The Alternate Current Transformer Vol. 2: The Utilisation of Induced Currents (1896) — part 6 of 36

The magnitude of this induced electromotive force is not in any way dependent upon the nature of the material of which this conductor is made. Faraday experimentally proved this (<< Exp. Bes.," § 198-201) by taking a double conductor com- posed of an iron and a copper wire twisted together and united at one end. On passing this double conductor through a magnetic field no induced current was detected in it by a galvanometer. This proved that the electromotive forces set up in each separate conductor were equal and opposite, and hence, since the lengths, field, and velocities were the same, no factor entered into the production of the effect, which depended on the nature of the conductor. From further

ELECTBO-MAQNETIC INDUCTION. 75

experiments with circuits partly metallic and partly electro- lytic fluids he inferred that in all bodies, whether what are commonly called conductors or non-conductors, or elec- trolytic conductors, identically the same electromotive force is brought into existence by moving the same lengths in the same way in the same magnetic fields.

When a metallic disc is rotated in a uniform magnetic field so that its axis of rotation is parallel to the direction of the field, there is set up a difference of potential between the centre and the edge. In this case we cd.n tap off a current by an external wire connected to the centre and the edge of the disc.

We can now show that, starting with the elementary law above stated, as to the magnitude of the induced E.M.F. in an element of a conductor, we can deduce the other principle of the relation of the induced E.M.F. to the rate of change of tbe induction through the circuit.

Let A B C D (Fig. 24) be a conducting rectangle, of which the plane is perpendicular to the induction lines of a uniform magnetic field of strength H, the same being shown in plan on the figure ; let the circuit be capable of revolving about an axis 0 0 in its own plane, and let it be displaced through any angle, d, as shown in elevation and plan in Fig. 20. If the frame is so displaced it is clear that the sides A C,BD "cut*' across lines of magnetic induction, but that the upper and lower sides do not. During this displacement the vertical sides alone will be the seat of electromotive forces. Imagine this frame to revolve round the vertical axis with a uniform angular velocity oi, and at any instant t to have a position such that its plane makes an angle 0 with the plane normal to the lines of force. Let the length of the side A G be L and that of A B be B : the actual velocity of the side

A C is ^!— , and the strength of the field, in a direction per- pendicular to its length and its direction of motion at that instant, is H sin ^. Hence the electromotive force of induction

in the side A C is — L H sin ^, and an equal and oppo-

sitely directed electromotive force acts in the side B D at the same instant. Hence the total electromotive force aotinff

76

slbctho-maqnstic mDVcnoN.

round the frame is equal to co H B L sin ^. . If the area of A B C D is denoted by A we may write the above as 0) n A sin 0. The angular velocity co may be expressed as

the time rate of change of 0, or as -— ; hence the expression

for the total electromotive force of induction round the frame

is HA sin ^'^^-, or- /^ (HAcos^).*

.f

A

i

. I

1

B

A

1

1

0

Ti

^

0

U

V

B

B -1 — -;

Fig. 24,

The expression A cos 6 denotes the apparent size of the frame as looked at from a considerable distance along the direction of the lines of induction, and the quantity H Acos ^ is the numerical value of the number of lines of magnetic induction passing through or traversing the frame in its posi- tion when its plane is inclined at an angle 6 to the normal position. We assume that these Unes are spaced out according

• We here suppose the (ircuit to be foimcd of a single loop of wire having a practically negligible self-induction. The above statements would require some modification for a circuit of many turns of wire.

ELnCTRO MAGNETIC INDUCTION. 77

to the rule proper for such distribution, viz., that the number passing through a unit of area whose plane is taken normal to the direction of these lines is numerically equal to the magnetic induction over that area.

