# Wheatstone Bridge

A Wheatstone bridge is an electrical circuit used to measure an unknown electrical resistance by balancing the two legs of a bridge circuit , one leg of which includes the unknown component. The primary advantage of the circuit is its ability to provide extremely accurate measurements ( as opposed to something like a simple voltage divider ). [1] Its operation is similar to that of the original potentiometer .The Wheatstone Bridge was invented in 1833 by Samuel Hunter Christie (sometimes “Christy”) and was improved and popularized by Sir Charles Wheatstone in 1843. One of the early uses of Wheatstone Bridge was for soil analysis and comparison.

## Surgery

In the figure, x is the fixed, yet unknown, resistance to be measured. R1 R2 ,and R3 are resistors ofknown resistance and the resistance of R2 is adjustable The resistance R 2 is adjusted until the bridge is “balanced” andno current flows through the galvanometer V g . At this point, the potential differencethe two midpoints

(BandD)will be zero. Hencethe ratio of the two resistances in the( R2  /  R1 )( Rx  / R3 ) is equal to the ratio of the two resistances. If the bridge is unbalanced, the direction of the current indicates whether R 2 is too high or too low.

at the point of equilibrium,

{\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}}}


Zero current detection with a galvanometer can be done with extremely high precision. Therefore,

if R1 R2 , and R3 are known to have high precision , then Rx can be measured to high precision Very small changes in Rx disrupt the equilibrium and are easily detected.

Alternatively, if R1, R2 and R3 are known , but R2 is not adjustable, calculate the value of x using the voltage difference or current flow through the meter , using Kirchhoff ‘s circuit laws . can be done for. This setup is often used in strain gauge and resistance thermometer measurements, as it is usually faster to read a voltage level from a meter than to adjust the resistance to zero the voltage.

## Etymology

### Quick Derivation at Equilibrium

At the point of equilibrium, the voltage and current between the two midpoints ( b and d ) are both zero. therefore,

{\displaystyle I_{1}=I_{2}}, , , and:{\displaystyle I_{3}=I_{x}}{\displaystyle V_{D}=V_{B}}
{\displaystyle {\begin{aligned}{\frac {V_{DC}}{V_{AD}}}&={\frac {V_{BC}}{V_{AB}}}\\[4pt]\Rightarrow {\frac {I_{2}R_{2}}{I_{1}R_{1}}}&={\frac {I_{x}R_{x}}{I_{3}R_{3}}}\\[4pt]\Rightarrow R_{x}&={\frac {R_{2}}{R_{1}}}\cdot R_{3}\end{aligned}}}

### Complete Derivation Using Kirchhoff’s Circuit Laws

First, Kirchhoff’s first law is used to find the currents at 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 law is used to find the voltage in the loops ABDA and BCDB :

\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 written as:

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

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

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


Because of: 3 = x and 1 = 2 in the above equation being proportional to Kirchhoff’s first law, 2 over x cancels out from the above equation. The desired value of R x is now given as:

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

On the other hand, if the resistance of the galvanometer is so high that IG is negligible , it is possible to calculate R x from the three other resistor values ​​and the supply voltage ( V S ), or the supply voltage from all four resistors . Value. To do this, one has to extract the voltage from each potential divider and subtract one from the other. The equations for this are:

{\displaystyle {\begin{aligned}V_{G}&=\left({R_{2} \over {R_{1}+R_{2}}}-{R_{x} \over {R_{x}+R_{3}}}\right)V_{s}\\[6pt]R_{x}&={{R_{2}\cdot V_{s}-(R_{1}+R_{2})\cdot V_{G}} \over {R_{1}\cdot V_{s}+(R_{1}+R_{2})\cdot V_{G}}}R_{3}\end{aligned}}}


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

## Importance

The Wheatstone Bridge illustrates the concept of a differential 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 explosive . The Kelvin bridge was specially adapted from the Wheatstone bridge to measure very low resistance. In many cases, the importance of measuring unknown resistance relates to measuring the effect of a physical phenomenon (such as force, temperature, pressure, etc.), allowing the use of Wheatstone Bridge in measuring those elements indirectly.

The concept was extended to alternating current measurement by James Clerk Maxwell in 1865 and further improved as the Blumlin Bridge by Alan Blumlin 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 different types of resistance when the fundamental Wheatstone bridge is not suitable. There are some modifications:

• Carey Foster Bridge , for measuring small resistors
• Kelvin bridge , for measuring small four-terminal resistors
• Maxwell Bridge , and Wien Bridge for measuring reactive components
• Anderson’s bridge , for measuring the self-inductance of a circuit, is an advanced form of Maxwell’s bridge.

#### What is a stone bridge measured by?

The Wheatstone bridge is a small circuit used for measurement. It was invented by Samuel Hunter Christie in 1833 but it was improved and popularized by Charles Wheatstone. In addition to other functions, it is used to find the value of an unknown resistance.

#### What is the formula of Wheatstone Bridge?

P/Q = R/S
In this formula, if the values of the three resistors are known in advance, the value of the fourth unknown resistance can be easily determined. In this formula, if the ratio of any two resistances is known, then the value of the third resistance can be found. The battery and the galvanometer can be interchanged in a Wheatstone set.

#### When is Wheatstone Bridge most sensitive?

Wheatstone bridge is most sensitive when all four resistances are of the same order.

#### Why is the cell key and then the galvanometer key pressed before using the Wheatstone bridge?

If the galvanometer key is pressed first, even after attaining the equilibrium position, there will be deflection in the galvanometer due to self-induction, due to which there is a possibility of the reading being wrong.