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[[Image:Wheatstonebridge.svg|right|thumb|300px|alt=A Wheatstone bridge has four resistors forming the sides of a diamond shape. A battery is connected across one pair of opposite corners, and a galvanometer across the other pair. |Wheatstone bridge [[circuit diagram]].]]
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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 (measuring instrument)|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.<ref>"The Genesis of the Wheatstone Bridge" by Stig Ekelof discusses [[Samuel Hunter Christie|Christie's]] and [[Charles Wheatstone|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.</ref>
 
== Operation ==
In the figure, <math>\scriptstyle R_x</math> is the unknown resistance to be measured; <math>\scriptstyle R_1</math>, <math>\scriptstyle R_2</math> and <math>\scriptstyle R_3</math> are resistors of known resistance and the resistance of <math>\scriptstyle R_2</math> is adjustable. If the ratio of the two resistances in the known leg <math>\scriptstyle (R_2 / R_1)</math> is equal to the ratio of the two in the unknown leg <math>\scriptstyle (R_x / R_3)</math>, then the [[voltage]] between the two midpoints ('''B''' and '''D''') will be zero and no [[Current (electricity)|current]] will flow through the [[galvanometer]] <math>\scriptstyle V_g</math>. If the bridge is unbalanced, the direction of the current indicates whether <math>\scriptstyle R_2</math> is too high or too low. <math>\scriptstyle R_2</math> 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 <math>\scriptstyle R_1</math>, <math>\scriptstyle R_2</math> and <math>\scriptstyle R_3</math> are known to high precision, then <math>\scriptstyle R_x</math> can be measured to high precision. Very small changes in <math>\scriptstyle R_x</math> disrupt the balance and are readily detected.
 
At the point of balance, the ratio of
:<math>\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}</math>
 
Alternatively, if <math>\scriptstyle R_1</math>, <math>\scriptstyle R_2</math>, and <math>\scriptstyle R_3</math> are known, but <math>\scriptstyle R_2</math> is not adjustable, the voltage difference across or current flow through the meter can be used to calculate the value of <math>\scriptstyle R_x</math>, 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 first law|Kirchhoff's first rule]] is used to find the currents in junctions '''B''' and '''D''':
 
:<math>\begin{align}
  I_3 - I_x + I_G &= 0 \\
  I_1 - I_2 - I_G &= 0
\end{align}</math>
 
Then, [[Kirchhoff's circuit laws#Kirchhoff's voltage law (KVL)|Kirchhoff's second rule]] is used for finding the voltage in the loops '''ABD''' and '''BCD''':
 
:<math>\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}</math>
 
When the bridge is balanced, then {{math|''I''<sub>''G''</sub> {{=}} 0}}, so the second set of equations can be rewritten as:
:<math>\begin{align}
  I_3 \cdot R_3 &= I_1 \cdot R_1 \\
  I_x \cdot R_x &= I_2 \cdot R_2
\end{align}</math>
 
Then, the equations are divided and rearranged, giving:
:<math>R_x = {{R_2 \cdot I_2 \cdot I_3 \cdot R_3}\over{R_1 \cdot I_1 \cdot I_x}}</math>
 
From the first rule, {{math|''I''<sub>''3''</sub> {{=}} ''I''<sub>''x''</sub>}} and {{math|''I''<sub>''1''</sub> {{=}} ''I''<sub>''2''</sub>}}. The desired value of {{math|''R''<sub>''x''</sub>}} is now known to be given as:
:<math>R_x = {{R_3 \cdot R_2}\over{R_1}}</math>
 
If all four resistor values and the supply voltage ({{math|''V''<sub>''S''</sub>}}) are known, and the resistance of the galvanometer is high enough that {{math|''I''<sub>''G''</sub>}} is negligible, the voltage across the bridge ({{math|''V''<sub>''G''</sub>}}) can be found by working out the voltage from each [[potential divider]] and subtracting one from the other. The equation for this is:
:<math>V_G = \left({{R_x}\over{R_3 + R_x}} - {{R_2}\over{R_1 + R_2}}\right)V_s</math>
 
where {{math|''V''<sub>''G''</sub>}} 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]], [[inductance]], [[Electrical impedance|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]] in about 1926.
 
==Modifications of the fundamental bridge==
[[File:Kelvin bridge by RFT.png|right|thumb|300px|[[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–Varley_divider|Kelvin Varley Slide]]
* [[Kelvin bridge]]
* [[Maxwell bridge]]
 
==See also==
{{Portal|Electronics}}
* [[Murray loop bridge]]
* [[Maxwell bridge]]
* [[Wien bridge]]
* [[Phantom circuit]] - a circuit using a balanced bridge
* [[Post Office Box (electricity)|Post Office Box]]
* [[Potentiometer]]
* [[Potential divider]]
* [[Ohmmeter]]
* [[Resistance thermometer]]
* [[Strain gauge]]
* [[E-meter]] - a variation used by [[Scientology]]
 
==References==
<references/>
 
==External links==
* [http://www.magnet.fsu.edu/education/tutorials/java/wheatstonebridge/index.html Wheatstone Bridge - Interactive Java Tutorial] National High Magnetic Field Laboratory
*[http://www.efunda.com/designstandards/sensors/methods/wheatstone_bridge.cfm efunda Wheatstone article]
*[http://books.google.com/books?id=z3lKAAAAMAAJ Methods of Measuring Electrical Resistance - Edwin F. Northrup, 1912, full-text on Google Books]
*[http://www.strainmatics.com Measuring strain using Wheatstone bridge principles]
 
{{DEFAULTSORT:Wheatstone Bridge}}
[[Category:Electrical meters]]
[[Category:Electrical circuits]]
[[Category:Measuring instruments]]
[[Category:English inventions]]

Latest revision as of 19:26, 23 December 2014

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