The Alternate Current Transformer Vol. 2: The Utilisation of Induced Currents (1896) — part 35 of 36

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 jnarked ni that obtained in the same way, but by the use of

ie,ooo ^,

'^"" rj^^^r

10.00. j^^

,000 U2 _

».m jff.

kJ~~mI

V-tL

ft.000 .

oZ

8 IS 16 20

MagnetUifig Force,

FlO. 211.

ts

-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. 691

secondary currents, increase the primary current, and, second, the existence ^of magnetic leakage when alternating currents «re 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 sameB.M.S. values of the primary electromotive force being employed.

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

and Jni.i/=:g Tj,

where ^ 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, Z^ and Z2 will be greater for peaked current and potential curves than for flat or rounded curves.

Accordingly, whilst the respective primary and secondary inductions Z^ and Z, are greater for electromotive force curves with large form factors, their difference, Z^ - Z|, 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 hi is the instantaneous value of the magnetising force due to the primary current t, then

BOS TBB INDXrCTION COIL AND TBANSFOBMBB. and on open secondary drcoit we have

Hence •,-B,h+N,S,^.^.

This last equation is true for all forms of primary electro* motive force carves. If the permeahilitj of the iron was constant, the value of

— I would be constant and equal to fi, but in practice it is not

found to be constant. We see, however, that jj^ 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

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

^, or the slope of the current curve, will be greater than for at

flatter curves of electromotive force. This is seen to be the case by reference to Figs. 208 and 204, which show the no- load primary-current curves and primary electromotive force curves of the same transformer tested by Dr. Boessler on the Oanz 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 hysteresia 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 reluctajice of the magnetic circuit, it is

TEE INDUCTION COIL AND TBANSFOBMEB. 5»3

not of mnoh nse to make assumptions whidi 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 pzunary electromotive force. For if X is the total power taken up in the core at no load^ and Jme^^ ^wt^* are the B.M.3* values of the primary electromotive force and current^

and P is the power factor, then F= , 7==, from

which Jm.iy^ can be obtained.

It is seeUi however, that the primary current curve at no> load is always a more irregular curve than the curve o£ primary electromotive force, and that for the same B.M.Sv. value of the primary electromotive force the B.M.8. value off the primary current at no load (the magnetising current) is less for pointed or peaked potential curves than for fiat curves.

§12. Iron Gore Lobs in Transformers and Induction Ooils. — 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 beizkg 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 TBAN8F0RMER.

anywhere in the iron plate, in a plane perpendicular to the lines of induction, then, if « 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 B.M.8. value of the electromotive force of induction induced in the primary circuit is equal to the value of the expression 4/Ni . n S B, where/ is the form factor of the curve of electromotive force.

Hence the mean-square value (m,e^) of the electromotive force of induction must be numerically proportional to/* n* B-; 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 /^n^>B*. In other words, the eddy-current loss will be equal to f /^ n^ B*^ 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 &ct that the hysteresis loss per unit of volume of the oore is proportional to the product of a constant, the freqaencji 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 oore per unit of volume,

H-iyiiBi',

where ij 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

T-V(iy«Bi-«.Hf7/2/^B«). . . . (162)

THE INDUCTION COIL AND TBANSFOBMEB. bm

The eddy-^snrxent loss varies as the square of the tnaximum value of the induction, and the hysteresis loss as the l'6th fpower of the same.

Since the B.M.S. value {Jm7^) of the primary electro- tnotiye force has been shown to be related to B by the • equation

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

This last equation shows us that the eddy-current loss is «iot affected by the form factor of the curve of primary electro-

a-

I

40

I

.»

y

^t<^

A^^

-,^^

u

2^

CCi

^^

— ^-^'^'*'

of "^ « 00 g" M V

Ilaodmwm Value €f Core InduetiofL

Fig. 212.— Curve I taken with Oanz Macliine. Curve II taken v/ith Wechfller 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. O. Boessler, 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 •deipree.

596 THE INDUCTION COIL AND TBANSFORMER.

This dependence of the core loss upon the fonn £AOtor of the ccmro of electromotive force is shown by the results 6f Dr. Boessler's experiments embodied in the diagrams in Figs. 206 and 212. From Fig. 212 it will be seen thai at a giyen induction the core loss is greater when the trans- former is worked off the Oanz alternator than off the Wechsler alternator ; and from Fig. 206 it is shown that for the same B.M.S. value of the primary potential difference the core loss is greater with the Wechsler than with the Oanz alternator.

