Wheatstone bridge

# Wheatstone bridge

Overview
A Wheatstone bridge is an electrical circuit used to measure an unknown electrical resistance
Electrical resistance
The electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...

by balancing two legs of a bridge circuit
Bridge circuit
A bridge circuit is a type of electrical circuit in which two circuit branches are "bridged" by a third branch connected between the first two branches at some intermediate point along them. The bridge was originally developed for laboratory measurement purposes and one of the intermediate...

, one leg of which includes the unknown component. Its operation is similar to the original potentiometer
Potentiometer (measuring instrument)
A potentiometer is an instrument for measuring the potential in a circuit. Before the introduction of the moving coil and digital volt meters, potentiometers were used in measuring voltage, hence the '-meter' part of their name...

. It was invented by Samuel Hunter Christie
Samuel Hunter Christie
Samuel Hunter Christie was a British scientist and mathematician.He studied mathematics at Trinity College, Cambridge where he was second wrangler. He was particularly interested in magnetism, studying the earth's magnetic field and designing improvements to the magnetic compass...

in 1833 and improved and popularized by Sir Charles Wheatstone
Charles Wheatstone
Sir Charles Wheatstone FRS , was an English scientist and inventor of many scientific breakthroughs of the Victorian era, including the English concertina, the stereoscope , and the Playfair cipher...

in 1843.
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Encyclopedia
A Wheatstone bridge is an electrical circuit used to measure an unknown electrical resistance
Electrical resistance
The electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...

by balancing two legs of a bridge circuit
Bridge circuit
A bridge circuit is a type of electrical circuit in which two circuit branches are "bridged" by a third branch connected between the first two branches at some intermediate point along them. The bridge was originally developed for laboratory measurement purposes and one of the intermediate...

, one leg of which includes the unknown component. Its operation is similar to the original potentiometer
Potentiometer (measuring instrument)
A potentiometer is an instrument for measuring the potential in a circuit. Before the introduction of the moving coil and digital volt meters, potentiometers were used in measuring voltage, hence the '-meter' part of their name...

. It was invented by Samuel Hunter Christie
Samuel Hunter Christie
Samuel Hunter Christie was a British scientist and mathematician.He studied mathematics at Trinity College, Cambridge where he was second wrangler. He was particularly interested in magnetism, studying the earth's magnetic field and designing improvements to the magnetic compass...

in 1833 and improved and popularized by Sir Charles Wheatstone
Charles Wheatstone
Sir Charles Wheatstone FRS , was an English scientist and inventor of many scientific breakthroughs of the Victorian era, including the English concertina, the stereoscope , and the Playfair cipher...

in 1843.

## Operation

In the figure, is the unknown resistance to be measured; , and are resistors of known resistance and the resistance of is adjustable. If the ratio of the two resistances in the known leg is equal to the ratio of the two in the unknown leg , then the voltage
Voltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...

between the two midpoints (B and D) will be zero and no current will flow through the galvanometer
Galvanometer
A galvanometer is a type of ammeter: an instrument for detecting and measuring electric current. It is an analog electromechanical transducer that produces a rotary deflection of some type of pointer in response to electric current flowing through its coil in a magnetic field. .Galvanometers were...

. If the bridge is unbalanced, the direction of the current indicates whether is too high or too low. is varied until there is no current through the galvanometer, which then reads zero.

Detecting zero current with a galvanometer
Galvanometer
A galvanometer is a type of ammeter: an instrument for detecting and measuring electric current. It is an analog electromechanical transducer that produces a rotary deflection of some type of pointer in response to electric current flowing through its coil in a magnetic field. .Galvanometers were...

can be done to extremely high accuracy. Therefore, if , and are known to high precision, then can be measured to high precision. Very small changes in disrupt the balance and are readily detected.

At the point of balance, the ratio of

Therefore,

Alternatively, if , , and are known, but is not adjustable, the voltage difference across or current flow through the meter can be used to calculate the value of , using Kirchhoff's circuit laws
Kirchhoff's circuit laws
Kirchhoff's circuit laws are two equalities that deal with the conservation of charge and energy in electrical circuits, and were first described in 1845 by Gustav Kirchhoff...

(also known as Kirchhoff's rules). This setup is frequently used in strain gauge
Strain gauge
A strain gauge is a device used to measure the strain of an object. Invented by Edward E. Simmons and Arthur C. Ruge in 1938, the most common type of strain gauge consists of an insulating flexible backing which supports a metallic foil pattern. The gauge is attached to the object by a suitable...

and resistance thermometer
Resistance thermometer
Resistance thermometers, also called resistance temperature detectors or resistive thermal devices , are sensors used to measure temperature by correlating the resistance of the RTD element with temperature. Most RTD elements consist of a length of fine coiled wire wrapped around a ceramic or glass...

