Theory and Calculation of Electric Circuits — part 7 of 15

CHAPTER VIII SHAPING OF WAVES BY MAGNETIC SATURATION 65. The wave shapes of current or voltage produced by a closed magnetic circuit at moderate magnetic densities, such as are com- monly used in transformers and other induction apparatus, have COCCI | COR CCE Ea Lp et PP A ed | PEEP YLT ee if tA LT AT TY tern | tp ft A rl SEARCHER PI es TTA TTT ET HA 4 {tt Z a eet SE Pt Lae TT eee oe 4eneee eee ee CO Cee TO eee PTT TT Tt et tt a Le et Be eee ee TTT IACI TT Ve — ~Lpber i ttt i tt Fia. 55. been discussed in “Theory and Calculation of Alternating-cur- rent Phenomena.” . The characteristic of the wave-shape distortion by magnetic 125

126 ELECTRIC CIRCUITS saturation in a closed magnetic circuit is the production of a high peak and flat zero, of the current with a sine wave of impressed voltage, of the voltage with a sine wave of current traversing the | circuit. . PTT TT TION EEE EE EE EE TT FECA CNN ee ee PT TT YLT TIN Nie EJ ae cones LI PT iyi TT TT NT AL | en seent0 | TY Prt7] Pret TIN IN TT Terr] PIAZTA TTT TEN ALT TAA Vi YT TTT TTT tE NN TTT Ye PATA TT ETT Tt tt SALT TA YZ VIAL TT ETT Ett tt iS | VTA 7 . Pa LT TT TE TT TTT TT ENT Fie. 56. In Fig. 55 are shown four magnetic cycles, corresponding re- spectively to beginning saturation: B = 15.4 kilolines per cm.?, H = 10; moderate saturation: B = 17.4, H = 20; high saturation: Pit tT TT eAAN ET ETE TE ET TT 7 aa ZAIN TT eee EVER Nee Se | | jel LAT | | TAT AT TT TTT TET Pty me INC IN TT TP er (4H P| PRS A ETT ZL TA ff ofa LEG /| gaan |_| HARE |- oa Pi ttt tT tT tT ttt tT AAT VTA TT ZEEE EEN aas , Fia. 57. B = 19.0, H = 50; and very high saturation: B = 19.7, H = 100. Figs. 56, 57, 58 and 59 show the four corresponding current waves I, at a sine wave of impressed voltage e, and therefore sine wave of magnetic flux, B (neglecting ir drop in the winding, or rather, é is the voltage induced by the alternat- ing magnetic flux density B). In these four figures, the maxi-

SHAPING OF WAVES BY MAGNETIC SATURATION 127 mum values of ¢, B and I are chosen of the same scale, for wave- shape comparison, though in reality, in Fig. 59, very high sat- uration, the maximum of current, J, is ten times as high asin Fig. ; 56, beginning saturation. As seen, in Fig. 56 the current is the usual saw-tooth wave of transformer-exciting current, but slightly peaked, while in Fig. 59 a high peak exists. The numerical values are given in Table I. PT TT tT tT tT ry EET TET TT SY RE | | ERE EH pm fp pV SHH Se eee L4+Ety yp rat | AL IN TTT Le HEY ELS Psst A LY aE 7 TIT TT ET EE TE EE TE NE AEE ZT By to NA eo rT LL TT tT TT ET EE TE RAL TI Fia. 58. HHP Pet PT TTT t ty POCA Net eas CECA ECNE is 2 OO sett tcf |S pray tp ert TKN TTT ee LL A | Tt | ee seer T OY Bary SAE ry jf. |} PET TENE AE IZ Y | A_| —|-+-t +t +t} + A C7] FT TT TTT TTT TT TE | KAAS TL Fra. 59. That is, at beginning saturation, the maximum value of the saw- | tooth wave of current differs little from what it would be with a sine wave of the same effective value, being only 4 per cent. higher. At moderate saturation, however, the current peak is already 42 per cent. higher than in a sine wave of the same effective |

128 ELECTRIC CIRCUITS value, and becomes 132 per cent. higher than in a sine wave, at the very high saturation of Fig. 59. Inversely, while the maximum values of current at the higher Tass I . | Begin- Ea High ee : ‘ ning sat- | ate sat- | satura- | satura- . ; Boisa | Beizd | Bois. | Ba18.7 ‘_ Bine wave of voltage, és, ae 3.08 3.48| 3.80 3.94 Maximum value of current, J...........| 10.00 | 20.00 | 50.00 |100.00 Effective value of current, X~/2:i1.....] 9.6 | 14.1 | 29.8 | 43.0 Form factor of current wave £ seeeeeeee] 1.04] 1.42 1.68] 2.32 Ratio of effective currents..............| 1.00] 1.47] 3.11] 4.48 EE ENG i PE TTT TTT TAT TENT MELE TT PTL TT TT TAL EE EE TN EEE PET TT TETAS EEL IN AEE TE P| eet SNA TTT NAT | pte le tgp Saannyns prmre tT NLL Sanne eyelet Cee SEER ae eee te NTT NOTE rT ELT T ttt ttt TIN LAA4 Fig. 60. ° Pitti tt hi tert te tt TE ET TT TT | fh FG. -ABGRHE / LE HEEB LL . PET TT TAA ET A TY EE ETT | Hitt At Pt I TP Et : P| et ey AP Se ET TE ATT ee | eT | | | dT VTL | PPA EP: | Pt FET TA pee mee AWE TET COCA és 28 CONC : See Kee iT vIerT TT Tt ttt tt NS Le Fia. 61. .

