an ampere meter for measuring alternating currents of very...

41

An Ampere Meter for measuring Alternating Currents of Very HighFrequency.

By E. B. Moullin .

(Communicated by R. V. Southwell, F.R.S.—Received May 21, 1928.)

Introduction.The measurement of alternating currents of very high frequency in general

presents great difficulty because all existing ammeters have a frequency correction which cannot be calculated. The dynamometer ammeter of ordinary construction is very satisfactory for currents whose frequency does not exceed a few hundred cycles per second, but is unsuitable for currents whose frequency is many kilocycles because the highly inductive winding presents an enormous impedance to the current and its presence in a circuit modifies completely the conditions obtaining therein. Further, the distribution of current over the cross section of the wire and the value of the current from turn to turn will alter with the frequency, with the result that a steady current calibration becomes invalid and the correcting factor cannot be calculated or even estimated roughly. So even if its presence can be tolerated in a circuit, such an instrument can be used only for relative measurements at one frequency and its indications cannot be reduced to absolute measure.

The ammeters in general use are thermal instruments depending on the thermal expansion of a suitable element or on the production of a thermoelectric E.M.F. in a junction placed close to a wire heated by the high frequency current. In either system the resistance of the heated element depends on the frequency of the current which heats it and so a steady current calibration cannot be used indiscriminately. The necessary configuration of the heated element and its situation with respect to surrounding alternating magnetic fields usually renders impossible the calculation of its resistance. I t is usual to make the heated wire of a high resistance material and with a small diameter so as to render the calibration sensibly constant up to a high value of frequency. If the calculated resistance of such a straight isolated wire at an assigned frequency has increased by only a small fraction of 1 per cent, above the steady current value, then it will seem reasonable to suppose that the calibration of the bent and unisolated heated wire is valid to at least 1 per cent, up to this frequency. For higher frequencies there will be a correction term whose value can be estimated only very roughly. To maintain

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42 E. B. Moullin.

the calibration, valid np to a frequency of some thousand kilocycles per second the heating wire must be so fine that it will not carry a current of more than, say, 1 ampere. To measure larger currents we are faced by the problem of providing a shunt which the shunting ratio is independent of frequency. Brief consideration will show it is very difficult to arrange a group of fine parallel wires so that each has precisely the same resistance, and further that the situation of each one is the same with respect to all the others and also the remainder of the circuit. If both conditions are not fulfilled the total current will not always divide equally among the component parallel paths and the calibration curve will be subject to a frequency correction. A common and successful method is to arrange the parallel wires as generators of a cylinder, double cone, or hyperboloid and to allow each wire to heat one of a group of thermocouples connected in series electrically.

Another method of shunting a fine wire thermal ammeter is by means of a current transformer with a closed iron core. This method has recently been developed successfully, and it is now possible to measure currents up to about 1000 amperes in value and having a frequency up to a few hundred kilocycles with an accuracy which probably is considerable.

Very recent developments in radio communication have more or less suddenly extended the important spectrum of electric currents from 1000 kilocycles to 30,000 kilocycles (A = 10 m.), and over this extended range of frequency the accuracy of ammeters at present is very doubtful.

The author has attempted to produce an ammeter of a form which can carry, unshunted, currents up to several hundred amperes and in which the frequency error can always be calculated. He will now describe the conception, analysis and constructional development of this instrument.

(2) Elementary System oj the .Consider two long straight solid circular cylinders of diameter d placed parallel

to one another at a distance apart D, and let them carry the same current oppositely directed, as shown in section in fig. 1.

The two cylinders will repel one another, and if the currents i are distributed uniformly over the cross section, the repulsive force F per unit length is given by the equation F = 2f2/D. If the current is alternating and of root mean square value I, the mean value of the repulsive force is given by F = 2I2/D.

If we can measure the mean force we have a measure of the mean square current flowing, and the system is an ammeter which can be calibrated by steady current and used to measure alternating current. A convenient method of



measuring the force is to observe the spread of the distance D when the movement of the cylinders is resisted by an elastic constraint. Let one cylinder

Measuring Alternating Currents o f Very High Frequency. 43

Fig. 1.

be fixed, and let a force F move the other wire through a distance x against an elastic constraint of value ;jl per unit displacement. Then

[ix = 2I2/(D + x), therefore [ix2 -j- pDa; — 2I2 = 0, therefore

x = H a/D 2 + 8I2/p - D}.The relation between x and I is a hyperbola which will not differ much from

the parabola x = 2I2/{jlD, if x is always small compared with D.In the mechanical development of this instrument clearly it would be unsuit

able to observe the movement of the cylinder by unaided vision or sufficiently to magnify the movement by a lever system. The movement should be observed by a high power optical microscope, and then it is convenient to restrict the greatest value of xto a small fraction of a millimetre. If the cylinders are thin wires it will be desirable to make D several millimetres and then the calibration curve will tend to be parabolic.

This ammeter will have a frequency error because the alternating current will not distribute itself uniformly over the cross section, and because the capacity between the two cylinders will cause the current to alter along the length of the parallel system. In reality the system will be complicated still further by the magnetic and electrostatic relationship of the system to external objects and fields. We shall consider each cause of frequency error separately and in the order stated above.

(3) Change in Force due to Ununiform Current Distribution.When the current is alternating the distribution will depart from uniformity

and from axial symmetry, but since tWo long parallel equal circular cylinders



44 E. B. Moullin.

form a system amenable to exact mathematical treatment, the alteration of force may be calculated. I t is illustrative to regard the changing distribution of current as arising from two causes. A distribution having axial symmetry would arise from the magnetic field within the wires and tend ultimately to a tubular current: this distribution having axial symmetry will not alter the force, which remains the same as if all the current was concentrated along the axes of the cylinders. But the external field of one wire will penetrate the other wire, and generate therein an eddy current system having symmetry about the diameter perpendicular to the line of centres and directed a \ every instant so as to tend to neutralise the penetrating field. Therefore the eddy current in the hemicylinder nearer the other wire will tend to have always the same direction as the main current in that wire, and the additional repulsive force arising therefrom will more than counterbalance the attraction from the opposite eddy current in the further hemicylinder.

We may obtain the expression for the force in the limiting condition, where the frequency tends to infinity, in a very simple manner. For in the limiting condition, there will be no flux inside the wire and the whole current will be concentrated on the surface : and in these circumstances the circumference of each wire must coincide with a line of force of the resultant field.

