., therefore do not exist in such machines to any appreciable extent, except where produced by other causes. Such are a pulsation of the magnetic reluctance of the field due to the armature slots, or a pulsation of the armature reactance, as discussed in Chapter XXV of “Theory and Calculation of Alter- nating-current Phenomena,” or a space resonance of the armature conductors with some of the harmonics. The latter may occur if the field flux distribution contains a harmonic of such order, that the voltages induced by it arein phasein the successive arma- ture conductors, and therefore add, that is, when the spacing of the armature conductors coincides with a harmonic of the field flux, and the armature turn pitch and winding pitch are such that this harmonic does not cancel.

Inversely, if two turns are displaced from each other on the

armature periphery by * of the pole pitch, or = ,and are connected in series, then in the resultant voltage of these two turns, the n™ harmonics are out of phase by n times - , or by + = 180°, that is, are in opposition and so cancel.

Thus in a unitooth Y-connected three-phase alternator, while each phase usually contains a strong third harmonic, the terminal voltage can contain no third harmonic or its multiples: the two phases, which are in series between each pair of terminals, are one-third pole pitch, or 60 electrical degrees displaced on the armature periphery, and their third harmonic voltages therefore 3 X 60 = 180° displaced, or opposite, that is, cancel, and no third harmonic can appear in the terminal voltage wave, or delta volt- age, but a pronounced third harmonic may exist—and give trouble—in the voltage between each terminal and the neutral, or the Y voltage.

62. By the use of a fractional-pitch armature winding, higher harmonics can be eliminated. Assume the two sides of the arma-

4 '

120 ELECTRIC CIRCUITS :

ture turn, conductor and return conductor, are not separated from each other by the full pitch of the field pole, or 180 electrical degrees, but by less (or more); that is, each armature turn or coil covers not the full pitch of the pole, but the part p less (or more), that is, covers (1 + p) 180°. The coil then is said to be (1 + p) fractional pitch, or has the pitch deficiency p. The voltages in- duced in the two sides of the coil then are not equal and in phase, but are out of phase by 180 p for the fundamental, and by 180 np for the n‘* harmonic. Thus, if np = 1, for this n™ har- monic the voltages in the two sides of the coil are equal and oppo- site, thus cancel, and this harmonic is eliminated.

Therefore, two-thirds pitch winding eliminates the third har- monic, four-fifths pitch winding the fifth harmonic, ete.

Peripherally displacing half the field poles against the other half by the fraction q of the pole pitch, or by 180 q electrical de- grees, causes the voltages induced by the two sets of field poles to be out of phase by 180 ng for the n“ harmonic, and thereby eliminates that harmonic, for which ng = 1.

By these various means, if so desired, a number of harmonics can be eliminated. Thus ina Y-connected three-phase alternator with the winding of each.phase covering 60 electrical degrees, with four-fifths pitch winding and half the field poles offset against the other by one-seventh of the pole pitch, the third, fifth, and seventh harmonic and their multiples are eliminated, that is, the lowest harmonic existing in the terminal voltage of such a ma- chine is the eleventh, and the machine contains only the eleventh, thirteenth, seventeeth, ninteenth, twenty-third, twenty-ninth, thirty-first, thirty-seventh, ete. harmonics. As by the distrib- uted winding these harmonics are greatly decreased, it follows that the terminal voltage wave would be closely a sine, irrespec- tive of the field flux distribution, assuming that no slot harmonics exist.

68. In modern machines, the voltage wave usually is very closely a sine, as the pronounced lower harmonics, caused by the field flux distribution, which gave the saw-tooth, flat-top, peak or multiple-peak effects in the former unitooth machines, are greatly reduced by the distributed winding and the use of frac- tional pitch. Individual high harmonics, or pairs of high harmon- ics, are occasionally met, such as the seventeenth and ninteenth, or the thirty-fifth and thirty-seventh, ete. They are due to the pulsation of the magnetic field flux caused by the pulsation of the

SHAPING OF WAVES 121 field reluctance by the passage of the armature slots, and occa- sionally, under load, by magnetic saturation of the armature self- inductive flux, that is, flux produced by the current in an arma- ture slot and surrounding this slot, in cases where very many - ampere conductors are massed in one slot, and the slot opening ” bridged or nearly so. :

The low harmonics, third, fifth, seventh, are relatively harm- less, except where very excessive and causing appreciable increase ; of the maximum voltage, or the maximum magnetic flux and thus hysteresis loss. The very high harmonics as a rule are rela- tively harmless in all circuits containing no capacity, since they are necessarily fairly small and still further suppressed by the inductance of the circuit. They may become serious and even dangerous, however, if capacity is present in the circuit, as the current taken by capacity is proportional to the frequency, and

even small voltage harmonics, if of very high order, that is, high .

frequency, produce very large currents, and these in turn may cause dangerous voltages in inductive devices connected in series into the circuit, such as current transformers, or cause resonance effects in transformers, etc. With the increasing extent of very high-voltage transmission, introducing capacity into the systems, it thus becomes increasingly important to keep the very high harmonics practically out of the voltage wave. ;

