The Alternate Current Transformer Vol. 2: The Utilisation of Induced Currents (1896) — part 11 of 36

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 ot 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 B, 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 •electromotiye force of which the maximum value is E, and the result is to generate in the primary circuit a primary eurrent, which, as we have seen, is also a simple periodic quantity, and is to be denoted by its maximum value. If 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 Ig.

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, Ii, moving under an effective electromotive force, B Ii, and brings into existence a counter electromotive force of self-induction, which causes the primary current Ii to lag behind E by an angle called the primary lag ^i. If n is the frequency of the vibrations and 2irn ■•/>, as before ; then, as we have before seen,

tan 01 B r^f and this counter electromotive force of self- B

induction is a periodic quantity of which the maximum

is liplu a^d of which the phase is 90* behind that of tiie

j^ective 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 4>iuzent If ; but, since it is also a simple periodic quantity, its

SIMPLE PERIODIC CURRENTS. 175

time-raie of cliange, and therefore the impressed electromotive force in the secondary, is in quadrature with the primary current. Since the induction through the secondary, due to tk current Ii in the primary, is M Ii, by the definition of M the maximum value of the rate of change of this induction for a pulsation ;> is M ^ Ii.

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 2> = 2irn.

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

notation to — . dt

8. This secondary impressed electromotive force gives rise to a secondary current, Ij, moving under an effective secondary electromotive force, S I2, and creating a counter electromotive force of self-induction in the secondary, represented by N j9 1^.

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

4. This secondary current, I,, 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 I,^. 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 B, 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 Ij,

176 SIMPLE PERIODIC CURRENTS.

the secondary 1^ and the difference of phase between E, I^. and Ig.

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'r and on it set off any length, 0 T, which we asmme 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 « B Ii. 0 Q is then the effective electromotive force in the primary circuit. From Q draw Q P at right angles to OQ, and set off QP equal to Lp times 0 Tor to Lpli; QP represents the electromotive force of self-induction in th& primary circuit. Join 0 P.

From 0 draw 0 C at right angles to 0 Q, and set off 0 C equal to Mp times 0 T. O C is then equal to M j9 1^, and 0 C represents the impressed electromotive force in the secondary circuit. On 0 C describe a semi-circle, and set off 0 B, making

an angle GOB with 0 C such that tan G 0 B«^, or tan

G 0 B a the ratio of the inductive to the ohmic resistance for the secondary circuit. Join B G. On the same scale on which 0 G represents the impressed electromotive force in the secon- dary circuit, viz., Mpl^, OB will then represent the effective electromotive force in the secondary, or will represent S Ig, and hence, if 0 D is taken equal to one S^ part of 0 B, 0 D will represent la, or the secondary current. Next draw -a line 0 £ perpendicular to 0 B, and therefore parallel to B G, and on it set off a length, 0 K, equal ioMp times 0 D or to Mpl^ 0 £ represents then the back inductive electromotive force of the secondary on the primary.

The impressed electromotive force which has to be applied to the primary to produce in it the primary current 0 T and

SIMPLE PERIODIC GURBENTS.

177

to induce in the secondary the secondary current 0 D has therefore to be equal and opposite to the resultant of three

electromotiTe forces, or to equilibrate three electromotive forces, viz., the effective electromotive force of the primary

178 SIMPLE PERIOBIO CURRENTS.

O Q, the electromotive force of self-induction in the primary B Q, and the back electromotive force in the primary due to the inductive effect of the secondary on the primary, viz., 0 E.

The resultant of 0 Q and Q P is 0 P. If, then, we draw P P' from P, and make it equal and parallel to 0 K, and join O P', 0 P' will be the resultant of 0 Q, Q P, and OK; and hence 0 P' will represent E, or the impressed electromotive force required to be applied to the primary to maintain the currents I^ and I^.

It is to be understood that in this diagram a unit of length stands for a volt, or unit of electromotive force ; and hence, on that assumption, E represents in volts the impressed E.M.F. — that is, the maximum of the simple periodic E.M.F. required to maintain the currents Ii Ij, of which 0 T, 0 D represent the maximum values. The relative phases are indicated by the positions of these lines. To obtain the actual values of the E.M.F. and currents at any instant we have only to take the projections of 0 P', 0 T, and 0 D on any line drawn through 0 suitably placed, and the magnitudes of these projections will give the required quantities. We must then suppose the whole diagram to be enlarged or diminished without distortion until the length of 0 P' is numerically equal to the maximum value in volts of the impressed E.M.F. E, and then 0 T and 0 1>, will represent the currents Ii and L in magnitude. We may consider the two right-angled triangles 0 Q P, 0 B C as pivoted together at 0, and revolving round 0 ; the fluctuations of the projections of 0 P', 0 T, 0 D on any line will give us the cyclic values of E, Ii, and Ij. We can next obtain some useful relations between these quantities from the geometry of the figure. In the triangle 0 B C,

0 0^^ = 0 B« + BC2. Hence I^^ M'^ //^ = S^ V + N' p^ V ;

or ii^Vs^T^.

1-2 Mi? ■ •

primary current impedance of secondary secondary current ~ M^

Mp might by analogy be called the mutual reactance.

To obtain the value of Ii in terms of E and the inductances and resistances, we project the lines OP' and OP on the

SIMFLE FERIODIC CURRENTS. 179

Teridoal line 0 E, and express the fact that 0 P' or E is in aU cases the resultant of 0 E and O P. Let the angle F 0 Q' be called <fy, 4> is the angle by which the primary current lags behind the total impressed electromotive force. Then COB is ^2, and TOE = COB = ^i, since QOG and EOB are both right angles.

Hence we have by resolution on 0 K

E cos (<^+ 6^) = M;? I2+ VTF+Fi? Ii cos {0,+$.,) ;

ibut, smce Ia = Ii -: — .^.-,

-we have by substitution

E cos {<!> + e,) = j-;^-^^-— + -v/R' +2>' L' cos {0, + e,)| I„

*nd therefore a relation established between E and l^ which is known when <j> is known.

Since tan 0^ ^^, and tan 0^ =-?i^,

it follows by an easy transformation that

RS-/.^LN

To find the value of <^, suppose that whilst E and Ij remain the same, the secondary circuit is suppressed. We should then only have an impressed electromotive force, E, creating a current, Ii, and from the diagram and from what has been before explained it is obvious that the effective and self-induc- tive electromotive forces in the circuit would then be repre- sented by 0 Q' and Q' P'. If we denote these by the symbols B'li and L'plj we may properly caU R' and L' the equivalent resistance and inductance ; that is to say, these quantities are the resistance and inductance which the primary circuit should have in order that, when there is no secondary circuit, the primary impressed electromotive force may generate in it the same current which it does when the secondary circuit is present and the primary has its natural resistance R and inductance L. We see, then, that the effect of bringing up the secondary and allowing it to be acted upon and react

n2

180 SIMPLE PERIODIC CURRENTS.

upon the primary is to increase the effective resistance and diminish the effective inductance of the primary ; in other words, the equivalent resistance of the primary circuit i^ greater and the equivalent inductance is less by reason of the presence of the secondary circuit.

We have then to find the value of cos {<t> + Ot). Cos (<^ + 6i) = cos <fi cos O2 - sin <^ sin 6^ ;

but cos6 = — r 6m^ = — _£-,

and cos 0^ = _ sin 6^ = ? ;

hence cos (</> + 0,) R^S-KN^

Substituting this in the equation connecting £ and I^, wfr arrive at

' R'S-L'Ny

Returning to Fig. 72, we see from it that P' Q' is parallet to P Q, and hence, if we draw P'V parallel to Q' Q, we have FQ' = PQ-PV

= PQ-PFcosFPV n=PQ-PP'sin^j,

or L>IiLp=I,-M;?I,|^: or. smce -=- = — — «

we have L' = L-— — ^ (42)