Writing N for this number of lines so piercing through the frame at any instant, we have, as the expression for the total electromotive force acting round the frame at any instant, the

quantity - — ; that is, the electromotive force of induction dt

is numerically equal to the rate of change (decrease) of the included lines of induction. It is customary to speak of this induced electromotive force as generated either by the '* cutting of lines of force " by the various elements of the conductor or by a change in the number of lines of force piercing through the aperture of the circuit ; but they are merely two different geometrical ways of viewing the same phenomena. The actual results are capable of receiving a physical explanation on the assumption that the act of inter- section of a line of force and a portion of a conducting circuit is productive of an electromotive force. We see that the total electromotive force is the resultant effect due to a summing- up of all the forces acting on each element of the circuit, each elemental E.M.F. being measured by the product of the length of that element, the field strength around it, and its normal velocity in that part of the field. The result is concisely expressed by the number which expresses the time rate of change of the whole number of the lines of induc- tion traversing the circuit. This same may be extended to any circuit of any form moving in any way in any field.

If a circuit of any form which is traversed by an electric current is placed in a magnetic field due to other neighbouring currents or magnets, there is a flux of induction through that circuit due partly to the current in the conductor and partly to the external field of the other currents or magnets. If there be M lines of induction due to the external field passing through it, and N lines of induction due to its own current, any variation of the external induction, of which the rate of

change at any instant is represented by - ~, will produce an impressed electromotive force in such a direction that taking

78

ELECTRO^MAGNETIG INDUCTION.

lines of induction out of the circuit induces an electromotive force in the clockhandwise ( + ) direction, as seen from that side of the circuit at which the lines enter. When a current is flowing in any conductor, the relation between the direction of the current and that of its own lines of induction is the same as the relation between the thrust and the twist of a corkscrew. Htnce, it is evident that, if we consider a circular current (Fig. 25) with the current flowing in it clockhandwise ( + ), as seen from one side, its own lines of induction pass

through the circuit in the positive dirieciion, or away from the eye.

Accordingly, a little reflection 6hows that, if the current in the conducting circuit is made to increase, an opposing electromotive force is created by the increasing induction of the current on its own circuit. The current in the act of increasing crowds its own circuit more full of lines of induc- tion, and creates an electromotive force of induction during the period of this increase equal numerically at any instant to its own rate of increase, and directed in opposition to the impressed external electromotive force which is driving the current.

CHAPTER IIL

THE THEORY OF SIMPLE PERIODIC CURRENTS.

§ 1. Vaiiable and Steady Flow.— In the following pages we shall be ohiefij concerned in considering the properties and uses of currents of electricity which are periodic in character ; that is, which are changing in strength from instant to instant in a cyclic or periodic manner. An electric current or an electromotive force may either be steady, in which state it remains oniformly at the same value, or it may be variable^ in which case it is changing in value from instant to instant. In this last case we can consider two separate conditions. The current strength or electromotive force may be periodic or non-periodic in value. A non-periodic variable current or electromotive force is one which changes in value from instant to instant accordingly to any assigned law or mode, but in which the same series of values are not regularly repeated. A periodic current or electromotive force is one which runs through a regular cycle of values, returning after a certain period to the same value. It is accordingly said to vary in a cyclic manner, because it changes through a cycle of values. We may take illustrations of these three states from the flow of fluids. A stream of fluid may exist in a steady state ; in this case the motion of each particle of the stream has settled down into a uniform condition as regards velocity. If we imagine a small short tube open at both ends, held anywhere in that flowing fluid, the same volume of fluid would flow through that tube in every unit of time. We may, however, find the fluid in such a condition that the velocity of each particle of the fluid at any point is changing, and the flow is then in a variable condition

^UNi^ rvsixr)

80 SIMPLE PERIODIC CURRENTS.

If that change is of such a character that the motion is regular in its mode of change, then the flo^Hs said to be periodic. Thus, in a non-tidal river the water flows in general uniformly in one direction ; it is in a steady state. At the time of a flood its speed at any point may be rapidly increasing, and in this case its flow is variable. In the case of a tidal river the flow of water is regularly reversed, a cycle of fluid motion is repeated at any point, and the motion is said to be cyclic or periodic in character.