Now it follows readily that equal circles centred at A and B, fig. 1, will be lines of force for equal and oppositely directed currents centred at A' and B', where A' and B' are mutually inverse points for both circles. Hence it follows that these currents produce a field outside these circles which is similar to that produced by the surface distribution we have postulated. Hence ultimately we have

F = 2I2/A'B'.

I t can be shown that A'B' = \ /D 2 — d2, therefore

F = 2I2/\ /D 2 — d in the limit. (1)

We may notice that the ultimate increase of force is small so long as D is considerably larger than d.

An expression for the force F at any frequency n may be derived. Writing z npd2/p, where p = 2nn and p is the specific resistance of the material,it may be shown that




where roughly — 4>i — z4/48 and — </>2 = z4/384 for values of z <C 1*5 and — </>x = 1 — a/2 /z and — </>2 = 1 — 2 \ / 2 /z within 5 per cent, for values of z > 3-5.*

Thusz4F = 2I2{ l + l ^ , 4

D \ ^ 96 D2— — with z <1-5 ,3072D4 J

and

F =_ 2 L 2/ + -D 2 D2 ‘ °

3 # 1 d f d2\ 18 D4 V2z D2 + D2'

with z > 3-5.

Formula (2) shows that the limiting value of F is approached very quickly so long as D/d is appreciably greater than unity. In Table I below are shown collected the value of F 0 and F^ and the value of F when z = 10, for various values of D/d.

Table I.

D Id. F2 = 0 .

Fz = 10.

F2 = 00 . ®oo /Fio*

3/2 2 P/D 1-296 F0 1-34 F0 1-0322 2 P/D 1-13 F0 1-15F 0 1-023 2 P/D 1 -0476 F0 1-06 F0 1-014 2 P/D 1-026 F0 1-031 F0 1-005

We may see from this table that so long as D/d is greater than 3 the limiting increase of F is less than 6 per cent., and that in all cases shown the limiting value has sensibly been reached when z = 10.

For small current ammeters it is desirable to use thin wires of copper or tubes of aluminium spaced well apart and then the correcting factor is small in value and does not become appreciable until the frequency is very high. For example, Table II shows the correcting factor at various frequencies for copper wires 1 • 1 mm. diameter and placed 4 mm. apart.

Table II.

Wave-length A or A = 104 m A = 3000 to A = 600 to A = 300 to A = 30 tofrequency n. 3 X 104. 105. 5 X 105. 106. 107.

Correcting factor for copper wires ........... 1-012 1-024 1-031 1-034 1-04

* Values of fa and may be found from * Phil. Mag.,’ vol. 19, p. 49 (1910), and‘ Phil. Trans.,’ A, vol. 222, p. 64 (1921).

The author is indebted to Mr. R. R. M. Mallock for these expressions.



46 E. B. Moullin.

Difficulties of mechanical construction make it desirable to replace at least one cylinder by a thin-walled tube. An exact expression for the force between hollow tubes or between a tube and a cylinder has not yet been derived because the analysis is extremely cumbersome and laborious. But the limiting value of the force cannot be affected by the absence of the solid core, so that for very high frequencies expression (1) is applicable to a tube. But it still remains to investigate the rate of approach to the limiting value. Consider a thin tube of mean radius B and thickness t and specific resistance p, which is in the field of a line current I sin pt situated at B as in fig. 2. Then if the field inside

F ig. 2.

the cylinder due to the eddy currents in the cylinder be ignored, it follows that the eddy currents induced by the current at B will be in phase quadrature with this current, and then the mean force on the cylinder will be zero. If AB is large compared with R, the force on the cylinder will be affected more by change of phase of the eddy currents than by their change of distribution round the cylinder. Consequently the mean value of the force must depend largely on a factor of the form p2R2(p2R2 -f- ocp2/f2).

—It follows from elementary analysis that the current density

g is given by

= M ±

and by

g — ^ —[ j-j) cos nQ cos pt, for very low frequencies,

g — 2 | cos nh sin pi, for infinite frequency. J




Let (f>0 be the potential outside the cylinder of the eddy currents in the cylinder. Then it may be shown that*

# ckt v' /’R2\w n cos pt 4- m sin . A+ •= 2ml ?(d J » ( * + * > ■sin

where m = 2npiRJp.Whence it follows that H the field at B is given by

1 _ tt _ 2ml f, /Rf\w cos pt m sin ptD \0 ¥ '0 = o~~ D i D2/ (m2 + n2)

C- p HI sin pt . dt I2 ^ /'It2x-n m2J0 2t z/pD i \D2 m2 + *

If the tube carries a current I sin pt, then the total force is

„ I2 L . m2 B2 , m2 R4) . , nVvS uF = D l 1 + l T ^ 2 55 + 4 + ^ 2 D‘ f appr°X- l£ D » E -

This expression gives the force between a line current and a tube, but needs further consideration before it may be applied to calculate the force between a solid cylinder and a tube. We will consider only frequencies which make z for the solid cylinder greater than, say, 10. Then the surface of the copper cylinder cannot differ appreciably from a line of force, and the field of all the current in it can differ inappreciably from that of a line current centred at an inverse point such as B' in fig. 1. Therefore in fig. 2 the solid cylinder must be supposed centred at a point 0 on AB produced such that OB . OA = R2.

We may best explain the process of calculating the force between a solid cylinder and a thin tube by means of a numerical example which refers to a specific instrument to be described later.

The solid cylinder is of copper 1 • 1 mm. diameter, and the thin tube is of aluminium 1*1 mm. external diameter and 0-025 mm. thick. When the frequency is 106 cycles, z for the solid copper cylinder is 11-8, and so for all greater frequencies we shall suppose the current in it may be considered concentrated at the point B' of fig. 1. The cylinders are mounted so that their centre line distance is 4-0 mm. Hence the distance D in fig. 2 is approximately 4 (1 — R2/D2) = 4 X 0-98 mm., whence R2/D2= l/5 0 . For the aluminium tube of mean radius 0-537 mm., we find that = 1 • 79n/10°. Hence may be calculated the following table :—

* See ‘ Camb. Phil. Soc. Proc.,’ vol. 23, p. 902 (1927), F. W. Carter.



48 E. B. Moullin.

Table III.

n. 5 X 10s. 104 * 6 *. 2 X 106. 3 X 106. 4 X 106. Infinite.

