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# Wheatstone bridge

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 Title: Wheatstone bridge Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Wheatstone bridge

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. Its operation is similar to the original potentiometer. It 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.

## Contents

• Operation 1
• Derivation 2
• Significance 3
• Modifications of the fundamental bridge 4
• References 6

## Operation

In the figure, \scriptstyle R_x is the unknown resistance to be measured; \scriptstyle R_1, \scriptstyle R_2 and \scriptstyle R_3 are resistors of known resistance and the resistance of \scriptstyle R_2 is adjustable. If the ratio of the two resistances in the known leg \scriptstyle (R_2 / R_1) is equal to the ratio of the two in the unknown leg \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 \scriptstyle V_g. If the bridge is unbalanced, the direction of the current indicates whether \scriptstyle R_2 is too high or too low. \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 \scriptstyle R_1, \scriptstyle R_2 and \scriptstyle R_3 are known to high precision, then \scriptstyle R_x can be measured to high precision. Very small changes in \scriptstyle R_x disrupt the balance and are readily detected.

At the point of balance, the ratio of

\begin{align} \frac{R_2}{R_1} &= \frac{R_x}{R_3} \\ \Rightarrow R_x &= \frac{R_2}{R_1} \cdot R_3 \end{align}

Alternatively, if \scriptstyle R_1, \scriptstyle R_2, and \scriptstyle R_3 are known, but \scriptstyle R_2 is not adjustable, the voltage difference across or current flow through the meter can be used to calculate the value of \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

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

\begin{align} I_3 - I_x + I_G &= 0 \\ I_1 - I_2 - I_G &= 0 \end{align}

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

\begin{align} (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{align}

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

\begin{align} I_3 \cdot R_3 &= I_1 \cdot R_1 \\ I_x \cdot R_x &= I_2 \cdot R_2 \end{align}

Then, the equations are divided and rearranged, giving:

R_x =

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

R_x =

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:

V_G = \left( - \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: