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For general use it has been found best to make the two losses approximately equal; but for special cases, where constancy of voltage is important, or where transformers operate with considerable load most of the time, the copper loss may be reduced at the sacrifice of increased iron loss. If, however, constant voltage is not important, and a transformer is likely to run at light loads a great part of the time, then the iron loss should be diminished at the expense of the copper loss. Ordinarily these differences require transformers to be specially designed for the purpose; but variations in frequency, voltage, or other conditions will alter the relation between the two losses. The hysteresis loss in watts, W., is : W = n Vf B1.6.

(70)

In this expression, V is the volume of the iron core in cubic centimeters, f is the frequency in cycles per second, B is the maximum Aux density in lines per square centimeter, and n is a constant depending upon the quality of the iron. For high-grade annealed sheet iron suitable for transformer and armature cores, the value of n is usually between 2 x 10-10 and 3 x 10-10. In calculations where the exact value is not known, an average value of 2.5 x 10-19 may be assumed. The ordinary values of B, the maximum Aux density, for various sizes of transformer and frequencies are given on page 157.

Ageing of Transformer Iron. About 1894 attention was called to the fact that the iron cores of transformers became changed after being used for some time, the hysteresis loss increasing considerably. Investigations showed that the core loss of some commercial types of transformers rose to two or more times the initial value after a few months' operation. It was found that this was due to heat, the same effect being produced by heating the iron in any other way. This phenomenon depends upon the mechanical and chemical character of the iron, but the exact effect of the different impurities has been found to be difficult to determine. By experience and the exercise of great care, manufacturers have been able to avoid this increase of hysteresis loss, so that transformers made at present have very little more core loss after long periods of use. Professor W. E. Goldsborough has given * the results of tests made on several of the most prominent types of transformer, and these show that the core loss remained practically constant for 800 hours at full load. These tests were made in a room where the temperature was about 25° C., and the full load was applied for ten and a half hours a day. This is fully as severe as ordinary practical service ; nevertheless, the temperature of the cores did not exceed 75° C., allowing the standard rise of 50° C. Up to this point it is found that the ageing effect does not occur ; but above 80° it begins, and at 100° it becomes considerable, increasing with the temperature up to about 200°, beyond which it falls again.

* Paper before National Electrical Light Association, May, 1899.

The conclusion is, that transformer cores should not be run, even for a short time, at temperatures exceeding 80° C. At full load the allowable rise is 50° C.; hence the room temperature must not be greater than 30° C., or 86° F. On the other hand, in very hot weather, or in an engine-room where the temperature may be 10 or 20 degrees higher than this, transformers should not be run at full-load, or may be operated for shorter periods of time, so that they do not attain maximum temperature. This matter is sufficiently important to demand attention, and every one installing or using transformers should guard against the possibility of the core temperature rising above the point at which ageing begins. This limit may be ascertained from the makers. The eddy or Foucault current loss in the core in watts, W., is We = b V f? t? Bạ.

(71)

In this expression t is the thickness of the laminations in centimeters; V, f, and B have the same significance as in (70), and b is a constant depending upon the specific resistance of the iron. In the iron ordinarily used the value of b is about 1.6 x 10-11.

In a paper read before the American Institute of Electrical Engineers, May, 1900, Mr. Fitzhugh Townsend claims that the eddy current loss is proportional to B16 instead of B?.

This equation assumes that the sheets of iron forming the core are properly insulated from each other, otherwise the loss is greater, because the eddy currents flow from one sheet to another as if the core were a solid mass of iron.

Since the eddy current loss in (71) varies as to, the square of the thickness of the iron plates, it may be reduced to a very small value by making them very thin. On the other hand, the insula

tion and unavoidable space between the plates is about 2 mils, so that in practice a thickness of 10 to 15 mils is generally adopted in order that the proportion of iron in the total volume of the core shall be high. Assuming an ordinary thickness of 12 mils for the plates and a distance between them of 2 mils, the actual iron

12 in the core is 7019 = .86 of the total volume. With these relations the eddy current loss constitutes about 20 per cent of the core loss, the hysteresis loss making up the remaining 80 per cent. In some cases the thickness of plates is reduced to 7 or 8 mils, so that the eddy current loss is only 8 or 10 per cent of the core loss; but this tends to increase the volume of the core as well as the weight of copper, and involves more labor in construction, hence the final gain is doubtful.