F„/F0 1-024 1-035 1-038 1-039 1-04 1-04

On comparing Table II and Table I I I we see that the one tubular conductor retards slightly the attainment of the limiting correcting factor, but to the accuracy of the calculation the two tables do not differ for frequencies above 1000 kilocycles, and then we may consider the limiting correcting factor of 4 per cent, has sensibly been reached.

We may approach the tube problem in another way. When the frequency is very high the current density decreases inwards from the surface of a solid in a manner which is sensibly exponential, and therefore the current may be said sensibly to be concentrated in a surface skin of finite thickness. If the depth of this skin is small compared with the radius of the cylinder, then the rate of penetration will differ insensibly from that appropriate to an infinite flat plate. If the density at the surface of such a plate is as, then it is well known that a — <ysz~mx cos (pt — mx), where m2 = Therefore at adepth 4/m from the surface the current density will have fallen to about 2 per cent, of its surface value, and so we say that the depth of penetration is about 4/m. Thus if t is the thickness of a tube, we may reasonably suppose that the limiting value of the force has sensibly been reached when t = 4/m, which is when z — 2\ / 2 d/t. If we apply this criterion to the aluminium tube specified previously we conclude that the limiting force has been reached when n — 108, but our more detailed work has shown it has been reached much sooner, namely, when n — 3 X 106. However, the simple criterion z = 2 \/2 d]t would find useful application to thick tubes.

We have now found how to calculate the change of force due to non-uniform current density round the cylinders. For the value of DJd contemplated it is always less than 4 per cent.

We will now investigate the change of force which arises from the current distribution along the parallel wire system.

(4) Change of Calibration due to Current Distribution along the ParallelWire System.

We are contemplating two long parallel wires at a distance apart which issmall compared with their length ; for example, two wires about 4 mm. apart



Measuring Alternating Currents Very High Frequency. 49

and each having a length of from 100 to 150 mm. We may apply with confidence to this system the analysis appropriate to a long uniform telegraph line and, following Kelvin, Heaviside and others, suppose the capacity and inductance per unit length are constant. In fact, we suppose the distribution of electric lines along any elementary section differs inappreciably from the distribution which would obtain if the whole of each cylinder were an equipotential surface, and we suppose the distribution of flux threading any elementary section differs inappreciably from that which would obtain if the current were the same at each point of the system. The system is depicted diagrammatically in fig. 3. Let the capacity and inductance per unit length be respectively C

Fig. 3.

and L, and let the potential of and the current at a point on the upper wire distant x, from the origin A, be v and i respectively. If the system is isolated in free space, then from symmetry, these quantities will be — v and — i at the similar point on the lower wire. I t is well known that the current distribution is little affected by resistance except when the frequency is close to a natural frequency of the system. The complete analysis is well known, and since we do not contemplate using the system near a natural frequency wejwill not complicate and obscure the main issue by taking account of resistance whose presence can make only a second order correction.

It follows readily that

therefore§ f i _ Tf<§li = „2?53a;2 3(2 3(2'

Let the current entering at A be — I cos and leaving at B be I cos pt. When x — l ..., v — 0, therefore di/dx — 0.Whence

i — — I (cos p<x.x + tan pa l. sin pax) cos pt.

= - C08 pt.cos pod

(3)

(4)

VOL. CXXI.— A. E



50 E. B. Moullin.

Similarly it follows that

I sin po ( —cos

. sin pt. (5)

Let the surrounding medium have unit permeability and dielectric constant k, and let c be the ratio of the electromagnetic to the electrostatic unit of current or charge. Then for a parallel cylindrical system we have LC = 2.

The force per unit length between the cylinders is a repulsion of value 2t2/V D 2—d2 due to the magnetic field and also an attraction of value 2g2/Ar\/D2 — d2 due to the electrostatic field.

Hence the electrostatic attraction Fe is given by

_ 2 . 4C2i,2c2 _ 2I2 sin2 —-e ~ kVv2- d 2 ~ \/W 2 - d2 cos2 pol ' &m

Hence we have for the mean total force

F =a/D 2

cos2 po. (I — x) — sin2 pa ( — x) cos2 pol

_ I2 sin 2 pol I tan polV i)2 — d2 2po cos2 pol A-/D2 — d2 pol

(6)

(6a )

approx., if pol < < 1.If pod = 7t/2, there is a current node at A and B and the entering current is

zero if the resistance is zero ; the force expressed in terms of the entering current must then clearly be infinite, as equation (6) shows. When pol tc/4 and tc/6 respectively we find by evaluating (6) that the total force is respectively 28 per cent, and 10 per bent, greater than for the same entering current of zero frequency. Equation (6a) may be interpreted better in terms of wavelength than of frequency. When nol = rc/2, the length of the system equals a quarter wave-length of current; since the system is immersed in a •dielectric of constant k the length of the current waves on the system is thereby made 1/A: of the length they would have in a dielectric of constant unity. Let the wave-length in free space, corresponding to a frequency n — p /2tc, be X. Then we may rewrite (6a ) as follows :—

F =a/ d 2 ( l + ^ f t

7l*kW\ (6b)



We are contemplating here a system whose length is about 15 cm. and immersed in a dielectric of constant 2. Then from (6a ) the force will be 2-2 per cent, greater than for zero frequency when the wave-length is 7 • 5 m., and the frequency is 40 million cycles per second.

We may emphasise in passing that the correction for longitudinal current distribution does not depend on the capacity of the system except in respect of the dielectric constant. Therefore it is advantageous to place the cylinders close together so as to reduce the inductance of the system, and this arrangement also has the advantage of yielding a large force for a given current.

Though we have presumed the instrument is to be used to measure currents whose wave-length is much longer than the system, yet it is still applicable when the length of the system is many wave-lengths of current. To obtain the proper correcting factor it is then advisable to include the resistance term in the analysis because attenuation may be sufficient to reduce appreciably the amplitude of successive current loops.