SHAPING OF WAVES BY MAGNETIC SATURATION 129 saturations are two, five and ten times the maximum current value at beginning saturation, the effective values are only 1.47, 3.1 and 4.47 times higher. Thus, with increasing magnetic satura- tion, the effective value of current rises much less than the maxi- mum value, and when calculating the exciting current of a satu- rated magnetic circuit, as an overexcited transformer, from the magnetic characteristic derived by direct current, under the as- PTT tt td Et tt Et TT PTT TTT TT ET TTT Tf | PET TT Tae eT TTT Pt Tt tT tthe TEE tT tT tT ee See aceeNe Oe | Pt tT tT ETAL ET TE NEAT PT TT TURAN TA pj det YR JA | Pet ey Ne NY rit tT tT YET TEENA Gg Pt Tt ATT TT EE te TG | Pt tt ea tt PE ET ET AS PT TY IAL [ ] exmonso {| IR PEPEE TT] tren CTS PET ET TT TT ee es ene || TT | PET TT Tt | fee ssemene TT PET TTT yet et ey TT Pt tT tT eT tT te tT tt . PLETE TTT TT TT tt Tt Fria. 62, , sumption of a sine wave, the calculated exciting current may be more than twice as large as the actual exciting current.

66. Figs. 60 to 63 show, for asine wave of current, J, traversing a closed magnetic circuit, and the same four magnetic cycles given in Fig. 55, the waves of magnetic flux density, B, of induced vol- tage, e, the sine wave of voltage, ¢, which would be induced if the

9

130 ELECTRIC CIRCUITS

magnetic density, B, were a sine wave of the same maximum value, and Fig. 63 also shows the equivalent sine wave, e:, of the (distorted) induced voltage wave, e.

As seen, already at beginning saturation, Fig. 60, the voltage peak is more than twice as high as it would be with a sine wave, tree] fy |

| HE] HET TE EE Td tT | | Se Barreo | tt COCCHI tala COO PTT ET erates |] | | See eee Pt tt dt tte ET EE EE EE TT eee Pee | | Pt tT TE EE EEE EE Pt tT tt tT ET EE EE -— On Pees i} ee eee ptt TEU LT pe tt - 7 eee laAncenNee ee Bee: SCN S24 Pees pC he LA | Zt TINE EN NIT ZL VILL REE PET TIA TE ET NET MAN PEt Ty TTT tT EN Fia. 63. and rises at higher saturations to enormous values: 18.5 times the sine wave value in Fig. 63.

The magnetic flux wave, B, becomes more and more flat-topped with increasing saturation, and finally practically rectangular, in Fig. 63.

The curves 60 to 63 are drawn with the same maximum values

SHAPING OF WAVES BY MAGNETIC SATURATION 131 of current, J, flux density, B, and sine wave voltage, eo, for better comparison of their wave shapes.

The numerical values are:

’ TaBie II ‘ . Begin- | Moder-| High /|Veryhigh ning sat-| ate sat- | satura- | satura- | Eases Sine wave of current, J, maximum.......|10.0 {20.0 {50.0 {100.0 . Flat-top wave of magnetic density, B, Mmaximum................000 cece eee | 15.4 17.4 19.0 {19.7 Peaked voHage wave e, maximum.......| 7.4 {18.8 {35.5 |73.0

• Ration... cee cece eee eee eee se] 1.00.) 2.56 | 4.80 | 9.88 Sine wave of voltage, eo, maximum, for same maximum flux.................| 3.08 | 3.48 | 3.80 | 3.94

Ratio... 0... cece eee eee eeeees | 1.00 1.138 1.23 1.28 Form factor of voltage wave, =: vesseee.{ 2.40 | 5.40 | 9.35 | 18.50 Equivalent sine wave of voltage, e,, maxi-

• MUM........ 0.02 eee eee ee eee cess ee| 3,95 | 6.383 | 9.58 | 13.80

Ratio...........0..0.s0eeeeeeeeees-| 1.00 | 1.60 | 2.42 | 3.50

& (maxima).....0.22- eee eee 1.282| 1.864! 2.520| 3.500 < (maxima)........................| 1.87 | 2.97 | 3.70 | 5.28

As seen, the wave-shape distortion due to magnetic saturation is very much greater with a sine wave of current traversing the closed magnetic circuit, than it is with a sine wave of voltage im- pressed upon it.

With increasing magnetic saturation, with a sine wave of cur- rent, the effective value of induced voltage increases much more rapidly than the magnetic flux increases, and the maximum value of voltage increases still much more rapidly. than the effective

| value: an increase of flux density, B, by 28 per cent., from begin- ning to very high saturation, gives an increase of the effective value of induced voltage (as measured by voltmeter) by 250 per . cent., or 3.5 times, and an increase of the peak value of voltage (which makes itself felt by disruption of insulation, by danger to life, etc.) by 888 per cent., or nearly ten times.

At very high saturation, the voltage wave practically becomes one single extremely high and very narrow voltage peak, which occurs at the reversal of current.

132 _ ELECTRIC CIRCUITS

‘ ° At the very high saturation, Fig. 63, the effective value, e:, of the voltage is 3.5 times as high as it would be with a sine wave of magnetic flux; the maximum value, e, is more than five times as high as it would be with a sine wave of the same effective value, e}, that is, more than five times as high, as would be expected from the voltmeter reading, and it is 18.5 times as high as it would be with a sine wave of magnetic flux.

