The values of the hysteresis losses in good soft iron, in microwatts per cubic centimetre for a frequency of 50 alternations per second, for various values of the maximum induction, are given in the following table :

63. Effect of Shape of P.D. Curve on Iron Losses in Transformer Cores.-We have seen that, as a rule, we cannot in general expect the wave-form of the P.D. applied between the primary terminals of a transformer to be a true sine curve. It may, in fact, have almost any shape of a periodic nature. It becomes a matter of interest, therefore, to inquire how the iron losses in transformer cores are affected by the shape of the P.D. curve. We proceed to attack the problem from a theoretical point of view.

Assuming that the iron losses are independent of the load on the transformer, we may proceed on the supposition that the secondary circuit is open. Let v be the instantaneous counter E.M.F. in the primary circuit, A the cross-sectional area of the core, N the number of primary turns, b the instantaneous value of the induction, and S the area of the v curve taken over half a period.

We then have, neglecting leakage—

v = ANdb

dt

Integrating this over half a period, we get

(31)

where B is the maximum value of the induction, and T is the periodic time.

If we assume that the hysteresis loss W1 is given by (Steinmetz's formula)—

W1 = kn Ba
W1

where a and k are constants, and n is the frequency, we get, by substitution

Again, the eddy-current loss is given by (see p. 102)—

×(mean square value of primary counter E.M.F.)

The total iron losses are therefore given by

W = W1+ W2

= kn

S

a

m

on (24N) (mean sq.value of primary counter E.M.F.)

+

A2N2

Now, on open secondary circuit, the primary P.D. is almost

exactly equal to the primary counter E.M.F., but opposite in phase; we may therefore write

If, then, the P.D. between the primary terminals, as measured by a hot wire or electrostatic voltmeter, is kept constant, the total iron loss is given by equation (34), whatever be the shape of the

wave.

The problem of finding the area of the wave which makes the iron losses a minimum is the same as that of making the area of the P.D. wave as small as possible while the R.M.S. potential' difference, V, remains constant. We have, therefore, to make—

Vdt a minimum

while

SV2dt

= a constant

(35)

From this we deduce that Vdt has no absolute minimum, but may be made as small as we please by making the maximum value of V correspondingly large and the P.D. curve narrow and peaky.

It is to be noticed that the hysteresis losses only are affected by the wave form of the primary P.D. as the eddy current losses are constant when the P.D. is constant.

It is not advisable to attempt to reduce the hysteresis loss too much in the manner here indicated, as, by so doing, the maximum P.D. becomes very great, and the insulation of the transformer must be correspondingly increased, a course which would produce a very costly appliance.

These theoretical results have been amply proved by experimental investigation, and we would draw the attention of our readers to the writings of Mr. Steinmetz,1 Dr. Fleming,2 Messrs. Beeton, Taylor, and Barr, Dr. Roessler, Mr. Evershed,5 and Mr. Feldman, on the subject.

1 Electrician, August 24, 1894, vol. xxxiii. p. 498.

2 Ibid., June 28, 1895, vol. xxxv. p. 304; January 10, 1896, vol. xxxvi.

3 Ibid., June 21, 1895, vol. xxxv. p. 257; June 28, 1895, vol. xxxv. p. 286; November 8, 1895, vol. xxxvi. p. 61.

▲ Ibid., November 22, 1895, vol. xxxvi. p. 124; December 6, 1895, vol. xxxvi. p. 222.

5 Ibid., March 27, 1891, vol. xxvi. p. 635.
6 Ibid., October 18, 1895, vol. xxxv. p. 809.

Perhaps the experimental investigation of greatest interest is that of Messrs. Beeton, Taylor, and Barr, on account of an ingenious method of obtaining various wave forms from a single alternator by the use of a piece of apparatus called an injector, by means of which resistance, self-induction, or capacity can be thrown in the circuit at any instant during the period at will, producing wave forms of the P.D. which could only have been obtained otherwise by a series of dissimilar alternators.

The conclusions arrived at experimentally are

1. That if the R.M.S. value of the applied P.D. is constant, and the area of the P.D. wave is constant, then, whatever be the shape of this wave, the total iron loss cannot vary.

2. That if the R.M.S. value of the applied P.D. is constant, but the area of the P.D. wave varies, then, whatever be the shape of the wave, the total iron loss will vary by an amount which is only dependent upon the area of the P.D. wave.

These results are in complete agreement with the preceding theory.

TESTS OF TRANSFORMERS.

64. Tests on Ferranti standard transformers furnish the following results :

1. Show that, if the primary and secondary coils of a transformer are so wound that there is no leakage, the equivalent self-induction of the primary circuit with closed secondary is given by

where L1 and L2 are the self-inductions of the primary and secondary coils respectively, r, is the total resistance of the secondary circuit, and p = 2′′n, n being the frequency, the secondary external circuit being non-inductive.

2. Assuming the result given in Question 1, show that in a transformer having no leakage, and non-inductive external secondary circuit, the primary current lags behind the applied by P.D. an angle ø given by

pL1r22

tan p = r1r2 + p2L2(r1L2 + r2L1)

where r1 is the resistance of the primary coil.

3. "Show that, in a transformer having no magnetic leakage, the impedance of the primary on closed non-inductive secondary circuit is greater than, equal to, or less than its impedance on open secondary, according as 2rr, is greater than, equal to, or less than pM, where r, is the resistance of the primary coil, r2 the total resistance of the secondary circuit, M the mutual induction of the primary and secondary coils, and p = 2rn, n being the frequency.