6.2 Current Transformers: Part 2.
10. Lowfrequency performance:
The effective secondary parallel reactance of a current transformer
in the lowfrequency region of its passband is dominated by the
inductance of the secondary winding. Hence, for the purpose of
calculating the lowfrequency response, the model can be simplified
as shown below.
Here the capacitance has been ignored, and the inductance L
i should be taken to be about 1 or 2%
less than
the measured inductance of the secondary coil. The relationship
between primary current and secondary voltage (neglecting losses)
given earlier as equation (
4.1)
is now reduced to:
The obvious implication is that there will be significant phase
and amplitude errors in the output unless X
Li
>> R
i. In fact,
the lower 3dB bandwidth
limit occurs when X
Li = R
i,
with an attendant phase error of +45°.
Phase and magnitude
errors in the
current sampling network of a bridge will compromise the accuracy
of the balance condition. We will see in
later
sections however, that compensation can be achieved by
modifying the voltage sampling network so that its output has
the same frequency response as that of the current sampling network.
Such compensation however, does not correct for the lowfrequency
reduction in output voltage, and this has implications with regard
to the sensitivity of bridges, and the accuracy of currentmeasuring
instruments. It is therefore principally the desire for a flat
amplitude response that dictates the minimum value of inductance
that can be tolerated, and we can understand this issue by considering
the simple RF ammeter circuit shown below:
Here the detector is sensitive only to the magnitude of the output
voltage, and the reading on the meter (assuming a scale calibrated
to allow for diode forward voltage drop) is given by:
V
meas = (√2) 
Vi
where:

Vi = 
I
(
jX
Li // R
i) / N
i.e.,
Vmeas
= (√2) I (jXLi // Ri)
/ N 
The measured output (i.e., the sensitivity) is therefore
always proportional to the magnitude of the secondary load impedance,
which (neglecting losses and the detector input resistance) consists
of the secondary winding reactance in parallel with the load
resistance.
The effect of placing
an inductance
in parallel with an impedance [discussed in Impedance Matching,
section 57]
is to move the
resultant impedance anticlockwise around a circle of constant
conductance as the inductive reactance decreases. The reactance
X
Li=2πfL
i, of
course, decreases as the frequency decreases, and the constant
conductance G
i is equal to 1/R
i.
We can therefore visualise the process using the Zplane diagram
below:
Where
Zi
= (
jX
Li
// R
i)
An expression for the
magnitude
of an impedance in its parallel form was given in [
AC Theory,
Section 18, eqation 18.2]. Using the present notation it
becomes:

Zi =  R
i
X
Li / √( R
i²
+ X
Li² ) 
At high frequencies, X
Li
becomes very large
and its contribution to
Zi
becomes
correspondingly small. In this capacitancefree model therefore,
the magnitude of
Zi
can be considered
to be equal to R
i at high
frequencies,
i.e., as X
Li→∞,

Zi→R
i. Consequently,
since the measured voltage for a given input current is proportional
to 
Zi,
we can define a dimensionless
amplituderesponse function for the system as:
ηLF
= Zi / Ri = XLi
/ √ ( Ri²
+ XLi² ) 
(η is "eta", this time unbold because the
quantity it represents is scalar). The reason for deriving this
expression, is that we can use it to obtain the minimum amount
of inductance required in order to keep the drop in meter sensitivity
within acceptable limits at the lowest frequency of operation.
All we need to do is rearrange it until we have X
Li
expressed in terms of R
i and
η
LF, as follows:
η
LF² = X
Li²
/ ( R
i² + X
Li²
)
η
LF² ( R
i²
+ X
Li² ) = X
Li²
X
Li²  η
LF²
X
Li² = η
LF²
R
i²
X
Li² (1  η
LF²)
= η
LF² R
i²
X
Li² = R
i²
η
LF² / (1 
η
LF²)
Now taking the square root to obtain X
Li,
we note that we only want the positive result (i.e., the magnitude
of X
Li), and so:
X
Li = R
i
η
LF / √(1 
η
LF²)
but since X
Li is an inductive
reactance,
we know it will be positive and so we can dispense with the magnitude
symbol in this instance (but not in general) to obtain:
XLi
= Ri ηLF / √(1  ηLF²) 
The results for some possible design values for η
LF are tabulated below:
Table 10.1. Current transformer, lowfrequency
desensitisation
and phase error.
Loss in %
=100(1ηLF)

Loss in dB
=20Log(ηfLF)

ηLF
=Zi/Ri

XLi
=RiηLF/√(1ηLF²)

φ
=Arcan(Ri/Xi)

