Twisted-pair
distributed
transmission-line
parameters
For more accurate modeling of twisted-pair
transmission-line
distributed
parameters,
you must take into account several important factors. Material
and structural make-ups, twist pitch, skin effect, and proximity effect all playa
role.
By T Kien
Truong,
The Boeing
Co
importance.
Using
MathCAD,
we will
explore
the
of
the
many
so-
of a lossy transmission
line.
frequency-dependent
Abstract
Both
called
unshielded-twisted-pair
twisted-pair
(STP)
used from
your
avionics
both
types
of
address
find
using
the
explain
(hence,
reflection)
for magnetic
this
geometry
only
textbooks
parallel
the
when
the
the
effect
you
the lumped
for
dlnates
or
transceiver).
parameters
that
The
coordinates.
after
Most
transmission
series
structural
make-up,
of twist,
the
parallel
environment
with
wire
a
can provide
for simple
the
usIng
and
uniform
replace
dielectric.
are
unsuitable
particularly
or for applications
in which
integrity
at every
point
along
the
the twisted
models
frequency
the
~eometry,
the .CartesIan
model
model.
of
can compute
model
Circuit
the
rather
modeling
line
will
be the
article.
parameters
line
length)
not
of frequency
analyses
You
distributed
transmission
characteristic
account
cable
the
and
line as a function
fidelity.
ladder
Transmission-line
take into
and the effects
coaxIal
attenuation
of an upcoming
of
usIng
the
model
distributed
are separated
wires,
~Ire
can
using
function
of bus i~peda~ce
the
the twisted-pair
topic
modeling.
of
high
than
parameters
(wave
material
the
With
ohms,
R,
and
a
polar
purely
make-up
energy
jwL)/
(a+
the
resistance
common
for
resistive
situation
and
is
line
distortionless
transmission
when Zo becomes
in
unit
inductance
L,
G.
line
conductor
loss,
= Ro +jXo
only
which
c,
,£:~{'[~fi~
specified;t{r~,.;b
reactance
for
a
given
the impedance
independent
transmission
is a
loss).
is that
true
.It
(per
the geometry
misconception
example,
a
transmission
storage,
and
of
conductance
after
jwC)
air
the bus is of high
shunt
of
These
at
high
the signal
series
loss, and radiation
ZO=12O.0.,
This
C,
quantity
parameters
are modeled
propagation,
both
ZO,
is a complex
resistance
capacitance
and
impedance,
distributed
shunt
coorpair
the
These
Zo = ~(R+
as parallel
With
accurate
with
response
Introduction
s~ch
for
calculation
frequency
line
dielectric
derivation
terms
in the transmission
twisted-pair
textbooks
SPICE
and
of current-carrying
transmission-line
most
dispersion
on skin depth resistance.
Most
standard
for frequency-dependent
twisted
impedance
but also the pitch
dispersion,
accurate
physical
dimension,
constants
the proximity
dielectric
analyze
parameters
the measured
bus
(current-mode
we
pair.
configuration
in
the wires
accurately
.These
and material
wire
can
but you
the
are necessary
parameters,
often
on twisted
model
change
coupling
article,
more
for these types,
guidance
allow
to
parameters
and anyone
simulation,
don't
more
for high-frequency
cable,
Concerning
models
of
references
lines
"constants"
that
usage
transmission
or coax
most
cannot
In
on
practical
example,
reflect
lines,
the equations
find
to safety-critical
universal
especially
Papers
can rarely
are widely
of the twisted-pair
microstrip
readily
pair
Ethernet
transmission
missing,
applications.
and shielded-
lines
Despite
calculation
are mostly
For
office's
applications.
accurate
(UTP)
transmission
behaviors
of
for
is
frequency.
a
R=G=O
line in which
r-;-;-;;;
..
