Paper No. 1631
MEASUREMENTS SECTION
621.317.733
BRIDGES WITH COUPLED INDUCTIVE RATIO ARMS
AS PRECISION INSTRUMENTS FOR THE COMPARISON OF
LABORATORY STANDARDS OF RESISTANCE OR CAPACITANCE
By C. W. OATLEY, M.A., M.Sc, Member, and J. G. YATES, M.A., Associate Member.
(The paper was first received 28th August, and in revised form 9th December, 1953.)
SUMMARY
The paper deals with the use of bridges with inductive ratio arms
for the accurate comparison of two resistances or two capacitances
whose impurities are sufficiently small to be neglected in their effect
on the comparison of the major components. The comparison of the
impurities is not considered.
If the transformers used in such bridges were perfect their effective
ratios would be equal to their turns ratios and the bridges would be
completely immune from the effects of stray capacitance to earth. In
practice, the results are modified by imperfect coupling, by the
resistances of the windings, by stray capacitances and by eddy currents
induced in the core. A detailed analysis of all these factors is given
and an estimate is made of the errors they are likely to cause. Finally,
the results of measurements on six transformers, designed in accordance
with the theoretical analysis to give low errors, are presented.
It is concluded that, for ratios up to 100/1, there is no difficulty in
constructing transformers whose effective ratios are equal to their
turns ratios within 1 part in 104 when used in a suitable circuit. For
higher ratios the difficulties increase, but even with 1000/1 the error
need not exceed 1 or 2 parts in 104. Suggestions are put forward for
the construction of transformers in which the errors are still smaller
than those already achieved.
precision. When the two impedances to be compared are equal
no problem arises, since the method of substitution can be used
with any convenient bridge and the final result is then independent of the accuracy with which other components of the
bridge are known. It is often necessary, however, to calibrate a
condenser or resistor against a standard of very different value,
and the accuracy of the measurement is then limited by the
precision with which the values of the ratio arms are known.
There are well-known methods of overcoming this difficulty by
the use of additional condensers or resistors of intermediate
value which can be connected in series or in parallel. Such
methods have been employed in some of the tests described
herein, but they are tedious and inconvenient for general use
even if the necessary components are available.
For work of this kind the use of transformer ratio-arms is
very attractive, since, if the accuracy of the ratio can once be
established for a particular transformer, there is every reason to
expect that it will remain constant with time. It thus becomes
important to investigate how closely, in actual transformers, the
effective ratio can be made equal to the turns ratio.
(2) GENERAL THEORETICAL CONSIDERATIONS
LIST OF PRINCIPAL SYMBOLS
(2.1) Statement of the Problem
Rn — D.C. resistance of the wth coil (i.e. the total resistance
The ideal circuits of the two bridges that have been investiwhen eddy currents are neglected), ohms.
gated are shown in Figs. l(a) and l(b). Similar components are
Ln = Inductance of the nth coil, henrys.
used in the two circuits and these bridges are referred to throughMmn — Mutual inductance of coils m and n, henrys.
out the paper as the A- and B-connections, respectively.
to — Angular frequency, radn/s.
Nn = Number of turns in the nth coil.
P =
N2IN{.
ZUZ2 = Impedances under comparison, ohms.
(1) INTRODUCTION
The valuable properties of a.c. bridges in which two windings
of a transformer are used as ratio arms were first pointed out by
Blumlein,1 who showed that if the coupling between the windings
were perfect the impedance ratio of these arms would be equal
to their turns ratio, and, moreover, the bridge would be independent of stray capacitance to earth even though no Wagner
earth were used. Since that time, bridges of this type have been
developed for a variety of purposes. For example, Kirke2 and
Wilhelm3 have considered their use at high frequencies;
Vanderlyn and Clarke4 have described an extremely flexible
general-purpose bridge in which two transformers are used;
while Watton and Pemberton5 have shown that bridges with
coupled inductive ratio arms are admirably suited to the measurement of extremely small capacitances.
The paper is concerned with the use of such bridges for the
comparison of two resistances or two capacitances with high
Written contributions on papers published without being read at meetings are
invited for consideration with a view to publication.
Mr. Oatley and Mr. Yates are in the Engineering Laboratory, University of Cambridge.
[91
Fig. 1.—Basic bridge circuits.
(a) A-connection.
(6) B-connection.
In practice the coupling between the windings cannot be made
perfect; the windings will have appreciable resistance and there
will be stray capacitances. The effects of these factors are
considered in Sections 2.4, 2.5 and 4.
In practice, also, Zx and Z 2 will be neither pure resistances nor
pure capacitances, but, in the type of work under consideration
and at frequencies in the audio range, the power factor of a
standard capacitor is unlikely to exceed 0 001, and the phase
angle of a standard resistor is of the same order. Hence, within
a few parts in 106, the effective series value of the major component of impedance will be equal to its effective parallel value
and both will be independent of the exact magnitude of the
92
OATLEY AND YATES: BRIDGES WITH COUPLED INDUCTIVE RATIO ARMS AS PRECISION INSTRUMENTS FOR
impurity. Thus, so long as means are provided in the bridge to
compensate the impurity so that a true balance can be obtained,
the effect of the impurity can be neglected.
For precision work at audio frequencies the advantages of a
nickel-iron core are overwhelming. The complications, arising
from the use of ferromagnetic cores are discussed in Section 3,
and the latter part of the paper is devoted to bridges in which
the coils are wound on a closed core of high-permeability alloy.
This result is well known for iron-cored coils in which virtually
the whole of the flux is carried by the iron, but the above proof
is general and applies equally to air-cored coils.
(2.3) Balance Conditions in the Ideal Case
Reverting to the bridge circuits of Fig. 1, assume, for the
moment, that the effects of coil .resistance and of stray capacitances are negligible. Fig. l(o) can then be redrawn in the
2.2) Theorems on Coupling
It will be convenient at the outset to prove three theorems
which will subsequently be needed. The first is well known,
although a proof of it is rarely to be found in standard textbooks.
Let direct currents i{ and i2 be established in two coils in
directions such that the total energy of the system is
If /, be assumed fixed, differentiation shows that W will be a
minimum when i2 = Mx2ixfL2 and it then has the value
Kun-y^-MfJ^)
. . . . (1)
Fig. 2.—Basic circuit equivalent to Fig. l(a).
Since Wmin cannot be negative, it follows that Mx2 cannot be form of a well-known equivalent circuit as in Fig. 2; there is
now no coupling between the coils. The condition for balance is
greater than y/iL^L-^).