Incidentally it follows herefrom, that the specifications of wave shape, that it should be within 5 per cent. of a sine wave, which is still occasionally met, has become irrational: a third harmonic of 5 per cent. is practically negligible, while a thirty-fifth harmonic of 5 per cent., in the voltage wave, would hardly be permissible. This makes it necessary in wave-shape specifications, to discriminate _ against high harmonics. One way would be, to specify not the wave shape of the voltage, but that of the current taken by a ; small condenser connected across the voltage. In the condenser current, the voltage harmonics are multiplied by their order. That is, the third harmonic is increased three times, the fifth harmonic five times, the thirty-fifth harmonic 35 times, etc. However, this probably overemphasizes the high harmonics, gives them too much weight, and a better way appears to be, to specify the current wave taken by a small condenser having a specified amount of non-inductive resistance in series.

. Thus for instance, if s = 1000 ohms = capacity reactance of .

the condenser, at fundamental frequency, r = 100 ohms = re-

122 ELECTRIC CIRCUITS . , sistance in series to the condenser, the impedance of this circuit, for the n™ harmonic, would be . £ 1000 . Z,=1—-j7 =100-——j (7) ; or, absolute, the impedance, zn = 10004 iB + 0.01 (8) and, the admittance, 0.001 n ES 9 y 1 + 0.01 n? ©) and therefore, the multiplying factor, . Yn 1.005 n = 28 eee 10 f yo 614+ 0.01 n? (10) this gives, for n f n f

• 1 1.0 13 8.0 3 2.9 15 8.4 5 4.5 25 9.3 7 5.8 35 9.6 . 9 6.7 45 9.8 11 7.4 00 10.0 Thus, with this proportion of resistance and capacity, the maxi- mum intensification is tenfold, for very high harmonics. By using a different value of the resistance, it can be made anything desired. A convenient way of judging on the joint effect of all harmonics of a voltage wave is by comparing the current taken by such a condenser and resistance, with that taken by the same condenser and resistance, at a sine wave of impressed voltage, of the same effective value. Thus, if the voltage wave e = 600 + 183 + 12, + 97 + 45 + 211 + 313 + 3003 + 245. = 600 { 1 + 0.03; + 0.02, + 0.015, + 0.0067,+ 0.00331:

• 0.00513 + 0.0525 + 0.0425 } °

SHAPING OF WAVES 123 (where the indices indicate the order of the harmonics) of effect- ive value

6 = VOOFIS TIFT O PEPE TOT BOT oe

= 601.7 . is impressed upon the condenser resistance of the admittance, y,,

• the current wave is 4 = 0.603 { 1 + 0.087; + 0.09, + 0.0877 + 0.0445, + 0.02471:

• 0.04:3 + 0.4623 + 0.375 }

= 0.603 X 1.173

= 0.707 while with a sine wave of voltage, of eo = 601.7, the current would be

to = 0.599, giving a ratio 1 i, 1.18, or 18 per cent. increase of current due to wave-shape distortion by higher harmonics.

64. While usually the sine wave is satisfactory for the purpose. for which alternating currents are used, there are numerous cases where waves of different shape are desirable, or even necessary for accomplishing the desired purpose. In other cases, by the internal reactions of apparatus, such as magnetic saturation, a wave-shape distortion may occur and requires consideration to avoid harmful results.

Thus in the regulating pole converter (so-called “split-pole converter’’) variations of the direct-current voltage are produced at constant alternating-current voltage input, by superposing a third harmonic produced by the field flux distribution, as discussed under “Regulating Pole Converter” in “Theory and Calcula- tion of Electrical Apparatus.”? In this case, the third harmonic must be restricted to the local or converter circuit by proper transformer connections: either three-phase connection of the converter, or Y or double-delta connections of the transformers with a six-phase converter. .

The appearance of a wave-shape distortion by the third har- monic and its multiples, in the neutral voltage of Y-connected

transformers, and its intensifications by capacity in the secondary

124 ELECTRIC CIRCUITS

circuit, and elimination by delta connection, has been discussed

in Chapter XXV of “Theory and Calculation of Alternating- current Phenomena.”’

In the flickering of incandescent lamps, and the steadiness of arc lamps at low frequencies, a difference exists between the flat- top wave of current with steep zero, and the peaked wave with flat zero, the latter showing appreciable flickering already ata - somewhat higher frequency, as is to be expected.

In general, where special wave shapes are desirable, they are usually produced locally, and not by the generator design, as — with the increasing consolidation of all electric power supply in

, large generating stations, it becomes less permissible to produce a desired wave shape within the generator, as this is called upon to supply power for all purposes, and therefore the sine wave as_- the standard is preferable.

One of the most frequent causes of very pronounced wave- shape distortion, and therefore a very convenient means of pro- ducing certain characteristic deviations from sine shape, is mag- __ netic saturation, and as instance of a typical wave-shape distor- tion, its causes and effects, this will be more fully discussed in the following.