Also, again, 0 Q represents R Ii and 0 Q' on the same scale R' I^, and

OQ' = OQ + QQ' -OQ + P'V -OQ + PFsinFPV -OQ + PFcos^a;

hence R'l.-RI^ + M;?!,-^ ;

Ml? I,

and. therefore. R'-R + -^LlL^ (43>

SIMPLE PERIODIC CURRENTS. 181

These formulsa (42) and (48) give us the effective inductance and resistance of the primary circuit as affected by the secon- dary. They were first given by Clerk Maxwell in a paper in the Philosophical Transactions of the Boyal Society in 1865i entitled •* A Dynamical Theory of the Electromagnetio Field " {Phil. Trans., 1866, p. 476) .♦

If we form from (42) and (48) the function B' S— L' N j?, we find it to be M*^* + R S— ;?*LN; and hence, by substitution in the expression already given connecting E and Ii, we arrive finally at the result

E

or Ii =

JW+flK

Following the usual nomenclature, we may oaU the expres- sion JB,'*+ji^J7^ the equivalent impedance of the primary •circuit, and we have as the final result for the induction coU of constant inductance

Primary current impressed electromotive force ^ strength ^ equivalent impedance of primary

circuit

Secondary current ^ ly; ^ w primary current

strength impedance of secondary drouit*

The angle of lag of primary current behind impressed

E.M.F.«<^, where tan <^s-C— _, and the angle of lag of the

secondary current behind the primary is seen to be 90°+ 6^ and

tan ^3=-^ ; hence we have the values and relative phases of S

the currents and the impressed electromotive force.

In the above equations we are to understand current strengths

and electromotive forces to be the maximum values during

the period. If ii and i^ be the actual values at any time t,

reckoning time from the instant of the zero value of the

electromotive force, then, from the principles previously

• See also Lord Rayleigh on ** Forced Harmonic OsciUafciona of VarioM Periods," PhU. May., May, 1886, p. 375.

182 SIMPLE PERIODIC CURRENTS.

explained in this chapter, it is obvious that t\ = Ij sinQpt-^),

pr ti= sin (pe--<^);

and

or U^—, , sin(2?<-^-ea-90°).

The student will find the above expressions for the primary and secondary currents can be deduced by analytical processes from the simultaneous equations.

L'iiL + M4^ + Rt=Esini>f, . . (44) dt dt ^ ^

N^ + MliL + Si>0, .... (45) dt dt ^ ^

which equations can be established for two circuits by analo- gous methods to that by which in §18 a current equation was arrived at for one circuit, subject to a simple periodic electromotive force.* It is easily seen that if n is very great, or the alternations extremely rapid, then

la M* If the primary and secondary circuits consist of two equal circuits, so interwound that for these circuits L =: M « N, then for very rapid alternations we see that the secondary current Ig is equal in magnitude, and exactly opposite in phase, to the primary current Ii, and the magnetic fields due to these currents respectively are also equal and opposite in direction at every instant.

§30. The Flow of Simple Periodic Currents into a Oon- denser. — The electrical capacity of a conductor of any kind is measured by the quantity of electricity required to charge it to unit potential. Two conducting surfaces so arranged as to have constant capacity are generally called a condenser. The most simple and familiar form of this appliance is the Leyden jar, in which two surfaces of tin foil are separated by a

*See Mascart and Jouberfs " Electricity," VoL I., p. 521.

SIMPLE FERIODIC CUIiRENTS, 18^

sheet of glass. Condensers for practical purposes are generally formed of sheets of tinfoil, separated by some dielectric, such as paraffined paper, mica, gutta-percha, tissue, or ebonite. If a quantity of electricity, Q, measured in microcoulombs, is given to one of the plates or sets of plates of a condenser, and if the other set are kept at zero potential, the charged plates will be raised to a certain potential, say Y volts, above that of the earth. The capacity of the condenser G in microfarads is then such that by definition

CV = Q.

If the potential difference of the condenser terminals at any instant is v, and if it is changing, and if q is the charge or quantity of electricity in the condenser at that same instant, then

Gv^q,

or C^" = l^;

but ?-? is the time rate at which the charge is changing or dt

is the value of the current i at that instant flowing into or out of the condenser. Hence

dt '

and C— =i.

dt

Suppose the condenser is being charged through a circuit whose resistance is B, then i is the current which is flowing through this resistance at the instant considered ; and the fall of potential down the resistance is R i volts. If a constant potential, Y, is being applied to the outer extremity of the resistance, so that the total difference of potential between one plate of the condenser and the outer end of the resistance attached to the other plate is Y volts, we have at any instant the equation

r + Rt = Y,

or v + GB^=^Y,

dt

or Cb1%i;=Y (40)

dt

184 SIMPLE FERIODIC CURRENTS.

This is the differential equation for the potential v of the condenser at any instant, e, when being charged through a resistance, B, by a constant potential, Y. The equation (46) is easily integrated as follows: — Multiply all through

--< by 6^° , where e is the base of the Napierian logarithms, viz.,

2-71828. Then we have, as a result, the equation

dt EC RC

or ^Lt^^^V^ie^^'

dt^ / dt^ /

Both sides of this last equation can then be integrated. Hence, integrating with respect to time, we obtain the equation t_ « i;e^^=V«*^ + a constant^

t or v=^Y + Ge^^.

The constant C is determined by the condition that when the time t is zero then v is also zero. Hence C= - V, and, therefore, ^

.t;=Vrif«"^'°) (47)

The student should compare equations (46) and (47) for the value of the instantaneous potential of a condenser charged by a constant voltage with the equations (26), page 127, and (27), page 132, for the instantaneous value of a current in an inductive circuit acted on by a constant pressure, and it will be seen that they are similarly constructed. The quantity

CB is called the time-constant of the condenser just as ,^

B

is called the time-constant of the inductive coil. The greater the condenser time-constant the larger will be the time taken by the condenser potential to arrive at a given fraction of the steady impressed voltage applied to charge it. Practically, the condenser is fully charged in a time equal to eight or ten times its true constant. The reader must note that if the capacity G of the condenser is measured in microfarads, then the resistance B must be measured in megohms to obtain the correct measure of the time-constant 0 B in seconds.

SIMPLE FERIODIC CURRENTS. 186

Suppose, for example, that a condenser of one microfarad is being charged through a resistance of one megohm by an impressed voltage of 100 volts. It is required to find the potential difference at the terminals of the condenser at the end of the 1st, 2nd, 8rd, 4th, nth second. Let t'l, v.^, v„ v^ be these potential differences.

Then v^ = 100 (1 - e-^) = 63 volts nearly,

where <? = 2-71828,

and rj = 100 (1 - tr^) = 86 volts nearly,

or tv = 100(1 -O-

Since r~^®is an exceedingly small number, in ten seconds the condenser potential is practically equal to 100 volts. We see, therefore, that in charging condensers through resistances sufficient time must always be allowed for the charge, depend- ing on the value of the quantity C K, and the charge is not practically complete unless contact with the source of im- pressed voltage endures at least for a time equal to 5 C B, or, better still, 10 CK.

Supposing, in the next place, that the impressed voltage acting on the condenser is periodic in character or alternating, and that the condenser, instead of being charged through a resistance, has its terminals shunted by a resistance. Let the capacity of the condenser be, as before, G microfarads, and

let the resistance of the shunting conductor be — ohms ; that

is, let K be the shunt conductance. Let the impressed voltage be periodic, and let its frequency be n, and write p for 2:r n, as usual. Then, as above, the current flowing into the condenser at the instant when its terminal potential

difference is t^ is C — . Also the current flowing through the dt

shunt is Et;. We have, therefore, as the equation for the total current flowing into the shunted condenser, the expression

■ ■ •C'J;+Kf = f (48)

(it

where i is the total current flowing into condenser and through the shunt at the instant when the terminal potential difference of the condenser is v. Let time be measured from

186 SIMPLE PERIODIC CUBBENT8.

the instant when the total current denoted by i is zero, then,, if t is a simple periodic current, it may be expressed in terms^ of its maximum value by the equation i = Isinj>f.