The permissible rise in temperature being 50° C., the resistance of the iron core increases about 20 per cent after a long run at full load, hence the eddy current loss is reduced in the proportion 120: 100, or in other words it is about 17 per cent less. But the eddy current loss is only about 20 per cent of the iron loss, so that the latter is reduced 3 or 4 per cent at full working temperature, hysteresis being constant.

Flux Densities in Transformer Cores. The hysteresis loss varies as B1.6 in (70) and the eddy current loss as B’ in (71) or B16, in which B is the flux density, consequently the latter is kept at a low value in transformers in order that the core loss shall be small. Different designers adopt various densities, average figures being given in the following table.

ORDINARY FLUX DENSITIES IN TRANSFORMER CORES.

MAXIMUM LINES PER SQUARE CENTIMETER.

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The density is decreased with higher frequency in order to keep the iron losses in (70) and (71) nearly the same for a given volume of core. These densities are much lower than those

allowed in the armature cores of generators and motors which are often as high as 15,000 or 16,000 lines per square centimeter. The reason for this is the higher efficiency of 97 or 98 per cent which is expected of transformers compared with 92 to 94 per cent for machines. The former often operate for long periods at very light load, while generators usually have one-half to full load while they are running, consequently the constant core loss is a more serious matter in transformers.

Exciting Current. When the secondary circuit of a transformer is open and the primary circuit is closed, a certain current flows in the latter. This is called the exciting current, being also known as the leakage current, open-circuit current, and magnetizing current. It consists of two components, one of which supplies the energy to make up the transformer losses, and the other produces the magnetization of the core. The former represents true power in watts being practically equal to the iron losses, and the latter is apparent power being wattless. The total value of the exciting current depends upon the design and size of the transformer, but is ordinarily about 5 per cent for 1 k. w., and about 2. to 1 per cent in sizes of 25 to 100 k. w. or larger.

The power factor of the exciting current at no load differs considerably in the various types, but is usually about 70 per cent. Since this is equal to the cosine of the angle of lag, it follows that the no-load primary current lags about 45° with respect to the primary impressed E.M.F. It is evident also that the energy component of the current is about equal to the magnetizing component. When a transformer is loaded even slightly, for example, to one-tenth of its normal capacity, the power factor rises to very nearly 100 per cent, and the primary current is practically in phase with the primary impressed E.M.F., provided the load is non-inductive, which is usually the case in electric lighting. If, however, induction motors or other forms of inductive load are present, the power factor will be less than 100 per cent, and the current will lag behind the E.M.F. as in any alternating-current circuit.

The above statements apply to transformers having closed magnetic circuits. In the so-called “hedgehog" type with open magnetic circuit consisting of a simple straight core, the no-load exciting current is about ten times as great, being more than onehalf of the full load value in a 3 k. w. size, with a power factor of

only .063, which is also about one-tenth as much as for closed magnetic circuit.* On account of its low-power factor this large exciting current does not involve directly any greater loss of power in true watts. But it uses up the current capacity of the generators, lines, etc.; the heating effect and drop for a wattless current being the same as for any other current having the same value in amperes. Furthermore, it reacts injuriously upon the regulation of generators and transformers, greatly increasing their drop in volts. For these reasons the closed magnetic circuit has been adopted almost universally. In fact, the greatest care is exercised in making the magnetic circuit as complete as possible, the effect of joints being reduced to a minimum, which also reduces magnetic leakage, as explained on page 150.

To prevent the flow of this exciting current, magnetic cut-outs have been devised to open the primary circuit automatically when the secondary circuit is open. It is objectionable, however, to open and close the primary (high voltage) circuit whenever the load is thrown off or on, so that this arrangement is seldom used in practice. Another plan is to open the primary lines at the station during the hours that the current is not required. This is customary in smaller systems, and is possible on certain circuits of large systems, but it cannot be applied generally in important plants.

Efficiency of Transformers. The efficiency of a transformer is the ratio of the watts output W, measured at the secondary terminals to the watts input measured at the primary terminals W. Since the losses occurring in a transformer are: W, the copper loss from (69), W, the hysteresis loss from (70), and W. the eddy current loss from (71), it follows that the output is equal to the input minus these losses, hence

W. W.- (W. + Wo+ W.)
Efficiency = w

(72)

W The efficiency is very high for transformers made by the best manufacturers, being about 98 per cent at full load for sizes of 25 k. w. or larger, and 94 or 95 per cent at one-quarter load. This is higher than the efficiency of almost any other practical apparatus, nevertheless it is found that the aggregate losses are large, and form a heavy item in the cost of alternating-current supply, because

* The Alternate Current Transformer by J. A. Fleming, London, 1896, p. 567.

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