The Alternate Current Transformer Vol. 1: The Induction of Electric Currents (1896) — part 34 of 35

The predetermination of the magnetic leakage of a trans- former is a matter of some difficulty, and can only be antici- pated in certain limited cases. We can obtain the relation between the leakage drop, the resistance drop, and the total drop if we assume an approximately simple periodic variation of the electromotive forces, currents and inductions, as follows : —

Let E: be the true resistance of the primary circuit and B2 that of the secondary circuit of the transformer, and let S be the cross-section of the magnetic circuit or core. Let N: be the number of primary turns, N2 the number of secondary turns, and a stand for the ratio of Nj to Na. Let b: be at any instant the induction density in that part of the core enveloped by the primary coil, and b.2 that part enveloped by the secon- dary coil; the difference between these inductions may be called the density of the leakage of induction, and be denoted by b. Hence

&-V-**

In other words, if S is the cross-section of the core, then S b = S &! - S bz and S b represents that part of the induction linked with the primary coil which is not linked with the secondary coil. If <?x is the primary terminal potential diffe- rence at any instant, and e2 that of the secondary terminal at the same instant, and % and i2 the currents at the same moment, then, by fundamental equations, we have

, ..... (146) and 0 = R2?2 + f2 + SN2i^. . . . (147)

Let us write ~ = a and b = b1-ba;

N2

we have by elimination from the fundamental equations the result

^ + ea + E2/2-E1^=SN2^. . . (148) a a at

THE INDUCTION COIL AND TRANSFOEMER. 577

If el varies in a simple periodic manner so that el = Ex sin p t, then, since ez is always opposite in phase and similar in form to vlt we must have

ez = - E2 sin p t.

Moreover, when the transformer is fully loaded, the currents *! and i2 are in step with the electromotive forces el and et, but *2 differs ISOdeg. in phase from i^.

Hence i1 =

We can also write b = - B sin^ t, because the leakage b is deter- mined by, and is in step very nearly with, the secondary current. Hence, by substitution of the above values in the equation (148) we arrive at the equation

?i - E2 - E2 12 - E^A sin p t = - S N2;> B cos pt. a a/

The quantity S N2j9 B cos p t is the instantaneous value of the- potential difference of the secondary circuit lost by leakage — that is to say, it is the measure of the amount by which the secondary terminal potential difference would be increased if there were no leakage. Hence the left-hand side of the above equation represents the same thing. The factors which mul- tiply the sin p t and cos p t respectively in the above equation give, therefore, the maximum value of the "leakage," and there- fore, when divided by /2, represent the mean-square value.

Hence the quantity i -Eg-K^-R or, which

\ a.

comes to the same thing, the quantity

represents the mean-square ( ^/mean2) value of the loss of potential difference of the secondary circuit due to magnetic leakage when the potential difference is measured in volts.

Tjl Tjl T T

The quantities -^i -4. -4r» -j= represent the magnitude of the currents and potentials as read in alternating-current

578 THE INDUCTION COIL AND TRANSFORMER.

ammeters and voltmeters. We may denote these mean-square values by the symbols (Ej), (E2), (Ij), (I2), and the values which these mean-square potential differences and currents have at full and at no .secondary load by the symbols (EJ,, (E2)f, (Ei),,, (E2)0, &c.

The total loss of secondary terminal voltage between full and no secondary load will be given by the difference between the values of the expressions

and {1 (EJ. - (E2)0 - R2 (I2)0 - 1

The value of (I2)0 is, of course, zero.

If the primary terminal potential difference is the same at no load as at full load, we have for the secondary drop due to magnetic leakage the expression

(E2)0 - (E,),- (B, (I2V + 1 B, (I,),- - B, (U) - (149)

a a

The secondary terminal volts at full load being denoted by (E2)/, and that at no load by (E2)0, we see that the quantity (E2)0— (E.,);- represents the total secondary drop due to all causes. The quantity

therefore represents that part of the drop due to the resistance of the primary and secondary circuits.

