The Alternate Current Transformer Vol. 2: The Utilisation of Induced Currents (1896) — part 9 of 36

found in the circuit. On applying the electromotive force E to the circuit, the current strength grows up in the wire as graphically represented by the curve, the law of growth being that the rate of growth at any instant, multiplied by the time- constant, is equal to the difference between the actual current at that instant and the maximum current strength finally attained, or, symbolically.

I-i=T

di di

the solution of the above differential equation being I-i=ld \

or

^{-i'\

(27)

This last equation gives us the value of the current strength at any time t seconds after closing the circuit, in terms of the time-constant, and the maximum current, I, which is finally attained.

The maximum current, I, would be produced at once in the circuit if its inductance were zero, so that we may finally for- mulate the law of growth of current in a circuit of constant inductance L, resistance B, and no sensible capacity, by saying that the current strength at any instant, added to the rate ofgrotrth of the current strength at that instant multiplied bt/ the iime-constant^ is equal to the current which would e^vist m the circuit if its in- ductance were zero.

SIMPLE PERIODIC CURRENTS. 133

§ 18. InstantaneooB Valne of a Simple Periodic Onrrent.— The application of these principles to the case of simple periodic currents will lead to another important equation. Let there be a circuit which has an inductance L and resistance B, and let a simple periodic electromotive force act upon it ; let 'the maximum value of this E.M.F. be E, and let p stand

for 2xn, where n is the frequency of the oscillation, or -

n

is the duration of one single complete period, p is a quantity of the nature of an angular velocity, and may be called the pulsation. Then, if t is the time which has elapsed from the commencement of the wave of E.M.F. and e is the actual value of the E.M.F. at that instant,

c=E sin j5t. In this case the impressed electromotive force varies from instant to instant, passing from zero to a maximum E, then to zero again, and then to a negative maximum - E. Accord- ingly, our fundamental equation for the current strength at any instant is expressed thus :

^(M + Rt = tf=EBin;)e. . . . (28) at

For, the total rate of expenditure of work on the circuit at any instant when the current has a value i is ei, and this must be equal to the rate at which electrical work is being dissi- pated as heat, or to Bt' by Joule's law, and to the rate at which work is being stored up in the magnetic field, which is

— (iLi»).

Hence i- (J Lt«)+Bt»=6i,

at

or, L~ + Bt=Esinj)« (29)

at

In order to solve this differential equation, and obtain the

value of the current i in the circuit at any instant under the

periodic electromotive force, we may adopt a well-known

algebraic device, and substitute for the value of sin pt its

equivalent in exponential terms. It is shown in treatises on

trigonometry that ^^ _^^

sin^. ^ "'''

2k

134 SIMPLE PEBIODIC CUBRENT8.

where k= v^ — 1, and e is now the number 2'71828, which is the base of the Napierian logarithms.

k0, -he-

Also that cos ^=i-J:f ;

2

hence cos ^+^sin O^e^^,

These are called the exponential values of the sine and cosine.

Taking the equation (29),

L^*+Ri = Esinp«, lit

we divide both sides by L, and, writing T as before for the

time-constant — , we get K

di , i E .

t

Multiply both sides hy e^ {e being here the exponential base, not impressed E.M.F.), and we have

^,eK'Le^=^eKmpt. c/e T RT ^

The left-hand side of this equation is the complete differential

of t^T, and may be written —U'ct j; and on substituting

the exponential value for sin;?t and putting * for ^ — 1, we have

The right-hand side of this last equation is the differential -with respect to t of

_E Ik

BTj I+^jpT l>-^pT I ' \ [p T -^

and this last becomes by simplification

E_ -^j_ ^ g-*^' )

2/cli^ [ l'+/./T l-/c2?T y

SIMPLE PERIODIC CUEBENT8. 136

Hence, equating both sides of equation (80), when integrated we have

1=

2R^t 1+kpT 1-kpT r

Substituting back into sine and cosine terms, and recol- lecting that

e^'* =cos pt+k Bin pt,

and ^-**«=cosjp t—k smp t,

we get finaUy

t= ^/ sinj?f— 7)TcosjpM

This equation gives us a value i for the current at any instant, and at a time t reckoned from the instant when the impressed electromotive force is zero. The value of t is accordingly called the imtantaneoua value of the periodic current, and the instantaneous value runs through a certain cycle of magnitudes, ranging from zero at one particular instant to a maximum value I at another instant.

We can, however, put the above equation in a more intel- ligible form. Replace T by --, and let 9 be an angle whose

R

tangent is equal to -^ ;

R

hence tanfl=3^«jpT.

R

It follows by an easy transformation that

coB^=-7=====- and sm ^= —

We have, then, for the value of the current t, the equation

. E (sin;? t—p T cosjp t) . "r I i+p»T J '

or, by substitution,

" vIhF^ 1"° ^ * "" ^~ ^ ^ ^ ' T

E

E_

186 SlMfLS PERIODIC CtJEMSNTS.

This is called iiie j^articiUar solution of the equation

L— +Ri = E sin 2Jt=e, dt

and it shows us three things : — First, that the phase of the current i is retarded behind that of the impressed electro- motive force by an angle 0 — such that tan d=^p T = — ?;

second, that the maximum value of the current is obtained by dividing the maximum value of the electromotive force by a quantity equal to v^R*+;;L; and, third, that the current curve is a simple periodic curve. The quantity VR^+p'L' is called the impedance of the circuit.

The mathematical student will, however, remember that the complete solution of the equation

dt involves a constant of integration, and this is obtained by adding to the particular solution above obtained the com- plementary function which is obtained by taking the solution of equation (29) when E sin j? t = 0. Now, since the solution of

L'ii+Rt-Ois dt

-^«

where C is a constant of integration and e in this last equation is the base of the Napierian logarithms, we have, then, the complete solution of the differential equation

L^'+Rt = Esini)t d t

given by the equation

t--— J^— =sin(i?f-(9) + C€"i^'. . (82)

The complementary function dies out rapidly as time in- creases at a rate depending on the value of . Physically,

R the meaning of this is that the current does not settle down into its regular periodic state until a shorter or longer time after the closing of the circuit depending on the value of the

time-constant ^.

SIMPLE! PEBIomC CURRENTS. 137

We shall return again to discuss the complete solution of the above equation (29), and show how to determine the value of the constant C in equation (82).

We have seen from the explanations on previous pages that the mean-square value of a simple periodic quantity is equal to its maximum value divided by V2. Hence, if we write Im for the impedance, we can put the equation, giving the instantaneous value of the current produced by a simple periodic impressed electromotive force of maximum value E, operating on a circuit of resistance E, inductance L, with a pulsation p, in the form

E t= sin(^j«-^);

or, if we denote the maximum value of the current during the

E phase byjbhe letter I, and since I = f^> we have

i = lsin(pt-0).

Hence, we may write this in words as follows :—

tJie maximum vahie\ f tlie maximum value of ilie

of the current j- = -j impressed electrofnotive force strength J { impedance

and t7i£ mean-square ] [ the maximum value

value of ttie

current strength

V2

We see, then, that, in the case of simple periodic electro- motive force, the quantity called the impedance appears to be related to the impressed E.M.F., just as does the resist- ance to the steady E.M.F. in the case of continuous currents, and the above may be called the equivalent of Ohm's law for simple periodic currents. Compare as below

For steady'

or

continuous

currents

For simple'

periodic or

alternate

currents

current \ __ (electromotive force ,^, , , strength ] " \ ^.SSia/i^^ ^^^"^ ^ ^^^)-

_ r mcan-sfpiare value of

mean-square

value of tfie

current

strength

the electromotive force impedance

138

SIMPLE PERIODIC CURRENTS.

Impedance is a quantity which is measured, like resistance, in ohms, and has for that reason been sometimes, but erroneously, called the virtual resistance*

§ 19. Geometrical Illustrations. — The current equation, expressing the current strength in terms of the impressed electromotive force, the resistance, inductance, and phase angles, which holds good when a circuit of constant induct- ance and no sensible capacity is subjected to moderately great pulsations of electromotive force has been in the previous pages arrived at algebraically from first principles.

Pfo. 54.

