The Alternate Current Transformer Vol. 1: The Induction of Electric Currents (1896) — part 16 of 35

in direction to the secondary current in the screen, and, secondly, a succeeding direct current. Let the current curve of this tortiary current in the secondary circuit be represented by the fine firm line in Fig. 100. The total quantities of electricity flowing in each part of the two portions of the tertiary current are equal. The resultant effect, then, of the action of the primary current when interrupted is to cause in the secondary circuit the true secondary current, which is an unidirectional flux (thick curve), and a superimposed tertiary current, which is a bi-directional flux, its algebraic total of quantity being, zero.

FIG. 100,

If we add together at each instant the ordinates of the two- current curves we get a resultant curve (dotted line) which represents the actual current curve in the secondary circuit. The total area (electric quantity) enclosed between the hori- zontal line and the dotted curve must be equal to the total area enclosed between the thick firm line and the horizontal, because we have added and substracted equal areas ; but the maximum ordinate of the dotted curve will be less than that of

270 MUTUAL AND SELF INDUCTION.

the thick firm line curve, and the form of the curve will be very different also. It is, then, clear that the superposition of a complete tertiary current, which is of itself but very little able to affect a galvanometer on a secondary current which gives a definite galvanometer indication, is not able to alter that galvanometer deflection, depending as it does 011 the total quantity of the discharge. The magnetising power and shock, however, depend upon the maximum value or suddenness with which the induced current rises to its maximum value, and this factor is very much affected by the overlaying of a secondary current by a tertiary. We see, then, that the experiences of Faraday and Henry may be completely recon- ciled, and that the detection of magnetic screening depends upon the nature of the detecting instrument in the secondary circuit.

The practical outcome of much of the foregoing discussion of magnetic screening is that the use of lead-covered cable for the conveyance of periodic currents of the usual frequency (60 or 100 alternations per second) is of HO advantage in respect of prevention of inductive disturbance in neighbouring tele- phone wires. Not only is the lead too poor a conductor, but the frequency of alternation is too small to render the magnetic screening effective. The only effective method of annulling the inductive disturbance is to carry the periodic current along a conductor which lies in the axis of, and is insulated from, a concentric enclosing tube or sheath, which acts as a return. This return must be itself insulated from the earth, and the condition to be fulfilled is that at any instant, and at any section the algebraic sum of the currents in the core and sheath must be zero ; reckoning current in one direction positive, and in the other negative.

The whole question of magnetic screening has been worked out .mathematically by several mathematicians, and besides the section in Clerk-Maxwell's Treatise (Vol. H. § 654, 2nd Ed.), the advanced student may be referred to memoirs by Prof. Charles Niven " On the Induction of Electric Currents in Infinite Plates and Spherical Shells " (Phil. Trans. Roy. Soc., 1881, p. 307), and also to Prof. H. Lamb " On Elec- trical Motions in a Spherical Conductor" (Phil. Trans. Eoy. Soc., 1883, p. 519).

MUTUAL AXL> SELF INDUCTION. 271

§ 9. Reaction of a Closed Secondary Circuit on the Pri- mary.— If a Bell telephone is placed in series with a coil of many turns of fine wire wound on a hollow bobbin, and if both are placed in series with the secondary circuit of a small induction coil, the strength of the secondary current can be so adjusted that the telephone emits a low murmur or rattle. This being the case, let a solid bar of copper be introduced into the bobbin of fine wire, and it will be found that the noise of the telephone is increased. If a bundle of fine iron wires is substituted for the copper rod it will, on the other hand, reduce the noise or stop it altogether. The explanation of this effect is to be found in the reaction which a closed secondary circuit has upon its primary in changing the resultant impedance of the primary. We have shown in Chapter in. (p. 180), that the re-active effect of the secondary is to increase the resistance and reduce the inductance of the primary circuit, and we have deduced two formulae given by Maxwell for the value of the equivalent resistance E' and the equivalent inductance I/ of a primary coil of resistance B and inductance L in the presence of a secondary coil of resistance S and inductance N, the mag- netic circuit having a constant resistance, and the mutual inductance being M. Hence, the equivalent impedance of the primary coil in presence of the secondary is -/E^+p2!/2, and that which we may call its isolated or intrinsic impedance is equal to Vw + p* L2. For brevity we may write the symbol Im for VK'+^L* and Im' for VB'a+#aL'a, also Itn2 for V s2 + j92 K2. The question then arises, which is the greater — Im' or Im ? To discover this, take for E' and L' the values given on page 180, and we have

