NMI TR 17
Two-stage Transformers with Partitioned Excitation Cores
Greig W Small
11
© Commonwealth of Australia 2013
First edition — August 2013
National Measurement Institute
Bradfield Road, Lindfield, NSW 2070
PO Box 264, Lindfield, NSW 2070
T
F
W
(61 2) 8467 3600
(61 2) 8467 3610
www.measurement.gov.au
ii
CONTENTS
1
Introduction ......................................................................................................................... 1
2
Partitioned Excitation Core ................................................................................................. 2
3
Excitation Windings with Unequal Turns .......................................................................... 4
3.1
Ratio of Unity ............................................................................................................ 4
3.2
Ratios other than Unity.............................................................................................. 5
4
Conclusion .......................................................................................................................... 7
5
References ........................................................................................................................... 7
iii
ABSTRACT
Partitioning the excitation core of a two-stage transformer into two or more sections with
their separate excitation windings may effect a significant economy of core material. It may
also allow the close approximation of non-integer transformer ratios with fewer turns than
would be required in a conventional design. Application of the technique to the design of
transformers with amplifier-assisted excitation and to an active coaxial choke is described.
Keywords: two-stage transformers, pre-excited transformers, partitioned excitation cores,
amplifier-assisted excitation, active coaxial chokes
iv
1
INTRODUCTION
Two-stage voltage transformers comprise a magnetic circuit, usually a toroidal core, with a
winding that is connected to a voltage source to provide the excitation of the transformer, and
a second core with a winding that couples both cores and is connected to a voltage source to
define the input, or primary, voltage. The second core is excited by the deficit in excitation of
the first core. The first core and its excitation winding may be shielded to contain any leakage
flux. Conducting shields may enclose, and be connected to the respective grounds of, the
primary and secondary windings to contain capacitance currents. A representation of the half
cross-section of such a transformer is shown in Figure 1 [1, 2].
Electrostatic
screens
Excitation
winding
and core
Magnetic
shield
Primary
winding
and core
Secondary
winding or
windings
Figure 1. Two-stage transformer with magnetic shield and inter-winding screens
In some low-frequency designs the second core winding may also be connected to the
excitation source, and a third core and winding added to reduce further the excitation error
[3]. For the purposes of this document, these and similar arrangements are to be regarded as
elaborations of two-stage transformers.
The excitation core is designed to carry the total excitation flux, which is proportional to the
applied voltage and inversely proportional to the turns of the excitation winding and to the
signal frequency. Efficient use of the core will see that the peak flux density approaches the
saturation flux density for the core material. In a toroidal core, the excitation field is inversely
proportional to the distance from the core axis so that the flux density is also approximately,
to the degree that the excitation is linear, inversely proportional to radial distance. To this
degree the peak flux density at the outer radius, r2, of the core is less than that at the inner
radius, r1, in the ratio r1/r2.
For small transformers this may be inconsequential, but for larger transformers operating at
lower frequencies and with requirements for higher voltages per turn the excitation core may
be both expensive and massive and it becomes important to design for efficient use of core
material. For a given total flux to be carried by the core, this points to a high aspect ratio of
core height to radial width. Factors that limit this approach are the mechanical implications of
a high aspect ratio and the greater length, and hence higher resistance and uncoupled
reactance, of the excitation winding with consequent increase in excitation error.
Alternatively, more efficient use of core material may be had by partitioning the core radially
into two or more sections, each with its own excitation winding (Figure 2).
NMI TR 17
1
Excitation
winding
and core
Excitation
winding
and core
Figure 2. Two-stage transformer with a partitioned excitation core
2
PARTITIONED EXCITATION CORE
Consider a core of height h1, inner radius r1 and outer radius r2 with flux density Bmax at
radius r1. Again, to the degree that the excitation is linear, the flux density B(r) at radius r is:
B(r) = Bmax r1/r
and the total flux ϕ is:
ϕ = Bmax h1
r2
r1 / r dr
r1
= Bmax h1 r1 loge(r2/r1)
(1)
Suppose the excitation were to be increased in proportion to the radius, so that the flux
density is maintained at Bmax. For the same total flux ϕ, the outer radius r2' is implied by:
Bmax h1 (r2'− r1) = Bmax h1 r1 loge(r2/r1)
giving
(r2'/r1) = 1 + loge(r2/r1)
(2)
The mass m' of this core relative to that, m, of the first is:
(m'/m) = ((r2'/r1)2 – 1)/((r2/r1)2 – 1)
(3)
Values of r2'/r1 and m'/m for a range of values of r2/r1 are given in Table 1.