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

In the Second Volume of this Treatioei we retum to the discussion of these matters, and enter into more details aa to the practical considerations to which they lead m the Construction of the Induction Coil and Transformer*

EVD OF VOLUMB I.

APPENDIX

Note A. (See page 19.)

The Magnetic Force at any Point in the Plane of a Oircolar Onrrent. — The magnetic force at any point in the plane of a circular conductor conveying a current may be found by an elegant geometrical method due to Mr. A. Bussell (see The Electrician, Vol. XXXI., p. 284).

Let P be any point in the plane of a circular current, and let F be the magnetic force at P due to a current of strength 'i in the wire. Let (f < be an element of length of the circle^ und let OB be the radius of the circle. Take any point B on the circumference of the circle {see Fig. 1), draw the diameter through 0 P, and at B draw a tangent to the circle. Then draw the radius 0 B, and through P draw P N perpen- dicular to the tangent. Join PB. Let OP-^o, OB^B, PB=r, PNo;?, and the angle BPN»^.

598 APPENDIX.

Then, by Ampire's law, the magnetic force P at P due to the element dsot the circuit at R is equal to -^-p ^ . Hence

J 7^ r But pd8^i^d<l>,

hence F=/ — ^;

and, since r« V R*-a*sin'<^-a0OS8^,

therefore,

1 V B,*- ff' sin' <^4-acos 6

r R»-a»

Hence,F-^^^J^V^R«^a«sin»<^rfi^ + g^.f^<5os^

The second integral, taken between the assigned limitSi is

zero. The first integral, j VR^ -a* sin* <^ (f <^ is called an

elliptic integral of the second order, and represents the Ipngtb of the circumference of an ellipse which has 0 for its centre^ P for one of its foci and 2 R for its major axis.

Hence, if we describe an ellipse on the diameter of the circle, with 0 as its centre and P as its focus, the magnetic force F at the point P due to a current i in the circle is equal

to the value of t^— « x the length of the circumference of this R* - a'

ellipse.

In practice we can easily describe this ellipse by means of two pins and a thread, and then measure the length of its circumference with a measuring-wheel, such as is used for measuring distances upon a map, or by laying a thread round the ellipse and measuring its length.

In this way a practical measurement can be made of the magnetic force due to a circular current at any point in its- plane.

Up to the limit of a=0'8R, the length I of the oircum* ference of the ellipse can be calculated approximately from the formula —

APPENDIX. 699

Note B. (See page 85,)

The Total Indaetion in a Oircnlar Solenoid. — If we consider a circular-sectioned ring to be closely wound over with turns of wire, we obtain a circular solenoid.

Let a be the mean radius of the circular cross-section of the solenoid, and let B be the radius of the circular axis of the solenoid. The whole solenoid may be considered to be resolved into elementary solenoids. Let (2 8 be the cross* section of one of these, and let x be the radius of the circular axis of the circular elementary solenoid.

Let N be the total number of turns of wire on the ring*

Then, for the elementary solenoid, there are o^ turns per nnit of length.

The magnetic force in the interior of the elementary solenoid is H, and ^_47rNI_2NI

27r X X

where I is the current in the solenoid.

Hence, if the medium is non-magnetic, the surface integral of magnetic force is the measure of the induction. Hence the magnetic induction in the interior of the elementary solenoid is

2NI

d S. Hence the whole induction Q through the whole

solenoid is obtained by integrating this last expression over

the area of the solenoid, viz., ira^. Therefore between proper

limits f ; <

Q = 2NI/'1?

J X

— taken over the circular cross-section can X

be shown to be equal to 2ir {R - -v/R* - a*}»

and hence Q - 4ir N I {R - VW^%

This, then, is the expression which should be employed for the total induction over the circular cross-section of the ring, if the mean radius of cross-section a is not very small compared with the radius of the circular axis B.

If — is a small quantity, then, since

fS

«00 APPENDIX.

we have R - V^R*— a' = ttq ^^^^Y ^^en ^ is small, and hence

or ^TT'times the current turns per unit of length of solenoid multiplied by the area of cross-section of the solenoid; and for a very large thin circular solenoid the magnetic force in the centre is Air times the .current turns per unit of length.

