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148 IEEE TRANSACTIONS ON PARTS, HYBRIDS, AND PACKAGING, VOL. PHP-7, NO. 4, DECEMBER 1971
Twisted Magnet Wire Transmission Line
PETER LEFFERSON
Abstract - Transm ission line principles are applied to twisted
magnet wire lines made of two wires to establish design limits.
Express ions are developed to predict the effects of wire film insulation
and of twisting. A design procedure is developed to realize a desired
characte ristic impedance for the design of radio frequency broad-band
transformers , signal combine rs, and pulse transformers.
INTRODUCTION
B
OAD-BAND transformers and signal combiners for HF
thru UHF and the digital field using twisted magnet wire
transmission line, have been discussed in the literature and are
widely used. [l]-[3]. The terminal impedances of these
passive devices are a function of: the transmission-line charac-
teristic impedance, line length, core material, and source
impedance at other te rminals. If the line impedance can be
controlled, these devices can be built with a wide frequency
range without requiring additional lumped elements. The
terminal reactance can be tailored in the same way to improve
matching into active devices.
This paper will consider only the twisted magnet wire
transmission line and a design procedure will be given as a
function of wire size, wire insulation thickness, insulation
relative dielectric constant, and wire twist.
Twisted magnet wire transmission lines using the wire size
range from 4 to 44 can be realized with characteristic
impedances ranging from 10 to 85 R. More practical examples
would be a 50-a line made with a pair of number 40 wires and
a 25,a line made with a pair of number 15 wires.
Basic Dimension Standard
The United States military specification for magnet wire,
MIL-W-583, will be used for a well-ordered base on which to
build a design procedure. Among other th ings, this specifica-
tion defines round magnet wire having film insulation. It lists
American Wire Gage sizes 4-44 in four insulation thickness
groups with their dimensions and tolerances. The minimum
insulation thickness for each wire s ize in groups 2-4 is
approximately the minimum thickness for the wire size in
group one multiplied by the group number. The maximum
thickness is about equa l to the minimum of the next higher
group. These data from MIL-W-583 is repeated in Table I.
Using the dimensions from this table and the expression for
Manuscript eceived une 9, 1971; evisedSeptember 3, 1971.
The author
s with the Milton Roy Company , St. Petersburg, Fla .
C.L. Ruthroff, Some broad band transformers , Proc. IEEE, vol.
47, pp. 1337 - 1342, Aug. 1959.
0. Pitzalis, Jr., Practica l design information for braod-band
transmission line transformers,
Proc. IEEE, vol. 56, pp. 738 - 739,
Am. 1968.
* 3R.E. Matick, Transmission line pulse transformers - Theory and
application,P roc. IEEE, vol. 56, pp. 47 - 62, Jan. 1968.
characteristic impedance of two parallel wires [4]
Z1 = p cash-4.
-
(1)
Vt req
u
The computer calculated characteristic impedances are plotted
in Fig. 1 for a relative dielectric constant of 1 O.D and d are
the wire diameters with and without the insulation film,
respectively. freq is the equivalent relative dielectric constant.
Fig. 1 shows the calculated impedances for the film-thickness
extremes.
WIRE TWIST AND PITCH ANGLE
The effect of twisting the line can be normalized for all
wire sizes if the twist per inch T is equated to the angle
included between each wire and a line drawn down the center
of the transmission line (pitch angle0). For bifilar w ire, twist
is given by
tan 0
j-z-----
I-iD
(2)
The relation of tw ist and pitch angle is plotted in Fig. 2 for
wire sizes 4-44 and the insulation thicknesses of classes one
and four.
Experimental data suggest that optimum performance is
obtained from lines having pitch angles between 20-45.
When the line twis t is loose, it becomes difficult to maintain
continuous line geometry as the line is wound on a form. This
is seen as a large reflection on a time-domain reflectometer.
The transmission line is subjected to excessive stress as the
twist angle approaches 50.5 and in the vicinity of 50.5 it will
break. Equation 2 and. he maximum pitch angle are developed
in the Appendix.
A line impedance rise of 1 or 2 S2can be anticipated when
the line is wound of a ferrite form.
FILM DIELECTRIC
The equivalent dielectric constant that must be considered
in order to give meaning to the theoretical characteristic
impedance of F ig. 1, is a function of the film insulation and
pitch angle. It is common to express a system having two
dielectrics as
freq = (YE~I+ /3Er* CX /3 = 1
=Erl (1 -P)+PEr*
= 91 + P (Er2 - frl>,
where erI and er2 are the relative dielectric constants of air (or
some other surround ing material) and the film insulation ,
41nternationa1 Telephone and Telegraph Co., Reference Data for
Radio Engineers, 5th ed.