Required Li
for
fmin=1.6MHz, Ri=50Ω

1 %

0.09 dB

0.99

7.02 Ri

8.1°

34.9μH

5 %

0.45 dB

0.95

3.04 Ri

18.2°

15.1μH

10 %

0.92 dB

0.90

2.06 Ri

25.8°

10.2μH

11%

1 dB

0.891

1.97 Ri

27°

9.8μH

29%

3dB

0.707

Ri

45°

5.0μH

50%

6 dB

0.5

0.58 Ri

60°

2.9μH

The figure X
Li ≥ 7R
i
for 1% max. error is corroborated by ref [
The ARRL Antenna
Book, 19th edition, ARRL publ, 2000.
ISBN: 0872598047. Bridge types p27.4].
The data in the table tell us that if we want a current reading
at the lowest frequency of operation (f
min)
to be within 1% of a highend reading of the same current, then
X
Li must be at least 7 times
larger than
R
i at the lowest frequency.
If we want
agreement within 5%, then X
Li
must be at
least 3 times larger than R
i,
and so on.
The more stringent design requirements at the top of the table
are appropriate for directreading RF ammeters, where scale accuracy
is important; but for the design of RF bridges, where we are often
in a position to turn up the generator level if the bridge sensitivity
starts to fall, a reduction in sensitivity of 3dB, or even 6dB,
may be perfectly acceptable.
Also shown in the
table are the
phase errors associated with the various design criteria. Notice
that even when the amplitude is controlled to within 1%, the phase
error at the minimum frequency is 8°. Underhill & Lewis
[
43] recommend that the maximum acceptable error for
an
impedance matching system should be ±7° (1.2:1 SWR),
and so a large secondary inductance, notwithstanding the
propagationdelay
issue, is not a way of avoiding the need for phase compensation
when designing bridges. We might, of course, decide to use a low
value of load resistance
and a large inductance;
but that
implies a low output voltage (i.e., insensitivity), which is fine
when monitoring a transmitter producing kilowatts, but not so
good when trying to tune an antenna using a few watts.
For those interested
in designing
accurate RF ammeters, note that using a large secondary inductance
is neither the only, nor necessarily the best, way of obtaining
a flat frequency response. By the inclusion of a capacitor, the
secondary loading network can be modified to produce an output
that is flat within 1% over the 1.6 to 30 MHz range (or greater)
using an inductance of around 10 μH. This configuration, which
does not appear to have been reported elsewhere, is referred to
in these documents as the MaximallyFlat Current Transformer [see
article of that name].
Ref:
[
43]
"Automatic
Tuning of Antennae". M J
Underhill
[G3LHZ]
and P A
Lewis.
SERT Journal, Vol 8, Sept 1974, p183184.
Gives criteria for achieving 1.2:1 SWR, i.e., 45 ≤ R ≤
56Ω,
17.5 ≤ G ≤ 22.5 mS, 7° ≤ φ
≤ +7°.
11. LF phaseerror demonstration.
A simple technique for demonstrating currenttransformer lowfrequency
phaseerror is illustrated below. It uses a method that works
well at low frequencies (1.6  3 MHz) but can give misleading
results at higher frequencies unless it can be shown that the
oscilloscope Yamplifiers have identical propagation delay and
that the delay does not vary with the settings of the gain controls.
In this case a waveform with the same phase as the primary current
is obtained by measuring the voltage across a 50 Ω load
resistor that terminates the generator. When this is compared
with the voltage appearing across a 50 Ω resistor terminating
the secondary winding, a phase difference is seen on the oscilloscope
screen. By adjusting the Yamplifier gain and shift controls,
both waveforms can be made to have exactly the same height and
be equally displaced about the central horizontal scale on the
graticule. The phase difference between the two waves is then:
Δφ =

