The Alternate Current Transformer Vol. 1: The Induction of Electric Currents (1896) — part 6 of 35

round the frame is equal to w H B L sin 0. If the area of A B C D is denoted by A we may write the above as w H A sin 6. The angular velocity o> may be expressed as

7 n

the time rate of change of 6, or as — ; hence the expression for the total electromotive force of induction round the frame

is HA sin 0', or- d t

(HAcos^. ^

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 A cos 0 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 lines are spaced out according

• We here suppose the circuit to be foi med 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.

ELECTRO-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

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-MAGNETIC 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. Hence, 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 direction, or away from the eye.

Accordingly, a little reflection shows 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 III.

THE THEORY OF SIMPLE PERIODIC CURRENTS.

§ 1. Variable and Steady Flow.— In the following pages we shall be chiefly 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 uniformly 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

80 SIMPLE PERIODIC CURRENTS.

If that change is of such a character that the motion is regular in its mode of change, then the flow is 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 difficulties 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 from 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 calling to aid the language of the differential calculus. If ds be a small area described on an equipotential surface, and if d q be the quantity of electricity which flows in a small time d t through that area d s, and if i is the strength of the current at the centre of that small area at any instant, then in the limit

§ 2. Current and Electromotive Force Curves. — 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 CURRENTS.

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 O, after the lapse of a time 0 T the current is positive, and is represented by a

/

FJG. 26.

line T I. After the lapse of a time 0 T' the current is negative, and is represented in strength by a line T' I'.

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-

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 0 X. In Fig. 28 is represented a curve such that there are five different values of the ordinate of the

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 carve is called a simple periodic curve, or simple sine curve, or simple harmonic curve. This curve may be described as follows : — Let a circle (say a coach wheel) roll with uniform speed along a straight line, 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

FIG. 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 line A B. Let it cut the dotted curve at the point 0. The mathematical student will see that if the point O is taken as origin, and 0 C is called a, and C M called y, then also, if the radius C P of the circle is E, and the angle M P C = P C N is called 6, it is clear that

and or,

y = R sin 6.

« = R sin (180 - £).

If / is the circumference of the circle, then £ = 2?rR, and, by

substitution,

I . 2-

^aT^-f

(21)

G 2

84

SIMPLE 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 I = A B is called the wave length, and R = S E 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 C.

§ 3. Simple and Compound Periodic Curves.— 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.

FIG. 39.

As an example, in Fig. 30 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 yv X yz, of a common abscissa, 0 X, into a third, Xt/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 carves. 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,

SIMPLE PERIODIC CURRENTS. 85

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 " Theorie 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

\

FIG. 31.

angle multiplied by a constant. Take such a case aa that of a zig-zag line, made up of lines inclined at an angle of GOdeg., 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. 31 is

« = z. \ sin x - - sin 3 x + — sin 5 x - &c.

9 25

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

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 hi 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 PERIODIC CURRENTS. 87

motion. It becomes important, then, to start by examining the simplest form of periodic motion. Suppose a circular disc (Fig. 32), having a pin at its centre, 0, to be pivoted so as to revolve round an eccentric point, C. 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-

FIG. 32.

FIG. 33.

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

FIG. 34.

For it is clear the point 0 (Fig. 33) 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 C as 0 moves round, and the point A

88

SIMPLE PERIODIC CURRENTS.

imitates the motion of B. If the angle 0 C D is called x and the radius 0 C is a, then the length B C is a sin x, and the displacement of A at any instant from its mean or middle position has the same value. 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. 34), 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

FIG. 35.

which the ordinate A Y is proportional to the sine of the abscissa XY, or the equation to the curve will be of the form y = a sin x, a being some constant quantity. Hence a simple periodic curve is also called a sine 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, F1 F2, are fixed in space, and two, M1 M2, 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 M1 or M2 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 M1 or M2, and as M1 or M2 moves up anJ down with a S.H.M., the free end of A will also execute similar vibrations. If M1 and M2 move together the displacement of A at any instant is equal to the sum of the displacements of M1 and M2. 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 be of the form

y = a sin x -t- a sin a/,

a and a' being the amplitudes and x x the phase angles of the two motions of M1 and M2 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.

§ 5. 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 PERIODIC CURRENTS.

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

• B2 cos 2;? t + A3 sin Sp t + B3 cos 3 /> t + &c.

The problem is, given any complex periodic curve, to find the A's and B'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, J sin ptsinqtdt,

and tcosptcosqtdt,

when integrated between the limits 0 and TT, are equal to zero, if p and q are unequal integers; and equal to J if p and q are equal integers. For, since

2 sin p t sin q t = cos (p - q) t - cos (p + q) t, and 2 cos p t cos q t = cos (p - q) t + cos (p + q) t ;

sm(p-q)t

r therefore, jsmptSmqtdt=

sin (p - q) t

and

/

Hence, if p and q are unequal integers, both these inte- grals between the limits t = Q and t = ir are zero. If p = q they both become equal to | . Again, if y is the ordinate of a periodic curve, and if I is the half-wave length, then the

integral -I yd I represents the mean value of y during half

U o

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

n is M /, and this area is also expressed by the integral / y d I.

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

SIMPLE PERIODIC CURRENTS.

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

V — A0 + Ax sin x + A2 sin 2 x.