'\I L / C , WhICh IS real.
lossless
or
for
R/L=G/C
a
In general, we have a lossy twisted-pair
transmission line for which the parameters R, L, C,
and G are nonzero and are themselves functions of
frequency. What we need is the characterization of
a bus cable that shows the variation of its
magnitude and phase with frequency. It is only if
the impedance remains fairly constant over our
frequency range of interest that we can treat the
characteristic impedance as a constant, We then
can specify the load impedance to match the
transmission line for that particular range of
The same is true for the permeability of the
conductor, which is a complex number and
frequency.
changes with frequency: 1.1=l.1rI.10=1.1'+jl.1",
UTP impedance calculation
However, I.1r can change drastically at the low
(kilohertz) level of frequency. At the megahertz
Below are the constants and specification for a
given 1200 UTP used in this analysis (all units in
level, the relative permeability of both copper
(0,999991) and silver (0.99998) is practically
unity, and thus we will use 1.10in the equations
mks).
(Reference 1).
In general, the permittivity of a dielectric is a
complex number and changes with frequency:
E=ErEO=E'+jE", However, between 1 and 100
MHz,
the permittivity
of
PTFE
(polytetrafluorethylen, or Teflon) is almost constant at
2.1, whereas its dissipation factor is 0.0002
(Reference 1).
Wirelength:
WireSpocing:
I := 50
D := 2.11' 10-3
If you use the virgin dielectric constant, er=2,1, for
PTFE, you would miscalculate the impedance
C
d
using
d
on
to
uc or
'
t
lome er:
Virgin PTFEdielectric constont:
:=
112
.'
10 -3
the
simple
parallel
wire-over-ground-plane
.
geometry that most textbooks lIst.
e r := 2.1
Measuredeffective UTPpermitivityet
Air permitivity:
:= 1.56
p
e 0 := 8.854 ,10-12
In this example, the conductor for the twisted pair
consists of 19 silver-plated strands of copper and
multiple layers of dielectric with different material
..
D.leIect rlc
permlob'III'ty:
11:= 4 ' It , 10-7
constants as the fi gure below illustrates:
Copperconductivity:
(JI := 5.8 .107
Silver (plating) conductivity:
(J2 := 6.17 ,107
Frequency:
3
4
7
f := 10 , 10 ..10
Nominalfrequency:
,
.-4
Dielectricloss tangent:
fo := 6.106
Numberof twists (/m):
T := 20
p f := 2.10
All equations, calculations, and plotting in this.
analysis use MathCAD
software
which is a
powerful and easy-to-use math w~rksheet. After
establishing the equations, you can substitute the
transmission-Iine
values with your own twistedpair parameters and watch all the calculated results
and plots change instantly.
,
..
The calculatIon for the helIcal pItch angle of twist
(Referenc~ 2) plus the b~s length (before twisting)
and velocIty of propagatIon are:
9 := atan{T'1t'O)
9=
7,55odeg
10:=T,I,1t'O,~
10=
50,44
jl-r~
v
1
,-
v
P'--
p
= 0,8
F:;:
Using the twist angle as a factor to compensate for
the equivalent dielectric constant and with q as the
twist correction factor, the expression for the
equivalent dielectric constant, eeq=I +q( er-1), is
based on the facts that eeq=l for all air (er=1) and
eeq=er for all dielectric (q=l). The parallel
equivalent of these layers should match the
measured overall dielectric constant of 1.56.
We can calculate the impedance of the twisted pair
(still assuming losslessness)as follows:
(
-3
q := 0,45
+ 1,10
180
q = 0,51
Rdc:={1,15)'-'
(19)
25
e eq = 1,56
-11
C eq = 3,47-10
~
2
r(d\2
(d )2
l1t' \ 2)
1
,a 2 J
Rdc=0,029
At high frequency, the sk,in ef~ect forces mo~t of
the propagation of electrical sIgnal to the sIlver
,- ( 11'e 0' e eq
eq ,C eq
)
L
eq
= 4 99 -10-7
,
Z eq := C eq
plating.
You commonly use silver for better
conductivity and better corrosion resistance.