In considering the motion, either of actual fluids or of electric currents, we can, then, distinguish three states — the variable, the periodic, and the steady condition. In the first case the strength or direction of the electric currents or of the fluid velocity is changing at every instant ; in the latter cases the flow has settled down into a permanent state. The questions involved in dealing with the variable or periodic states present rather more difliculties than do problems in steady flow, for the reason that the notions of time and inertia enter into these in a way in which they do not when that flow has reached a steady condition. We shall proceed to examine in an elementary manner some features of electrical flow when variable or periodic. We must, however, prepare the way by considering some purely geometrical properties of certain curves, and also some modes of motion which have special reference to the kind of electric current to be considered sub- sequently. When a mass of water is in motion, a particle of water selected for examination has at any instant a certain velocity in a certain direction. This may be represented graphically by a straight line drawn frjm that particle representing its velocity in direction and magnitude. Similarly, if electricity is flowing through the mass of a conductor in any manner, it is possible at any point to draw a vector or line representing at that instant the direction and magnitude of the current at the point from which the line is drawn. Lines drawn within the mass of a fluid at any points such that the flow at that instant is along or tangential to these lines are called flow lines. In the first place, let us make the supposition that the flow has reached a steady condition. The flow lines are then fixed. When this is the case each line of flow becomes the actual path of a fluid

SIMPLE PERIODIC CURRENTS, 81

particle, and is called a stream line. A surface may be supposed to be described in the mass of the fluid everywhere perpendicular or orthogonal to the stream lines; such a surface is called an equipotential or level surface. We may also suppose such a level surface drawn in the mass of a con- ductor through which a current is flowing. Let any area be drawn on the equipotential surface, and let it be divided up into units of area. If the quantity of fluid or of electricity flowing through each unit of area is the same, and if, more- over, it is the same for each unit during each succeeding instant of time, the current is said to be steady and to be uniformly distributed. The quantity flowing per unit of time through any area is the numerical measure of the mean strength of current over that section of the conductor, and the quantity flowing per unit of time through a unit of area is the measure of the mean density of current over that unit of area. If the distribution of current and strength is not uniform, we can only express them at any time and place by caUing to aid the language of the differential calculus. If ds be a small area described on an equipotential surface, and if dq be the quantity of electricity which flows in a small time d t through that area d «, and if i is the strength of the current at the centre of that small area at any instant, then in the limit

»4: (2°)

§ 2. Onrrent and ElectromotiTe Force Oorves. — To fix our ideas, let us now suppose the electric flow to take place through a thin cylindrical conductor, such as a wire, in which, at positions sufficiently remote from the ends, the stream lines will be parallel to the axis of the wire and the equipotential surfaces perpendicular to it. Consider any one section, and let the flow across this section be variable both in strength and direction — that is to say, let it vary in the quantity of electricity which flows across that section in each succeed- ing instant, and let the flow be first one way and then the other, changing in any manner, however irregular. We can represent graphically the state of things as regards electric flow at that section by means of a curve called a current curve.

82

SIMPLE PERIODIC CUERE!fTS,

Take a horizontal line (Fig. 26) to represent the uniform flow of time. At successive instants let ordinates be drawn to this line, representing the strength of current flowing past that section, and let them be drawn above ( + ) or below (-), according as the direction of the flow is to the right or to the left. Thus, if time begins to reckon from 0, after the lapse of a time 0 T the current is positive, and is represented by a

line TI. After the lapse of a time OT' the current is negative, and is represented in strength by a line T' T.

This current curve is obviously a single-valued function — that is to say, corresponding to a given instant of time the current can only have one value. The curve can never cut itself or double back.

We may here remind the student of the distinction between single and multiple-valued functions. A single-valued func-

X

Fig. 27. Single valued function.

Fig. 28. Multiple- valued functions.

tion is one which, when represented graphically by a continuous curve, presents only one value of the ordinate for each value of the abscissa.

In Fig. 27 is represented graphically a single-valued function, having only one value of the ordinate X Y corresponding to a given value of abscissa OX. In Fig. 28 is represented a curve such that there are five different values of the ordinate of th©

SIMPLE PERIODIC CURRENTS,

83

curve corresponding to one value of the abscissa 0 X. This curve represents a multiple- valued function.

