The Alternate Current Transformer Vol. 1: The Induction of Electric Currents (1896) — part 32 of 35

taneous value of the potential charging the condenser. Many other forms of apparatus have been described, but the principles are practically the same as those above referred to- In all cases a condenser is charged through a contact-maker, and the potential of the condenser determined by either a galvanometer or voltmeter. A few practical suggestions in connection with the construction of such contact-makers may be useful. In the first place, if a condenser and electrostatic voltmeter are used, care must be taken to see that both are very highly insulated. The use of the condenser is to act as a reservoir and supply the electrical leakage of the voltmeter. If a condenser of large capacity is employed, then the contact must be suitably prolonged, or else the condenser will not be charged completely during the contact. The contact springs sometimes give trouble by making imperfect contact with the disc, and too much pressure must not be applied to the brushes, or else they create a trail of metallic deposit on the insulating disc. The author has found a material called stabiiit a very suitable insulating material for the construction of the insulating disc of the contact-maker, and the contact piece may be a transverse slip of steel or hard brass let into it. The contact springs are best made of steel, tempered and well cleaned at the contact surfaces. The contact-maker is best constructed by attaching a circular disc of stabiiit or ebonite to the shaft of the motor or alternator and turning it up very accurately on the shaft. The metal contact slip is then let into the disc and a pair of insulated springs are carried on a rocking arm which moves round an axis co-axial with that of the motor or alternator. As the disc revolves the contact slip passes under the springs, and connects them together for an instant. These springs are connected to the circuit of the voltmeter and condenser, so that when the contact is made between the springs the con- denser and voltmeter in parallel with it are connected to the alternating circuit under test for a short instant. The instant during the period when the contact is made can be varied by rocking over the arm carrying the springs.

Prof. Kyan has suggested and employed a jet of salt water as a means of making an electric contact. Through this jet a steel needle passes at an assigned instant during the revolution.

THE INDUCTION COIL AND TRANSFORMER. 535

With care and proper construction the steel spring contact- maker works very well, and is much more convenient to use than a contact in which a liquid jet is employed.

M. Blondel has described several forms of instrument, which he calls oscillographs, for the direct representation by optical means of the form of alternating-current curves, enabling us to project on to a screen a luminous line having the form of the alternating current curve. For a description of these we must refer the reader to his Paper in the Comptes Rendus, Vol. CXVL, No. 10, March -6, 1893, p. 502, and to The Electrician, Vol. XXX., March 17, 1893, p. 571.

§ 3. Discussion of Transformer Diagrams. — The methods, some of which have been described in the previous section

FIG. 173.— Curve of Electromotive Force of Thomson-Houston Alternator on Open Circuit.*

enable us, as it were, to look inside the transformer and observe the nature and order of the electrical operations taking place in it. We shall proceed to discuss some of the experimental results which have been thus obtained. In the first place, it must be noted that the curve of primary potential difference, or as it is generally called, the curve of primary E.M.F., ~*In Figs. 173, 174, 175, the dots represent the actual position of observations.

536 THE INDUCTION COIL AND TRANSFORMER.

depends upon the construction of the alternator producing the electromotive force, and also upon the nature of the circuit, whether inductive or non-inductive, which that alternator is supplying. Any assumption that the curve of primary potential difference is always a simple sine curve is very far from true. The form of the curve of primary electromotive force is not even a fixed and independent attribute of the alternator. The form of the E.M.F. curve of the alternator may be quite different when taken on open circuit to that which it is when taken at the terminals of the alternator when this last is loaded with an inductive or non-inductive load of transformers. In Fig. 173 is shown the curve of electro-

FIG. 174. — Curve of Electromotive Force of -Thomson-Houston Alternator working on an Inductive Circuit.

motive force of a Thomson- Houston alternator at no load or on open circuit, and in Fig. 174 the E.M.F. curve of the same machine when actuating a load of transformers, the secondary circuits of which are lightly loaded. It will be seen that the second curve is quite different to the first, and that neither of them is even approximately a simple periodic curve. In Fig. 175 is shown the E.M.F. curve of a Mordey alternator at full load on a water resistance. It is, then, clear that no assumption must be made as to the con-

THE INDUCTION COIL AND TRANSFORMER. 537

stancy of the form of the curve of electromotive force of any alternator, but that the form of the curve of primary terminal potential difference of the transformer under test must always •be determined.