It is, however, possible to elucidate its meaning by geometri- cal methods. Let a circular disc (Fig. 54) be pivoted at the centre 0, and at any point P on the circumference let a plummet line be attached. In front of the circle is a fixed horizontal line XX'. Let the disc move round counter- clockwise at a uniform rate, the time of one revolution being T. As the disc goes round, the length of plummet line P M above X X' fluctuates. Since P M = 0 P sin P 0 M, it follows that, if the magnitude of P M be taken at small equal intervals of time during one revolution, and such heights be plotted oif as off-sets at equal distances above and below a datum line, the extremities of these ordinates will he on a simple periodic or sine curve. In other words*

SIMPLE PERIODIC CURRENTS, 139

PM grows and shrinks in height in accordance with a simple periodic law. "We can, therefore, represent any quantity which fluctuates in magnitude according to a simple sine law of growth by representing it as the pro- jection of a point on the circumference of a circle revolv- ing uniformly, taken on a horizontal or vertical fixed line drawn through the centre. Hence, if OF represents the maximum value of an electromotive force fluctuating periodi- cally, P M will represent its various magnitudes during the complete period. The magnitude of P M at any instant is known when we know 0 P, which is called the amplituds, or maximum value, and POM the pJmse angle of the motion. A diagram, in which the projection on any other line of a radial line revolving round one extremity is made to represent a simple periodic function, is called a clock-diagram. In clock- diagrams radial lines are taken to represent in magnitude the maximum values of the quantities which are to be represented as periodically varying. Any line through the centre may be taken as the line on which projections are taken, and the projections in this line give us the instantaneous values of the periodic quantity whose maximum value is represented by the radius. If different radu are drawn from one centre, repre- senting currents or electromotive forces, then the angular interval between these radii represent the phase difference of these quantities.

§ 20. Ghraphic Bepresentation of Periodic Onrrents. — On such a diagram let a radius be drawn to any scale repre- senting by its vertical projection the periodic fluctuation of an impressed electromotive force, varying according to a simple sine law, and acting on a circuit of given inductance and resistance with a fixed periodicity; the problem is to draw on the same diagram another radius, of which the vertical projection shall represent the actual current strength in the circuit at the corresponding instant. The impressed electromotive force at any instant balances, or is equal to, the sum of two others, viz., the effective electromotive force driving the current, which is equal to the product of the ohmic resistance of the circuit and the current at that instant in it; and the inductive or counter-electromotive

140

SIMPLE PERIODIC CURRENTS.

force, which is equal to the rate of variation of the flux of force or number of lines of force traversing the circuit. The phases, or times of maximum, of these two components are not identical. They differ by 90°, since the effective electro- motive force has the same phase as the actual current, and the inductive electromotive force, depending on the raie of variation of the current, comes to a maximum at the instant when the current is zero, or is changing sign.

By the proposition in § 10, these two periodic quantities can therefore be represented by sine curves, one of which is shifted backward relatively to the other, so that the crest of the wave of one coincides with the zero point of the other. We shall first proceed to show that the sum of two

Fig. 55.

simple periodic motions of the same periodic time, but different phases and amplitudes, will, when added together, produce a simple periodic motion of the same periodic time.

Let a parallelogram of cardboard, 0 A B C (Fig. 56), be cut out and pivoted by a pin at the angle 0, so as to turn freely clockhand-wise. Let a vertical line, OY, be drawn through 0, and in any position let the sides 0 A, 0 C, A B be projected on to OY. The projection of lines equal and equally inclined are equal; hence, since AB is equal and parallel to 0 C, the projection of A B — viz., a h — is equal to that of 0 0 — viz., Oc. But 06 = 0<i + a6 always for any position of the card ; hence 06 = Oa4-Oc. The projection of the diagonal is therefore equal to the sum of the projeo-

SIMPLE PERIODIC CURRENTS.

