# 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?

## 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:

#### Answer

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