This treatment can now be extended to three coils to obtain
'12
various relations between their self- and mutual-inductances.
(4)
L, - M 12
For example, let current ixflowin Lx and, as before, let the current
i2 in L2 have the value Mx2ix\L2 and flow in such a direction that and, for perfect coupling, this becomes
the total energy for coils 1 and 2 has the minimum value given
IL2 N2
Z2 L2 + yU,L 2 )
by eqn. (1).
(5)
L
+
TV,
Next suppose a current i3 to flow in L3 in such a direction
x
that the total energy of the system is
Similarly for the B-connection [Fig. 1 (/>)], it is easy to show that
the balance condition is
W =
~ ixi3(MX3 ~ M23Mx2\L2)
where (MI3 ^ M^M^l^i) is the positive difference between
MX3 and M23Mx2\L2. On differentiating, it is found that, for a
fixed value of /',, ff'is least when
M23MX2\L2)IL3
Z2+ML2
Zx +JOJ(LX -r Ml2)
M
and, for perfect coupling between coils 1 and 2, eqn. (2) shows
that this relation is satisfied when
and then has the value
^ '
Since W^in cannot be negative, it follows that
(6)
x3
=
(7)
" l
(2.4) Effects of Imperfect Coupling between Coils 1 and 2
The ratio A^/A^ is the theoretical ratio of the bridge, since
vVj and 7V2 are quantities which can be given precisely known
Finally, consider the physical conditions under which two values. It has been shown that this theoretical ratio will be
coils with self-inductances L, and L2 and turns Nx and N2, equal to the effective ratio if there is perfect coupling between
respectively, can have perfect coupling so that LXL2 = Mx22. coils 1 and 2, but although this is a sufficient condition it is not a
Let currents ix and i2 flow in the coils in directions such that the necessary one. In fact, a little consideration will show that,
magnetic fields oppose, and let i2 be equal to Mx2iJL2. Then, with either the A- or the B-connection, it will be possible to
since M22 •=• LXL2, eqn. (1) shows that the total energy of the construct coils which satisfy the balance condition Z2\ZX =
system will be zero. But, if H is the magnetic field strength N2\NX even though the coupling between coils 1 and 2 is
and fj, the permeability at an element of volume dv, the total zero. This fact is of considerable importance because it shows
that tests on the accuracy of a bridge with known impedances
energy is given by
Zx and Z 2 cannot by themselves yield conclusive information
dv
about the perfection of coupling.
8TT
Although perfect coupling is theoretically not essential, it is the
where the integration is to be taken throughout all space. Since condition that will always be aimed at in practice, partly to
pH2 cannot be negative, the integral can be zero only if H is achieve freedom from effects of stray capacitance to earth and
zero everywhere. To fulfil this condition the two coils must be partly because, with imperfect coupling, the bridge ratio can
physically coincident and their numbers of turns must be in the be determined only by measurements on another bridge. It is
therefore of interest to investigate how the bridge ratio is affected
inverse ratio of the currents. Thus, for perfect coupling,
by the small departures from perfect coupling which inevitably
arise in practice. It will be convenient to limit the discussion to
• • (3) the case in which the coils consist of solenoids with windings
,V2
L2
(M13 ~ M23M12/L2)2 > (L, - M}2\L2)L3
(2)
THE COMPARISON OF LABORATORY STANDARDS OF RESISTANCE OR CAPACITANCE
of a small number of layers, the two solenoids being of equal
length and having as nearly as possible the same diameter.
The conclusions reached apply with slight modifications to
toroidal windings also, and these two types are the ones most
likely to occur in practice. In the first instance, the presence
of a nickel-iron core is not assumed; the effects of such a core
will be dealt with later.
Let coil 1 consist of a single-layer solenoid; ideally coil 2
should be coincident with coil 1. If this were possible its
inductance would be p2Lu where/? is the turns ratio N2INU and
the mutual inductance would be pL{. In practice these values
cannot be achieved because coil 2 must be wound on top of
coil 1. In consequence of its greater cross-sectional area its
inductance will be increased to p\Lx + 8L) and the mutual
inductance between the coils will be reduced to p{Lx — 8M).
Thus, from eqn. (4), the effective bridge ratio for the A-connection will become
2
P (L{ + 8p + p(Ll
-8M)
L, + p(Lx - 8M)
'
P8L
-
=q.
(8)
So long as the windings have the same geometrical form (number
and spacing of layers, etc.) and the core, if any, has the same
effective permeability, 8L and 8M will both be proportional to
/,, so that the ratio error will be independent of the number of
turns in the primary winding.
A similar expression can be deduced for the B-connection,
but it has few useful applications.
(2.5) Effects produced by Coil Resistances
When the coils have resistance, the balance condition for the
A-connection with perfect coupling (Fig. 2) becomes
R2 + ,/OJ(L2 + Ml2)
Z2
(9)
If it is assumed that Zx and Z 2 are either pure resistances or pure
capacitances, Z2\ZX is a real number and, for balance,
'12
wrp
(10)
JL, + Mx
x2
In practice Z, and Z 2 will rarely have precisely equal phase
angles, and some form of phase adjustment must therefore be
provided in the bridge. Once this has been done it becomes
possible to balance the bridge whether eqn. (10) is satisfied or
not. However, the effective bridge ratio will be equal to p only
if eqn. (10) is satisfied, so it is important to investigate how the
ratio is affected by small deviations of RJR2 from its proper
value.
Let the resistance of coil 2 be R2 + 8R, where R2 satisfies
eqn. (10), and let the effective bridge ratio be q + j8q. For the
uses of the bridge which are here being considered the important
quantity is the deviation of q from the turns ratio p.