New York: Sames, 1969, ch. 22, p. 22.
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LEFFERSON: TWISTED MAGNET WIRE
149
TABLE I
MINIMUM INCREASE IN DIAMETER OF BARE ROUND WIRE AND THE MAXIMUM
OVERALL DIAMETBR DUE TO FILM INSULATION
Wire size
(AWG)
CLASS 1
CLASS 2 CLASS 3
CLASS 4
Minimum Maximum Minimum Maximum Minimum Maximum Minimum Maximbm
Diameter increase in overall increase in overall
increase in overall
increase in overall
nominal diameter diameter diameter diameter diameter
diameter diatieter diameter
(W (InI (In)
(InI (In) (InI
(InI (InI
(In)
4---
5---
6-e-
7---
8--m
9---
lO---
ll---
12---
13---
14---
15---
16---
17---
18---
19---
20---
21---
22---
23---
24---
25---
26---
27---
28---
29---
30---
31---
32---
33---
34---
35---
36---
37---
38---
39---
40---
41---
42---
43---
44---
0.2043
. 1819
.
1620
.
1443
. 1285
. 144
. 1019
. 0907
.0808
.0720
.0641
0571
: 0508
.0453
.0403
.
0359
.0320
. 0285
.0253
. 0226
. 0201
.
0179
. 6159
.
0142
.
0126
. 0113
.
0100
. 0089
.0080
. 0071
.0063
. 0056
.
0050
.0045
.0040
.0035
.
0031
. 0028
.0025
.0022
.0020
0.0019
. 0019
.0018
.0017
.0016
.bo16
.0015
.0015
.0014
.0014
.
0014
.0013
.0012
.0012
.OOll
.OOll
. 0010
.OOlO
.OOlO
. 0009
.
boo9
. 0009
.0008
.0008
.0007
.0007
.0006
.0006
.0006
.0005
.0005
.0004
.0004
.0003
.0003
.0002
.0002
.0002
.0002
.0002
. 0001
0.2093 0.0037 0.2111 0.0049
. 1867 .0036 . 1884 .0048
. 1665 .0035 . 1682 .0047
. 1485 .0034 . 1502 .0046
. 1324 .0033 . 1342 .0045
. 1181 .0032 . 1198 .0044
. 1054 .0031 . 1071 .0043
. 0941 .0030 . 0957 . 0042
.0840 .0029
.0855 .004b
.0750 .0028 . 0765 .0039
.0&70 .0027 .0684 .0038
. 0599 .0026
.0613 .0037
.0534 .0026
.0548 .0036
.0478 .0025 . 0492 .0035
.0426 .0024 .0440 .0034
-0382 t 0023
. 0395 .0033
.0341 .0022 .0353 .0031
.0306 .OOZl . 0317 .0030
.Q273 .0020
. 0284 .0029
.0244 .0019 .0255 .0028
.0218 .0019 . 0229 .0027
. 0195 .0018
.0206 .0026
. 0174 .0017
.0185 .0025
. 0156 .0016
. 0165 .0023
0139 .0015
. 0148 .0022
. 0126 .0014 . 0134 .0021
. 0112 .0013
. 0120 .0020
. 0100 .0013 . 0108 .0019
. 0091 .0012 . 0098 no018
.0081 .OOll . 0088 .0017
.0072 .OOlO
. 0078 .0015
.oo64 .0009 .0070 . ooi4
.0058 .0008 .0063 .0013
.0052 .0008
.0057 .0012
.0047 .0007 . 0051 .OOll
.0041 .0006 .0045 . 0010
.0037 .0006 .0040 .0009
.0033 .0005 . 0036 .0008
.0030 .0004 .0032 .0008
.oO26 ,0004 . 0029 .0007
.0024 .0004
.0027 .0006
0.2125 0.0064
. 1897 .0062
. 1695
. 1515
: 0059
0059
.1355 .0058
. 1211 .0057
: 0969084 .00540056
.0867 .0052
. 0776 .0050
. 0695 .0049
.0624 .0048
.0558 .0046
.0502 .0045
. 0450 .0044
.0404 :0042
.0362 .0040
.0326 .0039
. 0292 .0037
.0263 .0036
. 0237 .0035
. 0214 .0034
. 0192 .0032
. 0172 .0030
. 0155 .0029
. 0141 .0028
. 0127 ,0027
. 0115 .0026
. 0105 .0025
. 0095 .0024
. 0084 .0021
.0076 .0020
. 0069 .0019
; 0062 .0017
.0056 .0016
. 0050 .0015
.0044 .0013
.0040 .OOlO
.0037 .OOlO
.0033 .OOlO
.0030 .0009
0.2148
.
