SECTION XXXVI. RESISTANCE AND SELF-INDUCTION OF A ROUND WIRE WITH
CURRENT LONGITUDINAL. DITTO, WITH INDUCTION LONGITUDINAL. THEIR OBSERVATION AND MEASUREMENT.
When the effective resistance to sinusoidal currents is not much greater than the steady resistance, we may employ the formulae (446) [p. 64], to estimate the effective resistance and inductance. On the other hand, when it is a considerable multiple of the steady resistance, we may employ the simple formulae (456). But in intermediate cases, neither pair of formulae is suitable, and it therefore happens that in some practically realisable cases we require the fully developed formulae which are equivalent to (446), but are always convergent.
Let R be the steady resistance per unit length of round wire of radius a, conductivity k, inductivity /a ; and R' its effective resistance to sinusoidal currents of frequency q = nj2TT. Let also
z = fin/R = TTfxhia2 = 2ir2fxka2q (866)
Then the formula required for R' is
The law of formation of the terms is plainly shown, so that the series may be continued as far as is necessary to ensure accuracy. But so far as is written is quite sufficient up to 2= 10.
The corresponding formula for L', what the L of the wire becomes at the frequency q, is
U . 1 + 2?6(1+to(1 + 3^u(1+4^18 L Same denominator as in (876).
Here L = |/z, simply. R'jR increases continuously, and L'fL decreases continuously, as the frequency increases.
The following are the values of PJjR for values of z from \ to 10 :—
z. B'/B. z. B'/B.
1 1*02 6 2 01
2
1 1-08 7 2*14
2 1-26 8 2*27
3 1-48 9 2*39
4 1-68 10 2*51
5 1*85
The curve, whose ordinate is B'/B - 1 and abscissa z, is convex to the axis of abscissae up to about z= 2J, and then concave later.
Let us take the case of an iron wire of one-eighth of an inch in radius (about No. 4 B.W.G.), of resistivity 10,000, and inductivity 100. These data give us z = q/51, by (866). Take, then, £ = g750. Each unit of z means 50 vibrations per second. Then q = 50 makes = 1 *08 ; <2 = 500 makes z= 10 and B'/B= 2*51, or the effective resistance 2£ times the steady.
To obtain similar results in copper, with /x=1, &_1=1600, making fxk to be Yg- part of its former value, we require the radius to be four times as great, or the wire to be 1 in. in diameter. But if it be of the same diameter, ^ = 500 will only make £ = anc^ there will be only a slight increase in the effective resistance.
In the present notation the very-high-frequency formulae are
B' = IJn = Btyzfi ; (896)
and, by comparison with the table, we shall be able to see how large z must be before these are sensibly true. Using (896), 2=4 gives B'/B = 1
- 41, much less than the real value; z = 8 gives 2 instead of 2*274 ;
«=10 gives 2-234 instead of 2*507. On the other hand, (896) makes U too big, but not so much as it makes B! too small. Thus 10 makes Un/B = 2*234 instead of 2-21, which is what the correct formula (886) gives.
Probably z = 20 would make (896) fairly well represent the resistance, as it nearly does the inductance when z= 10. In the case of the iron wire above mentioned, £ = 50, or <?=2500, will make the effective resistance five times the steady.
If the wire be exposed to sinusoidal variations of longitudinal magnetic force by insertion within a long solenoidal coil, the effect, when small, on the coil-current, is the same as if the resistance of the coil- circuit were increased by the amount IB(, given by
B' = L1x iirfxkn2a2 = L^n x \z (906)
[Reprint, vol. I., p. 369, the last equation. Also p. 364, equation (36).] Here I is the length of the core and coil, having N turns of wire per unit length, and Xj = (2;raN)2fi
is the steady inductance, due to the core only, per unit of its length.
If CQ be the amplitude of the coil-current, the mean rate of generation of heat in the core is ^R'CQ1, per unit of its length. When the effect is large, use the formula
R[ , 1 + ^(1+£3Uo(1+3^T4(1 + "‘ .
557 i2—7 ir—7 (91 *)
1 + -2-(l+ + +
[vol. I., p. 364, equation (36), and the next one.] (I have slightly changed the notation to suit present convenience, and show the law of formation of the terms. The old y equals the new 16.Z2.) I did not give any separately developed expression for the L[ corresponding to Lx; being only a portion of the L of the circuit it was merged in the expression for the tangent of the phase-difference. [Vol. 1. , pp. 369 to 374, §§ 16, 17.] Exhibiting now L{ by itself, we have this formula:—
T' 1+f(1+^ln(1+qrTl(1+Z3T«(1 + -"
U_ 6 \ 2j.10 \ 33.14 \ 43.18 \ (926)
Lx Same denominator as in (916).
Notice that the numerators in (916) and (886) are the same, and that those of (926) and (876) are the same. At the frequency 500, using the same iron wire above described, we have, taking z = 10 in (916) and (926),
R[ = -188 Lxn, L[ = *225 Lx (936)
Or, with a little development,
R[ = 622 2^ = 243,000 N2, (946)
i.e., the extra resistance is 243 microhms multiplied by the length of the core, and by the square of the number of windings per unit length. At this particular frequency the amplitude of the magnetic force oscillations at the axis of the core is only one-fourteenth of the amplitude at the boundary. When it is the current that is longitudinal, it is the current-density at the axis that is only T*T its boundary-value.
Now, as cores may be so easily taken thicker, it is also desirable to have the high-frequency formulae corresponding to (916) and (926), which I now give. They are
<95j)
The value £=10 is scarcely large enough for their applicability. Thus (956) give (same iron wire),
R' = L(n = '223 Lxn, (966)
instead of (936), making R[ too big, and L[ too small, although the latter is nearly correct.
In one respect the reaction of metal in the magnetic field on a coil- current is far simpler than the reaction on itself when it contains the impressed force in its own circuit. If we have a sinusoidal current in a cod, subject to e = RC+ LC, e being the sinusoidal impressed force, C the current, R and L the steady resistance and inductance of the circuit; and we, by putting metal in its magnetic field, induce currents in it, and waste energy there, we know that the new state is also sinusoidal, subject to
e = R'C + L'C,
where Rf and L' have some other values. So far is elementary. This, however, is also elementary, that R' must be greater than R. For the heat in the coil per second is \RC, and the total heat per second is \R,'Cq. AS the latter includes the heat externally generated, R' is necessarily greater than R. But this simple reasoning, without any appeal to abstrusities, breaks down when it is the wire itself in which the change from R to R! takes place, and we then require to use reasoning based upon the changed distribution of current.
