5 Engineering Example 1
5.1 Undersea cable fault location
Introduction
The voltage ( [maths rendering] ), current ( [maths rendering] ) and resistance ( [maths rendering] ) in an electrical circuit are related by Ohm’s law i.e. [maths rendering] . If there are two resistances ( [maths rendering] and [maths rendering] ) in an electrical circuit, they may be in series, in which case the total resistance ( [maths rendering] ) is given by [maths rendering] . Or they may be in parallel in which case the total resistance is given by
[maths rendering]
In 1871 the telephone cable between England ( [maths rendering] ) and Denmark ( [maths rendering] ) developed a fault, due to a short circuit under the sea (see Figure 2). Oliver Heaviside, an electrical engineer, came up with a very simple method to find the location of the fault. He assumed that the cable had a uniform resistance per unit length. Heaviside performed two tests:
-
connecting a battery (voltage
[maths rendering]
) at
[maths rendering]
, with the circuit open at
[maths rendering]
, he measured the resulting
current [maths rendering] ,
- connecting the same battery at [maths rendering] , with the cable earthed at [maths rendering] , he measured the current [maths rendering] .
Figure 2 :
In the first measurement the resistances up to the cable fault and between the fault and the short circuit are in series and in the second experiment the resistances beyond the fault and between the fault and the short circuit are in parallel.
Problem in words
Use the information from the measurements to deduce the location of the fault.
Mathematical statement of problem
- Denote the resistances of the various branches by the symbols shown in Figure 2.
- Use Ohm’s law to write down expressions that apply to each of the two measurements.
- Eliminate [maths rendering] from these expressions to obtain an expression for [maths rendering] .
Mathematical analysis
-
In the first experiment the total circuit resistance is
[maths rendering]
. In the second experiment, the total circuit resistance is given by:
[maths rendering]
So application of Ohm’s law to each experimental situation gives:
[maths rendering] (1)
[maths rendering] (2)
Rearrange Equation (1) to give [maths rendering]
Substitute for [maths rendering] in Equation (2), divide both sides by [maths rendering] and introduce [maths rendering] and [maths rendering] :
[maths rendering]
Use a common denominator for the fractions on the right-hand side:
[maths rendering]
Multiply through by [maths rendering] :
[maths rendering]
Rearrange as a quadratic for [maths rendering] :
[maths rendering]
Use the standard formula for solving quadratic equations
with [maths rendering] and [maths rendering] :
[maths rendering]
Only positive solutions would be of interest.