Hence we have the following rule for determining the drop due to magnetic leakage : — Add together the product of the

secondary resistance and secondary current and - multiplied

a

into the product of primary resistance and primary current, after deducting from the last value the primary current at no load. Subtract this sum from the total observed drop, and the remainder is the secondary potential difference due to

Testing in this way a number of transformers, the author found that where the primary and secondary circuits were

THE INDUCTION COIL AND TRANSFORMER. 579

intermixed, the magnetic-leakage drop was small, but that where the primary and secondary circuits were separated, and in each consisting of one coil only, the magnetic leakage drop was large.

In the diagram in Fig. 202 are shown three curves, by which the three sources of secondary drop have been distinguished, the lines ac, ad showing the curves of drop due respec- tively to the primary and secondary resistance, and the curve « e showing the total drop.

In designing a transformer, it is not permissible to purchase small core loss at the expense of large secondary drop. In a proper specification for a transformer a limitation should be put upon the amount of secondary drop allowed, and it is usual to express it as a percentage of the normal potential difference or voltage of the secondary circuit when the trans- former is unloaded.

It is advantageous to so arrange the winding of the secondary circuit that if the drop is, say, 2 per cent., and the secondary-circuit voltage is 100, that the transformer shall give 101 volts terminal pressure at no load, and 99 volts at full load. In this way the full drop is divided and is not felt so much in working on 100-volt lamps as if the transformer were wound to give 100 volts at no load and 98 at full load.

§ 9. Effect of the Form of the Curve of Primary Electro- motive Force in the Transformer Efficiency and Currents. — It

has generally been assumed by many of those who have written on the subject of the alternate-current transformer that the efficiency, power factor and secondary drop were characteristics of the transformer only. It has already, in previous sections, been suggested that the form of the curve of primary potential difference or primary electromotive force had a considerable •effect in modifying the value of these quantities, and it will now be necessary to examine the matter a little more in detail. We will consider in the first place the effect of the form of primary terminal potential difference upon the form of the current curves and magnitude of the mean-square value of the currents.

The widely -different forms which the primary current of a transformer on open secondary circuit may have is shown in

pp 2

580 THE INDUCTION COIL AND TRANSFORMER.

the diagrams in Figs. 203 and 204. Fig. 203 shows the primary-current curve of a small transformer taken off a Ganz alternator having a peaked curve of electromotive force. The curves in Fig. 204 show the primary electromotive force and primary- current curves of the same transformer taken from a

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£

\

^

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a

^~<

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40 50 60 70 80 90 100 110 120 130 140

Degrees of Phase.

FlO. 203. — Primary Current Curve II and Primary Electromotive Forca Curve I of a Transformer taken off Ganz Alternator.

^4

0 10 20 30 40 50 60 70 SO 90 100

Degrees of Phase.

FIG. 204.— Primary Current Curve II and Primary Electromotive Force Curve I of the same Transformer taken off Wechsler Alternator.

Wechsler alternator. These and the following curves are from an interesting Paper by Dr. G. Koessler.*

• "Das Verhalten von Transformatoren unter den Einflusse von Wechselstromen Verschiedenen Periodischen Verlaufs." A Paper read at the third annual meeting of the Verband Deutscher Electrotechniker, Munich, July 6, 1895. See also The Electrician, Vol. XXXVI., 1895, p. ISO.

THE INDUCTION COIL AND TRANSFORMER 581

Not only do the forms of the current curves differ when taken with different-shaped electromotive force curves, but if the primary electromotive force is kept at the same 'mean- square value, and if the transformer is gradually loaded up, the mean-square values of the primary current corresponding to given secondary currents will differ if the curves of primary elec- tromotive force have different forms. This is shown in Fig. 205, where the ordinates represent the mean-square values of the secondary and primary currents of one and the same trans- former, taken off a Ganz and Wechsler machine respectively.

•

J

/,

X

'/

/

/

/

/n

/

/

^

s ,

.

/

/

'S

/

fT

x

/

/

4

/

/

/

'/

/

x

'

^

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x

/

/

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5

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/

x

x

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x

X

/

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,<-•

?

/

L«^

f

X

/

/

i

A

0-8 1-2 1-6 2-0 2'4 2'8 8"2 Secondary Current Amperes.

3'6 4'0 44

FIG. 205. — Current Diagram of a certain Transformer. Curve I, Primary Current taken with Ganz Alternator. Curve II, Primary Current taken with Wechsler Alternator. Curve III, twice value of Secondary Current. Transformation ratio of Transformer = 2 :1.