Table 1. Hypothetical minimum masses for several values of relative width
r2/r1
1.1
1.2
1.3
1.4
1.5
NMI TR 17
r2'/r1
1.095
1.182
1.262
1.336
1.405
m'/m
0.951
0.904
0.860
0.819
0.780
r2/r1
1.6
1.7
1.8
1.9
2.0
r2'/r1
1.470
1.531
1.588
1.642
1.693
m'/m
0.744
0.710
0.679
0.650
0.622
2
This result is the hypothetical limit to the reduction in core size and mass for the given height,
inner radius and maximum flux density. To approach this limit, consider partitioning the core
radially into n concentric cylinders of equal width, each cylinder being wound with nT turns
and all windings connected in parallel to the voltage source V of excitation. The spacing
between cylinders is, for the purposes of the argument, assumed to be zero, ignoring the
radial space that must be occupied by the excitation windings and any structure enclosing the
component cores. Clearly, with increasing values of n, the increasing numbers of turns and
the increasing space needed to accommodate the windings rapidly become self-defeating.
Now consider the core to be partitioned into two cores, each of which carries half the total
flux, with the same values of r1, h1 and Bmax. Let the outer radius of the inner core and the
inner radius of the outer core be r12, neglecting the space required by the excitation windings,
and the outer radius of the outer core be r2'.
The total flux ϕ1 in the inner core is:
ϕ1 = Bmax h1 r1 loge(r12/r1)
(4)
For ϕ1 to be equal to ϕ/2:
loge(r12/r1) = loge(r2/r1)/2
(r12/r1) = √(r2/r1)
so that
(5)
The total flux ϕ2 in the outer core is:
ϕ2 = Bmax h1 r12 loge(r2'/r12)
(6)
and for ϕ2 to be equal to ϕ1:
r12 loge(r2'/r12) = r1 loge(r12/r1)
(7)
That is
(r12/r1) loge((r2'/r1)/(r12/r1)) = loge(r12/r1)
from which
(r2'/r1) = exp[(1 + 1/(r12/r1)) loge(r12/r1)]
(8)
Some values of the relative widths r2/r1 of the single core and r2'/r1 of the partitioned core,
together with their relative masses m'/m, are given in Table 2.
A typical value is around 1.5, corresponding to a ratio of outer to inner diameters of 3:2; in
this case the saving in material with the partitioned core is nearly 13%.
Table 2. Relative widths and masses of single core and equivalent partitioned core
r2/r1
1.1
1.2
1.3
1.4
1.5
r2'/r1
1.098
1.191
1.279
1.364
1.445
m'/m
0.974
0.948
0.922
0.896
0.871
r2/r1
1.6
1.7
1.8
1.9
2.0
r2'/r1
1.523
1.598
1.670
1.740
1.807
m'/m
0.846
0.822
0.799
0.777
0.755
With the pair of cores each carrying half the flux of the single core, the excitation winding on
each core will require exactly twice the number of turns as that on the single core. The
partitioned core with its windings connected in parallel is then functionally equivalent to the
single core and its winding.
NMI TR 17
3
3
EXCITATION WINDINGS WITH UNEQUAL TURNS
Allowing the excitation windings to have different numbers of turns one from the other leads
to interesting possibilities, including the partitioning of the cores for other than the same flux
in both cores, and for effecting non-integer transformer ratios that might otherwise be
approximated only with high numbers of turns.
Let the numbers of turns of the excitation windings be N1 and N2, and of the common
secondary winding N12. As before, the excitation windings are connected in parallel to the
source of excitation voltage.
The effective turns ratio R is:
R = N12/N1 + N12/N2
= N12(1/N1 + 1/N2)
Let
N1 = 2N12 – a
and
N2 = 2N12 + b
(9)
(10)
where a and b are integers.
Then
R = N12(N1 + N2)/(N1N2)
= (4N122 + (b – a)N12)/(4N122 + (b – a)N12 + [(b – a)N12 – ab])
(11)
For values of b less than or equal to a, the denominator of equation (11) is less than its
numerator and R must be greater than 1. Other and more useful possibilities result from
setting b greater than a, are examined.
3.1
Ratio of Unity
Putting (b – a)N12 = ab in equation (11) results in R = 1 for all values of a. A ratio of 1 is
required for a conventional two-stage design where the N12 winding couples the additional
core and serves as the defining primary winding of the two-stage transformer.
For (b – a) = 1, N12 = a(a + 1) and from equation (10), N1 = a(2a + 1) and N2 = (a + 1)(2a +
1). The first several combinations of turns are given in Table 3.
Table 3. Combinations of turns with unity ratio
a
1
2
3
4
5
N12
2
6
12
20
30
N1
3
10
21
36
55
N2
6
15
28
45
66
a
6
7
8
9
10
N12
42
56
72
90
110
N1
78
105
136
171
210
N2
91
120
153
190
231
Since a, (a + 1) and (2a + 1) are mutually prime in all combinations, N12, N1 and N2 are also
mutually prime. Hence each of these combinations of turns may be scaled up but not down to
numbers of turns appropriate to the application.
For values of (b – a) other than 1:
(b – a)N12 = a2 + (b – a)a
(12)
To avoid combinations that are multiples of some other, N12, a and b must have no factors in
common. If (b – a) has any unpaired prime factor pj then, from equation (12), pj must be a
NMI TR 17
4
factor of a2 and therefore of a, of b, of a2/(b – a), and hence of N12. (b – a) must therefore be
a square, and a2 but not a must be divisible by (b – a). The smallest value a1 of a is therefore:
a1 = √(b – a)
If a is any multiple of a1 such that the multiplier m includes a prime factor pi of (b – a), then
pi will also be a factor of a2/(b – a) and hence of N12 as well as of a and b. Multipliers m must
therefore be mutually prime with (b – a). Combinations of turns for several values of (b – a)
and a meeting these constraints are given in Table 4.
Table 4. Combinations of turns with unity ratio and values of (b-a) other than 1
(b-a)→
a↓
2
6
10
14
18
N12
3
15
35
63
99
4
N1
4
24
60
112
180
N2
12
40
84
144
220
(b-a)→
a↓
3
6
12
15
21
N12
4
10
28
40
70
9
N1
5
14
44
65
119
N2
20
35
77
104
170
(b-a)→
a↓
4
12
20
28
36
N12
5
21
45
77
117
16
N1
6
30
70
126
198
N2
30
70
126
198
286
It is seen that larger values of (b – a) rapidly lead to large values for the numbers of turns,
while for smaller numbers of turns the proportional differences between N1 and N2 are large
and probably less useful than is the case for (b – a) = 1.
3.2
Ratios other than Unity
As noted above, combinations of different numbers of winding turns on two primary cores
permit the approximations to ratios that otherwise might require much larger numbers of
turns. For example, consider a transformer ratio bridge for the comparison of a resistor of
value 100 kΩ with a resistor of value RK/2, where RK is the von Klitzing, or quantised Hall,
resistance of which the presently accepted value is 25 812.807 Ω. The nominal transformer
ratio is 7.748 091 87.
From the continued fraction expansion of the ratio, the successive convergents (and their
errors in parts per million) are:
7:1
8:1
23:3
31:4
984:127
1 015:131
(−96 552)
(32 512)
(−10 509)
(246.27)
(−7.792)
(−0.034)
To approximate this ratio to a few parts per million with a conventional transformer requires
a primary winding of about 1000 turns. For a partitioned-core transformer with paralleled
primary windings of 51 and 79 turns and a secondary winding of 4 turns, the ratio error is
−1.541 parts per million.