Horizontal separation between waveforms
Horizontal length of one cycle

× 360° 
In the example above, the current transformer primary is a stub
of URM108 (PTFESilver, 50Ω) cable, and the secondary is
61 turns of 36swg wire (0.225mm diameter including insulation)
on an Amidon T502 (red) core. The published A
L
(inductance factor) for this core is 4.9nH/turns², giving
a nominal secondary inductance (A
LN²)
of 18.2μH (± about 20%), but the measured inductance
at 1.5915MHz (10
^{7} radians/sec) was
19.9±0.5μH.
The phasedifference measurement was made at 1.6MHz, at which
frequency the reactance of the coil (2πfL) is
+200±5Ω.
The situation shown on the oscilloscope screen is therefore X
Li=4R
i.
The expected phase angle is:
Arctan(50/200±5) = 14.04±0.34°
Using the horizontal shift, timebase, and trigger level controls,
the oscilloscope display was manipulated so that one cycle of
the waveform was 10cm long. The distance between the zerocrossings
of the two waves was then found to be 0.4 ±0.05cm. The
measured phase difference is therefore:
360 × 0.4 / 10 = 14.40 ±1.8°
Which agrees with the calculated value.
On a practical point,
notice that
oscilloscope probes were not used, and that the signals were taken
along 50Ω cables of equallength and identical dielectric
(for equal timedelay) and terminated at BNC Tpieces on the
oscilloscope
front panel. The problem with probes is that there will be capacitive
coupling between them, and this gives rise to phase errors. The
large transmitter terminator is shown hanging down from the
frontpanel;
but a length of cable
after the Tpiece, so that
the load
can be placed more conveniently, is of no electrical consequence.
The generator was a Kenwood TS430S radio transceiver with plug
10 removed from the RF module to give 1.6 to 30MHz transmitter
coverage. The output level during the measurement was about 16Vpp
(5.7V RMS, 0.64W in 50Ω). No attenuator was used between
the main transmitter line and the oscilloscope because it is important
that both sampling points have the same input impedance (so that
both cables are equally mismatched). For the instrument shown,
both Yamplifier inputs are 1MΩ // 20pF, which is typical.
12. Transformer core selection.
Having examined the basic design considerations for current
transformers,
we now turn our attention to the problem of choosing transformer
cores from manufacturer's data. Here it will be assumed that we
are primarily interested in designing current transformers for
bridges, and a maximum lowfrequency desensitisation of 3dB is
acceptable; in which case the minimum secondary reactance must
be no less than the load resistance. If the minimum frequency
of operation is (say) 1.8MHz, and we intend to follow the normal
practice of terminating the transformer with a resistor of about
50Ω, then the minimum secondary inductance L
i=X
Li/2πf=4.42μH. With
this figure in mind,
we may trawl the catalogues looking for cores that will fit reasonably
tightly over a stub of coaxial cable, and which have A
L
(inductance / turns²) values that permit us to obtain a
transformer
ratio appropriate for our purpose.
Amidon
supplies Micrometals
and Fairrite cores in small quantities (see also: links page
for other sources). Hence, although suitable cores are available
from various manufacturers, the use of Amidon products is convenient
for private experimenters. Using the Amidon catalogue [
38],
and supplementary data from the Micrometals and Fairrite websites,
it transpires that the choice of core is remarkably limited once
the frequency range and the hole diameter have been specified.
Ref: [
38]
Amidon Associates
Inc. (Technical data book) Jan 2000.
Technical data for iron powder and ferrite cores, including A
L values, frequency ranges, wire
packing tables,
Q curves, etc. Maximum flux density calculations and recommendations:
p135. Information is also available online from:
www.amidoncorp.com
.
The point is that we need the core to be a tight fit on the cable
in order to minimise leakage inductance and magnetic pathlength;
and so once a cable diameter has been chosen, the core hole diameter
is the next size up that allows room for the secondary winding.
The choice of core material is then that which gives sufficient
secondary inductance with the required number of turns. The relevant
information is summarised below:
Table 16.3. 50Ω PTFE Coaxial Cable data.
Type

Overall dia.
/ mm

Jacket *

Dielectric

Max VRMS
/ KV

Capacitance/m
C0 /
pF/m**

Velocity
factor

UR M72 
4.5

FEP

PTFE

1.4

94

0.72

UR M102 
9.7

FEP

PTFE

3.5

96

0.70

UR M107 
9.0

FEP

PTFE

3.5

96

0.70

UR M108 
4.5

FEP

PTFE

1.4

94

0.72

UR M109 
2.45

FEP

PTFE

0.7

93

0.72

UR M110 
1.8

FEP

PTFE

0.35

92

0.72

RG142 
4.95

PTFE / FEP

PTFE

1.4

95.8

0.695

RG303 
4.32

PTFE

PTFE

1.4

95.8

0.695

RG316 
2.49

PTFE / FEP

PTFE

0.9

95.1

0.695

RG393 
9.91

PTFE

PTFE

5.0

96.5

0.695

RG400 
4.95

PTFE

PTFE

1.9

96.5

0.695

Sources:
Uniradio Metric series data from:
BICC Cableselector E15.
PTFE Coaxial Cables. July 1979.
American Radio Guide series data taken from:
The ARRL Antenna
Book, 19th edition, ARRL publ, 2000. ISBN: 0872598047.
Coaxial cable data p2419.
* Coating materials
vary / options
exist  check manufacturer's data. NEVER use PVC in RF applications
or where high temperatures may occur.
FEP = Fluorinated Ethylene Polypropylene. ε
r'
= 2.1 (nonpolar).
** Capacitance per unit length may vary depending on manufacturer.
Inductance per unit length:
L0
=
50²
C0
.
Table 16.4. Toroidal Core Dimensions:
Core

Outside dia.
D /mm

Hole dia.
d /mm

Thickness
h /mm

Mean path
le /
cm

Core area
Ae / mm²

Turn length*
/ mm

T25

6.35

3.05

2.44

1.5

4.2


T30

7.80

3.83

3.25

1.83

6.5


T37

9.53

5.21

3.25

2.32

7.0


T44

11.18

5.82

4.04

2.67

10.7


T50

12.70

7.62

4.83

3.20

12.1

≈18.7

T68

17.53

9.40

4.83

4.24

19.6

≈21.8

T80

20.19

12.57

6.35

5.15

24.2


FT23

5.84

3.05

1.52

1.34

2.1


FT37

9.53

4.75

3.18

2.15

7.6


FT50

12.70

7.14

4.78

3.02

13.3

≈19.1

FT50A

12.70

7.92

6.35

3.68

15.2

≈21.5

FT50B

12.70

7.92

12.7

3.18

30.3

≈34.2

FT82

20.96

13.11

6.35

5.26

24.6


*Wire length required for 1 turn estimated as ≈
D  d + 2h + 4mm
Table 16.5. Maximum number of turns in singlelayer
winding:
AWG

Wire dia
/ mm

T25
(3.05 ID)

T30
(3.83 ID)

T37
(5.21ID)