FIG. 175. — Curve of Electromotive Force of Mordey Alternator on Water-Resistance Load.

2,000

0 60 120 180 240

Degrees of Phase.

FIG. 176.— Primary E.M.F. and Primary Current Curves of Mordey Transformer, on Open Secondary Circuit, supplied off Mordey Alternator, with no other load.

We have, in the next place, to consider the case of the transformer when the secondary circuit is open or unloaded, and to inquire what under those circumstances is the form and

538 THE INDUCTION COIL AND TRANSFORMER.

relative position of the primary terminal potential difference curve and the primary-current curve. A number of examples of such curves are given in the diagrams on pages 537 to 541. In Figs. 176 to 183 are shown the primary-current curves and primary E.M.F. curves for transformers on open secondary circuit made by the Brush Electrical Engineering Company

2,000

120 180 240

Degrees of Phase.

FIG. 177. — Primary E.M.F. and Primary Current Curves of Mordey Transformer, 011 Open Secondary Circuit, supplied off Mordey Alternator, furnishing Current also to other transformers lightly loaded.

2,000 1,000

l:00t. 2,00

,'*

Degrees of Phase.

FIG. 178. — Primary E.M.F. and Primary Current Curves of Thomson- Houston Transformer, on Open Secondary Circuit, supplied off Mordey Alternator, with no other load.

and the Thomson-Houston Company, the electromotive force being supplied by Mordey or Thomson-Houston alternators in various states of load. The Mordey-Brush transformers

THE INDUCTION COIL AND TRANSFORMER. 539

are 50 kilowatt size and the Thomson- Houston are 30 kilowatt size.

It will be seen that the primary current under these conditions always lags behind the curve of primary E.M.F. or primary terminal potential difference. The primary current, vrhen the secondary circuit of the transformer is open, is called

2,000

l.OOC

0

l.OOC

2,000

•6 •4 "2 0

•2-* •4 •6

Deyreet of Phase.

FIG. 179. — Primary E.M.F. and Primary Current Curves of Thomson- Houston Transformer, on Open Secondary Circuit, supplied off Mordey Alternator, furnishing Current also to other Transformers lightly loaded.'

2,000 1,000

1,000 2.COO

o eu 120 180

Degrees of Phase.

FIG. 180.— Primary E.M.F. and Primary Current Curves of Mordey Transformer, on Open Secondary Circuit, supplied off Thomson-Houston Alternator, with no other load.

the magnetising current of the transformer. Even if the curve of primary potential difference is nearly a true sine curve, the curve of primary current is not of a similar character, but is always more irregular. The form of the primary current curve depends not merely upon the form of

540 THE INDUCTION COIL AND TRANSFORMER.

the primary E.M.F. curve, but upon the nature of the iron used in the iron core and upon the structure of the trans- former generally, so that the primary -current curves of two transformers by different makers will have different forms of

3,000

2,000

1,000 0

1,000 2,000

8,000

300

360

0 60 120 ISO 240

Degrees of Phase.

FIG. 181.— Primary E.M.F. and Primary Current Curves of Mordey Transformer, on Open Secondary Circuit, supplied off Thomson-Houston Alternator, furnishing Current also to other Transformers lightly loaded.

2,000 1,000

I •

1,000 2,000

60

•240

120 180

Degrees of Phase

Fia. 182.— Primary E.M.F. and Primary Current Curves of Thomson- Houston Transformer on Open Secondary Circuit supplied off Thomson- Houston Alternator with no other load.

magnetising current curve, even if worked off the same alternator. This is well shown in the curves in Figs. 176 and 178, in which a Brush and Thomson-Houston transformer are

THE INDUCTION COIL AND TRANSFORMER. 541

worked off the same Mordey alternator. The curve of primary E.M.F. is the same in each case, but the curve of primary current is of a quite different form. This is brought about by differences in the reluctance of the iron circuit producing small differences in the form of the curve of magnetic induction in the core.