141

tions of the adjacent sides. As the card moves uniformly round the magnitudes of the projections fluctuate at each instant, according to a simple periodic law. Hence the sum of the simple periodic motions of which 0 A, 0 C are the amplitudes, and which have a fixed difference of phase represented by the angle A 0 C, is the simple periodic motion represented by 0 6 in amphtude and relative phase. If, then, a point be subjected to two simultaneous simple periodic motions of given amplitudes, and of which the phases differ by 90°, the actual motion will be represented, as to ampli- tude and phase, by the diagonal of the parallelogram of which these two form the adjacent sides. Eetuming in thought to electric motion, consider the motion of a particle

Fio. 56.

of electricity (if we may be allowed the expression) in the wire subjected to two simultaneous simple periodic motions of unequal amphtude and fixed difference of phase equal to 90^. The displacement at any instant due to the two together is equal to the sum of each separately. If the individual motions are represented by the vertical projec- tions of two lines, 0 A, 0 B, fixed like hands of a toy clock at right angles (Fig. 66), the resultant motion is that indi- cated by the projection of the diagonal 0 C on the same vertical. We have seen (in § 10) that, if the variation of a quantity is represented by a simple sine curve, the variation of its rate of cluinge is represented by a sine curve of different amplitude

142

SIMPLE PERIODIC CURRENTS.

shifted backwaxd by 90° of phase, or by a quarter of a wave length. It follows from this proposition that if we add together at every instant the motions or the ordinates repre- senting them on a diagram of two simple periodic motions, one of which is the curve representing the rate of change of the ordinate of the other, we shall get a new sine curve, of which the maximum value falls between that of the other two, and of which the amplitude is different, but wave length or periodic time the same. In Fig. 57 the thick line sine curve represents one wave of a simple periodic motion. The fine continuous line is a sine curve of equal wave length, of which the ordinate PM at any point represents or is pro- portional to the rate of change of the ordinate Q M of the

Fio. 57.

thick curve at the same instant. Adding together the ordinates of the thick and thin curves, we get a new dotted line sine curve, of which the ordinate B M is equal to Q M

• rate of change of Q M. If we substitute for the sine curve diagram a clockhand diagram (Fig. 56), then the projection of OB — viz., Ob — corresponds to the ordinate QM of the thick curve ; that of 0 A — ^viz., 0 a — corresponds to PM, the ordinate of the thin curve ; and that of 0 C, the diagonal of the rectangle 0 A, 0 6, corresponds to B M, and is the resultant of the motion 0 B, and the rate of change of that motion, viz., 0 A. If, then, 0 B represents the amplitude or maximum value of the actual periodic current in a circuit, a line, 0 A, drawn at right angles to 0 B, will represent to a suitable scale the rate of change of that current.

SIMPLE FEBIODIC CURRENTS. 143

We are, then, led to this converse proposition, that we can resolve any simple periodic curve into a pair of component periodic curves of equal periodic time, but of which the maxi- mum value happens for one before and for one after that of the original.

§ 21. Impressed and Effective Electromotive Forces. — If at any instant a current of which the instantaneous value is i is flowing in an inductive circuit of which the true resistance is B, the quantity Et represents the voltage necessary to make this current flow, and this part of the impressed electro- motive force is called the effective eUctroTnotive force in the circuit. If the circuit is an inductive circuit, there will be

another electromotive force equal in magnitude to L — , which

dt

acts either with or against the total applied or impressed

electromotive force. This is called the inductive electromotive

force. The electromotive force which is at any instant

applied to the circuit is called the impressed electromotive force.

The eflFecfcive electromotive force is always the resultant of

the impressed and inductive electromotive forces. Hence, if

these last two electromotive forces are represented in a clock

diagram in magnitude and relative phase by the two sides

of a parallelogram, the effective electromotive force will be

represented in magnitude and phase by the diagonal of that

parallelogram. A further condition is that, since the inductive

electromotive force depends upon the rate of change of the

current, it is always at right angles as regards phase with

the effective electromotive force. This last is always in

step or in synchronism as regards phase with the current.

Hence, if we require to draw a clock diagram of electromotive

forces for an inductive circuit in which a simple periodic

impressed electromotive force is acting, we see that the proper

construction is as follows : — Take any line, 0 P (Fig. 58), to

represent the magnitude of the maximum value of the

impressed electromotive force ; on 0 P describe a semi-circle,

0 M P ; let 0 P be supposed to revolve round the point 0 in

the contrary direction to the hands of a watch, and let the

projection of OP, at any instant on any line OY drawn

through 0, be taken. Then 0 P represents the maximum

144

SIMPLE PERIODIC CURRENTS.

Talae, E, of the impressed electromotive force, and 0 q, or the projection of 0 P, represents the magnitude of the instan- taneous value of the impressed electromotive force at an instant when 0 P has completed such part of one revolution as is represented by the angle POX.

