Statistical Variation of Wire Systems
Eric J. Lundquist
November 2009
Introduction
This paper discusses a comparison of the measurement and calculation of
wire parameters, the effects of randomly generated dielectric constant
parameters, and statistical variation integrated with ABCD and FDTD
simulation code. The analytical solution for wire pairs was also used to
estimate the effects of statistical variation on wire pairs and shielded wire
pairs.
The parameters being observed are: velocity of propagation, characteristic
impedance, and permittivity. After measurement, these results are
compared to calculations made according to a more simple, lossy,
insulated two-wire model. Conclusions are drawn from the comparison of
measured results and model calculations, which indicate a notable
similarity between results, and possible reasons for divergence of
measured results are explored. The multi-wire bundle used for
measurements is one that is typical to those used in aircraft.
Wire Bundle Measurements
In order to perform the measurements, wires were laid out on a large
board, as shown in Figure 1. Using a measuring tape, the length of the wire
bundle was found to be approximately 40’ 7.25” (40.6 feet), or 12.375
meters. The wire tips were stripped of insulation (about 1-2 cm)
preparatory to taking measurements. At this time, a piece of the insulation
was measured with a caliper in order to determine the insulation
thickness t. The wire tips were then matched and labeled by applying a
multimeter voltage and thus detecting the voltage at the opposite end of
the corresponding wire.
Figure 1 (left): Wire tips, after being identified and matched
Figure 2 (right): Wire layout on board, with connections made to generator and
scope
Once the wire ends were stripped, measurements began with the use of
the programs NDR2 and NDR4. Measurements were taken of each wire
with respect to a reference wire (see Table 1). A PNCODE generator was
connected, along with the scope, to the wire being measured, with the
reference wire being connected to ground.
Table 1: Measurements of wire length (l), velocity of propagation (VoP), and
characteristic impedance (Zo)
Wire
Length
(m)
12.375
Wire #
VoP (c)†
Zo (Ω)
A*
2
B**
C
0.602
200.9
D
0.602
99.5
E
0.589
87.9
F
0.602
102.3
G
0.600
167.0
H
0.602
209.1
I
0.600
135.2
J
0.602
115.1
K
0.602
102.3
L
0.594
96.4
M
0.600
109.9
N
0.602
126.9
O
0.602
149.9
P
0.602
163.7
Q
0.602
141.0
R
0.602
131.5
S
0.600
122.1
T
0.602
127.7
U
0.602
127.7
V
0.600
117.3
W
0.602
135.5
X
0.602
127.2
Y
0.602
187.7
Z
0.602
180.3
† Measured with a sampling point based on a 2 G-sample/s sampling rate,
with t=0.5 ns.
* Wire A was not measured due to incidental damage.
* Measurements for Zo made with reference to Wire B.
The frequency was set to 58 MHz, with an amplitude of 2500 mV.
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Figure 3 (left): Measurement station equipped with function generator and scope
Figure 4 (right): Connection configuration with PNCODE close-up
NDR2 was used to measure VoP. The program was opened and the
sampling point at which the wire reflection occurs was recorded. This
sampling point was converted to m/s and also displayed as a fraction of c
(as shown below), using the following equation:
VoP(m /s) 

2 l
S  (0.5e  9)
l=wire length
S=sampling point
(0.5e-9)=sampling interval for 2GS/s device=0.5 ns
NDR4 was used to determine Zo. A multi-turn variable 1-kΩ resistor was
connected to the opposite ends of the wires being measured, and the
variable resistance was carefully adjusted until the peak shown in NDR4
approached zero, thus matching the impedance of the wires. The resistor
was then disconnected and the value of Zo was determined from the
corresponding resistance measurement.
The measured results were placed in a spreadsheet and the statistics of
these measurements were also obtained.
Wire Bundle Calculations
The desired parameters were also calculated using classic RLGC
equations, as well as a model for effective permittivity (eff), in order to
compare and verify the effectiveness of these models. The effective
permittivity model takes into account an insulated two-wire model, which
is outlined as follows:
4
eff 
12 (d1  d2 )
1d1  2 d2
(2)

Figure 5: Cross-section of wire dimensions
The RLGC parameters were programmed, which depend on the size and
shape of the conductors, the insulating material surrounding them, and
the material the conductors are made of.
For a parallel-wire
configuration, the RLGC parameters can be calculated as follows:
R' 
Rs
a
 
ln (d / 2a)  (d / 2a) 2  1 

 
 
G' 
ln (d / 2a)  (d / 2a) 2  1 


 
C'
ln (d / 2a)  (d / 2a) 2  1 


L' 
 / m
(3)
 H / m
(4)
( S / m)
(5)
( F / m)
(6)
Intrinsic resistance Rs:
Rs 
  f  uconductor
f

c
c
 
(7)
Conductor Radius: a (meters)
Distance between the centers of the conductors: b (meters)
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Conductivity:  c  S / m
Table 2: Important Propagation Constants of Wiring calculated from the RLGC
model
Parameter
Equation
Characteristic Impedance
(ohms)
Complex Propagation
Constant
Attenuation Constant
(Np/m)
Phase Constant
(radians/m)
Velocity of Propagation
(VoP) (m/s)
R ' j L '
L'