(5) The Effect of External Surrounding Objects and Fields.Hitherto we have supposed the system isolated in space and far from all

other conducting bodies or alternating fields ; of course such a condition is unrealisable. The magnetic field of the loop will induce currents in neighbouring conductors, and these currents will modify the force between the cylinders. Also the cylinders will be situated in the magnetic field of the remainder of the circuit and this again will modify the force. Further, each cylinder will have capacity to earth and to other neighbouring bodies, and the conditions imposed in section (4) will be violated. All these circumstances would produce errors whose magnitude could not be calculated, and uncertainty as to their importance would reduce largely the value of the instrument. Since the effect of external conditions cannot be prevented it is necessary to impose external conditions whose effect is constant and calculable precisely. This can be done by enclosing the cylinders inside a long thick copper tube and placing them symmetrically and parallel to the axis as shown in fig. 4. The magnetic field of the two cylinders will then induce eddy currents in the thick copper shield tube which will have a value in the limit such that there is no external resultant field at any point, and then the cylinders cannot induce currents in conductors outside the shield tube. Similarly external magnetic fields will induce, in the shield tube, eddy currents whose value in the limit is such that there is no resultant field at any point inside the tube, and then the force between the cylinders is not altered by external magnetic fields. The




52 E. B. Moullin.

Fig. 4.

eddy currents which the cylinders induce in the shield tube will modify the force between the cylinders, but we may calculate the limiting value (which is

approached rapidly at high frequencies) of this effect very readily, provided we make R so large compared with d that we may ignore the non-uniform distribution of current.round the cylinders and regard them simply as line currents. The surface distribution of current on the inside surface of the screen tube must be such that there is no external resultant field : in this condition the circumference of the screen tube is a bounding line of force of the internal field. Now the circle of radius R will be a line of magnetic force for equal and opposite line currents situated at B, and at B' the inverse of B ; similarly for currents at .A, and at A' the inverse of A. Hence the field within the cylinder is the same as that produced by currents -j- I at A and — I at B, together with image currents — I at A' and -j- I at B \

The resultant field H at B is given by

H = 21 (1/D — 1/A'B — 1/B'B).Hence

212 / 4 OA . 0A'\D \ ' A'B . B'B /

2P j 1 __________AR2________ \D 1 (OA + OA') (OA'- OA) j

_ 2I2 f 4R2«2 )D 1 R4 - a4 J (V

1 hus the effect of the screen is to reduce the force between the cylinders, and therefore tends to counteract the increase due to the non-uniform distribution round the cylinders.




We will now investigate the rate of approach to the limiting value of the effect of the screen on the force between the cylinders, and also on the screening effect from external fields. If fa is the potential within the screen tube, of thickness t, of the eddy currents in the screen, it may be shown (see ‘ Camb. Phil. Soc.,’ Carter, ibid.) that

therefore

therefore

f a =n cos pt + sin pt . n

»(«* + . ) ■ Sm M9’

i • j j iwhere n is odd and to — — *— ,P

r — a

mean F = 41 * ^ to2D" i \R2/ m2 + w2’

In a particular instrument to be described later, the screen tube is of copper 40 mm. radius and 3 mm. thick, and D = 2a = 4 mm. Whence we find that m — 0-0Zn. Therefore

tti I2 L to2 11F = v W T J , l 1 - x ioolapprox-

Hence the limiting reduction of force is 1 *0 per cent., and has sensibly been attained when to = 4 which is when n — 135 cycles per second. If the screen is situated in a uniform alternating field H sin pt, then it may be shown that the eddy currents produce within the screen a field given by

H? = ----- — —- sin + — cos pt).n r + 1 V to /

The penetrating field will arise from the inductance coils completing the ammeter circuit, and will be in phase with the current in the cylinders forming the ammeter ; therefore the quadrature term of the eddy current field will produce no net force on the cylinders and so the net penetrating field is in effect reduced by the screen in the ratio 1/(1 -j- to2).

We will now consider the capacity effect of the screen tube. Since the cylinders are enclosed completely inside an earthed screen they can have no capacity to external objects nor can external fields penetrate the screen. First suppose the far end of the loop is connected to the earthed screen tube as shown in fig. 5 ; and that the potential of A is Y sin pt and that of B is —V sin pt. With this symmetrical arrangement we have not altered the system of fig. 3



54 E. B. Moullin.

except slightly to increase the capacity and decrease the inductance by corresponding amounts which are readily calculable. The longitudinal current

Fig. >.

distribution will be unaltered, and as before the total force will not be increased by 1 per cent, so long as the wave-length is greater than 50 But since A and B are connected to a circuit which inevitably will have large earth capacity it would be impossible to arrange that the potentials of A and B were always equally above and below the screen potential. If the ammeter is used to measure high frequency currents it will be disposed at a point in the circuit such that the potential of either A or B is zero. We will suppose that of B is zero, and then the current entering at A will leave by the point D, and the wire BD will carry a current only because a current is induced in it by the magnetic field of the current in AD. We will suppose the resistance of the circuit through BD and the case may be ignored, then the whole of the wire BD is at zero potential and must therefore carry sufficient current to make the flux zero which passes between BD and the screen. Let a flux L per unit length pass between wire AD and the screen when AD carries unit current which returns by the case and is disposed so as to produce no external field, and let a portion M of this flux pass between wire BD and the case. Then since the wires are placed symmetrically on each side of the axis, we have at every instant

i1M = - * 2L.

The flux (f> passing between wire AD and the screen is given by<f> = i1L + i2M = ix (L2 - M2)/L.

Hence the wire AD and the case, in the presence of the wire BD, behave as a uniform line having inductance (L2 — M2)/L per unit length and whose capacity is the reciprocal of this expression.

It is well known that

L = 2 log, ? (l — — _ y2) approx., if R > > and R > > r,



and it follows readily that.M = 2]ogjR -+ (R2 +

8 2a ( 2 a 2)

Measuring Alternating Currents Very High Frequency. 55

2 loge — approx., if R > > and R > > LaTherefore

LM

log.(R/r) log.; (R/2a)'

In the example considered R = 38, r and 2.Whence L/M = 2, and i2 = and F = 2 ^ /D = Irt2/D.Hence if the point B becomes in effect connected to the case by virtue of

the large earth capacity associated with the point B, then the force would be halved in the limiting condition for the example considered. Therefore it is quite unsafe to attach the point D to the screen tube and the only safe method is permanently to connect B to the screen. But now corresponding points on the two cylinders will have different potentials with respect to the screen and also the current will not be the same at corresponding points ; the problem is complicated and must be solved by full analysis.

(6) Analysis of Two Cylinders placed Symmetrically within EarthedCylindrical Screen.