To observe these changes qualitatively is easy enough. But to do so quantitatively and accurately is another matter. It cannot be done with intermittences. A convenient little machine giving a strictly sinusoidal impressed force of good working strength, adjustable from zero up to very high frequencies, is a thing to be desired. But we may employ very rapid intermittences with an approximation to the theoretical results. I have obtained the best results with a microphonic contact, without interruptions, but it was difficult to keep it going uniformly. Slow intermittences give widely erroneous results, i.e., according to the sinusoidal theory, which does not apply, making the changes in resistance and induction much too large. Here, of course, the silence—the best minimum to be got—is a loud sound.
I should observe, by the way, that a correct method of balancing is presumed. In Prof. Hughes’s researches, which led him to such remarkable conclusions, the method of balancing was not such as to ensure, save exceptionally, either a true resistance or a true induction balance. Hence, the complete mixing up of resistance and induction effects, due to false balances. And hidden away in the mixture was what I termed the “ thick-wire effect,” causing a true change in resistance and inductance [vol. II., p. 30]. In fact, if I had not, in my experiments on cores and similar things, been already familiar with real changes in resistance and inductance, and had not already worked out the theory of the phenomenon of approximation to surface conduction [first general description in vol. I., Art. 30, p. 440; vol. II., p. 30], on which these effects in a wire with the current longitudinal depend, it is quite likely that I should have put down all anomalous results to the false balances.
Of course, we should separate inductance from resistance. Perhaps the simplest way is that I described [vol. II., p. 33, Art. xxxiv.] of using a ratio of equality, reducing the three conditions to two, ensuring independence of the mutual induction of sides 1 and 2, and also of sides 3 and 4 (allowing us to wind wires 1 and 2 together, and so remove the source of error due to temperature inequality which is so annoying in fine work), and requiring us merely to equalise the resistances and the inductances of sides 3 and 4, varying the inductance to the required amount by means of a coil of variable inductance, consisting of two coils joined in sequence, one of which is movable with respect to the other, thus varying the inductance from a minimum to a maximum—an arrangement which I now call an Inductometer, since it is for the measurement of induction. The oddly-named Sonometer will do just as well, if of suitable size, and its coils be joined in sequence. The only essential peculiarity of the inductometer is the way it is joined and used. This method of equal ratio was adopted by Prof. Hughes in his later researches (Royal Society, May 27, 1886); he, however, varies his induction by a flexible coil, which I hardly like. Lord Rayleigh has also adopted this method of separating induction from resistance, and of varying the inductance. (Phil. Mag., Dec., 1886.) I found that the calibration could be expeditiously effected with a condenser, dividing the scale into intervals representing equal amounts of inductance. Lord Rayleigh does, indeed, seem to approve somewhat of Prof. Hughes’s method, with its extraordinary complications in theoretical interpretation (very dubious at the best, owing to intermittences not being sinusoidal). But if it be wished to employ mutual induction between two branches to obtain a balance, there is the M63 or M64 method I described [vol. II., Art. XXXIV.]. which is, like the method of equal ratio, exact in its separation of resistance and inductance, with simple interpretation. I have since found that there are no other ways than these, except the duplications which arise from the exchange of the source of electricity and the current indicator. Using any of these methods, we completely eliminate the false balances ; now we shall have perfect silences, independent of the manner of variation of the currents, whenever the side 4 [in figure, p. 33, vol. II. ], containing the experimental arrangement, is equivalent to a coil, w ith the two constants R and X, and can therefore equalise a coil in side 3 (presuming that the equal-ratio method is employed). But if in the equation V=ZC of the experimental wire, Z is not reducible to the form of R + L(d/dt), it is not possible to make the currents vary in the same manner in the sides 3 and 4, and so secure a balance. That is, we cannot balance merely by resistance and self induction, the departure of the nearest approach to a balance from a true balance being little or great, as the manner of variation of the current in side 4 differs little or much from that of the current in its ought-to-be equivalent side 3. The difference is great when a coil with a big core is compared with a coil without a core; and, as in all similar cases, as before remarked, at a moderate rate of intermittence, we must not apply the sinusoidal theory to the interpretation. If we want to have true balances when there is departure from coi1-equivalence, we must specialise the currents, making them sinusoidal. Then we can have silences, and correctly interpret results. We appear to have false balances. But they are quite different from the before-mentioned false balances, as they indicate true changes in resistance and inductance, owing to the reduction of Z to the required form, in which, however, the two “constants ” are functions of the frequency.
102 \ - '5 • ELECTRICAL PAPERS.
t'\ /; >,■ :• :*.77^.
SECTION XXXVII. GENERAL THEORY OF THE CHRISTIE BALANCE. DIFFERENTIAL
EQUATION OF A BRANCH. BALANCING BY MEANS OF REDUCED COPIES.
It is not easy to find a good name for Mr. S. H. Christie’s differential arrangement. There are objections to all the names bridge, balance, lozenge, parallelogram, quadrangle, quadrilateral, and pons asinorum, which have been used. It seems to be a nearly universal rule for words, used correctly in the first place, to gradually change their meaning, and finally cause us to talk nonsense, according to their original signification. Thus the Bridge is the conductor which bridges across two others. But it has become usual to speak of the differential arrangement as a whole as the Bridge; and then we have the four sides of the bridge, which is absurd. Quadrilateral is the latest fashion. It has four sides, truly. But there are six conductors concerned; so we should not call the differential arrangement itself the quadrilateral.
I propose to simply call it the Christie, without any addition, just as telegraphers speak of the Morse, or the Wheatstone, meaning the apparatus taken as a whole. Thus we can refer to the Christie, the quadrilateral, and the bridge, the latter two being parts of the former. This will suppress the farrago.
In the usual form of the Christie we have four points, A, Bx, B2, C, united by six conductors, numbered from 1 to 6 in the figure. The quadrilateral has the four sides, 1, 2, 3, 4. The bridge-wire is 5, joining Bj to B2, and 6 is the battery-wire. The battery-current goes from A to C by the two distinct routes ABjC and AB2C. Some of it crosses the bridge, up or down; except under special circumstances, when the bridge-wire is free from current, which is the useful property.