Note C. (See page 52,) . The Oalibration of the Ballistic Galvanometer. — A galvano- meter consists generally of two parts — a magnet and a coil of wire. ' The magnet may be fixed and the coil suspended and movable; or the magnet suspended and. movable and the coil fixed. If the arrangements are such that the movable sjrstem when disturbed is brought to rest without vibration, or with very few vibrations, the galvanometer is called a damped or aperiodic galvanometer. If, however, the resistance to motion is so small that the movable part, when disturbed or made to oscillate, continues for a long time to oscillate with very slowly decreasing ampUtude of vibration, the galvano^ meter is called a ballistic galvanometer.

If a body is suspended so as to vibrate about a vertical axis, and if when given an angular displacement about 'that axis it returns when released to its original position, and oscillates about it, the body will in general execute vibrations of decreasing amplitude. If I is the moment of inertia of the body about that axis, and if /a is the torque or couple which tends to restore the body to its original position if displaced round the axis by a unit angle, then, if at any instant the angular velocity of the oscillating body is a>, the equation of motion, neglecting for the moment resistance to motion, is

where 6 is the angle of displacement at the instant when the angular velocity is w. .

Since <d = _- , we have as the equation of motion, at

df 1 '

. APPENDIX. eOl

and the time t of one complete vibration of the body will be

This gives us the periodic time .of oscillation of the body, assuming that the resistance to motion due to friction is nothing.

If there is any small resistance to the motion, such as air resistance or other causes of energy dissipation/ the amplitude of the swings of the vibrating body will gradually decrease*. If the experiment is tried of setting the needle or coil of a ballistic galvanometer in motion, and observing the amplitudes Xi, A'a, a?8, <&c., of successive swings to the right and lefb, ii will be found that x^, a^, x^, &c., form a descending geometrical progression, and that the values of log x^, log x^, log x^, &c.| form a descending arithmetic progression.

If the difference of the logarithms of the amplitudes of two successive swings, one to the right and one to the left, is ■taken, this difference, denoted by A, is called the logarithmic decrement of the galvanometer, and it will be found that this difference is approximately constant for any two successive swings right and left during the progress of the decay of the excursions. That is, log x^ - log a: » A, also log or, - log o^ - A, .&C. Hence, if the logarithm of the amplitude falls off or decreases by an amount A in the course of the passage-of the needle or coil of the galvanometer from the extremity of one swing to the right to the extremity of one swing to the left, it may fairly be assumed that, if the coil or needle is started from rest by a sudden blow, the excursion x^ it would make if there were no resistance is related to the excursion x^ it does actually make with the actual resistance elisting by the relation

loga:o-logj:i«-,

or log,o?^ = ^,

Xi 2

or ^=107.

If the base of the logarithms is 10, or the logarithms are ordinary ones, then by the exponential theorem we have

.602 APPENDIX.

where M is the tnodtdus 2*808, or the multiplier for converting ordinary logarithms into Napierian logarithms. M is the logarithm of 10 to the base e, the base of Napierian logarithms.

If - is small, we may neglect j. and higher powers in com- parison with -, and write

• A.

10«=1 + M^, nearly.

Hence 5i = l + M ^, nearly,

Xi 2

or

fl?o=a;i(l+M-j, nearly.

In other words, we can find what the excursion x^ would be if there were no air resistance by multiplying the actual first

excursion o^^ by a factor ( 1 + M - j called the correcting fiictor.

Hence, if the movable system of a ballistic galvanometer receives a sudden blow or impulse when at rest, and if it then makes an excursion the angular magnitude of which is x^, and if such excursion is slightly resisted by air friction, we can eliminate the results of this and correct for the frictional resost-

ance by multiplying Xi by the correcting factor f 1 +M- j.

If the logarithmic decrement is measured directly in Napierian logarithms, then the correcting factor is simply

H)

where A equals the Napierian logarithmic decrement, or the logarithm of the ratio of one swing to the next.

Returning to the equation of motion of the needle or coil, since we have at any moment

at

let us consider a small magnetic needle of magnetic length I hanging in the centre of a coil so wound that when a current flows through the coil there is a uniform magnetic field due to the current in all the region within which the needle lies. Let the normal position of the needle when at rest be such

AFFENDIX. eoa

that the magnetic axis of the needle is at right angles to the direction of the field due to the current in the coil.

Then let M be the magnetic moment of the needle, H the strength of the controlling field, the direction of H being at right angles to the field due to the current in the coil.

