The Alternate Current Transformer Vol. 2: The Utilisation of Induced Currents (1896) — part 34 of 36

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THE INDUCTION COIL AND TBANSFOBMEB. 671

In Fig. 201 the three curves show the progress of increase of the power-factor as the load on the secondary circuit is progressively increased* The upper curve represents the growth of power factor (F) for a 6,500-watt Westinghouse transformer. Beginning at 0*8, it rises up to unity at about one-tenth of full load. Hence at and after this load the apparent watts are the same as the true watts, and the real power taken up in the transformer is quite' accurately given by the product of the primary terminal pressure and the primary current, mean-square ( Jmenn^j values being under-

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Fio. 201.— Relation of Power Factor to Secondary Output.

stood. For an open magnetic circuit transformer like the " Hedgehog '' the case is quite different. The power factor begins at a value of 0-08 or 0-06, and it never rises up above 0-8. Hence at no stage of the load is the real power taken up by the transformer equal to the " apparent watts." A transformer like the 4,000-watt Kapp appears to occupy an intermediate position, and although it has a medium power factor to start with, its power factor rises up to unity at about half-load. The importance of this fact in alternate- current station working is very great. It shows us, if we have a station wholly supphed with transformers of the type of Mordey, Westinghouse, Thomson-Houston, Ferranti, &c., that the apparent power supplied to the transformers is equal to the real power at any hour when all the transformers are more than one- tenth loaded.

572 THE INDUCTION COIL AND TRANSFORMER,

The reciprocal of the power factor of a transformer on open secondary circuit is a measure of the reluctance of the mag- netic circuit of the transformer. In the case of a transformer with an air-iron magnetic circuit (open-circuit type) the reluc- tance of the iron circuit is large and the reciprocal of the power fekctor large also, and may be a number approximating to 16 or 17. In the case of a closed iron circuit transformer like the Mordey transformer, with very short magnetic circuit and very small reluctance, the reciprocal of the power factor is very small, and will be a number approximating to 12 to 14. The introduction of any bad magnetic joint into the iron circuit, or the employment of iron of small permeability, im- mediately decreases the magnitude of the power factor of that transformer. Any joint or break in the magnetic circuit accordingly increases the value of the reciprocal of the power factor, and although this alone will not affect the total core loss in the transformer, it is an indication of the increased reluctance of the magnetic circuit. The advantage of a large power feustor is that it involves a small value of the magnetising current of the transformer. In the case of an alternating current station large magnetising current involves additional waste of power in the passage of this current through the distributing mains. This point will be discussed at greater length in connection with the subject of alternating current distribution.

§ 8. Magnetic Leakage and Secondary Drop. — ^If a trans- former has the mean-square value of the potential difference of its primary terminals kept perfectly constant, whilst at the same time secondary currents of various magnitudes are taken from its secondary coil by altering the resistance of the external secondary circuit, we find that the mean-square value of the potential difference between the secondary tenninals of the transformer changes with every change in the secondary load.

The secondary terminal potential difference (S.P.D.) becomes less as the secondary current and load increases. The diffe- rence between the secondary terminal potential difference at no load and at any load is called the secondary drop of the transformer due to that load.

THE INDUOTION COIL AND TRANSFORMER. 573

We may represent the variation of secondary drop with secondary load by a diagram as follows : Let a horizontal line be taken on which are set off distances representing the fractions of the fall secondary load, and let vertical ordinates set up at these points represent the value of the secondary potential differences at these loads. For convenience sake we may make these ordinates represent the magnitude of the secondary terminal potential difference diminished by a certain constant amount which is less than the least difference found with full load. For instance, suppose the secondary terminal potential difference at no load is 100 volts and at full load is 97 volts, we may make the vertical ordinates represent the terminal potential difference minus 90 volts. The curve

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Output in Secondary Watts, a 6 is the hnrlzontiil line thronf|rh a; Carre ao Is the Carre of Drop dne to secondary rts'staDoe ; Curve ad, that due to primary reitotaoce ; and Curve ae U the Curve of total Drop.