Now
R2 + 8R + yo>(£2 + Ml2)
=•- q + j8q
Rx + jio{Lx + Ml2)
(11)
93
In this case the effect of 8R is a property of the transformer
itself and is quite independent of the values of the impedances
Z, and Z 2 which are being compared. With the B-connection
the case is very different, and some assumption must be made
about the nature of Z, and Z 2 . Suppose, first, that they are
nearly pure resistances such that
ZXS
and Z 2 = S(q + j8q)
With the same notation as before, and perfect coupling being
assumed, it is easy to show that the balance conditions are now
J
(13)
= 0
On the other hand, if nearly pure reactances are being compared
so that
Z, ••- jX and Z 2 = jX(q -f j8q),
the balance conditions are
p= q
8R = X8q
(14)
(3) COMPLICATIONS RESULTING FROM THE USE OF A
FERROMAGNETIC CORE
(3.1) Effect on Leakage Flux
In an iron-cored transformer the total flux may be divided
into two parts: that which resides entirely in the iron and which
therefore necessarily links all the turns of all of the coils, and
that which resides wholly or partially in air. Again, for the
present purpose, it is convenient to regard the flux in air as
resulting from two separate agencies; the current in the magnetizing coil and the free magnetic poles on the surface of the
iron, although the free poles are themselves produced by the
current in the coil. This division is instructive because the flux
density at any point resulting directly from current in the coil
can be calculated and measured with precision, while that
resulting from the free poles cannot be calculated and is not
easy to measure. Nevertheless, it is easy to estimate the order
of magnitude of the flux density caused by the free poles (see,
e.g., Ref. 6). The flux density in the iron is nearly constant and
equal to the value that it would have if the magnetizing coil were
uniformly distributed round a toroid of the same material, crosssection and effective path-length as exist in the actual transformer. Thus the magnetic field strength throughout the iron
of the transformer must be nearly equal to that which would
exist in the toroid. At points outside the transformer coil this
field strength is much greater than would be produced by current
in the coil in the absence of free poles, while at points inside the
coil it is considerably less—probably about one-third as great.
Thus it appears that, over the relevant volume of space, the
field strength due to the free poles is likely to be of the same
order of magnitude as that which the current in the coil produces
at points inside the coil. The two fields will aid each other at
some points and oppose at others. Further information concerning them will be derived from measurements described in
Section 6.1.
Equating real and imaginary quantities gives
RX(R2 -f 8R) + o>2(L, + Ml2)(L2
R]
Ml2)2
which, by use of eqn. (10), can be put in the form
r
q- p 1+
8RIR2
M 12 )2//?2
(12)
(3.2) Effect of Eddy Currents in the Core
As a result of eddy currents in the core the effective resistance
of each of the transformer coils will be increased and its effective
inductance decreased. Let the two coils, 1 and 2, in Fig. 3
be perfectly coupled so that their inductances are L, and L2 =
P2LX respectively. Let Rx and R2 be their d.c. resistances.
Suppose one particular filament of eddy current to flow in a
94
OATLEY AND YATES: BRIDGES WITH COUPLED INDUCTIVE RATIO ARMS AS PRECISION INSTRUMENTS FOR
Fig. 3.—Equivalent circuit for a filament of eddy current.
path of resistance R and self-inductance L which has mutual
inductance M with coil 1. Then, by eqn. (2), it will have mutual
inductance pM with coil 2. The changes in inductance and
resistance of coils 1 and 2 brought about by this particular
filament of eddy current may easily be shown to be
AL,= -
M2a)2L
V-Oi2 + R2
M2co2R
2
R
Then
ALj/AL2 =
p2M2co2L
R2
A/?, =
P2M2O)2R
= \\p2
05)
and the same will be true for changes caused by all other filaments of eddy current. Thus the total changes in inductance
may be written L\ and p2L\, and the changes in resistance R[
and pzR[.
It is convenient to devise for coils 1 and 2 an equivalent
circuit from which all mutual inductances have been removed.
The impedances between A and C and between B and C have
already been considered; they have the values
ZBC = (R2 + P2R[) + jtopKLi - L\)
Since the total inductance between A and B in the absence of
eddy currents would be L,(l + p2 + 2p) and its coupling to the
eddy-current filament shown in Fig. 3(a) is M(l + p), it follows
that
*i(l + P2) + Joi[Lx(\ + p2) - L[(\ + p)2]
Equating impedances, the equivalent circuit is found to have
the values shown in Fig. 4.
when using eqn. (12) to estimate the error arising from failure
to adjust the resistances accurately, the total effective resistances
and inductances given in Fig. 4 must be used.
The above treatment assumes perfect coupling. When this
is not achieved the errors arising from eddy currents will depend
on the paths followed by these currents, and no simple general
formulae can be found for them. However, the equations derived
above can be used to set upper limits to the errors in particular
cases, and these limits prove to be so high that the errors cannot
safely be assumed negligible. The application of these conclusions to transformers of the type described in Section 5.1 is
discussed in Section 10.
When the B-connection is used the same result is obtained.
In this case the equivalent circuit of Fig. 4 is still applicable to
coils 1 and 2, but eddy currents caused by current in coil 3 must
now be considered. Each such element of current will induce
electromotive forces into coils 1 and 2, but, with perfect coupling
between these coils, the e.m.f.'s will always be in the ratio 1 : p.
The overall effect of these e.m.f.'s as well as those induced
directly by current in coil 3 can therefore be represented in the
equivalent circuit of Fig. 4 by adding generators of e.m.f. E and
pE, respectively, at A and B.
(3.3) Effect of Hysteresis
Because hysteresis exists, the flux in the core is not proportional
to the current in the magnetizing winding; in fact it is a nonlinear function, not only of the current at a particular instant,
but also of previous values of the current. Under these conditions the inductance of each of the coils varies with time throughout the cycle, but it is still possible to define instantaneous values
for self- and mutual inductance. Furthermore, if two coils are
perfectly coupled at any instant of the cycle, the condition
LXL2 = M22 will be satisfied throughout the cycle although
Lj, L2 and Ml2 are varying. Also, L2\LX will always be equal to
the turns ratio p.
With the A-connection, let coils 1 and 2 be perfectly coupled
and let their resistances be adjusted so that R2\RX — p. At any
instant let the current through the coils be /. Then
P.P. across coil 2
P.D. across coil 1
(In + Ml2)4 + R2i
di
+R
.
•\/L2{y/L2
)
Fig. 4.—Complete equivalent circuit for the A-connection.
Returning now to the balance conditions expressed by
eqns. (5), (7), (12) and (13), one observes two important facts:
first that, with perfect coupling, the bridge ratio is in no way
affected by eddy currents, and secondly that, to avoid errors
caused by resistances of the windings, it is only necessary to
adjust the d.c. resistances of these windings to have the correct
ratio p, since the additional resistances resulting from eddy
currents will already be in the correct ratio. On the other hand,
Thus, at every instant, the potential of the junction of the two
coils divides the p.d. across the bridge in the ratio 1 : p.
The a.c. source will normally have appreciable internal
impedance, so that, even if it has a sinusoidal e.m.f., it will not
produce a sinusoidal p.d. across the bridge because the current
through coils 1 and 2 will be non-sinusoidal. However, balance
will be attained if the impedances Zx and Z 2 divide the voltage
across the bridge in the ratio 1 : p.
Unless Zj and Z 2 have phase angles which are independent of
frequency it will not be possible to attain a true balance under
the conditions stated. Thus it is a wise precaution to use a
selective detector tuned to the fundamental frequency of the
source. It should be noted, however, that this refinement is
necessitated by imperfections in the impedances Zx and Z 2 and
not by defects in the bridge.
When the coupling is imperfect the ratio of the instantaneous
p.d.'s across the coils will not, in general, be equal to p, and,
THE COMPARISON OF LABORATORY STANDARDS OF RESISTANCE OR CAPACITANCE
because of the changing permeability, it may vary from point
to point of the cycle. However, the above treatment shows that
the error averaged throughout the cycle should be of the same
order of magnitude as it would be if there were no hysteresis.
Similar results can be deduced for the B-connection.
(4) EFFECTS OF STRAY CAPACITANCE
One of the great merits of bridges of the type considered is
that, in the ideal case, they completely eliminate errors resulting
from stray capacitances. With practical transformers, in which
the coupling is imperfect and the windings have appreciable
resistance, the elimination is not complete and it is therefore
desirable to make an estimate of probable errors.
The junction of coils 1 and 2 will be earthed, and capacitances
between various parts of the apparatus and earth will therefore
appear across one or other of these coils. Capacitances between
windings of the two coils or between different portions of the
winding of one coil will have effects essentially similar to those
caused by capacitances to earth; the case now to be investigated
is that in which an arbitrary impedance Z is connected across
coil 2.
If, in the first instance, perfect coupling between coils 1 and 2
is assumed and the equivalent circuits discussed in Section 3.2
are used, Figs. 5(a) and 5(6) will represent the relevant portions
95
So far, perfect coupling between coils 1 and 2 has been assumed.
To remove this restriction suppose the self- and mutual-inductances of coils 1 and 2 to be modified slightly in such a way that
the impedances between AO and OB in Fig. 4 are still in the
ratio 1 : p, but so that Ml2 is no longer equal to LXL2. In the
absence of stray capacitance the bridge ratio will still hep, but,
when stray capacitance is present, eqn. (17) must be modified
by the addition in the denominator of a term j8X, where 8X is
of the order of magnitude of cofyX^i^i) ~ ^tf]> t o allow for
the alteration in ZB. The approximate relation now becomes
= pf
P
! - •
?
8X
jpRx
jR8X\
=
j\
JC\
j\
/
so that, in this case, stray capacitance affects the bridge ratio
even if no dielectric loss is associated with it.
(5) DETAILS OF TRANSFORMERS TESTED
(5.1) Constructional Details
Experiments have been carried out on six specially constructed
transformers. In each case the core was built up from interleaved
E and I Mumetal stampings 0 004 in thick. The dimensions of
the stampings are shown in Fig. 6; the number used for each
/Jin
»£••
ft'"
r
t
^ In
Fig. 5.—Circuits to illustrate effect of stray impedance.
(a)) A-connection.
(b)
b) B-connection.
Fig. 6.—Dimensions of stampings used for transformer cores.
of the circuits for the A- and B-connections respectively. In transformer was such that the centre limb, on which the windings
these diagrams ZA and ZB represent the corresponding impe- were placed, had a square cross-section. The coils were wound
dances shown in Fig. 4 and it is assumed that the d.c. resistances in consecutive layers on a thin square-section cardboard tube
of coils 1 and 2 have been adjusted so that R2 = pRx. The which was just large enough to fit over the core. The layers
effective bridge ratio is equal to the voltage ratio VCB\VAC, and were insulated from each other with thin paper; all the layers
of any one transformer were made as nearly as possible of the
it is easily shown that, in both cases, this ratio is given by
same length (about 2 cm) and kept directly over each other.
Enamelled wire was used and, except where the number of
pZ
turns per layer was 10 or less, the gauge was chosen so that the
4- (p + \)ZB
space between turns was as small as possible. Where a 10-turn
Let this be equal to q + j8q. Then, substituting for ZA and ZB winding was used for coil 1 the turns were evenly spaced and the
the values given in Fig. 4 and putting Z = R — jX, we have
gauge was the same as that used for the associated coil 2.
Single-turn windings are discussed in Section 5.2.
p(R-jX)
<7 + j8q =
Connections were taken to terminals on the insulating lid of
•
• (17)
R. + R- jX
the metal box in which each transformer was mounted for
When, as is usually the case, X is large compared with R and protection. In order to make the ratio of the d.c. resistances of
both are much larger than pRu this can be written approxi- coils 1 and 2 equal to their turns ratio, Eureka wire was inserted
between one end of the appropriate coil and the corresponding
mately as
terminal. Initially this adjustment was carried out to within
. . . (18) about 1 part in 1 000, but, because of the high temperaturecoefficient of resistance of copper and of the possibility that
For the purposes now under consideration the term jpRJX is passage of current through the transformer may cause uneven
unimportant, since it will merely necessitate an adjustment of heating, it is unlikely that the resistance ratio remained accurate,
the phase-balancing device in the bridge. Eqn. (18) thus indi- to within less than about 1 %.
cates that stray capacitance is likely to involve appreciable error
In Table 1, which gives details of the windings, the convention
only if it is associated with considerable dielectric loss.
is adopted that coil 1 has fewer turns than coil 2 so that p i§
96
OATLEY AND YATES: BRIDGES WITH COUPLED INDUCTIVE RATIO ARMS AS PRECISION INSTRUMENTS FOR
always greater than unity. The windings are tabulated in the
order in which they were applied, beginning at the core.
Table 1
WINDING DETAILS OF TRANSFORMERS TESTED
Transformer No.
Coil No.
No. of turns
No. of layers
1
3
2
1
100
100
1
2
2
1 (sheet)
2
3
2 (half)
1
2 (half)
100
50
10
50
2
1
1
1
3
3
2 (half)
1
2 (half)
100
500
100
500
2
4
1
4
4
3
2 (half)
1
2 (half)
100
500
10
500
1
4
1
4
5
3
2
1
100
1000
1
I
.
1 (sheet)
4
2
4
Fig. 7.—Effective circuit for single-turn transformer coil with loops in
parallel.
Fig. 8.—Perspective view of single-turn coil made from Eureka sheet
with copper connections.
and B were nearly but not quite touching. This arrangement
was used in transformers Nos. 1 and 5.
6
The chief disadvantage of the arrangement is that it cannot
conveniently be used unless the single-turn winding is outside
all the others, and, as is shown hereafter, this is not usually the
best disposition. The following alternative was therefore devised.
(5.2) The Problem of the Single-Turn Coil
A single-layer winding of 10 uniformly spaced turns was first
For many purposes a transformer with a turns ratio as high applied, and above it a similar layer of 9 turns, with a layer of
as 1 000/1 is extremely convenient. The use of an excessively thin paper between them. The two windings were then conlarge number of turns in coil 2 is to be avoided since this leads nected in opposition so that they became equivalent to a disto poor coupling with coil 1, so the possibility of reducing coil 1 tributed single turn. This method was used in transformer
to a single turn becomes attractive. However, if there is appre- No. 6. Its chief disadvantage is that the resistance of the singleciable leakage of flux from the core the e.m.f. induced in such a turn winding becomes inconveniently large and the resistances
turn will depend on its position on the core.
of the other windings have to be increased artificially to keep
To investigate the magnitude of this effect a single-layer them in the correct ratio.
winding was placed on the core, and over it two separate singleturn loops, onefixedin the centre and the other movable between
(6) ESTIMATION OF ERRORS IN PRACTICAL
the centre and the end of the core. An e.m.f. was applied to the
TRANSFORMERS
single-layer coil, and the two loops were connected in opposition
(6.1) Investigation of Leakage Flux
so that a measurement could be made of the difference between
the e.m.f.'s induced in them. In this way it was estimated that
In order to gain further information about the magnitude of
the flux linked with a single loop will vary by about 1 part in errors due to imperfect coupling, the following tests were carried
1 000, according to the position of the loop. Thus, if a single- out. Five single-layer coils (A, B, C, D and E) were wound one
turn winding is to be used it must be of such form that the e.m.f. above the other on a square-section cardboard former of the
induced in it is an accurate average of the e.m.f.'s that would be same dimensions as those used in the transformers described
induced in a number of loops evenly spaced along the length above. The winding lengths were also the same as in the transoccupied by the other windings of the transformer.
formers and each coil had 90 turns.
Before inserting any iron core, the self-inductance of each
An attempt was first made to construct a winding in which
about 20 single loops were connected in parallel. This would be coil and the mutual inductances between each pair were measured
electrically equivalent to the circuit of Fig. 7, and, if £,, E2, etc., on a bridge, with an error within about 0-3 /xH. The results
are nearly equal, it is easy to show that the effective e.m.f. are shown in Table 2, which makes use of the fact that MAA = LA,
between the bars A and B will be a true average, provided that etc.
the resistances of these bars are small compared with R. This
Bearing in mind the limited accuracy of the figures and the
arrangement was finally abandoned because it was found to be fact that the space between adjacent layers was unlikely to be
unsatisfactory mechanically, but it was replaced by another in quite constant, one is able to form from this Table a simple
which the same principles apply. A strip of thin sheet Eureka physical picture of the leakage flux. Suppose unit current to be
of the same width as the other windings was soldered to two flowing in, e.g., coil C, giving a flux linkage of 86- 3 /xH with this
stout copper wires, A and B, and formed round the outside of coil. As one proceeds inwards to coils B and A, thefluxlinkage
these windings, as shown in Fig, 8. Its length was such that, decreases by about 8 JUH at each step because of the decreasing
when bound tightly in position with thin string, the wires A area of cross-section of the coils. On the other hand, as one
2 (half)
1
2 (half)
500
(10-9)
500
THE COMPARISON OF LABORATORY STANDARDS OF RESISTANCE OR CAPACITANCE
Table 2
MUTUAL INDUCTANCE OF FIVE SINGLE-LAYER COILS
Coil
A
B
c
D
E
A
72-2
(-0-6)
71-6
(-10)
70-6
(-1-4)
69-2
(-1-2)
71-6
(7-6)
79-2
(-1-2)
78-0
(-1-D
76-9
(-1-5)
75-4
70-6
(7-4)
78-0
(8-3)
86-3
(-1-0)
85-3
(-1-8)
83-5
69-2
(7-7)
76-9
(8-4)
85-3
(8-5)
93-8
(-1-3)
92-5
68-0
(7-4)
75-4
(8-1)
83-5
(9 0)
92-5
(8-4)
100-9
B
C
D
E
680
The figures in parentheses represent the changes in mutual inductance between
successive entries in each column.
97
such variation would affect all readings in the same way and
differences would still be accurate to within 10%.
To illustrate the method of calculation, consider the case in
which an e.m.f. is applied to coil A and the quantity (EB — EC)IEB
is found by measurement to be 2-8 parts in 105. Table 2 shows
that, if there were no free poles on the iron, the difference would
be given by
(EB - EC)IEB = (71-6 — 70-6)/130 x 103
or 0-8 part in 105. It is therefore concluded that the free poles
produce a flux which is responsible for a fractional difference in
the e.m.f.'s of 2 0 parts in 105.
Similar measurements were made with all possible combinations of three windings and also a certain number in which the
input e.m.f. was applied to two windings in series. A representative selection of results is given in Table 3.
Table 3
proceeds outwards to coils D and E, the flux linkage decreases
by about 1 • 4 /tH at each step because the field outside C is,
in general, opposite in direction to that inside, and thus produces
opposing flux in the spaces between C and D, and C and E. The
average flux density parallel to the windings just inside C must
therefore be about 8/1-4 times as great as that just outside C.
Averaged over the whole Table, the ratio is 8-1/1 -2 for all coils,
and the discrepancies are nowhere much greater than the probable
experimental error.
It should perhaps be emphasized that this simple result is
produced only because the total winding depth is small compared with the cross-sectional dimensions of the coils—a
condition which applies to the transformers under consideration.
Thus, when unit current flows in any of the coils, the average
field it produces just inside and just outside that coil is nearly
the same for all coils.
A Mumetal core, similar to that used in the transformers,
was now inserted in the five coils whose mutual inductances
had been measured. As a result, the inductance per layer and
the mutual inductances between pairs of layers were all increased
to about 130 mH. Furthermore, as explained in Section 3.1,
the flux outside the core was now the vector sum of two components, namely that due to the coil and that due to free poles
on the core. The component due to the coil is unaffected by
the presence of the iron, and the deductions from Table 2 with
respect to it still hold good.