Let the needle be set oscillating by a sudden impulsive couple acting upon it when at rest. Let d be the angular displacement of the magnetic axis of the needle at any instant /•

The restoring couple acting on this needle at that instant is M H sin 6, and, by the equation of motion,

li!:!=MHsin^. dt

Hence, I(fcu4^ = MHsin^f?^;

' dt

but — — = CO.

dt

Hence I (o dcu = M H sin 0 ^ ^.

Integrate the above equation from the instant when the needle leaves its position of rest with a finite angular velocity 12 until it reaches a displacement d and has a zero angular velocity. We have

ir wdto^Unj^inOde, or JIi2« = MH(l-cos^),

= 2MH8in*^-;

therefore 12 = 2 /^sin|. ... (1)

2

If the impulse which starts the needle from rest with il finite velocity is that due to the flow of a quantity of electricity through the coil surrounding the needle, which flow is all over before the needle has had time to move sensibly from rest, we can obtain a relation between the quantity of electricity so sent through and the excursion of the needle.

For let i be the current in the coil at any instant, and let O be a constant depending upon the form of the galvanometer

tM APPENDIX.

toils, such that O t id the magnetic field due to the onrrent* Then, since M is the moment of the needle, M 0 1 is the whole oouple acting on needle, and the impulse of this couple is "HGidtin Bk time d t. If the whole impulse is over before the needle has had time to move, then the whole impulse of the couple must be equal to the total gain of angular momen- tum I dot taking place in .the time d t.

fience UQidt^ld<a;

but it dq is the quantity of electricity which has flowed through the galvanometer coils in the time dl, we have

• ' dt

and MGtie»M6ig«I(2oi.

But since the whole impulse is over before the needle has time to leave its position of rest, we have, by integrating from &> = 0 to CO » 12, the equation

MGQ = in,

where 12 is the angular velocity with which the needle leaves its position of rest. But by equation (1),

therefore ^"'V'^"°|'

or Q — fcsin-.

Hence we see that, when a quantity Q of electricity is dis- charged suddenly through the coils of a ballistic galvanometer, the whole discharge being over in a very short time compared with the period of free vibration of the needle, the quantity of electricity is proportional to a constant, ifc, called the ballutie comtanty multiplied by the sine of half the angle of excursion of the needle.

If the galvanometer has a logarithmic decrement A, which is moderately small, say not more than 10 per cent., then to a close approximation

Q = *sm|(l+^).

APPENDIX. ea^

To determine k for any galvanometer, the easiest way is to charge a condenser having a capacity of c microfarads to a potential of r volts by placing it in contact with a battery. Qf known voltage, and then discharging this quantity c v micr<>- coulombs through the ballistic galvanometer. If this quantity. cv gives a '* throw " ^, we have

cv •

Hence k is determined.

Such a calibrated ballistic galvanometer can immediately be employed to measure a magnetic field. For if a loop of wire having N turns is placed in a field of induction so that the induction is linked with the circuit, and if the loop is connected with the ballistic galvanometer, then on suddenly withdrawing the loop from the field we shall get a " throw " of the galvanometer which can be interpreted to mean so much quantity in microcoulombs passing through the galvano- meter< Then, by the principles explained on page 81, the total change in induction linked with the circuit is equal to the product of the resistance of the circuit and the quantity set flowing through it, or

Induction x linkages « resistance x quantity.

If induction is measured in microwebers— one weber bein^ 10^ C.G.S. lines or units of induction, and one microwebec^ therefore 100 C.G.S. units — we have the rule —

Microwebers x linkages = microcoulombs x ohms ;

and if one microweber of induction passing through the loop linked once with this circuit, is suppressed, it will cause one microcoulomb of electric quantity to flow through the circuit if its resistance is one ohm.

In employing the ballistic galvanometer to measure mag- netic induction, it must be observed that the condition of application of the above principles is that the whole change of induction must be completed in a very short time compared with the periodic time of the needle of the galvanometer.

To suggest, as is sometimes done, that the induction in tha field magnets of a dynamo can be measured by surrounding

-606 AFFENDIX.

them with a loop of wire connected to a ballistic galvanometer, and then short-circuiting or breaking the field circuit, is to presuppose a most unlikely event, viz., that the time during which such change of induction takes place is small compared with the periodic time of an ordinary ballistic galvanometer.