1920
.1717
, 1537
.
1377
. 1233
.1106
.
0991
.0888
. 0796
.0715
.0644
.0577
.
0520
.0468
.0422
.
0379
.0342
.0308
. 0279
.0252
.0228
.0206
.
0185
.
0166
.
0152
.
0137
.
0124
.0113
.0102
.
0091
.0082
.0074
.0067
.0060
.0053
.0047
.0043
.0038
.0035
.0032
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150
IEEE TRANSACTIONS ON PARTS, HYBRIDS, AND PACKAGING, DECEMBER 1971
130-
120-
llO-
LO-
O
Z_W-
c
z 8Q-
2
E 70-
I
ua-
E
B so-
ti
24D-
1
Jo-
M-
10 -
I
I I I II I I I I I I I I, I I I I1 I I I II I I I I I I II II 1 I I1 I I I
5
10 15
20
25
30
35
40 44
WIRE SIZE (AWG)
Fig. 1. Characteristic impedance for bifilar magnet wire transmission line based on MIL-W-583 dimensions and with relative
1DODt
loo-
5
E -
w
P lo-
t -
F -
c -
1.0 -
dielectric constant of one.
I CLASS 1
- - CLASS 4
0.1 6 1
0 10
20 30 40 50
60
PITCH ANGLE IN DEGREES
Fig. 2. Bifilar transmission line twist per inch versus pitch angle for wire sizes 4-44 having class-l and 4 insulation thickness.
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LEFFERSON: TWISTED MAGNET WIRE
151
E
.q FROM C- MEASUREMENTS
E
4 ; 10 1; 2b 25 ;D 3; 4b i-0
PITCH ANGLE IN DEGREES
Fig. 3.
Measured equivalent dielectric constant versus pitch angle for a sample of no. 29 magnet wire having class-2
insulation thickness.
respectively . erl and er2 are published constants, but 0 is a domain re flectometer. The film relative dielectric constants
function of pitch angle.
were calculated from capacitance measurements of wire
The behavior of fi can be seen by observing ereq as the
samples in a mercury bath.
pitch angle is increased, If one assumes that the internal,
It was observed that the measured relative dielectric
external, and mutual inductance of the line are not changed by
constant can be greater than the published value for the film
twist, ereq can be calculated from measurem ents of the actual material. This might be expected because as the twist is made
characteristics impedance by
very tight, the wire dimension and the film thickness change
SO
calculated Zo for flat parallel wires
that they no longer fit the model for which the theoretical
Greq =
with E req = 1
parameters were calculated.
measuredZO
(4)
When the /3 is calculated from the equivalent dielectric
constant measurements, it is found to fit the expression .
It can also be calculated from capacitance measurements of
the line by
p = 0.25 + 4 x 1o-4 8 2.
(7)
measured capacitance
In Fig. 4 this expression is used to predict the characteristic
(5) .
ereq =
calculated capacitance for the same ength of
impedances for three magnet wire sizes, as samples. It is seen
to consistently predict the correct impedance sufficiently
flat parallel wires with ereq =l
closely to yield a voltage standing-wave ratio (VSWR) within
It can be calculated from the electrical line length by
l.l:l.
Greq = measure
measured electrical length
d h srcal length of the twisted liney . (6)
Laboratory data show agreement among all three of these
within the measurement accuracy of wire dimension, charac-
teristic impedance, and relative dielectric constant of the film.
Five percent random impedance fluctuation along a twisted
line is common. Fig. 3 illustrates an example using number 29
wire having a class-2 thickness of polyester film insulation. The
example shows Ereq
calculated from capacitance, electrical
length, and impedance measurements of four line samples
twisted to different pitch angles.
The slope of fi versus 8 in (7) is a function of the softness
of the wire insulation. Equation (7) holds for most modern
magnet wire insulating films but it w ill change if the insulation
is very soft. Polytetrafluoroethy lene (Teflon) covered hook-up
wire is a good example that shows how far it can change. The
relative dielectric constant versus pitch angle is given in Fig. 5
for a sample of no. 24 19-strand wire. A fit is found by using
The characteristic impedance and electrical length data
were taken with the Hewlett-Packard model 1415A time
The following steps form a useful outline for designing a
twisted magnet bifilar wire transmission line for a required
characteristic impedance.
p=o.zs+ 1 x1o-3 02 .