It is not practicable to repeat the measurements of Table 2
with the iron core in position because the values obtained
would depend appreciably on the maximum flux density in the
iron, and it is hardly possible to keep this constant from one
measurement to another. Interest lies in relatively small
differences of self- and mutual inductance, and these would be
quite meaningless. The expedient was therefore adopted of
applying a constant e.m.f. to one winding, and measuring, as a
fraction of one of them, the difference between the e.m.f.'s
induced in two other windings. This was done for all combinations of windings, the measurements being made with the aid
of a high-gain amplifier and an accurate attenuator. The input
e.m.f. was observed with a valve voltmeter, and the output from
the amplifier with an oscilloscope. In order to relate these
measurements to those of Table 2 it is desirable to work in terms
of quantities which are independent of the number of turns in
each winding, and this can conveniently be achieved by using
the ratios of the quantities in Table 2 to the inductance of a
single layer (or the mutual inductance between two layers)
when the iron core is in position. For this inductance the value
of 130mH is taken, and although this figure might vary by
about 10% with differing conditions of excitation of the iron,
E.M.F.'s DUE TO LEAKAGE FLUX
Test No.
Winding
connected
to external
e.m.f.
c
Windings
connected in
opposition
Fractional difference of e.m.f.'s
(reckoned positive when inner
winding has the greater e.m.f.)
Due to
coil
Due to
free poles
Total
Parts in 10' Parts in 10' Parts in 10'
A and B
-5-7
-5-9
-5-7
+ 2-4
+ 2-7
+ 2-6
2
B
C
DandE
+0-9
+ 1-2
+ 1-4
+ 2-2
+ 1-9
+ 1-8
-3-3
-3-2
-31
+ 3-1
+3-1
+ 3-2
3
A
C
E
BandD
+ 1-8
-5-6
-13-2
+4-5
+ 6-4
+ 6-9
+ 6-3
+0-8
-6-3
4
B
C
E
AandD
-4-1
-11-3
— 18-9
+ 7-8
+8-5
+9-6
+ 3-7
-2-8
-9-3
c
A and E
-100
-180
-2-9
+9-5
+ 10-7
+ 121
+ 6-6
+0-7
-5-9
A andE
in
series
BandC
CandD
BandD
-3-6
-3-8
-7-3
+ 3-3
+ 3-5
+ 70
-0-3
-0-3
-0-3
1
D
E
A
5
B
D
6
Once again it is possible to form a simple physical picture of
the leakage flux due to the free poles. In general this flux is in a
direction opposite to that of the main flux through the coils.
In magnitude it is equal to about 2-7 parts in 105 of the main
flux for each space between adjacent windings. Here also the
simplicity of the result arises from the relatively small winding
depth used in these transformers. Moreover, the result holds
only for the component of leakage flux parallel to the axis of the
coils and averaged over a complete winding layer. There is no
suggestion that the leakage flux is parallel to this axis, although
it may be so over the greater part of the winding length; on this
point the experiments give no information.
(6.2) Errors Resulting from Imperfect Coupling
The results of Section 6.1 can now be used to decide the
optimum arrangement of windings and to estimate the errors to
which they will lead. The results can be summarized as follows.
When a coil consisting of a small number of complete layers is
98
OATLEY AND YATES: BRIDGES WITH COUPLED INDUCTIVE RATIO ARMS AS PRECISION INSTRUMENTS FOR
excited by the application of an e.m.f. the effective flux existing
outside the core, resolved parallel to the axis, will consist of
three components. If these are expressed in terms of the core
flux included between adjacent layers (with 90 turns per layer
closely spaced) their magnitudes are:
Flux due to free pole (opposing main flux) 2-7 parts in 105
Flux due to coil, inside exciting coil (aiding
main
flux)
6-2 parts in 105
Flux due to coil, outside exciting coil (opposing
main
flux)
0-9 part in 105
90 turns per layer represents the windings actually used. With
different numbers of turns per layer the space between layers
will be different and all three figures will change in the same ratio,
but the general conclusions reached will still hold good.
Dealing first with the B-connection, let coil 3 consist of a
small number of complete layers. Then the fractional leakage
flux per layer-space for windings outside coil 3 will be
(0-9 + 2-7) = 3-6 parts in 105, opposing the main flux. For
windings inside coil 3 the figures are (6-2 — 2-7) = 3-5 parts in
105, aiding the main flux. If, therefore, coils 1 and 2 are both
to be placed on the same side of coil 3 it is immaterial whether
they lie inside or outside. In either case, if both are single-layer
coils, the ratio error due to leakage flux will be about 3 • 5 parts
in 105. On the other hand, if coils 1 and 2 are placed symmetrically one on each side of coil 3, the total flux between them
will be almost zero, since the fields inside and outside coil 3
are equal in strength but opposite in direction. In these circumstances the ratio error will be negligible. These conclusions are
borne out in detail by the figures given in Table 3, which also
shows (e.g. Test No. 5) that it is not sufficient merely to place
coil 3 between coils 1 and 2; the disposition must be symmetrical,
and this will not be easy to achieve if different gauges of wire
are used for coils 1 and 2.
A much better arrangement is to place coils 1 and 2 adjacent
to each other and to split coil 3 into two halves connected in
series and disposed on either side of coils 1 and 2. Since each
part of coil 3 has only half the normal number of turns, the
components of fractional flux per layer-space are seen to be
3-1 parts aiding the main flux and (2-7 + 0-45) parts opposing
it. The flux between the two halves of coil 3 will thus be nearly
zero, and, if coils 1 and 2 are in this space, the bridge error will
be negligible whether they are spaced symmetrically or not.
This is confirmed by the results in Table 3 (Test No. 6).
Similar arguments may be employed in dealing with the
A-connection, and, using the notation of Section 2.4, SLjL and
SM/L may be found. Thus, when coils 1 and 2 are both singlelayer coils and coil 2 is immediately outside coil 1, it is found,
with the standard inter-layer space, that
SL/L, = 3-5 X 10-5
S M / L , = 3-6 X 10"5
Substitution in eqn. (8) gives for the effective bridge ratio the
approximate value
4 1 + 3-5 x 10-5(2/7 - 1)/O + 1)]
The error therefore varies from 7 parts in 105 when p is large to
—3-5 parts in 105 when p is very small.
If, on the other hand, coil 2 is split into two single layers in
series, placed one on either side of coil 1, the corresponding
values are
SL/L, = - 3-5 x 10-s
hMILx = 3-5 x 10"5
and eqn. (8) gives for the effective ratio the value
p[\ - 3 - 5 x 10-5/(p + 1)]
The error therefore varies from zero when p is very large to
— 3-5 parts in 105 when p is very small.
