The Alternate Current Transformer Vol. 1: The Induction of Electric Currents (1896) — part 9 of 35

the exponential value for aio.pt and putting k for V — 1, we have

i^)..(-i-^)l (30)

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

E

2fcBT (

— T~- — rjr

and this last becomes by simplification E •£ ( #»* f e

SIMPLE PERIODIC CURRENTS. 135

Hence, equating both sides of equation (30), when integrated we have

f- E / ^ ^ \

2B/H 1+kpT 1-kpT )

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

ekp = cos pt --lcsinpt, and e~*pt=cospt — ksiupt,

we get finally

._ E f shi£>£— 2>T cos p t )

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 i is accordingly called the instantaneous 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. Keplace T by — , and let 6 be an angle whose fi

tangent is equal to — t ;

hence tan 6=—£=pT.

It follows by an easy transformation that

R LP

We have, then, for the value of the current i, the equation i==E (sin p t—p T cosp t\ .

or, by substitution, •p

136 SIMPLE PERIODIC CURRENTS.

This is called the particular solution of the equation

L— +Et=E sinjp t = f, 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 6 — such that tan Q=p T = _ ^;

B

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'lft+p'T?; and, third, that the current curve is a simple periodic curve. The quantity /V/K2-fp2L2 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 p t = 0. Now, since the solution of

at

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— +Bi = Esin^« d t

given by the equation

-'. . (32)

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

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 — . B

SIMPLE PERIODIC 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 (32).

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 /2. 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

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

-p

phase by the letter I, and since I = |— , we have

i = Isin(pt-0). Hence, we may write this in words as follows : —

tJw iji'iximitm value] f the maximum value of the

-I i>

of the current ]- = -j impressed electromotive force strength J [ " impedance

and the mean-square ] f t/w ma<rt-,/mm raZtM

flafofc? o/ f/te - = 4 j= .

current strength j (_ ** *

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 1

or current ~\ (electromotive force

continuous f strength J=\ ^isTan^e (Ohmslaw).

currents J

For simple^ mean-ware] , IMrtl)I_,ylMW ridue Oj periodic or I value oj the I = | , ele^.omotive foi^e

alternate ( current .

currents J strength } <• 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.

FIG. 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 off as off-sets at equal distances above and below a datum line, the extremities of these ordinates will lie 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 0 P 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 amplitude, or maximum value, and POM the phase 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 he 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 radii 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. Graphic Representation of Periodic Currents. — 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 ohrnic 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 rate 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. 55), be cut out and pivoted by a pin at the angle 0, so as to turn freely clockhand-wise. Let a vertical line, 0 Y, be drawn through 0, and in any position let the sides 0 A, 0 C, A B be projected on to 0 Y. The projection of lines equal and equally inclined are equal ; hence, since A B is equal and parallel to 0 C, the projection of AB — viz., a b — is equal to that of OC — viz., Oc. But 06 = 0 a + ab always for any position of the card; hence Ob = 0a4-0c. The projection of the diagonal is therefore equal to the sum of the projec-

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 B in amplitude 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. Returning in thought to electric motion, consider the motion of a particle

FIG. 56.

of electricity (if we may be allowed the expression) in the wire subjected to two simultaneous simple periodic motions of unequal amplitude 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, OB, fixed like hands of a toy clock at right angles (Fig. 56), 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 change is represented by a sine curve of different amplitude

142

SIMPLE PERIODIC CURRENTS.

shifted backward 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

FIG. 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 R 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., 0 b— 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, OB, corresponds to E M, and is the resultant of the motion 0 B, and the rate of change of that motion, viz., OA. If, then, OB 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 PERIODIC 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 E, the quantity Ei represents the voltage necessary to make this current flow, and this part of the impressed electro- motive force is called the effective electromotive force in the circuit. If the circuit is an inductive circuit, there will be another electromotive force equal in magnitude to L — , which

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 effective 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, OP (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.

value, E, of the impressed electromotive force, and Og, 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 0 P passes through the angle POX,

i= Lpl COS(pt — 9)

FIG. 58.

the angle P 0 X is the same fraction of four right angles of 2?r that t is of T. Hence the angle POX is, in magni- tude, equal to ^- . For shortness, 2u-/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 e = 1E sinpt,

SIMPLE PERIODIC 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 0 P describe a semi-circle, 0 M P, on that side of 0 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 LJJ to B, where L is the inductance and K the resistance of the circuit. Through 0 draw 0 K parallel to MP. Produce MO to N, and make ON equal to OM. Draw N K parallel to 0 P and join 0 K. Then, on the same scale on which 0 P represents E, the maximum magnitude of the impressed electromotive force, 0 K will represent the maximum magnitude of El or the effective electromotive force, and 0 N will represent ~Lp I or the maximum magni- tude of the inductive electromotive force. By the geometry of the figure we see that, if the angle P 0 K is called 0, the projection Ok of OK on OY is equal to OK sin(pt— 6), and also the projection 0 n of ON on 0 Y is equal to

0 N cos (p t - 0} . Hence 0 n = — (0 k) . Moreover, 0 K is at

the resultant of 0 P and 0 N, and 0 N is at right angles to 0 K ; therefore 0 N and 0 K fulfil all the conditions requisite for being the representation of the maximum values of the inductive and effective electromotive forces. For OK 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 K ; 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, 0 K, and 0 N, 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 K 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 current, 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 R I, where R is the resistance of the circuit and I is the maximum value of the current, and R I is therefore the maximum value of the effective electromotive force in the circuit, we see that the magnitude of 0 N must be It pi, and that of OP must be A/R2 + p2L2I, 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 OK represents the effective electro- motive force RI, and ON represents the inductive electro- motive force L p I. If we take one Rth part of 0 K, we have the value of the current in the circuit. The angle of lag 6 by

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

cos Q =

Impedance of circuit. ~ -/R2+^y2L2'

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. 59), 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 C A B is

SIMPLE PERIODIC CUEEENTS. 147

-p an angle which has a cosine equal to 2— , we see that

this angle, which we may call 0, is the angle of lag of current behind electromotive force, and, moreover, that pL__ reactance of circuit ~~ K "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 power 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

FIG. 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 P M, Q M, 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 mean 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 common centre 0, separated by a fixed angle, P 0 Q. At equal small intervals of time corresponding to equal angular motions let the projections Op, 0 q 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 period.

Denote by X the length of 0 P, and by Y the length of 0 Q, and let the angle P 0 Q be ft, 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, Oq, which are the vertical projec- tions of 0 P, 0 Q.

. 61.

Let Op be denoted by p, and 0 q by q Then p = X cos a,

and # = Y cos (a+/3);

and therefore p q = X Y cos a cos (a +/:?).

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 being a very large number. At each stage let the value of p q be measured as above, then the mean value of the product p q is one ?zth part of the sum of all the n values so taken. Call this mean value of the product M. Then,

XYjcos /2+cos Sacos(Sa+/3) + cos2Sacos(2Sa-f-/3)x

n ( .... +COS71— 18aCOS(w-l Sa + /3)j

By trigonometry we have

cos (n— 1 8 a) cos (n— 1 8 a+/3) = J cos (2n-l S a+f3) + i cos /3, since cos A+B + cos A— B = 2 cos A cos B.

M

SIMPLE PERIODIC CURRENTS. 149

Accordingly every term, except the first in the cosine series for M, splits up into the sum of two others, one of which is always | cos /3. Rearranging the terms, we get for the value of M as follows : —

M =— f| n cos £+icos £+cos (2 8 a+0)-f cos (4 3 a

-fcos

)) J

The cosine series in the inner bracket consists of a series of cosines of angles in arithmetic PL egression 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 8 a very small, and n 8 o equal to 2?r. For when

n 8 a = ITT, 2 7i-l 8 a+P = 4ir+0, and cos (47T+/3) = 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. : —

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 i 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*, ..... (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 is obvious, then, that if the lag is 90°, this mean product is zero, and that no work is done at all.

When 0 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 no

F.G. 62. — Electromotive Force Curve.

sensible inductance. Let the curve in Fig. 63 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 PEBIODIC CURRENTS.

151

set off as the ordinates of a new curve represented by the dotted line in Fig. 64. This dotted curve is, then, the curve of power 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

Ordinates | Products ..

FIG. 64.