Consider the system depicted in fig. 6 where the potential and current at a point A, distant x from the origin, on the lower wire are and ix respectively, and at the corresponding point on the upper wire are v% and i2. Consider the

777777T7 ; 7ffC V 7777777T 77r7 7 77777777777^777,

Fig. 6.



56 E. B. Moullin.

flux contained between two transverse planes perpendicular to the axis and placed a distance 8x apart and having traces HEAD and GFBC in the figure. Let Ku , K]2, K22 and K21 be the capacity coefficients per unit length for the cylinders. Then Ku = K22 from symmetry and K21 = K12 from general properties of these coefficients. Let qxbe the charge on AB at time t and q28x be the corresponding charge on EF.

Then

qx8x = + K12v2) Sx and — (K22v2 + K21Vi) 8x.Therefore

- d idx

andK ll | 1 + Kl2^ S c | . + K ^ ,

di2_ __ dv2 | — pdx

dvj

The flux through ABCD = -f- (L^ + Mi2) 8x, where L and M have been defined in the previous section. Therefore

ThereforeE.M.F. round ABCD = - + M ^ ) 8x,

-°P> = L ^ i + M ^-2.cx ct dt

The flux through EFGH = — (L + Mix) 8x,

thereforeE.M.F. round HGFE - Sv2 = - ^ {— (Li2 + Mi j Sx},

ox dt dt

A cross relationship evidently must exist between L, M, C and K since in this system the electric and magnetic lines are everywhere orthogonal: at this stage it is important to discover what this relation is.

We will calculate first the value of C and K. If = v2 then q = —and q = (C -f- K) v. Consider the system of line charges shown in fig. 7 where

Fig. 7.




OA . OA' = OB . OB' = R2, then these charges will make the screen tube of radius R an equipotential

= 2 q (A + t-A— + 1AB + x AA' — x AB' + ’

therefore

o i / 1 , 1 , 1b = 2 ? J, 1 7 + ^ - +

1 \AB + x AA' — x AB' + x

j dx and q = (C + K)

If vi = — v2, then qx = — q2and q — (C — K) This system would berepresented in fig. 7 by interchanging the signs of the charges at B and B'.

Then

fR / 1 1v — 2q \ (-----k ttt—;-----kJr \ x

1 1AB' + x AA' — x AB -f- x

"Whence, by addition,rR / i i \

dx

dx and q = (C — K)

2C 2K

* i 1 1_L - f ---- -—x AA' — x

r __iI \AB' --k % AB + x

To calculate L and M we must place currents -j- I and — I at A and B with image currents at A' and B'.

I t then follows at once that

fR/ 1= 2 L + AA' j

and

thereforeJr \AB + x AB' + '

C/K = — L/M, which is the relation we are seeking.

Collecting the five equations we have, as follows :—

3 ^ 2 __ P ^ ^ 2 I XT

d£ “ dt dt *

01?-| T 0i'-| | -m/f 0 9 0^9a^ = L si! + M s f ’ - a *

L §£a 4. ]\i ^ dt- dt '

( 8)

(9)

( 10)

- C/K = L/M. (8)



58 E. B. Moullin.

Whence we get from (9) that

K di,C Bx

therefore

k a H0

= c_dx

K2Bv1d ’

k a ^c a®

a*a»

c 3z2a*ha*2

c2 - k 2 a2v, , k a2*'C dx . St ' Snd C dx2

&Hdx2

K2 dv2at '

c2 — K2 av

Whence we get from (10) and (11)

and

k . b\ _c a«2 a#2

k _ a \ _ a^>C dx2 dx2

- lv3 / L 3 $ , M <**\

C ' 3(2 h 3(2

dsc. 3 ( 11)

And hence by means of (8)

- — { + 5 (L2 - M2)L \Bx. dt2 J/§ !h10a?2

andE /§ £ , 5 (L2 _ M2) a% | = ras,L ( 0a?2 1 M ( dt2 I 10a?2

- (L2 — M2) ]-' a*2)

— (L2 — M) ^ > M 1 1 dt2 J

Since it is impossible M2 should equal L2, we have therefore,

(12)

a*2therefore

^ + | (L2 - M2 ) ^ = 0 and g f + E B ^ J B Q . 0, 13

c3x2 L <L

8%0 2M2) S = ( ^ . a n d g p

(13)

ISiow equation (13) had the same form as equation (3) and show that each wire, in the presence of the other wire and the screen tube, behaves as if it had a uniform inductance (L2 — M2)/L per unit length and a capacity C per unit length.

Hence

h = (A sin p$x + B cos p$x) cos pt + (D sin p$x + E cos sin ptand

<2 (h sin p$x -f- G cos p$x) cos pt -f- (H sin p$x -f- N cos sin pt. (14)




These equations are perfectly general and are adaptable to a variety of terminal conditions ; for example, the two wires may carry currents from distinct circuits which have no other connecting link. The terminal conditions most interesting to the present discussion are those for which the two wires are connected together at the far end and have one near end connected to the screen tube. Expressed in symbols these conditions are as follows :—

x — 0, v1 = V sin pt, ix — — I cos pt, v2 — 0. x — l, vi — and h — — i%>

ix = (A sin pfix — I cos pfix) cos pt

and

also

— (— sin pfix + I cos p&x) cos pt, since — (^ )' (3 / \ ox . x= o

t KV sin pfix -f- G cos p $ x ) cos pt, sincedx I,x= 0

C pY cos pt,

■ KpY cos pt,

— sin p$l + I cos pfil — — sin + 6 cos pfil, since P P

when x — l. Therefore(C + K) Y tan p$

also— cos p$x -j- I sin pfix — — cos pfix — G sin p$x,P P

since($h\ = i^h )

\dx.L=lCX l x = l ’therefore

therefore

G = I (G — K ) y cotP

21 = — {(C — K) cot p$l — (C + K) tan p$l}.P

Whencey — V / L /, K2\ tan 2

- I ” V C 1 C2/ 1 - K sec 2p$l/C’(15)

andc K/C - sec 2 j

1 — K sec 2pfilJCtherefore,

i , = i («25.yP’.M -_») - K C0!i pM > ) cos P t . (16)cos 2 pfil — K/C



60 E. B. Moullin.

Now3 qx _ _ 33 t

therefore13 (sin pG> (21 — x)-f K sin pfor/C} .

cos 2p$l - K/C 81

After very laborious reduction it may be shown that if the previous equation for i 2 is referred to the origin x = then equation (16) results. Conse quently a single equation suffices to represent the current flowing up one cylinder and back by the other cylinder.