It is thus seen that the peaked electromotive force curve gives a primary current with smaller mean-square value than a rounded curve.

The most important fact, however, is that the iron core loss in the transformer, and therefore its efficiency, is sensibly affected by the form of the curve and primary electromotive force. In Fig. 206 are shown two curves, the ordinates of which represent the total power given to a transformer for Certain values of the secondary output represented by the

582 THE INDUCTION COIL AND TRANSFORMER.

abscissae, the transformer being tested in the two cases on the Ganz and Wechsler alternators.

There is between the two power lines a nearly constant difference of ordinate, showing that the cause is to be sought in the difference between the iron core losses in the two cases.

^.

•^

^

^

<;

'^

^

^

^

^>

I

.^

•z-

^

-&

2

^

&

**

^

•^

^

^

^

' 1

100 200 30 4UO 500

Secondary output in Watts.

FIG. 206. — Energy Diagram of Transformer. Curve I, Ganz Alternator. Curve II, Wechsler Alte nator.

100 200

Secondary output in Watts.

FlG. 207. — Efficiency Curve of Transformer. Curve I taken on Ganz Alternator. Curve II taken on Wechsler Alternator.

It follows that both the efficiency curves and power-factor curves of the transformer plotted in terms of the secondary output will differ if the form of the primary electromotive force curve is varied.

In Figs. 207 and 208 are shown the forms of the efficiency and power-factor curves of the same transformer when taken off the Ganz machine with peaked electromotive force curve and the Wechsler machine with a rounded curve.

We find also that the secondary drop is considerably affected by the form of the primary electromotive force curve. In

THE INDUCTION COIL AND TRANSFORMER. 583

Fig. 209 are shown the secondary drop curves of the same transformer taken on the above-mentioned alternators, the primary electromotive force having in each case the same constant mean-square value.

It is clear, therefore, that a peaked electromotive force curve of the type given by the Ganz alternator causes a less iron core loss but a greater magnetic leakage than does a curve of a more rounded form similar to that of the Wechsler machine. Numerous tests and experiments made by the author with the Mordey and Thomson-Houston alternators had established this fact prior to the appearance of the Paper by Dr. G. Roessler,

120 160 200 240 280 320 Secondary output in Watts.

400 460 480

FIG. 208.— Power Factor Curves of a Transformer. Curve I taken on Ganz Alternator. Curve II taken on Wechsler Alternator.

200 300

Secondary output in Wattt.

FIG. 209.— Curve' of Secondary Drop of Transformer. Curve I taken on Ganz Alternator. Curve II taken on Wechsler Alternator.

from which the above transformer diagrams, taken off the Ganz and Wechsler alternators, are copied. It is generally true that a sharp-peaked electromotive force curve gives a less hysteresis loss in the iron than does a rounded or sine electromotive force curve having the same mean -square value.

§ 10. The Form Factor and Amplitude Factor of a Periodic

Curve.— The above differences are closely connected with the magnitude of the form factor of the curve of primary electro- motive force. This quantity is denned as the ratio of the

684 TEE INDUCTION COIL AND TRANSFORMEE.

square root of the mean of the squares of the equispaced ordinates of a curve to the true mean value of the equispaced ordinates. If we denote the first function, viz., the mean- square value, by the letters R.M.S. (root mean square), and the second function by the letters T.M. (true mean), then the form factor of any single-valued periodic curve is defined as follows : —

The form factor = The R.M.S. value of equispaced ordinates = The T.M. value of equispaced ordinates

Take, for instance, in the case of a simple sine curve, the R.M.S. value of the equispaced ordinates is equal to the value of the maximum ordinate divided by ^/2. The T.M. value

of the equispaced ordinates is equal to the value of the maxi-

o mum ordinate multiplied by -.

7T

Since

-^=0-707 and f= 0-637,

the ratio of the R.M.S. value to the T.M. value for a simple

sine curve is

0-707 0-637

1-1.

For several other simple forms of curve the form factor, R.M.S. value, and T.M. value are as below, the maximum ordinate in each case being taken as unity : —

T.M. value of

R.M.S. value

Curve.

ordinate as

of ordinate as

Form factor

fraction of max.

fraction of max.

