# Wheatstone bridge

Wheatstone bridge circuit diagram. The unknown resistance Rx is to be measured; resistances R1, R2 and R3 are known and R2 is adjustable. If the measured voltage VG is 0, then R2/R1Rx/R3.

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 a wheatstone bridge 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 in 1833 and improved and popularized by Sir Charles Wheatstone in 1843. One of the Wheatstone bridge's initial uses was for the purpose of soils analysis and comparison.[2]

## Operation

In the figure, ${\displaystyle \scriptstyle R_{x}}$ is the unknown resistance to be measured; ${\displaystyle \scriptstyle R_{1},}$ ${\displaystyle \scriptstyle R_{2},}$ and ${\displaystyle \scriptstyle R_{3}}$ are resistors of known resistance and the resistance of ${\displaystyle \scriptstyle R_{2}}$ is adjustable. If the ratio of the two resistances in the known leg ${\displaystyle \scriptstyle (R_{2}/R_{1})}$ is equal to the ratio of the two in the unknown leg ${\displaystyle \scriptstyle (R_{x}/R_{3}),}$ then the voltage between the two midpoints (B and D) will be zero and no current will flow through the galvanometer ${\displaystyle \scriptstyle V_{g}.}$ If the bridge is unbalanced, the direction of the current indicates whether ${\displaystyle \scriptstyle R_{2}}$ is too high or too low. ${\displaystyle \scriptstyle R_{2}}$ is varied until there is no current through the galvanometer, which then reads zero.

Detecting zero current with a galvanometer can be done to extremely high accuracy. Therefore, if ${\displaystyle \scriptstyle R_{1},}$ ${\displaystyle \scriptstyle R_{2},}$ and ${\displaystyle \scriptstyle R_{3}}$ are known to high precision, then ${\displaystyle \scriptstyle R_{x}}$ can be measured to high precision. Very small changes in ${\displaystyle \scriptstyle R_{x}}$ disrupt the balance and are readily detected.

At the point of balance, the ratio of

{\displaystyle {\begin{aligned}{\frac {R_{2}}{R_{1}}}&={\frac {R_{x}}{R_{3}}}\\[4pt]\Rightarrow R_{x}&={\frac {R_{2}}{R_{1}}}\cdot R_{3}\end{aligned}}}

Alternatively, if ${\displaystyle \scriptstyle R_{1},}$ ${\displaystyle \scriptstyle R_{2},}$ and ${\displaystyle \scriptstyle R_{3}}$ are known, but ${\displaystyle \scriptstyle R_{2}}$ is not adjustable, the voltage difference across or current flow through the meter can be used to calculate the value of ${\displaystyle \scriptstyle R_{x},}$ using Kirchhoff's circuit laws (also known as Kirchhoff's rules). 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

Directions of currents arbitrarily assigned

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

{\displaystyle {\begin{aligned}I_{3}-I_{x}+I_{G}&=0\\I_{1}-I_{2}-I_{G}&=0\end{aligned}}}

Then, Kirchhoff's second rule is used for finding the voltage in the loops ABD and BCD:

{\displaystyle {\begin{aligned}(I_{3}\cdot R_{3})-(I_{G}\cdot R_{G})-(I_{1}\cdot R_{1})&=0\\(I_{x}\cdot R_{x})-(I_{2}\cdot R_{2})+(I_{G}\cdot R_{G})&=0\end{aligned}}}

When the bridge is balanced, then IG = 0, so the second set of equations can be rewritten as:

{\displaystyle {\begin{aligned}I_{3}\cdot R_{3}&=I_{1}\cdot R_{1}\\I_{x}\cdot R_{x}&=I_{2}\cdot R_{2}\end{aligned}}}

Then, the equations are divided and rearranged, giving:

${\displaystyle R_{x}={{R_{2}\cdot I_{2}\cdot I_{3}\cdot R_{3}} \over {R_{1}\cdot I_{1}\cdot I_{x}}}}$

From the first rule, I3 = Ix and I1 = I2. The desired value of Rx is now known to be given as:

${\displaystyle R_{x}={{R_{3}\cdot R_{2}} \over {R_{1}}}}$

If all four resistor values and the supply voltage (VS) are known, and the resistance of the galvanometer is high enough that IG is negligible, the voltage across the bridge (VG) can be found by working out the voltage from each potential divider and subtracting one from the other. The equation for this is:

${\displaystyle V_{G}=\left({R_{2} \over {R_{1}+R_{2}}}-{R_{x} \over {R_{x}+R_{3}}}\right)V_{s}}$

where VG is the voltage of node D relative to node B.

## Significance

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 by Alan Blumlein around 1926.

## Modifications of the fundamental bridge

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: