Wheatstone bridge

Last updated May 03, 2024

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.

Operation

In the figure, $R x$ is the fixed, yet 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. The resistance $R 2$ is adjusted until the bridge is "balanced" and no current flows through the galvanometer $V g$ . 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 $(R 2 / R 1)$ is equal to the ratio of the two resistances in the unknown leg $(R x / R 3)$ . If the bridge is unbalanced, the direction of the current indicates whether $R 2$ is too high or too low.

At the point of balance,

{\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}}

Detecting zero current with a galvanometer can be done to extremely high precision. 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.

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 $R x$ , 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

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, $I 1 = I 2$ , $I 3 = I x$ , $V D = V B$ .

Because of $V D = V B$ , then $V DC = V BC$ and $V AD = V AB$ .

Dividing the last two equations by members and using the above currents equalities, then

{\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}}

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 $I G = 0$ , so the second set of equations can be rewritten as:

{\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}}

Then, equation (1) is divided by equation (2) and the resulting equation is rearranged, giving:

R_{x}={{R_{2}\cdot I_{2}\cdot I_{3}\cdot R_{3}} \over {R_{1}\cdot I_{1}\cdot I_{x}}}

Due to $I 3 = I x$ and $I 1 = I 2$ being proportional from Kirchhoff's First Law, $I 3 I 2 / I 1 I x$ cancels out of the above equation. The desired value of $R x$ is now known to be given as:

R_{x}={{R_{3}\cdot R_{2}} \over {R_{1}}}

On the other hand, if the resistance of the galvanometer is high enough that $I G$ is negligible, it is possible to compute $R x$ from the three other resistor values and the supply voltage ( $V S$ ), 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}&=\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}}

where $V G$ 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 Blumlein bridge by Alan Blumlein in British Patent no. 323,037, 1928.

Modifications of the basic 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

Related Research Articles

In electrical engineering, impedance is the opposition to alternating current presented by the combined effect of resistance and reactance in a circuit.

In fluid dynamics, Stokes' law is an empirical law for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid. It was derived by George Gabriel Stokes in 1851 by solving the Stokes flow limit for small Reynolds numbers of the Navier–Stokes equations.

Two-terminal components and electrical networks can be connected in series or parallel. The resulting electrical network will have two terminals, and itself can participate in a series or parallel topology. Whether a two-terminal "object" is an electrical component or an electrical network is a matter of perspective. This article will use "component" to refer to a two-terminal "object" that participates in the series/parallel networks.

A resistor–capacitor circuit, or RC filter or RC network, is an electric circuit composed of resistors and capacitors. It may be driven by a voltage or current source and these will produce different responses. A first order RC circuit is composed of one resistor and one capacitor and is the simplest type of RC circuit.

In electronics, a voltage divider (also known as a potential divider) is a passive linear circuit that produces an output voltage (V_out) that is a fraction of its input voltage (V_in). Voltage division is the result of distributing the input voltage among the components of the divider. A simple example of a voltage divider is two resistors connected in series, with the input voltage applied across the resistor pair and the output voltage emerging from the connection between them.

A current mirror is a circuit designed to copy a current through one active device by controlling the current in another active device of a circuit, keeping the output current constant regardless of loading. The current being "copied" can be, and sometimes is, a varying signal current. Conceptually, an ideal current mirror is simply an ideal inverting current amplifier that reverses the current direction as well, or it could consist of a current-controlled current source (CCCS). The current mirror is used to provide bias currents and active loads to circuits. It can also be used to model a more realistic current source.

Kirchhoff's circuit laws are two equalities that deal with the current and potential difference in the lumped element model of electrical circuits. They were first described in 1845 by German physicist Gustav Kirchhoff. This generalized the work of Georg Ohm and preceded the work of James Clerk Maxwell. Widely used in electrical engineering, they are also called Kirchhoff's rules or simply Kirchhoff's laws. These laws can be applied in time and frequency domains and form the basis for network analysis.

A bridge circuit is a topology of electrical circuitry 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 bridging points is often adjustable when so used. Bridge circuits now find many applications, both linear and non-linear, including in instrumentation, filtering and power conversion.

In electrical engineering and electronics, a network is a collection of interconnected components. Network analysis is the process of finding the voltages across, and the currents through, all network components. There are many techniques for calculating these values; however, for the most part, the techniques assume linear components. Except where stated, the methods described in this article are applicable only to linear network analysis.