By comparing Figs. 176 and 181 it will be seen that the form of the curve of primary current is also dependent upon the form of the curve of primary E.M.F. , and for the same transformer the curves of current may be considerably altered by supplying it off a different alternator, or off the same

60

860

120 180 240

Degrees of Phase.

FIG. 183.— Primary E.M.F. and Primary Current Curves of Thomson- Houston Transformer, on Open Secondary Circuit, supplied off Thomson. Houston Alternator, furnishing Current also to other Transformers lightly loaded.

alternator in different states of load. In some alternators, such as the Mordey alternator, the armature reaction is very small and the form of the curve of electromotive force given by the machine is not very different whether the machine is worked on open circuit or on fuU load, on water resistance or on an inductive load. In the case of a machine with large armature reaction, the form of the curve of electromotive force will, under these various conditions, be greatly altered, and hence the form of the primary-current wave of trans-

542 THE INDUCTION COIL AND TRANSFORMER.

formers on open secondary circuit connected to it will be -quite different also.

We pass on next to consider the form and position of the •curve of secondary terminal potential difference or secondary E.M.F. when the transformer secondary circuit is open.

3000 2000

1000 - 2000 -

TIG. 184.— The Primary E.M.F. Curve (firm line) and Secondary E.M.F. 'Curve (dotted line) of a Thomson-Houston Transformer taken off a Thomson-Houston Alternator. The Secondary Curve is drawn to a scale which makes its Maximum Ordinate equal to that of the Primary Curve, and the Curves are seen to be identical in form.

FIG. 185. The same Primary and Secondary E.M.F. Curves, delineated in Fig. 184, are here drawn with the Secondary Curve (dotted) reversed and superposed on the Primary Curve to show its exact coincidence with •the Primary Curve.

It is found that this curve of secondary electromotive force is, under these conditions, an exact copy on a reduced scale of the curve of primary E.M.F., and that it is in exact opposition to it in phase. In Fig. 184 are shown the curves of primary and secondary terminal potential difference of a Thomson-

THE INDUCTION COIL AND TRANSFORMER. 543

Houston transformer at no load. The primary terminal potential difference curve or primary E.M.F. curve is repre- sented in Fig. 184 by a firm line, and the secondary E.M.F. curve by a dotted line. The secondary curve has been drawn to such a scale that the ordinates of the secondary curve are equal to those of the primary curve. In Fig. 185 the curve of secondary E.M.F., represented by a dotted line, has been reversed and drawn over the primary to show the exact coinci- dence of the two curves.

FIG. 186.— Primary E.M.F. Curve (I), Primary Current Curve (II) and Secondary E.M.F. Curve (III) of 10-light Westinghouse Transformer on Open Secondary Circuit

This constitutes one of the most valuable properties of the transformer, viz., that it copies varying or periodic potential difference exactly to a reduced or increased scale. Hence, if we have a pair of terminals between which there is a periodi- cally-varying potential difference having a /mean-square value of, say, 2,000 volts, and we attach the primary circuit of a suitably- wound transformer to these terminals, we can produce a periodically-varying potential difference of lower

644 THE INDUCTION COIL AND TRANSFORMER.

or higher value, and the curve of which is an exact copy to a reduced or increased scale of the original. We shall see later on that useful applications can be made of this fact.

If the secondary circuit of the transformer is closed by a non-inductive resistance, such as incandescence lamps, then the Curve of secondary terminal potential difference or secondary electromotive force undergoes a displacement and is brought forward or lags behind the curve of primary electromotive force. The reason for this is to be found in the magnetic

7

0-3 =

FiG. 187.— Primary E.M.F. Curve (Ij, Primary Current Curve (II) and Secondary E.M.F. Curve (II!) of 10-light Westinghouse Transformer, loaded to one- tenth of full load.

leakage across the magnetic circuit which then takes place, and which will be discussed in a later section. The act of closing the secondary circuit of the transformer and produc- ing a secondary current also effects a displacement in the position of the primary- current curve. As the transformer is loaded up the primary- current curve is displaced backwards, so that the lag in phase between the primary current and primary electromotive force is decreased. At full load the

THE INDUCTION COIL AND TRANSFORMER. 545

primary current and secondary electromotive force are nearly in opposition of phase. This is seen to be the case by examining the series of curves in Figs. 186 to 189, which were taken by Prof. Eyan from a small Westinghouse trans- former, in which magnetic leakage is not by any means absent.

