The Alternate Current Transformer Vol. 2: The Utilisation of Induced Currents (1896) — part 8 of 36

arranged as a shunt on the magnet. This current, however, is, by assumption, not strong enough to illuminate the lamp. On raising the key and stopping the steady current through the lamp the electric inertia of the coil sends a momentary powerful current through the lamp, which causes it to flash up. Again, if a small shunt-wound dynamo be occupied in supplying current to a few incandescent lamps, and the two hands be employed to raise simultaneously the brushes from the armature, the momentary rush of current from the field- magnet due to this extra currejvt will disagreeably impress the phenomenon upon the mind of the observer if the experiment

Fig. 47.

is made with any but a very small dynamo. With a large dynamo this experiment is very dangerous to perform.

Neither of these experiments is well fitted to illustrate the extra current at the closing of the circuit or the effect of electric inertia on starting the current in a helix. The arrangement most suited to exhibit the whole effect is that of the differential galvanometer as used by Edlund, or that employing Wheatstone's bridge, due to Maxwell.

In Edlund*s arrangement^ a differential galvanometer is •employed, of which the two coils GiG^ are so placed and

• JSet Poggendorff's Annalen, 1849.

SIMPLE PERIODIC CURRENTS.

115

I70und that when equal and oppositely-directed currents are sent through them the needle is unaJSected. The coils are then connected, as shown in Fig. 48, to a battery, B, an electromagnet or helical coil, L, and a wire, B, of equal resistance to L, but wound double. The galvanometer coils are so connected to the circuits L and B that when the steady current from the battery flows through the divided circuit the needle remains at zero. On closing the circuit it is then found that the needle makes a sudden deflection in a direction indicating a brief current passing in coil Gs, and on breaking the circuit it makes another deflection, indicating a transitory current passing through Gj. In other words, the balance is

Fio. 4a

•destroyed at the instant of breaking and making, but restores itself again when the currents become steady. This experi- ment, therefore, most clearly shows that the electromagnetic helix L, although of exactly the same electrical resistance as the coil B, differs from it in possessing a peculiar quality, which it has in virtue of being in the form of a coil or helix, and to which the name self-induction or imlitctame has been given. We are able to define this term as follows : — The self-induction or inductance of a circuit is, speaking generally, a quality of it in virtue of which a finite and steady electromotive force applied to it cannot at once generate in it the full current due to its resistance, and when the electromotive force is with- drawn time is required for the current strength to fall to zero.

i2

116

SIMPLE PERIODIC CURRENTS.

It must, however, be noticed that not only does the inductance of a circuit depend upon the geometrical form of the circuit, but it depends upon the magnetic permeability of the region which surrounds the cuxuit and on the magnetic permeability of the conducting circuit itself. If, in the arrangement with the dif- ferential galvanometer, the steady balance is obtained by using* a copper wire helix wound on a cardboard tube and balanced against a non-inductive but equal re^istance, it is found that the insertion of a soft iron core into the helix greatly increases the '^kick'' on making contact, indicating the passage of a greater quantity of electricity through the opposite galvano- meter coil, and therefore a greater delay in the time of establishing the steady balance.

H'H'H'I'l

Fio. 49.

Maxwell's method of exhibiting the effect of inductance is a preferable arrangement.

Four conductors are arranged in a rectangle joining the points a, />, r, d, and the diagonals are completed by a galvano- meter and battery (Fig. 49). P, Q and R are non-inductive resistances, and E is an electro-magnetic helix. If B and E are equal in actual resistance and P ; Q=R : E, then the permanent closing of the battery circuit does not finally affect tlie galvanometer indication, and these circuits (battery and galvanometer) are then said to be nmjwfate rircuits.

When, however, the battery key is first put down the galvanometer receives an impulse in one direction ; when the

SIMPLE PERIODIC CURRENTS, 117

key is kept down the galvanometer soon returns to zero, or to its original position. On raising the key the needle receives an impulse in the opposite direction. Examination of these impulses shows that if the current enters the quadrangle at d^ on closing the key the potential rises at h faster than it does at a, and that on raising the key the potential digs down at h faster than at a; but that, if the ** balance" is properly ob- tained, the points a and h reach finally the same potential when the key is kept closed.