A current source is an electronic circuit that delivers or absorbs an electric current which is independent of the voltage across it.

A magnetic circuit is made up of one or more closed loop paths containing a magnetic flux. The flux is usually generated by permanent magnets or electromagnets and confined to the path by magnetic cores consisting of ferromagnetic materials like iron, although there may be air gaps or other materials in the path. Magnetic circuits are employed to efficiently channel magnetic fields in many devices such as electric motors, generators, transformers, relays, lifting electromagnets, SQUIDs, galvanometers, and magnetic recording heads.

A Widlar current source is a modification of the basic two-transistor current mirror that incorporates an emitter degeneration resistor for only the output transistor, enabling the current source to generate low currents using only moderate resistor values.

<span class="mw-page-title-main">Nodal analysis</span> Method in electric circuits analysis

In electric circuits analysis, nodal analysis, node-voltage analysis, or the branch current method is a method of determining the voltage between "nodes" in an electrical circuit in terms of the branch currents.

A phase-shift oscillator is a linear electronic oscillator circuit that produces a sine wave output. It consists of an inverting amplifier element such as a transistor or op amp with its output fed back to its input through a phase-shift network consisting of resistors and capacitors in a ladder network. The feedback network 'shifts' the phase of the amplifier output by 180 degrees at the oscillation frequency to give positive feedback. Phase-shift oscillators are often used at audio frequency as audio oscillators.

In electronics, diode modelling refers to the mathematical models used to approximate the actual behaviour of real diodes to enable calculations and circuit analysis. A diode's I-V curve is nonlinear.

The Wien bridge is a type of bridge circuit that was developed by Max Wien in 1891. The bridge consists of four resistors and two capacitors.

A Kelvin bridge, also called a Kelvin double bridge and in some countries a Thomson bridge, is a measuring instrument used to measure unknown electrical resistors below 1 ohm. It is specifically designed to measure resistors that are constructed as four terminal resistors.

A potentiometer is an instrument for measuring voltage or 'potential difference' by comparison of an unknown voltage with a known reference voltage. If a sensitive indicating instrument is used, very little current is drawn from the source of the unknown voltage. Since the reference voltage can be produced from an accurately calibrated voltage divider, a potentiometer can provide high precision in measurement. The method was described by Johann Christian Poggendorff around 1841 and became a standard laboratory measuring technique.

In electronics, the Carey Foster bridge is a bridge circuit used to measure medium resistances, or to measure small differences between two large resistances. It was invented by Carey Foster as a variant on the Wheatstone bridge. He first described it in his 1872 paper "On a Modified Form of Wheatstone's Bridge, and Methods of Measuring Small Resistances".

<span class="mw-page-title-main">Anderson's bridge</span> Circuit in electronics

In electronics, Anderson's bridge is a bridge circuit used to measure the self-inductance of the coil. It enables measurement of inductance by utilizing other circuit components like resistors and capacitors.

References

↑ "Circuits in Practice: The Wheatstone Bridge, What It Does, and Why It Matters", as discussed in this MIT ES.333 class video
↑ 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.
↑ 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. discusses Christie's and Wheatstone's contributions, and why the bridge carries Wheatstone's name.
↑ Maxwell, J. Clerk (1865). "A dynamical theory of the electromagnetic field". Philosophical Transactions of the Royal Society of London. 155: 459–512. Maxwell's bridge used a battery and a ballistic galvanometer. See pp. 475–477.

External links

Media related to Wheatstone's bridge at Wikimedia Commons
DC Metering Circuits chapter from Lessons In Electric Circuits Vol 1 DC free ebook and Lessons In Electric Circuits series.
Test Set I-49

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

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

[2] 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.

[3] 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. discusses Christie's and Wheatstone's contributions, and why the bridge carries Wheatstone's name.

[4] Maxwell, J. Clerk (1865). "A dynamical theory of the electromagnetic field". Philosophical Transactions of the Royal Society of London. 155: 459–512. Maxwell's bridge used a battery and a ballistic galvanometer. See pp. 475–477.

[1]

[2]

[3]

[4]

v t e Bridge circuits
Resistance	Kelvin bridge Wheatstone bridge
Resistance–Capacitance	Schering bridge Wien bridge
General	Bridged T circuit Carey Foster bridge Fontana bridge Lattice filter Murray loop bridge Transformer ratio arm bridge
Inductance	Maxwell bridge Anderson's bridge Hay's bridge
Other	Diode bridge H-bridge