To summarize, it may be said that with single-layer coils the
error should be less than 1 part in 104 with any reasonable
arrangement of the windings whether the A- or the B-connection
is used. By careful arrangement of the windings it should be
possible to reduce the error to less than 1 part in 105. If one of
the coils has so many turns that several layers become necessary
the error will, of course, be greater, but it should still be possible
to keep it below 1 part in 104 if the total number of layers does
not exceed about 10.
(6.3) Errors Resulting from Coil Resistance
With the A-connection, the error resulting from coil resistance
can be calculated from eqn. (12), provided, as explained in
Section 3.2, that R is taken to be the error in the d.c. resistance
of coil 2, while, in place of Lu i?j and R2, the quantities (Lj — L[),
[Ri + R[(X + p)] and [R2 + pR\{\ + p)] of Fig. 4 are used to
allow for the effects of eddy currents. The d.c. resistances of
coil 2 for the various transformers are given in Table 4. The
remaining quantities will vary with frequency, and a value of
1 kc/s has been assumed, since tests to be described were made
at about this frequency. The quantity (L{ — L[) has a value of
about 130 mH for a 100-turn winding and will be proportional
to the square of the number of turns. For all of the windings
the effective value of Q was about 3. Using these values and
assuming the d.c. resistances to be adjusted to 1 %, it appears
that the error due to resistance will be rather less than 1 part
in 105 for all transformers except No. 6. In this transformer the
resistances are abnormally high because of the method adopted
to provide the single-turn coil, and if their d.c. values are adjusted
to only 1 %, an error of about 5 parts in 105 will result.
With the B-connection the error will depend on the magnitude
and nature of the impedances being compared, but eqn. (13)
shows that when these are nearly pure resistances it is likely to
be much more serious than with the A-connection unless the
resistances are very high. For example, if transformer No. 2
were used for the comparison of nominal 100-ohm and 10-ohm
resistors, an adjustment error of 1 % in the d.c. resistances of the
windings would lead to a final error of 2 parts in 104. With
lower resistances the error would be correspondingly more
serious, and for this reason the B-connection cannot be considered satisfactory for the comparison of resistances. According
to eqn. (14) the B-connection should be satisfactory for the
comparison of pure capacitances, and errors due to winding
resistance should be negligible in this case.
(6.4) Errors Resulting from Stray Capacitance
A measurement of the capacitance between two adjacent
layers of wire in a transformer of the type under consideration
gave a value of 200 jUfiF with a power factor of 0-1, and this
may be taken as typical of the stray capacitances that are likely
to arise in practice. At 1 kc/s the corresponding reactance and
series resistance are about 106 and 105 ohms respectively, and
these values can be substituted in eqn. (18) to estimate the order
of magnitude of the bridge error. This gives 2 parts in 104 for
transformer No. 6 and 1 part in 105 for Nos. 3, 4 and 5; for
the two others the error is quite negligible. The additional error
caused by the combination of stray capacitance and imperfect
coupling is also negligible.
(7) DIRECT TESTS OF THE TRANSFORMERS
(7.1) Preliminary Considerations
Direct tests of the effective ratios of the transformers were
carried out by using them to compare two resistances whose
ratio could be determined with high accuracy. Similar tests
99
THE COMPARISON OF LABORATORY STANDARDS OF RESISTANCE OR CAPACITANCE
the stray capacitances from the bridge through this transformer
to earth. At first the transformer used was one in which the
insulating material had a high power factor, and reversing the
connections caused a very small change in the effective bridge
ratio. When a better isolating transformer was used this difficulty disappeared and a reversal of connections necessitated only
a slight resetting of the phase adjustment of the bridge.
To avoid errors resulting from drift in the values of the
resistances, the d.c. and a.c. tests were made as quickly as
possible after one another. In experiments on transformers
with ratios of 10/1 or 100/1 it was possible to avoid using resistances smaller than 100 ohms, and, since short, thick leads and
(7.2) General Procedure
There was available for the test a wide range of fixed and soldered joints were used when possible, it is believed that errors
decade resistors, all constructed of constantan wire and adjusted from resistance of leads were negligible. At least two indeto 0-1 %. To obtain two resistors whose ratio was known with pendent sets of readings were taken with eaeh transformer, and
high precision, use was made of the well-known fact that if n the consistency of the results suggests that the5 overall error of a
resistances are equal within 0-1 %, their resistance in series is n1 complete test was within about 3 parts in 10 . For the 1 000/1
times their resistance in parallel to within 1 part in 106. Thus, transformers the error was somewhat greater, partly because
the resistance in parallel of ten 1 000-ohm resistors was compared with such a high ratio the sensitivity of the bridge is inherently
with a resistor A, nominally of 100 ohms, and the ratio of A to less. However, it4 is believed that the results are correct to less
the resistance of the ten 1 000-ohm resistors in series was then than 1 part in 10 .
known with high precision. The comparison was made on a
(7.3) Results and Comments
sensitive d.c. Wheatstone bridge, both unknowns being shunted
with much higher variable resistances (10 to 100 kilohrns) to
The results are shown in Table 4 and- it will be seen that they
achieve balance. The effect of any inequality in the ratio arms largely confirm the findings described in earlier Sections. At
with capacitances were not made because condensers of the
necessary precision were not available.
There is no reason to suppose that the accuracy to be obtained
with transformers of the type described varies rapidly with
frequency, but preliminary experiments suggested that about
1 kc/s would be a good compromise. At higher frequencies
errors from stray capacitance become increasingly important,
while at lower frequencies the voltage that can be applied across
a single-layer coil may become inconveniently small. However,
the exact frequency is not at all critical.
Table 4
RESULTS OF TESTS ON SIX TRANSFORMERS
Transformer No.
Resistance of Coil 2 (ohms)
No. of turns in Coil 1
No. of turns in Coil 2
Measured ratio, A-connection
Measured ratio, B-Gonnection
1
2-24
1
100
99-997
100-012
2
1-92
10
100
99-996
10-0001
was eliminated by repeating the measurements with the unknowns interchanged. Finally, the ten 1 000-ohm resistors and
the resistor A were used as known resistances to check the
transformers with a nominal ratio of 100/1. To provide fine
adjustment of the resistance ratio by a known amount, both the
resistor A and one of the 1 000-ohm ones were shunted with
much higher resistances. A variable air-condenser was also
connected across one of the resistances to provide for phase
adjustment.