Let us suppose 0 P to start from the position 0 X, and let time be reckoned from that instant of starting. Then, if T be the time of one complete revolution, and if t be the time in which OP passes through the angle POX,

e=E sin p t

R i = R I 5/>7 (pi- 0)

Lpi = .Lpl cos(jpi — 9)

Fio. 68.

the angle P 0 X is the same fraction of four right angles of 2ir that t is of T. Hence the angle POX is, in magni-

tude, equal to-^-.

For shortness, 27r/T is written p. There-

fore p is a quantity of the nature of an angular velocity. Hence, if the magnitude of 0 q, which is the projection of 0 P, is denoted by e, and if OP is denoted by E, we see that tfaE Bin ptf

SIMPLE FERIODIC CURRENTS. 145

or the instantaneous value of the impressed electromotive force runs through a cycle of values represented by the ordinates of a sine curve.

Next, on O P describe a semi-circle, 0 M P, on that side of O P which is towards the direction in which 0 P is rotating. Take a point M on this circumference, such that 0 M is to M P in the ratio of L^ to B, where L is the inductance and B the resistance of the circuit. Through 0 draw 0 E parallel to MP. Produce MO to N, and make ON equal to OM. Draw N E parallel to 0 P and join 0 E. Then, on the same scale on which 0 P represents E, the maximum magnitude of the impressed electromotive force, OE will represent the maximum magnitude of BI or the effective electromotive force, and ON will represent L^^I or the maximum magni- tude of the inductive electromotive force. By the geometry of the figure we see that, if the angle POE is called 6, the projection Ofe of OE on OY is equal to OE sinQjt— d?), and also the projection On of ON on OY is equal to

0 N cos (p t - ^). Hence 0 n = — (0 A-). Moreover, 0 E is

d t

the resultant of 0 P and 0 N, and 0 N is at right angles to OE; therefore ON and OE fulfil all the conditions requisite for being the representation of the maximum values of the inductive and effective electromotive forces. For OE is obviously the resultant of 0 P and ON. 0 N is in such a direction that its projection or instantaneous value is numerically determined by the rate of change of the projection or instantaneous value of 0 E ; and we know, by fundamental principles, that the electromotive force of self-induction is determined by the rate of change of the current in this circuit ; that is, by the rate of change of the effective electro- motive force. By considering the relative positions of OP, OE, and ON, it will be seen that they are in the right directions to represent these three quantities. For, if the system of lines be supposed to revolve round 0, then, when the projection of 0 E is above 0 X — that is, when the current in the circuit is increasing — ^the projection of 0 N is negative and is decreasing. This means that the electromotive force of self-induction is in such a direction as to oppose the ourrent. Also, when the effective electromotive force or

L

146 SIMPLE PERIODIC CURRENTS.

current is in the same direction, but decreasing, then the inductive electromotive force is positive, or in the same direction as the current, and is increasing. Accordingly, if the magnitude of 0 K is KI, where E is the resistance of the circuit and I is the maximum value of the current, and B I ia therefore the maximum value of the effective electromotive force in the circuit, we see that the magnitude of 0 N must be Lj) I, and that of 0 P must be VR« + p«L* I, and this last, we know, is the value of the impressed electromotive force E« Accordingly, on whatever scale 0 P represents the impressed electromotive force E, then OE represents the effective electro* motive force EI, and ON represents the inductive electro- motive force It pi. If we take one Eth part of 0 E, we have the value of the current in the circuit. The angle of lag 0 by

which the current is behind the impressed electromotive force in phase is an angle, such that —

o^« a Eesistance of circuit. E

cos vsss _^ _

Impedance of circuit. VH^-^p^U'

The diagram shows us, therefore, not only how to represent the current and impressed electromotive force in an inductive circuit properly as regards phase and magnitude, but tells us practically how the angle of lag should be measured.

