**Wheatstone** **bridge**

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A ** Wheatstone bridge** is an electrical circuit used to measure an unknown electrical resistance by balancing two legs of a

**bridge**circuit, one leg of which includes the unknown component. The primary benefit of the circuit is its ability to provide extremely accurate measurements (in contrast with something like a simple voltage divider).

^{[1]}Its operation is similar to the original potentiometer.

The **Wheatstone** **bridge** was invented by Samuel Hunter Christie (sometimes spelled "Christy") in 1833 and improved and popularized by Sir Charles **Wheatstone** in 1843. One of the **Wheatstone** **bridge**'s initial uses was for soils analysis and comparison.^{[2]}

## Operation[edit]

In the figure, *R _{x}* is the fixed, yet unknown, resistance to be measured.

*R*_{1}, *R*_{2}, and *R*_{3} are resistors of known resistance and the resistance of *R*_{2} is adjustable. The resistance *R*_{2} is adjusted until the **bridge** is "balanced" and no current flows through the galvanometer *V _{g}*. At this point, the potential difference between the two midpoints (

**B**and

**D**) will be zero. Therefore the ratio of the two resistances in the known leg (

*R*

_{2}/

*R*

_{1}) is equal to the ratio of the two resistances in the unknown leg (

*R*/

_{x}*R*

_{3}). If the

**bridge**is unbalanced, the direction of the current indicates whether

*R*

_{2}is too high or too low.

At the point of balance,

Detecting zero current with a galvanometer can be done to extremely high precision. Therefore, if *R*_{1}, *R*_{2}, and *R*_{3} are known to high precision, then *R _{x}* can be measured to high precision. Very small changes in

*R*disrupt the balance and are readily detected.

_{x}Alternatively, if *R*_{1}, *R*_{2}, and *R*_{3} are known, but *R*_{2} is not adjustable, the voltage difference across or current flow through the meter can be used to calculate the value of *R _{x}*, using Kirchhoff's circuit laws. This setup is frequently used in strain gauge and resistance thermometer measurements, as it is usually faster to read a voltage level off a meter than to adjust a resistance to zero the voltage.

## Derivation[edit]

### Quick derivation at balance[edit]

At the point of balance, both the voltage and the current between the two midpoints (**B** and **D**) are zero. Therefore, , , , and:

### Full derivation using Kirchhoff's circuit laws[edit]

First, Kirchhoff's first law is used to find the currents in junctions **B** and **D**:

Then, Kirchhoff's second law is used for finding the voltage in the loops **ABDA** and **BCDB**:

When the **bridge** is balanced, then *I*_{G} = 0, so the second set of equations can be rewritten as:

Then, equation (1) is divided by equation (2) and the resulting equation is rearranged, giving:

Due to: *I*_{3} = *I*_{x} and *I*_{1} = *I*_{2} being proportional from Kirchhoff's First Law in the above equation *I*_{3} *I*_{2} over *I*_{1} *I*_{x} cancel out of the above equation. The desired value of *R*_{x} is now known to be given as:

On the other hand, if the resistance of the galvanometer is high enough that *I*_{G} is negligible, it is possible to compute *R*_{x} from the three other resistor values and the supply voltage (*V*_{S}), or the supply voltage from all four resistor values. To do so, one has to work out the voltage from each potential divider and subtract one from the other. The equations for this are:

where *V*_{G} is the voltage of node D relative to node B.

## Significance[edit]

The **Wheatstone** **bridge** illustrates the concept of a difference measurement, which can be extremely accurate. Variations on the **Wheatstone** **bridge** can be used to measure capacitance, inductance, impedance and other quantities, such as the amount of combustible gases in a sample, with an explosimeter. The Kelvin **bridge** was specially adapted from the **Wheatstone** **bridge** for measuring very low resistances. In many cases, the significance of measuring the unknown resistance is related to measuring the impact of some physical phenomenon (such as force, temperature, pressure, etc.) which thereby allows the use of **Wheatstone** **bridge** in measuring those elements indirectly.

The concept was extended to alternating current measurements by James Clerk Maxwell in 1865 and further improved as Blumlein **bridge** by Alan Blumlein in British Patent no. 323,037, 1928.

## Modifications of the fundamental **bridge**[edit]

The **Wheatstone** **bridge** is the fundamental **bridge**, but there are other modifications that can be made to measure various kinds of resistances when the fundamental **Wheatstone** **bridge** is not suitable. Some of the modifications are:

- Carey Foster
**bridge**, for measuring small resistances - Kelvin
**bridge**, for measuring small four-terminal resistances - Maxwell
**bridge**, and Wien**bridge**for measuring reactive components - Anderson's
**bridge**, for measuring the self-inductance of the circuit, an advanced form of Maxwell’s**bridge**

## See also[edit]

- Diode
**bridge**, product mixer – diode bridges - Phantom circuit – a circuit using a balanced
**bridge** - Post office box (electricity)
- Potentiometer (measuring instrument)
- Potential divider
- Ohmmeter
- Resistance thermometer
- Strain gauge

## References[edit]

**^**"Circuits in Practice: The**Wheatstone****Bridge**, What It Does, and Why It Matters", as discussed in this MIT ES.333 class video**^**"The Genesis of the**Wheatstone****Bridge**" by Stig Ekelof discusses Christie's and**Wheatstone**'s contributions, and why the**bridge**carries**Wheatstone**'s name. Published in "Engineering Science and Education Journal", volume 10, no 1, February 2001, pages 37–40.

## External links[edit]

- Media related to
**Wheatstone**'s**bridge**at Wikimedia Commons *DC Metering Circuits*chapter from*Lessons In Electric Circuits Vol 1 DC*free ebook and*Lessons In Electric Circuits*series.- Test Set I-49