Fia. 202. — Secondary Drop Curves of 6,500- watt Weatinghouse Transformer.

defined, as in Fig. 202, by the extremities of these ordinates is called the secondary terminal volt curve, and it shows in a graphical manner the gradually diminishing secondary terminal potential difference as the transformer is loaded up. This '* secondary drop " arises from two causes. The first is the loss of potential due to resistance, and the second is the loss of secondary potential due to magnetic leakage. Let the resistance of the secondary coil of the transformer be represented by B,* and the secondary current (mean-square value) be represented by (I9). Then Bs(Ia) is the loss of voltage due to secondary resistance. If the primary terminal potential difference is kept constant, then,

S74 THE INDUCTION COIL AND TRANSFORMER.

over and above the loss of secondary voltage due to the tesistanoe of the internal secondary circuit, there is a portion of the secondary drop which is due to loss of voltage by tiie resistance of the primary circuit, and, in addition to this, &e loss above mentioned, which is due to magnetic leakage, ^Furthermore, the secondary drop is to a considerable ext^t •dependent, as will be explained presently, upon the form of ihe curve of primary terminal potential difference. Hence ihe difference between the potential difference of the secondary -terminals of the transformer at no load and full load, primary potential difference being constant, is dependent on £Dur things, viz., upon —

(1) The resistance of the primary circuit ;

(2) The resistance of the secondary circuit ;

^8; The magnetic leakage of the transformer as affected by,

(a) Its construction.

(h) The form of the curve of primary terminal poten- tial difference. The effect called the magnetic leakage in a transformer may be generally described as follows : The primary current •creates in the iron core a certain total induction, or in usual language creates a certain number of lines of induction in the •core which are linked with the primary circuit. The mag- netising effect of the secondary current is at any instant opposed to that of the primary, and hence creates an induction in the core in an opposite direction. The resultant, or actual induction in the core at any place is due to the difference of the opposed magnetising forces acting on the core. When the transformer has its secondary circuit open the magnetic induction in the core is that due to the primary current only, which is then generally called the magnetising current When the secondary circuit is closed and a secondary current produced, the rise of induction in that part of the core enveloped by the .secondary circuit is delayed, and its maximum value is reduced. The simplest way in which the effect of increasing the secondary current of the transformer can be regarded is as follows: Let us denote by the letter Z^ the maximum value of the total magnetic induction in the core which would be produced by the primary current if it acted alone, and by Z, the same due to the secondary current, these values being the inductions just within that part of the core enveloped by

THE INDUCTION COIL AND TRANSFOBMER. 676

the primary and secondary coils respectively. The whole of the induction Zj which is linked with the primary coil turns is not, however, linked with the secondary. Let a fraction, say fi Zi, of this primary induction escape linkage with the secon- dary coil, and a similar fraction, say /? Zj, of the secondary mduction will escape linkage with the primary coil. Then the total induction linked with the primary coil is Zj - Z2(l -)8), because the induction caused by the primary current is opposed in direction to the induction caused by the secondary current, and the inductions, like the two currents, are opposite in phase and reach their maxima nearly coincidently. Also, for the same reasons, the total induction linked with the secondai^ circuit is

Z,(l-/3)-Z,.

fi is called the coefficient of leakage.

The value of the total induction linked with the primary circuit is therefore the product of the number of primary turns Ni and the resultant induction Zi-Za(l-/?), and, similarly, the value of ths total induction linked with the secondary circuit is given by the product of the number of secondary turns N, and the resultant induction Z^ (l-P)" Z^ Hence we have the relation,

The total linkage of primary circuit

and induction traversing it The total linkage of secondary circuit and induction traversing it

_N,{Z,^Z,(l-j8)}^n,

N,{Z,(l-ie?)-.Z,}- •

It will be shown presently that this fraction T represents the ratio of the mean-square value of the primary terminal potential difference to that of the secondary terminal potential difference.

This ratio, which is denoted by T, is called the transforma- tion ratio of the transformer. Since the difference between Zj and Z] remains nearly constant as Z^ and Z^ increase, it is easily seen that the transformation ratio increases as Zi and Z, increase, subject to the condition that Z^— Zj is nearly constant at all loads.

Hence, if the mean-square value of the primary electro- motive force is kept constant, that of the secondary potential

676 THE INDUCTION COIL AND TRANSFORMER.

difference decreases as the currents, and therefore the* inductions, in the core increase, and this effect is called the^ *' secondary drop."

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 Uie 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 Bi be the true resistance of the primary circuit and Ba 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, N, the number of secondary turns, and a stand for the ratio of N^ to N,. Let h^ be at any instant the induction density in that part of the core enveloped by the primary coil, and h^ that part enveloped by the secon- dary coil; the difference between these inductions may be called the density of the leaksige of induction, and be denoted by 6. Hence

fc =» 5j — ftj.

In other words, if S is the cross-section of the core, then 8 6 a S 61 ~ S 6a and S h represents that part of the induction linked with the primary coil which is not linked with the secondary coil. If e^ is the primary terminal potential diffe- rence at any instant, and e^ that of the secondary terminal at the same instant, and i^ and t^ the currents at the same momenti then, by fundamental equations, we have

^«R,t, + SN,^ (146)

and 0-Rat;+^,+SN,^. . • . (U7)

N Let us write "yT"^^ *^^ 6«6i-6,;

we have by elimination from the fundamental equations the- result

^ + ^, + R,t,-E,!!-SN,^. . . (148) a a at

THE INDUCTION COIL AND TRANSFORMER, 677

If 0j varies in a simple periodic manner so that «i"-£i sinpt, then, since e^ is always opposite in phase and similar in form to v^f ^0 must have

#2 = — £2 sin p U

Moreover, when the transformer is fully loaded, the currents ti and tg are in step with the electromotive forces «| and e^ but t| differs ISOdeg. in phase from i^.

Hence ii^li^mpt

fa- -IgSiniJ*.

We can also write 5 » - B sin^v t, because the leakage 5 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

(5l. - E, - Rj I« - eJA sini) t - - S Nap B cos i>«.

The quantity S N^p 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 potentiajl 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 pt and oonpt respectively in the above eqi:iatioii give, therefore, the maximum value of ^e 'leakage," and there- fore, when dividecl by V^2, represent the mean-square value.

Hence the quantity ^^-Ea-Rglj-Rji^-L, or, which, comes to the same thing, the quantity

VaV2 Jl ^^/2 a^ s/2>'

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

E E T T

The quantities --ii —^ -j^ -^ represent the magnitude '

of the currents and potentials as read in alternating-current .

pp

678 THE INDUCTION COIL AND TRANSFORMER.

ammeters and voltmeters. We may denote these m6an*«quare values by the symbols (E^), (E^), (Ii)» (l2)> and the values which these mean-square potential differences and currents have at full and at no secondary load by the symbols (E^)^ (Ej);, (Ei)o, (E,)o, &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

}1(E,),-(E^,-R,(I,)/-Jb,(I,),) and ji(E,),-(E,).-R,(I,).-iRi(I.)„|.

la a )

The value of (19)11 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

(E,)o - (E,),- (r, (I>)r + -Ri (I,)/- - Ri (Ii)o).(149)

The secondary terminal volts at full load being denoted by (Es)/) and that at no load by (E9)o, we see that the quantity (Egjo— (El)/ represents the total secondary drop due to all causes. The quantity

{Ri(I.),+ iR,(I,),-lR,(Ix)o} . . (160)

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

magnetic leakage.

Testing in this way a number of transformers, the author

found that where the primary and secondary drcuits were

THE INDUCTION COIL AND TRANSFORMER. 679

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 carves, by which the three sources of secondary drop have been distinguished, ihe lines ac, ad showing the curves of drop due respec- tively to the primary and secondary resistance, and the curve 4i 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 spedfioation 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 ihe 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 Ourrents. — 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 ^ect 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 ourrent 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 TRAN8F0BMEB.

the diagrams in Figs. 208 and 204. Fig. 203 shows the: primary-current curve of a small transformer taken off a Oanz 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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70 80 90 100 110 1«0 160 140 Degreet €f Phat^ Fkl 203. — Primary Current Cui-ve II. and Primary Electromotive Foroe Curve I of a Transformer taken ofif Granz Alternator.