/.

ordinate.

ordinate.

Sine

0-637

0-707

1-1

0-7854

0-835

1-063

Triangle

0-5

0-58

1-16

Rectangle

1-0

1-0

1-0

0-785

0'816

1-039

Parabola with axis

vertical

0-666

0-730

1-096

Two semi - parabolas

meeting at a cusp. .

0-33

0-447

1-35

The form factor of any curve can easily be obtained geome- trically as follows : On one side of a straight line (see Fig. 210) plot a wave diagram of the curve, and on the other side of the

THE INDUCTION COIL AND TRANSFOKMEB. 585

line plot a polar diagram of the same curve, with its pole on the line of reference. Then, by the proposition on page 193, the radius of the semicircle, so drawn that its area is equal to the area of the polar curve, is the E.M.S. value of the ordinates, and the height of the rectangle described on the base line, so that its area is equal to that of the wave curve, gives the T.M. value of the ordinates. Hence the ratio of the radius of the semi- circle to the height of the rectangle is the form factor of the curve, which is represented by the wave or , polar diagram.

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u

f

1!

0°

9

6

3

"

/X

\

NS

— ,

^

S

/

^

'

FIG. 210.

This form factor is an important quantity in the design of alternators, and by suitably proportioning the width of arma- ture coils and field poles the form factor can be varied within wide limits.

It is evident that for the same R.M.S. value the form factor will be greater if the curve is a sharp-peaked curve than if it is a rounded curve like a semicircle or sine curve.

If some of the ordinates of any curve are increased so as to form a peak, this operation, geometrically considered, increases the R.M.S. value faster than it increases the T.M. value, and so increases the form factor of the curve.

586 THE INDUCTION COIL AND TEANSFORMER.

The amplitude factor of a periodic curve is defined as the ratio between the root mean-square (B.M.S.) value of the ordinates and the value of the maximum ordinate, or

The amplitude) _E.M.S. value of equispaced ordinates _f factor j ™ Value of maximum ordinate

For the same E.M.S. value the amplitude factor is less for a sharp-peaked curve than for a rounded or flat curve.

These two factors — the form factor/ and the "amplitude factory — are important quantities in the case of periodic curves.

§11. General Analytical Theory of the Transformer and Induction Coil. — It remains, then, to indicate the manner in •which the various periodic and fixed quantities concerned in the action of the transformer are connected and how they can be determined.

For convenience we may collect together the symbols employed to represent the various quantities with which we are concerned.

e1 = The value of the primary terminal potential dif- ference or primary E.M.F. at any instant. Ex = The maximum value of the same. me1 = The true mean (T.M.) value of el during the

period.

,Jme^ = Th& root-mean-square (R.M.S.) value of el during the period.

the same quantities for the secondary terminal .- potential difference.

Ej = The resistance of the primary circuit. E2 = The resistance of the secondary circuit. Nj = The number of turns on the primary coil. N2 = The number of turns on the secondary coil. 61==The density of magnetic induction in the core

inside primary coil.

Bj = The maximum value of induction density br Zl = The total induction produced in the core due to

primary coil. 62, B2, Z2 are the same quantities for the secondary circuit.

THE INDUCTION COIL AND TRANSFORMER. 58T-

**! = Primary current at same instant that the primary

terminal potential difference is ev Ij = Maximum value of ir ij = True mean value of il during the period.

Root-mean-square value of ^ during the period. m i2, J^ijz are the same quantities for the secondary

circuit.

/4 = Magnetic force due to the primary current iv Hj = Maximum value of 7ir Ji.2 = Magnetic force due to the secondary current. H2 = Maximum value of A2. X = Total power loss in watts in the iron core. Y = Hysteresis loss in watts in the iron core per cubic

centimetre. U = Eddy-current loss in watts in the core per cubic

centimetre.

V = Total volume of the iron core. S = Cross-sectional area of iron core. I = Mean length of magnetic circuit. f= The form factor = E.M.S -r- T.M. value. /7 = The amplitude factor = E. M.S. -5- maximum value. n = The frequency. p = 2;r n = the angular velocity. T = periodic time = n~l.

