Wheatstone bridge

Wheatstone bridge circuit diagram. The unknown resistance R_x is to be measured; resistances R₁, R₂ and R₃ are known and R₂ is adjustable. If the measured voltage V_G is 0, then R₂/R₁ = R_x/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, $\scriptstyle R_x$ is the unknown resistance to be measured; ${\displaystyle \scriptstyle R_{1},}$ ${\displaystyle \scriptstyle R_{2},}$ 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 ${\displaystyle \scriptstyle (R_{x}/R_{3}),}$ 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 $\scriptstyle R_2$ 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 ${\displaystyle \scriptstyle R_{1},}$ ${\displaystyle \scriptstyle R_{2},}$ 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

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

Alternatively, if ${\displaystyle \scriptstyle R_{1},}$ ${\displaystyle \scriptstyle R_{2},}$ and $\scriptstyle R_3$ 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 ${\displaystyle \scriptstyle R_{x},}$ 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:

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

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 I_G = 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:

$R_x = {{R_2 \cdot I_2 \cdot I_3 \cdot R_3}\over{R_1 \cdot I_1 \cdot I_x}}$

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

$R_x = {{R_3 \cdot R_2}\over{R_1}}$

If all four resistor values and the supply voltage (V_S) are known, and the resistance of the galvanometer is high enough that I_G is negligible, the voltage across the bridge (V_G) 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 V_G 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

Kelvin 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:

• Carey Foster bridge, for measuring small resistances
• Kelvin bridge, for measuring small four-terminal resistances
• Maxwell bridge, for measuring reactive components.

See also

• Phantom circuit – a circuit using a balanced bridge
• Post Office Box
• Potentiometer
• Potential divider
• Ohmmeter
• Resistance thermometer
• Strain gauge

References

1. "Circuits in Practice: The Wheatstone Bridge, What It Does, and Why It Matters", as discussed in this MIT ES.333 class video
2. "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

• DC Metering Circuits chapter from Lessons In Electric Circuits Vol 1 DC free ebook and Lessons In Electric Circuits series.
• Wheatstone Bridge – Interactive Tutorial National High Magnetic Field Laboratory
• Test Set I-49