Hence C^ + Kt'=»Isin;>e (49)

at

On comparing these equations, (48) and (49), for the instantaneous value of the condenser potential difference when a periodic current is flowing through it with the equations (29), page 188, and (81), page 185, it will be seen that they are structurally the same, and we can at once write down the solution of equation (49) by imitating the solution of (29), writing C instead of L, K instead of R, v instead of t, and I instead of E. Hence the solution of

C — Kv = lsia2)t dt

is v = — / sin {pt—0) + B, constant, . (50)

C n where S is an angle such that tan ^ = ~*

The quantity /K'* + Cj9 has not yet received an acknow- ledged name, but it is analogous to the impedance in the case of the current flow in inductive circuits. It has been suggested that this quantity should be called the adimttance of the condenser.

From equation (50) we see that the terminal voltage of the condenser lags behind the charging current in phase; or otherwise that the current is in advance of the potential difference. If the shunt wire is removed this is equivalent to making K equal to zero, and under these circumstances we have

t;«^sin(/>«- 90), or v = y.-C(yspt (51)

Vjp

This shows that in the case of a condenser having a di- electric W7.th no true conducting power, the charging current is exactly 90deg. in advance in phase of the terminal potential difference of the condenser.

SIMPLE PERIODIC CURRENTS, 187

It is clear, therefore, that as regards producing lag or lead of current capacity acts in the opposite direction to inductance^ and may be considered to be equivalent to a negative inductance^

It is also evident that if a shunted condenser is placed in series with an inductive circuit a proper relation may be formed between the inductance, resistance, capacity, and conductance^ such that the combination of inductive resistance and shunted condenser acts to an impressed simple periodic electromotive force just as if it were totally non-inductive. This annulment of inductance has important applications in practice in tele- graphy, and it is therefore desirable to define these conditions carefully.

§ 31. A Shimted Oondenser in Series with an Inductive Besistance. — Let a condenser of capacity C have its terminals closed by a circuit whose conductance is K, and hence its resistance K'~'^ Let this shunted condenser be placed in series with an inductive resistance whose inductance is L and resistance B. At any instant let vi be the potential difference of the terminals of the condenser and v.2 those of the inductive resistance when a periodic current having at this same instant a value i is flowing through the system. Let v be the value of the oyer-all potential difference. Then suppose that v is a simple periodic potential diiference, or that

Then th,e current i will diifer in phase from v, and may be represented by the expression

t = I sin (p t - <^).

Capital letters represent maximum values as usual. We caa then form the fundamental equation for the shunted condenser in series with an inductive circuit as follows : — We have for the shunted condenser terminal potential r^ the equation

C^^+Kn=,-

at

. . (52)

and for the inductive resistance the equation

L - +lit = ta, . . . , at

. . (58)

and also Vi + ra=v, ....

. . (54>

188 SIMPLE PERIODIC CUEBENT8.

since the over-all potential v is the sum of the separate potential steps Vi and Vs* Eliminate t^ and v^ from these equations, and we get

GJj^ + {CR + KJj)^ + (KE + l)i=.C^ + Kv, . (55)

also, since v « Y sin /> t,

we have C --? = C jo V cos jp «,

at

and K V - K V sin^ t.

Hence we have

C^J + Kt;= JK' + Cp'Ysmipt-e). . (56)

This last follows at once from the lemma on page IGl. Then, since t = I sm (p e - <^),

we obtain _! = Ip cos {pt- <i>\

at

a.nd -— =-17)" sin {p t - <^),

and by substitution of these last values in equation (55), we arrive at the equation

(KR + l-CL^*)Isin(jD«-<^) + (CR2) + KL^)Icos(j5«-^) « ^KM^ V V sin (j5 < - 6>) . . (67)

or (CR + KL)^'+(KR + l-CLy)t

• v/K' + C'p^ V sin {pt-O). . . (58) Since it is shown on page 161 that any function of the form A sin ^ + B cos 0 can always be expressed in the form