An electromagnetic helix with or without a core of soft iron, behaves itself, therefore, towards an external electro- motive force to which it is submitted as if it had an internal counter-electromotive force which gradually disappears — allow- ing the full current due to its resistance to be estabUshed in it more or less slowly, and behaves also, at the removal of this external electromotive force, as if a direct internal electro- motive force suddenly made its appearance within it, this also gradually dying away.

The reader will see, therefore, that every electric circuit can not only dissipate electric energy in virtue of its resistance, but can conserve energy in virtue of its inductance. The resistance is measured by the rate of dissipation of energy which takes place when unit current (one ampere) flows through the circuit, and this rate of dissipation varies as the square of the current strength. The inductance is measured by the electromagnetic momentum associated with the circuit when unit current flows in it. Since, dynamically considered, the rate of change of momentum is a numerical measure of the force producing it, we must define electromagnetic momentum as that quantity the rate of variation of which numerically measures the electromotive force. We have already seen that if lines of magnetic induction (or force) per- forate through and are linked with a circuit, then any variation of the number of these lines of induction or linkages gives rise to an induced electromotive force in the circuit equal in numerical magnitude to the rate of change of the included lines of induction. When an electric circuit is re- moved from all other circuits and magnets and is traversed by a current, the turns of this circuit are linked with and include the lines of magnetic induction created by itself. Hence we

118 SIMPLE PEBIODIC CURRENTS.

are able to connect the quantity we have called the electro- magnetic momentum with the number of lines of magnetic induction which are linked with the circuit and which are created by the current flowing in that circuit. If a unit cur- rent is flowing in any circuit, there are a certain number of lines of magnetic induction at any instant linked with or perforating that circuit, and the number of these linkages defines the inductance of that circuit.

§ 12. Electromagnetic Momentum. — The justification for the use of the term electromagnetic momentum is as follows : — When a heavy body is in motion it possesses at any instant momentum, in virtue of its inertia. Numerically the momentum of a heavy particle is obtained by taking the product of its mass and its velocity, each measured in appropriate units. The time rate of change of a body's momentum in any direc- tion is, by the second law of motion, the measure of the force acting upon it in that direction, or, in the notation of tha calculus,

d {m r) _r.

-JT--''

We have seen that the induced electromotive force in a circuit depends on the time rate of change of the magnetic induction through it, and hence the magnetic induction at any instant through a circuit bears the same relation to the induced electromotive force in it that « body's momentum does to the mechanical force acting on it. Maxwell has accordingly employed the term electromagnetic momentum to represent the flux of magnetic induction or the number of lines of magnetic induction passing through a circuit, because it is upon the rate of change of this quantity that the induced electromotive force depends. Faraday very early recognised that induction effects depend on a cliange of some quantity. He makes frequent mention of the electrotonic state, and he spoke of a conductor in a magnetic field, when traversed by lines of induction, as in the electrotonic state, and he considered that when the electrotonic state was either assumed or disappeared its com- mencement or end was marked by the production of the induced electromotive force. Maxwell identified Faraday's electrotonic state with the total induction passing through

SIMPLE FERIODIC CUKRENTS. 119

the circtdt or linked with it* Consider, then, the operations which go on when a conducting circuit — say a simple loop of wire — ^is subjected to a steady electromotive force. The instant that force is applied, a current begins to flow in the circuit ; the instant that current begins, lines or rings of induction spread out from the circuit ; and the loop at any instant encloses a certain number of lines of induction which are increasing at that instant at a certain rate. A counter or opposing electromotive force exists in that circuit numerically equal to the time rate of increase of this induction. In circuits which do not enclose or surround iron or other magnetic metal, or which are immersed wholly in a medium of constant perme- ability, the magnetic induction at any point in the neighbour- hood of the circuit is numerically proportional to the strength of the current at that instant flowing in the circuit. This is the &.ct which hes at the root of the operation of most galvano- meters, viz., that the field at any point in the neighbourhood of the coil is simply proportional to the strength of the cur- rent flowing in the coil. If, then, i represent the strength of the current at any instant in the circuit, and L be a certain constant quantity such that Lt represents the induction through the coil or circuit due to the current t in it, then L i is the measure of the electromagnetic momentum of that circuit. This quantity L is a coefficient which, in this case, is dependent only upon the geometrical form of the circuit, and, under the assumption that there is no magnetic material in or near the circuit through which the lines of induction can pass, it is a constant quantity.