Then the fundamental equations are as follows : — When the secondary circuit is open and the transformer, therefore, at no load, we have

l ..... (151)

The above equation holds good also when the transformer is loaded up, provided we then interpret bl to mean the resultant induction density in the iron core as affected by the current in the secondary coil.

In all good modern closed iron circuit transformers the value of Ka ^ is so small at all times during the period, when compared with ev that we may without sensible error write

Hence eldt=SK1dbl . . r . . (152)

588 THE INDUCTION COIL AND TEANSFOEMEE.

If we integrate this last equation throughout one-quarter period we have already seen that

SN -D H. i. ... . T

n 1B1= I eLdt =

J n

4

but T = *.

n

Hence 4S N1B1w = »» .^ (153)

But if/ is the form factor of the curve of primary potential difference, then

Therefore, from equations (153) and (154), we have

/W7*? = 4/«N1SBlf . . . (155)

which gives us the E.M.S. value of the primary potential difference in terms of the maximum value of the induction density in the core within the primary coil.

If the secondary circuit of the transformer is closed, and a secondary current is being taken from the transformer, then the currents, inductions and potentials are determined by the two equations,

and 0 = K2t2 + c2 + N2 S ' (157)

If the secondary circuit is open, and hence R2 i2 equal to zero,

and Rj ^ practically negligible in comparison with N2 S i-^, we may write (156) and (157)

XT

and %= -«j

and, therefore, as already shown,

Bp .... (158) and m.<>2 = 4wN2SB2 ..... (159)

THE INDUCTION COIL AND TRANSFORMER. 589

In the previous section we have shown that the total induction S B1 linked with the Nx primary turns may be expressed as (Zx - Z2 + Z2y3), where Zl is the maximum value of the total induction due to the primary current alone, and Z2 that due to the secondary current alone.

Hence, as before, writing Zl-Z2 + Z2(3 for S Bx, and Zl-Z2-Z1(3 for S B2, we have, by substitution in equations (158) and (159), the results

and m . e2 = 4 n N2 (

It is an experimental fact that the curve of secondary potential is an exact copy of the curve of primary potential at no load, and very nearly also at any load ; hence the form factors of the curves of primary and secondary potentials are the same. Writing /for this form factor we have

and

Hence the transformation ratio of the transformer T is given by the equation

T_ ^'^7_N2(Z1-Z2-Z1/?) ,

• '

Accordingly we see that the transformation ratio of the transformer is never exactly equal to the ratio of the turns unless the leakage coefficient (3 is zero, and that the trans- formation ratio diminishes as Zx and Z2 increase with load, because their difference Zj— Z2 always remains approximately constant at all loads, and is the mean core induction.

The leakage coefficient /? is a function of the form factor/, such that /? is greater as / is greater. Thus, peaked primary potential curves give greater secondary drop than rounded potential curves, even if they have the same E.M.S. value.

From equation (155) we see that the maximum value of the core induction B, either within the primary or secondary coil, is smaller in proportion as the form factor / is greater, if the B.M.S. value of the primary potential remains constant.

£90 THE INDUCTION COIL AND TfiANSFOKMER.

Hence the maximum value B of this core induction is less for pointed or peaked primary electromotive force curves than for rounded or flat curves, the R.M.S. value of the primary terminal potential difference being constant.

This has been experimentally proved by Dr. Roessler in his researches on the influence of the form of the potential and current curves on transformer action.

In Fig. 211 are shown three curves. The curve marked I is the magnetisation curve of a small transformer measured with the ballistic galvanometer in the ordinary way. The curve marked II is the curve of induction as obtained with alter- nating currents, using a Wechsler alternator, and the curve marked III that obtained in the same way, but by the use of

16,000 14,000 12,000 10,000 | 8,000 6,000 4,000 2,000 0

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,

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ftf

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r/-

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if

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(/

1

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4 8 12 16 20 24 28 32

Magnetisitiy Force.

FIG. 211.

a Ganz alternator. The values of the maximum induction B for the alternating currents are obtained from the primary electromotive force curves, as already described.

It is seen that the curves of induction as obtained by the alternating-current machines lie below that obtained by the continuous currents and ballistic galvanometer.

