Locating phase-to-phase cable fault

A phase-to-phase (short-circuit) fault between two cores of a multi-core cable can be located by the Murray loop test, a Wheatstone bridge method that needs one sound core.

Connections

   +--------------(G)--------------+
   |                               |
   c ---/\/\------+------/\/\----- a
   |     P        |        Q       |
   |           battery             |
   |              |                |
   |           core 2              |
 sound core 3                faulty core 1
   |                               x  <- fault (core 1 to core 2)
   |                               |
   +====== link at far end ========+

1. Core 1 and core 2 are shorted together at the fault. Core 3 is sound.
2. At the far end, link core 1 to sound core 3 to form a loop.
3. Connect ratio arm $P$ to the test end c of core 3 and ratio arm $Q$ to the test end a of core 1, and connect the galvanometer across a and c.
4. Connect the battery between the junction of $P$ and $Q$ and core 2 (the other shorted core). Core 2 takes the place of earth in the earth-fault test. Current then flows through the fault into the loop.
5. Adjust $P$ and $Q$ for zero galvanometer deflection.

Calculation

Let $R_x$ be the resistance of core 1 from the test end to the fault, and $R_L$ the whole loop resistance (core 1 + core 3). The arm next to $P$ is $R_L - R_x$ and the arm next to $Q$ is $R_x$. At balance:

$$\frac{P}{Q} = \frac{R_L - R_x}{R_x} \;\Rightarrow\; R_x = \frac{Q}{P+Q}R_L$$

For cores of equal size and cable length $L$:

$$x = \frac{Q}{P+Q}\times 2L$$

Example

A 3 km cable balances at $P = 2.5\ \Omega$ and $Q = 1\ \Omega$:

$$x = \frac{1}{3.5}\times 6 = 1.714\ \text{km}$$

The fault resistance lies in the battery circuit, so it does not affect the balance. The Varley loop test can be used in the same way when fine adjustment by a series resistor is preferred.