Let us generalise the Christie thus:—Let the sole characteristics of a branch be that the current entering it at one end equals that leaving it at the other, with the additional property that the electromagnetic conditions prevailing in it are stationary, so that the branch becomes quite definite, independent of the time.
Thus, all six branches may be any complex combinations of conductors and condensers satisfying these conditions. The communication between the two ends of a branch need not be conductive at all; for example, a condenser may be inserted. As an example of a complex combination, let branch 3 consist of a long telegraphic circuit, symbolised by the two parallel lines starting from 3 and ending at Y3, where they are connected through terminal apparatus. This branch then consists of a long series of small condensers, whose + poles are all connected together by one wire, and the - poles by the other wire. There is also conductive connection (by leakage) between the two wires. There is also electromagnetic induction all along the line. But, as the current entering the line from Bx to 3, and that leaving it, from 3 to C, are equal, the telegraphic line comes under our definition, provided it be stationary in its properties. Observe that this does not exclude the presence of other conductors, between which and the line in branch there is mutual induction, providing this does not disturb our fundamental property of a branch. We may, indeed, remove the original restriction, but then it will no longer be the Christie, for more than four points will be in question. Suppose, for example, there is mutual induction of the electrostatic kind between branches 1 and 2, which is most simply got by connecting the middles of 1 and 2, taken as resistances, through a condenser. Then there are six points, or junctions, concerned, and a slight enlargement of the theory is required.
Let us now inquire into the general condition of a balance, or of no current in the bridge-wire due to current in 6, which, therefore, enters the quadrilateral at A and leaves at C, and which may arise from impressed force in 6 itself, or be induced in it by external causes. First, as regards the self-induction balance in the extended sense. This does not mean that each side of the quadrilateral must be equivalent to a coil, but merely that the four sides are independent of one another in every respect, except in being connected at A, Bp B2, C. Thus we can have electrostatic and electromagnetic induction in all six branches, but independently of one another. Under these circumstances it is always possible to write the differential equation of a branch in the form V=ZC, where C is the current (at the ends), V the fall of potential from end to end, and Z a differential operator in which time is the independent variable. When the branch is a mere resistance E, then Z=E, simply. When it is a coil, independent of all other conductors, then
Z=E + Lp,
where L is the inductance of the coil, and p stands for djdt. When it is a condenser, then Z=(Sp)~1, where S is the capacity, if the condenser have also conductance K, or be shunted by a mere resistance, then
Z=(K+Sp)~\
These are merely the simplest cases. In general, Z is a function of p, p2, etc., and electrical constants.
Now let the positive direction of current be from left to right in sides 1, 2, 3, 4, and suppose we know their differential equations
Vl = ZlCv V2 = Z2C2, etc.
To have a balance, so far as the current from 6 is concerned, the potentials at B^ and B2 must be always equal, except as regards inequalities arising from impressed forces in other branches than 6, with which we are not concerned. Therefore
V^-V, and Vs=Vv \ (u) or, and Zst\ = Z4Cr j
But, CX = C9 C2 = C4.
So, using these in (lc), we get
Z1C1 = Z2C2, \ / 2C\
Eliminate the currents by cross-multiplication, and we get
ZxZ^Z2Zv (3c)
which is the condition required. It has to be identically satisfied, so that, on expansion, the coefficient of every power of j? must vanish.
If we take Z=R + Lp (as when each side is a coil, or equivalent to one), we obtain the three conditions given in my paper “On the Use of the Bridge as an Induction Balance, equations (1), (2), (3) [vol. II., Art. xxxiv., p. 33].
As another example, take Z = (K + Sp)'1 (shunted condensers), and we obtain three similar conditions. But it is needless to multiply examples here. We have only to find the forms of the four Zs, expand equation (3c), and equate to zero separately the coefficient of every power of p. It does not follow that a balance is possible in a particular case, but our results will always tell us how to make it possible, as by giving zero values to some of the constants concerned, when one branch is too complex to be balanced by simpler arrangements in other branches.
The theory of a balance of self and mutual electromagnetic induction
I propose to give by a different and very simple method in the next Section. At present, in connection with the above generalised self- induction balance, let us inquire how to balance telegraph lines of different types, or when they can be simply balanced. It is clear, in the first place, that if we choose sides 1 and 3 quite arbitrarily, we have merely to make side 2 an exact copy of side 1, and 4 an exact copy of side 3, in order to ensure a perfect balance. Imagine the bridge-wire to be removed; then we have points A and C joined by two identical arrangements. The disturbances produced in these by the current from 6 must be equal in similar parts; hence, if Bj and B2 be corresponding points, their potentials will be always equal, so that no current will pass in the bridge-wire when they are connected. But we can also get a true balance when the “line” AB2C is not a full- sized, but a reduced copy of the line ABjC. It is not the most general balance, of course, but is still a great extension upon the balance by means of full-sized copies. The general principle is this:—
Starting with sides 1 and 3 arbitrary, make 2 and 4 copies of them, first simply qualitatively, as it were; thus, a resistance for a resistance, a condenser for a condenser, and so on. This is like constructing an artificial man with all organs complete, but in no particular proportion. Then, make every resistance in sides 2 and 4 any multiple, say s times the corresponding resistance in sides 1 and 3. Make every condenser in sides 2 and 4 have, not s times, but s-1 times the capacity of the corresponding condenser in sides 1 and 3. And, lastly, make every inductance in sides 2 and 4 be s times the corresponding inductance in sides 1 and 3. This done, s being any numeric, AB2C is made a reduced (or enlarged) copy of AB^, and there will be a true balance. That is, the potentials at corresponding points will be equal, so that the bridge-wire may connect any pair of them, without causing any disturbance. Now let a telegraph line be defined by its length Z, and by four electrical constants R the resistance, S the electrostatic capacity, L the inductance, and K the leakage-conductance, all per unit length. It is not by any means the most general way of representation of a telegraph line, but is sufficient for our purpose. Let C be the current, and V the potential-difference at distance x from its beginning. We require the form of Z in V=ZC at its beginning. This will depend somewhat upon the terminal conditions at the distant end, so, in the first place let V=Q there. Take
C=cos mx. A + sin mx. B, (4c) V= -(R + Lp)m~\sin mx. A — cos mx. B), (5c) - m2 = (K+ Sp)(R + Lp), (6c)
p standing for dfdt as before. These are general, subject to no impressed forces in the line. A and B are arbitrary so far. But at the end x = l, we have F=0 imposed, which gives, by (5c),
BjA =ta.n ml, (7c)
so that at the x = 0 end, we have, by (4c), (5c), and (7c),
T R L*p B rr / j> T \i tan ml <n \ C= rn ' A ’ 01 Z=(R + Lp)l—^- (80)
This is the Z required. From the form of m2, we see that if the total resistance Rl and total inductance Ll in one line be, say, s times those in a second, whilst the total capacity SI and total leakage-conductance KI in the second line are s times those in the first, then the values of ml are identical for the two lines. If these lines be in branches 3 and 4, we therefore have
^3 _ (~^3 Lspjl^ /q v
Z~(RH + L<J>% ’ {C)
so that we may balance by making sides 1 and 2 resistances whose ratio RJR2 is s; or, if coils be used, by having, additionally, LJL^ = s; or, if condensers are used, (K2 + S^p)l(Kx + Sxp) = s; and so on.