T, T and L =L- *L

s-+

Forming from these the function E'2 + p2L'2, we have

272 MUTUAL AND SELF INDUCTION.

If S = x, or the secondary circuit is open, the right-hand; side of the above equation is zero, and we find that the impedance of the primary circuit is not altered by the pre- sence of the open secondary, as of course it should not be.

If S is not infinite, that is if the secondary circuit is closed, then the above equation shows us that, if the quantity 2 R & is greater than the quantity jo2 (2 L N - M2), then Im' is greater than Im, or the impedance of the primary circuit is increased by closing the secondary. But if 2 R S is less than p* (2 L N - W) , then Im' is less than Im, or the impedance of the primary is- decreased by closing the secondary circuit.

If <*! stands for ^-, and o.2 for J-^—, and also if /? stands for

K JN

M • , , , it is not difficult to show* that to make Im' greater

than Im we must have

% oa less than - — - .

When the secondary circuit has a certain critical value it is possible to show experimentally that above this value closing the secondary circuit increases the primary impedance, but below this value closing the secondary circuit decreases the primary impedance.

The following experiment was made in the laboratory of Prof. Elihu Thomson t : — A small induction coil had its primary circuit arranged in series with nine incandescent lamps joined in parallel, thus exciting it with an alternating current of about ten amperes. When the secondary circuit was closed by means of a vacuum tube of high resistance, a marked fall occurred in the candle-power of the lamps used as a resistance in the primary circuit. The impedance of the

• See Mr. E. C. Rimington, "On the Behaviour of an Air Core Transformer when the Frequency is Below a Certain Critical Value," Proc. Physical Soc., London, October 27, 1893.

t See The Electrician, Vol. XXXII., p. 225.

MUTUAL AND SELF INDUCTION. 273

primary was thus increased. When the secondary circuit was closed through a water resistance, the lamps brightened up, thus showing that the primary impedance was decreased.

Hence the closing of the secondary circuit does not always decrease the primary impedance. Mr. Rimington (loc. cit.) quotes an experiment with an air core transformer or induction coil consisting of two circuits without iron core, in which closing the secondary circuit had the effect of decreasing the primary current by about 3 per cent., thus showing an increased primary impedance. Generally speaking, however, the closing of the secondary circuit so that the total secondary circuit resistance is small has the effect of decreasing the primary circuit impedance.

Hence also holding a conductor or conducting circuit of low resistance near a primary coil has the effect of decreasing the impedance of that coil and therefore increasing the flow of primary current through it under the influence of a constant impressed primary electromotive force.

The explanation of our experiment with the induction coil and the copper rod is now simple. The introduction of the copper rod into the fine wire helix is equivalent to approxi- mating to a primary coil a closed secondary circuit. The impedance of the fine wire circuit to the alternating current from the secondary circuit of the induction coil is hence reduced ; it gets more current, and the telephone is made to emit a louder sound. If, however, a core of divided fine iron wire is introduced into the fine wire helix, the result is simply to increase the impedance of that circuit, and therefore to reduce the current actuating the telephone. When considering in particular the theory of the induction transformer as applied to electric distribution we shall see the above principles have important practical bearings.

In a Paper recording some experimental results on the self-induction and resistance of compound conductors* Lord Eayleigh has given some comparisons of the results of theory and experiment on Maxwell's formulas above alluded to. By the use of a resistance and inductance bridge, very similar to one designed by Prof. Hughes, the measurements of the inductance and resistance of a circuit can be made separately

• See Phil. Mag., December, 1886, p. 469.

274 MUTUAL AND SELF INDUCTION.

with ease. A pair of wires was wound on one bobbin ; each wire had a resistance of nearly •! ohm, and a diameter of -037in. Each coil consisted of nine double convolutions. In certain arbitrary units the resistance of one of these copper wires to steady currents was 1-75, and its inductance 11-2°. These values were obtained when the other coil was on open circuit. On closing the unused coil, the resistance of the first rose to 2-67 and its inductance fell to 4-7°. To compare this with the theory.

The formulae are E' = E +

B*-

M2N

L'-L-

Now E = S = 1-75 x -0492 xlO9 absolute C.-G.-S. units of resistance,

and L = N = ll°-2 x 1553 centimetres,

M = 11° X 1553 centimetres,

and p = 27r n = 2 x 3-1415 x 1050.

The periodic current used had a frequency of 1050 per second; hence

Therefore Efc=E (1 + -6) = 1-6R,

and L'- = L (l-'6)= -4L;

but 1-6 x 1-75 = 2-8 = E',

and •4xll°-2 = 4°-5=L'.

These calculated values compare very favourably with the observed values, viz. :

E' = 2-67, L'= 4-7°,

and experimentally confirm the truth of Maxwell's formulae for the increased resistance and diminished inductance of a circuit when placed near a closed secondary circuit.

§ 10. Hughes's Induction Balance and Sonometer. — In 1879 Prof. Hughes constructed and described a very perfect induc- tion balance, with which he was able to conduct researches of an exceedingly interesting character. In order to have a perfect induction balance he found it necessary to make all the

MUTUAL AND SELF INDUCTION.

275

four coils exactly similar.* Four boxwood bobbins (see Fig. 101) are each wound over with 100 metres of No. 32 copper wire. These coils are arranged in pairs at a considerable distance apart, so that the coefficient of mutual induction between the separated pairs is negligible. Two of the coils, A andB, are joined in series with each other and with a battery and interrupter I, and the other two coils, C and D, are employed respectively as secondary coils to these two. These secondary coils are in series with each other and with a telephone receiver T, and are so joined up that the direction of the induction of A on C is oppo- site to that of B on D. One pair of coils is placed hi a fixed position, and the other pair can be slightly moved to or from

FIG. 101.

each other by means of a micrometer screw. The coils are first adjusted so that the inductions are equal and opposite, and on listening at the telephone the opposing secondary currents produce at best but a very slight sound, which can be perfectly abolished by adjusting the distance of one pair of coils. When this is the case, if we insert in the opening of the bobbin of one of the primary coils a disc or piece of metal d, the balance is destroyed, and we hear sounds more or less intense. In order to get some comparative measurements, Prof. Hughes designed

• " On an Induction Current Balance." By Prof. D. E. Hughes. Proc. Koy. Soc., No. 196, May 5, 1879.

T 2

276

MUTUAL AND SELF INDUCTION.

a companion instrument, called a sonometer. In this instrument a pair of primary coils are, as before (see Fig.. 102), joined in series with each other and with a battery. The coils are fixed at the extremities of a bar. Between these primary coils slides a single secondary coil, and the primary coils are so wound that their inductions on this secondary coil are equal and opposite. When this secondary coil is exactly between the two primary coils, a telephone placed in series with the secondary coil gives out no sound when the primary current is rapidly interrupted. If, however, the secondary coil is slid from one primary and towards the other, the differential action creates an induced current detected by the telephone. By reading off on the bar the extent of displacement necessary to create in the telephone a sound of a certain magnitude an arbitrary reading can.

FIG. 102.

be obtained corresponding to every different value of the secondary current. A switch is provided, by means of which the same telephone can be shifted rapidly from the induction balance secondary circuit to the sonometer secondary circuit. The experiments first performed consisted in placing within one primary coil of the induction balance certain equal-sized discs of different metals, and then so arranging the sono- meter secondary coil that the noise in the telephone produced by the current in the secondary of the sonometer was judged by the ear to be equal to the sound produced in the telephone when it was shifted to the secondary circuit of the induction balance, and in which the inductive balance had been broken

MUTUAL AND SELF INDUCTION. 277

down by the insertion of the disc of metal. Discs of various metals the size and shape of an English shilling were made, and, when inserted in the induction coil, the sonometer bar readings, reckoned from the centre or absolute zero of sound given in certain arbitrary degrees, were as follows : —