The setup for the equivalent dielectric constant as a
function of the twist pitch is as follows:
~
Z eq = 119,83
eq
Unlike
1
~
e eq := 1 + q , ( e r -1 )
L
~{ f) :=
2
' 9'-
(1t'eo'eeq)
For this lossy twisted-pair line, you calculate the
skin depth and dc resistance as follows (taking into
account the proximity factor and the conductor
area ratio as a good approximation):
)
1t
C eq :=
As the bus-core figure illustrates, the diameter of
each strand is about one-fifth of the overall conductor diameter. The cross-sectional area of each
strand is therefore 1125the overall conductor area.
coaxial cable, the twisted-pair
~ := 0 , 5 " 60
bus has two
adjacent conductors that give ris~ t~ th~ pro~imity
effect in which the current dIstributIon
IS not
e r := 2,1
{ ~) := 0,45 + 1,10-3,{ ~) 2
q
uniform as the next figure illustrates.
e eq { ~) := 1 + q { ~) , (e r -1
Lumping all of the wire strands together, the
proximity correction factor for two wires that are
right next to each other is P=1.15 (Reference 3),
This setup results in the following
)
plot:
y=a+j13=.J(R+ jwL)o(G+
The MathCAD
constant is:
jwC).
20
expression for the propagation
IX!) 1
y(f)
:= (1.RdC.d.~
+ j .2.7t.f.Le~
.J( (2-1t.foPf.Ceq + j .2.1t.f.Ce~ )
Depending on whether the cases are lossless, low
loss, or distortionless, the attenuation and phase
factors can be approximately constant or linear
with frequency. The phase velocity , or velocity of
propagation (vp=ro/13), is constant only if13 is
You can calculate the effect of the twist
the impedance as follows:
approximately
following
Z("
plot
a linear
function
shows
the
of
frequen.cy.
attenuatIon
~e
JD
acos,\~
) :=.
In
1t
)
~ ( o -.o~
II:\ \
\ e a e eq( " ) )
decibels/meter:
140
a(f)
(~
:= Re(y(f».
In( 10)
)
12
Z(~)
0.0155
10
8
0.010
a(!)
60
0
0.00
~
0
0
5.1rf>
f
The phase shift in degrees/meter is:
13(f) := Im(y (f» .~
1t
1.107
The impedance of the twisted pair is fairly constant
with a fixed twist angle. Nonuniformity in
conductor diameters and dielectric thickness,
together with variation in the twist angle, can
significantly affect the bus impedance. Intuitively,
you can see why the impedance increases when the
twisted wires are separated for current-mode
transceiver coupling: As.the wires are separated to
form an eye, the capacItance decreases, and the
inductance increases.
The following plot shows the impedance variance
with O.OOl-in. manufacturing tolerances on the
c~nductor
diameter,
d, and the overall
dielectric
diameter, t:
):;f:J;;'
"
Using
the
theoretical
,,~~,
constant,-,?,C
propagation
calculate the characteristic impedance
impedance as functions of frequency:
and input
"
,
130
"""
'z
,"
Y2(f)
Zd1(t)
e-E7-e
Z d2( t )
:=
(R(f)
+j
'2-7[ ,pC)"
125
120
Z
( f ) ,-
O
,-
~
R(f)
0,0021
115
0,00215
(): 20
a 2(f)
,
'2-7['f'L,
,
+ J -2 '7[ 'f'C
0,0022
(
:= Re y2(f»
t
,
W1th thlS plot m mmd, you can see that the bus
terminator need not exactly match the specified
bus impedance because the impedance likely
varies by a small percent- Because the resistive
termination swamps any small presence of reactive
load, it has to match only to a nominal busimpedance value in the middle of the above
impedance band. Even at 5% mismatch, the
reflection
coefficient
would be (126-120)/
(126+ 120)=2.4%, which results in reflected power
:= Zo(f)-
'"",CC""'"""
"""""'C',c'
c"'C',"C'C',""",c
'
," "c ""c'
In( 10)
+ Zo(f)'Sin~y2(f).I)
.