This quantity L is called the constant coefficient of self- induction of the circuit, or, more shortly, tfie inductance of the circuit.

The inductance^ or the coefficient of self-induction, is thus defined: — In the case of circuits conveying electric currents which are wholly made of non-magnetic material and wholly immersed in a medium of constant magnetic permeability, the total magnetic induction through the circuit per unit of current flowing in that circuit when removed from the neigh- bourhood of all other magnets and circuits is the numerical measure of the inductance or of the coefficient of self- induction. Otherwise, the ratio of the numerical values of the electro-

120 SIMPLE PERIODIC CURREI^TS.

magnetic momentum of such circuit and the current flowing in it when totally removed from all other currents and magnets is the numerical value of the indvxitance of that circuit.

§ 13. Electromagnetic Energy. — Let us confine our atten- tion first to one circuit of constant inductance or self-induction in which a current is being generated by a constant electro- motive force appUed to it. Each, increment of strength of the current creates an electromotive force opposing the impressed or external electromotive force.' Hence this external electro- motive force has to do work against an opposing force of its own creating all the time the current is rising in strength. When a mechanical force overcomes a resistance through a certain distance, mechanical work is being done, and, accord- ingly, we may ask — What is the electromotive force doing all the while it is increasing a current against an opposing electro- motive force? The answer is, it is doing electrical work. The result of causing a current having a strength i at any instant to flow for a small time, </t, against an opposing E.M.F. at any instant equal to ^, is that a quantity of work, represented by e i d «, is done in- the time d t. If e is tli^i instantaneous value of the opposing electromotive force of self-induction, it is measured at any instant by the rate of

d I change of electromagnetic momentum L t, or by L — .

dt

Hence the work done in raising the current from a strength

t to a strength i+dl against the counter-electromotive force of

di self-induction is L — idt = Jjidi, and if this is integi'ated dt

between limits zero and I, we get the whole quantity of work

so done against self-induction alone in bringing up a current

from zero to its full value, I, in the conductor, but

/

Exactly in the same way it may be shown that the work done in bestowing a velocity V upon a mass M is measured by the quantity J M V^.

The total work done against the electromotive force of self- induction in creating a current I in a conductor of constant

SIMPLE PERIODIC CURRENTS. 121

inductance L is, then, numerically equal to half the square of the final current strength, multiplied by the value of the con- stant inductance or coefficient of self-induction.

The equivalent of this work is found in the magnetic field formed round the conductor, and hence the formation of a magnetic field represents so much energy, measurable in foot- pounds per cubic inch, or in any other similar units, such as •ergs or kilogrammetres, per cubic centimetre of field.

Next let us consider the case of two circuits. Let the con- stant coefficient of self-induction of the first be L, and let it be traversed at any instant by a current i. Let the inductance of the other be N, and let it be traversed by a current i\ Let the coefficient of mvtual induction be M.

The definition of this last quantity is as follows : — If both circuits be traversed by unit currents, and if there be no other field than that due to these currents, the number of lines of induction which traverse both circuits, or are linked with both circuits, is called the constant coefficient of mutual induction. It will be a quantity constant for a given form and position of the two circuits on the assumption that the lines of induction flow in a medium of constant magnetic permeability. Hence, if we consider the work done, r^E, in raising the currents i and V by small increments, di and d I', in a small time, dt, we find it consists of four parts — a part, L i d /, representing work done by the current i against its ovm counter-electro- motive force, and a similar part, N i' d i', for the other circuit, then a portion, Midi', representing the work done by the current t in its own circuit against the induced electromotive force, due to the increment of the current i' in the other, and lastly, a similar part, M i' d i, for the second circuit. Hence, we have

dE=:Jjidi+Uidi'+Ui' d i+'S i' di'.