Silver (chemically pure) 125 ! German Sitver 50

Gold 117 ! Iron (pure) 40

Silver coin 115 | Copper (alluy) 40

Aluminium 112 j Lead 38

Copper 100 Antimony """ 35

Zinc 80 Mercury 30

Bronze 76 i Bismuth 10

Tin 74 !Zinc(alloy) 6

Iron (ordinary) 52 | Carbon 2

This list does not agree in order entirely with that of any of the lists of electrical conductivity. In some degree it evidently has reference to conductivity, because, roughly speaking, the best conductors come at the top and the worst at the bottom ; but whilst it is headed by silver, which has the highest conductivity per unit of volume, we find aluminium, which has the highest conductivity per unit of mass, occupying a position above that of copper. The disturbing effect of the metal on the inductive balance is not, however, simply proportional either to the conductivity per unit of mass or per unit of volume. In more recent experiments a graduated zinc wedge pushed in more or less between one pair of coils of the induction balance was employed to obtain comparative numbers representing the disturbance produced when discs of various metals are inserted in the other coil. The elementary theory of the induction balance is of course contained in all that has gone before in this and the last chapter. It is, generally speaking, dependent for its action on effects similar to those producing magnetic shielding. If the discs are slit so as to prevent circumferential electric currents in their mass, their action in disturbing the inductive balance is mitigated or annulled. If the metal disc is replaced by a copper coil with open extremities no effect is observed on the inductive balance. If the ends of the coil are joined, the coil behaves as if it were a metallic disc and causes loud sounds in the telephone. The effect due to the iron disc is a mixed one. It in part acts like any other metal disc, but it differs from them in one respect. If any non-magnetic disc is placed edgeways in the centre of

278 MUTUAL AND SELF INDUCTION.

the primary bobbin it has a diminished effect in disturbing the balance ; in the case of iron the disturbance is increased by turning the disc edgeways. In order to have before us a typically simple case, imagine an induction balance made of two very long primary helices and each embraced near the centre by a small secondary coil. Let the primary coils be traversed by a simple periodic current. We have then in the interior of the primary coil a uniform magnetic field varying synchronously with the primary current in a simple periodic manner, and the rate of change of the magnetic field at any instant will be a measure of the electromotive force acting in the secondary circuit. Suppose into one primary helix is inserted a thin copper tube ; this will form a closed secondary circuit, and secondary periodic currents will be induced in it, flowing round the cylinder in directions parallel to the turns of the primary helix. As this copper cylinder possesses a very sensible time constant, the phase of these secondary currents in the copper cylinder will be nearly opposite to that of the primary current. The resultant magnetic field in the interior of the cylinder is therefore that due to the resultant of these two simple periodic currents which are nearly opposed in phase. Hence the absolute magnitude of the interior field and its rate of variation will be less than if the copper cylinder was removed. It results, therefore, that the induction through the secondary helix and the electromotive force impressed on it will be diminished by the presence in the primary coil of this copper cylinder. The diagram given on page 177, showing a geometrical construction for the magnitudes of the primary and secondary currents in an induction coil without iron, shows us why the primary and secondary currents are thus more or less opposite in phase. Since, in a general way, the higher the conductivity of the tube or disc introduced into the primary the greater the time constant, and the greater the lag in phase of the currents induced in this metallic circuit behind the phase of the inducing primary, it follows that the resultant interior field acting to produce inductive electromotive force in the secondary helix will be diminished by the introduction of discs of very high conductivity more than by discs of very low conductivity.

MUTUAL AND SELF INDUCTION. 279

From the principles discussed under the head of magnetic shielding it would appear that the differences between various metals inserted as discs in the induction balance would be less marked at very high speeds of interruption than at very low ones. With respect to the action of iron, two effects have to be considered which are the results of very different actions. The introduction of the iron into the primary coil reduces the magnetic resistance of the circuit of induction of that coil, and this cause, if it operated alone, would destroy the inductive balance by raising the inductive electromotive force in that secondary circuit corresponding to the primary into which the iron is introduced; but the iron disc, like every other disc, gets circumferential induced currents created in it, and these, if they acted alone, would destroy the inductive balance by lowering the inductive electromotive force in that secondary coil.