Zo(f)-cos~y2(f)-I) + ZL-sin~y2(f).I)
Although MathCAD can calculate all the complex
math functions, it is useful to find an easy way to
calculate some real numbers for
simple
comparison- You can calculate the bus impedance,
Z, and propagation constant, y, for a particular
frequency from the distributed RLCG parameters
as follows:
of only 0.12%.
r = .J(R + jOJL)(G + jaJC)
Armed with this information, you can first analyze
the input impedance as a function of frequency at a
bus
length of 50m and a 5% mismatched load at
1260.
I := 50
Z L := 126
=
(
-OJL)( , "..1'"'\ (1+
J
JLV'-'J
~
jOJL
)( 1 ~
+ jOJC )
f := 1 '106 , 1.1 '106 " 10 .106
For high-quality, low-Ioss cable at high frequencies (more than a few megahertz), in which
oJL> > R and aJC> > G, you can use the following
simplification for the characteristic impedance:
R(f)
:= 0,06 + 4,69 -10-4,[r
Z=
G(f)
:= 6,17 .10-12,f
L := 4,99 '10-7
C := 3-47 .10
-II
',""
) -'c'
(ZL-COS~Y2(f)-I)
Zinif)
,,
',:",,;",,:'
+J
G( f)
,
",
"
+-t-+
,
'2'7[-f'L)-(G(f)'+j
~
V(G+
=f§
jaJC)
C
Also, by using the approximation
(l+a)(l +b)=(l +a+b) and Jl+-;;
= 1 + c /2
for a,b,c « I, you can manually evaluate the
propagation constant as follows:
)
STP bus
References:
As for STP, the shield now contains the electric
field, and this distribution represents additional
shunt capacitance from each twisted wire to the
shield. For the differential mode of transmission,
the equivalent effect is increased distributed
capacitance, C; decreased impedance; and longer
propagation delay. The extra dielectric insulation
between the shield and the already-insulated UTP
also increases the distributed conductance, G,
which represents dielectric loss.
I.
Note that the shield has no effect on the magnetic
field. With differential mode of propagation, the
shield carries no net current (except for commonmode noise current that results from the
transmission-line imbalance).
The parameters for STP are therefore similar to
those of UTP, except for the increased capacitance
and conductance. The dc resistance, Rdc, and skindepth resistance, Rac, also change due to the
proximity effect (change in surface current
distribution) from the shield.
Summary
Using MathCAD, this article has constructed the
frequency-dependent equations of the many socalled constants of a lossy transmission line. You
need to be careful to find out all of the
assumptions; be careful not to take a given
component constant at its surface value.
Manufacturers' specifications usually associate
with a specific test condition. It is common to find
published data measured only at one frequency (I
MHz typical). This article plots the changes in the
impedance, attenuation, and phase as functions of
frequency, twist pitch, load mismatch, material
make-up, and dimensional tolerance of the twisted
pair. With accurate calculations for the distributed
parameters, you can build an accurate model for
the twisted pair that reflects real transmission-line
behavior for a range of frequencies.
"Reference Data for Radio
Howard Sams & Co, 1956.
2. Lefferson, Peter, "Twisted
Transmission Line," 1971.
Engineers,"
Magnet
Wire
3. Magnuson, Phillip, "Transmission Line & Wave
Propagation," 1981.
4. Brown et al, "Lines, Waves, and Antennas,"
1973.
Author's Biography
T Kien Truong has
been an engineer
with the Boeing Co
(Seattle) since 1985.
He is a graduate of
Iowa State Univ- -:
ersity (Ames, IA).
His professional experience
includes
working with antenna and RF , fiber-optic sensors, and faulttolerant computer design. In his free time, Kien
enjoys the freedom of telemark skiing and sailing.
You can reach him at Boeing Commercial
Airplane Group, PO Box 3707, M/S 19-HM,
Seattle, W A 98124.