03)
DESIGN PROCEDURE
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152
IEEE TRANSACTIONS ON PARTS, HYBRIDS, AND PACKAGING, DECEMBER 1971
Fig. 4.
Comparison of measured to estimated characteristic impedance for 3 wire sizes versus pitch angle.
~~ MEASUREhiENTS
t
,1
1
1
I
I I
I
0
I,,
5 10
15
20 25
,
t
0 35
40 45
50
PITCH ANGLE IN DEGREES
Fig. 5.
Measured equivalent dielectric constant versus pitch angle for a sample of Teflon-covered no. 24 19-strand wire.
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LEFFERSON: TWISTED MAGNET WIRE
153
Fig. 6.
Bifilar wire transmission line twisted to its maximum pitch angle of 50.5.
1) Choose a film insulation thickness group and film
relative dielectric constant. (These are usually limited by their
availability and other physical and electrical considerations.)
2) Calculate the relative dielectric constant assuming pitch
angle of 30 from (3) and (7):
ereq = err + 0.61 (LIZ - err)
where err = 1 for air.
3) Calculate the required characteristic impedance for
relative dielectric constant of 1 from
Zl = N&q;
where
Zl
impedance when Ereq = 1;
Z desired impedance.
4) Use Fig. 1 to choose the correct wire size.
5) Use Fig. 2 to determine the required twis t per in.
CONCLUSION
The characteristic impedance of any twisted magnet wire
transmission line can be found if the dimensions and the
dielectric constant are known. Figs. 1 and 6 give the impedance
at the expected tolerance limits for standard wire sizes having
an insulation dielectric constant of 1. This is modified by the
equivalent d ielectric constant, which is a function of the
relative dielectric constant of the film insulation
and
increases as the line is twisted. The twist for various pitch
angles [determined by (2)] is shown in Fig. 2 for wire sizes
4-44. The equivalent relative dielectric constant is related to
the pitch angle by (3) and (7).
The electrical length of a twisted line is related to the
square root of the equivalent dielectric constant in (6).
Using these basic expressions the following statement can
be made.
1) The impedance increases as the wire dimensions de-
crease.
2) The impedance decreases as the wire twist is increased.
(The pitch angle increases.)
3) The relative dielectric constant for the line is controlled
more by the film dielectric constant as the pitch angle
increases.
4) For small pitch angles the line impedance can be
reduced by immersing the line in a material other than air.
5) For large pitch angles the line impedance can be
reduced by using a wire insulating film having a high dielectric
constant. The useful pitch angle range is bounded by the
limitation on line uniformity at about 20 and the strength of
copper at about 4.5.
For a film dielectric constant of 3.5 and with air as the
surrounding medium, the useful impedance range for twisted
magnet wire transmission lines is lCM50a.
Over 60 magnet wire transmission lines of many different
forms have verified the conclusions of this paper.
APPENDIX
The cen ter lines of two twisted wires form a helix as shown
in Fig. 6. The helix diameter is one wire diameter. It forms a
sine wave in the longitudinal plane.
A = (D/2) sin (2 71Td)
(9)
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154
IEEE TRANSACTIONS ON PARTS, HYBRIDS, AND PACKAGING, DECEMBER 1971
where
lines is D and twist can be expressed as
D wire diameter;
T twist per inch;
Tmax = (sin 0 )/(2D).
00
d distance along the transmission line.
The pitch angle comes out of this as
The maximum pitch angle of 50.5 is found by equating 9 and
tan 0 = 71TD
(lo) 11. It is not a function of wire diameter.
The actual wire length can also be seen n Fig. 6 where one
where 43 is the pitch angle defined in Fig. 6.
twist is unwrapped. The wire length for one twist is
The maximum pitch angle occurs when the line has been
wound to the point where the distance between wire center
nDJ1 + l@n 01.
(12)
Peter Lefferson received the B.E.E. and M.E. degrees from the University of Florida, Gainesville, in 1962
and 1965, respectively.
He worked for NASA as an Instrumentation Engineer for one year. He spent five and one half,years with
Electronic Communication Inc., in design of UHF transceivers. Presently, he is with Milton Roy Company,
St. Petersburg, Fla.
A h i d li d li i d U i id d d A i i D l d d N b 20 2009 11 47 f IEEE X l R i i l