Amongst single-valued functions, or single ordinate curves, there is one which is particularly important, because it proves to be the constituent element of every single-valued function. This curve is called a simple periodic curve^ or simple sine curve, or simple harmonic curve. This curve may bo described as follows : — Let a circle (say a coach wheel) roll with uniform speed along a straight Hne, A B : a point P on its circumference will mark out a curve called a cycloid, represented in Fig. 29 by the thick line, A E P B. If the point P be projected at every instant on the vertical diameter of the circle, then the point M will mark out a curve (represented by the dotted curve) as the circle rolls along which has been

r

^>^

t^'rv

1 y*^

K'

\ n\

Fia. 29.

sometimes called ** the companion to the cycloid." It is also called a harmonic curve, a sine curve, or a simple periodic curve. Draw a line 0 S N through the centre of the circle and parallel to the base hne A B. Let it cut the dotted curve at the point O. The mathematical student will see that if the point O is taken as origin, and 0 C is called .r, and C M called y, then also, if the radius C P of the circle is 11, and the angle M P G = P G N is called 6, it is clear that

j' = R(180-^/) and 2/ = 1^ sin d.

or,

2/ = 11 sin (180--^).

If / is the circumference of the circle, then Z = 27rll, and, by su1)-;titution,

//=- / -si,i/^;:.r (21)

/

o 2

84

81MFLE PERIODIC CURRENTS.

This last is the equation to the dotted curve 0 E M B, and it is the equation to a simple periodic or sine curve. The quantity ^»AB is called the wave length, and B«SE is called the amplitude of the harmonic curve. It will be seen that this simple periodic curve is a smooth wavy curve which has points of maxima above and below the axis 0 G.

§ 3. Simple and Oompound Periodic Ourves. — If on one common axis we draw two simple periodic curves of any wave lengths and any amplitudes, and having any relative position with regard to each other, we may obtain another curve, called a complex periodic curve, by adding together the ordinates of the two simple curves.

Fia. 30.

As an example, in Fig. 80 are shown two simple sine curves, represented by the firm lines, of which one has double the wave-length and about two and a-quarter times the amplitude of the other. If these curves are superimposed, and a new curve, represented by the dotted line, formed by adding the ordinates X ^p X t/g, of a common abscissa, 0 X, into a third, X^3, then we obtain, by repeating this at all points, a new curve, which is called a complex periodic curve, because it is compounded of two simple sine curves. The dotted curve is the complex sine curve, and the two firm-line curves are its two components.

We may in this way add together any number of simple periodic curves and obtain an exceedingly complicated complex periodic curve, which is, however, always, like a simple periodic curve, a single- valued function. It is clear, also,

B Fio. 31.

angle multiplied by a constant. Take such a case as that of a zig-zag line, made up of lines inclined at an angle of 60deg., like the teeth of a saw (Fig. 31). We can, by Fourier's theorem, express the equation to this periodic line in terms of a series of sine or cosine terms. Thus the equation to the zig-zag line in Fig. 81 is

i/ = ~l sino;- - sin 8a; + — - sin 5a?-&o. ^ .

Hence, by adding together the ordinates of a number of sine curves suitably chosen and placed, we can obtain a complex periodic curve which imitates in form any given single valued periodic curve, however complex it may be, provided only that it is periodic, and that the curve does not cut itself.

This very remarkable theorem has applications in all departments of physics. In acoustics it shows that any

SIMPLE PERIODIC CURRENTS. 86 I

]

that just as we can compound simple periodic curves into a complex one, so we can resolve a complex single-valued function into a set of simple periodic components, suitably situated with respect to one another.

§4 Fourier's Theorem. — One of the most attractive and important of all mathematical discoveries is that of Jean Baptiste Fourier, who in his " Th6orie Analytique de la Chaleur," published in 1882, gave a demonstration of the above theorem, viz., that any periodic curve, however com- plex, provided it is a single-valued function, can be resolved into a series of simple periodic curves, of suitable amplitudes and wave-lengths, and be placed in a certain relative position to each other. In mathematical language, any single- valued periodic function can be expressed analytically as a sum of a series of terms the first of which is an arbitrary constant, and each of the following terms is the sine or cosine of an

86 SIMPLE PERIODIC CURRENTS.

continuous sound may be resolved into a series of simple harmonic sounds. In alternating current investigations it demonstrates that any curve of current, however complex, can be resolved into a series of simple periodic currents. If, then, any single function is graphically represented — that is to say, any such curve as in Fig. 30 — we see that this curve may be described by a point which moves horizontally with a uniform velocity, whilst at the same, time it executes in a vertical direction a movement which is the sum of a number of simple harmonic motions superimposed upon one another. The combination of these two rectangular motions causes the point to describe the curve considered.