But if there be apparatus at the distant end of the line, it must also be allowed for. Let V = YC be the equation of the terminal apparatus; that is, this equation connects (4c) and (5c) when x=l. Using it, instead of the former we shall arrive at
p_(R + Lp) tan ml + mY/(R + Lp) > m ' 1 - tan ml.mYj(R + Lpf { ’
instead of (8c). Now, just as before, adjust the constants of lines 3 and 4, so that m3l3 = mil4, and, in addition, make YJY4 = s. Then, supposing each side of the quadrilateral to be a telegraph line, the full conditions of balance by this kind of reduced copies are
- Wi - % _ ^2 _ ^1
JRoln LtXn Stit K.L Yo
The difference from the former case is that we now have in sides 2 and 4 reduced copies of the terminal apparatus of lines 1 and .3. It will be observed that the equalities in the first line of
(11c) make side
2 a reduced copy of side 1, and that those in the second line make side
4 a reduced copy of 3, whilst the equalisation of the two lines of (11c) makes the scale of reduction the same, so that AB2C is made a reduced copy of ABjC.
If one of the four sides, say side 3, of the quadrilateral be a telegraph line, we must have at least one other telegraph line, or imitation line, namely, in side 4. But, of course, sides 1 and 2 may be electrical arrangements of a quite different type. Further, notice that only two of the sides, either 1 and 2, or 3 and 4, can be single wires with return through earth, so that if the other two are also to be telegraph lines they must be looped, or double wires. In certain cases precisely the same form of Z as that above used will be valid, but this is quite immaterial as regards balancing by means of a reduced copy.
The balance expressed by equation (3c) is exact—that is, it is independent of the manner of variation of the current. The balance by means of reduced copies is also exact, but is only a special case of the former. But there is always, in addition, the periodic or S.H. balance, when the currents are undulatory. Then merely two conditions are required, to be got by putting p2 = - n2, where w/27r is the frequency, in ZXZ± - Z^Zp which will reduce it to the form a + bp, in which a and b contain the frequency. Now, a = 0 and b- 0 specify this peculiar kind of balance, which is, generally speaking, useless. Whilst, however, the balance of ABjC and AB2C by making the latter a reduced copy of the former is, when applied to the Christie, only a special case of (3c), it is, in another respect, far more general; for it will be observed that any pair of corresponding points may be joined by the bridge-wire, although the result may be an arrangement which is not the Christie.
SECTION XXXVIII. THEORY OF THE CHRISTIE AS A BALANCE OF SELF AND
MUTUAL ELECTROMAGNETIC INDUCTION. FELICI’S INDUCTION BALANCE.
As promised in the last Section, I now give a simple, and, I believe, the very simplest, investigation of the conditions of balance when all six branches of the Christie have self and mutual induction. Referring to the same figure (in which we may ignore the extensions of branches 3 and 4 to Yg and Y4), we see that as there are six branches there are twenty-one inductances, viz., six self and fifteen mutual.
ELECTROMAGNETIC INDUCTION AND ITS PROPAGATION. ] 07
This looks formidable. But since there are only three independent currents possible there can really be only six independent inductances concerned, viz., three self and three mutual, each of which is a combination of those of the branches separately.
Thus, let Cv C3, and C6 be the currents that are taken as independent, and let them exist in the three circuits ABjB2A, CB^C, and AB2CA (via branch 6), with right-handed circulation when positive.
Then the other three real currents G2, C4, and Cb are given by
n _n p n _ n p p _ /» r> .
L/2— ^6 V4~L'6~ 3’ — '-'l 3 *
if the positive direction be from left to right in sides 1, 2, 3, and 4, from right to left in 6, and down in 5, which harmonises with the positive directions of the cyclical currents C\, C3, and C6. B.
Next, let mv m3, m6, and m13, m36, m61 be the inductances, self and mutual, of the three circuits. Thus, m1 = induction through ABjB2A due to unit current in this circuit; and m13 = the induction through CBgBjC due to the same, etc. We have to find what relations must exist amongst the resistances and the inductances in order that there may never be any current in the bridge-wire, provided there be no impressed forces in 1, 2, 3, 4 or 5.
We obtain them by writing down the equations of E.M.F. in the two circuits ABXB2A and CB2BjC on the assumption that there is no current in the bridge-wire, which requires C1 = C3; and this we do by equating the E.M.F. of induction in a circuit, or the rate of decrease of the induction through the circuit, to the E.M.F supporting current, which is the sum of the products of the real currents into the resistances, taken round the circuit.
Thus,
= E1Cl — -^2(^6 — ^i)>l 0 2c)
-p(m3C3 + m31Cj + m&3Cf) = R3C3 - R4(C6 - C3),f‘
where p stands for djdt. But C\ = C3t which, substituted, makes
{(-^1 + -^2) {mi ^'13)-?^} ^1 = (^2 — # .(13c)
{ (-^3 + ^4) + (m3 + mis)Pl C1 = (^4 - m36P)C6’J
which have to be identically satisfied. Eliminate the currents by crossmultiplication, and then equate to zero separately the coefficients of the powers of p. This gives us
K + ™13 + Wiie)^4 - ™36^1 = (m3 + m31 + ™36)E2 ~ m\A' (m\ + = (”<« +
which are the conditions required. First the resistance balance; next the vanishing of integral extra-current due to putting on a steady impressed force in branch 6; and the third condition to wipe out all trace of current, and make branches 5 and 6 perfectly conjugate under all circumstances.
