The Alternate Current Transformer Vol. 1: The Induction of Electric Currents (1896) — part 10 of 35

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 Cardew 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 — — of the maximum value of the periodic electromotive force.

166 SIMPLE PERIODIC CUEEENTS.

The Cardew voltmeter is not adapted to measure any but 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 il 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

A/2

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 — - .

SIMPLE PERIODIC CURRENTS. 157

Let us suppose a coil of constant inductance L and resist- ance B to be traversed by a simple periodic current of fre- quency TO ( where 2a-w = p). Let an electro-dynamometer be inserted in series with it, and let a Cardew 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

TT T circuit is equal to -^- cos d, 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 is

•p very nearly proportional to — — , and the dynamometer reading

v2

in that circuit is proportional to —= ; therefore, the product of v2

Tjl T

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 GTJEEENT8.

agreeing in phase with, the periodic variation of potential difference at the terminals of the inductive 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 can 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

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 E 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 PERIODIC CURRENTS.

159

fine wire coil /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 inductance in comparison with the circuit Q Q'. 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 S respectively. Consider simply a divided circuit (Fig. 68) in which B and S are the branches. Let x 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 i is a simple periodic function of I, and we then write i = I sin p t, where p = 2?r n, n being the frequency. Applying the funda- mental equation of §15 (p. 126) to each circuit, we see that

dt

also

K. 3 ' 1 T » X T> XT « V

Accordingly, L — + B a; = N -^

but, by the principle of continuity, i = x + y

always, since there can be no accumulation of electricity at P or P' ;

hence x = i—y,

160 SIMPLE PERIODIC CURRENTS.

and, hence, I/(t'~y) +R(t-y) = N^+Sy, dt dt

or

at

But i = I sin ^? t.

and

or

dt

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. 38, and it is

e being here the base of Nap. logs, and not impressed E.M.F. In the case before us P =

and

L + .N

The integrals of e Pt sin p t d t and eFtcosptdt are required* They are as follows : —

and

hence it follows that

RI p (P sin^t-^cosp t) Ijpl ePt(Pcospt +psin.j)t) = e LTN ~

SIMPLE PERIODIC CURRENTS.

Therefore we have by substitution

---

(R + S)2 + (L

S)L/, _

161

\

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

This equation for y gives us the strength of the current in the fine wire coil, and it shows us that the phase of the currents 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 0 + B cos 6, where A and B are constants, may otherwise be written

^A2 + Ba sin (6 + </>), B A

Draw any rectangle (Fig. 69) 0 P, 0 Q, and draw a pair of rectangular axes, OX, OY, through 0. Project the points Q, R, P on 0 Y. Then, by geometry, if P 0 X = 6 and P 0 R = <£,

where

toni-tl.

= 0 P sin $ + 0 Q cos 6,

162 SIMPLE PERIODIC CURRENTS.

hence OP sin 0 + 0 Q cos 0 = OR sin (8 + <j>)

= VO P2 + 0 Q2 sin (6 + <£). But tan*-°-|;

hence A sin 6 + B cos 6 = J&* + B2 sin (8 + <£), . (85)

where tan <£ = — .

A

Returning then to the equation for y, the coefficients of sin 2) t and cos p t in the equation are respectively R (R + S)

• L (L + N) f, which represents the A, and (R + S) L p - R (L + N) p, which represents the B, in the above. Squaring each of these expressions, and adding the results, we obtain as a result

{(B + S)2 + (L + N)2 p*} (R2 +P* L2) ;

hence we finally arrive by substitution at the equation for y

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

(39)

and it is obvious that, if S L is greater than R N, tan 0 is positive, and tan ff is negative. If SL = RN, then there is no lag, and the branch currents x and y agreein phase with the' main current i.

The general result is, therefore, this — When an impressed electromotive force acts on a circuit which branches into two,

SIMPLE PERIODIC CURRENTS. 163

having each self but no mutual induction, there is a difference of phase between the currents in the main line 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 phase 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 Kayleigh 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 K and L be the resistance and inductance of any branch, and p the pulsation = £TT n. Let E' 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

where and

If we take, as usual, tan 9 = P _ , and write (Im) for impedance, R

• See Lord Rayleigh " On Forced Harmonic Oscillations of Various Periods " (PhU. May., May, 1886, p. 379).

Al2

164 SIMPLE PERIODIC CURRENTS.

where (Im)2 = R2+;/L2, we can write the above relations

Let B"+/>9 L'2 be written (IM)2. This is the compound or equivalent impedance of the system of parallel conductors. It is obvious that

hence (IM)2 =

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

then (IM)2

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

hence tan.--

SIMPLE PERIODIC CURRENTS. generally, and in the case considered will be

165

tan c

.hence after reduction tan e = ^~r.

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

Compare this equation with that which determines the angle by which the phase of the branch current y in S is .a/iead of the main current i. It is, as we have seen,

(SL-RN)p

In the expressions for tan e and tan 6 put N = 0, and they both become equal to

This shows that, when N = 0, the current y in the branch S is as much ahead of the main current i as i 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 S is entirely unaffected by being joined in parallel with an inductive circuit R; but if N is not quite zero, then the current in branch S is affected, as regards its lag, by the fact of being

166 SIMPLE PERIODIC CURRENTS.

joined in parallel with an inductive circuit. The nature of this affection will be dependent on whether SL— RN is

positive or negative — that is, whether — or — is the greater —

R b

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

S circuit is greater. If — is greater than — , then the current R b

y in S is ahead of the main current i, but lags behind the im- pressed electromotive force. If — - is less than ~, then the R b

current y hi S lags behind the main current i in phase, and,. a fortiori, behind the impressed electromotive force.

§ 27. Wattmeter Measurement of Periodic Power. — Returning to the wattmeter problem, let one of these divided circuits, viz., the one of resistance R, 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 R and the wattmeter fine-wire circuit S. The electro-dynamic action in the wattmeter is then one between a current in S, which we have called y, and one hi the thick- wire circuit, which is the same as that in the inductive circuit R, which we have called x.

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

The current x is the same as the current in R ; 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 R.

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

SIMPLE PERIODIC CURRENTS. 167

From this the real watts taken up in the circuit are known and, since the force required to bring 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 may 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 fact 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 coil. 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 ( y'mean'2) value of the current hi

1 E

the fine-wire circuit is equal to —7= --when subjected

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 the 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 E.

To dissect its action, first let us suppose the circuit R 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 hi 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 x and y, and to

168 SIMPLE PERIODIC CURRENTS.

diminish the cosine of this angle of phase difference. Hence, 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 8 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 S, so far 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. Correcting 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 maximum value of the potential dif- ference between the two points M M' of the divided circuit, of which E. is the resistance of the inductive circuit consisting of the thick wire of the wattmeter in series with the circuit in which the poiver 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 difference. 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 00, C A are in the ratio respectively of the resistance to the reactance of these circuits. Otherwise the angle A 0 B 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 E circuit. Take one Sth 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 PERIODIC CURRENTS.

169

Similarly, set off 0 X equal to one Eth part of 0 C, and (0 X represents the maximum current X in E. On 0 X, 0 Y describe a parallelogram 0 Y I X, and draw the diagonal 0 I, and produce it to 0 D. Then 0 I represents the maximum value of the main current I just before division. Join A D ; A D and 0 D will represent the product of the current I and the equivalent reactance and resistance of the two circuits K and S in parallel respectively.

To prove this last proposition, we must refer again to the paper by Lord Rayleigh on " Forced Harmonic Oscillations -of Various Periods " (Phil. Hag., 1886).

FIG. 71.

If E' 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 Eayleigh's paper that

E' = 'B' and L/ = A5^ B5'

where

170 SIMPLE PEEIODIC CURRENTS.

R and L being the resistance and inductance of any branch „ and the mutual inductance being zero.

Apply this theorem to the case under consideration, viz, the two inductive resistances (E, L) (S, N) in parallel, and we have

R S

~R2+/L2 Effecting the multiplication we have

R (S2 +p2 N2) + S (R2 +p* L2)

(b*+>»Na)(Ha+7>*Ls) L (S2 + j>2 N2) + N (R2 + jo2 L2)

Turnhig back to Fig. 71, we see from the geometry of the figure that, if the angle B 0 D is as before called 6, BOD = DAB.

We have then _°H = 0 B - A B tan 0.

cos 0

But since A B=p N Y and 0 B = S Y by construction, therefore 0 D = S Y cos 0-p N Y sin 0.

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

(SL-RN)jp

~R(R + S)+L(L + N)/'

hence, eliminating the sin and cos terms, and substituting for Y the value obtained from equation (36), page 162, we get

where I is the maximum value of the main current, and Y that of the current hi the S circuit.

On comparing this value for 0 D 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 C represent S Y and R X, 0 D represents R' I. Similarly, it may be shown that A D =p L' I. For the angle C 0 D = & = angle CAD, and

AE*=AC-QC tang'. cos 6

or AD=pLXcos 0'-RXsiu2';

a similar substitution, with help of equation (39), page 162, enables us to see that

and this is equal to the value found by analysis for p L' 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 oi their maximum values, and the cosine of the difference of

From Fig. 71 this mean value is therefore QX2OY cosine BOG.

This, however, is not the measure of the power expended hi the R circuit. The true watts are proportional to the mean product of x and a current equal to one Sth 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 £? . 0 X . cosine C 0 A,

S

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

NowOY is taken as one 8th part of the effective electro- motive force in the S circuit ; and on the same scale on which

472 SIMPLE PERIODIC CURRENTS.

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

°X-OYcosineBOC,

2

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

~ 0 X cosine A 0 C,

2 S

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

f\ -p Tn1 O A

the fine-wire circuit being 0 Y or - instead of - or — -

b b b

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

OB 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

cosCOA_ cos CO A _& cos C 0 B = cos (C 0 A - B 0 A) ~ '

But from the diagram

cosine C 0 A = — — --_-r.— -,

and cosine B 0 A =

JB

TE^L'

SIMPLE PERIODIC CURRENTS. 17?

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

or

If we put Ts = — , where Ts is the time-constant of the S,.

or fine-wire circuit, and TR = _ , where TR is the time-constant

it of the B circuit, we have

F- 1+?2T*2 , (41)

2'

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 when TS = 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 TR is greater than Ts, then F is a proper fraction. The wattmeter reads too high, and the real watts are less than the observed. If TK is less than Ts, then the observed readings are too low. If TR = Ts, 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 Circuits of Constant 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, which is called the Primary (Pr.), a change of magnetic induction.

174 SIMPLE PERIODIC CURRENTS.

takes place through the other, called the Secondary (Sec.). We have, then, to regard the primary and secondary as linked together by loops of induction, and the closed lines ol induction, together with the two circuits, must be considered as forming three links of a chain. We shall suppose the self and mutual inductance to be known and to be constant, as also the resistance of each circuit. The primary inductance and resistance will be denoted by L and E, and those of the secondary by N and S, and the mutual inductance by M. The primary circuit is to be subjected to a simple periodic electromotive force of which the maximum value is E, and the result is to generate in the primary circuit a primary current, which, as we have seen, is also a simple periodic quantity, and is to be denoted by its maximum value, Ij. The change of induction through the secondary follows the •change of current, and gives rise to an impressed electromotive force in the secondary circuit, which, being represented by the rate of change of the simple periodic induction, is also a simple periodic quantity, and gives rise to a simple periodic current in the secondary, to be denoted by its maximum value I2.

The general description of the phenomena produced in such a system of primary and secondary circuits connected by an air magnetic circuit is a follows : —

1. The application of a simple periodic impressed electro- motive force, E, produces a simple periodic current, I1} moving under an effective electromotive force, KIU and brings into existence a counter electromotive force of self-induction, which causes the primary current Ij to lag behind E by an angle called the primary lag 0:. If n is the frequency of the vibrations and 2Trn=p, as before; then, as we have before seen,

tan #!= * and this counter electromotive force of self- induction is a periodic quantity of which the maximum is Lpln and of which the phase is 90° behind that of the effective electromotive force or current, or is in quadrature with it.

2. The field round the primary, and therefore the induction through the secondary, is in consonance with the primary current Ia j but, since it is also a simple periodic quantity, its

SIMPLE PERIODIC CUEEENTS. 175

time-rate of change, and therefore the impressed electromotive force in the secondary, is in quadrature with the primary current. Since the induction through the secondary, due to a current ^ in the primary, is M I1} by the definition of M the maximum value of the rate of change of this induction for a pulsation p is M p Ilt

It is useful to note that in all dealings with simple periodic quantities, if X is the maximum value of a simple periodic quantity which runs through its cycle n, times in a second, the maximum value of its time-rate of change is denoted by p X, where p = 2-!rn.

If, then, as usual, simple periodic quantities are denoted by the letter signifying their maximum values, prefixing p to any one gives us the value of the maximum of its first differential coefficient with regard to time, or p is here equivalent in

notation to — dt

3. This secondary impressed electromotive force gives rise to a secondary current, I2, moving under an effective secondary electromotive force, S I2, and creating a counter electromotive force of self-induction in the secondary, represented by N p I2.

The secondary current lags behind the secondary impressed electromotive force by an angle 0., such that

4. This secondary current, I2, reacts, in its turn, on the primary, and it creates what is called a back electromotive force, or reacting inductive electromotive force, on the primary circuit. The phase of this must be in consonance with that of the electromotive force of self-induction in the secondary, and it is represented by the quantity M I2^>. This is obviously in quadrature with the phase of the secondary current or secondary effective electromotive force.

5. There is, then, a phase difference between the primary and secondary currents, and also between the primary impressed electromotive force and the primary current.

The general problem is, then : Given the value of the induc- tances L, M, N, and the resistances R, S, and that of the im- pressed electromotive force E and the frequency n, find from these seven quantities other four, viz., the primary current IJf

176 SIMPLE PERIODIC CURRENTS.

the secondary I2, and the difference of phase between E, Ilf. and Ia.

We shall attack the problem geometrically, as this method exhibits far better than the algebraic method the relation between the various quantities involved. The method adopted is to construct an electromotive force diagram, in which all lines represent on any scale volts ; and moreover, as each of the quantities considered is a periodic quantity, the lines all represent the maximum value of each quantity, and the value at any instant can be obtained by taking the projections of all lines on any straight line through the centre of the diagram, suitably placed.

Let 0 (Fig. 72) be taken as a centre ; draw any line 0 Q', and on it set off any length, 0 T, which we assume as the magnitude of the maximum of the primary current. All other lines will be in proper proportion to this. Produce 0 T to 0 Q so that 0 Q = R Ii- 0 Q is then the effective electromotive force in the primary circuit. From Q draw Q P at right angles to 0 Q, and set off Q P equal to L p times 0 T or to L p Ix ; Q P represents the electromotive force of self-induction in the primary circuit. Join 0 P.