T44
(5.82 ID)

T50
(7.62 ID)

T68
(9.40 ID)

T80
(12.6 ID)

18

1.024

4

5

9

10

16

21

30

20

0.810

5

7

12

15

21

28

39

22

0.643

7

11

17

20

28

36

51

24

0.511

11

15

23

27

37

47

66

26

0.404

15

21

31

35

49

61

84

28

0.320

21

28

41

46

63

79

108

30

0.254

28

37

53

60

81

101

137

32

0.201

37

48

67

76

103

127

172

34

0.160

48

62

87

97

131

162

219

36

0.127

62

78

110

124

166

205

276

38

0.099

79

101

140

157

210

257

347

40

0.079

101

129

177

199

265

325

438

Table 16.6. Core Materials:
Core
material

Type

Initial permeability.
μi

TC*
2070°C
/ ppm/°C

Tuned circuit freq.
range
/ MHz

Broadband xformer
freq. range / MHz

Available
sizes.

Carbonyl
Iron

2 (red)

10

95

2  30

No data**

T25, 30, 37, 44, 50, 68, 80 
1 (blue)

20

280

0.5  5

No data

15 (red/wh)

25

190

0.1  2

3 (grey)

35

370

0.05  0.5

No data**

NiZn
Ferrite

67

40

1300

10  180

200  1000

FT23, 37, 50, 50A, 50B, 82 
61

125

1500

0.2  10

10200

43

850

12500

0.01  1

1  50

* Temperature coefficient of initial
permeability.
These figures are approximate (see manufacturer's graphs for more
accurate figures at the temperature of interest).
** No data in catalogue, but known to be useful at least over
the 1.8  30MHz range (Broadband transformer requirements are
less stringent than for highQ inductors).
Table 16.7. Iron powder cores, A
L
Values / nH/turns², ±20%
Type

T25

T30

T37

T44

T50

T68

T80

1 (μi=20)

7.0

8.5

8.0

10.5

10.0

11.5

11.5

2 (μi=10)

3.4

4.3

4.0

5.2

4.9

5.7

5.5

3 (μi=35)

10.0

14.0

12.0

18.0

17.5

19.5

18.0

15
(μi=25) 
8.5

9.3

9.0

16.0

13.5

18.0

17.0

Table 16.8. Ferrite cores, A
L
Values
/ nH/turns², ±25%
Type

FT23

FT37

FT50

FT50A

FT50B

FT82

43 (μi=850)

188

420

523

570

1140

557

61 (μi=125)

24.8

55.3

69

75

150

73.3

67 (μi=40)

7.8

17.7

22

24

48

22.4

Let us propose, at this point, that it has been decided that the
current transformer will be fitted over a coaxial cable of 5mm
diameter such as RG303. This immediately limits the choice to
T44, T50, T68, FT50, FT50A, and FT50B. We may however expect
the T44 (5.8mm hole diameter) to be a very tight fit when wound
with wire of about 0.2mm diameter, and the T68 (9.4mm hole diameter)
to be a loose fit unless wound with wire of about 1mm diameter
(max. 21 turns). We will therefore reject the smaller cores on
the grounds that their A
L
values are not
significantly different from T50 versions; and try to avoid using
the larger cores unless special requirements dictate otherwise.
Turning our attention to the ferrites, we can also observe that
the highpermeability type43 material has a huge temperature
coefficient (1.25%/°C  which will affect lowfrequency phase
accuracy) and will be too lossy for medium to highpower applications
(see loss factor vs frequency curves in ref [
38] ).
[
38]
Amidon Associates
Inc. (Technical data book) Jan 2000.
Technical data for iron powder and ferrite cores, including A
L values, frequency ranges, wire
packing tables,
Q curves, etc. Maximum flux density calculations and recommendations:
p135. Information is also available online from:
www.amidoncorp.com
.
The lowpermeability type67 material also has no great advantage
over powdered iron (it is slightly less lossy than iron, but the
benefit is marginal). We will therefore consider five primary
candidate cores: T502, T503, FT5061, FT50A61, and FT50B61,
and two secondary candidates: T682 and T683. We can now perform
calculations to find the numbers of turns on each of these cores
that will give at least 4.42 μH (i.e., X = 50 Ω @ 1.8 MHz);
but for the sake of those who wish to design RF ammeters, or who
want to maintain bridge sensitivity at LF, we will also tabulate
results for various multiples of this inductance (but see also:
the maximallyflat current transformer). When the A
L
value is specified in nH/turns², the inductance of a winding
on a particular core is given by:
L / nH = A
L N²
and the required number of turns for a given inductance is:
N = √(L / A
L ),
where L is in nH,
and A
L is in
nH/turns².
Table 16.9. Turns required for target inductance.
(Approximate
wire lengths, computed from core dimensions, are shown in brackets
below the numbers of turns).
X @ 1.8 MHz
→
L →
Core AL/nH

50Ω
4.42μH

100Ω
8.84μH

150Ω
13.26μH

300Ω
26.5μH

350Ω
30.9μH

700Ω
61.9μH

T502

4.9

30.0
(55cm)

42.5
(79cm)

52.0
(96cm)

73.6
(1.36m)

79.5
(1.47m)

112.4
(2.08m)

T682

5.7

27.8
(61cm)

39.4
(87cm)

48.2
(1.05m)

68.2
(1.48m)

73.7
(1.61m)

104.2
(2.27m)

T503

17.5

15.6
(30cm)

22.5
(43cm)

27.5
(52cm)

38.9
(72cm)

42.1
(78cm)

59.5
(1.11m)

T683

19.5

15.1
(33cm)

21.3
(46cm)

26.1
(57cm)

36.9
(81cm)

39.8
(87cm)

56.3
(1.22m)

FT5061

68.8

8.1
(15cm)

11.4
(22cm)

14.0
(27cm)

19.8
(38cm)

21.3
(40cm)

30.2
(57cm)

FT50A61

75

7.7
(18cm)

10.1
(22cm)

13.3
(28cm)

18.8
(41cm)

20.3
(43cm)

28.7
(63cm)

FT50B61

150

5.4
(21cm)

7.7
(27cm)

9.4
(34cm)

13.3
(48cm)

14.3
(51cm)

20.3
(69cm)

Previously it was mentioned that for operation up to 30 MHz, the
winding length should be kept well below 1.2 m, and for operation
up to 54 MHz, the winding should be shorter than 67 cm. It should
also be said that, while transformers can be expected to give
unacceptable phase errors at high frequencies if these lengths
are exceeded; we cannot be sure that the phase performance will
be acceptable if they are not. Some form of HF phase compensation
will usually be necessary, and the fewer the turns the better
it will be. We should also note, that although it is not always
stated in the manufacturer's data, there is a tolerance associated
with core A
L values; and so
turns should
be calculated for an inductance of about 20% higher than the minimum
acceptable value, and the actual inductance of the winding should
preferably be measured for the purpose of calculating frequency
compensation networks. The origin of the '10 to 40 turns' ruleofthumb
is therefore apparent from the data in the table above. Notice
also, that in the case of the FT50series beads, increasing the
bead thickness increases the wire length faster than it increases
the inductance, which means that the thicker (A and B) beads have
no advantage over the standard FT50 unless power transmission
is important. Since, in designing currentmonitoring transformers,
we are only interested in abstracting about 1 Watt from the generator,
power throughput is not a major consideration (and core saturation
is impossible in this application with any of the available materials).
We should recall however, that core loss occurs in the absence
of any secondary winding, and a substantial part of this will
appear as an additional resistive component in parallel with the
primary impedance. Consequently, the low permeability beads are
to be preferred when monitoring the outputs of highpower transmitters.
Also there may be situations in which we desire to use a transformer
identical to the current transformer for voltage sampling, in
which case the more stringent flux density considerations for
the voltage transformer will dictate the design of the current
transformer.
It is, of course,
always instructive
to see what others have done before embarking on a design, and
so we will finish this section by reviewing some actual transformers.
A current transformer described in the ARRL antenna book [19th
edition 2000, ch.27, p10] uses an Amidon T503 (12.7mm diam. μ
i=35) powdered iron core with 31
turns of 24
AWG wire on the secondary and a secondary load of 50 Ω. The
A
L value of
17.5 nH/turn² for this
core predicts an inductance of 16.8 μH for the secondary winding,
and a reactance of +190 Ω at 1.8 MHz. Using the lowfrequency
analysis developed in
section 10,
η=190/√(50²+190²)=0.967,
corresponding to
a drop in output of 3.3% or 0.29dB at 1.8 MHz. The transformer
is usable from 1.854 MHz, and is stated to give an output that
is flat within 0.3 dB from 1.850 MHz with the generator working
into a 50 Ω load. This transformer can be used with signal
power levels up to 1.5 kW from 3.5 to 30 MHz, with reduced power
handling at 1.8 MHz. Increasing the core size to T682 (17.5 mm
diam, μ
i=10) and using
40 turns of 2630
AWG wire on the secondary gives 1.5 kW signal handling at 1.8 MHz,
but reduces the upper frequency limit to around 30 MHz. The A
L value of the T682 core is
5.7 nH/turn²,
corresponding to an inductance of 9.12 μH for 40 Turns, and
a reactance of +103 Ω at 1.8 MHz. Thus
η=103/√(50²+103²)=0.9, giving a
drop in output
of 10% or 0.92 dB at 1.8 MHz. Clearly, both of these transformers
give good performance in highpower applications, but the output
voltages are somewhat low at the 100 W level (1.414A primary current),
being 2.28 V RMS for the 31 turn version and 1.76 V RMS for the
40 turn version; which leads to the need for a linearised detector
[see
diode detectors] or a fixedgain arrangement
with
a nonlinear meter scale in some applications. In order to increase
the voltage output it is necessary to reduce the number of turns
and use a higher permeability (i.e., ferrite) core, thus reducing
the power handling capability. A test transformer, consisting
of 10 turns wound on an FT5061 core (12.7mm diam. μ
i=125)
and loaded with 50 Ω, gave a relative amplitude response
that was consistent with the ideal transformer with secondary
inductance model from 1.8 to at least 30 MHz, but is suitable
only for power levels of around 100 W (continuous) in 50 Ω
systems. The calculated output of this transformer however is
7.07 V RMS when the primary current is 1.414 A. The A
L
value for the FT5061 core is 68 nH/turn², giving an inductance
of 6.8 μH for 10 turns, and a reactance of +77 Ω at
1.8 MHz.
In this case η=77/√(50²+77²)=0.84,
corresponding
to a drop in output of 16% or 1.53 dB at 1.8 MHz.
13. Currenttransformer propagation delay.
It is often stated in the technical literature that the
selfcapacitance
of a singlelayer coil arises from the capacitance between adjacent
turns. This idea was put forward in the early part of the 20th
Century; but the experimental 'evidence' in favour was shown to
be fraudulent by R. G. Medhurst in 1947[see:
Selfresonance and selfcapacitance of solenoid coils, by DWK]. Bizarrely, the
selfsame theory resurfaced as new research in 1999, and got through
the peerreview process despite citing the paper that refuted
it. In fact, there is good reason for supposing that a coil has
very little selfcapacitance as such, because the properties of
inductors arise from the propagation of electromagnetic energy
along the winding wire. The DC conception of energy stored in
a magnetic field is a special limiting case of energy stored in
an EM wave, which is detained in the coil due to the finite time
it takes to make its convoluted journey along the wire. The idea
of capacitance between the turns then falls down because the electric
vector of the travelling wave is perpendicular to the coil axis,
i.e., there is very little Efield component parallel to the axis
and so very little capacitance in the conventional sense. 'Self
capacitance' is instead a convenient fiction, which allows us