If the Christie consists of short wires, which are not nearly closed in themselves, then, as I pointed out before [vol. II., Art. 34, p. 37], the theory of the balance expressed in terms of the self and mutual inductances of the different branches becomes meaningless, because the inductances themselves are meaningless. Under these circumstances, equations (14c) are the conditions of a balance, from which alone can accurate deductions be made. Even if we have the full equations in terms of the twenty-one inductances of the branches, they will express no more than (14c) do. We could not, for instance, generally assume any one of the inductances to vanish, as it would produce an absurdity, viz., the consideration of the amount of induction passing through an open circuit. Hence it is quite possible that (14c) may be useful in certain experiments, in which such short wires are used that terminal connections become not insignificant.
At the same time it is to be remarked that such cases are quite exceptional. I would not think, for example, of measuring the inductance of a wire a few inches long, in which case (14c) would, at least in part, be applicable, if I could get a long wire and swamp the terminal connections. Still, however, equations (14c) and the way they are established are useful in another respect. In general, I have not found any particular advantage in Maxwell’s method of cycles. It has seemed to me to often lead to very roundabout ways of doing simple work, from what I have seen of it. This applies both when the steady distribution of current in a network of conductors is considered, due to steady impressed forces, as in the original application; and also when the branches are not treated as mere resistances, but transient states are considered, provided the branches be independent, so that, as I remarked before, the equation of a branch may be represented by V = ZC, where Z takes the place of B, the resistance in the elementary case. But in our present problem there is such a large number of inductances that there is a real advantage in using the above method, an advantage which is non-existent in a problem relating to steady states. We greatly simplify the preliminary work by reducing the number of inductances from 21 to 6. But, of course, on ultimate expansion of results we shall come to the same end.
If we use the first and third of (14c) in the second, it becomes
{«,, +,««, + m61(l + |i)}. (|s - ^) = 0; (IBe)
and, as either of these factors may vanish, we have in general two entirely distinct solutions. If the second factor vanish, the whole set of conditions may be written
R1_R2^ ml6 7»j 4- ml3 R-R- mm m3 + ml3 ’ whilst, if it be the first factor that vanishes, we shall have
_ mi + m\3 _ 1 _|_ m3 mis (17c)
R2 R4 w»61 in63
expressing the full conditions. Both (16c) and (17c) are included in (14c).
Suppose now that we make the branches long wires, or coils of wire, or many coils in sequence, etc., and can therefore localise inductances in and between the branches. We require to expand the six m*s. Their full expressions will vary according to circumstances. When all the twenty-one inductances are counted, they are given by
mi= A + Lb + L2 + 2(ilT15 - M25 - M12), >
m3 = L3 + L4 + 1/^ + 2(M45 — M34 — -M35),
= Lq + L2+ + MSi), mi3 ~ ~ + (M^3 — My4 — il/j5 — M23 + -^24 "*■ “^25 -^53 ~ -^54)>
W16 ~ ~ L.2 + (-^12 + ^14 + -^16 — ^24 — -^26 -^62 ^54 + -^5e)> l,i36 ~ ~ ^4 + (^30 + -^34 + -^36 ~ “^42 — ^46 ~ ^52 — -^54 — '
Here X stands for the inductance of a branch, and M for the mutual inductance of two branches. These are got by inspection of the figure, with careful attention to the assumed positive directions of both the cyclical and the real currents.
In the use of these, for insertion in (14c), we shall of course equate to zero all negligible inductances. As an example of a very simple case, let coils be put in branches 4 and 6, between which there is mutual induction, and let the other four branches be double-wound or of negligible inductance. Then all except Z4, L& and M46 are zero, giving
mx =0, m3 = L4, m6 = Z4 + 2Mm,
W13 = 0, W?16 = 0> m36 = — -^4 — M^Q'
Insert these in the second of (14c), and we get
+ — M4GR2, or — L4 = (1 + R2/R1)M46. ...(19c)
The third condition is nugatory. Hence (19c), with a resistance balance, but without the need of measuring R3 (or, equivalently, i?4), gives us the ratio of the M of two coils to the L of one of them in terms of the ratio of two resistances.
As another example, let all the il/’s be zero except M12 and M34, whilst all the L’s are finite. We shall then have, besides the resistance balances, the two conditions } (20c) 4* J
0 = (LYL4 - L2L3) + (L2 - LJM34 + (L3 - L4)Ml2,
0 = (LlR4 + L4R\ ~ L2R3 — L3R2) + (R3 — R4)M12 + (R2 — R\)M.34.
If we now take Rx = R2,, Lx = L2; that is, let sides l and 2 be equal, we reduce the three conditions (14c) to i?3 = /?4, Ls = Lr This is obvious enough in the absence of mutual induction; but we also see that induction between sides 1 and 2, and between 3 and 4, does not in the least interfere with the self-induction balance Whilst remarkable, this property is of great utility. For it allows us to have the equal wires 1 and 2 close together, preferably twisted, and then this double wire may be doubled on itself, and the result wound on a bobbin. We ensure the equality of the wires at all times, doing away with the troublesome source of error arising from the disturbance of the resistance balance from temperature changes, which occur when 1 and 2 are separated, and also doing away with interferences from induction between 1 and 2 and the rest. We also do away with the necessity of keeping coils 3 and 4 widely separated from one another.
Passing to a connected matter, Maxwell, Vol. II., Art. 536, describes the well known mutual induction balance with which Felici made such measure the L of each coil by itself, and it is well to previously adjust the coils to have equal L and R. But the present use of the inductometer is not to measure self induction, but mutual induction. Therefore make 1 and 2 the coils wrhose M is wanted, 3 and 4 the coils of the inductometer. If within range (it is well to have inductometers of different sizes, for various purposes), we immediately measure ilf12, and have the full advantages of Felici’s balance.
