Wheatstone bridge: Difference between revisions

Latest revision as of 15:52, 4 September 2025

Template:Short description Template:More citations needed

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. The unknown resistance Template:Mvar is to be measured; resistances Template:Math, Template:Math and Template:Math are known, where Template:Math is adjustable. When the measured voltage Template:Mvar is 0, both legs have equal voltage ratios: Template:Math and Template:Math.

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 the circuit 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 (sometimes spelled "Christy") in 1833 and improved and popularized by Sir Charles Wheatstone in 1843.^[2] One of the Wheatstone bridge's initial uses was for soil analysis and comparison.^[3]

Operation

In the figure, Template:Math is the fixed, yet unknown, resistance to be measured. Template:Math, Template:Math, and Template:Math are resistors of known resistance and the resistance of Template:Math is adjustable. The resistance Template:Math is adjusted until the bridge is "balanced" and no current flows through the galvanometer Template:Math. At this point, the potential difference between the two midpoints (B and D) will be zero. Therefore the ratio of the two resistances in the known leg Template:Math is equal to the ratio of the two resistances in the unknown leg Template:Math. If the bridge is unbalanced, the direction of the current indicates whether Template:Math is too high or too low.

At the point of balance, $\begin{aligned} \frac{R_{2}}{R_{1}} & = \frac{R_{x}}{R_{3}} \\ \Rightarrow R_{x} & = \frac{R_{2}}{R_{1}} \cdot R_{3} \end{aligned}$

Detecting zero current with a galvanometer can be done to extremely high precision. Therefore, if Template:Math, Template:Math, and Template:Math are known to high precision, then Template:Math can be measured to high precision. Very small changes in Template:Math disrupt the balance and are readily detected.

Alternatively, if Template:Math, Template:Math, and Template:Math are known, but Template:Math is not adjustable, the voltage difference across or current flow through the meter can be used to calculate the value of Template:Math, using Kirchhoff's circuit laws. 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

File:Wheatstonebridge current.svg

Directions of currents arbitrarily assigned

Quick derivation at balance

At the point of balance, both the voltage and the current between the two midpoints (B and D) are zero. Therefore, Template:Math, Template:Math, Template:Math.

Because of Template:Math, then Template:Math and Template:Math.

Dividing the last two equations by members and using the above currents equalities, then $\begin{aligned} \frac{V_{D C}}{V_{A D}} & = \frac{V_{B C}}{V_{A B}} \\ \Rightarrow \frac{I_{2} R_{2}}{I_{1} R_{1}} & = \frac{I_{x} R_{x}}{I_{3} R_{3}} \\ \Rightarrow R_{x} & = \frac{R_{2}}{R_{1}} \cdot R_{3} \end{aligned}$

Alternative derivation at balance using voltage divider expressions

ADC and ABC form two voltage dividers, with Template:Mvar equal to the difference in output voltages. Thus $\begin{aligned} V_{D C} & = V_{B C} \\ I_{2} R_{2} & = I_{x} R_{x} \\ V_{A C} \frac{R_{2}}{R_{1} + R_{2}} & = V_{A C} \frac{R_{x}}{R_{3} + R_{x}} \\ \frac{R_{2}}{R_{1} + R_{2}} & = \frac{R_{x}}{R_{3} + R_{x}} \\ \frac{R_{1} + R_{2}}{R_{2}} & = \frac{R_{3} + R_{x}}{R_{x}} \\ 1 + \frac{R_{1}}{R_{2}} & = 1 + \frac{R_{3}}{R_{x}} \\ \frac{R_{1}}{R_{2}} & = \frac{R_{3}}{R_{x}} \end{aligned}$

Full derivation using Kirchhoff's circuit laws

First, Kirchhoff's first law is used to find the currents in junctions B and D: $\begin{aligned} I_{3} - I_{x} + I_{G} & = 0 \\ I_{1} - I_{2} - I_{G} & = 0 \end{aligned}$

Then, Kirchhoff's second law is used for finding the voltage in the loops ABDA and BCDB: $\begin{aligned} (I_{3} \cdot R_{3}) - (I_{G} \cdot R_{G}) - (I_{1} \cdot R_{1}) & = 0 \\ (I_{x} \cdot R_{x}) - (I_{2} \cdot R_{2}) + (I_{G} \cdot R_{G}) & = 0 \end{aligned}$

