Varley loop test for cable faults

The Varley loop test is a Wheatstone bridge method for locating earth faults and short-circuit faults in underground cables. It differs from the Murray loop in that the ratio arms $P$ and $Q$ are fixed, a variable resistor $S$ is adjusted for balance, and the loop resistance is measured in a separate step.

Connections

                battery + switch K
                       |
           P           B           Q
   A ----/\/\----------+---------/\/\---- C
   |                                      |
   +-----------------(G)------------------+
   |                                      |
   S (variable)                           |
   |                                      |
   D  <- K position 1                     |
   |                                      |
 faulty core (X to fault)          sound core (Y)
   |                                      |
   +======== link at far end =============+
   K position 2: earth (current enters through the fault)

1. Link the faulty core and the sound core at the far end to form a loop.
2. Connect the fixed ratio arms $P$ and $Q$, with the variable resistor $S$ between A and the test end D of the faulty core. Connect the galvanometer across A and C.
3. The battery is connected to the P–Q junction B. A two-way switch K connects its other terminal either to D, the test end of the loop (position 1), or to earth (position 2).

Step 1: loop resistance (K at position 1)

The bridge is an ordinary Wheatstone bridge, with $S$ in one arm and the whole loop $R_L = X + Y$ in the other. Adjust $S$ to $S_1$ for balance:

$$\frac{P}{Q} = \frac{S_1}{R_L} \;\Rightarrow\; R_L = \frac{Q}{P}S_1$$

Step 2: fault position (K at position 2, battery to earth)

The current now enters the loop through the fault. One arm is $S + X$ (variable resistor plus faulty core up to the fault), and the other arm is $Y = R_L - X$. Rebalance with $S = S_2$:

$$\frac{P}{Q} = \frac{S_2 + X}{R_L - X}$$ $$P R_L - P X = Q S_2 + Q X \;\Rightarrow\; X = \frac{P R_L - Q S_2}{P + Q}$$

With $P = Q$ (the usual case):

$$X = \frac{R_L - S_2}{2}$$

Distance to the fault

If $r$ is the conductor resistance per metre:

$$x = \frac{X}{r} \quad\text{or}\quad x = \frac{X}{R_L}\times 2L$$

where $L$ is the cable length (both cores of equal size).

Example

With $P = Q$, $R_L = 2\ \Omega$ for a 2 km cable (4 km of loop), and $S_2 = 0.6\ \Omega$:

$$X = (2 - 0.6)/2 = 0.7\ \Omega,\qquad x = 0.7/2\times 4 = 1.4\ \text{km}$$

The variable resistor allows finer balance than the Murray loop. The fault resistance is in the battery circuit, so it does not affect the result.