JA^ + B^ sin (^ + a), where tana=-^,

it follows at once that the maximum value of the current I is given by the equation

p V'(K'+CV) ,.Q.

and it is not difficalt to show that this last equation (59) may he written in the form

y^

SIMPLE FEmODIC CURRENTS. 18^

K

Let R* stand for — - — ^=3-*,

and L^ for

K2 + CV

If the condenser and shunt are removed it is clear that the maxininm value of the current I will be given by

Hence the effect of adding the shunted condenser is as i£ the resistance of the inductive circuit were increased by an

amount equal to ==- — ;---^, and the inductance of the K^^ + C^p*

inductive circuit diminished by an amount equal to

C

K^ + C^p^'

n Hence, if the quantity =- — ——5 is equal to L, which is K^ 4- C* p*

the inductance of the circuit, then the total effective induct- ance of the circuit will vanish, and the inductive resistance and shunted condenser together will act as if there were no resultant inductance at all. This condition is fulfilled, when we have

which equation gives us the required magnitude of the capacity

to annul the inductance. Uniting ^ instead of K, so that r is-

r

the resistance of the shunt, we have

L

(GO)

C(1-CL;;2)

as the final equation, which tells us what must be the resistance of the shunt to be put across the terminals of a condenser of capacity G in order that the combination may just annul the inductance of a resistance in series with it whose resistance is B and inductance L. If L is measured in henries, C must be in farads, and E and r be given in ohms.

It will be seen. that the annulment is only exact for one particular frequency, and that change of frequency means a change in the value of r requisite to neutralise the induct-

190

SIMPLE PERIODIC CUBEENTS,

ance. In telegraphic work it is usual to employ a shunted condenser to annul more or less completely the self-induction of relay magnets.

§ 32. Representation of Periodic Currents by Polar Diagrams. Two methods of graphically representing simple periodic currents or electromotive forces, viz., clock diagrams and wave diagrams, have been hitherto used. Each of these methods lias some peculiar advantages of its own. There is, however, a third method which has sometimes especial utility, and this is by a polar diagram.

Let a straight line 0 P (Fig. 78) revolve round one of its -extremities 0, and let the angle of inclination $ which the

revolving line makes with another fixed straight line OA passing through this centre of revolution be called the angle of displacement of the revolving line. Let any point P be taken on the revolving line, and let the distance of this point from the centre be denoted by r. The length r is called the radius vector. Let r increase or diminish as 0 P revolves, but so that the length OP = r varies according to any law connecting it with the angle of displacement A OP. The extremity P of the line 0 P will then trace out a curve called a polar diagram.

Suppose that 0 P runs through a cycle of values beginning with a zero value when ^=0, and^ after reaching a maximum

SIMPLE FEBIODIC CURRENTS. 191

Talne, becoming zero again after the line 0 P has completed one half revolution, or when ^ — 7, the polar diagram will he a closed curve passing through the origin. Since the radius O P runs through a cycle of values from zero to a maximum, and returns to zero again during a revolution of the radius through two right angles, or an angle tt, the radius r may smtahly represent the value of any periodic quantity which completes a cycle of values in the time represented by one complete half revolution of the radius vector. Suppose, then, that 0 P varies as sin 6, where 6 is the displacement phase angle — that is, let

r=Esin^.

It is then clear that in this case the polar curve is a circle. For if we draw the line 0 B through 0 perpendicular to 0 A, and draw PB at right angles to OP, since AOP=^ and OP=r, and since by suppositiop r=R sin d, it follows that the angle P B 0 is always equal to 0, and that, therefore, the length 0 B is a constant length equal to E. In other words, the locus of the point P is a circle passing through 0, P, and B. Hence the circle is the curve which represents in a polar diagram a simple periodic quantity, and if the length 0 P is taken to represent the magnitude of a simple periodic current or electromotive force at any instant corresponding to a phase angle A 0 P, the circle passing through 0 will be the polar curve representing this simple periodic current or electro- motive force.