When the bridge is balanced, then Template:Math, so the second set of equations can be rewritten as: $\begin{aligned} I_{3} \cdot R_{3} & = I_{1} \cdot R_{1} (1) \\ I_{x} \cdot R_{x} & = I_{2} \cdot R_{2} (2) \end{aligned}$

Then, equation (1) is divided by equation (2) and the resulting equation is rearranged, giving: $R_{x} = \frac{R_{2} \cdot I_{2} \cdot I_{3} \cdot R_{3}}{R_{1} \cdot I_{1} \cdot I_{x}}$

Due to Template:Math and Template:Math being proportional from Kirchhoff's First Law, Template:Math cancels out of the above equation. The desired value of Template:Math is now known to be given as: $R_{x} = \frac{R_{3} \cdot R_{2}}{R_{1}}$

On the other hand, if the resistance of the galvanometer is high enough that Template:Math is negligible, it is possible to compute Template:Math from the three other resistor values and the supply voltage (Template:Itco), or the supply voltage from all four resistor values. To do so, one has to work out the voltage from each potential divider and subtract one from the other. The equations for this are: $\begin{aligned} V_{G} & = (\frac{R_{2}}{R_{1} + R_{2}} - \frac{R_{x}}{R_{x} + R_{3}}) V_{s} \\ R_{x} & = \frac{R_{2} \cdot V_{s} - (R_{1} + R_{2}) \cdot V_{G}}{R_{1} \cdot V_{s} + (R_{1} + R_{2}) \cdot V_{G}} R_{3} \end{aligned}$ where Template:Math 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^[4] and further improved as Template:Vanchor by Alan Blumlein in British Patent no. 323,037, 1928.

Modifications of the basic bridge

File:Kelvin bridge by RFT.png

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, and Wien bridge for measuring reactive components
Anderson's bridge, for measuring the self-inductance of the circuit, an advanced form of Maxwell's bridge

References

Template:Reflist

External links

Template:Commons category inline
DC Metering Circuits chapter from Lessons In Electric Circuits Vol 1 DC free ebook and Lessons In Electric Circuits series.
Test Set I-49

Template:Bridge circuits Template:Authority control

pl:Mostek (elektronika)#Mostek Wheatstone'a

↑ "Circuits in Practice: The Wheatstone Bridge, What It Does, and Why It Matters", as discussed in this MIT ES.333 class video
↑ Script error: No such module "Citation/CS1".
↑ Script error: No such module "Citation/CS1". discusses Christie's and Wheatstone's contributions, and why the bridge carries Wheatstone's name.
↑ Script error: No such module "Citation/CS1". Maxwell's bridge used a battery and a ballistic galvanometer. See pp. 475–477.

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

[2] Script error: No such module "Citation/CS1".

[3] Script error: No such module "Citation/CS1". discusses Christie's and Wheatstone's contributions, and why the bridge carries Wheatstone's name.

[4] Script error: No such module "Citation/CS1". Maxwell's bridge used a battery and a ballistic galvanometer. See pp. 475–477.

[1]

[2]

[3]

[4]