Integrating this between the limits zero and I for one circuit, and zero and I' for the other, we find the whole energy repre- sented by the two currents I and 1' flowing in the circuits to be

E=JLP+MI^^-iN^^ . . . (24)

The electro-kinetic energy is said to be a quadratic function of the currents and the inductances.

122 SIMPLH FERIODIG CURRENTS.

§ 14. Tlie Unit of Inductance.— T/i^ Henry. — The practical unifc of inductance is called one Jmiry. The henry is the unit of inductance which is in consistent relation with the ohm, the volt, the ampere, the watt, and the joule. A cir- cuit has an inductance of one henry when there are 10^ G.G.S. lines of magnetic induction linked with the circuit, or when there are lOVlinkages of current and magnetic lines of induction, under the condition that one ampere of current traverses the circuit, and that no other lines of induction than those due to itself perforate or are linked with the circuit. If the circuit is a coiled circuit of wire, and the wire makea n turns round a total number N lines of magnetic force or induction, then there are n N linkages of circuit and induction. Suppose that we have a circular solenoid formed by winding thin, closely placed, covered wire on a wooden ring of circular cross section. Let the mean cross section of the circular solenoid be S, and let the induction density in the interior of the solenoid be B, when one ampere is sent through the wire windings. Then there are B S lines of induction in all round the interior of the solenoid. Let there be N turns of wire in all on the ring, then there are N S B linkages of current and magnetic lines of force. The inductance of this solenoid, or^ its self-induction measured in henrys, is

If we consider the above circular solenoid or very long straight solenoid to be wound on a wooden or non-magnetic core, the value of the induction B in the interior is numerically the same as that of the magnetic force in the interior, viz.»

~ -Y- units, where A is the ampere current in the coil^

N the number of windings, and L the mean length of the coil. Hence the self-induction of such a coil in henrys is

.J — - N2, or is proportional to the square of the total

number of windings N.

An enormous number of wire windings are, therefore^ necessary to obtain any sensible fraction of a henry of in- ductance in a circuit in which the path of the Unes of magnetic force is wholly in air, or in some body of unit magnetic permeability.

SIMPLE FEBIODIC CURRENTS. 123

In the case of sach air or non-ferric magnetic circnits the inductance is a constant quantity which depends only on the geometrical form of the circuit.

The moment, however, that we introduce an iron core we alter the state of affairs. The inductance is then no longer the same for all values of the induction, because the induction varies with the magnetising force, but not proportionately to it. Hence, we cannot speak generally of the inductance of such an electric circuit when linked with an iron, or partly iron, magnetic circuit, except to define its value corresponding to one particular value of the current. We can, however, always refer to the instantaneous value of the inductance when we have occasion to mention a particular value which it has when varying from instant to instant. For very low or very high degrees of magnetisation, however, the inductance of such a circuit will be constant, but very different.

The following table taken from figures obtained by Mr. A.

E. Kennelly and Prof. Ayr ton* will furnish the reader with an

idea of the approximate magnitude of the inductances of

various well-known instruments, measured in henrys and

fractions of a henry : —

Cardew voltmeter about 1 microhenry.

Ordinary telegraph sounder 25 — 50 millihenrys.

Astatic mirror galvanometer, about \ « henrva

5,000 ohms / ^

Mirror speaking galvanometer, 2,250 \ ^^ ,

ohms .....j OT> nenrys.

Single coils of Morse receiver 93 millihenrys.

Induction coil (giving 2in. spark)! c^ ,

secondary cirSiit .....] 61 • • lieniys.

Shunt dynamo (100 volts, 35 amp8.)\ c i,

armature / ° nenrys.

Field magnets of the above dynamo \ 13*6 h

in series / 10*0 nenrys.

^"^^^n^^""" ^^^' ^^ °^"' ""} 12 millihenrys.