We see also from (15) that Z = 0, when 2 n, 2rc, etc., and Z = ,when cos 2 p$l = K/C.

When Z = 0, the system is in resonance and there is a current node at = it follows that this happens when X = 4?, in contrast with X = 2 for the symmetrical system of figs. 3 or 5. Hence one striking result of earthing the point B in fig. 5 is to halve the lowest frequency at which resonance occurs, and a study of equations (15) and (16) shows that the general effect of earthing the point B is approximately the same as earthing the point D together with doubling the length of the line.

We will now consider the force on the system, and refer to fig. 4 ; a current \ at A and a current — i 2 at B will produce a repulsion given by

Similarly charges qx and — q2 will produce an attraction given by

Since we may express i2 and q2 asr

t a = - I <

cos p(kr — — cos — x)V

cos 2p$l —

V J

cos pt,

and

q2 = ip<




It follows after considerable reduction that we may express F as

ai)

KfC21 \ 2 /*• + rE") cos

BA7,11" -

2KC

/ 0 OJ K \ 2(cos 2 p$l- - )

therefore

F .dx —

J2 C0S 2Pfa C0S P - X)* 2RBB' / K 21 cos 2 pfil-

cos 2 pfi (l —

cos 2p$l — ^C /

l{(1 + ~ )COS 2 p ^ - ^ l / I ------ Ll\ c2/ c / I d ba'

l - 2Ki y- — q— C2~ysin + c1 sin

DD' 2p(3

(17)

By evaluating (17) we may obtain the correcting factor for any assigned value of p. Since it is not intended to use the instrument at frequencies near that which makes cos 2p$l = K/C, we shall derive by expansion an approximate expression in which p3$3l3, etc., are ignored. Then

F . dx = ^ { i 1 )2- 2 p 2p2J2}~2

f( i K \2_ C ! + K 2 | / 1 _ J _ vIV C / C2 F 1 / Vd BA'/’

K \ 2 /^ 'BB'

approx.

m

m

| jl + 2p2lfl2 __C____ C ^ + K 2\/ f lC - K C2

K2\ \ / l 1 ) 1 1K2/ J I d BA' j BB'J appr0X‘

BA' BB'> + k

= I2Z (1 — ----------Vd BA' BB'/

D BA'/ D

1 + 1 +

1 \ C2 — 2CK — K2 (C - K)2

2K2 \8 rcW

2p2 2]

BA'1 —

(C - K)2/ X2n approx., (18)

since BA' is very nearly equal to BB'. Here, as before, Jc is the constant of the dielectric and X is the wave-length in three space corresponding to the frequency n — p/2tc.

In a particular instrument to be considered later, and for which various



62 E. B. Moullin.

relevant dimensions have been given already, D/BA' — 1/200 and K/C - __ 0-53 ; inserting these values we find that

F = ^ x 0-99 j l + |7t2F ^ j . (19)

Comparing this expression with (6b ) we find the effect of the tube has been to make the correcting factor for longitudinal current distribution 12*5 per cent, greater than for an unscreened system of twice the length. We are contemplating a system of length 15 cm. immersed in a dielectric of constant 2. Then from (19) the force will be 5-4 per cent, greater than for zero frequency when the wave-length is 10 m. and the frequency is 30 million cycles ; also the force will not exceed that for zero frequency by as much as 1 per cent, so long as the wave-length is greater than 25 m. and the frequency less than 12 million cycles.

(7) Constructional Details of an Ammeter for Small Currents.The force between the wires is 2I2/D dynes per centimetre of length. We have

been contemplating that D should be about 4 mm., and then the force will amount to 0-2 dyne per centimetre for a current of 2 A. I t is evident from this figure that the elastic control force must be very small in order to obtain a measurable deflection. The system which has been adopted with success may best be described with reference to fig. 8, which shows a longitudinal and transverse section of the instrument. One cylinder of the system is fixed rigidly to an ebonite frame which is fixed to the underside of the cover-plate of the thick copper screen tube. The other cylinder stands on two vertical legs of phosphor-bronze suspension strip which is soldered at one end to the cylinder and at the other end to a copper block fixed to the ebonite frame. These two pieces of strip serve to conduct the current to and from the cylinder, and their stiffness supplies the main part of the elastic restoring force. Current is led to the fixed cylinder by a wire passing through a large ebonite disc fixed to one end of the screen tube. Having passed along the fixed cylinder it passes to the bottom of one of the suspension strip legs and thence through the moving cylinder to a point on the end of the screen tube.

The whole screen tube is filled with paraffin oil which serves a three-fold purpose. The moving cylinder is a hollow aluminium tube with sealed ends. The upward thrust of the oil is much greater than the weight of the tube and in this manner the suspension strips are put in tension and therefore have no tendency to buckle. The oil immersion greatly assists in cooling the suspension strips, where necessarily the current density is very high. Also the




oil immersion damps the motion of the cylinders and renders robust and portable what would otherwise be a very fragile instrument.

To the middle of the moving cylinder is attached a vertical quartz pointer whose free end passes out through the screen tube, and is viewed through a window in the metal chamber fixed to the cover plate of the screen. The pointer magnifies the motion of the moving cylinder about five times. The movement of the pointer is observed by a microscope with a 1-inch objective and 10-inch tube length, and which has an eye-piece scale 1 cm. long divided into 100 parts. The maximum excursion of the moving cylinder is consequently about 0 • 20 mm., which is about 5 per cent, of the distance between the centres of the cylinders.

Since there is a considerable upward force of buoyancy on the moving cylinder it is evident the initial distance between the cylinders will depend on the amount by which



64 E. B. Moullin.

the plane of the ebonite frame departs from the vertical, and we have here a method of setting the initial distance to a standard value. The whole screen tube is carried on horizontal trunnions so that it can be rotated slightly about an axis parallel to the axis of the tube. This slight rotation is produced by the adjusting screw shown in the end section elevation of fig. 8. To adjust the instrument for use the screen is rotated and the microscope pointer cross traversed simultaneously until the centre line of the pointer and that of a fixed fiducial pointer fixed inside the window is brought to lie on the two terminal divisions of the eye-piece scale ; when this adjustment has been effected the cylinders are at their standard distance. Then the microscope is cross traversed until the fixed fiducial pointer has disappeared from view and the quartz pointer has been brought to the zero of the eye-piece scale. The instrument, may then be calibrated with steady current.

