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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. 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.

## Operation

In the figure, $R_4$ is the 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. If the ratio of the two resistances in the known leg $(R_2 / R_1)$ is equal to the ratio of the two in the unknown leg $(R_x / R_3)$, then the voltage between the two midpoints (B and D) will be zero and no current (electricity) will flow through the Galvanometer (G) or $V_g$. No current flow through G $V_g$, the bridge is said to be "balanced". If the bridge is unbalanced, the direction of the current indicates whether $R_2$ is too high or too low. $R_2$ 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 $R_1$, $R_2$ and $R_3$ are known to high precision, then $R_x$ can be measured to high precision. Very small changes in $R_x$ disrupt the balance and are readily detected.

At the point of balance, the ratio of

\begin{align} \frac{R_2}{R_1} &= \frac{R_4}{R_3} \\ \Rightarrow R_4 &= \frac{R_2}{R_1} \cdot R_3 \end{align}

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 $R_4$, 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, Kirchoff's current law 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}

The potential difference between points B and D will be near zero (0 VDC) volts using a digital multimeter.

Then, Kirchhoff's voltage law (KVL) 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_4) - (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:

$R_4 = {{R_2 \cdot I_2 \cdot I_3 \cdot R_3}\over{R_1 \cdot I_1 \cdot I_x}}$

From the first rule, I3 = Ix and I1 = I2. The desired value of R4 is now known to be given as:

$R_4 = {{R_3 \cdot R_2}\over{R_1}}$

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 balance state equation for this is:

$V_G = \left({{R_4}\over{R_4 + R_3}} - {{R_2}\over{R_1 + R_2}}\right) *\ V_s$

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

## Example

### Balance State

In a balanced state, where all four resistances are known. The Wheatstone bridge can be analysed as two series resistances in a parallel circuit. For example, you have a Wheatstone Bridge balanced state circuit with four known resistance values: R1 = 5Ω, R2 = 10Ω, R3 = 2Ω, and R4 = 8Ω. Voltage source (Vs) equals 12 VDC. Find Vg which is the circuits Vout.

1. $V_G = \left({{R_4}\over{R_4 + R_3}} - {{R_2}\over{R_1 + R_2}}\right) *\ V_s$

2. $V_G = \left({{8}\over{2 + 8}} - {{10}\over{5 + 10}}\right) *\ V_s$

3. $V_G = \left({{8}\over{10}} - {{10}\over{15}}\right) *\ V_s$

4. $V_G = \left({{.8}} - {{.66667}}\right) *\ V_s$

5. $V_G = .1333 * 12$

6. $V_G = 1.6001$

### Unbalance State an unbalanced state using a variable potentiometer (Rv) and unknown resistor value (Rx)

In an unbalanced state, only three resistances are known (R1, R2, along with a variable potentiometer (R3/4 or Rv). This Wheatstone bridge is still two series resistors in a parallel circuit; its unbalanced because we have to calculate the fourth resistance. Calculating the resistance and adjusting the pot (Rv) balances the voltage potential within the circuit to zero across the galvanometer. For example, you have a Wheatstone Bridge unbalanced state circuit with three known resistance values: R1 = 10Ω, R2 = 100Ω, Rv = 25Ω, and Rx = ???Ω. Voltage source (Vs) equals 12 VDC. Find Rx (unknown resistor value) using the circuit ratio method:

1. $\left({{VR_2}\over{VR_1}} = {{VR_x}\over{VR_v}}\right)$

Next, we re-arrange the ratio to find VR2. WB redrawn to show left side Rv (variable pot) and Rx (unknown resistor value)

2. $VR_2 = \left({{I_A * R_2}\over{I_A * R_1}} = {{I_B * R_x}\over{I_B * R_v}}\right)$

Here IA and IB cancel each other out and is re-written as.

3. $VR_2 = \left({{R_2}\over{R_1}} = {{R_x}\over{R_v}}\right)$

Now, we rearrange the equation to solve for Rx.

4. $R_x = \left({{R_S}} * {{R_2}\over{R_1}}\right)$

Finally, we plug in the numbers from above.

5. $R_x = \left({{25}} * {{100}\over{10}}\right)$

6. $R_x = 250$Ω

## 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, electrical 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.

## Application

The practical applications for the Wheatstone Bridge circuit is used for many biomedical applications to include but not limited to:

• MRI/CT - table positioning, accurate movement of the CT scan imaging device and equal patient weight distribution. High accuracy is needed to perform imaging functions while preventing over-travel of the patient placed within the scanning tube.
• Infusion pumps/Syringe pumps - monitors and controls the amount of fluid flow of intravenous medication that was to be received via the tubing.
• Mammography - monitors the amount of physical force that is applied to the patients breast by the machine itself when attempting to take an image.
• Conventional radiography - monitors the amount of x-ray dose received to the AEC cells and the patient.
• Scales, weighing/Patient lifts - with the incorporation of a load cell into the bottom metal plates these scales routinely require re-zeroing which uses the above circuit.
• Remote robotic surgeries - used so physicians are able to precisely measure both depth of force and drill bit rotational force during remote hip surgeries
• Dialysis machines - ensures uniform fluid flow and circulation of proper rate, proportion and frequency according to the parameters set by its accompanying electronic controller device.
• Ventilator gas tester - ensures uniform gas flow (i.e I:E, PEEP, TV, fIO2, SIMV, AC, etc...).
• Pressure Meter - ensures uniform negative and positive pressure flow (i.e. mmHg, cmH20, inH20, PSI, etc...).
• Vital signs simulator
• Strain gauge circuits
• Pressure transducer circuits
• and more