to cling to the concept of lumped inductance and still account
for the fact that the coil has a selfresonant frequency (well,
many in fact, but the lumpedcomponent theory becomes inaccurate
as we approach the lowest one). The first selfresonance occurs
when the wire used to wind the coil is one electrical halfwavelength
long, because this is the condition that allows the wave to arrive
back at its starting point in phase with itself. The effective
propagation velocity at this frequency is also surprisingly close
to the speed of light, but the issue is complicated because the
coil is a dispersive transmissionline, i.e., the velocityfactor
changes with frequency. Hence, selfcapacitance measurements,
which depend on resonating the coil against a known capacitance
and noting the difference between the actual f
0
and the f
0 predicted from the
inductance,
give different answers depending on the measurement frequency.
Fortunately, provided that we keep well away from the opencircuit
resonance frequency (which can be calculated fairly accurately
just from the length of the wire), the velocity is reasonably
constant, which means that the apparent selfcapacitance is reasonably
constant (even if it nolonger predicts the selfresonance frequency),
and we can just about get away with the concepts of lumped inductance
and capacitance for the purposes of circuit design.
In order to deduce an
expression
for the propagation delay that occurs in a transformer secondary
winding, we can start by considering the effect of a magnetic
disturbance halfway along the wire. This disturbance applies
equalandopposite electromagnetic forces to the conductors leading
away from it, resulting in two equalandopposite waves that propagate
towards the transformer terminals (see diagram right).
The two waves combine at the terminals to produce a finite output
voltage because they cause opposing displacements of the potentials
at the terminals.
Now consider what
happens when the
distances travelled by the two waves are not equal (i.e., when
the magnetic disturbance occurs at an arbitrary point in the coil).
We can work out what happens by observing that the transformer
is a linear system (neglecting coresaturation effects, which
only occur under extreme conditions). This means that there is
no mixing between components at different frequencies, and we
can analyse the behaviour at an arbitrary frequency to deduce
(within reason) what happens at all frequencies. So we may regard
our two waves as sine waves of the same frequency, and the sum
of two such waves is always another sine wave. If the distance
to the terminal in the (arbitrarily defined) upstream direction
is less than the distance in the downstream direction; then the
upstream wave arriving at the terminals will be advanced relative
to the downstream antiwave; but the two waves will combine to
produce an average phase that is the same as it would have been
had the disturbance occurred at the exact mid point. Hence, regardless
of the point of disturbance, the output of the transformer has
a phase dictated by the average distance to the terminals, and
this is always half the length of the winding wire (i.e.,
w/2).
In fact, transformers
would not
work if the output phase was not substantially the same for magnetic
changes at all points on the wire; because a change in magnetisation
of the core causes nearly simultaneous magnetic disturbances at
all points in the coil. Hence we can consider the output to arise
from an infinite number of magnetic disturbances, all of which
produce their effect at the terminals in phase and therefore add
together. If the arrival phases were not all the same, there would
be considerable cancellation, resulting in a loss of output. As
it transpires, we already have a name for this loss of coupling
caused by imperfect linking between the core and the winding;
it is called
leakage inductance.
So the output of the
transformer
is retarded by an amount equal to the time it takes for an
electromagnetic
wave to travel half the length of the winding wire. In fact, this
retardation occurs in both primary and secondary windings; but
in a current transformer with a singleturn primary, the additional
primaryside delay is relatively small. If we call the propagation
time t
p, and the velocity v,
noting that
the units of velocity are [distance]/[time], we have:
v =
w
/ (2 t
p)
The velocity of propagation of an electromagnetic wave is:
v = 1 / √ (μ ε)
where μ is the permittivity and ε is the
permeability
of the environment. We can break this relationship down further
by noting that permittivity is the product of relative and freespace
permittivities, and likewise for permeability. Hence:
v = 1 / √ (μ
0
μ
r
ε
0 ε
r
)
But 1/√(μ
0
ε
0)
is the velocity of light, c. Also, the quantity √(μ
r ε
r)
has a name
with that some people may be familiar, it is the
refractive
index, n (and 1/n is the velocity factor). Thus we have:
c / n =
w
/ (2 t
p)
i.e.,
t
p = n
w / (2
c) . . .
. (
13.1)
Now, consider the transformer operating at a frequency f. The
timepercycle (period, t) is 1/f. The
timedelay
in the
transformer is the negative of the propagation time, and so the
delay expressed as a fraction of one cycle is:
Δt / t = t
p / t =
f t
p
There are 2π radians in one cycle, and so the delay expressed
as an angle in radians is:
Δφ = 2πf t
p .
. . . (
13.2)
The phase shift is negative. The timedelay manifests itself as
a capacitive effect. This capacitance is fictitious in the sense
that it does not exist for static electric fields; but it is genuine
in the sense that energy put into the transformer does not emerge
immediately and so is temporarily stored.
Precise measurements
of currenttransformer
HF phase error vs. frequency are given in [Evaluation and optimisation
of RF current transformer bridges, section 16a]. When the effects
of transformer secondary inductance and ferrite permeability dispersion
are removed from the data, Δφ vs. f is a
straightline
graph. This is as predicted by equation (
13.2);
i.e., the experimental data support the view that, when other
contributions are controlled, the remaining phase error is due
to propagation delay.
In order to represent the time delay as a parasitic capacitance,
we note that the observable phase shift can be reproduced by placing
a capacitance C
i' in parallel
with the
secondary terminals.
The phase shift depends on the load resistance and is given by:
Δφ = Arctan(R
i
/ X
Ci')
i.e.:
Tan(Δφ) = 2πf C
i'
R
i
Hence, using (
13.2):
2πf C
i' R
i =
Tan(2πf t
p)
But for small angles, Tan(x)→x (when x is in radians). Hence,
to a reasonable approximation:
C
i' R
i
= t
p
Using equation (
13.1)
then
gives:
Ci'
= n w
/ (2 Ri c) 
(13.3)