In other words, for a given induction, the magnetising force required is greater with alternating than with continuous currents. There are two reasons for this : first, the existence of eddy currents in the core, which, acting like smaller closed

THE INDUCTION COIL AND TRANSFORMER, 591

secondary currents, increase the primary current, and, second, the existence of magnetic leakage when alternating currents are used. The curves show, however, that when the peaked form of primary electromotive force curve given by the Ganz machine is used, the induction corresponding to a given mag- netising force is less than when the Wechsler machine with rounded electromotive force curve is employed, the sameR.M.S. values of the primary electromotive force being employed.

The root-mean-square values of the primary and secondary currents, viz., ^/m . if and Jm . i^, are connected with the maximum values of these variables by the equations

and

where g is the quantity already called the amplitude factor.

It has been shown that for peaked or pointed curves the amplitude factor is smaller than for flat or rounded curves. Hence for the same root mean-square value of the primary and secondary currents the maximum values of these quantities are greater for peaked current curves than for rounded or flat curves. Hence the inductions created by these currents respectively are greater — that is, Zx and Z9 will be greater for peaked current and potential curves than for flat or rounded curves.

Accordingly, whilst the respective primary and secondary inductions Zl and Z0 are greater for electromotive force curves with large form factors, their difference, Zt - Z2, which is the resultant core induction B, is less. Hence we see that the secondary drop, or increase of transformation ratio produced by loading up the transformer must be greater when the primary electromotive force curve is peaked than when it is rounded, the same mean-square value of this last being preserved constant.

We have, then, to discuss the form of the curves of primary current under variations of form factor of the electromotive force curve.

If /<! is the instantaneous value of the magnetising force due to the primary current iv then

10

592 THE INDUCTION COIL AND TRANSFORMER. and on open secondary circuit we have

Hence ^-B^N.S,..,

^4^8^.*... . . (161)

This last equation is true for all forms of primary electro- motive force curves.

If the permeability of the iron was constant, the value of

— 1 would be constant and equal to a, but in practice it is not dhi db,

found to be constant. We see, however, that ^- is the slope

of the geometrical tangent to the hysteresis curve at the instant considered — that is, it is the trigonometrical tangent of the angle which the geometrical tangent to the hysteresis curve makes with the positive direction of the axis of time, and this is not found to be a constant quantity as we travel

round the hysteresis curve. The value of - — 1, however, is not

dh^

greatly affected by the form of the curve of e1. Hence, for curves of primary electromotive force which have a peaked form, and therefore a large maximum value, the value of

— ?i, or the slope of the current curve, will be greater than for d t

flatter curves of electromotive force. This is seen to be the case by reference to Figs. 203 and 204, which show the no- load primary-current curves and primary electromotive force curves of the same transformer tested by Dr. Koessler on the Ganz and the Wechsler alternator.

The exact predetermination of the form of the primary current at no load from the curve of primary electromotive force is, at any rate as yet, an impossible matter. It would be an easy thing to predetermine if the hysteresis curve always had the same form, but as this last is affected to a considerable extent by variations in the quality of the iron and of the reluctance of the magnetic circuit, it is

THE INDUCTION COIL AND TRANSFORMER. 593

not of much use to make assumptions which are not justified in practice.

A knowledge of the power factor of transformers of any particular type will always enable us to make an approximation to the value of the magnetising current if the total power taken up in the core is known and the mean-square value of the primary electromotive force. For if X is the total power taken up in the core at no load, and ,Jm^t J^T* are the E.M.S. values of the primary electromotive force and current,

•v-

and F is the power factor, then F = - — j from

which ,Jm . ^2 can be obtained.

It is seen, however, that the primary current curve at no load is always a more irregular curve than the curve of primary electromotive force, and that for the same B.M.S. value of the primary electromotive force the B.M.S. value of the primary current at no load (the magnetising current) is less for pointed or peaked potential curves than for flat curves.

§ 12. Iron Core Loss in Transformers and Induction Coils. _ It has already been explained that two distinct causes of energy dissipation exist in the iron cores of transformers and induction coils — viz., the magnetic hysteresis loss and the eddy-current loss. The former of these is not affected or diminished by any amount of lamination of the core, but the latter can be reduced to a very small percentage of the total loss by constructing the core of iron plates of thickness not greater than 0-014 inch, the plates being separated from each other by very thin paper, or a layer of paint or varnish. In the chapter in the Second Volume of this Treatise devoted to the Construction of the Transformer, the various practical details connected with the core construction, and the pre- determination of the core loss for plates or wires of given size are considered.