1-81 3'5 4-05 6'06 676 T'O

•906 1-75 2-47 3-03 338 35

1-64 6-12 12-25 18-37 22 '86 25'4

datum line of the curve must be reckoned plus, and ordinates drawn below must be reckoned minus, 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 45°.

r»«r too/2'4? 3-03 3-3S 3'5 3'38 3'03 2 -47 1 '75 '91 0 '91 1'75 2'47

iDdies^0 1-gl 35 4.956-06 6.76 7 6-76 g-06 4 '95 3 "5 l'8l 0 Products ..0 5-5 11-83 17'3 20'48 20'48 17'3 11'83 5'5 0 318 3 "18 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.

153

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 90°, 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 cycle is zero ; hence, the rate of doing work, or the power, is zero. Eeturning to Fig. 64, the first case, it is easy to see

FIG. 65. Lag 90°.

'3 3'38 3'°3 2 47 * 73

rbl 3.5 4.^ ^

Products ..0 .613 10 Cl 1223 1061

6'13

that the rate of doing work, or the power, is measured by the mean ordinate of the shaded work areas considered as indicator diagrams. Since the dotted curves are perfectly symmetrical, if we draw a line Y Y' at half the height of the maximum ordinate of the dotted curve, it will cut the two hummocks into two parts, and the area of the upper part, or mountain above the line Y Y', would just fill up the valley between the bottom parts of the hummocks. Since the rate of doing work is equal to the work done in one complete cycle divided by the

154 SIMPLE PERIODIC CURRENTS.

time of duration of that cycle, it obviously follows that this mean ordinate X Y measures the power or rate of doing work and is

T? T

equal in magnitude to ; hence this product is a measure of

2

the rate at which the electromotive force does work. Eefer- ring to the second case, we see that the mean ordinate X Y is no longer equal to half the maximum ordinate of the positive or upper loop, but is equal to half the difference between the magnitudes of maximum ordinates of the positive and nega- tive loops of the power curve, and is therefore less than

From the proof given previously we have seen that it

' F T is equal to — - (cosine of lag).

In the third case considered, of a lag of 90 degrees, it is easy to see why the resultant rate of doing work is zero. In the first quarter of a stroke the electromotive force propels the current, and this last is in the direction of the E.M.F., but in the second quarter of a stroke the current is negative or opposite to the electromotive force ; in other words, the current is moving against the force and does work against the E.M.F., and the same push and re-push is repeated in the second half of the period. Hence, on the whole, though there is an impressed electromotive force and a current flowing, no resultant work is done and no energy dissipated.

S 24. The Experimental Measurement of Periodic Currents and Electromotive Forces. — At this stage it will be an advan- tage to direct attention to the practical means of measuring the •V/rnean square value of periodic currents and electromotive forces, and also the mean value of the power given to an inductive circuit of any kind. There are, amongst others, two instruments especially useful for measuring periodic currents. One of these, the Cardew voltmeter, depends upon the prin- ciple that when a wire traversed by a current, either steady and unidirectional or steadily periodic, is placed in an enclosure the walls of which are approximately at a constant tempera- ture, the wire will itself, after a short time, attain a constant temperature. This constant temperature is reached when there is a state of equilibrium between the rate at which heat

SIMPLE PERIODIC CURRENTS. 155

is radiated by the wire and the rate at which the walls of the enclosure radiate heat back to it.

The wire has a definite length corresponding to each tem- perature, and means are provided for measuring this elongation with great accuracy. The total amount of heat generated in the wire per second is dependent upon the rate of generation at each instant. The instantaneous rate of heat development is, by Joule's law, equal in mechanical units to the product of the resistance of the wire and the square of the value of the instantaneous current flowing in it.

If the wire is traversed by a simple periodic current, and we construct from the current curve diagram another curve whose ordinates are equal to the square of the correspond- ing current ordinates, we have a curve every ordinate of which is proportional to the instantaneous rate of generation of heat in a wire traversed by the periodic current. Since the horizontal line measures time, it is obvious that the whole area of the outer curve, or heat curve, represents the total work done per semi-period by the current in producing heat, and that the same total work would be done by a steady current which had a value equal to the square root of the mean of the squares of all the ordinates of the periodic current curve.