## Hanna Curves For Silicon Steel

ence in inductance between Y and Y' is 4 per cent, for a difference in air gap of 33 per cent.

An example will show how easy it is to make a reactor according to this method.

Example. Assume a stack of silicon-steel laminations having a cross section Y% in. by Ji in., and with iron filling 92 per cent of the space. The length of the flux path l0 in this core is in. It is desired to know how many turns of wire and what air gap are necessary to produce 70 henrys when 20 ma direct current are flowing in the winding.

This problem is solved as follows:

In Fig. 66 the abscissa corresponding to LP/V = 53 X 10~4 is NI/lc = 25 for silicon steel. The ratio of air gap to core length le/lr is between 0.0005 and 0.001.

The total air gap is nearly 0.001 X IVi or 7.5 mils; the gap at each joint is half of this value, or 3.75 mils.

The conditions underlying Hanna's method of design are met in most applications. In receivers and amplifiers working at low audio levels, the alternating voltage is small and hence the alternating flux is small compared to the steady flux. Even if the alternating voltage is of the same order as the direct voltage, the alternating flux may be small, especially if a large number of turns is necessary to produce the required inductance; for a given core the alternating flux is inversely proportional to the number of turns. D-c resistance of the coil is usually fixed by the regulation or size requirements. Heating seldom affects size.

38. Reactors with Large A-C Flux. With the increasing use of higher voltages, it often happens that the a-c flux is no longer small compared to the d-c flux. This occurs in high-impedance circuits where the direct current has a low value and the alternating voltage has a high value. The inductance increases by an amount depending on the values of a-c and d-c fluxes. Typical increase of inductance is shown in Fig. 68 for a reactor working near the saturation point. Increasing a-c flux soon adds to the saturation, which prevents further inductance

Fig. 68. Increase of inductance with a-c induction.

increase and accounts for the flattening off in Fig. 68. Saturation of this sort may be avoided by limiting the value of the d-c flux.

To illustrate the effect of these latter conditions, suppose that a reactor has already been designed for negligibly small alternating flux and operates as shown by the minor loop with center at G, Fig. 69. Without changing anything else, suppose that the alternating voltage across the reactor is greatly increased, so that the total a-c flux change is from zero to Bm. (Assume that the reactor still operates about point G.) The hysteresis loop, however, becomes the unsymmetrical figure 0BmD'0. The average permeability during the positive flux swing is represented by the line GBm, and during the negative flux swing by OG. The slope of GBm is greater than that of the minor loop; hence, the first effect exhibited by the reactor is an increase of inductance.

The increase of inductance is non-linear, and this has a decided

Fig. 69. Change of permeability with a-c induction.

effect upon the performance of the apparatus. An inductance bridge measuring such a reactor at the higher a-c voltage would show an inductance corresponding to the average slope of lines OG and GBm. That is, the average permeability during a whole cycle is the average of the permeabilities which obtain during the positive and negative increments of induction, and it is represented by the average of the slopes of lines OG and GBm. But if the reactor were put in the filter of a rectifier, the measured ripple would be higher than a calculated value based upon the bridge value of inductance. This occurs because the positive peaks of ripple have less impedance presented to them than do the negative peaks, and hence they create a greater ripple at the load. Suppose, for example, that the ripple output of the rectifier is 500 volts and that this would be attenuated to 10 volts across the load by a linear reactor having a value of inductance corresponding to the average slope of lines OG and GBm. With the reactor working between zero and Bm, suppose that the slope of OG is 5 times that of GBm. The expected average ripple attenuation of 50:1 becomes 16.7:1 for positive flux swings, and 83.3:1 for negative, and the load ripple is or an increase of nearly 2:1 over what would be anticipated from the measured value of inductance.

This non-linearity could be reduced by increasing the air gap somewhat, thereby reducing Hic. Moreover, the average permeability increases, and so docs the inductance. It will be apparent that decreasing Hde further means approaching in value the normal permeability. This can be done only if the maximum flux density is kept low enough to avoid saturation. Conversely, it follows that, if saturation is present in a reactor, it is manifested by a decrease in inductance as the direct current through the winding is increased from zero to full-load value.