Thus, an oversaturated closed magnetic circuit reactance, which consumes é = 50 volts with a sine wave of voltage, é&, and thus of magnetic density, B, would, at the same maximum mag- netic density, that is, the same saturation, with a sine wave of current—as would be the case if the reactance is connected in ser- ies in a constant-current circuit—give an effective value of ter- minal voltage of e: = 3.5 X 50 = 175 volts, and a maximum peak voltage of e = 18.8 X 50 X +/2 = 1330 volts.

Thus, while supposed to be a low-voltage ‘reactance, é) = 50 volts, and even the voltmeter shows a voltage of only e: = 175, which, while much higher, is still within the limit that does not endanger life, the actual peak voltage e = 1330 is beyond the danger limit.

‘ Thus, magnetic saturation may in supposedly low-voltage cir- cuits produce dangerously high-voltage peaks. .

A transformer, at open secondary circuit, is a closed magnetic circuit reactance, and in a transformer connected in series into a circuit—such as a current transformer, etc.—at open secondary circuit unexpectedly high voltages may appear by magnetic. saturation. .

67. From the preceding, it follows that the relation of alternat- ing current to alternating voltage, that is, the reactance of a closed magnetic circuit, within the range of magnetic saturation, is not constant, but varies not only with the magnetic density, B, but for the same magnetic density B, the reactance may have very differ- ent values, depending on the conditions of.the circuit: whether constant potential, that is, a sine wave of voltage impressed upon the reactance; or constant current, that is, a sine wave of current traversing the circuit; or any intermediate condition, such as brought about by the insertion of various amounts of resistance, or of reactance or capacity, in series to the closed magnetic cir- cuit reactance.

The numerical values in Table III illustrate this.

I gives the magnetic field intensity, and thus the direct current,

SHAPING OF WAVES BY MAGNETIC SATURATION 133 which produces the magnetic density, B—that is, the B-H . curve of the magnetic material. An alternating current of maxi- mum value, J, thus gives an alternating magnetic flux of maxi- mum flux density B. If I and B, were both sine waves, that is, if PT TT TTT Ty tt tt te Pt wT TT TP ET EE al CECE EE EERE PT der TT ETT tT a 4 eee | Vij PvE TTT tet tt tT t PATE EEE ET ett TTT PP EE AE EE HH ig Het ett Se EaEn PTT yer tT TT ft TT Py TTA TT toa Py YT TTT tTyT tt et yt PI7ZLT TTT Tt ey te tt et Cece CECA Plaerrrri tty ttt tt 7A TTT EP tet tt et tt tt . FETT tae bad TT TT TT Pb kb bb & bh % to io yo ho fo wo Fig. 64, during the cycle current and magnetic flux were proportional to each other, as in an unsaturated open magnetic circuit, eo, as given in the third column, would be the maximum value of the induced voltage, and 2) = ; the reactance. This reactance varies with

134 ELECTRIC CIRCUITS the density, and greatly decreases with increasing magnetic satu- . ration, as well known. However, if e9 and thus B are sine waves, I can not be a sine wave, but is distorted as shown in Figs. 56 to 59, and the effective . ° value of the current, that is, the current as it would be read by an alternating ammeter, multiplied by +/2 (that is, the maximum value of the equivalent sine waves of exciting current) is given as . 4; The reactance is then found as z, = e This is the reactance Say ae Pt TET ETT TE YE | Pt ttt et tT Yi tt PLETE TE ITA TT PET TT TT YT Tt tel pi tt tt yt tt tt py PT TTY yyy et Saree Aa eee ee PL tT Yett tT ttt tl Pt itt tet tT tT pivAt tt | Tt | 7 | VI | net eee TT | | | hee to Zo Fya. 65. . of the closed magnetic circuit on constant potential, that is, on a sine wave of impressed voltage, and, as seen, is larger than 2». If, however, the current, J, which traverses the reactance, is a sine wave, then the flux density, B, and the induced voltage are not . sines, but are distorted as in Figs. 60 to 63, and the effective value of the induced voltage (that is, the voltage as read by alternating voltmeter), multiplied by 4/2 (that is; the maximum of the equiva- lent sine wave of voltage) is given as e; in Table III, and the true maximum value of the induced voltage wave is e. ;

SHAPING OF WAVES BY MAGNETIC SATURATION 135 The reactance, as derived by voltmeter and ammeter readings : under these conditions. that is, on a constant-current circuit, or with a sine wave of current traversing the magnetic circuit, is te = a thus larger than the constant-potential reactance, 2». Much larger still is the reactance derived from the actual maxi- mum values of voltage and current: 2, = r SENSE ~ EEE TRAE Efe eee PT tT tT TNE EEE ET Tt | PL TT TIN: TEE TE TT TT PTT Ty tT ALEELET TEE fp Nd PTT TET EINE EEE TT et tT TT TT TN feet PE SERRE REENSNEEE SERRE ONNEeE PT Ee TT TT TUL NN AE TL FEET eee SERRA PLT tT] Pritt et A, FET e ie tbh Fia. 66. . Itis interesting to note that z,,, the peak reactance, is approxi- mately constant, that is, does not decrease with increasing mag- . Netic saturation. (The higher value at beginning saturation, for I = 20, may possibly be due to an inaccuracy in the hysteresis eycle of Fig. 55, a too great steepness near the zero value, rather than being actual.) It is interesting to realize, that when measuring the reactance of a closed magnetic circuit reactor by voltmeter and ammeter readings, it is not permissible to vary the voltage by series resist- ance, as this would give values indefinite between x, and 2,, de-

136 ELECTRIC CIRCUITS

pending on the relative amount of resistance. To get z,, the generated supply voltage of a constant-potential source must be varied; to get x., the current in a constant-current circuit must be varied. As seen, the differences may amount to several hun- dred per cent.

As graphical illustration, Fig. 64 shows: ;

As curve I the magnetic characteristic, as derived with direct current.

Curve II the volt-ampere characteristic of the closed circuit reactance, J, é9, as it would be if J and B, that is, eo, both were sine

waves.

Curve III the volt-ampere characteristic on constant-potential alternating supply, 71, é.

Curve IV the volt-ampere characteristic on constant-current_ alternating supply, as derived by voltmeter and ammeter, I, é1, and as

’ Curve V the volt-ampere characteristic on constant-current alternating supply, as given by the peak values of J and e.

Fig. 65 gives the same curves in reduced scale, so as to show V completely.

Fig. 66 then shows, with B as abscissw, the values of the react- ANCEB TM, Lp, Le, ANd Lm.

Tasie III

I B cry nF | oy 3] a 4 e = P E

2.0| 7.30 fee betas cocheckecectes: 1.00

3.0/10.00]...../0.6670]...... corcberesr ice ceeeee{ 1,09

4.0] 11.60|...../0.5750|......[.cc eee ceeeeee cece eeafeeeeeeteeeeea] 187

5.0] 12.50].....)0.5000|......[....ccefeccceeleceeese eeeeeclieesen| 1646

7.5|14.30/...../0.3810]......[...0..[..0c eee | 1,08

10.0 | 15.40 |3.08 |0.3080| 9.0 | 0.342 | 3.95 | 0.305 | 7.4 | 0.74 | 2.37 | 2.40 15.0] 16.70 |.....[0.2280|......)s.cccec[seces[eceeeceleceeesfeceeee] 8.27 20.0|17.40|3.48|0.1740] 14.1 | 0.247 | 6.33 | 0.316 | 18.8 | 0.94 | 4.20 | 5.40 80.0] 18.30].....]0.1220]......[....c.clecceecfeceesesfeeeeesfeeeess| 6.00 40.0|18.70].....[0.0080|......[...cccefecececfeceeeeefeceeesfeceeee| 7685 50.0] 19.00 | 3.80 |0.0760] 29.8 | 0.127 | 9.68 | 0.912 | 35.5 | 0.71 | 9.60 | 9.35 76.0) 19.35 ].....)0.0520 |... pec ec cele w eee de eee e ee leseeaafeceses| 14.10 100.0 | 19.70 | 3.94 | 0.0304] 43.0 | 0.002 |13.80 | 0.138 | 73.0 | 0.73 | 18.50 [18.5

• 125.0/19.85].....[0.0820 |... bocce el cece sleccecealasseneleneeee] 22.80 150.0 |19.96 }.....)0,0270 2.0.00). .seesteceeescereespeereestesees | 27,00

68. Another way of looking at the phenomenon is this: while with increasing current traversing a closed magnetic circuit, the magnetic flux density is limited by saturation, the induced voltage

SHAPING OF WAVES BY MAGNETIC SATURATION 137 peak is not limited by saturation, as it occurs at the current rever- sal, but it is proportional to the rate of change of the magnetic flux density at the current reversal, and thus approximately pro- portional to the current.

Thus, approximately, within the range of magnetic saturation, with increasing current traversing the closed magnetic circuit (like that of a series transformer):

The magnetic flux density, and therefore the mean value of in- duced voltage remains constant;

The peak value of induced voltage increases proportional to the current, and therefore; -

The effective value of induced voltage increases proportional to the square root of the current.

Thus, if the exciting current of a series transformer is 5 per cent. of full-load current, and the secondary circuit is opened, while the primary current remains the same, the effective voltage consumed by the transformer increases approximately ~/20 = 4.47 times,

. and the maximum voltage peak 20 times above the full-load voltage of the transformer.

. As the shape of the magnetic flux density and voltage waves are determined by the current and flux relation of the hysteresis cy- cles, and the latter are entirely empirical and can not be expressed mathematically, therefore it is not possible to derive an exact mathematical equation for these distorted and peaked voltage waves from their origin. Nevertheless, especially at higher satu- tation, where the voltage peaks are more pronounced, the equa- tion of the voltage wave can be derived and represented by a Fourier series with a fair degree of accuracy. By thus deriving the Fourier series which represents the peaked voltage waves, the harmonics which make up the wave, and their approximate val- ues can be determined and therefrom their probable effect on the system, as resonance phenomena, etc., estimated.

The characteristic of the voltage-wave distortion due to mag- netic saturation in a closed magnetic circuit traversed by a sine wave of current is, that the entire voltage wave practically con- tracts into a single high peak, at, or rather shortly after, the mo- ment of current reversal, as shown in Figs. 63, 62, etc.