These two effects conflict, and it is an interesting confirmation of theory to find that Prof. Hughes says it is possible to intro- duce into one primary coil of the induction balance a disc of iron and some soft iron wires in such positions that these opposite actions nullify each other, and, though each mass of iron separately would destroy the induction balance, yet the two together being introduced complete silence in the tele- phone is the result. The sensibility of the induction balance to minute differences of electric conductivity and magnetic permeability is very remarkable. If into one coil of a carefully- adjusted balance we place a good sovereign, or shilling, and into the other a bad one, the telephone detects the base coin with unerring certainty by the loud noise given out. In the same way, if two pieces of soft iron are introduced into the two primary coils, and a balance is obtained, the mere magnetisa- tion of one of them will be at once detected, because that magnetised piece becomes thereby less permeable, and destroys the balance. We may present the rough general theory of the induction balance in another way. Let the " coin " be simply regarded as a closed circuit, between which and the primary circuit surrounding it there is a certain coefficient of mutual induction. The two primary coils forming one primary circuit have, on the whole, no action on the two secondary coils forming one secondary circuit, and we may therefore

280 MUTUAL AND SELF INDUCTION.

consider the primary circuit as if it were in a position conjugate to the secondary. The coin, however, is acted upon inductively by the primary circuit, and the eddy currents or secondary currents generated in it react on the secondary circuit, causing in it tertiary currents, which affect the telephone. Looking at it from this point of view, we might construct an induction balance thus. Let A (see Fig. 103) be a single primary coil, and B a secondary coil, having a telephone in series with it. Place the coil B in a position conjugate to A — that is, with its axis at right angles to that of A. Then let variation of current in A produce no current in B. Now hold a sheet of copper anywhere, say at C, and the telephone will be caused to sound. For A, though it

FIG. 103.

cannot affect B inductively directly, yet it can produce a secondary current in C held at a non-conjugate position, and these secondary currents in C will create other tertiary currants in B. The experiment thus appears to indicate a sort of reflection of inductive power.*

This was experimentally shown by Mr. Willoughby Smith in his Paper on " Volta-Electric Induction " (see Journal of Society of Telegraph-Engineers, Vol. XII., page 465).

• The full theory of the induction balance has been given by Prof. Oliver Lodge. See Proc. Phys. Soc. London, Vol. III., p. 187. Also in the same volume is a Note by Prof. J. H. Poynting " On the Graduation of the Sonometer."

MUTUAL AND SELF-INDUCTION. 281

An interesting experiment due to Mr. Willoughby Smith is to employ a simple Bell telephone receiver, unconnected with any circuit, as an induction finder. If a coil of wire is traversed by an electric current, either rapidly intermittent or alternating, then a Bell telephone held anywhere in the magnetic field emits a sound. The pulsating field disturbs the magnetism of the telephone magnet, and enables us, therefore, to detect rapid electromagnetic disturbances at the place where it is held. It is obvious, then, that the induction balance, combined with a telephone, is an apparatus of extreme sensitiveness. It can render evident the smallest differences of weight, nature, degree of purity or temperature of two conductors of identical dimensions, such as two coins placed in identical conditions in respect of the two systems of coils.

It enables us to detect very small masses of metal in a badly conducting body, and may be employed with much advantage in verifying the insulation of the different windings of a coil, the ends of which are open. At the same time, however, it lends itself better to qualitative than to quantitative work, as it is difficult to interpret rigorously the results obtained.*

§ 11. The Transmission of Kapidly Intermittent or Alter- nating Currents through Conductors. — Some experiments by Prof. Hughes in 1886 on the self-induction of metallic wires were the means of directing the general attention more closely than before to the nature of the propagation of electric currents of high frequency through metallic conductors, and although mathematical writers, particularly Maxwell and Oliver Heaviside, had previously considered the problem theoretically, the experimental results drew the attention of many to this question to whom the more recondite mathematical investigations were unknown. Prof. Hughes's

• For further information on the use and theory of the induction balance the student may consult, with advantage, Mascart and Joubert's " Elec- tricity," Vol. II., § 986 ; also Hughes, Phil. Mag. [5], Vol. II., p. 50, 1879. On the differential telephone, see Chrystal, Phil. Trans. Roy. Soc. Edin., Vol. XXIX., p. 609, 1880. 0. Lodge, Proc. Physical Soc. London, Vol. III., p. 187, on intermittent currents and the theory of the induction balance.