§ 16. Omrent Orowth in Inductive Oircuits. — We see, therefore, that when electric energy is spent on a conductor in the production of a current, in addition to the energy taken up in the performance of any chemical or external

• See The Electrician, Vol. XXVI., p. 290, also pp. 267 and 305.

124 SIMPLE PERIODIC CURRENTS.

mechanical work, part of it is dissipated as heat by an irre- Tersible process, and part is associated with the circuit in a recoverable form, and is taken up in the establishment of the energy of the magnetic field, which then exists round the conductor. This last portion of the energy, however, dissipates itself as soon as the impressed electromotive force is with* •drawn.

A mechanical operation analogous to that of starting a current in a wire may be found in the process of starting from rest, or increasing the speed of, a heavy fly-wheel which runs in bearings with friction. On applying a twisting force or torqiie to the axle of the wheel we get up its speed. To maintain the speed, force has to be continually applied to the wheel, and the work so done against friction is frittered away irreversibly into heat in the bearings. The friction is analogous to the electrical resistance ; it may be called the frictional resistance.

When the speed of the wheel is constant there is, however, associated with the wheel a certain quantity of energy in a kinetic form measured by ^Ia>^, where I is the moment of inertia, and cu the angular velocity of the wheel. As soon as the maintaining force is withdrawn this accumulated energy dissipates itself in heat by friction, or is utilised in some other way. During the time that the speed of the wheel is being increased, force must be applied to it for two purposes : firstly, to increase its angular momentum, and, secondly, to overcome the friction at the bearings. Suppose that, instead of revolv- ing on bearings with friction, the fly-wheel revolves in a more or less viscous fluid, and that the bearings are truly frictionless ; in such case the frictional resistance to motion would be fluid resistance, and would for low speeds be approximately proportional to the angular velocity. If I is the moment of inertia and w the angular velocity of the wheel at any instant, then it is shown in treatises on dynamics that the product of the moment of inertia and the rate of change of

the angular velocity at the instant, or I--, is the numerical

dt

measure of the torque or twisting force acting on the wheel

to increase its angular velocity, friction being neglected. If

we call the constant frictional coefficient B, so that B w is at

SIMPLE PERIODIC CURRENTS, 125

any instant the measure of the force necessary to maintain the motion against friction, the total torsional or twisting force acting on the wheel to maintain its angular velocity against the force of friction, and to increase it against the force of inertia, .

is F-Boi+li?*.

at

A precisely similar equation may be found connecting the electromotive force, electric current, electrical resistance, and inductance in the case of current starting in a wire. The above equation gives us a value for the instantaneous angular velo- city, or enables us to find the angular velocity after any time when F, B, and I are given. When a current of strength i is flowing steadily in a linear conductor, such as the wire under consideration, the energy associated with it in the form of a magnetic field is measured by the quantity \ L t^, where L is the quantity called the inductance of the circuit. Since this quantity L bears to electromagnetic energy a relation similar to that which the moment of inertia of a wheel does to the energy of its rotation, it might be called the coefficient of electromagnetic inertia ; but, as this would be a cumbersome - name, it has been called the inductance, or, frequently, the self-induction of the circuit. The numerical product of the moment of inertia and the angular velocity of the wheel is called the angular momentum, and, analogously, the product ^ of the inductance of a circuit and the current flowing at that instant through it is called the electromagnetic momentum.

The rate at which the angular momentum of a wheel is increasing or diminishing at any instant is a measure of the rotational force, or the couple acting on it at that instant. So also the rate of change of the electromagnetic momentum of a circuit is the measure of the electromotive force acting on it as far as mere change of current strength is concerned, and omitting, for the present, that part of the electromotive force required to overcome the true resistance. We have, then, the following parallel between a fly-wheel, with moment of inertia I, revolving frictionlessly, and having an angulajf velocity a> at any instant, and an electric circuit of inductance L, having a current of strength i flowing in it at any instant : —

126 8IMFLE FERIODIC CURRENTS.

