Development of Wire Diagnostic System: Algorithm Development to Ascertain the True...

8/8/2019 Development of Wire Diagnostic System: Algorithm Development to Ascertain the True CharacteristicImpedance of

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Development of Wire Diagnostic System: Algorithm

Development to Ascertain the True Characteristic

Impedance of a Wire.

Dr. Eric BechhoeferGoodrich Fuels and Utility Systems

Vergennes, VT 05491802-877-4875

[email protected]

Dr. Jun YuDepartment of Mathematics and Statistics , University of Vermont

Burlington, VT 05401(802) 656-8539

[email protected]

AbstractElectronic1,2 wiring interconnect system (EWIS)

traditionally have been treated as a commodity as apposed

to a system. The EWIS, being responsible for the transfer of

power and information of aircraft systems, represents a point

of failure that is usually considered as a Maintainers last

resort. We attempt to develop a diagnostic capability for the

detection and progression of EWIS damage of data using

Time Domain Reflectometry (TDR). TDR measures changes

in EWIS characteristic impedance. Damage, such as chafe,

nicks and corrosion, change the characteristic impedance of

the EWIS, suggesting that detection is dependent upon as

true representation of impedance as possible . Wire

characteristic impedance is corrupted by: multiple reflections

(an artifact resulting from the interaction of the TDR

waveform and changes in impedance on the EWIS), andattenuation in the high frequency component of the TDR

signal.

A transmission line problem was developed using RLC

circuit model. We present systems of differential equations

to solve for the wire characteristic impedance. This led to a

system of wave equations in terms of line voltage and

current, which we solve by the method of characteristics.

The inverse scattering method was then successfully

implemented to remove multiple reflections. We then control

environmental effects including skin effect, capacitance and

wire resistance to account for attenuation in the TDR signal.

Formulas based on hypothesis as well as derived from theelectromagnetic field theory for the capacitance as a function

of the distance of the wires were developed and tested.

These solutions were then implemented in a hand held

device for the prognostics of EWIS.

1 0-7803-8155-6/05/$17.00 2005 IEEE2 IEEEAC paper #xxxx, updated September

TABLE OF CONTENTS

1.INTRODUCTION.......................................................11042.THE RLCCIRCUIT..................................................11053.CALCULATING CHARACTERISTIC IMPEDANCE11064.CONTROLLING SIGNAL ATTENUATION.............11075.EXAMPLES ................................................................11076.DISCUSSION...........................................................1109REFERENCES ................................................................1109BIOGRAPHY................................................................1109

1. INTRODUCTION

The study of the transmission line problem, especiallyinvolving multiple scattering, is important to many

applications. Inverse scattering method provides a powerful

tool to analyze multiple reflections of the waves in the

transmission line. This type of the problem has been applied

to discrete transmission line models [1] and [2], elastic waves

in layered media [3], and electromagnetic scattering [4].

While [1], [3] and [4] dealt with lossless inverse scattering

problems; [2] studied the problems in lossy and absorbing

media. In our present paper, we focus on developing a

diagnostic procedure for analyzing changes in EWIS

characteristic impedance and for detecting wire damages.

This procedure incorporates inverse scattering algorithm

with formulas derived from data-analysis modeling and

electromagnetic field theory to account for various

environmental effects.

We start, in Section 2, with a general problem formulation for

the wire response, based on Maxwells equations.

Representing a transmission line by the distributed

parameter equivalent circuit, we study wave propagation on

the line in terms of voltage and current. For a more detailed

derivation, see Chapter 7 of [5], for example. In Section 3, we


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describe the implementation of the inverse scattering

algorithm for calculating characteristic impedance in

Goupillaud medium. The center piece of the algorithm is the

so called layer peeling method described, for example, in

[6]. In Section 4, we show that the theory can be used to

derive a correction for multiple reflections and the effect of

capacitive and resistance in a transmission line. Examples are

then presented in Section 5 and finally, we do discussion in

Section 6.

2. THERLCCIRCUITWhen a fast rise time pulse sent down a wire, the measured

voltage response is depended on a number of physical

phenomena. These phenomena can be modeled as a series

RLC circuit.

In the series RLC circuit, the inductor and capacitor are

connected in series. The source-resistor circuit can be

written as a Thevenin equivalent, which reduces to a circuit

that has a voltage source, resistor, inductor, and capacitor

connected in series.

The governing equation for the capacitor voltage vC(t), is

given by:

LC(d2vC)/dt

2+ RC(dvC)/dt + vC = vT, (1)

whereL, Cand R are constants. The Thevenin voltage vTis

always a known driving force, where the initial conditions

are:

vC(0) = Va, (dvC(0))/dt = I0/C (2)

The second-order differential equation characterizes theresponse of the series RLC circuit in terms of the capacitor

voltage. Once the solution of the capacitor voltage is found,

it is possible to solve for every other voltage or current,

including the inductor current, using the element and

connection constraints. This is the solution strategy used

to solve a model of a transmission line.

We consider a lossless transmission line that is switched to

a direct current (DC) source. The model is described by a

distributed inductance and capacitance, with inductance and

capacitance per unit length designated as Ld and Cd,

respectively. The conditions on the line at a time dtafter the

connection of the source, when the source voltage haspenetrated a distance dx into the line can now be described.