With the same maximum value of magnetic density, B, and thus of flux, &, the area of the induced voltage wave, and thus the mean value of the voltage, is the same, whatever may be the wave of magnetism and thus of voltage, since = f edt, and the area of

138 ELECTRIC CIRCUITS ; - the peaked voltage wave of the saturated magnetic circuit, e, thus is the same as that of a sine wave of voltage, é&. Neglecting then . the small values of voltage, e, outside of the voltage peak, if this voltage peak of ¢ is p times the maximum value of the sine wave, éo, its width is ; of that of the sine wave, and if the sine wave of voltage, €, is represented by the equation €o COS p (11) the peak of the distorted voltage wave is represented, in first ap- proximation, by assuming it as of sinusoidal shape, by Peo COS DH (12) That is, the distorted voltage wave, e, can be considered as represented by peo cos p@ within the angle T wv

• 2p <¢< 2p (13) and by zero outside of this range.

The value of p follows, approximately, from the consideration that the peak reactance, z,, is independent of the saturation, or constant, since it depends on the rate of change of magnetism with current near the zero value, where there is no saturation, and the ratio a thus (approximately) constant.

Or, in other words, if below saturation, in the range where the magnetic permeability is a maximum, the current, 7, produces the magnetic flux, , and thereby induces the voltage, e’, the reactance is

1? v= = (14)

This is the maximum reactance, below saturation, of the mag- netic circuit, and can be calculated from the dimensions and the magnetic characteristic, in the usual manner, by assuming sine waves of 7 and B.

The peak reactance, 2, of the saturated magnetic circuit is ap- . proximately equal to 2’, and thus can be calculated with reason- able approximation, from the dimensions of the magnetic circuit and the magnetic characteristic.

If now, in the range of magnetic saturation, a sine wave of cur-

SHAPING OF WAVES BY MAGNETIC SATURATION 139 rent, of maximum value J, traverses the closed magnetic circuit, the peak value of the (distorted) induced voltage is

• € = Ink (15) where ' moval (16) is the maximum reactance of the magnetic circuit below satura- tion, derived by the assumption of sine waves, e’ and 7. If B is the maximum value of the magnetic density produced by - the sine wave of current of maximum value, J, and, é, the maxi- mum value of the sine wave of voltage induced by a sinusoidal variation of the magnetic density, B, the ‘form factor’ of the peaked voltage wave of the saturated magnetic circuit is € — tut P= en (17) - thus determined, approximately. As illustrations are given, in the second last column of Table ITI, the form factors, p, calculated in this manner, and in the last column are given the actual form factors, po, derived from the ‘curves 60 to 63. As seen, the agreement is well within the un- certainty of observation of the shape of the hysteresis cycles, except perhaps at J = 20, and there probably the calculated value is more nearly correct.

69. The peaked voltage wave induced by the saturated closed magnetic circuit can, by assuming it as symmetrical and counting the time from the center of the peak, be represented by the Fourier series. € = a, cos ¢ + a3c0s 3 ¢ + a, cos 5 d+ a,cos7¢+... (18) = Zan COS nd where x. ‘ a, = f e cos nod (19) Jo = 2 avg(e cos ng)3 (20) The slight asymmetry of the peak would introduce some sine terms, which might be evaluated, but are of such small values as to be negligible. (a) For the lower harmonics, where n is small compared to p,

140 ELECTRIC CIRCUITS cos n@ is practically constant and = J] during the short voltage peak ¢ = peo cos p¢, and it is, therefore, a, = 2 avg(e)? . = 2avg(peo cos pp)e 2 5? = 5, AVE (Peo cos po)o 4 = 2 9 AVE COS = — eo. Tv (b) For the harmonic, where n = 9, it is | dy = 2avepeo cost pd) 2 z . = 5 2VE (peo cos* pd)o = 2 e9 avg cos® = é.

. (c) For still higher harmonics than n =p, cos n¢@ assumes negative values within the range of the voltage peak, and a, thereby rapidly decreases, finally becomes zero and then negative, at n = 3 p, positive again at n = 5 p, etc., but is practically negligible.

Thus, the coefficients of the Fourier series decrease gradually, with increasing order, n, from * €& aS maximum, to é) for n = p, and then with in- creasing rapidity fall off to negligible values. Their exact values can easily be derived by substituting (12) ‘ into (19), 2p a, = f peo cos p¢ cos n@d@ (21) here the integration is extended to op only, as beyond this, the voltage, ¢, is not given by equation (12) any more, but is zero. (21) integrates by a, = 22e0 /sin(p + n)o , sin(p ~ n)o /2 "ef ptn p—n /o oF n _ 3 n _ 2 |fa + 4) final! -3) , | ; 7 pt+n p-n

SHAPING OF WAVES BY MAGNETIC SATURATION 141 : _ n _ 7" mm... but since sin (1+ 5) = sin (1 — 5), itis . _ 3 n 4 eosin 5(1 - 4) an = (22) r(1 - =) and _ 8 n sins(1 — — 4e 2 ( *) 6 ae co ns (23) ; 1 7 as the equations of the voltage wave distorted by magnetic saturation.

70. These coefficients, a,, are very easily calculated, and as in- stances are given in Table IV, the coefficients of the distorted voltage wave of Fig. 62, which has the form factor p = 9.35.