But if there is metal about the coils 1 and 2 (of course there should be none, or very little, about the inductometer, or it should be carefully divided), we cannot get telephone balances. If the departure from balance is serious, and it is not practicable to remove the metal, we may give up the telephone and use a suitable galvanometer, one whose needle will not move till all the current due to a make has passed, and then move if it can. But if the metal be iron, and we want to measure the steady M in presence of the iron (not finely divided), of course we must not remove the iron and measure something else than what we want to know. Then the galvanometer is indispensable. We lose the advantage of the telephone, but Felici’s balance has still its peculiar merits left, in a very great measure.
Apart from the question of measurement, Felici’s balance is highly instructive, as to which see Maxwell’s treatise, to which we should add that the telephone should always be used if possible. Besides the experiments referred to, the balance is useful for studying the influence of iron in the field on the M of two coils, increasing or decreasing it, according to position. Use non-conducting iron [vol. II., Art. 36, later]. Here we have another proof to that there mentioned, that there is no appreciable waste of energy in finely divided iron when the range of the magnetic force is moderate, although very perfect silences, like those when there are no F. currents, and no iron, are not always obtainable.
As regards Felici’s balance when employed for observing differential effects, e.g., Prof. Hughes’s magical experiments with coins, and so forth, I cannot recommend it, for several reasons. The theory is complex, in the first place, so that scientific interpretation of results is difficult. Next, considerable accuracy in adjustment of the coils, in two equal pairs, similarly placed, is required. Lastly, the independence of resistance, etc., ceases when there are F. currents to disturb; and as we are not able to trace the variations of resistance, we may, in sensitive arrangements, when balancing one set of F. currents and reactions against another set, be interfered with by unknown temperature variations.
Perhaps the easiest way is to take a long wire, double it on itself and then double again, giving four equal wires. Wind two side by side to make one pair of coils (1 and 2), and the others in the same manner, to make the other pair. Of course we have increased sensitiveness by the closeness of the wires.
But it is far better not to use four coils, but only two, viz., coils 3 and 4 in the equal-sided self-induction balance, with 1 and 2 made permanently equal, as before described. The temperature error is then under constant observation, and we know at once when the resistance balance of coils 3 and 4 (apart from F. currents) is upset. Interpretation is also an easier matter, both in general reasoning and in calculations.
SECTION XXXIXa. FELICI’S BALANCE DISTURBED, AND THE DISTURBANCE
EQUILIBRATED.
Referring to the last figure, in which imagine the galvanometer to be replaced by a telephone, and the key by an automatic intermitter, let us start with a perfect balance due to the M of one pair of coils being cancelled by the M of the other pair, and consider the nature of the effects produced by the presence of metal in or near either pair of coils. .i.i
First, let 3 and 4 be the coils of an inductometer, and 1, 2 other coils of any kind, separate from one another. The simplest action is that caused by non-conducting iron. It acts to increase or decrease the M of either or both pairs of coils according to its position with respect to them, and its effect can be perfectly balanced by a suitable increase or decrease of the M (mutual inductance) of the inductometer coils. Suppose, for example, the disturber is a non-conducting iron bullet, and is brought into the field of the coils 1, 2. If it be inserted in either coil, it increases their M. This is mainly because it increases the L of the coil in which it is inserted. If the two coils have their axes coincident, as in the figure, the bullet will cause their M to be increased by placing it anywhere on the axis, or near it. But if the bullet be brought between the coils laterally, so as to be, for instance, between the numerals 1 and 2 in the figure, the result is a decreased M. Here the L of each coil is little altered, and the decrease of M results from the lateral diversion of the magnetic induction by the bullet from its normal distribution. By pushing it in towards the axis a position of minimum M is reached, after which further approach to the axis causes M to increase, ending finally on the axis with being greater than the normal amount.
If the disturber be a non-conducting core (round cylinder), the greatest increase of M is, of course, when it is pushed through both coils, which are themselves brought as close together as possible, and when the core itself is several times as long as the depth of the coils. M is then multiplied about four times when the coils are about of the shape shown, with internal aperture about ^ the diameter of the coils. If the coils be wound parallel on the same bobbin, the increase is much greater. If the whole space surrounding the coils be embedded in iron to a considerable distance, we shall approach the maximum M possible. The effective inductivity of the non-conducting iron is considerably less than that of solid iron, which counterbalances the freedom from F. currents.
Using solid iron, no silence is possible, owing to the F. currents, although there is a more or less distinctly marked minimum sound for a particular value of M. The substitution of a bundle of iron wires reduces this minimum sound, and when the wires are very fine, it is brought to comparative insignificance; but only by very fine division of iron are the F. currents rendered of insensible effect. It will be, of course, remembered that the range of the magnetic force variations must be moderate, so as to render the variations in the magnetic induction strictly proportional to them, otherwise no perfect balance is possible with non-conducting iron.
On the other hand, non-conducting (i.e., very finely divided) brass (or presumably any other non-magnetic metal) does nothing. Diamagnetic effects are insensible. The above remarks apply, for the most part, equally well to the self-induction balance, except that iron always increases the L of a coil.
So far is very simple. It is the effect of the conductivity (in mass) of the disturbing matter that makes the interpretation of results troublesome. If the disturber be non-magnetic, we have a secondary current due to the action on the secondary circuit of the current induced in the disturber by the primary current; at least I suppose that this is the way it might be popularly explained. If the disturber be not too big, the M of the inductometer which gives the least sound (instead of silence) is sensibly the old value which gave silence before its introduction. If it be magnetic, there is usually increased M also. Changing the M of the inductometer to suit this, the minimum sound is still far louder than with an equally large non-magnetic disturbing mass (metallic) because the F. currents are so much stronger in iron. To this an exception is Prof. Bottomley’s manganese-steel of nearly unit inductivity, in which the F. currents should be, and no doubt are, far weaker than in copper, on account of the comparatively low conductivity. If this be not so, then it must be found out why not. Again, if the iron be independently magnetised so intensely as to reduce the effective inductivity sufficiently, then, as I pointed out in 1884, the F. currents should be made less than in copper.