So, we can estimate C
i'
provided that we
can put a number to the refractive index n (or the velocity factor
1/n). This is an intriguing problem, because it forces us to consider
the medium in which the electromagnetic energy is travelling.
Practically none of the energy will be inside the wire because
a conductor cannot support an electric field (this being the reason
for the skin effect). Hence the energy must be distributed around
the outside of the conductor, and to some extent within the magnetic
core material.
For a transformer
core, we can estimate
the refractive index as √(μ
i
ε
r), where
μ
i is the
initial permeability. The permittivity of magnetic materials is
somewhat harder to come by, but data given by Snelling [
48]
suggest that it is in the 10 to 100 range for NiZn ferrites. For
the experiments described in [Eval & Opt.] type 61 ferrite,
which has μ
i=125, was
used. If we take
ε
r=20 as a fair
approximation,
this gives n=50, or a velocity factor 1/n=0.02. Thus, if the energy
propagating along the wire were concentrated in the core, we would
expect an enormous propagation delay.
Ref:
[
xx]
"RF Autotransformers
 Transmission Line Devices modelled using SPICE", Nic
Hamilton, G4TXG, Electronics World, Nov. 2002, p5256. Dec. 2002
p2026.
Part 1: Limitations of the conventional transformer model. Transmission
line model. Part 2: Core losses. Winding resistance.
[
48]
Soft Ferrites: Properties
and Applications. E C Snelling. 2nd ed. Butterworth. 1988.
ISBN 0408027606.
Permittivity of ferrites: p127  129.
In fact, the
propagation delay is
modest. It is difficult to separate the phase shift due to delay
from other sources of HF phase error, but from the author's data
it appears that the refractive index is around 1.2 (velocity factor
= 0.83). This suggests that the energy is primarily concentrated
in the wire insulation and the air around the wire. It also indicates
that we should use wire with very thin (and preferably nonpolar)
insulation. Previously, the use of plasticcoated wire was criticised
on the basis that it stuffs the core with insulator and makes
less room for the conductor. Now we criticise it on the basis
that it increases propagation time. Use enamelled wire (or perhaps
even bare wire, if you can be sure that adjacent turns won't touch).
Hence, for the
purpose of estimating
the secondary selfcapacitance of transformers wound with enamelled
wire on type 61 ferrite (and probably accurate enough for other
core types as well), equation (
13.3)
becomes:
Ci'
= 1.2 w
/ (2 Ri c) 
(13.3a)

Combining the constants, using c=299729458 m/s, results
in the simple formula:
Ci'
/pF = 2 (w
/mm) / (Ri /Ω) 
(13.3b)