The relation of the impedance and resistance of an inductive circuit may be represented geometrically as follows : Draw a right-angled triangle, ABC (Fig. 69), and take the base A B to represent the resistance of the circuit, and the hypotenuse, AC, to represent the impedance. Then the side BC will represent the magnitude of the quantity p L. This has been called the reactance of the circuit, and since the angle 0 AB is

SIMPLE PERIODIC CURRENTS. 147

■D

an angle which has a cosine equal to —p=====.. we see that

this angle, which we may call 6^ is the angle of lag of current behind electromotive force, and, moreover, that

pL_reactance of circuit ^^ "" R "" resistance of circuit'

§ 22. The Mean Value of the Power of a Periodic Current. Having now seen how the fluctuation of current strength is related to that of the impressed E.M.F. in an inductive circuit under the conditions of a simple sine law of variation, we pass to the consideration of the measurement of the potoer taken up in or supplied to circuits traversed by periodic currents.

Let the thin line curve in Fig. 60 represent the curve of impressed electromotive force in an inductive circuit, and the

Fio. 60.

thick line the corresponding curve of current. Then at any instant the rate at which energy is being expended on the circuit is equal to the product of the ordinates PM, QM, which at any point M on the time line represent the electro- motive force and current respectively. The mean rate of expenditure of energy, or the viean power being taken up in the circuit, is then the mean of all such products taken at equal and very near intervals of time during one complete period. This is not by any means identical with the product of the mean current and mean electromotive force. To arrive at an expression for this mean power, we must pave the way by a preliminary proposition on the mean product of two simple periodic quantities. An elegant geometrical method of

l2

148 SIMPLE PERIODIC CURRENTS.

obtaining this has been given by Mr. Blakesley. We shall^. however, give here an algebraical proof of this proposition. Let there be two radii OP, OQ (Fig. 61), which revolve in equal periodic times round a conmion centre 0, separated by a fixed angle, P 0 Q. At equal small intervals of time corresponding to equal angular motions let the projections 0^, 0 g of these lines be taken on a vertical line through 0. It is required to find the mean value of the product Op, Oq during one complete peri6d.

Denote by X the length of 0 P, and by Y the length of 0 Q, and let the angle P 0 Q be /J, and P 0 p be a. )3 is the angle of phase difference, and X and Y are the maximum values of the periodic quantities Op, 0^, which are the vertical projec- tions of 0 P, 0 Q.

Fi3. 61.

Let Op be denoted by p, and Oqhj q Then p^Xcosa,

and 9 = Ycos(a+j3);

and therefore ;? <y = X Y cos a cos {a+P).

Let the pair of radii 0 P, 0 Q be supposed to turn round one complete revolution, proceeding by n small steps, each step increasing the angle a by a very small amount, 8, a and n bt-ing a very large number. At each stage let the value otpq be measured as above, then the mean value of the product p q is one nth part of the sum of all the n values so taken. Call this mean value of the product M. Then, ,j ^XY jcos )3+cos 6a cos (Sa+py+cos 2 3aCOs(28a+j8)| ^ "^ M I .... +COS?l— l8aC0S(«-iSa+^)j

By trigonometry we have cos (n-1 S a) cos(n-l 5 a+y8) = J cos (2"n^l 8 a+P) + J cos P, since cos A+13 + cos A-B = 2 cos A cos B.

SIMPLE PERIODIC CURRENTS. U9

Accordingly every term, except the first in the cosine series for My splits up into the sum of two others, one of which is always J cos p. Rearranging the terms, we get for the value of M as follows : —

M =— fi n cos )3+ J^cos jS+cos (2 S a+/3)+cos (4 S a+fS)

.... +C08 (2"n^l 5a+jS))J

The cosine series in the inner bracket consists of a series of cosines of angles in arithmetic piogression taken all round the circle. Hence, since the cosine of any angle is numerically equal to that of the cosine of its supplement, but of opposite sign, these cosine terms will cancel each other out pair and pair, when n becomes very great and 5a very small, and nSa equal to 2ir, For when

»8 a = Itt, 2 w- 1 8 a+P = irr+fi, and cos (47r+jS)-COS)3. The first and last terms of the series are in this case identical, and for every term there will exist one of equal magnitude and opposite sign. The sum of the series of cosine terms in the inner bracket is accordingly zero.

The value of M reduces then to that of the first term, viz. : —

^=-2-008)8.

The mean value of the product of two simple harmonic or periodic functions of equal period but different amplitude and phase is equal to half the product of their maximum values, and the cosine of their difference of phase.