To obtain an idea of the disturbance in the secondary circuit due to a conducting mass, let it be a simple linear circuit, and call it the tertiary. Let the suffixes x and 2 refer to the primary and secondary circuits, and 3 to the tertiary. Then the equations of E.M.F. are
e = *1" ^i3P^~ 3» ^
0 = ^2^2 -^23-P^3> j (21®)
0 = + -^32^2 + ^3^3 J
where Z=R + Lp, and e is the impressed force in the primary. Here Mu is missing, it being supposed to be properly adjusted to be zero. From these,
C = -^13^23 P^e (22c)
2 Z1(Z2Zs-M^y-Z2M^
H.E.P.—VOL. II. H
is the secondary current’s equation. The secondary current therefore varies as the product of the M of the tertiary and primary into the M of the tertiary and secondary. It is therefore made greatest by making coils 1 and 2 in the figure coincident (practically) by double-winding, and putting the disturber in their centre. In this case, let R and L be the resistance and inductance of the primary and also of the secondary circuit, r and I those of the tertiary, and m the former M13 or J/23, now equal. Then (22c) becomes, if z=r + lp,
<23‘>
But m is very small compared with L, so m
Let the impressed force be sinusoidal; then p2= - w2, making
n _ rrfirPe /o^-\
(r + lp){(R?-LW) + 2RLp} V '
Let R = Ln, which condition is readily reached approximately. Then
i-ft-fSS" <*>
gives the secondary current in amplitude, (m?nl2RL)(r2 + l2n2)~l per unit impressed force, and phase. If the tertiary could have no resistance, the secondary current would be of amplitude m2/2RLl per unit impressed force, and in the same phase with it.
Now seek the conditions of balance by means of a fourth linear circuit placed between coils 3 and 4 in the figure, supposed to be exactly like coils 1 and 2. Let the suffix 4 relate to this fourth circuit. Then (21c) become
e= Z1C1 + M13pC3 + 3fupC±,'
0 = Z2C2 + M23pC3 + M2ipC#
0 = M3^pCX + ^3^3 j 0 = MA1pC-y + M42pC2 +
Here, besides M12, M34 is also missing, because of the distance between the two disturbers. From these,
&C2 = (M23M3lZ4 + M2AM4lZ3)e (28c)
is the equation of C2, where A is the determinant of the coefficients in (27c). For a balance, the coefficient of e must vanish. This gives
r L M M {_3 3 31 32 /OQ/.\
r4 L, M41M42 ' ( )
If the coils of each “ transformer” are coincident and equal, M3l = M32, and M41= and, the JkTs being small, (28c) becomes
Q —MjiZj - MgZ3
2 ZLZ3Z4 ’ (}
where Z is that of either the primary or secondary circuit.
We do not need to balance the disturber in one pair of coils by means of a precise copy of it in the other pair, similarly placed. It may be a reduced copy, according to (29c).
SECTION XXXIXV. THEORY OF THE BALANCE OF THICK WIRES, BOTH IN THE
CHRISTIE AND FELICI ARRANGEMENTS. TRANSFORMER WITH CONDUCTING CORE.
This brings me to the subject of balancing rods against one another, either in the Christie or in the Felici differential arrangements, when placed in long solenoids ; and to the similar question of balancing thick wires in the Christie, when the current in them is longitudinal. As I pointed out before [vol. n., p. 37], if a wire be so thick that the effect of diffusion is sensible, it cannot be balanced in the Christie against a fine wire, but requires another thick wire in which the diffusion effect also occurs. I refer to true balances, independent of the manner of variation of the current, in which, therefore, the resistance of the one wire, though different at every moment, is yet precisely that of the other wire (or any constant multiple of it). Perhaps the best way to define the resistance is by Joule’s EC2. In the sinusoidal case a mean value is taken. According to this heat-generation formula, there always is a definite resistance at a particular moment, but what it may be will require elaborate calculation to find. This definition of the resistance to suit the instantaneous value of the dissipativity does not agree precisely with the sinusoidal E', which represents a mean value; but the sinusoidal E' has important recommendations which outweigh this disadvantage. Suppose, now, we want to balance an iron wire against a copper wire, the wires being straight and long, though not so long as to require the consideration of electrostatic capacity. For simplicity, first let the ratio be one of equality, so that sides 1 and 2 in the Christie are any precisely equal admissible arrangements, which may be mere resistances. Let the iron wire be in side 3, the copper wire in side 4. We have to make side 3 an electrical full-sized copy of side 4. For definiteness, imagine Ys and F4 to be short-circuits, that one of the two parallel lines leading to either is the wire under test, whilst the other is a return tube, thin and concentric.
First, in accordance with the description of how to make copies [vol. ii., p. 104], make the resistances of the two returns equal. Next, make the inductances due to the magnetic field in the space between wires and returns equal, by proper distance of returns, or by inducto- meters in sequence with sides 3 and 4. There is now left only the wires themselves to be equalised. First, their steady resistances require to be equal. Next, their steady inductances (^/x x length). These two conditions will give balance to infinitely slow variations of current, and can be satisfied with wires of all sorts of sizes and lengths. But we require to make them balance during rapid variations of any kind. For instance, a very short impulse will cause a mere surface current in the wires, that is, in appreciable strength, if they be thick; and still the wires must balance. The full balance is secured by a third condition, viz., that the time-constants of diffusion shall be equal. This time-constant is fikirc2, where /x is the inductivity, k the conductivity, and c the radius of a wire. Or, [d/R, the quotient of the inductivity by the resistance per unit length (or any multiple that we may find convenient of this quotient).
Thus, if the iron has inductivity 100, that of copper being 1, whilst k for copper is about six times the value for iron, the copper wire must have a radius of about four times that of the iron. This is indispensable. Fixing thus the relative diameters, the rest is easy, by properly choosing the lengths. In a similar manner, we may have the resistances in any proportion; as, for instance, to obviate the necessity of having wires of very different lengths, keeping, however, the proper ratio of diameters.
The following will be more satisfactory as a demonstration. If Z is the V/C operator, then ZJZ2 = ZJZ± is the condition of balance [vol. II., p. 104]. So we have merely to examine the form of the Z oi & straight wire. This is [vol. II., p. 63].