We have next to consider the position of the curve of magnetic induction. The curve of induction is obtained, as already described, by integrating one or other of the

r I sco

too

\

1

0-4

FIG. 188.— Primary E.M.F. Curve (I), Primary Current Curve (II) and Secondary E.M.F. Curve (III) of 10-light Westinghouse Transformer loaded to half load.

curves of electromotive force. The process of obtaining a second curve, by taking as ordinates the area up to successive absciss® of a first curve and plotting these areas as new ordinates to the limiting abscissae, is a process which always has the effect of smoothing out irregularities in the original curve, so that if the first curve is one not far removed in form from a simple sine curve the second or integration curve will be more nearly still a simple sine curve.

NN

546 THE INDUCTION COIL AND TRANSFORMER.

An analytical proof of this is as follows : If the ordinate y of a periodic curve is represented by a Fourier series, as it can always be if periodic and single valued, then y may be expressed by the series

y = A sin p t + B cos p t + C sin 2 pt + D cos 2 p t + &c., -where A, B, C, &c., are constants. Hence,

yd« = l P

— sin 2»«- — cos 2ni + &c..

/

D

'Bsmpt- A cospt + —

C

• — cos 2/j

1800

160

A

\

r

FIG. 189.— Primary E.M.F. Curve (), Primary Current Curve (\) and Secondary E.M.F. Curve (\) of 10-light \Vestinghouse Transformer at full load.

It will be seen that the result of the integration has been to effect a change of phase of all the components and to weaken the higher harmonics by diminishing the coefficients which denote their amplitudes. Hence the process of forming a new periodic curve by taking as ordinates the area of a first

THE INDUCTION COIL AND TKANtFJBMER. 547

periodic curve up to successive abscissae always has the effect of wiping out irregularities of form of the primary curve, and yielding a curve more nearly a simple sine curve. It follows, therefore, that the curve of magnetic induction is always less irregular than the curve of primary or secondary electromotive force from which it is derived. From what has been already said, it will be seen that the curve of magnetic induction in the core has its maximum value at the moment when the electromotive force curve from which it is derived has its zero value. We may, then, sum up the general facts about transformer indicator diagrams by saying that when a trans- former is at work we have —

1st. A varying potential difference between the primary •terminals Avhich follows a certain wave form depending — (a) On the nature of the alternator; (6) On the state of the load of that alternator, whether

full or light, inductive or non-inductive ; (c) On the nature and construction of the transformer

connected to the alternator.

No assumptions must be made as to the form of this curve, "but in every case its true form at the terminals of the trans- former under test must be determined. The curve of primary electromotive force has widely different forms in the cases met -with in practice.

2nd. If the transformer has its secondary circuit open, we have a primary current flowing into its primary circuit which is called the magnetising current, and which lags in phase behind the curve of primary electromotive force. As the trans- former secondary circuit is loaded up this curve of primary current is brought more into step with the primary electro- motive force under the conditions that the load on the secondary circuit is a non-inductive load.

The curve of primary current is an irregular periodic curve the form of which is affected by the form of the curve of primary electromotive force and by the nature of the trans- former, and may have very different forms as these two operating causes are changed.

3rd. We have a curve of secondary terminal potential difference which is in exact opposition to the curve of primary terminal potential difference when the transformer

548 THE INDUCTION COIL AND TRANSFORMER.

is on open secondary circuit, and which is an exact copy of the curve of primary potential difference to a reduced scale. This curve of secondary potential difference may be shifted forward in phase as the transformer secondary is loaded up, so as to come more nearly into opposition with the curve of primary current.

4th. We have a curve of magnetic induction, and this induction is not the same in different parts of the core, or the same on open secondary circuit as at full load. The form of the curve is always more nearly a simple periodic curve than is the form of the curves of primary and secondary terminal potential difference.

Each of these curves heing a periodic single-valued curve, can be expressed by a Fourier series and analysed into con- stituent harmonics. Thus the ordinate ^ of the curve of primary potential difference corresponding to any instant t reckoned from the beginning of the phase can be expressed by the series

«i = E! sinp t + F! cos^> t + E3 sin 3 p t + F3 cos 3p t

• E5 sin 5p t + F5 cos 5p t + &cv

The constant or first term of the Fourier series is zero, because the curve is always symmetrical above and below the axis of time. Moreover, only the odd harmonic constituents are present, viz., the harmonics whose wave lengths are one- third, one-fifth, &c., of the fundamental wave length, and if by any form of harmonograph we mechanically resolve any of these transformer curves, we find that they can be quite adequately represented by the first three odd terms of the Fourier series — that is to say, we can build up any transformer curve by adding together the ordinates of three simple periodic curves the wave lengths of which are in the ratio of 1;3; 5, the amplitudes and relative positions being suitably chosen. The reason for the absence of the even harmonic constituents — viz., those whose wave lengths are |, £ that of the fundamental — is to be found in the peculiar symmetry of these transformer curves. On looking at any transformer curve it will be seen that it is of such a character that, if the portion below the time AXIS be considered to be reversed, we should get a repetition of the same form. Thus, a curve of electromotive force

THE INDUCTION COIL AND TRANSFORMER. 549

in Fig. 190, when so treated, becomes rectified into that in Fig. 191.

On considering, then, the form of any curve, it will be seen that the harmonic constituents must be such that if we move forward 180deg. along the time axis the value of the ordinate becomes negative but remains the same in magnitude.

If e represents the ordinate of any curve at any point corresponding to an instant t, and if p as usual is 2?r n, where

FIG. 190. — A Transformer Curve showing the typical symmetry of all transformer curves.

Fm. 191.— The same Transformer Curve shown in Fig. 190, but with the second half of the wave rectified to show the typical symmetry ol curve.

w is the frequency, then we can represent the value of e by the series

• E3 sin 3pt + ~Fs cos The harmonic constituents must be such that if we put (pt + v) for pt the value of e becomes — e.

It is easily seen that, since sin (pt + ir)=-sin pt and sin {8(ji* + 7r)} = -sin8j? t, &c., whereas sin {2 (pt + ir)} = sin 2 pt, the essential condition is that only the odd

550 THE INDUCTION COIL AND TRANSFORMER.

harmonics must be present. Hence the expansion of the ordinate of the real transformer curve can only contain the 1st, 3rd, 5th, &c., terms. As a matter of experience it is found that any transformer curve met with in practice can very

FIG. 192.— The Harmonic Analysis of the Curve of E.M.F. of a Thomson- Houston Alternator at no Load. The thick Curve C is the Curve of E.M.F., and the Curves marked HI, H3, H5, are the Harmonic Constituents-

with Wave Lengths in the ratio of 1, 3, 5.

Fio. 193.— The Harmonic Analysis of the Curve C of E.M.F. of a. Thomson-Houston Alternator partly loaded up on water resistance. The Harmonic Constituents of the Curve are represented by the Curves marked HL H3, HB.

nearly be represented by the first three odd terms of the series, and hence any observed transformer carve can be very quickly analysed into its constituents by the arithmetical process explained on page 92.

By the use of mechanical harmonographs or analysers thia can, of course, be very easily done, and an illustration ia

THE INDUCTION COIL AND TEANSFOEMER. 551

given in Figs. 192, 193 and 194 of the E.M.F. curve of a Thomson-Houston alternator so analysed. The curves given in Figs. 192, 193 and 194 were analysed for the author by Mr. G. U. Yule, with his mechanical harmonograph. It will be noticed that when the curve C is symmetrical, the harmonic constituents start from the same point, and have no lag relatively to one another.

FIG. 194.— The Harmonic Analysis of the Curve C of E.M.F. of a Thomson-Houston Alternator, partly loaded up on Inductive Resistance. The Harmonic Constituents of the Curve are represented by the Curves marked Hj, H3, H5, with wave lengths in the ratio of 1, 3, 5.