@@ Line 2: / Line 2: @@
 {{More citations needed|date=March 2019}}
-[[Image:Wheatstonebridge.svg|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]]. The unknown resistance ''R''<sub>''x''</sub> is to be measured; resistances ''R''<sub>1</sub>, ''R''<sub>2</sub> and ''R''<sub>3</sub> are known, where ''R''<sub>2</sub> is adjustable. When the measured voltage ''V''<sub>''G''</sub> is 0, both legs have equal voltage ratios: ''R''<sub>2</sub>/''R''<sub>1</sub> =&nbsp;''R''<sub>''x''</sub>/''R''<sub>3</sub> and ''R''<sub>''x''</sub>=&nbsp;''R''<sub>3</sub>''R''<sub>2</sub>/''R''<sub>1</sub>.]]
+[[Image:Wheatstonebridge.svg|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]]. The unknown resistance {{mvar|R<sub>x</sub>}} is to be measured; resistances {{math|''R''<sub>1</sub>}}, {{math|''R''<sub>2</sub>}} and {{math|''R''<sub>3</sub>}} are known, where {{math|''R''<sub>2</sub>}} is adjustable. When the measured voltage {{mvar|V<sub>G</sub>}} is 0, both legs have equal voltage ratios: {{math|1= ''R''<sub>2</sub>{{thinsp|/}}''R''<sub>1</sub> = ''R''<sub>''x''</sub>{{thinsp|/}}''R''<sub>3</sub>}} and {{math|1= ''R<sub>x</sub>'' = ''R''<sub>3</sub>''R''<sub>2</sub>{{thinsp|/}}''R''<sub>1</sub>}}.]]
 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 the circuit is its ability to provide extremely accurate measurements (in contrast with something like a simple [[voltage divider]]).<ref>"Circuits in Practice: The Wheatstone Bridge, What It Does, and Why It Matters", as discussed in this MIT ES.333 class [https://www.youtube.com/watch?v=-G-dySnSSG4 video]</ref> Its operation is similar to the original [[Potentiometer (measuring instrument)|potentiometer]].
-The Wheatstone bridge was invented by [[Samuel Hunter Christie]] (sometimes spelled "Christy") in 1833 and improved and popularized by Sir [[Charles Wheatstone]] in 1843.<ref>{{cite journal |last=Wheatstone |first=Charles |date=1843 |title=XIII. The Bakerian lecture.—An account of several new instruments and processes for determining the constants of a voltaic circuit |journal=Phil. Trans. R. Soc. |volume=133 |pages=303–327 |doi=10.1098/rstl.1843.0014}}</ref>  One of the Wheatstone bridge's initial uses was for [[soil analysis]] and comparison.<ref>{{cite journal |title=The Genesis of the Wheatstone Bridge |first=Stig |last=Ekelof |journal=Engineering Science and Education Journal |volume=10 |issue=1 |date=February 2001 |pages=37–40 |doi=10.1049/esej:20010106  |doi-broken-date=7 December 2024 |url=https://edisciplinas.usp.br/pluginfile.php/5618117/mod_resource/content/1/The%20genesis%20of%20Wheatstone%20bridge.pdf}} discusses [[Samuel Hunter Christie|Christie's]] and [[Charles Wheatstone|Wheatstone]]'s contributions, and why the bridge carries Wheatstone's name.</ref>
+The Wheatstone bridge was invented by [[Samuel Hunter Christie]] (sometimes spelled "Christy") in 1833 and improved and popularized by Sir [[Charles Wheatstone]] in 1843.<ref>{{cite journal |last=Wheatstone |first=Charles |date=1843 |title=XIII. The Bakerian lecture.—An account of several new instruments and processes for determining the constants of a voltaic circuit |journal=Phil. Trans. R. Soc. |volume=133 |pages=303–327 |doi=10.1098/rstl.1843.0014}}</ref>  One of the Wheatstone bridge's initial uses was for [[soil analysis]] and comparison.<ref>{{cite journal |title=The Genesis of the Wheatstone Bridge |first=Stig |last=Ekelof |journal=Engineering Science and Education Journal |volume=10 |issue=1 |date=February 2001 |pages=37–40 |doi=10.1049/esej:20010106  |doi-broken-date=12 July 2025 |url=https://edisciplinas.usp.br/pluginfile.php/5618117/mod_resource/content/1/The%20genesis%20of%20Wheatstone%20bridge.pdf}} discusses [[Samuel Hunter Christie|Christie's]] and [[Charles Wheatstone|Wheatstone]]'s contributions, and why the bridge carries Wheatstone's name.</ref>
 == Operation ==
@@ Line 13: / Line 13: @@
 At the point of balance,
-: <math>\begin{align}
+<math display=block>\begin{align}
    \frac{R_2}{R_1} &= \frac{R_x}{R_3} \\[4pt]
    \Rightarrow R_x &= \frac{R_2}{R_1} \cdot R_3
@@ Line 31: / Line 31: @@
 Dividing the last two equations by members and using the above currents equalities, then
-: <math>\begin{align}
+<math display=block>\begin{align}