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10 60 00 70 80 Degree* nJPhfU^ 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. Roessler.*

• "Das Verhalten von Transformatoren unter den Einflusee voik Wechselstrdmen Verschiedenen Periodischen Verlaufs." A Paper read at the third annual meeting of the Verband Deutscher Electrotechniker^ Munich, July 6, 1895. Sw also Tht Eleoirician, Vol. XXX VI., 1895, p. 160u

TUE INDUCTION COIL AND TRANSFORMER 581

Not only do the forms of the current carves differ when taken with different-shaped electromotive force curves, but if; the primary electromotive force is kept at the $ame 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 Oanz and Wechsler machine respectively.

0 0-4 08 12 16 2-0 24 28 8-2 86 4-6 4 4 Seeondary Cwrrent Amper§».

FiQ, 205. — Current Diagram of a certain Transformer. Cm*ve I, Primary Current taken with Ganz Alternator. Curve II, Primary Current t^kea 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 « 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 TBANSFORMER

abscissad, the transformer being tested in the two cases on the Oanz 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.

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

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Fio. 207. — Efficiency Curve of Transformer. Curve I taken on Qani 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 Oanz machine with peaked electromotive force curve and the Wechsler machine with a rounded curve.

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

THE INDUCTION COIL AND TBAN8F0RMER. 583

Elg. 209 are sho-wn 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 Qanz 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. Boessler,

40 80 120 160 SOO 240 280 820 8G0 400 400 480 Secondary output in WatU.

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

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Fio. 209. — Curve of Secondary Drop of Transformer. Curve I taken on Qanz 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 Foxm Factor and Amplitude Factor of a Periodie Oiurve. — The above differences are closely connected with the magnitude of the form factor of the curve of primary electro* motive force. This quantity is defined as the ratio of the

684 THE INDUCTION COIL AND TRANaFOEMEE.

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 B.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 fnotor- ^^^ ^'^'^' ^^^"^ of equispaced ordinates . The T.M. value of equispaced ordinates

Take, for instance, in the case of a simple sine curve, the B.M.S. value of the equispaced ordinates is equal to the value of the maximum ordinate divided by JS. The T.M. value of the equispaced ordinates is equal to the value of the maxi-

mum ordinate multiplied by —

Since

J- =0-707

V2

and ? =0-687,

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

. 0-707 1 1 eme curve IS - — =al*l. 0-687

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

Curre.

T.M. value of ordinate as

ordinate.

R.M.S. value

of ordinate as

fraction of max.

ordinate.

Form factor

Sine

0-637

0-7854

0-5

1-0

0-785

0-666

0-33

0-707

0-836

0-68

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0-730

0-447

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Triangle

1-063 1-16

RectanKle

10

Semi-ellipse

1*039

Parabola with axin

1-096

Two semi - parabolas meeting at a cusp..

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

TBB INDUCTION COIL AND TRANSFORMER 685

dine plot a polar diagram of the same onrve, trith its pole on <the line of reference. Then, hj the proposition on page 198, the radius of the semioirele, so drawn that its area is equal to the area of the polar curve, is . the B.M.S, value of the ordinates, and the height of the rectangle described on the base line, so that its area is equal tb that of the wave curve, ^ves 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 lorm factor of the curve, which is represented by the wave or ^lar diagram.

Fia. 210.

This form f^tor is an important quantity in the design of ^dtemators, 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 B.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 B.M.S. value faster than it increases the T.M. Talue, and so increases the form flEkctor of the curve.

686 THE INDUCTION COIL AND TBANSFOEMEE.

The amplitude fcustor 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) _B.M.8> value of equispaced ordinates _ factor j Value of maximum ordinate ^

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

^ These two feMtors — the form £B.ctor / and the amplitude factory — are important quantities in the case of periodic curves.

§ 11. General Analytical Theory of the Transformer and Induction OoiL — ^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.

e^ = The value of the primary terminal potential dif- ference or primary E.M.F. at any instant. Ej==The maximum value of the same. m«^=: The true mean (T.M.) value of e^ during the period. Jme^^^The root-mean-square (B.M.S.) value of ^^ during the period.