Since the copper screen tube offers no impediment to the earth's magnetic field, the steady current calibration will generally be affected by the vertical component of this field. For this reason a small permanent magnet is carried on a jointed arm fixed to the case. The position of this magnet is varied imtil the deflection produced by any given current is independent of its direction, and then the earth’s field has been neutralised and the calibration may be made.

Since the frequency correction depends on d2/J32, it is important to be assured that D is not less than the value intended, more especially as this distance may get altered unintentionally in the process of assembly. The effect of the earth’s vertical field provides a simple method of measuring D after the instrument has been completed. For let the control magnet be removed and a steady current calibration be made with the current passing through the instrument in the direction which gives the smaller deflection for a given current. The current for zero deflection can then be found from this curve ; let it be I0 and the value of the vertical component of field be H. Then D = 2I0/H.

Fig. 9 shows the calibration curve for the instrument described in fig. 8. The eye-piece scale has 100 divisions and the possible accuracy of reading the scale is rather better than 1 per cent, at full deflection ; since the scale is sensibly parabolic this corresponds to an accuracy of ^ per cent, of current. I t has been shown in section 2 that the equation of the deflection curve is a hyperbola differing little from a parabola. The maximum excursion of the moving cylinder is about 5 per cent, of the initial distance between the cylinders, and from this information it is possible to test whether the curve



of fig. 9 fits the hyperbola whose equation is given in section 2. But since the fractional accuracy of reading is poor in the early part of the scale, such

Measuring Alternating Currents of Very High Frequency. 65

*

SCALE BEADING

Fig. 9.

a test is hardly justifiable. If for the curve of fig. 9, the deflection from 10 to 100 is plotted against the square of the current, the result is a straight line which does not pass quite through the origin and would make zero current correspond to a positive deflection of one division. Thus, though strictly speaking the scale is hyperbolic, for practical purposes it may be considered parabolic.

Fig. 10 is a calibration curve with steady current and with the earth’s magnetic field not neutralised; it illustrates the process ofjdetermining I0.

1-4

1-2

1-0!i1

/s0-6

( 0 4 IN

A.

\\ 0-2

h-U

D

\ O

SCALE RE ADING

2 0 10 0 10 2 0 3 0 4 0 50 6 0

Fig. 10.

Fig. 11 is an external view of the complete instrument.The heating of the strips and cylinders tends to set up convection currents

in the paraffin oil, and these tend to increase the deflection. On switching onVOL. cxxi.—A. F



66 E. B. Moullin.

the current the deflection attains rapidly a definite value and then tends slowly to creep an extra two divisions at full scale. The current should he

Fig. 11.

deduced from the deflection which results on switching on and before the slow creep has taken place.

The instrument described has also a small temperature coefficient, the sensitivity tending to increase slightly with rise of temperature ; this may be due in part to the change of density of the paraffin oil. If the instrument is to be used in circumstances where the daily changes of temperature are considerable, it is desirable to calibrate before use. The ammeter here described is probably too sensitive for average requirements, and it is thought that this instrument is probably the most sensitive which can be made in a portable form. In one which would require, say, 5 amperes for full deflection the small inconveniences from convection currents of oil and temperature coefficient probably would disappear. For the control force would then be 25 times as great, while the convection currents and buoyancy forces would remain sensibly the same.




(8) The Effect of Irregularities at the Ends of the System.The arrangement of conductors in fig. 8 departs slightly from the ideal

uniform and symmetrically arranged line postulated in fig. 3, mainly because the suspension strip is not in the same straight line with the two cylinders. We must consider the probable effect this will have on the calculated correction factors. The magnetic field will be modified slightly near the sharp corners and this will make a local alteration of current distribution in the cylinders. But since the total correcting factor for proximity effect is only about 3 per cent., an appreciable alteration of this factor over some 5 per cent, of the total length can make only a second order change in the net result.

The effect on the longitudinal current distribution is a little more difficult to predict. That part of the system which is formed by the suspension strip may be regarded as having uniform inductance and capacity per unit length. I t is true this portion is perpendicular to the axis of the screen, but since the screen is far away we may reasonably ignore the small effect it has in making the inductance differ from point to point. We may reasonably consider the system as one uniform line, formed by the cylinders, joined to another uniform line formed by the suspension strips ; the second line may have slightly different constants per unit length, consequently the effective length of the whole line will be slightly longer or shorter than the actual length ; this will make a slight uncertainty in the value of A/A0, but not one sufficient to make appreciable uncertainty in the correcting factor in cases of practical interest whenA > 50 A0.

I t follows from equation (3) that if the uniform line is connected in series with a second uniform line of length Vand having constants 1/ and C' per unit length, then

i — I cos pa ( l + —— ) / cos (pcd -f- 0),\ pa / /

wheretan 0 = paLT/L.

Since V < l and pod is small, 0 is small and then

I cos pa ( l + —■- V \ L /COS pa ( l + Vapprox.

The effect of the second uniform line is apparently to increase l in the ratio (1 + LT/LZ). Since L and L' vary as the logarithm of the distance between the wires it is not possible for L'/L to depart appreciably from unity. Therefore the effective length of the line cannot differ much from the sum of the



68 E. B. Moullin.

lengths of the aluminium tube and the suspension strips. Thus there is a very small uncertainty in the ratio of J/X, but one which will have a second order effect on the correcting factor in cases of practical interest where X > 50 l.The correcting factor will further be slightly modified because the force between the suspension strips contributes very little to the deflection which arises almost entirely from the force between the aluminium tube and the fixed cylinder.

For this reason we must integrate in formula (23) between — 0 and some value of x which is less than ( l + I').In the particular instrument described l = 10 cm. and V = 6 cm. If this ratio of fV is used, then formula (25) is not altered appreciably except that l inside the brackets must be replaced by l + V.In general the precise value of the correcting factor for longitudinalcurrent distribution is slightly uncertain within very narrow limits, but yet the correction itself is so small even for the shortest wave-length in use at present, that the slight uncertainty is quite unimportant and negligible.

We will now deduce a table of correction factors for the instrument illustrated in fig. 8, and for which the relevant dimensions are as follows : 1*1mm., D = 4 mm., R = 39 mm., I = 10 cm., = 6 cm., and = 2.