§ 4. Derivation of Curves of Power and Hysteresis.— From the curves of current, electromotive force, and induction obtained as above described we can construct two other curves which give us the variation of the total power supplied to the transformer, and the total loss in the iron core per cycle. These curves are obtained as follows : — Let us assume that a set of transformer curves has been taken when the trans- former is on open secondary circuit. Taking the curves of primary current and primary terminal potential difference, we multiply together (as explained on page 153, § 23, of Chapter III.), the corresponding ordinates of the two curves for abscissae taken at equidistant points on the time axis, and set up a new ordinate representing the value of the product ei, where e is the primary potential difference and i the primary current at the same instant. This product set off as an ordinate defines another curve called the power curve, and the true mean ordinate of this power curve gives us the

552 THE INDUCTION COIL AND TRANSFORMER.

mean power taken up in the transformer at no load. To obtain the true mean ordinate of the power curve we have to integrate the whole area included between the power curve and the time axis, and to reckon those areas which lie above the time axis as positive and those which lie below as negative. The total area of the positive and negative parts algebraically added, and divided by the length of the axis representing one complete period, gives us the true mean ordinate of the power curve. Hence, we can, from the transformer diagram taken on open secondary circuit, determine the mean power taken up in the transformer. The amount dissipated in heat in the copper of the primary circuit is generally an exceedingly small fraction of the total loss, and hence the mean power obtained as above is practically the value of the power taken up in the iron core.

The analytical expression of this fact is as follows : Taking the fundamental equation for the transformer on open secondary circuit, viz.,

%-BA+SNrll

at we multiply the equation all through by i\ and obtain

at£?,

d t

or e1It =

If this last equation is integrated between the limits 0 to

m

_ , where T is the complete periodic time, and each integral

Q

multiplied by — , we obtain an expression for the mean power given to the transformer during one half-period. Thus,

The first term on the left hand side represents the true mean power given to the transformer in one half-period. The second term represents the power dissipated as heat in the primary circuit in one half-period, and the third term repre- sents the power dissipated on eddy currents and hysteresis in

THE INDUCTION COIL AND TRANSFORMER. 553

the core in the same time. If a horizontal line is taken, and from an origin distances are set off right and left to represent the varying values of the primary current i during the period, and vertical ordinates corresponding to these abscissas taken to represent the values of the induction density b in the core at the same instant, then a curve will be defined which will be a cyclic curve, and will give us the total core loss per cycle when the numerical value of its area is multiplied by the factor NjS. If the, iron core is well laminated, eddy current loss will be practically absent ; and the value of this area, therefore, will give us the true hysteresis loss in the iron.

4000

/

f

7

1000

/

/

L

/

/

/

y

'

I

t

-^

X

3000 4000

4

3

2

i

\

i

•>

3 4

Magnetuiny Force. FIG. 195.— Hysteresis Curve of a Ganz Transformer.

Hence, such a curve is called the hysteresis curve of the core. In Fig. 195 is shown the hysteresis curve of the Ganz trans- former, so obtained from the current and induction curves of the same transformer as given in Fig. 170. • In order to obtain the correct numerical value of the hysteresis loss per cycle it must be noted that if all the quan- tities S, i and b are measured in C.G.S. measure, the value oi the integral S^J idb will give us, when taken round one complete cycle, the value of the core loss in ergs during one

554 THE INDUCTION COIL AND TRANSFORMER.

complete period. And this value has to be divided by 107 to reduce it to joules. Since the integral SN1lidb can be written l(N i) d(Sb), we see that the core loss per cycle in

joules can at once be obtained by taking the area of a loop curve, the horizontal ordinates of which represent the periodic values of the primary ampere-turns, and the vertical ordinates the corresponding total core induction during one complete period, taken in a unit equal to 108 C.G.S. units of magnetic induction.