To determine A2, multiply all through by sin 2 x and inte- grate between the limits x = 0 and x = IT,

I y sin 2 x dx = I A0 sin 2.i- <lx + f A^in a; sin 2 ar<Z.r

/7T A9 sin2 2 a; ^ x. o -

All the integrals on the right-hand side of the equation

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

2

Hence

y sin 2 a; d x.

In other words, A2 is equal to f «•«•<? the mean value of the product of y and sin 2 x throughout the half period. In

_^£

— c-^

s

^x

f>

"\

\

' /

/

N

/

\

/

k

^

/

2 4 6 8 10 12 14 16 18 20 22 2<

Time

FIG. 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.

SIMPLE PERIODIC CUEEENTS.

A single example will make this clear.* There is a certain complex periodic curve, one period of which is represented hi Fig. 36. The problem is to find the simple harmonic or sine curves of which it is composed. Call y the ordinate of the curve. 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 p t 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, pt is successively 15°, 30°, 45°, 60°, &c.

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

y pt

y pt

y pt

13-3660 0

11-7940 135

4-8030 270

14-0355 15

10-8660 150

5-9645 285

14-3300 30

97060 165

7-5000 300

14-3295 45

8-3660 180

9-3940 315

14-1340 60

6-9645 195

108660 330

13-8295 75

5-6700 210

12-2940 345

13-4640 90

4-6705 225

13-3660 360

13 0355 105

41330 240

12-5000 120

4-1705 255

Proceed then to make a second table as follows :—

I.

t

II.

y

III. ft

IV.

sinpt

V.

y x sinpt

VI. cospt

VII.

y x coap t

0

1

2 3 4

13-3660 14-0335 14-3300 14-3295 &c.

0

15 30 45

0

0-2588 0-5000 0-7071

0 3 6324 7-1650 10-1324

I

0-9659 0-8660 0-7071 &c.

13-3660 13-5569 124098 10-3124

Similarly in Column VIII. put the values of sin 2 pt ; in Column IX. put the values of y x sin 2 p t ; and in Columns X. and XI. put cos 2 p t and y x cos 2 pt. Then the value of the constant term A0 + B0 is the mean or average value of all the 24 numbers in Column II.

• The example above given is taken almost verbatim from a letter by Prof. John Perry in The Electrician of February 5, 1892, Vol. XXVIII., p. 362.

SIMPLE PERIODIC CURRENTS. 93

AJ 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.

A9 is the same for Column IX., and B0 for Column XI.

Any number of columns may be calculated corresponding to the multiple angles, 3 p t, 4 fj 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 = A0 + B0 H- v/Aj2 + B^2 sin (pt+6)

sin (2pt + ff)+ &c.,

by grouping together the sine and cosine terms.

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

2/ = 10 + 5 sin (>* + 30°) -sin (2^-60°), and this shows us that the given periodic carve is made up of two sine curves of amplitudes, 5 and 1 respectively, which differ in phase by 3CK 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 Currents and Electromotive Forces. — Eeturning, 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. 37) 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.G.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

94 SIMPLE PERIODIC CURRENTS.

lines of the field, then the magnetic induction or number of lines of force through the frame is the product of H, and the apparent size of the frame, as seen along the direction of the lines of force of the field, is equal to H I k cos x.

If the area of the frame is A square centimetres, the magnetic induction through it is HAcosz. The effective electromotive force acting to produce a current in the circuit

FIG. 37.

is numerically equal to the time rate of change (decrease) of the magnetic flux or induction, or to

dt dt

This last equation is merely a symbolic statement of the fact that, if such a frame of area A revolve round an axis perpendicular to the lines of force in a, uniform magnetic field,

SIMPLE PERIODIC CURRENTS. 95

H, with an angular velocity ^f, then the integral electromotive force acting round the frame at any instant corresponding to

an angular displacement x is H A — sin x.

d t

If the angular velocity remains constant, the effective electromotive force will be simply proportional at any instant to the sine of the angular displacement of the frame from its initial position. Such a frame produces by its uniform revo- lution a simple periodic variation of electromotive force in its own circuit. If we suppose such a frame to have a closed circuit, then this periodically varying electromotive force will produce in the circuit an electric current which varies in strength very nearly as the sine of the angle of the displace- ment of the frame from its zero position when no lines of force penetrate through its area. Hence, graphically repre- sented, the current varies according to a simple harmonic law, or is a simple sine current. We can then synthesise by

the superposition of such simple harmonic electric currents any form of variable current, however complicated. Let a series of such sine inductors be joined up on one circuit (Fig. 38), each capable of being regulated as to angular velocity, and imagine these to revolve in magnetic fields of equal strength. These sine inductors are originally set with the plane of their frames at certain different but fixed angles to the planes at right angles to the fields of force in which they revolve, and they must be supposed to maintain these relative positions during their revolution. Accordingly, the effective electromotive force in the whole circuit, when they are all joined up in series and set revolving at fixed speeds, is represented by a function

e = A sin x + A' sin x + A" sin x" + &c. ;

and by Fourier's theorem any possible periodic variation of e which, graphically described, is a single-valued function, can

96 SIMPLE PERIODIC CURRENTS.

be produced by suitable values of the speeds and phase angles of these sine inductors.