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Wheatstone bridge
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<h2 id="operation">Operation</h2> <p>In the figure, <i>Rx</i> is the fixed, yet unknown, resistance to be measured. <i>R</i>1, <i>R</i>2, and <i>R</i>3 are resistors of known resistance and the resistance of <i>R</i>2 is adjustable. The resistance <i>R</i>2 is adjusted until the bridge is "balanced" and no current flows through the <a href="/facts/Galvanometer/Y3rFC18f">galvanometer</a> <i>Vg</i>. At this point, the <a href="/facts/Potential_difference/EN2po6N2">potential difference</a> between the two midpoints (B and D) will be zero. Therefore the ratio of the two resistances in the known leg (<i>R</i>2 / <i>R</i>1) is equal to the ratio of the two resistances in the unknown leg (<i>Rx</i> / <i>R</i>3). If the bridge is unbalanced, the direction of the current indicates whether <i>R</i>2 is too high or too low. </p><p>At the point of balance, </p> R 2 R 1 = R x R 3 ⇒ R x = R 2 R 1 ⋅ R 3 {\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}}} <p>Detecting zero current with a <a href="/facts/Galvanometer/Y3rFC18f">galvanometer</a> can be done to extremely high precision. Therefore, if <i>R</i>1, <i>R</i>2, and <i>R</i>3 are known to high precision, then <i>Rx</i> can be measured to high precision. Very small changes in <i>Rx</i> disrupt the balance and are readily detected. </p><p>Alternatively, if <i>R</i>1, <i>R</i>2, and <i>R</i>3 are known, but <i>R</i>2 is not adjustable, the voltage difference across or current flow through the meter can be used to calculate the value of <i>Rx</i>, using <a href="/facts/Kirchhoff%2527s_circuit_laws/fGAAFXB3">Kirchhoff's circuit laws</a>. This setup is frequently used in <a href="/facts/Strain_gauge/w12yKexN">strain gauge</a> and <a href="/facts/Resistance_thermometer/wxYpjdfR">resistance thermometer</a> measurements, as it is usually faster to read a voltage level off a meter than to adjust a resistance to zero the voltage. </p> <h2 id="derivation">Derivation</h2> <h3>Quick derivation at balance</h3> <p>At the point of balance, both the <a href="/facts/Voltage/EN2po6N2">voltage</a> and the <a href="/facts/Electric_current/Dc1xdnUX">current</a> between the two midpoints (B and D) are zero. Therefore, <i>I</i>1 = <i>I</i>2, <i>I</i>3 = <i>I</i><i>x</i>, <i>V</i>D = <i>V</i>B. </p><p>Because of <i>V</i>D = <i>V</i>B, then <i>V</i>DC = <i>V</i>BC and <i>V</i>AD = <i>V</i>AB. </p><p>Dividing the last two equations by members and using the above currents equalities, then </p> V D C V A D = V B C V A B ⇒ I 2 R 2 I 1 R 1 = I x R x I 3 R 3 ⇒ R x = R 2 R 1 ⋅ R 3 {\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}}} <h3>Alternative Derivation at Balance using Voltage Divider Expressions</h3> <p>ADC and ABC form two <a href="/facts/Voltage_divider/PYYlGlYU">voltage dividers</a>, with V G {\displaystyle V_{G}} equal to the difference in output voltages. Thus </p> V D C = V B C I 2 R 2 = I x R x V A C R 2 R 1 + R 2 = V A C R x R 3 + R x R 2 R 1 + R 2 = R x R 3 + R x R 1 + R 2 R 2 = R 3 + R x R x 1 + R 1 R 2 = 1 + R 3 R x R 1 R 2 = R 3 R x {\displaystyle {\begin{aligned}V_{DC}&=V_{BC}\\I_{2}R_{2}&=I_{x}R_{x}\\V_{AC}{\frac {R_{2}}{R_{1}+R_{2}}}&=V_{AC}{\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}}} <h3>Full derivation using Kirchhoff's circuit laws</h3> <p>First, <a href="/facts/Kirchoff%2527s_first_law/fGAAFXB3">Kirchhoff's first law</a> is used to find the currents in junctions B and D: </p> I 3 − I x + I G = 0 I 1 − I 2 − I G = 0 {\displaystyle {\begin{aligned}I_{3}-I_{x}+I_{G}&=0\\I_{1}-I_{2}-I_{G}&=0\end{aligned}}} <p>Then, <a href="/facts/Kirchhoff%2527s_circuit_laws/fGAAFXB3">Kirchhoff's second law</a> is used for finding the voltage in the loops ABDA and BCDB: </p> ( I 3 ⋅ R 3 ) − ( I G ⋅ R G ) − ( I 1 ⋅ R 1 ) = 0 ( I x ⋅ R x ) − ( I 2 ⋅ R 2 ) + ( I G ⋅ R G ) = 0 {\displaystyle {\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}}} <p>When the bridge is balanced, then <i>I</i><i>G</i> = 0, so the second set of equations can be rewritten as: </p> I 3 ⋅ R 3 = I 1 ⋅ R 1 (1) I x ⋅ R x = I 2 ⋅ R 2 (2) {\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}}} <p>Then, equation (1) is divided by equation (2) and the resulting equation is rearranged, giving: </p> R x = R 2 ⋅ I 2 ⋅ I 3 ⋅ R 3 R 1 ⋅ I 1 ⋅ I x {\displaystyle R_{x}={{R_{2}\cdot I_{2}\cdot I_{3}\cdot R_{3}} \over {R_{1}\cdot I_{1}\cdot I_{x}}}} <p>Due to <i>I</i>3 = <i>I</i><i>x</i> and <i>I</i>1 = <i>I</i>2 being proportional from Kirchhoff's First Law, <i>I</i>3<i>I</i>2/<i>I</i>1<i>I</i>x cancels out of the above equation. The desired value of <i>R</i><i>x</i> is now known to be given as: </p> R x = R 3 ⋅ R 2 R 1 {\displaystyle R_{x}={{R_{3}\cdot R_{2}} \over {R_{1}}}} <p>On the other hand, if the resistance of the galvanometer is high enough that <i>I</i><i>G</i> is negligible, it is possible to compute <i>R</i><i>x</i> from the three other resistor values and the supply voltage (<i>V</i><i>S</i>), or the supply voltage from all four resistor values. To do so, one has to work out the voltage from each <a href="/facts/Potential_divider/PYYlGlYU">potential divider</a> and subtract one from the other. The equations for this are: </p> V G = ( R 2 R 1 + R 2 − R x R x + R 3 ) V s R x = R 2 ⋅ V s − ( R 1 + R 2 ) ⋅ V G R 1 ⋅ V s + ( R 1 + R 2 ) ⋅ V G R 3 {\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}}} <p>where <i>V</i><i>G</i> is the voltage of node D relative to node B. </p> <h2 id="significance">Significance</h2> <p>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 <a href="/facts/Capacitance/0pH046TL">capacitance</a>, <a href="/facts/Inductance/oIxj7JxX">inductance</a>, <a href="/facts/Electrical_impedance/I06NSXtN">impedance</a> and other quantities, such as the amount of combustible gases in a sample, with an <a href="/facts/Explosimeter/7NulIopw">explosimeter</a>. The <a href="/facts/Kelvin_bridge/zVzNa1Mq">Kelvin bridge</a> 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 <a href="/facts/Physical_phenomenon/4CBrlVeW">physical phenomenon</a> (such as force, temperature, pressure, etc.) which thereby allows the use of Wheatstone bridge in measuring those elements indirectly. </p><p>The concept was extended to <a href="/facts/Alternating_current/qxICS3xa">alternating current</a> measurements by <a href="/facts/James_Clerk_Maxwell/jtYyze61">James Clerk Maxwell</a> in 1865<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> and further improved as Blumlein bridge by <a href="/facts/Alan_Blumlein/ka01T9Lu">Alan Blumlein</a> in British Patent no. 323,037, 1928. </p> <h2 id="modifications-of-the-basic-bridge">Modifications of the basic bridge</h2> <p>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: </p> <ul><li><a href="/facts/Carey_Foster_bridge/wmzQ7BbL">Carey Foster bridge</a>, for measuring small resistances</li> <li><a href="/facts/Kelvin_bridge/zVzNa1Mq">Kelvin bridge</a>, for measuring small <a href="/facts/Four-terminal_sensing/VdkPQnPQ">four-terminal</a> resistances</li> <li><a