‘ TasBLp IV . ¥ n sin=(1 — — p = 9.35 a, = 22° ving - 5) ?) n 17 pi

n=l 3 5 7 9 11 13 15 17 19 o = 1,270 |1.242)1.188)1.114) 1.018, 0.906 0.7860. 65810. 528 0.406

0 n =21 23 | 25 | 27 | 29 a | 3 = = 0.292 |0.189 o10}0.01 0.00 —0.060 —0.082)

0

As seen, after n = 9, the values of a, rapidly decrease, and be- come negative, though of negligible value, after n = 27.

In Fig. 67 the successive values of n are shown as curve.

In reality, the peaked voltage wave of magnetic saturation, as_- shown in Figs. 61 to 63, is not half a sine wave, but is rounded off at the ends, toward the zero values. Physically, the meaning of the successive harmonics is, that they raise the peak and cut off the values outside of the peak. It is the high harmonics, which sharpen the edge of the peak, and the rounded edge of the peak in the actual wave thus means that the highest harmonies, which give very small or negative values of a,, are lower than given by equations (23), or rather are absent.

142 ELECTRIC CIRCUITS ; Thus, by omitting the highest harmonics, the wave is rounded off and brought nearer to its actual shape. Thus, instead of fol- lowing the curve, a,, as calculated and given in Fig. 67, we cut it off before the zero value of a,, about at n= 23, and follow the curve line, a’,, which is drawn so that zt = 9.35, that is, that the voltage peak has the actual value. Pa EE EE EE EE TE ET TT | APNCCECEERE Et af Ne SE 7c BENS Cann Ht tt EE} jo SNE

• TE ETT TEX ETT TTT Ta | : (| -R XY | ef | TET TEIN; EE TT Pt tT T_T TTT ERM EEE TE fal AANA | aj Sy et PEE TEP PED TRE pr : EET EEE EET TELE AREER | heb tbh he 8 | Fia. 67. ; The equation of the peaked voltage in Fig. 62 then becomes €@ = € {1.270 cos ¢ + 1.242 cos 3d + 1.188 cos 5d + 1.114 cos7¢

• 1.018 cos 9¢ + 0.906 cos 11¢ + 0.786 cos 13¢ + 0.658 cos 15
• 0.529 cos 17¢ + 0.400 cos 19¢ + 0.240 cos 21¢}. Or, in symbolic writing, 6 = @9{1.270, + 1.2423 + 1.188, + 1.114, + 1.018, + 0.90611
• 0.78613 + 0.65815 + 0.52917 + 0.40019 + 0.24021}

SHAPING OF WAVES BY MAGNETIC SATURATION 143

= 1.270 e {11 + 0.9783 + 0.953, + 0.8777 + 0.800, + 0.71311

• 0.61713 + 0.51715 + 0.41617 + 0.31519 + 0.1892:}.

It is of interest to note how extended a series of powerful har- monics is produced. It is easily seen that in the presence of ca- pacity, these large and very high harmonics may be of consider- ; able danger. In any reactance, which is intended for use in series to a high-voltage circuit, the use of a closed magnetic circuit thus constitutes a possible menace from excessive voltage peaks if saturation occurs.

Pt ttt ta ttt TE EET TE Et PTET TT eee tT ET TTT TE TE TT TT Pt tt eta eT eT ee EE TE TE ETT TE Pt tT tt th EE EE EEE TE Tt ET TT Pt tT tT iE thE TET EE ET TE TT +H -HIRVAA-FAG RR || HACCP REA AA RY Pt tt tT ae EE ET IN; A EE BRREE?! SSSR ERNE PT TET YT TTT tT TN AN ATH TT TT PTT TIA TE ET TE TE EVAR TE EE TT CCEOAEC CEC EECA AE PTT VIET TT TET TAN ET TV Rive ELT ttt tt PUN LT Fig. 68. ;

71. Such high-voltage peaks by magnetic saturation in a closed magnetic circuit traversed by a sine wave of current can occur . only if the available supply voltage is sufficiently high. If the total supply voltage of the circuit is less than the voltage peak pro- duced by magnetic saturation, obviously this voltage peak must be reduced to a value below the voltage available in the supply circuit, and in this case simply the current wave can not remain a sine, but is flattened at the zero values, and with it the wave of magnetic density. |

Thus, if in Fig. 62 the maximum supply voltage is ZH = 19.0,

the maximum peak voltage can not rise to e = 35.5, but stops at

144 ELECTRIC CIRCUITS eS E, and when this value is reached, the rate of change of flux: density, B, and thus of current, J, decreases, as shown in Fig. 68, in drawn lines. In dotted lines are added the curves correspond- ing to unlimited supply voltage. The voltage peak is thereby reduced, correspondingly broadened, and retarded, and the cur- ; rent is flattened at and after its zero value, the more, the lower the maximum supply voltage.

The reactance is reduced hereby also, from x, = 0.192, in Fig. 62, to x, = 0.140.