In subsequent chapters we shall be examining effects which are due to periodic or fluctuating electric currents. Fourier's theorem gives us, when applied to these cases, a simplification of immense value, in that it enables us to see that, however complicated may be the fluctuation of current in a conductor, it can always be resolved into the sum of a series of simultaneous currents varying in a simple manner, and each of which can be graphically represented by a simple harmonic curve. The general consideration of periodic currents must, then, be preceded by an examina- tion of the elementary theory of electric currents of a periodic character, in which the variation is of the most simple kind.

Fourier's theorem applies also to many other physical phenomena of great importance. In acoustics it shows, for instance, that however complicated may be the motion of an air particle in a mass of air through which sound waves are being transmitted, it can be resolved into the sum of a series of motions such as would be produced by the action of tuning forks, each of which gives rise to a motion in the air particles approximately of the nature of a simple harmonic vibration. Helmholtz actually realised this in his synthesis of vowel sounds.

Physically interpreted, Fourier's theorem means that any variation of motion which can be represented by the changing ordinate of a single-valued periodic curve can be expressed as the sum of a series of simultaneous motions, each one of which is called a simple harmonic, or simple periodic, or simple sine

SIMPLE FERIODIO CUEREKTS,

87

motion. It becomes important, then, to start by examining the simplest form of periodic motion. Suppose a circular disc (Fig. 82), having a pin at its centre, 0, to be pivoted so as to revolve round an eccentric point, G. Let a T bar, moving in guides and having a slot in the cross-piece, be so fixed that the centre pin 0 is constrained to move in the slot. Further-

0..-

Fia. 32.

Fio. 33.

u

more, let the point C round which the disc moves be fixed to some support in the line of the bar A B produced. If the eccentric is compelled to move round G, the extremity of the bar A will move backwards and forwards with a motion called a simple harmonic motion or a simple periodic motion.

FiQ. 34.

For it is clear the point 0 (Fig. 88) is compelled to move in a circle round G as a centre, and hence the distance of the point A from C at any instant is the length of the bar A B plus the length B C, which is the projection of 0 C on the line A C. The point B, therefore, executes a simple vibration to and fro along the line A G as 0 moves round, and the point A

88

SIMPLE PERIODIC CURItENTS,

imitates the motion of B. If the angle 0 C D is called x and the radios 0 C is a, then the length B 0 is a sin a;, and the displacement of A at any instant from its mean or middle position has the same valne. The motion of A is called a simple harmonic motion, and the above eccentric and T bar is a mechanical device for compelling a point to describe a simple harmonic motion (abbreviated into S.H.M.). If such a harmonic motion be executed by point A (Fig. 84), whilst at the same time a strip of paper, S S', is caused to move uniformly in a direction perpendicularly to the line A B, a tracing point fixed to A will describe on the paper a curve of

'n\ 'r~\

Fio. 36.

which the ordinate AY is proportional to the sine of the abscissa XY, or the equation to the curve will be of the form y^asiax, a being some constant quantity. Hence a simple periodic curve is also called a siyie curve.

By combining together two similar pieces of mechanism it is possible to construct a machine which can add together graphically two simple harmonic motions in the same line, but of which the phase angles x and the amplitudes a are different. Machines for doing this have been devised by Lord Kelvin, Mr. Stroh, and others. Apart from complications the general principle is as follows.