The capacitance of the energized segment of the line Cd dx

has charged to a voltage of VA, hence a charge ofdQ = Cd

dx VA has been transferred.

DC

Figure 1 Lumped RC Circuit

The current flowing into the line is:

i = dQ/dt = Cd(dx/dt)Va = CdVau, (3)

where u is the velocity of propagation along the line of the

disturbance caused by being switched to the source. The

flow of the current establishes a magnetic flux f, which is

associated with the line inductance: d f=Ld dxi,. i = VA/Z0.

Faradays Law states that the rate of change of the flux must

equal the line voltage, e.g.: VA = (df)/dt = LdCdVAu2. From

this, the velocity of propagation on the line is simply:

u = (Ld Cd)-.5

. (4)

Substituting the velocity of propagation back into the

equation for current on the line gives:

i = Va(Cd/Ld).5. ( 5)

Note that this is the same relationship for resistance, namely:

i=Va/R. The characteristic impedance of the line is: Z0 =

(Ld/Cd).5, with units of Ohms (O). When a driving voltage

propagates along a line, there is an associated current pulse,

namely, i = Va/Z0. Given this, until a condition on the line

affects the characteristic impedance, (discontinuities such as

a termination), the input impedance of the line isZ0.

The Inverse Scattering ProblemThe process

of determining the true transmission line impedance from the

measured waveform is the inverse scattering problem. It is

an extraordinarily difficult problem to solve found in manyfields that studies waves. Some examples include the study

of elastic pressure waves (geology, seismology), sound

waves (acoustics), transmission lines (electrical engineer),

and the wave behavior of matter (quantum mechanics).

Fortunately, the inverse scattering problem for simpler, one-

dimensional system of layered media (a s is the case of

lossless transmission lines ) is relatively easy to solve. In

this case, there is a pair of coupled first-order partial

differential equations that govern two-state variables,

current i(x,t) and voltage v(x,t):

v(x,t)/x = -Z i(x,t)/ t. (6)

i(x,t)/x = -Z-1 v(x,t)/ t

whereZis the local impedance at the pointx.

It can be shown that when there is a mismatch in

characteristic impedance, some small amount of energy is

reflected from the discontinuity. Where there are a number

of these mismatches close to each other on the transmission

path, their reflections interact and generate a number of

reflection artifacts. These artifacts corrupt the measured


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6.DISCUSSION

We have gained a better understanding of wire response by

solving a system of differential equations for the wire

characteristic impedance. The inverse scattering algorithm

and normalization for frequency attenuation has been

successfully implemented in a hand held diagnostic tool,

which allows for an improved characterization of a EWIS.While not all fault features (such as a chafe of a single wire)

can be identified, we believe that we have the best

representation of wire characteristic impedance possible.

Identification of wire features, such as chafes and nicks, is

important when trouble shooting and diagnosing EWIS

damage and failure. We feel that moving to a condition

based management (CDB) paradigm (i.e. tracking changes in

EWIS characteristic impedance mean and variance, etc) will

allow for identification of gross system degradation that is

not identifiable via machine recognition. Finally, we realized

that this is a complicated problem due to the geometry of the

wire configuration and additional work in detection

algorithms is needed in order to find certain soft faults,

such as chafe in single wire. We have identified a number of

directions along which we would like to explore further, such

as the effect of inductive cross coupling due to the

numerous wires in close proximity within a harness.

REFERENCES

[1] A.M. Bruckstein and T. Kailath, Inverse Scattering for

Discrete Transmission-Line Models, SIAM Review, Vol. 29,

No. 3, pp. 359-389, 1987.

[2] J. Frolik, Forward and Inverse Scattering for Discrete

One-Dimensional Lossy and Discretized Two-Dimensional

Lossless Media, Ph. D. Thesis, University of Michigan,

Ann Arbor, MI, 1995.

[3] J. Berryman and R. Greene, Discrete Inverse Methods

for Elastic Waves in Layered Media, Geophysics, Vol. 45,

No. 2, pp. 213-33, 1980.

[4] P. Smith, Digital Realization of Forward and Inverse

Models of Electromagnetic Scattering, Electronics Letters,

Vol. 25, pp. 816-7, 1989.

[5] N.N. Rao, Elements of Engineering Electromagnetics.

Englewood Cliffs, Prentice Hall, 1991.

[6] G.M.L Gladwell, Inverse Problems in Scattering, An

Introduction. Dordrecht, Kluwer Academic Publishers, 1993.

[7] M.A. Heald, and J.B. Marion, Classical Electromagnetic

Radiation. Fort Worth, Harcourt Brace College Publishers,

1995.

BIOGRAPHY

Dr. Bechhoefer is retired Naval aviator

with a M.S. in Operation Research and

a Ph.D. in General Engineering, with a

focus on Statistics and Optimization.

Dr. Bechhoefer has worked at

Goodrich Aerospace since 2000 as a

Diagnostics Technical Lead. He haspreviously worked at The MITRE

Corporation in the Signal Processing

Center.

Dr. Yu is a Professor and Associate

Chair in the Dept. of Mathematics and

Statistics at the University of Vermont

with a Ph.D. in Applied Mathematics

from University of Washington in

Seattle (1988), focusing on Theory of

Partial Differential Equations and

Nonlinear Waves.

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