If the galvanometer is one with a movable coil of the d'Arsonval type, then it is better to standardize the galvano- meter by means of a coil of known length, turns and resistance, placed in the axis of a long cylindrical coil, the turns per unit of length of which are known. The reason for this is that if the galvanometer circuit is always closed the movement of the coil in the strong magnetic field induces currents in the galvanometer which resist its motion. Hence a powerful damping action comes into play from this cause alone. To ascertain what induction change caused a given galvanometer throw we proceed thus : —The standardizing secondary coil should be joined up in series with the galvano- meter coil and with the secondary or exploring coil, and it should be placed in the axis of a long coil, of which the field can be calculated. If, then, under the influence of . an unknown induction linked or unlinked with the exploring coil we obtain a galvanometer throw 0^ we can find out what. was the induction causing this throw h^y interrupting or reversing a known current in the long standard field coil, and thus Unking or unlinking a known induction with the standard secondary coil. If the current through the standard field coil is altered until it gives a similar throw 6 on being interrupted or reversed, we know at once that the value of the unknown induction linked or unlinked with the secondary or exploring coil which gives an equal throw, must be the same as that calculable induction linked with' the standardizing secondary coil.

INDEX.

Action at a Distance, 10 Actionn taking place in Transformera, 518 Admittance of Condenaer, 186 Alternatiug Current Curve Tracers, 522 Alternating Current Curve Tracing, 530 Alternating Current Flow in Conducting

Circuits, 492 Alternating Current Flow, Theory of, 293 Alternating Currents, Diritribution of, in

Conductors, 306 Alternating Currents, Propagation of,

through Conductors, 292 Alternative Path, Ex}>eriiiient of the, 401 Ampdre'a Diacoveries on Electromagnetism,

307 Ampere's Fundamental Law, 15 Amplitude Factor of a Periodic Curve, 583 Analysis of Compound Periodic Curve, 90 Analytical Theory of Transformer, 586 Apparent Power, 157 Apparent Power given to Circuit, 205 BaUiatlo Galvanometer, 51, Api)endix Note

C.,600 Ballistic Galvanometer, Use of, 51 Bernstein's Researches on Induction, 251 Blasema's Researches on Induction, 246 Branch Circuits, Impedance of, 163 Cardew Voltmeter, 155 Circuit, Inductive, 110 Curcuit, Magnetic, 33 Circuit Non-Inductive, 110 Ourcular Current, Magnetic Force at the

Centre of, 18 Circular Solenoid, Magnetic Force in In- terior of, 23 Classification of Transformers, 517 Clock Diagram for Inducing and Induced

Circuits, 177 Clock Diagrams, 141 Closed Circuit Transformer, 615 Closed Magnetic Circuit, 38 Coefficient of Mutual Inductance, 121

Coefficient of Self-induction, 119

Coil Induction, Theory of, 232

Complex Periodic Functions, 202

Compound Periodic Curve, 84

Compound Periodic Curve, Harmonic Analysis of, 91

Condenser Equation, 183

Condenser, Flow of Current into» 182

Condenser, Time-Constant of, 184

Conducting Power for Lines of Force, 40

Conduction Current, 335

Confirmation of Maxwell's Theory by Hertz, 477

Correcting Factor of Wattmeter, 168

Current and Electromotive Force Curves, 81

Current Diagram of Transformer, 561

Current Equation, 126

Current Flow in Circuits having Capacity, Inductance and Resistance, Initial Con- ditions oi, 199

Current Flow in Inductive Circuits, Initial Conditions of, 194

Current Growth in Inductive Circuits, 123

Current Sheets 339

Curve of Hysteresis, 60

Curve of Induction, Determination of, 528

Curves, Logarithmic, 130

Curves of Current and E.M.F. of Various Transformers on Open Secondary Cii-cuit, 538, 539, 540

Curves of Magnetisation, 51, 55, 58

Curves of Power and Hysteresis of Trans- former, 551

Curves of Primary and Secondai-y Alternat- ing Electromotive Force of Transformer, 542

Cycle, Magnetic, 60

Delineation of Periodic Curves of Current and Electromotive Force, 519

Derived Curves, 103

Description of Simple Periodic Curve, 96

Description of Transformer Diagrams, 535

608

INDEX,

Determination of Values, **v," 359 Dielectric Constants of Various Substances,

361-365 Discovery of Electromagnetic Induction, 2 Discovery of Induced Currents, 2 Displacement Current, 354 Displacement Currents and Displacement

Waves, 338 Direction of Magnetic Force, 14 Dove's Experiments on Induction, 262 Effect of Closing Secondary Circuit of

Induction Coil, 271 Effect of Form of Curve of Electromotive

Force on the Efficiency of Transformers,

579 Efficiency Curves of a Transformer taken

on Various Alternators, 682 Efficiency Curves of Transformers, 557 Efficiency of Transformers, 565 Electrical Oscillations, 372 Electrical Researches of Faraday, 1 Electric Currents produced by Magnetism, 5 Electric Displacement, 334 Electric Elasticity, 334, 338 Electric Surgings, 412 Electric Waves in Wires, 461 Electrodynamic Induction, 3 Electromagnetic Energy, 120 Electromagnetic Gyroscope, 324 Electromagnetic Induction, 14 Electromagnetic Induction, Discovery of, 2 Electromagnetic Medium, 335 Electromagnetic Momentum, 118 Electromagnetic Radiation, 478 Electromagnetic Repulsion, 307 Electromagnetic Repulsion, Theory of, 316 Electromagnetic Rotations, 320 Electromagnetic Theory, 332 Electromagnetic Waves in Air, 469 Electromotive Force Curves, 81 Electromotive Force Diagram for Inductive

Circuit, 144 Electromotive Force of Induction, 72 Electromotive Foi*ce of Rotating Coil, 76 Electromotive Intensity, 337 Electro-Optic Phenomena, 478 Electrostatic Induction, 1 Electrotonic State, 7, 118 Elihu Thomson's Experiments of Electro- magnetic Repulsion, 313 Energy Dissipation by Hysteresis, 64 Equation for Charge of a Condenser, 183 Equation for Periodic Current in Inductive

Circuit) 135 Equation for Rise of Current in Inductive

Circuit, 127 Equipotenttal Surface, 46 Ether, 333

Experimental Determination of Electro-

magnetic Wave Velocity, 499 Experimental Determination of Instan- taneous Value of a Periodic Current, 620 Experimental Determination of the Form

of Periodic Curves of Transformer^ 627 Experimental Measurement of Periodic

Currents and Electromotive Forces, 154 Experimental Proof of Existence of Oscilla*

cory Discharge, 381 Experimental Researches on Alternating

Current Transformers, 558 Faraday's Copper Disc Experiment^ 6 Faraday's Discoveries, 1 Fai'aday, Discovery of Self-Induction by,

111 Faraday's Electrical Researches, 1 Faraday'sExperiment with the IronRing, 3 Faraday's Law of Induction, 31 Faraday's Ring Coil, 9te Frontispiece Faraday's Theories, 6 Flow of Simple Periodic Currents into

Condenser, 182 Flux of Magnetic Force, 26 Force, Magnetic, 43 Form Factor of a Periodic Curve, 583 Form Factor of Various Curves, 584 Fourier's Theorem, 85 Function of the Condenser in an Induction

Coil, 390 Galvanometer, Ballistic, 51, Appendix Kote

C.,600 General Analytical Theory of the Trans^

former. 586 General Dwscription of the Action of

Transformer, 514 Geometrical Illustrations, 138 Graphic Representation of Periodic Cur*

i-ents, 139 Harmonic Analysis of Transformer Dia- grams, 550 Harmonic Curve, Description of, 87 Hedgehog Transformer, Current Diagram

of, 562 Helmholz's Researches on Induction, 252 Henry, Discovery of Electro-magnetism.

by, 11 Henry, Joseph, 11 Henxy Joseph, Researches of, 207 Henry's Discovery of Induced Currents of

Higher Orders, 219 Henry's Electromagnets, 11 Henry's Experiments on Self and Mutual

Induction, 207 Henry's Experiments with £lectromagneta»

12 Henry's InvestigationSi 11 Henry, The, 122

INDEX

H«rtE^8 Confirmation of Maxwell's Theory,

449 Herte*8 Electrical Papem, 461 Hertz's Experiments of Propagation of

Alternating Currents in Conductors, 492 Hertz's Researches on the Propagation of

Electromagnetic Induction, 418 Hertz's Besonator, 426 High Frequency, Effect of, in changing

Distribution of Current in CoDductor,

294 Historical Introduction, 1 Hopkinson's Researches on Hysteresis, 65 Hughes' Experiments on Transmission of