SIMPLE PERIODIC CURRENTS. 89

Let a cord pass over four pulleys (Fig. 85), two of which, F^F^, are fixed in space, and two, M^ M*, can be made to rise and fall in vertical lines with a simple harmonic motion by being attached to T bars and eccentrics. If the cord has one end, B, fixed, and the other end, A, free, it is easy to see that, if either the pulley M^ or M' rises and falls along a vertical line and the cord is just kept tight, the free end A will be displaced by an amount equal to twice the displacement of M^ or M*, and as M^ or M* moves up and down with a S.H.M., the free end of A will also execute similar vibrations. If M^ and M^ move together the displacement of A at any instant is equal to the sum of the displacements of M^ and M^. By providing the end A with a tracing point, and moving under it uniformly a sheet of paper in a direction perpendicular to the direction of motion of A, it will describe a curve of which the equation will bo of the form

y =» a sin a; + a sin x\

a and a' being the amplitudes and xx' the phase angles of the two motions of M^ and M^ respectively. This apparatus, or one of similar principle, has been devised and employed by Lord Kelvin in his researches on the tides. It will be evident from the foregoing explanation that a machine can be con- structed capable of causing a tracing point to move to and fro across a uniformly flowing sheet of paper, with a motion compounded of any number of simple harmonic motions of different amplitude and phase taking place in the same straight line.

§ 6. Mathematical Sketch of Fourier's Theorem. — Without going into a complete proof of Fourier's theorem, for which we must refer the advanced student to mathematical text- books, we propose to indicate to the student how it is prac- tically employed in the analysis of any complex curve into a series of simple harmonic constituents. At a later stage the student will find that this analysis is of use in discussing certain current and electromotive force curves obtained from transformers.

We start with the assumption, for the propriety of which we must refer the reader to more advanced treatises, that if y

90 SIMPLE PEEIODIC CUBBENT8.

is the magnitude of the ordinate of any complex periodic single- valued curve, we can always express y as follows : —

y = Aq + B^ + Ai sin p e + Bj cos /) t + Aj sin 2 ;? «

• Bj cos 2 /) « + A3 sin 8 /) t + B3 cos 8 /? t + &c.

The problem is, given any complex periodic curve, to find the A's andB's in the above equation for its ordinate at any point. To do this we need a preliminary lemma in the integral calculus. It is as follows : —

The integrals, jsmptBinqidt^

and jcosptQOsqtdt,

when integrated between the limits 0 and ir, are equal to

w

zero, if p and q are unequal integers; and equal to -, if p and q are equal integers. For, since

2 sin p « sin 2 1 = cos (i) - ^) t - cos {p + q)t, and 2 G08 p t cos q t =: cos {p -q)t + cos {p + q)t;

., . r. . , sin(p-q)t ^n(p + q)t

therefore, jsmptsmqtdt=^-^ ^^^_^^ - 2(y-fg) '

• f , sin (» - ^) t sin (« + g) *

and jcosptcosqtdt^ ^^^^^^ + ^^^^^j ■

Hence, if p and q are unequal integers, both these inte- grals between the limits t = 0 and t^v are zero. K^«=j

they both become equal to ^ . Again, if y is the ordinate of

a periodic curve, and if Hs the half-wave length, then the

integral ^l ydl represents the mean value of y during half I J 0

the period ; because it is obvious that, if the mean or average

value of y is called M, the area enclosed by the periodic

curve and the base line between the two limiting ordinates

is M /, and this area is also expressed by the integral I ydL

Hence the above equality results. From these two simple lemmas it follows that we can easily determine the values of

SIMPLE PERIODIC CURRENTS,

91

the constants in the harmonic expansion. Let us assume a simple case as an example. Let

y = Aft + Aj sin a: + Aj sin 2 x.

To determine Aj, multiply all through by 8in2.r and inte- grate between the limits :i; — 0 and x^^v,

\ ysin2a?daf=i I k^sm^xdx-k- I Aisina?sin2arrf;c

• I Ag sin^ 2 X d x.

All the integrals on the righi^-hand side of the equation

A V vanish except the last, which is equal to - 2- .

Hence

A,

I y^in^xdx.

' Jo

In other words, Aj is equal to tivke the mean value of the product of y and sin 2 a? throughout the half period. In

16

_^«<

14 12

/^

r^

N

7^

10

\

y

8 F

e

^

/

\

V

J

'

4 S

N

-^

10 12 14 16 16 20 22 24 Time

Fio.36.

the same way all the other constants may be found. The process of analysing a complex function into its simple harmonic constituents is then reduced to little more than mere arithmetic.