When checking the transformers with a nominal ratio of 10/1
the procedure, while the same in principle, was slightly different
in detail. For transformers with a nominal ratio of 1 000/1
two resistors were used and each was compared with a third
one, of suitable intermediate value, making use of the 100/1
and 10/1 transformers which had previously been calibrated.
For the a.c. measurements the bridge was fed through a
transformer from an oscillator at about 1 kc/s. The voltage of
the source was adjusted so that the core of the bridge transformer was operating at a reasonable flux density, but tests
showed that wide variations of input voltage had a negligible
effect on the results. The output from the bridge was fed to a
high-gain selective amplifier and thence to the Y-plates of an
oscilloscope. The X-plates were fed directly from the oscillator
so that the trace on the screen was an ellipse when the bridge
was unbalanced and a horizontal straight line when it was
balanced. This arrangement facilitated independent adjustment
of resistance and capacitance.
Since one side of the oscillator was earthed, reversing the
connections between it and the isolating transformer changed
3
99-6
100
1 000
10-0003
10-0005
4
109
10
1000
100-002
100-004
5
101-2
1
1 000
1000-36
100006
6
1 798
1
1 000
100011
first sight the difference between the nominal ratios and those
measured by means of the B-connection are much smaller than
might be expected. This is probably partly fortuitous, but in
any case the resistances used in these tests were relatively high,
so that errors resulting from imperfect adjustment of the
resistances of the windings are smaller than they might otherwise be.
(8) CONCLUSIONS
In the light of the above results it seems safe to conclude that
transformers of the type described with ratios up to 100/1 and
used with the A-connection can be replied upon to have an
effective ratio differing by less than 1 part in 104 from the turns
ratio. To achieve this result the windings must be arranged in
accordance with the principles laid down in Section 6.2, their
d.c. resistance ratio must be adjusted to be equal to the turns
ratio within 1 % and the power factors of stray capacitances must
be kept as small as possible.
The B-connection does not appear to have any advantages
over the A-connection and is less reliable for the comparison
of resistances unless these are high. It should be satisfactory
for the comparison of capacitances, but this conclusion has not
been tested experimentally.
For ratios higher than 100/1 the difficulties of making good
transformers increase progressively, but even with a ratio of
1 000/1 there is no difficulty in making transformers that can be
relied upon to within a few parts in 104.
It seems highly probable that with further development transformers can be built to have effective ratios equal to their turns
100
OATLEY AND YATES: BRIDGES WITH COUPLED INDUCTIVE RATIO ARMS AS PRECISION INSTRUMENTS
ratios within 1 part in 105 for ratios up to 100/1 and within
1 part in 104 for ratios up to 1 000/1. The most hopeful line of
attack would appear to be to use rather larger cores. The
leakage flux would then be a smaller proportion of the total flux
and there would also be room for single-layer windings of lower
resistance. It is not obvious that the use of toroids with uniformly distributed windings would be advantageous. Such an
arrangement would reduce the flux produced by free poles to a
negligible quantity, but the leakage flux due to the coils would
remain, and the cancellation achieved in the transformers
described herein might not be possible. It should perhaps be
emphasized that this cancellation is not fortuitous and must
occur with any transformer of approximately the dimensions
of those described and having the windings arranged correctly.
However, these are speculations which must be tested by experiment, and to carry out tests with the necessary precision would
require facilities which are hardly to be found outside one of the
national standardizing laboratories.
thetical circuit of resistance R4 and inductance L4, coupled to
coils 1 and 2 by mutual inductances M14 and M24 respectively, if
L[IR\ = L'JR^ = - L4\R4
and if
•
.
• (21)
L'2=-
• • • (22)
with similar equations for L\, R\ and R2.
For the transformers described in Section 5.1 the figures
given in Section 6.2 show that the imperfection of coupling can
be expressed as
L 1 L 2 - M 1 2 1 = 3-6 x 10-5L2 m m
(2 3)
Moreover, for these transformers, the ratio Ml2IL2 differs from
\\p by a negligible amount. From these facts and from
expression (2) in Section 2.2, applied to coils 1 and 2 and the
hypothetical eddy-current coil 4, it follows that
VM 2 4
pj
>3-6 x
(24)
Measurements of the variation of inductance and resistance
(9) REFERENCES
with frequency show that, for these transformers, at 1 kc/s
(1) BLUMLEIN, A. D.: British Patent No. 323037.
R'2 < — L'2u) a n d — L'2\L2 ~ 1 /4
(2) KIRKE, H. L.: Radio Section Chairman's Address, Journal
I.E.E., 1945, 92, Part III, p. 2.
Thus, from eqn. (22),
(3) WILHELM, H. T.: "Impedance Bridges for the Megacycle
Range," Bell System Technical Journal, 1952, 31, p. 999.
(4) CLARK, H. A. M., and VANDERLYN, P. B.: "Double-Ratio
A.C. Bridges with Inductively-Coupled Ratio Arms,"
Proceedings I.E.E., 1949, 96, Part II, p. 365.
(5) WATTON, W. L., and PEMBERTON, M. E.: "A Direct-Capaci-
tance Aircraft Altimeter," ibid., p. 379.
(6) MOULLIN, E. B.: "Principles of Electromagnetism" (Oxford
University Press, 1950).
(10) APPENDIX
From the equations given in Section 3.2 for the changes,
caused by a filament of eddy current, in the inductances and
resistances of coils 1 and 2, it follows that
, = AL2/Ai?2 -
-
•
• (20)
Thus the total changes L[, R[ and L2, R2 caused by all the eddy
currents can be ascribed to current flowing in a single hypo-
Substituting this result in expression (24) it follows that, given
the appropriate magnitude and distribution of eddy currents,
the difference between Ml4/M24 and \\p might be as great as
1%. The ratio error to which such a difference would give
rise might be of the order of 1 % and would be far too large to
be tolerated.
The above analysis merely sets an upper limit to the error,
which will, in general, be much smaller. Presumably a large
error could occur if a low-resistance eddy-current path were
closely linked with the leakage flux between coils 1 and 2, and
this might happen if, for any reason, an electrostatic shield were
inserted between these windings. In any case, the possibility
of appreciable error gives importance to an independent test.
For the transformers under consideration the results given in
Table 3 constitute such a test and show that the errors resulting
from eddy currents are negligible.