LECTURE V. SINGLE-ENERGY TRANSIENT OF IRONCLAD CIRCUIT. 22. Usually in electric circuits, current, voltage, the magnetic field and the dielectric field are proportional to each other, and the transient thus is a simple exponential, if resulting from one form of stored energy, as discussed in the preceding lectures. This, how- ever, is no longer the case if the magnetic field contains iron or other magnetic materials, or if the dielectric field reaches densities beyond the dielectric strength of the carrier of the field, etc. ; and the proportionality between current or voltage and their respective fields, the magnetic and the dielectric, thus ceases, or, as it may be expressed, the inductance L is not constant, but varies with the current, or the capacity is not constant, but varies with the voltage. The most important case is that of the ironclad magnetic cir- cuit, as it exists in one of the most important electrical apparatus, the alternating-current transformer. If the iron magnetic circuit contains an air gap of sufficient length, the magnetizing force con- sumed in the iron, below magnetic saturation, is small compared with that consumed in the air gap, and the magnetic flux, therefore, is proportional to the current up to the values where magnetic saturation begins. Below saturation values of current, the tran- sient thus is the simple exponential discussed before. If the magnetic circuit is closed entirely by iron, the magnetic flux is not proportional to the current, and the inductance thus not constant, but varies over the entire range of currents, following the permeability curve of the iron. Furthermore, the transient due to a decrease of the stored magnetic energy differs in shape and in value from that due to an increase of magnetic energy, since the rising and decreasing magnetization curves differ, as shown by the hysteresis cycle. Since no satisfactory mathematical expression has yet been found for the cyclic curve of hysteresis, a mathematical calcula- tion is not feasible,, but the transient has to be calculated by an '^''"r '*_/ ? :,": \ : 52 SINGLE-ENERGY TRANSIENT OF IRONCLAD CIRCUIT. 53 approximate step-by-step method, as illustrated for the starting transient of an alternating-current transformer in "Transient Elec- tric Phenomena and Oscillations," Section I, Chapter XII. Such methods are very cumbersome and applicable only to numerical instances. An approximate calculation, giving an idea of the shape of the transient of the ironclad magnetic circuit, can be made by neglect- ing the difference between the rising and decreasing magnetic characteristic, and using the approximation of the magnetic char- acteristic given by Frohlich's formula: which is usually represented in the form given by Kennelly: T/> p = - = a + crOC; (2) that is, the reluctivity is a linear function of the field intensity. It gives a fair approximation for higher magnetic densities. This formula is based on the fairly rational assumption that the permeability of the iron is proportional to its remaining magnetiza- bility. That is, the magnetic-flux density (B consists of a compo- nent 3C, the field intensity, which is the flux density in space, and a component (B' = (B — 3C, which is the additional flux density carried by the iron. (B' is frequently called the " metallic-flux density." With increasing 3C, (B' reaches a finite limiting value, which in iron is about &x' = 20,000 lines per cm2. * At any density (B', the remaining magnetizability then is (B^' — (B', and, assuming the (metallic) permeability as proportional hereto, gives and, substituting gives a, = cftco'rc^ * See "On the Law of Hysteresis," Part II, A.I.E.E. Transactions, 1892, page 621. 54 ELECTRIC DISCHARGES, WAVES AND IMPULSES. or, substituting 1_ 1 *** / t*« ,—fc / (/ • gives equation (1). For OC = 0 in equation (1), ^ = - ; for 3C = oo » = - ; that is, uv a: cr in equation (1), - = initial permeability, - = saturation value of Oi (7 magnetic density. If the magnetic circuit contains an air gap, the reluctance of the iron part is given by equation (2), that of the air part is constant, and the total reluctance thus is p = ft + ffK, where 3 = a plus the reluctance of the air gap. Equation (1), therefore, remains applicable, except that the value of a is in- creased. In addition to the metallic flux given by equation (1), a greater or smaller part of the flux always passes through the air or through space in general, and then has constant permeance, that is, is given by 23. In general, the flux in an ironclad magnetic circuit can, therefore, be represented as function of the current by an expression of the form where , . = & is that part of the flux which passes through 1 -f- ut the iron and whatever air space may be in series with the iron, and a is the part of the flux passing through nonmagnetic material. Denoting now L2 = nc 10-8, i where n = number of turns of the electric circuit, which is inter- linked with the magnetic circuit, L2 is the inductance of the air part of the magnetic circuit, LI the (virtual) initial inductance, that is, inductance at very small currents, of the iron part of the mag- SINGLE-ENERGY TRANSIENT OF IRONCLAD CIRCUIT. 55 netic circuit, and =- the saturation value of the flux in the iron. 72,CJ>' d That is, for i = 0, — r- = Z/i ; and for i = oo , <£' = T . i 0 If r = resistance, the duration of the component of the transient resulting from the air flux would be _ L2 nc 10~8 *V-7" T~ and the duration of the transient which would result from the initial inductance of the iron flux would be The differential equation of the transient is: induced voltage plus resistance drop equal zero ; that is, Substituting (3) and differentiating gives na 10~8 di . .,_ a di ' . (i+Wdi + ncl0rSdt+ and, substituting (5) and (6), t(l + bi)2 Z5 d* ' hence, separating the variables, Tidi + Tidi + dt = Q The first term is integrated by resolving into partial fractions 1 1 6 6 i(l + 6i)2 " i 1 + 6i (1 + 6i)2>. and the integration of differential equation (7) then gives If then, for the time t = tQ, the current is i = i0, these values substituted in (8) give the integration constant C: T1log- + !T2logio + T- + ^o + C = 0, (9) 56 ELECTRIC DISCHARGES, WAVES AND IMPULSES. and, subtracting (8) from (9), gives 1 + 6i 5 ' (10) This equation is so complex in i that it is not possible to cal- culate from the different values of t the corresponding values of i; but inversely, for different values of i the corresponding values of t can be calculated, and the corresponding values of i and t, derived in this manner, can be plotted as a curve, which gives the single-energy transient of the ironclad magnetic circuit. Tra sient o Ironclad Inductive Circuit : t=2.92- i + t-.6i j l+.6i (dotted: t = 1.0851g i— .50?) 2 3 4 5 Fig. 29. 6 seconds Such is done in Fig. 29, for the values of the constants a = 4 X 105, c = 4 X 104, b = .6, n = 300. SINGLE-ENERGY TRANSIENT OF IRONCLAD CIRCUIT. 57 58 ELECTRIC DISCHARGES, WAVES AND IMPULSES. This gives T = 4 Assuming i0 = 10 amperes for t0 = 0, gives from (10) the equa- tion : 4 T = 2.92 - 1 9.21 log10 ^ + . 921 .6 i Herein, the logarithms have been reduced to the base 10 by division with logwe = .4343. For comparison is shown, in dotted line, in Fig. 29, the transient of a circuit containing no iron, and of such constants as to give about the same duration: t = 1.0S5log™i- .507. As seen, in the ironclad transient the current curve is very much steeper in the range of high currents, where magnetic sat- uration is reached, but the current is slower in the range of medium magnetic densities. Thus, in ironclad transients very high-current values of short duration may occur, and such transients, as those of the starting current of alternating-current transformers, may therefore be of serious importance by their excessive current values. An oscillogram of the voltage and current waves in an 11,000-kw. high-voltage 60-cycle three-phase transformer, when switching onto the generating station near the most unfavorable point of the wave, is reproduced in Fig. 30. As seen, an excessive current rush persists for a number of cycles, causing a distortion of the volt- age wave, and the current waves remain unsymmetrical for many cycles.