If the current or electromotive force is periodic, but not simply periodic, then the polar curve representing it will be a unicursal curve passing through the origin 0, but will not be a circle. The polar diagram of an alternating current or electromotive force enables us very easily to determine the square root of the mean-square value of the periodic quantity represented by it. It is more difficult to do this with the ordinary wave-curve diagram.

Suppose, for example, that in Fig. 74 we have a wave diagram drawn representing a half wave of an alternating electromotive force which is not a simple sine curve. The ordinates represented by the dotted lines are proportional to the instantaneous values of the periodic quantity taken every 20deg. If we wish to find the Jme&n^ value of the

192

8IMFLE PERIODIC CUBRENTS.

ordinaies of this oarve there is no other way of doing this than by drawing a number of numerous equidistant vertical ordinates, measuring their lengths, squaring these magnitudes, and taking the square root of the mean of these squares. This is a troublesome arithmetical process, and in proportion as the wave curve is more irregular or complex, so much the more numerous must be the ordinates to obtain a correct result for the mean-square value. It is, however, a much easier process if the periodic quantity is represented by a polar curve. Let Fig. 75 represent the same periodic function as in Fig. 7i, drawn in a polar form. The dotted radii are proportional to the instantaneous values of the periodic quantity, and are-

tb 26 y& 46 S'Ci 6d 7d Ab «0 lOOnb i20ij

Fig. 74.

placed at angular intervals of lOdeg. Let any radius 0 P {ses Figs. 75 and 76) be denoted by r, and let the corresponding phase angle P 0 A be denoted by the letter 6. If we consider the radius to move forward by a small angle d 0 and to increase in length by a small amount dr, then it is obvious that the small increment of area d A swept out by the radius r is equal to^f^dd. To obtain the whole area, and included by the polar curve 0 P Q E, we have to integrate this quantity ^f^d$ from 0 to fl-| or to obtain the integral

A^^j^'f^dO.

SIMPLE FEBIODIC CUREENTS. 193

On the base line drawn through the polar centre 0 let a semicircle A G B (see Fig. 75) be described equal in area to the area included by the polar curve OPQ* Let OA = B be the radius of this semicircle. Then

" ^- Vij','''"'-

The expression on the right hand side of this last equation is obviously the square root of the mean of the squares of the instantaneous values of the periodic quantity r. Hence, we oan obtain at once the Vmean^ value of a periodic quantity current or electromotive force as follows. On any straight

line describe a semicircle and draw radii of this semicircle at angular intervals equal to those phase intervals for which the instantaneous values of the current or electromotive force are observed. Bet off on these radii lengths equal to the respec-, tive values of these instantaneous quantities and join the extremities of all these lines so set off by a curve. This curve is the polar diagram, and it is a closed curve. Take the area included by the polar diagram with an Amsler's or other plani- meter, and from a table of areas of circles find the circle whose area is double that of the polar curve. Then the radius of this circle is the square root of the mean of the squares of the periodic quantity. If, for instance, we have a periodic current

o

194 SIMPLE FERIODIC CURBENTS.

curve plotted down in polar form, this operation will give us the dynamometer value of the ourrenti or the value which would be read on a dynamometer*

The polar curve, therefore, lends itself very easily to the determination of the mean-square value of a periodic current or electromotive force, of which the instantaneous values are known throughout a semi-period, whether those values are laken at equidistant intervals of time or not. The reader virill, therefore, note that if we plot down a periodic quantity ia rectangular co-ordinates (Fig. 74) or wave form, and find the rectangle A P P' B of equivalent area to the half wave, the altitude AP of this rectangle gives us the tnie mean ordinate of the periodic curve; but if we plot down the periodic quantity to a polar diagram (Fig. 75) and find the

Fig. 76.

semicircle AGB of equivalent area, the radius OA of this semicircle gives us the square root of the mean-square value of the periodic quantity represented by the radii of the polar curve.