Angular kinetic energy of the wheel, or energy of rotation = ^Iw'

Electromagnetic energy of the circuit = ^Lt*

Angular momentum of wheel = I«

Electromagnetic momentum of circuit = Ls

Rate of change of angular momentum of wheel =coup]e\ _ t^ or torsionid force causing rotation / ~ dt

Bate of change of electromikgoetic momentum = electro- \ _ t^ motive force employed in clianging current strength/ dt

The symbol ( = ) must in the above be understood as equivalent to the phrase *' is measured by."

In the electric circuity over and above the electromotive force which is required to change the electromagnetic momentum, there is an amount required to overcome the frictional resistance of the wire, and which is defined and measured by Ohm's law E »Bt. Hence, at any instant, if E is the impressed electromotive force acting on the circuit, we may divide E into two parts, one part equal to Bt by Ohm's law, which is sometimes called the effective electromotive force, and which is that part of the impressed electromotive force which is operating to overcome the true resistance of

the circuit, and another part equal to L-i, which is the

a t

part operating to change the strength of the current at that

instant, producing a small change, d i, in the current strength

t in a time d t. Hence, in mathematical language, we have.

E = Rt+L|* (26)

This is the fundamental equation for varying or periodic currents, when the periodicity is not so rapid as to affect the uniform distribution of the current over the cross section of the wire, and when the electrostatic capacity of the circuit

may be neglected. The part L-i is often called the counter-

dt

electromotive force of self-induction, and the above equation

might be read in words —

'Electromotive Force employed in chang- ing strength of cur- rent, or the Inductive ElectromotiveForce.

Total ^

['Electromotive Force'

Impressed

employed in over-

Electro-

.ESS-

coming resistance, or

motive

the Effective Electro-

Force.

[ motive Force.

SIMPLE FERIODIC CURRENTS. 127

We might arrive at tliis fundamental equation otherwise thus : — The total rate of expenditure of energy in the circuit is at any instant measured by the product of the current at that instant existing in the wire and the difference of potential between its ends. The energy expended in the circuit is at any instant being partly dissipated at a rate equal to Bt^y B being the ohmic resistance and t the current, and partly being stored up in the field at a rate equal to the rate of change of the quantity JLt*, Hence we have : —

Bate of dissi-] ( Bate of absorption pation of energy }-+-! or storage of energy as heat J [in the magnetic field.

Bate of 8upply\ of energy / °

and this in symbols

at

or E^Bi + Jj— (26)

dt

which is our fundamental equation.

At this stage we must particularly caution the student to note one thing. The quantity L, which is caUed the induct- ance of the circuit, is a constant and definite numerical quantity for any given form of circuit only as long as this circuit consists of non-magnetic material and is immersed in a non-magnetic medium. If, however, the circuit embraces or is embraced by iron, as in the case of an electromagnet, or is immersed in a medium which is not diamagnetic but magnetic like iron, then it is no longer a constant quantity, but the inductance varies from instant to instant with the strength of the current flowing in the circuit. In this chapter we suppose ourselves dealing only with circuits of constant inductance, and in which the value of L is fixed by the form of the circuit alone.

§ 16. Equation for Establishment of a Steady Onxrent.— We return to our discussion of equation (26) (§ 15). When a cur- rent is flowing in a conductor, we may picture it as surrounded by its lines of magnetic induction properly mapped out. That is, so that the number of the lines of induction passing perpendicularly through a small unit of area taken at any

128

SIMPLE PERIODIC CURRENTS,

point in the field is equal to the numerical value of the mean strength of the magnetic field over the area. If the circuit has the form of a loop (Fig. 60) lying on a horizontal plane, with the current circulating round it in the opposite direction to that in which rotate the hands of a watch, then the lines of induction must be considered as springing out from the upper surface, and turning outwards and over the conductor, so as to re-enter the loop from the under surface. The closed circuit is, therefore, linked with a certain number of lines of induction, which, if the circuit is composed of non-magnetic material, are proportional in number to the strength of the current at that instant. Any increase in strength of the

^ -h

TJK<^

Fig. 50.

current causes more lines of induction to grow out from the circuit, and packs the loop fuller of lines of induction. By Faraday's law, any increase of the number of lines of induction traversing or linked with a circuit creates an induced electro- motive force numerically equal to the rate of increase of that number at that instant. Hence, if 100 million lines of in- duction— C.G.S. measure — are put or inserted at a uniform rate in one second into a circuit, it will create an induced E.M.F. of one volt in it. If lines of induction are thrust into a circuit, the direction of the current induced is counter clockwise, as seen from that side of the circuit at which they are thrust in (see Fig. 50).