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:

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

The bridge is balanced and , so the second set of equations can be rewritten as:

Then, the equations are divided and rearranged, giving:

From the first rule, and . The desired value of is now known to be given as:

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

This can be simplified to:

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

## 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
Capacitance
In electromagnetism and electronics, capacitance is the ability of a capacitor to store energy in an electric field. Capacitance is also a measure of the amount of electric potential energy stored for a given electric potential. A common form of energy storage device is a parallel-plate capacitor...

, inductance
Inductance
In electromagnetism and electronics, inductance is the ability of an inductor to store energy in a magnetic field. Inductors generate an opposing voltage proportional to the rate of change in current in a circuit...

, impedance
Electrical impedance
Electrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...

and other quantities, such as the amount of combustible gases in a sample, with an explosimeter
Explosimeter
An explosimeter is a device which is used to measure the amount of combustible gases present in a sample. When a percentage of the lower explosive limit of an atmosphere is exceeded, an alarm signal on the instrument is activated...

. The Kelvin bridge
Kelvin bridge
A Kelvin bridge is a measuring instrument invented by William Thomson, 1st Baron Kelvin. It is used to measure an unknown electrical resistance below 1 Ω. Its operation is similar to the Wheatstone bridge except for the presence of additional resistors...

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
Alternating current
In alternating current the movement of electric charge periodically reverses direction. In direct current , the flow of electric charge is only in one direction....

measurements by James Clerk Maxwell
James Clerk Maxwell
James Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...

in 1865 and further improved by Alan Blumlein
Alan Blumlein
Alan Dower Blumlein was a British electronics engineer, notable for his many inventions in telecommunications, sound recording, stereo, television and radar...

## 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:
• Carey Foster bridge
Carey Foster bridge
In electronics, the Carey Foster bridge is a bridge circuit used to measure low resistances, or to measure small differences between two large resistances. It was invented by Carey Foster as a variant on the Wheatstone bridge...

, for measuring small resistances
• Kelvin Varley Slide
• Kelvin Double bridge
• Maxwell bridge
Maxwell bridge
|- align = "center"|| A Maxwell bridge is a type of Wheatstone bridge used to measure an unknown inductance in terms of calibrated resistance and capacitance...

• Murray loop bridge
Murray loop bridge
Murray loop bridge is a bridge circuit used for locating faults in underground or underwater cables. It has been used for more than 100 years....

• Maxwell bridge
Maxwell bridge
|- align = "center"|| A Maxwell bridge is a type of Wheatstone bridge used to measure an unknown inductance in terms of calibrated resistance and capacitance...

• Wein's bridge, see Wien bridge
Wien bridge
The Wien bridge is a type of bridge circuit that was developed by Max Wien in 1891. The bridge comprises four resistors and two capacitors.Bridge circuits were a common way of measuring component values by comparing them to known values...

• Phantom circuit
Phantom circuit
In telecommunication and electrical engineering, a phantom circuit is an electrical circuit derived from suitably arranged wires with one or more conductive paths being a circuit in itself and at the same time acting as one conductor of another circuit....

- a circuit using a balanced bridge
• Post Office Box
Post Office Box (electricity)
The Post Office Box was a wheatstone bridge style testing device with pegs and spring arms to close electrical circuits and measure properties of the circuit under test....

• Potentiometer
Potentiometer
A potentiometer , informally, a pot, is a three-terminal resistor with a sliding contact that forms an adjustable voltage divider. If only two terminals are used , it acts as a variable resistor or rheostat. Potentiometers are commonly used to control electrical devices such as volume controls on...

• Potential divider
• Ohmmeter
Ohmmeter
An ohmmeter is an electrical instrument that measures electrical resistance, the opposition to an electric current. Micro-ohmmeters make low resistance measurements. Megohmmeters measure large values of resistance...

• Resistance thermometer
Resistance thermometer
Resistance thermometers, also called resistance temperature detectors or resistive thermal devices , are sensors used to measure temperature by correlating the resistance of the RTD element with temperature. Most RTD elements consist of a length of fine coiled wire wrapped around a ceramic or glass...

• Strain gauge
Strain gauge
A strain gauge is a device used to measure the strain of an object. Invented by Edward E. Simmons and Arthur C. Ruge in 1938, the most common type of strain gauge consists of an insulating flexible backing which supports a metallic foil pattern. The gauge is attached to the object by a suitable...