In a reactor having high a-c permeability the equivalent length of core Ic/ij. is likely to be small compared to the air gap lg. Hence, it is vitally important to keep the air gap close to its proper value. This is, of course, in marked contrast to reactors not subject to high a-c induction.

If a choke is to be checked to see that no saturation effects are ¡¡resent, access must be had to an inductance bridge. With the proper values of alternating voltage across the reactor, measurements of inductance can be made with various values of direct current through it.

If the inductance remains nearly constant up to normal direct current, no saturation is present, and the reactor is suitable for the purpose. If, on the other hand, the inductance drops considerably from zero direct current to normal direct current, the reactor very probably is non-linear. Increasing the air gap may improve it; otherwise, it should be discarded in favor of a reactor which has been correctly designed for the purpose.

Filter reactors subject to the most alternating voltage for a given direct voltage are those used in choke-input filters of single-phase rectifiers. The inductance of this type of reactor influences the following:

Value of ripple in rectified output.

Transient voltage dip when load is suddenly applied, as in keyed loads.

Peak current through tubes during each cycle.

Transient current through rectifier tubes when voltage is first applied to rectifier.

It is important that the inductance be the right value. Several of these effects can be improved by the use of swinging or tuned reactors. In a swinging reactor, saturation is present at full load; therefore the inductance is lower at full load than at no load. The higher inductance at no load is available for the purpose of decreasing voltage regulation. The same result is obtained by shunt-tuning the reactor, but here the inductance should be constant from no load to full load to preserve the tuned condition.

In swinging reactors, all or part of the core is purposely allowed to saturate at the higher values of direct current to obtain high inductance at low values of direct current. They are characterized by smaller gaps, more turns, and larger size than reactors with constant inductance ratings. Sometimes two parallel gaps are used, the smaller of which saturates at full direct current. When the function of the reactor is to control current by means of large inductance changes, no air gap is used. Design of such reactors is discussed in Chapter 9.

The insulation of a reactor depends on the type of rectifier and how it is used in the circuit. Three-phase rectifiers, with their low ripple voltage, do not require the turn and layer insulation that single-phase rectifiers do. If the reactor is placed in the ground side of the circuit one terminal requires little or no insulation to ground, but the other terminal may operate at a high voltage to ground. In single-phase rectifiers the peak voltage across the reactor is Ed0, so the equivalent rms voltage on the insulation is 0.707Edc. But for figuring Bmax the rms voltage is 0.707 X 0.67Eic. Reactor voltages are discussed in Chapter 4.

39. Linear Reactor Design. A method of design for linear reactors is based on three assumptions which are justified in the foregoing:

(a) The air gap is large compared to lc/p, f- being the d-c permeability.

(b) A-c flux density depends on alternating voltage and frequency.

(c) A-c and d-c fluxes can be added or subtracted arithmetically.

From (a) the relation B = n/l becomes B = //. Because of fringing of flux around the gap, an average of 0.855 crosses over the gap. Hence Bic = 0.4:TNIdc/0-85le. With lg in inches this becomes

The sum of Bac and Bj,. is Bm.iX, which should not exceed 11,000 gauss for 4% silicon steel, 16,000 gauss for grain-oriented steel, or 10,000 gauss for a 50% nickel alloy. Curves are obtainable from steel manufacturers which give incremental permeability ma for various combinations of these two fluxes. Figure 70 shows values for 4% silicon steel.

By definition, inductance is the flux linkages per ampere or, in cgs units,

Bdc = 0.6NIdc/lg gauss

Transposing equation 34

4>N BacAcN

If this is substituted in equation 37

3A9N2AC X 10~8

henry s

provided that dimensions are in inches. The term Ac in equation 38 is greater than in equation 36 because of the space factor of the laminations; if the gap is large Ac is greater still because the flux across it fringes. With large gaps, inductance is nearly independent of ha, at least with moderate values of Bm.,x. With small gaps, permeability

Fig. 70. Incremental permeability for 4% silicon steel with high a-c induction.

largely controls. There is always a certain amount of gap even with punchings stacked alternately in groups of 1. Table IX gives the approximate gap equivalent of various degrees of interleaving laminations for magnetic path lc of 5.5 in.