Returning to the consideration of the electrical problem, it is now clear that for simple periodic or sine variation the mean value of the product of the current at any instant and the simultaneous value of the impressed electromotive force in an inductive circuit is obtained by multiplying together half the product of their maximum values, and the cosine of the angle of lag. If t be the current at any instant, and e the impressed E.M.F., I and E being their maximum values, then the mean value of e i during a complete period is

— cos 6- (33)

150

SIMPLE PERIODIC CURRENTS.

and this is a measure of the mean rate of expenditure of energy on that circuit, or the mean power taken up. It ia obvious, then, that if the lag is 90°, this mean product is zero, and that no work is done at all.

When 6 has intermediate values between 0° and 90°, the real rate of dissipation or transformation of energy in the

"P T circuit will be intermediate between -— - and zero. In order

2

to understand how this can be, and how it is that a circuit may be traversed by a current and yet take up no power, we must examine a little more closely the nature of the phenomena.

§ 23. Power Curves.— Let the periodic curve in Fig. 62 represent a sinusoidal variation of electromotive force acting on a circuit which we shall for the moment assume has na

F.o. 62. — Electromotive Force Curve.

sensible inductance. Let the curve in Fig. 68 represent the corresponding current. The length of each ordinate of the second curve is equal in magnitude to that of the corresponding

Fig. 63.— Current Curve.

ordinate of the first curve divided by the value of the resist- ance of the circuit. Let the lengths of corresponding ordinates of these two curves be multiplied together, and the product

SIMPLE PERIODIC CURBENT8.

151

Bet off as the ordinates of a new curve represented by the dotted line in Fig. 64. This dotted curve is, then, the curve of potcer or activity, and represents the variation of the pro- duct of the current and the electromotive force taken at every instant.

In multiplying together the ordinates of the first and second curves, we must pay attention to the algebraic sign of each ordinate. Ordinates of each curve drawn above the horizontal

Fio. 64.

1-81

85

4-95

6-06

6-76

7-0

•906

1-75

2-47

8-03

888

8 6

1-64

612

12-26

18-37

22 86

28-4

Ordinates | Producta ..

datum line of the curve must be reckoned plus^ and ordinates drawn below must be reckoned minusy and in taking the product the algebraic law of signs must be regarded. It will be seen that the dotted line curve consists of a wavy line of two loops lying wholly above the mean datum line. If the area of the two hummocks enclosed by the dotted curve and the horizontal line is indicated or integrated, say, by an Amsler's planimeter.

152

SIMPLE PERIODIC CURRENTS.

the area represented by the shaded part so obtained is a measure of the total work done in one complete period of the current oscillation, and, since this area lies wholly above the datum line, it must be reckoned as positive^ or as work done by the electromotive force ; in other words, it represents the total energy transformed from electrical energy into heat in one complete period.

Next let us suppose that the same periodic electromotive force acts upon a circuit having inductance as well as resist- ance, and that therefore, as already shown, the current is retarded in phase behind the electromotive force. Let the thin curve in Fig. 65 represent the periodic impressed electro-

Fig. 65. Lag^S^.

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Producto ..0 5-5 1183 178 2048 2048 178 1188 65 0 818 818 0

Ordlnatet I ^'^ 803 S3S 3-5 3-88 303 247 176 -91 0

motive force, and the thick curve the current retarded by 45° in phase behind the other. Proceed as before to obtain the power curve by multiplying the heights of corresponding ordinates, the multiplication of the ordinates being shown below the figure. We find that the power curve representing the variation of the activity is a wavy curve, consisting of four sections, two large hummocks above the datum line, which are positive areas, and two small ones below, which are

SIMPLE PERIODIC CURRENTS.

168

negative areas. The algebraic sum taken with regard to sign of all these four areas, represented by the shaded parts, is a measure of the total work done in one complete period by the electromotive force. Going one step more, we may imagine that the current lag is 9€P, and in Fig. 66 we have drawn the power curve in this case, obeying the same instructions. We see that the power curve consists of four loops, two positive and two negative, and that the area of these hum- mocks are equal. Hence, the total area or indicated value in this last case is zero, and the work done in one complete cyde is zero ; hence, the rate of doing work, or the power, is zero. Returning to Fig. 64, the first case, it is easy to see

Fig. 65. Lag 90°.

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