In other words, if p is the form factor of the distorted voltage wave, which would, with unlimited supply voltage, be induced by the saturated magnetic circuit of maximum density, B, and é is the maximum value of the sine wave of voltage, which a sinu- soidal flux of maximum density, B, would induce, the distorted

voltage peak is . € = pe (24) | and the maximum value of the equivalent sine wave of the dis- torted voltage, or the effective voltage read by voltmeter, is : aq = Vpe (25) | If now the maximum voltage peak is cut down to E, by the | limitation of the supply voltage, and 3 = q, the form factor be- ; comes , ,_E_p Ss SS 26 Pe 4 (26) ‘ and the effective value of the distorted voltage, times ~/2 , that is, the maximum of the equivalent sine wave, is ey = Vp'e = Sb " (7) 1 Pp &o V4 ; =V eo, thus varies with the supply voltage, E. The reactance then is al, = O32 (28) I vq Thus, fore = 35.5, E = 19.0, it is q = 1.87,

SHAPING OF WAVES BY MAGNETIC SATURATION 145 and as & = 3.80; p = 9.35, it is p= F = 5.0, : ey = ou = 9.58 = 7.0 Vq 1.7 ’ . a= v7 = A = 1.40,

These values, however, are only fair approximations, as they are based on the assumption of sinusoidal shape of the rt | ft tt ttt td oe ‘In the preceding, the ey |} Hh . assumption has been made, that the magnetic flux passes Pt tT tT tT | tT tt Ty

Agni p entirely within the closed|_# 4 4 $ » 2 ub Bis magnetic circuit, that is, that Pt | feb Tt oy Ty there is no magnetic leakage flux, or flux which closes SERRE through non-magnetic space ; ; outside of the iron conduit. Pt tet Ey

If there is a magnetic leak- F | | tt tt LAY age flux—and there must rT i | | | tt VU always be some—it somewhat reduces the voltage peak, the AL more, the greater is the pro- Yi, A portion of the leakage flux to BERRA the main flux. The leakage rt | tL Wt tL flux, in open magnetic cireuit, Ho AL is practically proportional to the current, and that part of af -\ Ab the voltage, which is induced A by the leakage flux, therefore, I | it | ee

is a sine wave, with a sine Ae fiv[ | wave of current, hence does not Fia. 69. contribute to the voltage peak. .

Such high magnetic saturation peaks occur only in a closed magnetic circuit. If the magnetic circuit is not closed, but con- tains an air-gap, even a very small one, the voltage peak, with a sine wave of current, is very greatly reduced, since in the air-gap magnetic flux and magnetizing current are proportional.

10

146 ELECTRIC CIRCUITS Thus, below saturation and even at. beginning saturation, an air-gap in the magnetic circuit, of one-hundredth of its length, makes the voltage wave practically a sine wave, with a sine wave of current, as discussed in ‘““Theory and Calculation of Alternating- current Phenomena.” Pt tT tT det Td EE EE TT TT Pt tT tT THT TE TdT rE TE TE TT PT tt tT tie EEE TE EE ET Hf Pt ttt tt | | i tet tT EE EE pt tt ft diel TT EE ET Tt Pt | tt iE TE EE EE EE ET : Pt tt | dE EE P| bbe pei HEH EE EEE ET | ae HAE == HEH At-4 4-H-RR-- Pitt TET EE TE EE Pt ttt EE PET PP EE Pree Pt tT | tT UE TALE ENTE TT pa Rb A CTE NE | CB SEIEISSECTT LENE ECC SSSA Ft tT IAT ET EE TT INU FLT I7IE TTT Ett tt is Fia. 70. The enormous reduction of the voltage peak by an air-gap of 1 per cent. of the length of the magnetic circuit is shown in Figs. 69 and 70. , In Fig. 69, with the magnetic flux density, B, as abscisse, the

|

SHAPING OF WAVES BY MAGNETIC SATURATION 147 m.m.f. of the iron part of the magnetic circuit is shown as curve I. This would be the magnetizing current if the magnetic circuit . were closed. Curve II show the m.m.f. consumed in an air-gap of 1 per cent. of the length of the magnetic circuit of curve I, and curve III, therefore, shows the total m.m.f. of the magnetizing current of the magnetic circuit with 1 per cent. air-gap.

Choosing as instance the very high saturation B = 19.7, the

same as illustrated in Fig. 63, and neglecting the hysteresis— which is permissible, as the hysteresis does not much contribute to the wave-shape distortion—the corresponding voltage waves are plotted in Fig. 70, in the same scale as Figs. 56 to 63: for a sine wave of current, curves Fig. 69 give the corresponding values ; of magnetic flux, and from the magnetic flux wave is derived, as the voltage wave. The waves of magnetism are not plotted. ; é is the sine wave of voltage, which would be induced by a sinu- soidal variation of magnetic flux; ¢ is the peaked voltage wave induced in a closed magnetic circuit of the same maximum values of magnetism, of form factor p = 18.5 (the same as Fig. 63), and é; is the voltage wave induced in a magnetic circuit having an air-gap of 1 per cent. of its length. As seen, the excessive peak of e has vanished, and e, has a moderate peak only, of form - factor p = 1.9.

Even a much smaller air-gap has a pronounced effect in reducing the voltage peak. Thus curves IV and V show the m.m.fs. of the air-gap and of the total magnetic circuit, respectively, when containing an air-gap of one-thousandth of the length of magnetic circuit. ¢, in Fig. 70 then shows the voltage wave corresponding to V in Fig. 69: of form factorp = 7.4. _

Thus, while excessive voltage peaks are produced in a highly saturated closed magnetic circuit, even an extremely small air- gap, such as given by some butt-joints, materially reduces the peak: from form factor p = 18.5 to 7.4 at one-thousandth gap length, and with an air-gap of 1 per cent. length, only a moderate peakedness remains at the highest saturation, while at lower - saturation the voltage wave is practically a sine.