Table IX. Equivalent Gaps with Interleaved Laminations

0.014-in. Laminations Equivalent Air Cap in Inches Alternately Stacked (Total) with Careful Stacking

In groups of 1 0.0005

In groups of 4 0.001

In groups of 8 0.002

In groups of 12 0.003

In groups of 16 0.004

### Butt stacking with zero gap 0.005

Example. An input reactor is required for the filter of a 1,300-volt, 34-amp, single-phase, full-wave, 60-cycle rectifier. Let N = 2,800 turns, net Ac = 2.48 sq in., gross Ac = 2.76 sq in., lc = 9 in., le = 0.050 in. The 120-cycle voltage for figuring Bac is 0.707 X 0.67 X 1,300 = 605 volts.

Figure 70 shows

40. Linear Reactor Chart. In the preceding section, it was assumed that the core air gap is large compared to lcfn, where h is the d-c permeability. In grain-oriented steel cores the air gap may be large compared to lc/na, because of the high incremental permeability of these cores. When this is true, variations in h do not affect the total effective magnetic path length or the inductance to substantial degree. Reactor properties may then be taken from Fig. 71. In order to keep the reactor linear, it is necessary to limit the flux density. For grain-oriented silicon-steel cores, inductance is usually linear within 10 per cent if the d-c component of flux Bdc is limited to 12,000 gauss and the a-c component Bac to 3,000 gauss.

Dotted lines in quadrant I are plots of turns vs. core area for a given wire size and for low-voltage coils, where insulation and margins are governed largely by mechanical considerations. Core numbers in Fig. 71 have the same dimensions and weight as in Table VIII.

If the cores increased in each dimension by exactly the same amount, the lines in quadrant I would be straight. In an actual line of cores, several factors cause the lines to be wavy:

(a) Ratios of core window height to window width and core area deviate from constancy.

(b) Coil margins increase stepwise.

(c) Insulation thickness increases stepwise.

A-c flux density in the core may be calculated by equation 36, and Bdc by equation 35. If Bm materially exceeds 15,000 gauss, saturation is reached, and the reactor may become non-linear or noisy.

Instructions for Using Fig. 71.

1. Estimate core to be used.

2. Divide required inductance by area {Ac) of estimated core to obtain a value of L/sq in.

3. In second quadrant, locate intersection of L/sq in. and rated Iic-

4. On this intersection, read total gap length (lg) and number of turns (N). Gap per leg = lg/2.

5. Project intersection horizontally into first quadrant to intersect vertical line which corresponds to estimated core. This second intersection gives d-c resistance and wire size.

Example. Required: 15 henrys at Idc = 50 ma.

Estimate core No. 1.

L/sq in. = 84.3, lg = 0.015 in., N = 6,000, DCR = 800 ohms.

(Example shown starting with dotted circle.)

A similar chart may be drawn for silicon-steel laminations, but to maintain linearity lower values of flux density should be used.

41. Air-Gap Flux Fringing. In Section 39, equation 38 was developed for inductance of a linear reactor with an air gap. It is assumed that 85 per cent of the core flux is confined to the cross section of core face adjoining the gap. The remaining 15 per cent of the core flux "fringes" or leaves the sides of the core, thus shunting the gap. Fringing flux decreases the total reluctance of the magnetic path and increases the inductance to a value greater than that calculated from equation 38. Fringing flux is a larger percentage of the total for larger gaps. Very large gaps are sometimes broken up into several smaller ones to reduce fringing.

If it is again assumed that the air gap is large compared to lc/fi, the reluctance of the iron can be neglected in comparison with that of the air gap. For a square stack of punchings, the increase of inductance due to fringing is

Equation 39 is plotted in Fig. 72 with core shape VAc/S as abscissas and gap ratio lg/S as parameter.1