The Alternate Current Transformer Vol. 2: The Utilisation of Induced Currents (1896) — part 10 of 36

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 vdll 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

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

the rate at which the electromotive force does work. Refer- ring to the second case, we see that the mean ordinate X Y i& 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

E I

-— . From the proof given previously we have seen that it

E I 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 quartei' 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.

§ 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 '•mean 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. 165

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.

This square root of the mean of the squares of all the ordi- nates of a simple periodic curve has, however, been shown in §9 to be numerically equal to the value of the maximum ordinate of the periodic curve divided by v^2. It follows that the total heat generated per second in the wire is a numerical measure of the V'mean square value of the current or of half the square of the maximum value of a simple periodic current. A fine wire stretched out in the manner of a Gardew voltmeter wire has a very small inductance, and, when acted upon by a simple periodic electromotive force, the current produced in it is very nearly proportional to this impressed electromotive force. It follows, then, that, when a Cardew voltmeter is subjected to a simple periodic electromotive force, the needle takes a defi- nite position, corresponding to a definite expansion of the wire, which is that which it would take if the wire were subjected

to a steady electromotive force equal to -7;; of the maximum

value of the periodic electromotive force.

IW SIMPLE PEBIODIG CURRENTa.

The Carddw voltmeter is not adapted to measure any bat very small currents. The instrument generally employed to measure periodic currents of moderate and large magnitude is some modification of Weber's electro-dynamometer. In the best-known practical form of Siemens there are two coils of wires in series, one fixed and the other movable, and so placed that the currents in the movable coil circuit are traversed at right angles by the lines of force due to those in the fixed coil. When a simple periodic current traverses the coils in series, a force is brought into existence due to the electro- dynamic action, which is proportional to the instantaneous value of the square of the current strength. From instant to instant, however, the current strength varies. If the time of free vibration of the movable coil is very large compared with that of a complete period of the electrical vibrations, and ii the movable coil is brought back by a restoring force due to a spring or bifilar suspension or gravity, &c., into a fixed normal position, then, during one complete electrical period, we may consider that the movable portion receives a number of small impulses which are in magnitude represented by the square of the ordinates of the current wave. Hence, the total impulse on the movable coil is equal to the magnitude of the inte- grated area of a sine curve whose ordinates are respectively the squares of those of the current curve, and the mean force on the movable coil will obviously be proportional to the mean ordinate of this force curve. If the movable coil is so heavy that its time of free vibration is very long compared with the time in which the periodic forces on it run through a complete cycle, it will experience a displacement exactly that due to the mean of the forces acting upon it — that is, to the square root of the mean of the squares of these instantaneous currents —

or to — -, where I is the maximum value of the current during

the period. The periodic force on the movable coil is equiva- lent to a steady force when this periodic force runs through all its values in a time very short compared with the time of free vibration of the coil. Hence, if a simple periodic current has a maximum value I, when it is sent through an electro-dynamometer it will cause a deflection equal to that

which would be caused by a steady current equal to — .

V2

SIMPLE PERIODIC CURRENTS. 167

Let ns suppose a coil of constant inductance L and resist- ance R to be traversed by a simple periodic current of fre- quency n (where 2vn^p). Let an electro-dynamometer be inserted in series with it, and let a Gardew voltmeter be connected to the extremities of the inductive circuit.

We have before seen that if E and I are the maximum

values during the period of the impressed E.M.F. and current

in an inductive circuit, then the power taken up in that

E I circuit is equal to -^ cos 6, where 0 = angle of lag of current

behind the E.M.F. But the reading of the Cardew volt- meter when connected to the ends of an inductive circuit i&

E very nearly proportional to — — , and the dynamometer reading

in that circuit is proportional to -p; therefore, the product of

E I these readings is proportional to -^, and takes no account of

the difference of phase. The product of the ^mean square values of the current and of the electromotive force in an inductive circuit is generally called the apparent power or apparent watts given to that circuit, but it is not a measure of the true power given to the circuit. For this reason we can derive no information from the use, in this manner, of these instruments. The observed readings, and hence their product, does not take into account the difference of phase between the current and impressed E.M.F. in the inductive circuit. A very small error, in practice negligible, is also introduced by disregarding the inductance of the wire of the Cardew instrument. On this account, strictly speaking, currents in the wire cannot be taken as accurately propor tional to potential differences at the extremities, but this is in. ordinary usage a negligible error.

§ 25. Method of Measuring the True Value of the Power given to an Inductive Circuit. Theory of the Wattmeter. — If a current traversing an inductive circuit under a periodic impressed electromotive force is made to pass through another circuit which acts electro-dynamically upon a movable circuit conveying another current proportional in strength to, and

158

SIMPLE PERIODIC CURRENTS.

agreeing in phase with, the periodic variation of potential difference at the terminals of the indactive circuit, such an arrangement will, if it can be realised, afford a means for obtaining a true numerical measure of the power taken up in the inductive circuit.

An electro-dynamometer having its fixed coil composed of thick wire and its movable coil of fine wire, each circuit being independent, is most usually called a wattmeter. The examination of the circumstances under which the wattmeter oan and cannot be used to measure the power expended in a 'Circuit subject to simple periodic electromotive force, leads to some interesting considerations.

If the thick and thin wire coils of a wattmeter are traversed by two independent steady unidirectional currents, the force

R L

<ir'15666666666^<^

Fig. 67.

on the movable coil is at any instant proportional to the pro- duct of the strength of these two currents. If each of these currents are simple periodic currents the force varies with the product of the instantaneous values, and the compound curve formed by taking as ordinates the products of the correspond- ing values of these separate current strengths at each instant is itself a simple periodic curve, provided that the two com- ponent currents have constant amplitudes, equal period, and fixed difference of phase. Let a wattmeter be supposed to be joined up to an inductive circuit (Fig. 67) ; let R and L be the resistance and inductance of this inductive circuit between the points Q Q' ; let the thick wire coil Th of the wattmeter be joined in series with this inductive resistance, and let the

SIMPLE FERIODIC CURRENTS.

159

fine wire ooil/of the wattmeter of resistance S and inductance N be joined to the points P P' ; let the thick wire coil be of negligible resistance and iQductance in comparison with the circuit QQ'. If a simple periodic electromotive force operates on the double circuit between the points P and P', we shall have a current flowing in B and S. It is required to calculate at any instant the currents in B and 8 respectively. Consider simply a divided circuit (Fig. 68) in which B and S are the.branches. Let a be the current at any instant in B, and y that in S, and let i be the strength of the current in that part of the circuit just before it divides ; in other words, i is the main current, which is divided into x in the inductive resistance and y in the fine wire coil of the wattmeter. Let e be the potential difference between the points P P' at the same instant, and let X, Y, I, and E be the maximum values

of all these quantities respectively. We assume that t is a simple periodic function of I, and we then write i = I sinpt, where j)B23rn, n being the frequency. Applying the funda- mental equation of §15 (p. 126) to each circuit, we see that

also

L^^ + Bo;-.; dt

Accordingly, L~ + Ba7=N^ + Si/; at dt

but, by the principle of continuity,

i = .r + y always, since there can be no accumulation of electricity at P or F ; hence a? = {— y,

160 SIMPLE FERIODIC CURRENTS.

ana,hence, L^ii=l^ +B(»-y)-Nll' + Sy, at at

or L4^+Et-(L + N)4^+(B + S)y.

at at

But i^lsmpU

and — = Ij?cos2>e,

at

(L + N)^ + (R + S)2/ = ILi)oosp< + IRBmpe, a t

or ^ + R±|j,-Ii4c08^t+ RI 9iBJ>t.

dt L + N L + N L + N

This differential equation is of the type

where Q is a function of t. The solution of this will be found in '^ Boole's Differential Equations," p. 88, and it is

y = ,-P'

NQe^'dt + constj,

e being here the base of Nap. logs, and not impressed E.M.R

•p , a

In the case before us P = -t — r?>

L + N

The integrals of e^^Binptdt and e ^* cos j9 1 i ^ are required* They are as follows : —

and L'*cosptdt.fJiE^±^P^E±3;

hence it follows that

le"Qdt^[e^ =^ainptdt+fe'*}^coBptdt J J li+N J L + N

BI J (P anpt-p coa pt) Lp I g''*(Pco8pt+y8inj>f)

-L+N P*+/ L + N P*+2>»

and P+^»,(«+S);+(L+g)V.

^ (L + N)»

SIMPLE PERIODIC CURRENTS.

Therefore we have by substitution

I r[R(R + S) + L(L + N)joT8ini?«

161

y

or

: I r[R(R + S) + L(L + N);>T8ini7« \

(R + 8)» + (L + N)yj+f(R + S)Li.-R(L + N)p]cos;4^^^^

„^ I f[R« + p»L« + RS + LN/?«]siniJ« \

" (R + 8)» + (L + N)vt +[SL;9-RN2)]cos;)eJ'

Since the original equations are symmetrical in x and y^ B and S, L and N, the value for x is given by changing R to S and L to N in the equation for i/.

This equation for y gives us the strength of the current in ihe fine wire coil, and it shows us that the phase of the eorrents x and y in the branch circuits differs from that of the main current i by an amount which depends on L, N, R and S. In order to exhibit this in a simple form we may direct atten- tion to a simple trigonometrical transformation.

Fig. 69.

Trigonometrical Lemma, — The function A sin ^ + 6 cos 0^ where A and B are constants, may otherwise be written

^/A» + B« sin (^ + </>),

B

where

tan <^«

A

Draw any rectangle (Fig. 69) 0 P, 0 Q, and draw a pair of rectangular axes, OX, 0 Y, through 0. Project the points Q, R, P on 0 Y. Then, by geometry, if P 0 X-^ and P 0 R-</>, Or^Op + Oq,

-OPsin^ + OQcos^,

«0 R sin B 0 X»0 R sin (9+^) ;

M

162 SIMPLE PERIODIC CURRENTS.

henoe 0 P sin 0 + 0 Q cos 0»O B sin (0 + <^)

= ^OI« + OQ«sin (0 + </>). But tani^ = 2.Q;

hence A sin 0 + B cos 0 = ^/a* + B'' sin (0 + <^), . (86)

where tan <fr = -r—

A

Beturning then to the equation for y, the coefficients of

sin ^ t and cos pt in the equation are respectively B (B + S)

• L (L + N) p\ which represents the A, and (R + S) L j?-B

{L + N) Pf which reprei^ents the B, in the above. Squaring

each of these expressions, and adding the results, we obtain

as a result

{(B + 8)» + (L + N) V} (R" +P" Ii") ;

hence we finally arrive by substitution at the equation for y

— vxfc+p_ij sm{pt+e), . (86)

v^(B + Sf +i>«(L + N)« ^^ ^'

where tan g^B ^(B4-8)L;.-.B(L + N);>

wnere ^ R (B + 8) + L(L + N);)»' ^^

tan0= (SL-BN)p ^ ^

E(R + S) + L(L + N)i?« ^ ^

In this form the equation for y shows us that the phase of y is dhead of that of t, or that the main current lags behind the current in the branch S, provided that S L is greater than B N ; and, since the expression for the current x is perfectly sym- metrical, we can write it down at once, and it is

where tan ^ = (BN-SL)p ,q.

and it is obvious that, if S L is greater than B N, tan 6 is positive, and tan ^ is negative. If SL«BN, then there is no lag, and the branch currents x and y agreein phase with the main current t .

The general result is, therefore, this — ^When ah impressed electromotive force acts on a oirouit which branches into twOy

SIMPLE PERIODIC CURRENTS. 163

having each self but no mutual induction, there is a difference of phase between the currents in the main Une and branches ; that is, they do not come to their maximum values at the same instant. The main current lags behind the impressed electromotive force in phase, and the two branch currents respectively lag behind and are pressed ahead of the phase of the main current.

The question then arises, under what circumstances does the branch current which is in advance in phase of the main current get so much ahead that it comes into consonance with the phaae of the impressed electromotive force ?

To settle this question we shall have to discuss briefly the question of the compound impedance of branch circuits.

§ 26. Impedance of Branched Circuits. — Lord Bayleigh has treated the problem of the impedance of branched circuits under the assumption that any number of circuits are connected in parallel, posessing each self-induction, but having no mutual induction.''^

The problem is : Given the resistance and inductance of each branch, to find the compound resistance and inductance^ or equivalent resistance and inductance, of the system for simple periodic currents of given frequency.

Let B and L be the resistance and inductance of any branch, iuid p the pulsation «:27rn. Let B' and L' be the compound or equivalent resistance and inductance of the system of parallel conductors.

The solution of the problem given, for which we refer the reader to the original paper, is

If we take, as usual, tan 6 = ?~, and write (Im) for impedance,

B

• See Lord Rayleigh ** On Forced Harmonic Oscillations of Various Periods " {PhiL Mag., May, 1886, p. 379).

m2

164 SIJUPLE PERIODIC CURRENTS.

where (Im)-E'+;)'L, we can write the above relations

B

A-2 6-

(Im)« L

Let R"+2>L'' be written (IM). This is the compound or eqtiivalent impedance of the system of parallel conductors. It is obvious that

(Im)« (Im)« '

hence (IM)* = _-- ^

\ (lm)V \ (Imy^

Consider the case of a pair of conductors in parallel (Fig. 70)^ having resistances B and S and inductances L and N, but no mutual inductance.

Let VK2+/>«L«-(Imi),

and VS«+;>='N«-(Im,),

and

then (IM)s.

and ^R'^+p^L'^^ilil);

1

,(Imi)^ ■"(Im,)V ^(lm,) ^[Im^»J ^ or (T^)- ^ (I"',)(Im,)

The lag c of the main current just before branching, con- sidered with respect to the impressed electromotive force, will be given by the equation

tan<-^;

hence tanc-^- '^f

^(lm)«

8IMFLE FERIODIC CURRENTS. 165

generallyi and in the case considered will be

tanc=.— ^ "^-

hence after reduction

(S2+jt>gN«);f>L+(RHjp^L«)pN

This is the equation which determines the lag of phase of the current t behind the impressed electromotive force in the main branch before dividing into the branch currents x and ^ in B and S respectively.

tan € = — ,,

Fio. 70.

Compare this equation with that which determines the angle by which the phase of the branch current ^ in S is ^luad of the main current t. It is, as we have seen,

(SL-^RN)y "^^~R(H+S)+L(L+N)p«

In the expressions for tan c and tan d put N » 0, and they both become equal to

SLp

R(R + S) + //^L«-

This shows that, when N =0, the current y in the branch S is as much ahead of the main current i as t is behind the im- pressed electromotive force, and hence that y agrees in phase with the impressed E.M.F. acting on the double circuit; in other words, the current in the branch 8 is entirely unaflfected by being joined in parallel with an inductive circuit R; but if N is not quite zero, then the current in branch 8 is affected, as regards its lag, by the fact of being

166 SIMPLE PEmODIC GXTKRENT8.

joined in parallel with an induotive diouit. The natare of

this affection will be dependent on whether SL— BN is

L N positive or negative — that is, whether _ or — is the greater —

that is, whether the time constant of the B circuit or the

S circuit is greater. If -. is greater than — , then the current B D

y in S is aliead of the main current t, but lags behind the im-

L N

pressed electromotive force. If :„- is less than — -, then the

B o

current y in S lags behind the main current i in phase, and^

a fortiori, behind the impressed electromotive force.

§ 27. Wattmeter Measurement of Periodic Power. —

Betuming to the wattmeter problem, let one of these divided circuits, viz., the one of resistance B, be a circuit in which it is desired to measure the electrical power. In the ordinary way of using the wattmeter, the fine- wire coil, which we will assume has a resistance S, is placed in parallel with the inductive circuit, the thick-wire coil united in series with the inductive circuit. The main current i is thus divided between the inductive circuit B and the wattmeter fine- wire circuit S. The electro-dynamic action in the wattmeter is then one between a current in 8, which we have called y, and one in the thick- wire circuit, which is the same as that in the inductive circuit B, which we have called a-.

We have above arrived at expressions for the values of x and y. The question then arises how &r the indications given by the instrument, and which are due to the electro-dynamic action of the currents x and t/, and proportional to their nume- rical product, are proportional to the real power taken up in the circuit B.

The current x is the same as the current in B ; hence the error, if any, will result from the current y in S differing in phase or in proportionality from the potential difference between the ends of the circuit B.

In the ordinary mode of calibrating the wattmeter the instrument would be applied to measure a power in a non- inductive circuit traversed by a known current, and having a known potential difference at its ends.

SIMPLE FBBIODIC CURRENTS. 16^

From this the real watts taken up in the curcuit are known, and, since the force required to hring back the movable coil to its initial position is proportional to the product of the numerical values of the currents in the fixed and movable coils, we have at once the desired constant of the instrument.

If a wattmeter so calibrated is applied to measure power in an inductive circuit, there are two causes of error which maj or may not neutralise each other, and which may cause the measured watts as determined by the instrument to be greater than, equal to, or less than, the real watts or power taken up in the circuit.

The first of these causes of error is due to the fa^t that the fine-wire circuit of the wattmeter always has a sensible induc- tance— ^that is, N is not zero. It may be made very small by arranging the chief part of the wire resistance of the fine- wire circuit as a non-inductive resistance in series with the small inductive resistance which forms the movable coU. It follows that, if E be the maximum potential difference during the period between those points to which the fine-wire circuit is attached, the mean-square ( v^mean*) value of the current in

1 E

the fine- wire circuit is equal to -— = ^when subjected

v2 y S +^ -N"

to a simple periodic E.M.F. of angular velocity p. This quantity is not proportional merely to E, but depends also on the value of p. One effect of the impedance of the fine- wire circuit is to make the mean-square current in it under periodic E.M.F. less than it would be if produced by a steady E.M.F. equal to the mean-square value of the periodic E.M.F. But, in addition, the impedance causes a lag in phase of tho current in the fine-wire circuit behind the phase of the poten- tial difference between its ends. This is the second cause of error, and the effect of this lag is dependent upon the nature, whether inductive or non-inductive, of the circuit B.

To dissect its action, first let us suppose the circuit B is non-inductive — that is, let L be zero. The current a? in it will, therefore, coincide in phase with that of the potential difference at the points of junction. The current in S, viz., y, will, however, lag in phase behind that of the potential difference at the junction. The effect of this lag in S will be to increase the phase difference between z and y, and to

168 SIMPLE PERIODIC CURRENTS.

diminish the cosine of this angle of phase difference. Hence^

XY

the effect is to diminish the product — - cos 8, which measures

the true mean product of x and y, X and Y being their maximum values and B their difference of phase. Since by assumption X agrees in phase with E, any reduction of the above product reduces the instrumental reading, and makes it less than the true-power reading. If, however, we have to deal with a circuit possessing inductance, and in which, therefore, there is a current x, of which the phase lags behind that of the potential difference of the junc- tions, then the lag in the current y in the circuit 8, so fax from increasing the difference of phase of x and y, may operate to bring them nearer into accordance, and to increase the instrumental reading, and more than make up for the decrease due to the first-named cause of error.

§ 28. Oorrecting Factor of a Wattmeter. — The action of these two causes of error may be illustrated and explained best by the graphic method by a construction which at the same time shows us how to obtain geometrically the value of the compound resistance and impedance of a branched circuit.

Describe a circle with centre 0 (Fig. 71), and take any line OA to represent the maxunum value of the potential dif- ference between the two points M M' of the divided circuit, of which B is the resistance of the inductive circuit consisting of the thick wire of the wattmeter in series with the circuit in which the poicer is being measured, and S that of the fine wire of the wattmeter. Then, as before, the vertical projection of 0 A as it revolves represents the periodic variation of this potential diiference. On 0 A describe a semi-circle, and set off on 0 A, as a base, two right-angled triangles OCA, 0 B A, of which the sides OB, B A, and 0 G, C A are in the ratio respectively of the resistance to the reactance of these circuits. Otherwise the angle A OB is one whose tangent is p times the time-constant of the S circuit, and A 0 C is one whose tangent is p times the time-constant of the B circuit. Take one S"* portion of OB, and set off OY equal to it, then, as in § 21 (p. 144), 0 Y represents the maximum value Y of the current in the S circuit.

SIMPLE FEBIODIC CURRENTS.

169

Sinularly, set off 0 X equal to one R*^ part of 0 0, and OX represents the maximum current X in B. On 0 X, 0 Y describe a parallelogram 0 Y I X, and draw the diagonal 0 1, and produce it to 0 D. Then 0 1 represents the maximum value of the main current I just before division. Join AD ; AD and OD will represent the product of the current I and the equivalent reactance and resistance of the two circuits B and S in parallel respectively.

To prove this last proposition, we must refer again to the paper by Lord Bayleigh on ** Forced Harmonic Oscillations of Various Periods " {PhU. Mafj., 1886).

Fio. 71.

If B' represents the equivalent resistance of a number of resistances joined in parallel between two points, and L' repre- sents the equivalent inductance of the system, then it is shown in Lord Bayleigh's paper that

A B

. ^ B where ^^^E'+p'U'

" »'-lWL»'

170 SIMPLE PERIODIC CURRENTS.

B and L being the resistanoe and inductance of any branchr and the mutual inductance being zero. ' -^PP^y ^^ theorem to the case under consideration, viz, the two inductive resistances (R, L) (S, N) in parallel, and we have-

R . S

B-

L N

Effecting the multiplication we have

R(S'-hp'N^) + S(R'-4-/>'L«) ^- (b'»+7>*]S«)(R^+;y^L«) '

L(S^+y'N') + N(R'4-/>'L') ^- (S«+^«N*)(R'^"+/)'^L«) '

. ^, A R(S^+;^»N') + S(R^ + ;/^L')

and «-A«+i>*B»- (R + S)«+^«(L + N)^ '

B L(S«4-i>'N«)4-N(R^-hp^L'^)

^^A«+i7«B'= (R + 8)»+p«(L + N)« ""

Turning back to Fig. 71, we see from the geometry of the figure that, if the angle B 0 D is as before called $, BOD-DAB.

We have then _^= 0 B- A B tan 6.

cos(y

But since A B =2? N Y and 0 B = S Y by construction, therefore 0 D = S Y cos O^p N Y sin ^.

In § 25 we have found the value of tan 6 to be

(SL-RN);> , ^^-R(R + S)4.L(L + N)i^' hence, eliminating the sin and cos terms, and substituting for Y the value obtained from equation (86), page 162, we get

RS(R+S)+p«(SL«-fRN')T ^^^ (R + S)^+(L + N)V where I is the maximum value of the main current, and Y that of the current in the 8 circuit.

On comparing this value for OD with the value above calculated for R' we see that 0 D - R' I.

SIMPLE PERIODIC CURRENTS. 171

So that, on the same scale on which 0 B and 0 0 represent 8 Y and R X, 0 D represents R' I. Similarly, it may be shown that A D=p L' L For the angle C 0 D= tf'«angle 0 A D,and

A5;=AC-0Ctan^', cos^'

or AD=pLXcos ^'-RXsin^';

and, since tan 6^=, ^^ (^/"^^ ^ ■

S(R + S) + N(L-i-N)/'

a similar substitution, with help of equation (89), page 162, enables us to see that

yL(S»+/N')+i>N(R«+;7*L^)j ^^" (R + S)«+(L + N)V

and this is equal to the value found by anailysis for p JJ I.

This diagram shows us, then, what is the effect of the inductance of the wattmeter fine- wire circuit, and what must be the correction applied to the readings to get the real power expended in the inductive circuit.

The actual reading of the wattmeter is proportional to the true mean value of the product of x the current in the induc- tive circuit R and y the current in the fine-wire circuit S ; and this, as previously shown, is equal to half the product of their maximum values, and the cosine of the difference of phase.

From Fig. 71 this mean value is therefore

Q^-Q^cosineBOa 2

This, however, is not the measure of the power expended in the R circuit. The true watts are proportional to the mean product of X and a current equal to one S^ part of e, having a phase difference equal to the angle C 0 A, viz., that of the angle of lag of the current in R and the potential difference 0 A of its ends. Hence the real power or watts are proportional

to J? . 0 X . cosine C 0 A,

S

sinee E is the maximum of e, viz., the instantaneous potential difference between the extremities of the branch circuits.

Now O Y is taken as one S^ part of the effective electro- motive force in the 8 circuit ; and on the same scale on which

172 SIMPLE PERIODIC CUREENTS.

O A represents the impressed E.M.F. 0 B represents the effec- tive E.M.F. in that circuit. Hence, in taking the reading of the wattmeter, which is proportional to the quantity

5^^l2Z cosine BOO,

fis the watts, we are making an error; the quantity really required is the value of

1^ 0 X cosine A 0 0,

2 b

which is numerically equal to the real power. We see that two errors come in — one due to the maximum current in

OB fj 0 A

the fine-wire circuit being O Y or _ - instead of - or -— ^

and the other due to the phase difference being taken as the angle COB instead of C 0 A.

To correct the instrumental reading or observed watts to true value or real watts, we have to multiply the observed readings by two factors.

First, the ratio of ^^ or ^^'-^P'^\

0 13 b

which is the correction due to the self-induction of the fine- wire circuit or to the potential part of the wattmeter having a sensible inductance. The second is the ratio of the cosines of the angles C 0 A and C 0 B, or

cos C 0 A cos C 0 A

cosCOB cos(COA-BOA)

But from the diagram

cosme C O A = — ---^^^^=^» ^/R--f/L*

and cosine B 0 A = — ^

^k.

__ R

R b ph jpN

JW +/ L'2 V S* -I-/ xX'* JR' +/ U J6' -hi/'* H*

aiMFLE FEBIODIO CURRENTS. 173

Combining these two corrections into a single product, we. get as the fall correcting factor : —

or BS^+p'N'B p

BS*+i?»LNS

N If we put Ts-* ^, where Tg is the time-constant of the 8,. o

or fine- wire circuit, and Ta = ^ , where Tr is the time-constant.

B

of the B circuit, we have

F^l±tll-, (41)

and the real watts or power taken up in the circuit B is obtained by multiplying the observed watts by F. F becomes unity for two cases when L and N are both zero, and also whenTs-TR.

Hence, the ordinary wattmeter, applied as usual to measure the electrical power in a circuit traversed by a simple periodic current, gives absolutely correct readings only in two cases. First, when the fine-wire circuit and the circuit being measured have no inductance ; second, when the fine -wire circuit and the circuit being measured have equal time-constants.

But if Ta is greater than Tg, then F is a proper fraction. The wattmeter reads too high, and the real watts are less than, the observed. If T^ is less than Tg, then the observed readings are too low. If Ta - Tg, then the observed readings are correct. Hence the wattmeter may read too high, too low, or correct. Generally speaking, it reads too high, since the time-constant of the measured circuit will most often be in excess of that of the fine- wire circuit.

§ 29. Mutual Induction of Two Olrcnits of Oonstant In- ductance.— As an illustration of the above principles, it is useful to consider the case of the mutual induction of two circuits in one of which a simple periodic electromotive force operates. We suppose two circuits to be so placed relatively to each other that when a change of current occurs in one, whick is called the Primary (Pr.), a change of magnetic induction.

174 SIMFLE FERIODIC CURRENTS.