282 MUTUAL AND SELF INDUCTION.

experiments* on the self-induction of metallic wires were made with a combined resistance and induction bridge of somewhat novel form. Suppose that a quadrilateral arrange- ment be formed of four conductors P, Q, E, S, only one of which, P, has any sensible self-induction, and let the diagonals be completed by a telephone T, and battery B, with interrupter I. In the first place, let the resistance- balance be obtained for steady currents. This can be achieved by placing the telephone with the interrupter as a conjugate circuit to the battery (see Fig. 104), and altering one resistance, say, 11, until a balance is obtained. By a suitable

FIG. 104.

adjustment of the four resistances complete silence can be obtained in the telephone.

Next, let the interrupter be removed to the battery circuit,. all the other arrangements remaining the same (see Fig. 105). It will be found that the balance is destroyed, and that no mere change in the value of the resistance E will enable a per- fect balance to be obtained. The reason for this is that, on

• These experiments formed the subject of Prof. Hughes's Inaugural Discourse to the Society of Telegraph-Engineers on the occasion of his election to the office of President. See Journal of the Society of Telegraph Engineers, January 28, 1886, " The Self-induction of an Electric Current in Relation to the Nature and Form of its Conductor."

MUTUAL AND SELF INDUCTION. 283

closing the battery circuit, the inductance of P introduces a counter electromotive force into P and the potential rises at c faster than at d, and on breaking the circuit the potential at c dies down faster than at d ; and hence at each make and break the telephone is subjected to an alternate flux of current which causes it to emit a sound. Supposing that an attempt is made to get rid of this sound by shifting the point c so as to alter E, the steady balance will be destroyed, and the telephone will be traversed by a current during the time when all the currents have become steady ; but no such change in the value of B will prevent a variation of current taking place through

the telephone during the complete period from the first instant when the battery circuit is closed to the instant when it is opened again.

The only way in which a balance can be obtained in this last arrangement is by introducing into the telephone circuit an elec- tromotive force which shall be capable of being made to balance at every instant the inductive electromotive force due to the inductance of P. Prof. Hughes does this very ingeniously by introducing a pair of mutually inductive coils into the battery

284

MUTUAL AND SELF INDUCTION.

and telephone circuits, and the final arrangement is as shown in Fig. 106. M! and M2 are a pair of coils, one of which, M2, is in the battery circuit and is fixed, and the other, Mlf is in the telephone circuit, and can be placed so that, whilst its centre coincides with that of M2, its axis makes any required angle with that of Mlt In this way the mutual inductance between Ma and M2 can be varied from zero when the coil axes are at right angles to a definite maximum value when they are co-linear.

It is found that, when the coils M: M2 are in certain positions, the inductive electromotive force set up in the

a

-- \

FIG. 106.

telephone circuit by the induction of Mx on M2 can be made to neutralise the electromotive force of self-induction due to the inductance of P, when, in addition, a certain value is given to the resistance E. Under these circumstances the bridge can be balanced and the telephone completely silenced, both when the interrupter is in the battery circuit and also in the tele- phone circuit ; in other words, the bridge can be balanced both for steady and for variable currents.

In the arrangement adopted by Prof. Hughes the resist- ances Q, E, and S, were sections of one and the same fine

MUTUAL AND SELF INDUCTION.

285

German silver, 1 metre long, and having a total resistance of 4 ohms (see Fig. 107). The ends of this wire were joined ta

61

FIG. 107.

the conductor P under investigation, and the rest of the apparatus was arranged as described.