SIMPLE PERIODIC CURRENTS, 129

Applying this to the case before us, it is easily seen that any increase of current strength in the circuit in Fig. 51 crowds the space with more lines of force, and therefore creates in it an electromotive force of self-induction opposed to the impressed electromotive force which is acting to increase the current; and, so long as the current is increasing, this counter E.M.F. is at each instant proportional to the rate of growth of the current strength.

We can cast our equation (26) —

dt into another form, thus :—

R ^ ReZ«'

E

where g is the mftyimnTn value which the current can attain.

n

Fio. 51. — Lines of force being crowded into a circuit, inducing a counter- dockwiae E.M.F., as seen from the side at which they are put in.

Let US call this value I. The quantity p, or the ratio of the

inductance to the resistance of the circuit, is called the time- cons- ant of the circuit ; let this quantity be denoted by T, We

then have I-i=T^,

dt

which, in words, is a statement that if a steady E.M.F. is made to act on any circuit whose time-constant is T, the amount by which at any instant the current falls short of its fall value is equal to its rate of growth at that instant, multi- plied by the time-constant,

K

130

SIMPLE FERIODIC CURRENTS.

§ 17. Logaxithmic Onrres. — A curve such that the rate of growth or shrinkage of the ordinate or slope of the curve is proportional to the ordinate itself is called a logarithmic curve.

Let a curve (Fig. 52) he described by the extremity P of an ordinate, P M, which moves uniformly along 0 X, parallel to itself, and let P M shnnk in height at a rate proportional to its height at any instant. The differential equation to such a

curve is then

y= -A

dt

and since e-^* (where ^ = the base of Napierian logarithms = 2*71828) is a function which fulfils this condition of having

Fio. 62.

a differential coefficient proportional to itself we can write the solution of the above

y = tf '^ + a constant,

for it is at once seen that by differentiating the equation

t yz=e ^ + B, constant,

we obtain

and therefore

dp di'^ "A

e A

y=-A'

dy

dt

Returning to our equation for the current, we can write, as an equivalent for the equation

I-t = T^', dt

SIMPLE FBRIODIC OUBBENTS, 131

the equation I - f = - T^SlzS),

at

T I-i

Integrating this we have as a solution

" rp = l^S; (I - 0 + * constant.

The constant has to be determined by the condition that, when t=»0, » = 0, which gives constant = -log I. Hence the complete solution is

-i-log(I-t)-logI,

or I-t=l6"T.

This last equation expresses the fact that the amount by which the current falls short of its full value, I, at any time, t, after applying the E.M.F., is a fraction of its full value equal to e' f . When t = 0, or at the instant of closing circuit, I - 1 « I,

or the current t = 0; when e = T, I-i = -, or the deficit from

e

full current is equal to ^-=^ x the maximum current. Hence

we may define the time-constant of a circuit as the time reckoned from the instant of closing the circuit in which the current

rises up to a value equal to ?-I— of its full value, or to

about 0*682 of its maximum value. Approximately we may define the time-constant as the time from closing the circuit in which the current rises up to two-thirds of its maximum

value - .

The rise of current strength in a wire of inductance L and resistance B, when a steady external electromotive force, E, is applied to the circuit, can be represented by a current curve, as shown in Fig. 53. Let OX be a time line on which we mark off time as lengths reckoned from 0 ; let lines drawn vertically to this represent the current strength at any instant in a circuit of tinie constant T, inductance L, and resistance B ; and let

ic2

132

SIMPLE PERIODIC CURRENTS,

E OY»I^ represent the maximum current which is finally