href="/facts/Maxwell_bridge/e5B6R9QB">Maxwell bridge</a>, and <a href="/facts/Wien_bridge/9YF0z5yQ">Wien bridge</a> for measuring <a href="/facts/Electrical_reactance/LnvyrUnM">reactive</a> components</li> <li><a href="/facts/Anderson%2527s_bridge/lS8m8QEK">Anderson's bridge</a>, for measuring the self-inductance of the circuit, an advanced form of Maxwell's bridge</li></ul> <h2 id="see-also">See also</h2> <ul> <li>Electronics portal</li></ul> <ul><li><a href="/facts/Diode_bridge/jN3TqC6M">Diode bridge</a>, <a href="/facts/Frequency_mixer/RopD2DxD">product mixer</a> – diode bridges</li> <li><a href="/facts/Phantom_circuit/MI3KPDcf">Phantom circuit</a> – a circuit using a balanced bridge</li> <li><a href="/facts/Post_office_box_(electricity)/dHKAtjQG">Post office box (electricity)</a></li> <li><a href="/facts/Potentiometer_(measuring_instrument)/tsnDbrkx">Potentiometer (measuring instrument)</a></li> <li><a href="/facts/Potential_divider/PYYlGlYU">Potential divider</a></li> <li><a href="/facts/Ohmmeter/4PCYujxC">Ohmmeter</a></li> <li><a href="/facts/Resistance_thermometer/wxYpjdfR">Resistance thermometer</a></li> <li><a href="/facts/Strain_gauge/w12yKexN">Strain gauge</a></li></ul> <h2 id="external-links">External links</h2> <ul><li> Media related to Wheatstone's bridge at Wikimedia Commons</li> <li><a href="http://www.ibiblio.org/kuphaldt/electricCircuits/DC/DC_8.html"><i>DC Metering Circuits</i></a> chapter from <a href="http://www.ibiblio.org/kuphaldt/electricCircuits/DC/index.html"><i>Lessons In Electric Circuits Vol 1 DC</i></a> free ebook and <a href="http://www.ibiblio.org/kuphaldt/electricCircuits/"><i>Lessons In Electric Circuits</i></a> series.</li> <li><a href="http://radionerds.com/index.php/I-49">Test Set I-49</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>"Circuits in Practice: The Wheatstone Bridge, What It Does, and Why It Matters", as discussed in this MIT ES.333 class video <a href="https://www.youtube.com/watch?v=-G-dySnSSG4" target="_blank">https://www.youtube.com/watch?v=-G-dySnSSG4</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Wheatstone, Charles (1843). "XIII. The Bakerian lecture.—An account of several new instruments and processes for determining the constants of a voltaic circuit". Phil. Trans. R. Soc. 133: 303–327. doi:10.1098/rstl.1843.0014. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Ekelof, Stig (February 2001). "The Genesis of the Wheatstone Bridge" (PDF). Engineering Science and Education Journal. 10 (1): 37–40. doi:10.1049/esej:20010106 (inactive 7 December 2024).{{cite journal}}: CS1 maint: DOI inactive as of December 2024 (link) discusses Christie's and Wheatstone's contributions, and why the bridge carries Wheatstone's name. <a href="https://edisciplinas.usp.br/pluginfile.php/5618117/mod_resource/content/1/The%20genesis%20of%20Wheatstone%20bridge.pdf" target="_blank">https://edisciplinas.usp.br/pluginfile.php/5618117/mod_resource/content/1/The%20genesis%20of%20Wheatstone%20bridge.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Maxwell, J. Clerk (1865). "A dynamical theory of the electromagnetic field". Philosophical Transactions of the Royal Society of London. 155: 459–512. Bibcode:1865RSPT..155..459M. Maxwell's bridge used a battery and a ballistic galvanometer. See pp. 475–477. <a href="https://archive.org/details/dynamicaltheoryo00maxw/page/458/mode/2up" target="_blank">https://archive.org/details/dynamicaltheoryo00maxw/page/458/mode/2up</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>