    \frac{V_{DC}}{V_{AD}}&=\frac{V_{BC}}{V_{AB}} \\[4pt]
   \Rightarrow \frac{I_2R_2}{I_1R_1} &= \frac{I_xR_x}{I_3R_3}\\[4pt]
@@ Line 37: / Line 37: @@
 \end{align}</math>
-=== Alternative Derivation at Balance using Voltage Divider Expressions ===
+=== Alternative derivation at balance using voltage divider expressions ===
-ADC and ABC form two [[voltage divider]]s, with <math> V_G </math> equal to the difference in output voltages. Thus
+ADC and ABC form two [[voltage divider]]s, with {{mvar|V{{sub|G}}}} equal to the difference in output voltages. Thus
-: <math>\begin{align}
+<math display=block>\begin{align}
 V_{DC} &= V_{BC} \\
 I_2 R_2 &= I_x R_x \\
@@ Line 51: / Line 51: @@
 === Full derivation using Kirchhoff's circuit laws ===
 First, [[Kirchoff's first law|Kirchhoff's first law]] is used to find the currents in junctions B and D:
-: <math>\begin{align}
+<math display=block>\begin{align}
    I_3 - I_x + I_G &= 0 \\
    I_1 - I_2 - I_G &= 0
@@ Line 57: / Line 57: @@
 Then, [[Kirchhoff's circuit laws#Kirchhoff's voltage law (KVL)|Kirchhoff's second law]] is used for finding the voltage in the loops ABDA and BCDB:
-: <math>\begin{align}
+<math display=block>\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
@@ Line 63: / Line 63: @@
 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}
+<math display=block>\begin{align}
    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)}
@@ Line 69: / Line 69: @@
 Then, equation (1) is divided by equation (2) and the resulting equation is 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>
+<math display=block>R_x = {{R_2 \cdot I_2 \cdot I_3 \cdot R_3}\over{R_1 \cdot I_1 \cdot I_x}}</math>
-Due to {{math|1=''I''<sub>3</sub> = ''I''<sub>''x''</sub>}} and {{math|1=''I''<sub>1</sub> = ''I''<sub>2</sub>}} being proportional from Kirchhoff's First Law,  {{math|''I''<sub>3</sub>''I''<sub>2</sub>/''I''<sub>1</sub>''I''<sub>x</sub>}} cancels out of the above equation.  The desired value of {{math|''R''<sub>''x''</sub>}} is now known to be given as:
+Due to {{math|1=''I''<sub>3</sub> = ''I''<sub>''x''</sub>}} and {{math|1=''I''<sub>1</sub> = ''I''<sub>2</sub>}} being proportional from Kirchhoff's First Law,  {{math|''I''<sub>3</sub>''I''<sub>2</sub>{{thinsp|/}}''I''<sub>1</sub>''I''<sub>x</sub>}} cancels out of the above equation.  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>
+<math display=block>R_x = {{R_3 \cdot R_2}\over{R_1}}</math>
-On the other hand, if the resistance of the galvanometer is high enough that {{math|''I''<sub>''G''</sub>}} is negligible, it is possible to compute {{math|''R''<sub>''x''</sub>}} from the three other resistor values and the supply voltage ({{math|''V''<sub>''S''</sub>}}), or the supply voltage from all four resistor values. To do so, one has to work out the voltage from each [[potential divider]] and subtract one from the other. The equations for this are:
+On the other hand, if the resistance of the galvanometer is high enough that {{math|''I''<sub>''G''</sub>}} is negligible, it is possible to compute {{math|''R''<sub>''x''</sub>}} from the three other resistor values and the supply voltage ({{itco|{{math|''V''<sub>''S''</sub>}}}}), or the supply voltage from all four resistor values. To do so, one has to work out the voltage from each [[potential divider]] and subtract one from the other. The equations for this are:
-: <math>
+<math display=block>
 \begin{align}
 V_G & = \left({R_2\over{R_1 + R_2}} - {R_x \over {R_x + R_3}}\right)V_s \\[6pt]

Wheatstone bridge: Difference between revisions

Latest revision as of 15:52, 4 September 2025

Contents

Operation

Derivation

Quick derivation at balance

Alternative derivation at balance using voltage divider expressions

Full derivation using Kirchhoff's circuit laws

Significance

Modifications of the basic bridge

See also

References

External links

Navigation menu

Wheatstone bridge: Difference between revisions

Latest revision as of 15:52, 4 September 2025

Operation

Derivation

Quick derivation at balance

Alternative derivation at balance using voltage divider expressions

Full derivation using Kirchhoff's circuit laws

Significance

Modifications of the basic bridge

See also

References

External links

Navigation menu

Search