^ .are the same quantities for the secondary terminal / 1 1 potential difference.

B^ = The resistance of the primary circuit. B^ = The resistance of the secondary circuit. Nj = The number of turns on the primary coiL N^ = The number of turns on the secondary coil. 6j»The density of magnetic induction in the core

inside primary coil. Bj = The maximum value of induction density 6^. Z^sThe total induction produced in the core due io

primary coil. b^j Bg, Z2 are the same quantities for the secondary drcuii.

THE INDUCTION COIL AND TRANSFORMER. 691

ii» Primary current at same instant that the primary

terminal potential difference is e^, . I^» Maximum value of t. m . tj » True mean value of t^ during the period. ^j;;7^2 = Root-mean-square value of t\ during the period. i^f I21 tn i^, Jmt^ are the same quantities for the secondary circuit. ^» Magnetic force due to the primary current i^. Hj = Maximum value of /i^- /12- Magnetic force due to the secondary current. H2 = Maximum value of h^. X~ Total power loss in watts in the iron core.

Y » Hysteresis loss in watts in the iron core per cubio

centimetre. U » Eddy-current loss in watts in the core per cubio centimetre.

V = Total volume of the iron core.

8 = Gross-sectional area of iron core.

Z a Mean length of magnetic circuit.

/•-The form factor = R.M.S-^T.M. value.

g^The amplitude factor » E.M.S. -s- maximum value.

n = The frequency.

/) = 2irn = the angular velocity. T = periodic time = n-K Then the fundamental equations are as follows : — When the secondary circuit is open and the transformer, therefore, at no load, we have

«,-RiH + SN,^ (161)

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

In all good modem closed iron circuit transformers the value of B^ t^ is so small at all times during the period, when compared with 0^, that we may without sensible error write

Hence s^dt^SJii^db^ • • r • • (152)

588 THE INDUCTION COIL AND TRANSFOBMEB,

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

T •'o ^

but T = ^.

n

Hence ^S^^B^n^m.ej^ (158)

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

y^Jm^^ ..... (Ui)

{Therefore, from equations (158) and (154), we have

V«rr^« = 4/nNiSBi, . . . (155)

which gives us the B.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,

., = Bi.\ + N,S^ (166)

and O^Bgij+^jirNaS^ .... (157)

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

and Bj t^ practically negligible in comparison with N|S -jt, wo may write (166) and (167) ^^

^ ^ dt

and «2== "f-^f ^V^'

lit

and, therefore, as already shown,

m.tfi = 4nNjSBp .... (158) and in.^2=4nNjSB2 (169)

THE INDUCTION COIL AND TRANSFORMER. 68^

In the preTioas section we have shown that the total induction SB^ linked with the N^ primary turns may be expressed as (Z^ - Z^ + Z, ^), where Z^ is the maximum value of the total induction due to the primary current alone, and Z, that due to the secondary current alone.

Hence, as before, writing Z^-Zj+Z^iS for S Bp and Zj - Zj- Z^ )3 for S Bg, we have, by substitution in equations (158) and (169), the results

m,tfi = 4nNi(Zi-Zj + Zji8), and «i . ej=4nN2 (Z^ - Zj - Z^ P).

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 Jm . e^ ^fm , e^*

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

^_Vi;rv^N,(Z,-Z,-Z,/3) .g^

^/;;^:^ n,(z,-z,+z2/8)- • • ^ ~^

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

The leakage coefficient )8 is a function of the form factor /,^ such that P is greater as / is greater. Thus, peaked primary potential curves give greater secondary drop than rounded potential curves, even if they have the same B.M.S. value. ^

From equation (165) we see that the maximum value of the core indiietion B, either within the primary or secondary coil, is smaller in proportion as the form factor / is greater, if the B.MiS^ value (^ the primary potential remains constant.

.«90 THE INDUCTION COIL AND TBAN8F0BMEB.

Hence the maximum value B of this core induction is less for pointed o^ 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 rese^rchep on the influence of the form of the potential and •current curves on transformer action.