Supposing, however, that the core is properly laminated, and in planes parallel to the lines of induction in the core, there will still be a certain dissipation of energy, by reason of eddy electric currents set up in the iron as the induction changes its direction. If we consider a small circuit described

QQ

594 THE INDUCTION COIL AND TBANSFOBMEB.

anywhere in the iron plate, in a plane perpendicular to the lines of induction, then, if e is at any instant the electromotive force set up in this circuit by reason of the variation of the induction through it, the mean rate at which energy is being dissipated in this circuit must be equal to some constant, multiplied by the value of the mean of the square of e. But we have seen that if B is the maximum value of the induction in the core, then the E.M.S. value of the electromotive force of induction induced in the primary circuit is equal to the value of the expression 4/Nx .n S B, where/ is the form factor of the curve of electromotive force.

Hence the mean-square value (w.ez) of the electromotive force of induction must be numerically proportional to/2 «2 B2; also the same holds good for the eddy-current electromotive force and rate of energy dissipation, and the eddy current loss per unit of volume in the core measured in watts must be proportional to the product of some constant £ and the quantity /- ;i2 B2. In other words, the eddy-current loss will be equal to £/2 w2 B2 watts per unit of volume of the core, where / is the form factor of the curve of primary electro- motive force, n the frequency, and B the maximum value of the induction.

Mr. Steinmetz has shown that the hysteresis loss in iron cores can be represented by an arbitrary formula, expressing the fact that the hysteresis loss per unit of volume of the core is proportional to the product of a constant, the frequency, and the maximum value of the induction raised to a power very near to 1-6. Hence, if H is the hysteresis loss in the core per unit of volume,

where 17 is called the hysteretic constant of the iron.

This law, although only an empirical one, deduced entirely from observation, yet appears to be sufficiently exact to guide practice within the limits of the range of induction density employed in transformers.

Hence the total loss T in a transformer core of volume V is given by the expression

/2B2). . . . (162)

THE INDUCTION COIL AND TRANSFORMER. 595

The eddy-current loss varies as the square of the maximum value of the induction, and the hysteresis loss as the l-6th power of the same.

Since the E.M.S. value (J^) of the primary electro- motive force has been shown to be related to B by the equation

we can substitute for B, in equation (162), its value in terms of Vmej^i and we arrive at the equation

This last equation shows us that the eddy-current loss is not affected by the form factor of the curve of primary electro-

» s s s"

Maximum Value of Core Induction.

FIG. 212.— Curve I taken with Ganz Machine. Curve II taken with Wechsler Machine.

motive force, but that the hysteresis loss is affected by it, because the form factor / appears in the hysteresis term of the expression for T but not in the eddy current term.

Hence variation in the form factor of the curve of primary electromotive force will alter the total core loss in the trans- former, and make it less in proportion as the form factor is greater. This has been pointed out both by the author and by Dr. G. Eoessler, and is amply confirmed by experiment.

Hence the total core loss in a transformer is not an absolute and fixed quantity, but depends upon the form of the wave of primary electromotive force to a not inconsiderable decree.

QQ2

596 THE INDUCTION COIL AND TRANSFORMER.

This dependence of the core loss upon the form factor of the curve of electromotive force is shown by the results of Dr. Roessler's experiments embodied in the diagrams in Figs. 206 and 212. From Fig. 212 it will be seen that at a given induction the core loss is greater when the trans- former is worked off the Ganz alternator than off the "Wechsler alternator ; and from Fig. 206 it is shown that for the same R.M.S. value of the primary potential difference the core loss is greater with the Wechsler than with the Ganz alternator.

Broadly speaking, pointed or peaked potential curves give rise to greater core loss than rounded or flat primary potential curves.

In the Second Volume of this Treatise, we return to the discussion of these matters, and enter into more details as to the practical considerations to which they lead in the Construction of the Induction Coil and Transformer.

END OF VOLUME L

APPENDIX.

NOTE A. (See page 19.)