Z=L0p + Rf, (31c)
where / is the operator given by
/=82 = - <32c) JiKsc)
L0 is the inductance other than that due to the wire itself, and R is its steady resistance. Using this form of Z in our general equation of balance, we see that if we take s3c3 = s4c4, that is, make the diffusion time-constants equal, we make /3 = /4, so that the balance is given by
= = rj (33c)
-^4 -^04 r4
where the additional r3 and r4 are for the two return-sheaths, or other resistances that may be in sides 3 and 4. Of course Z1 and Z2 may be Rx and R2, the resistances of sides 1 and 2, when they are mere resistances. In virtue of the equality of the diffusion time-constants, we may express the full conditions by adding to (33c) this:—
-Gjp, (34c)
where l3 and Z4 are the lengths of the two wires.
Although this balance is true, yet there will be one practical difficulty in the way. As is very easily shown by sliding a coil along an iron wire or rod, the inductivity often varies from place to place. But if the wire be made homogeneous, the evil is cured.
Next, let it be required to balance a long iron against a long copper rod in long magnetising solenoids forming sides 3 and 4. Here the form of Z for the circuit of the solenoid is
Z=R + L0p + Lpf~ \ (35c)
where R is the total resistance (as ordinarily understood) of the circuit of the solenoid, L0 the total inductance ditto, due to the magnetic field everywhere except in the core, L that due to the core itself when the field is steady, and / as before, in (32c).
To balance the iron against the copper we therefore require, first, the equality of time-constants of diffusion, or the iron rod should be one-fourth the radius of the copper; this being done,
_ R _ A)3 _ Ls
^2 A L, v '
will complete the balance. The value of L (i.e., Ls or Z4) is
L = (2ircN)2nl, (37c)
if N is the number of turns per unit length, and I the length of the solenoid. As for L0, that is adjustible ad lib. nearly. The only failure will be due to want of homogeneity.
Lastly, balance two rods, one of iron, the other of copper, against one another in Felici’s arrangement, when each pair of coils consists of long coaxial solenoids, making two primaries and two secondaries, properly connected together. Let Rv R2 be the total resistances of the primary and tlie secondary circuits; Lov Z02, the total inductances, not counting the parts due to cores; M0 the total mutual inductance, not counting the parts due to cores; Lv Z2, and M those parts of the inductances, self and mutual, of the first pair of coils, due to the cores; and lv l2, m the same for the second pair. The equations of E.M.F. in the primary and secondary are then, if F and / are the two core-operators, as per (32c), and Cv C2 the primary and secondary currents,
e = RXCX + LolpCx + M0pC2 + F~lp(LxCx + MC2) +f~1p(llCl + /%Qc\
0 = R2C2 + L02pC2 + MQpCx + F~'p(L2C2 + MCX) +f~1P(l2C2+mCJ.f K 1
The first terms on the right are the e.m.f.’s used in the solenoid circuits against their resistance; the two following terms taken negatively the KM.F.’s of induction not counting cores; and the last two taken negatively those due to the cores. To have a balance, C2 must vanish. The second equation then gives
M0 + MF~l + mf-1 = 0 (39c)
So MQ = 0, or the mutual inductance of the circuits due to other causes than the cores, must vanish. Then, further,
F=f, and M= -m (40c)
So the diffusion time-constants of the cores must be equal, and the steady mutual inductance of one pair be cancelled by that of the other pair of coils, so far as depends on the cores, as well, as before said, as depends on the rest of the system. (When not counting cores is spoken of, it is not meant that air must be substituted. Nothing must be substituted.) The latter part is capable of external balancing. The balancing of the former part requires the value of
M=(27rc)2N1N2fxl, (41c)
where J\\, N2 are the turns per unit length in the two coils of a transformer of length I, to be the same for the two transformers.
The condition (39c) of course makes the primary equation independent of the secondary. It is then the same as if the secondary coils were removed.
This leads us to show the modification made in the equation of a transformer by the conductivity of its core. In (38c) we have merely to ignore the/terms, thus confining ourselves to one transformer, when the equations are given by the first lines. Now if the solenoids be of small depth, and there be no L external^, L01 and L02 become insignificant, and also M0, provided the cores fill the coils. We have then
e = + F~lp(L1C1 + MC2y 0 = R2C2 + F~1p(L2C2 + MCl
which only differ from the equations when cores are non-conducting by the introduction of F. The first approximation to F is unity (when very slow variations take place). It may be written thus:—
F~l = A-Bp, (43c)
when A and B are positive functions of p2, whose initial values are A = 1, B = 0. When the impressed force is sinusoidal, p2 = - n2, and A and B are constants. Then (42c) become
e = R& + (J+ MC2)Bn2 + Ap(LxCx + MC2\\ /A . x 0 = B2C,+(i2C2+MC\)Bn2 + Ap(L,C2+Mc\)) (i4c)
From these, by elimination, we have
e p T 7? Bn2(R2 + L2Bn2) + A2L2n2 + AR2p x
c=R (Vw*+(w ’ ( )
showing the effective resistance and inductance of the primary as modified by the secondary and conducting core.
But it is very easy with iron cores, without excessive frequency, to make simpler formulas suit. Let z = irc2k[m; then, if this is 10 or over [see vol. II., p. 99], we have
A=Bn=(2z)~l (46c)
approximately, which may be used in (44c), (45c) at once.
In an iron rod of only 1 cm. radius, and [x=!00, k =1/10,000, the value of z is one-fifth of the frequency. If of 10 cm. radius, it equals twenty times the frequency. With large values of z we have
e =]{ 4- hOi l RjliP (47c)
Cx JH- 2l%n2{R2(R2 + 2l2n)}~1 + {R2 + l2n)2 + Ifr* v 7
if Lv L2, and M, when divided by (2z)h, become lv Z2, and m. This gives the primary current. And
£■=-(+ Rih + RJfx\ + (48c)
C2 V 2mn M J 2mnv
gives the secondary current.
We can predict beforehand what these should lead to ultimately, from the general property that a secondary circuit, at sufficiently high frequencies, shuts out induction, or tends to bring L2C2 + MCX to zero, giving the ratio of the currents at every moment. The coefficients of p in (47c) and (48c) tend to zero, and the current in the primary to be the same as if its resistance were increased by the amount Jff2Z1/i2. The core need not be solid. A cylinder will do as well, since the magnetisation does not penetrate deep. It should, however, be remembered that although at low frequencies it is the core that contributes the greater part of the inductance, so that the rest is then negligible, yet when that due to the core actually becomes negligible, the rest becomes relatively important, and should therefore be allowed for.