In order to investigate the relation between the resistances and inductances which holds good when the bridge is balanced for

FIG. 108.

steady and also for variable currents, a diagram must be drawn (Fig. 108) representing the network of conductors. Then call

286 MUTUAL AND SELF INDUCTION.

the current at any instant in the inductive branch P, x, that in the branch Q, y, and that in the telephone circuit z. The current in the branch battery is then x + y. Let L be the inductance of P, and M the mutual inductance of the coils placed in the circuits B and T, and let all the other circuits, Q, E, and S, have no sensible inductance. Let e be the electromotive force of the battery at any instant t. Then the currents in the various branches at that instant are as follows : — In the branch P the current is x „ „ R „ ,, x-vz „ „ S „ „ y-z

„ „ Q „ „ y

B „ „ x+y T „ „ z

Applying Kirchhoff's corollaries to each of the three meshes of the network, we have three equations, viz.,

-L-jf---!*^ (94) s-U-- - • • (95)

T*-SF^=-My^ • • (96)

and these three equations enable us to find at any time t the current in any branch.* If we suppose the bridge to be balanced for variable currents, then z is zero, and on making this limitation we find the above equations reduce to the two,

Qy-Ps-L^f-Ba-Sy, • . - (97)

and -M^I-M^RZ-ST/. . . . (98)

Furthermore, let us assume that the currents vary according to a simple periodic law. In this case, if X is the maximum value of x, then we can write

• The general method of finding the current equations for any network is given in Maxwell's "Treatise on Electricity," 2nd Edition, Vol. II., § 755. Also see " Problems on Networks of Conductors," by J. A. Fleming, Phil. Mag., September, 1E85, Vol. XX., p. 221 ; or Proceedings Phys. Soc., Lond., 1885.

MUTUAL AND SELF INDUCTION. 287

where p as usual = 2«- n, n being the frequency of the alterna.

dx tion. Hence ^ =p!£cospt,

cPx and -75 =

Adopting the fluxional notation, it is convenient to write x for 7-7 and x for -r^- Hence, for simple periodic variation of a current a;, we always have the condition — x = p* x.

If we differentiate with respect to t the two equations (97) and (98), and eliminate x by the help of the equation x = -pzx, we obtain two other equations, which, together with the original two (97) and (98), give us the necessary four equations for elimi- nating the four variables x, y, x, y. We have thus,

Qy-Px-~Lx = Ea;-Sy. . . . (99)

-Mac -My = Ra?-Sy. . . . (100)

= Ex-Sy. . . . (101)

=Rx-Sy. . . - (102)

The student who has mastered the elements of determinant analysis will recognise that the variables x, y, x, y can be eliminated from these equations, and the relation which must always hold good between the constants can be found by equating to zero the determinant of these four equations. We have then

-L, 0, -(P + B), (Q + S)

-M, -M, B, S

-(P + R), (Q + S), Lp2, 0

-E, S, M/>2, Mp2 =0.

This determinant writes oat into the sum of three terms, viz. : —

288 MUTUAL AND SELF INDUCTION.

This long equation reduces to the simpler form

-(QB-SP)2] = 0.

In order that the sum of the two left hand terms in the above equation may always he zero, each factor in the square brackets must be separately zero, and it will be seen that each of these factors equated to zero are equivalent to the two- equations : —

QB-SP = ML/, ..... (103) and M(P + Q-f-B + S) = SL ..... (104)*

These equations express the relation which holds good between the resistances of the branches and the self and mutual induction coefficients of a Hughes bridge when the bridge is balanced for variable currents.

It will be seen that the ordinary relation of the resistances for steady balance, viz., P;Q = B;S is departed from, and that we have for the resistance of branch P, when traversed by variable currents, the value

p_QB-MLp»_QB

. S iS S

and for the inductance of branch P under these circumstances, the value

L_M(P + Q + B + S) < m t (10G)

S In some of his experiments Prof. Hughes interpreted his

O T?

results on the assumption that P was always equal to 5L- , and

b

L was equal to M ; but the complete investigation shows that this is not the case. A very full theoretical and practical examination of the induction bridge has been given by Prof. H. F. Weber, for which the student is referred to the pages of the Electrical Review, Vol. XVIII., p. 321, 1886, and VoL XIX., p. 30, 1886.1