73. Even a small air-gap in the magnetic circuit of a reactor greatly reduces the wave-shape distortion, that is, makes the voltage wave more sinusoidal, and euts off the saturation peak. The latter, however, is the case only with a complete air-gap. A partial air-gap or bridged gap, while it makes the wave shape

148 ELECTRIC CIRCUITS more sinusoidal elsewhere, does not reduce but greatly increases the voltage peak, and produces excessive peaks even below satura- tion, with a sine wave of current, and such bridged gaps are, there- fore, objectionable with series reactors in high-voltage circuits. In shunt reactors, or reactors having a constant sine wave of im- pressed voltage, the bridged gap merely produces a short flat zero of the current wave, thus is harmless, and for these purposes the bridged gap reactance—shown diagrammatically in Fig. 71 : —is extensively used, due to its constructive advantages: greater PT TdT ttt TET TL yy PTET TT ttt te TTT YY Tt Pt ttt tT Ty tt TT TAA TE Se e=- (eee Cee Ltt iererit td | | | AA TET TT yeah Pi Tt TT yA | Tt : PT TT tt tt vee TT VT PTT TT tt ya tye tet tT PT tt tt am TAT TT TV TT fii li] MA TT TT TA TT Pi TT WAAL fo ey LIL ITWA TT TY tt yt tA TT PIMA Tt EY ee HAA HH A AY | er eee res Fig. 71.- rigidity or structure and, therefore, absence of noise, and reduced magnetic stray fields and eddy-current losses resulting therefrom. Assuming that one-tenth of the gap is bridged, and that the length of the gap is one one-hundredth that of the entire mag- netic circuit, as shown diagrammatically in Fig. 71. With such a bridged gap, with all but the lowest m.m.fs. the narrow iron bridges of the gap are saturated, thus carry the flux density S + H, where S = metallic saturation density, = 20 kilolines per cm.’ in these figures, and H the magnetizing force in the gap. For one-tenth of the gap, the flux density thus is H + S, for the other nine-tenths, it is H, and the average flux density in the gap thus is

SHAPING OF WAVES BY MAGNETIC SATURATION 149 B=H+018S=H+2, or, if p = bridged fraction of gap, B=H+ pS. .

Curve II in Fig. 71 shows, with the average flux density, B, as

abscisse, the m.m.f. required by the gap, H=B-018 =B-2, while curve I shows the m.m.f. which an unbridged gap would require.

Adding to the ordinates of II the values of the m.m.f. required for the iron part of the magnetic circuit, or the other 99 per cent., gives as curve III the total m.m.f. of the reactor.

The lower part of curve III is once more shown, with five times the abscisse B, and 1000, 100 and 10 times, respectively, the

ordinates H, as III,. IIIs, IIIs.

74. From B = 2 upward, curve III is practically a straight line, and plotting herefrom for a sine wave of current, J and thus m.m.f.,H, the wave of magnetism, B, and of voltage, e, these curves become within this range similar to a sine wave as shown as B and e in Fig. 72. Below B = 2, however, the slope of the B-H curve and with this their wave shapes change enormously. The B wave becomes practically vertical, that is, B abruptly reverses, and corresponding thereto, the voltage abruptly rises to an ex- cessive peak value, that is, a high and very narrow voltage peak appears on top of the otherwise approximately sine-shaped voltage wave, é.

Choosing the same value as in Fig. 60, B = 15.4 or beginning saturation, as the maximum value of flux density: at this, in an entirely closed magnetic circuit the voltage peak is still moderate. On the B-H curve III of Fig. 71, the fiux density, B = 15.4, requires the m.m.f., H = 14.4 If then Band H would vary sinusoidally, giving a sine wave of voltage, éo, the average value of this voltage wave, é, would be proportional to the average rate of magnetic change, or to Z = “ = 1.07, and the maximum

. value of the sine wave of voltage would be 5 as high, or, xB 1.07 x o"“oH 2 = 1.68, |

, 150 ELECTRIC CIRCUITS . The maximum value of the actual voltage curve, e, occurs at the moment where B passes through zero, and is, from curve III,, ; = [2] = 2%. e=([z],- 5 = 580. a This, then, is the peak voltage of the actual wave, while, if it were a sine wave, with the same maximum magnetic flux, the maxi- mum voltage would be ey) = 1.68.

The voltage peak produced by the bridged gap and the form

factor thus is e _ 580 P= 1.68 ~ 34%

that is, 345 times higher than it would be with a sine wave.

: Obviously, such peak can hardly ever occur, as it is usually

. beyond the limit of the available supply voltage. It thus means,

that during the very short moment of time, when during the current reversal the flux density in the iron bridge of the gap changes from saturation to saturation in the reverse direction, a voltage peak rises up to the limits of voltage given by the sup- ply system. This peak is so narrow that even the oscillograph usually does not completely show it.

However, such practically unlimited peaks occur only in a perfectly closed magnetic circuit, containing a bridged gap. If, in addition to the bridged gap of 1 per cent., an unbridged gap of 0.1 per cent.—such as one or several butt-joints—is present,

giving the B-H curve IV of Fig. 71, the voltage peak is greatly . reduced. It is aB_-w 154 = 39H ~ 2 15.95 ~ 1) B 1000 ‘= lal.- 100 ~ 1% hence, the relative voltage peak, or form factor, e p= e = 6.6.

That is, by this additional gap of one one-thousandth of the length of magnetic circuit, the peak voltage is reduced from 345 times that of the sine wave, to only 6.6 times, or to less than 2 per cent. of its previous value.

As seen from the reasoning in paragraph and Fig. 67, the