The correction factor for the screen tube has been found in section 5 and its limiting value is given by formula (8) and has been found to be 1*0 per cent., which has sensibly been attained when the frequency exceeds 125 cycles per second. The correcting factor due to the proximity of the aluminium tube and solid copper cylinder is shown in Table III, its limiting value is 4 per cent., and has sensibly been attained when the frequency exceeds 2 million cycles per second.

I t may be calculated that K/C = — 0-53, and hence the correcting factor is infinite when 2 ( l-j- V) = 0*68rc, which is when X = 12, ( -f- if = 2. I t will therefore occur for this instrument when X = 1 • 8 m.

Table IV.

Frequency n. 5 X 105. 10*. 107. 2 X 107. 3 X 107.

Wave-length in free aether................ 600 m. 300 m. 30 m. 15 in. 10 m.Correcting factor for screen tube .... 0-99 0-99 0-99 0-99 0-99Correcting factor for cylinders ........ 1025 1-035 1-04 1-04 1-04Correcting factor for longitudinal 1-000 1-000 1-006 1-024 1 -054Net factor ....................................... 1015 1-025 1-036 1-054 1-084

Fig. 12 shows the results of Table IV plotted as a curve giving the fractional amount by which the observed deflection should be decreased before using it.



to determine the value of the current from the steady current calibration curve.

10

8

6

4

2

0 10 2 0 30

Fig. 12.


rRTQUCNCViN M tU tQ N S Of CYCLES P IR SECOND

(9) The Resistance and Reactance of the Ammeter.We shall define the resistance R of the ammeter as that quantity which when

multiplied by the square of the entering current equals the rate of energy dissipation in the system. The resistance will differ from the steady current value because of the distribution of current round the cylinders, and because of the longitudinal current distribution along the cylinders, and because of the eddy currents generated in the screen tube, and because of energy loss in the oil and ebonite dielectric. All except the last item may be calculated.

It may be shown by integrating the expression for a, given in the note in section (3), that the proximity of the cylinders increases their resistance in the ratio D /y 'D 2 — d2, when the frequency is very high.. This fraction amounts to 1*04 for the specific example. (Note.—This expression is well known, see for example Butterworth, ‘ Phil. Trans.,’ vol. 222, p. 72 (1921).) The resistance of an isolated solid circular cylinder is given by the equation

— i V 2 -f- !)•

For a copper wire 1 • 1 mm. diameter at 3 X 107 cycles we find that 95 and so Rft = 24, R0 = 0*5 ohm per metre. The resistance of the aluminium tube at this frequency will be about 0*7 ohm per metre. Hence the total resistance of the tube and cylinder will be about 0-12 ohm. The energy loss in the shield tube may be derived from the expression for given in section (5), and may be shown to be quite negligible. The resistance of the very thin suspension strip will not increase appreciably with frequency. The steady current resistance of the complete instrument is 0*2 ohm, and hence it is clear that the resistance of the strips dominates the whole. The effective



70 E. B. Moullin.

resistance at 10 m. wave-length will be of the order of 0-3 ohm, and then the power expenditure will be 0 • 7 W for a current of 1 • 5 A.

The reactance of the ammeter may be calculated from equation (15) and at 10 m. wave-length will not differ appreciably from the inductive reactance of the system.

I t may be shown that

2 (L — M) 2 =41 log ( l

= 0-128 p,H for the instrument considered.

Hence if n — 3 X 107 cycles, the reactance of the instrument is 24-5 ohms and then it will require 37 volts for a current of 1-5 amperes.

(10) An Ammeter for Large Currents.

I t would be an advantage to reduce the inductance of the ammeter. If the left-hand cylinder in fig. 1 was removed and replaced by an infinite conducting plane with its face passing through 0 and perpendicular to AB, the force on the right-hand cylinder would not be altered but the inductance of the system would be halved. I t will now be impossible to calibrate with steady current and we may have to depend entirely on calculated values. Alternatively we may make the diameter of the remaining cylinder so great that the limiting value of the force has sensibly been reached at frequencies which are low enough to permit of calibration by ordinary alternating current instruments. This cannot readily be done for instruments to measure small currents, because the moving parts must be kept light since the operating forces are very small. In these instruments we wish to depend on a steady current calibration, and it is advisable to arrange the distances so that the correcting factor is small and will therefore not be affected appreciably by small irregularities of construction. But if we can construct the ammeter so that it can be calibrated at moderate frequencies, then there is no longer reason to keep the limiting correcting factor small, but merely to arrange that the limiting value is approached closely at low frequencies. The system described as a modification of fig. 1 is unrealisable in practice but may be approached closely by the system illustrated in fig. 13, if R is made very much larger than r. By replacing the system by appropriate image currents it may be shown that the limiting value F of the force per unit length is given by the equation

F = 2dl2l\/(R2 + — — 4RV.



We will make the system definite by assigning values to this expression. Thus, let R — 8 cm., r — 0-8 cm. and 7*1 cm. Then it will be found

Measuring Alternating Currents o f Very High Frequency. 71»

Fig. 13.

that F = 227 dynes for a current of 100 A. If R has been infinite and (R — D) had remained 0-9 cm., it may be found that the force for 100 A would have been 243 dynes, and thus the curvature of the screen reduces the force by only 7 per cent. Thus we may apply formula (2) with confidence to calculate the approximate value of the correcting factor. On doing this we find the force will be within 5 per cent, of its limiting value when z 7, which is when n = 1600 cycles if the cylinders are of copper. Therefore, by making a calibration curve at various frequencies up to, say, 1000 cycles, it should be possible to infer the correcting factor for all frequencies above a few kilocycles. In the system now proposed the inside cylinder would be connected at one end to the outside return tube, which is also the screen, then in our previous notation A0 = 21.Hence if l = 40 cm., the total force would be about 9000 dynes and the correcting factor for longitudinal current distribution would be 1 per cent, when the frequency was 2'5 X 10° cycles if the cylinder was oil immersed. It may be calculated that the inductance of the system is 0-037 p. H and so would require 58 V to pass 100 A at 2-5 X 106 cycles, and for this current the powei absorbed would be of the order of half a watt. The method of supporting the moving cylinder and of observing the deflection would be of the same general character as described already.



an ampere meter for measuring alternating currents of very...

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