Although the foregoing permits us to estimate HF phase error,
it is important to be aware that there is a subtle difference
between parasitic capacitance and time delay. A lumped capacitance
in parallel with the secondary winding provides a reasonable basis
for simulating the behaviour of the transformer; but it is nevertheless
an approximation, and it is obviously a fiction because it depends
on R
i. The point is that a
secondary parallel
capacitance moves the load impedance clockwise around a circle
of constant conductance as the frequency increases, whereas a
timedelay moves the output phase around a circle centred on the
graph origin (0,0). In other words, a pure timedelay affects
only the output phase, without affecting the magnitude, whereas
a parallel capacitance affects both.
It will transpire
that the simple
relationship between delay and selfcapacitance derived above
is sufficient for the purpose of building accurate bridges. Against
the risk of being accused of lack of scientific rigour however,
it must be said that a transformer winding is a transmission line,
and a pure time delay with no attendant impedance transformation
only occurs when the line is matched. We can always estimate the
phaseshift from a knowledge of the linelength and the
velocityfactor,
but to quantify the impedance transformation it is necessary to
know the characteristic resistance.
In the approximation
that the line
is lossless, the characteristic (surge) resistance is:
R
surge = √(
L
/
C)
Where
L is the inductance per unit length and
C
is the capacitance per unit length. It is tempting to think that
the inductance of the line will be the same as the inductance
of the transformer winding, but this is not the case. Thinking
back to the idea of a wave and an antiwave created by a magnetic
disturbance; note that the upstream line and the downstream line
are in a state of close magnetic coupling, via the core, and that
the mutual inductance is negative. This means that the inductance
of the line overall is largely cancelled, just as it is when the
two conductors of an ordinary transmission line are brought into
proximity. Indeed, it is this ability to cancel its own inductance
when energy is abstracted from the secondary that allows a transformer
to work. In an ideal transformer, the cancellation is perfect,
and so we can deduce that the distributed inductance of the
transmission
line is the
leakage inductance of the winding.
Author's note:
Became embroiled in difficult theory at this point.
>>> writing in progress.
The matters discussed here are explored in the experimental research articles given in this section.
>>>>> ?? Speculative [no, not any
more].
The leakage
inductance of a winding
on small toroidal transformer is usually about 1% of the total
(it depends on the permeability of the core). Hence for a winding
with an inductance of 10 μH, we expect the leakage inductance
to be of the order of 100 nH. If we wish to determine the surge
resistance, of course, we also need to determine the distributed
capacitance; and this is another perplexing problem. The capacitance
will certainly be greater than the 8.85 pF/m of free space, and
it will depend on the proximity of other objects (such as the
Faraday shield). The best we can say (via a somewhat recursive
argument) is that it will be of the order of the 'selfcapacitance'
C
i'. In one of the author's
experimental
currenttransformers, with a secondary inductance of 9 μH and
a leakage inductance of about 9 0nH, it was estimated that C
i' was about 9 pF. Hence, for this
transformer,
a fair guess for √(
L/
C)
is that it is of the
order of 100 Ω. This astonishing result suggests that the
empiricallyderived wisdom that small RF transformers give nearly
ideal performance when terminated in 50 Ω (or thereabouts)
is influenced by the underlying transmissionline behaviour. It
means that the impedancetransformation occurring within the winding,
seen as a deviation from ideality, is small or negligible. The
line is verynearly matched when the load is a few tens of Ohms;
and the HF phaseshift, having minimal associated change in output
magnitude, looks like an almostpure timedelay.
>>>>>
Is this true? The measurements of [Eval & Opt.  sec.16a]
suggest that it is. Sheath helix (slowwave) theory, on the other
hand suggests that the time delay should, in principle, vary with
frequency. Further investigation is required, but it appears that
the superposition of the axial slowwave and the superluminal
helical wave results in a wave that does have a phase velocity
close to c.
>>>
Gap capacitance, leadwire capacitance and pitchangle effect
in coils of low N.
14. Effective secondary capacitance:
Although propagation delay makes a substantial contribution to
the effective secondary capacitance required for modelling purposes;
it is by no means the whole of the story. There are various other
effects, some of which increase the apparent capacitance, and
some of which reduce it and can make it negative overall. Failure
to recognise and quantify these influences gives rise to
inconsistencies
of performance
>>>>>
The topics for the next few sections are to be based on the
experimental
work described in [Eval & Opt.].
14a. Faradayshield protrusion capacitance.
>>> writing in progress
Throughline mismatch.
Secondary load reactance.
Perturbation series for C
i.
15. HF Neutralisation:
circuits that fake a transformer with no selfcapacitance.
Load port compensation capacitor.
Herzog's compensation.
Quadrature current injection. Neugebauer and Perrault
Parasitic Capacitance Cancellation in Filter Inductors.
T C Neugebauer and D J Perreault. 35th Annual IEEE Power Electronics
Specialists Congerence, 2004. The parasitic capacitance of a
powersupply
filter inductor can be cancelled by use of an auxiliary winding
and a capacitor.
Quadrature voltage addition
Phaseshift compensation.
Delaying the voltage sample.
© D W Knight 2008.
David Knight asserts the right to be
recognised
as the author of this work.