If the frequency is n, then n times the above integral gives the loss in the core per second ; and this should have the same numerical value as the mean ordinate of the power curve which measures the same quantity. The practical rule, therefore, for obtaining the core loss in the transformer due to the hysteresis and eddy current loss which may be present is as follows : Draw two axes at right angles ; on the hori- zontal axis set off right and left from the origin distances which represent the primary ampere-turns for the different instants during one complete period. At these points set up ordi- nates which represent the total induction in the core measured in units each equal to 108 C.G.S. units of magnetic induction, and complete the looped curve defined by these ordinates. The area of this curve, measured in terms of the area of a rectangle one side of which is the length taken to represent one ampere-turn and the other side is the length taken to represent 108 C.G.S. units of induction, will give the value of the core loss per cycle in joules, and multiplication of this value by the frequency n will give the mean loss of power in the core in watts. The number so obtained will agree closely with the value of the mean ordinate of the power curve in those cases in which the copper loss in the primary circuit when the transformer is not loaded can be neglected.

The form of this hysteresis loop will depend upon the manner in which the magnetic induction in the core varies with the magnetising force, and, as we shall see presently, the area of this hysteresis loop depends, amongst other things, upon the form of the curve of primary impressed electromotive force.

THE INDUCTION COIL AND TBANSFOEMEK. 555-

§ 5. The Efficiency of Transformers. — If the secondary circuit of the transformer is closed through a resistance, and the transformer is therefore loaded up, the power given to the primary circuit in part reappears in a transformed form in the external secondary circuit. As by far the most frequently presented case in practice is that in \vhich the resistance which closes the secondary circuit consists of incandescence lamps or other practically non-inductive resistances, we shall, therefore, in the first instance assume that the external secondary circuit is an inductionless resistance. Under these conditions the secondary current is, to a close approximation, in step or synchronism with the secondary electromotive force, and the mean power given to the external secondary circuit is measured by the product of the mean-square value of the secondary current strength and the mean-square value of the potential difference of the secondary terminals. If we denote by P2 the power thus given up to the external secondary circuit, and similarly by Pt the power given up to the primary circuit, the ratio of P2 to P1 is called the efficiency of the trans- former. This efficiency is generally expressed as a percentage, and will be denoted by the symbol e. Hence

.-100%.

The difference between Pl and P2 is represented by the power lost in the core and dissipated in the copper circuits of the transformer. If the symbol Cj stands for the mean-square value ( Vmean2) of the primary current, and C2 for that of the secondary currents at any time, and if Rx and E.2 are the resis- tances of these circuits when warm and at that time, then the power wasted in the primary and secondary circuits respec- tively is (V E2 and C22 R2, and if H is the core loss, viz., the hysteresis and eddy-current loss, then

P1-P2=C12B+C22E2 + H,

on the assumption that there are no eddy-current losses or energy dissipations in the copper circuits, or in the iron case or framework of the transformer.

One of the most important measurements, therefore, which it is necessary to make in connection with transformers is the

556 THE INDUCTION COIL AND TRANSFORMER.

measurement of the power given to the primary circuit. We shall defer to a later chapter on transformer testing a full discussion of the various methods which can be employed for determining either the magnitude of the quantity Px or of the difference P: - P2 in the case of a transformer at any load. It may suffice to state at present that one way in which this measurement can be made is by means of a properly con- structed wattmeter, which measures directly the power Pj given to the primary circuit. The objection to this method is that any error made in evaluating Px appears to the same extent and percentage in the ratio of P2 to P1? and therefore in the efficiency. Hence other methods have been devised for measuring directly the difference Pa - P2. However the value of the efficiency may be determined, the results are best set down in the form of an efficiency curve as follows : Each transformer is constructed to give safely a certain output of power to the secondary external circuit, which is called its full load, and is stated generally in .watts or kilowatts. The load on the secondary circuit in any other cases can be expressed as a fraction of the full load. To draw an efficiency curve for any transformer, a horizontal line is taken, on which are marked off the decimal fractions of the full load,

and at these points are set up ordinates which represent the

p

percentage efficiencies at these loads, viz., the value of 100—?,

PI

where Px is the power given to the primary circuit and P2 is the power given to the external secondary circuit. The ex- tremities of these ordinates delineate the efficiency curve.