Another application is in the amplifier-assisted excitation of a two-stage transformer. In this
example, a unity-gain amplifier drives the two excitation windings. A sensing winding,
coupling both the excitation cores and the additional core, provides the input signal to the
amplifier (Figure 3). The loop gain must be less than unity to ensure that this arrangement is
stable, but close to unity so that the pre-excitation error is small. As the primary winding need
NMI TR 17
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provide only the deficit in excitation, this configuration may be advantageous where the
primary voltage source is unsuitable or inconvenient as the source of pre-excitation. Let:
(b – a)N12 = ab + c
(13)
where c is an integer. Exhaustive examination of combinations of (b – a), a and c would be
unwieldy, and only the case of (b – a) = 1 will be considered here. Then:
N12 = a2 + a + c,
N1 = 2a2 + a + 2c
N2 = 2a2 +3a + 2c + 1
and
(14)
Values of N12, N1, N2 and the ratio for the first several values of a and c are given in Table 5.
Feedback
winding and
screen
+1
Figure 3. Two-stage transformer with amplifier-assisted excitation
Table 5. Values of turns N12, N1, N2 and ratio for (b – a) = 1 and several values of a and c
c→
a↓
1
2
3
4
5
6
1
N12
3
7
13
21
31
43
N1
5
12
23
38
57
80
2
N2
8
17
30
47
68
93
Ratio
0.975 0
0.995 1
0.998 6
0.999 4
0.999 7
0.999 9
N12
4
8
14
22
32
44
N1
7
14
25
40
59
82
3
N2
10
19
32
49
70
95
Ratio
0.971 4
0.992 5
0.997 5
0.999 0
0.999 5
0.999 7
N12
5
9
15
23
33
45
N1
9
16
27
42
61
84
N2
12
21
34
51
72
97
Ratio
0.972 2
0.991 1
0.996 7
0.998 6
0.999 3
0.999 6
It may be seen that, for example, with a = 4 and c = 2, the numbers of turns are 22, 40 and 49
(or any common multiple thereof) and the feedback gain, apart from the excitation error, is in
deficit of unity by 0.001.
Where the excitation core would have been partitioned for considerations of economy, this
implementation avoids the need for an attenuator. In other cases (for example, amplifierNMI TR 17
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assisted coaxial chokes in bridge circuits), the use of a resistive attenuator as an alternative
may prejudicially introduce additional noise and additional impedances in series and
admittances to ground. Where economy of core material is not a consideration, it may be
more convenient to partition the excitation core axially rather than radially.
The design of a coaxial choke of this configuration with excitation of 102 and 115 turns and
feedback of 54 turns (a = 6, c = 12, in deficit by 0.001) is shown in Figure 4 [4].
+1
Coaxial
cable
Figure 4. Amplifier-assisted coaxial choke employing a partitioned excitation core
The impedance presented to the coaxial cable by the second core is enhanced by the factor
1/δ, where δ is the deficit in excitation. For δ = 0.001 the enhancement factor is 1000,
equivalent to winding 31 turns of the cable on the same core without amplifier assistance. In
this configuration of active coaxial choke the feedback loop is completed and defined locally
to the device and independently of the coaxial cable, as distinct from other implementations
in which the feedback loop is completed by the outer circuit of the cable.
4
CONCLUSION
The partitioning radially of the excitation core of a two-stage transformer into two or more
separately excited sections may lead to a saving of more than 10% in core material. This may
be a significant economy, particularly in transformers operating at lower frequencies and with
a requirement for fewer turns per volt. Partitioning the core also introduces the possibility of
unequal turns of excitation windings, allowing close approximations to awkward ratios with
appreciably fewer turns than in a conventional design. A particular example is the amplifierassisted excitation of the transformer with a unity gain amplifier and a ratio of feedback turns
to excitation turns fractionally less than unity.
5
REFERENCES
[1] Thompson, AM and Small, GW (1971) AC bridge for platinum resistance thermometry.
Proc. IEE 118, 1662–66
[2] Small, GW (1996) The Efficacy of High Permeability and Conducting Shells as
Magnetic Shields on Toroidal Cores. TIP Technical Memorandum No. 144, TIP3004
(4 November)
[3] Small, GW; Budovsky, IF; Gibbes, AM and Fiander, JR (2005) Precision three-stage
1000 V/50 Hz inductive voltage divider. IEEE Trans. Instrum. Meas. 54(2) 600-3
[4] Small, GW and Marais, EL (2012) An active coaxial choke for precision impedance
measurements. CPEM 2012 Digest, Washington DC (1 – 6 July 2012) (submitted)
NMI TR 17
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