# Wheatstone Bridge

The Wheatstone Bridge is a useful circuit which can be used to measure an unknown resistance and is connected in a bridge topology.

It consists of three known resistances, an unknown resistance, a voltage source and a galvonometer (a device which detects current with high precision, in particular when it is zero).

The unknown resistance can be determined when the current flowing through the galvonometer is zero, i.e. the ratios of R1 to R2 and R3 to the unknown resistance Rx are equal for an equal voltage at nodes X and Y and hence no potential difference and no current flow between them. We can also say that the bridge is then balanced.

We can express this mathematically with the formula below which is the case at balance:

And then rearrange to solve for the unknown resistance:

## Derivation:

We can also mathematically derive the equation for balance from first principles using KCL and KVL, for this let's first assign branch currents to the circuit:

KCL at nodes X and Y:

KVL around loops AXY and BXY:

Assuming the bridge is balanced the Ig becomes 0 which simplifies the KVL equations as follows:

From the KCL equation at node Y with Ig = 0 we have that Ix = I2 which is expected as no current flows through Rg and similarly from the KCL equation at node X we have I1 = I3 which results in the following simplifications:

Firstly substitute [2'] into the KCL equation at node Y:

And now as I1 = I3 we can simplify this further by dividing both sides by I1 to arrive at the expected result:

### Example:

Q. Assuming the wheatstone bridge circuit shown below is balanced, what is the value of the unknown resistance Rx?

As the bridge is balanced, we can directly apply the derived formula to calculate Rx:

## Strain Gauges:

A strain gauge is a mechanical device which is capable of measuring the amount of mechnical strain exerted on it. It is a sensor where the force exerted on it is reflected by a change in resistance which we can connect in a wheatstone bridge to measure.

If a wheatstone bridge is setup with a strain gauge as the unknown resistance which is balanced at rest then any changes in strain will increase or decrease this resistance which will generate a non-zero output voltage across the bridge while unbalanced that is a function of strain.

For this circuit we effectively have two parallel voltage dividers and so we can apply the voltage divider to each branch and subtract the node voltages at x and y to determine the voltage across the bridge:

### Questions:

1. Determine the unknown resistance Rx for the balanced bridge below:

500Ω

2. What is the voltage output across the bridge below?

2kΩ