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 specially designed for measuring resistors which are manufactured as four terminal resistors.

**Background**

Resistances above about 1 ohm can be measured using a variety of techniques, such as using an ohmmeter or a Wheatstone bridge . In such resistors, the resistance of the connecting wires or terminals is negligible compared to the resistance value. For resistors of less than one ohm, the resistance of the connecting wires or terminals becomes important, and traditional measurement techniques will include them in the result.

To overcome the problems of these undesirable resistors (known as ‘ parasitic resistance ‘), very low value resistors and especially precision resistors and high current ammeter shunts are made as four terminal resistors. These resistors have a pair of current terminals and a pair of potential or voltage terminals. In use, a current is passed between the current terminals, but the volt drop across the resistor is measured at the potential terminals. The measured voltage drop will be entirely due to the resistor itself as the parasitic resistance of the current carrying leads from the resistor is not included in the potential circuit. A bridge circuit designed to work with four terminal resistors is used to measure such resistances.is required. That bridge is the Kelvin Bridge.

**Operating principle**

The operation of the Kelvin bridge is very similar to that of the Wheatstone bridge, but uses two additional resistors. Resistors *R1 *_{and }*R2* are connected to the four known terminals or external potential terminals of the standard resistor *R *_{S} and unknown resistor *R *_{X} ( identified as *P1* and _{P′1 in the diagram }*) . *The resistors *R *_{s} , *R *_{x} , *R *_{1} and *R *_{2} are essentially Wheatstone bridges. In this arrangement, the upper part of *R *_{s and}The parasitic resistance at the bottom of *R *_{x is outside the potential measurement portion of the bridge and is therefore not included in the measurement. }However, the link between *R *_{s} and *R *_{x} ( *R *_{equal} to ) *is* included in the measurement part of the circuit’s capacitance and can therefore affect the accuracy of the result. To overcome this, a second pair of resistors *R *_{1} and *R *_{2} form the second pair of arms of the bridge (hence the ‘double bridge’) and the interior of *R *_{s} and *R *_{x} ( identified as *P* ). potential terminals._{2} and *p *_{2} in the diagram). The detector D is connected between the junction of *R *_{1} and *R *_{2} and the junction of R *1 *_{and} R *2*

The equilibrium equation of this bridge is given by the equation

{\displaystyle {\frac {R_{x}}{R_{s}}}={\frac {R_{2}}{R_{1}}}+{\frac {R_{\text{par}}}{R_{s}}}\cdot {\frac {R'_{1}}{R'_{1}+R'_{2}+R_{\text{par}}}}\cdot \left({\frac {R_{2}}{R_{1}}}-{\frac {R'_{2}}{R'_{1}}}\right)}

In a practical bridge circuit, the ratio of *R* ‘ _{1} to *R* ‘ _{2} is arranged the same as the ratio of R1 to R2 (and in most *designs *_{, }*R1* = *R* ‘ _{1} and *R2 *_{=} R ‘ _{2} ) As a result, the last term of the above equation becomes zero and the rest of the equation becomes

{\displaystyle {\frac {R_{x}}{R_{s}}}={\frac {R_{2}}{R_{1}}}}

*rearrange R *_{x} to be the subject

{\displaystyle R_{x}=R_{2}\cdot {\frac {R_{s}}{R_{1}}}}

The parasitic resistance *R *_{par} is removed from the equilibrium equation and its presence does not affect the measurement result. This equation is similar to the functionally equivalent Wheatstone bridge.

In practical use the magnitude of supply B can be arranged to provide current through Rx and Rx at or close to the rated operating currents of the small rated resistor. This contributes to small errors in measurement. This current does not flow through the measuring bridge itself. This bridge can also be used to measure the resistances of the more traditional two terminal design. The bridge potential connections are only connected as close to the resistor terminals as possible. Any measurement will then exclude all circuit resistances that are not within the two possible connections.

**Accuracy**

The accuracy of the measurements made using this bridge depends on several factors. Of prime importance is the accuracy of the standard resistor ( *Rs *_{) . }Also of importance is the ratio of how close *R1* to *R2 is *_{,} the ratio of *R* ‘ _{1} to *R* ‘ _{2 }_{. }As shown above, if the ratio is exactly the same, then the error due to parasitic resistance ( *R *_{par} ) is completely eliminated. In a practical bridge, the goal is to make this ratio as close as possible, but keep it *exactly*_{}_{}_{}It is not possible to make equal. If the difference in the ratio is small enough, then the last term of the above equilibrium equation becomes so small that it becomes negligible. The measurement accuracy is also increased by increasing the current flowing through *R *_{s} and *R *_{x as large as the ratings of those resistors allow. }This gives the greatest potential difference between *R2* and *R* ‘ _{2} (the innermost potential connection gives those resistors and consequently enough voltage for changes in *R* ‘ _{1} and *R* ‘ _{2} to have its greatest effect _{.}_{}_{}_{}

There are some commercial bridges reaching accuracy better than 2% for resistance ranges from 1 microohm to 25 ohms. One such variant is shown above.

Laboratory bridges are typically constructed with high-accuracy variable resistors across the two potential arms of the bridge and achieve an accuracy suitable for calibrating standard resistors. In such an application, the ‘standard’ resistor ( R_{s} ) would actually be a sub-standard type (that is, a resistor that has accuracy 10 times better than the required accuracy of a standard resistor calibrated). For such use, the error introduced by mismatching the ratio in the two possible arms would mean that the presence of parasitic resistance R par could have a significant impact on the high accuracy required. To reduce this problem, the current connection to the standard resistor ( R_{x} ) ; Sub-standard resistor ( R_{s}) and the connections between them ( R across ) are designed to have as low resistance as possible, and the connections in both the resistors and the bridge are similar to those of a bus bar rather than a wire. Some ohmmeters include Kelvin bridges to obtain a larger measurement range. Instruments for measuring sub-ohm values are often referred to as low-resistance ohmmeters, milli-ohmmeters, micro-ohmmeters, etc.