G ' jC '
C'
Z0 
 lossless 
  ( R ' j L ')(G ' jC ')   j 
(8)
(9)
  Re{  }
(10)
  Im{  }
(11)
VoP 

c

(lossless)

r
Using these derivations, the statistics of the dielectric constants were
obtained.
Results
This section displays the results of both the measurements and the
corresponding calculations of the insulated two-wire model.
Table 3: Statistics of measured velocity of propagation (VoP), and characteristic
impedance (Zo)
STATISTICS
VoP(c)
Mean
0.601
6
Ave.
Zo
136.0
(12)
Std. Dev.
0.0031
33.4
Table 4: Statistics of the calculated effective permittivity (eff), velocity of
propagation (VoP) in terms of speed of light (c), and characteristic impedance
(Zo)
2.7790
VoP
(c)
0.6000
152.2734
0.1311
0.0153
30.3582
STATISTICS
eff
Mean
Std.
Deviation
Zo
Figure 6: Calculated velocity of propagation vs. distance between wire centers
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Figure 7: Calculated characteristic vs. distance between wire centers
Figure 8: Calculated signal amplitude vs. distance between wire centers
The calculated results showed a close similarity to the measured results,
though the standard deviation was greater in the calculated velocity of
propagation (VoP) and smaller for the calculated characteristic impedance
(Zo) than the measured results. The VoP shows a 393% increase in the
calculated standard deviation; whereas the Zo shows a 9% decrease in the
calculated standard deviation. Additionally, the VoP was noticeably more
uniform across the measured results than in the calculated results. This is
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likely due to the many surrounding wires consisting of conducting
materials which affect the effective permittivity of the measured wired
parameters.
Analytical Calculations
For verification purposes, the desired parameters were calculated using
classic RLGC equations, as well as a model for effective permittivity (eff),
in order to compare and verify the effectiveness of these models. The
effective permittivity model takes into account an insulated two-wire
model, which is outlined as follows:
eff 
12 (d1  d2 )
1d1  2 d2
The RLGC parameters were programmed, which depend on the size and
shape of the conductors, the insulating material surrounding them, and

the material the conductors are made of. RLGC parameters were
calculated using the parallel-wire configuration.
A code was designed to take a value for r (dielectric constant of the wire
insulation) and then calculate all the parameters and display the statistics.
Values for r were produced between the values of 1 and 4 (standard
range for insulating materials) using a uniform distribution density
function.
Calculation Results
The values produced for VoP, Zo, and the signal attenuation show a range
of possibilities which vary according to the random values of the
insulating dielectric constant.
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Figure 9: Randomized Dielectric Constant vs. Zo
Figure 10: Randomized Dielectric Constant vs. VoP
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Figure 11: Randomized Dielectric Constant vs. Signal Amplitude
The calculated results showed that the velocity of propagation (VoP) is
centered around a mean of about 1.8e8 m/s, and the characteristic
impedance (Zo) appears close to 200 ohms. Zo values remain higher than
200 ohms unless values greater than 4 for the dielectric constant are used,
or if other parameters are changed.
Statistical Variation in ABCD and FDTD Software
In order to determine the effects of statistical variation in wire systems,
which has been shown to be nearly inevitable in real physical systems,
these effects have been incorporated into software simulation.
In ABCD software, the alpha and beta components of the propagation
constant were varied statistically, with the result that the reflectometry
plot showed ±0.02 deviation in reflection from the expected value, as
shown below.
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Figure 12: Reflectometry plot without statistical variation
Figure 13: Reflectometry plot using same parameters, with statistical variation
In FDTD, the same comparison process was applied, using a different
configuration of wire lengths and properties, only altering the RLGC
components of each spatial cell. This resulted in increased reflections on
the same order of approximately ±0.02 in the reflection coefficient, as
shown below.
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Figure 14: Reflections in FDTD without statistical variation
Figure 15: Reflections in FDTD for same configuration, with statistical variation
In the FDTD method, in can also be observed that the signal propagation
is rough or “bumpy” when statistical variation is incorporated along the
line and the simulation movie is plotted visually.
Thus, both methods were in agreement that the statistical variation can
alter and change the results of the simulation within ±2%.
Shielded Wire Pair
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In order to simulate the effects of variation in a shielded wire pair, the
analytical equation for wire pairs was used and variation applied to either
Z0, the distance between the wires, or the wire diameters. The dimensional
tolerance was estimated to be approximately 0.001”, according to
specifications found in STP technical descriptions.i This value was used as
the threshold for the statistical and sinusoidal variation that took place in
the calculations shown below.
Figure 16: Characteristic Impedance of Wire Pair with Statistically Varying
Dimensions
Figure 17: Characteristic Impedance of Wire Pair with Sinusoidally Varying
Dimensions
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Through these simulations, it was found that the variation was only on the
order of 0.3 ohms when using the maximum dimensional variation.
Furthermore, when both dimensions were varied to the same degree, the
effects of the variation cancelled each other out and the resulting
characteristic impedance remained the same. Thus, the effects of variation
were found to be much smaller than anticipated or previously simulated.
Conclusion
This paper discussed the comparison of measured and calculated wire
parameters, the effects of randomly generated dielectric constant
parameters, and statistical variation integrated with ABCD and FDTD
simulation code.
It was shown that calculated results showed a close similarity to the
measured results, though the standard deviation was greater in the
calculated velocity of propagation and smaller for the calculated
characteristic impedance than the measured results. The velocity of
propagation was shown to increase by up to 400% in the calculated
standard deviation; whereas the characteristic impedance was shown to
decrease by up to 10% in the calculated standard deviation. Additionally,
the velocity of propagation was noticeably more uniform across the
measured results than in the calculated results. This is likely due to the
many surrounding wires consisting of conducting materials which affect
the effective permittivity of the measured wired parameters. Both ABCD
and FDTD simulations methods showed that the statistical variation can
alter and change the reflectometry results of the simulation within ±2%.
i
T Kien Truong, “Twisted-pair transmission-line distributed parameters,” Boeing, p. 5.
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