92

SIMPLE PERIODIC CURRENTS.

A single example will make this olear.* There is a certain complex periodic curve, one period of which is represented in Fig. 86. The problem is to find the simple harmonic or sine curves of which it is composed. Call y the ordinate of the cu rve. Divide the whole period into twenty-four equal parts. Let T

be the whole periodic time, and let p stand for ■— . Let t be

any fraction of the periodic time, so that pt is the angular magnitude of the abscissa corresponding to any ordinate y» Since we have divided the period into twenty -four equal parts each of these corresponds to an angular interval of 15^. Hence, p t is successively 15**, 30°, 45**, 60**, &c.

Measure from the curve lihe value of y corresponding to each of these intervals, and tabulate them as follows : —

y

pt

y

pt

y

pt

13-8660

0

11-7940

185

4*8080

270

14-0355

16

10-8660

150

5-9645

285

14-3300

80

9-7060

165

7-5000

300

14-3295

45

8-3660

180

9-8940

315

14-1340

60

6-9645

195

10 8660

380

18-8295

75

5-6700

210

12-2940

845

13-4640

90

4*6705

225

13-8660

360

13 0355

105

4-1830

240

12 5000

120

4-1705

255

Proceed then to make a second table as follows :

—

I. t

XL

y

III. pt

IV. ampt

V. y X ainp t

VI. cospt

VII. yxcoBpt

0

1

2 3

4

18-8660 14-0335 14-3300 14-8295

0 15 80 45

0 0-2588 0-5000 0-7071

0

8 6324

7-1650

10-1324

1

0-9659 0-8660 0-7071

13-3660 18-5569 12 4098 10-8124

Similarly in Column VIII. put the values oi Bin 2pt; in Column IX. put the values of ^ x sin 2 ^^ t ; and in Columns X. and XI. put cos 2 pt and y x cos 2 pt. Then the value of the constant term A^ + B^ is the mean or average value of all the 24 numbers in Column K.

• The example above given is taken almost verbatim from a letter by Prof. John Perry in The BUetrician of February 6, 1892, Vol. XXVIIL, p. 362.

SIMPLE PERIODIC CURRENTS, 93

A^ is twice the average of all the 24 numbers in Column V.

Bj is twice the average of all the 24 numbers in Column VII.

Aj is the same for Column IX., and B^ for Column XI.

Any number of columns may be calculated corresponding to the multiple angles, 8 p ^ 4 ;> t, &c., for higher terms of the Fourier series.

When we have all the sine and cosine terms it is easy to express y in the form

y-Ao + Bo+ v/V + B7 sin {pt-^-O)

• ^AT+B? sin (2p « + ^) + &c., by grouping together the sine and cosine terms.

In the example calculated above it is found that the value of y is approximately

y - 10 + 6 sin (2? t + 80°) - sin (2 ;? t - 60°),

and this shows us that the given periodic curve is made up of two sine curves of amphtudes, 5 and 1 respectively, which differ in phase by SO-'. The student will find it to be a useful exercise to take two or three simple periodic curves and add their ordinates into a complex periodic curve, and then by the Fourier analysis to re-discover the simple harmonic constituents again, and see if he can find the amplitudes correctly.

§ 6. Simple Periodic Onrrents and Electromotive Forces. — Returning, then, to electric currents, we may consider how a complex periodic current is made up of simple periodic currents superimposed. It is necessary to examine, in the first place, how a simple periodic current or electromotive force may be generated. Let A B C D (Fig. 87) be a rectangular frame or conductor, able to revolve round a vertical axis, 0 0', in a uniform magnetic field. The adjacent figure represents the same in plan. If the frame revolve round the axis 0 0', the total electromotive force acting round the circuit at any instant is numerically equal to the time rate of change of magnetic induction or number of lines of magnetic force passing through the circuit. If H is the field strength in C.6.S. units, I the length of the side A C, and k the length of the side C D, and x the angle which at any instant the plane of the frame makes with a plane drawn at right angles to the

04

SIMPLE PERIODIC CURRENTS,