# Wheatstone measuring bridge

The Wheatstone measuring bridge (short: Wheatstone bridge ) is a measuring device for measuring

It is made up of four resistors that are connected together to form a closed ring or square, with a voltage source in one diagonal and a voltage measuring device in the other.

It was invented by Samuel Hunter Christie in 1833 , but named after the British physicist Sir Charles Wheatstone [ˈwiːtstən], who recognized its importance and promoted its dissemination.

Basic structure of the Wheatstone Bridge

## description

Basic structure of the Wheatstone Bridge, redrawn

A graphically different arrangement shows more clearly that two resistors each form a voltage divider ; two voltage dividers are parallel to each other. The voltmeter establishes a cross-reference between these, which gives the circuit the name bridge circuit . The directly measured size of the arrangement is the voltage difference between the voltage dividers, also called diagonal voltage or bridge cross voltage.

The original Wheatstone bridge was used to measure resistance values ​​using the trimming method. First of all, the three known resistances must be changed until the diagonal voltage is zero. The fourth, unknown value can then be calculated from their resistance values. Due to the availability of inexpensive electronic measuring devices (which work with other methods; see resistance measuring device ), this measuring method is rarely used. Precision measurements are an exception here.

A method that is also often referred to as a Wheatstone bridge (alternatively a deflection resistance measuring bridge) is the modification of the deflection method, in which even small deviations in the resistance can be determined, which would hardly be resolved when measuring the entire resistance .

Illustrative example: A bridge with a temperature measuring resistor in one of the voltage dividers is in the balanced state at a reference temperature. If the temperature at the measuring resistor changes, the diagonal voltage changes approximately proportionally to the temperature change. The knockout procedure has a firm place in modern measurement technology.

## basis

At the two parallel voltage dividers, the voltage across any resistor (e.g. ) is compared with the corresponding voltage in the parallel branch (then across ). If these voltages are equal (but not equal to zero), the bridge is said to be balanced . As long as a negligibly small current flows in the bridge branch (this always applies to calibration, otherwise if ), the voltage dividers are unloaded, and the following applies: ${\ displaystyle R_ {1}}$${\ displaystyle R_ {3}}$${\ displaystyle R_ {5} \ gg R_ {1} \;, \ R_ {2} \;, \ R_ {3} \;, \ R_ {4}}$

{\ displaystyle {\ begin {aligned} U_ {1} & = U_ {0} {\ frac {R_ {1}} {R_ {1} + R_ {2}}} \\ U_ {3} & = U_ { 0} {\ frac {R_ {3}} {R_ {3} + R_ {4}}} \\ U_ {5} & = U_ {1} -U_ {3} = U_ {0} \ left ({\ frac {R_ {1}} {R_ {1} + R_ {2}}} - {\ frac {R_ {3}} {R_ {3} + R_ {4}}} \ right) \\ & = U_ { 0} {\ frac {R_ {1} R_ {4} -R_ {3} R_ {2}} {(R_ {1} + R_ {2}) (R_ {3} + R_ {4})}} \ end {aligned}}}
Equivalent circuit diagram for the output voltage

When measuring this voltage, it is important to note that it is associated with a significant source resistance due to the voltage dividers. With the ideal source of the supply voltage (with between the upper connection of and the lower connection of ) the following can be seen directly from the circuit: ${\ displaystyle R_ {q}}$${\ displaystyle U_ {0}}$${\ displaystyle R_ {0} = 0}$${\ displaystyle R_ {4}}$${\ displaystyle R_ {3}}$

${\ displaystyle R_ {q} = (R_ {1} \ parallel R_ {2}) + (R_ {3} \ parallel R_ {4}) = {\ frac {R_ {1} \ cdot R_ {2}} { R_ {1} + R_ {2}}} + {\ frac {R_ {3} \ cdot R_ {4}} {R_ {3} + R_ {4}}}}$

This means that for a symmetrical bridge with . ${\ displaystyle R_ {1} = R_ {2} = R_ {3} = R_ {4} = R}$${\ displaystyle R_ {q} = R}$

Together with a non-ideal voltage measuring device with an internal resistance , this can lead to a considerable measurement error, since the measured voltage is smaller than the open-circuit voltage by a factor ; see real voltage source . ${\ displaystyle R_ {5} <\ infty}$${\ displaystyle U_ {5 {,} \ mathrm {M}}}$${\ displaystyle U_ {5}}$${\ displaystyle R_ {5} / (R_ {5} + R_ {q}) \!}$

## Matching procedure

Bridge for resistance measurement

The balanced state is defined by ; then ${\ displaystyle U_ {5} = 0}$

${\ displaystyle R_ {1} \, R_ {4} = R_ {2} \, R_ {3}}$

or

${\ displaystyle {\ frac {R_ {1}} {R_ {2}}} = {\ frac {R_ {3}} {R_ {4}}}}$

This equation says: if three resistances are known, a fourth can be calculated. This provides a measuring method for measuring resistance, which is also called the zero balancing method of the Wheatstone bridge.

If the resistance to be measured is at the position of , then applies ${\ displaystyle R_ {m}}$${\ displaystyle R_ {1}}$

${\ displaystyle R_ {m} = {\ frac {R_ {2}} {R_ {4}}} \ \ cdot R_ {3}}$

and with the circuit shown, a four-digit value is set and the measuring range is sensibly a power of ten factor, e.g. B. 1: 1 or 1:10 or 100: 1. The area of ​​application covers roughly the range . ${\ displaystyle R_ {3}}$${\ displaystyle R_ {2}: R_ {4}}$${\ displaystyle R_ {m} = 1 \; \ Omega \; \ dots \; 1 \; \ mathrm {M} \ Omega}$

The last equation is independent of the supply voltage . However, please note: ${\ displaystyle U_ {0}}$

• ${\ displaystyle U_ {0}}$should be so large that when the bridge is almost balanced, an adjustment of one step to the lowest value still causes a noticeable change in the bridge transverse tension.${\ displaystyle R_ {3}}$${\ displaystyle U_ {5}}$
• ${\ displaystyle U_ {0}}$ should be so small that the inevitable heating of the resistors does not change them noticeably.

The bridge can also be operated with audio frequency instead of DC voltage and headphones can be used as an indicator, which is also a very sensitive indicator. However, the direction in which the adjustment has to be carried out is then no longer recognizable, since the phase position cannot be recognized with the ear.

### Measurement with a sliding wire potentiometer

Wiring diagram Wheatstone bridge circuit (practical)

The variant introduced by Gustav Kirchhoff (1824–1887) only requires a precision resistor and a sliding wire potentiometer. The resistance wire is stretched on a board or wound on a pipe. The ends of the wire are connected to the supply voltage and the sliding contact picks up the partial voltage of the potentiometer. The length ratio corresponds to the resistance ratio in the basic structure. In the balanced state, the unknown resistance is calculated as follows: ${\ displaystyle a / b}$${\ displaystyle R_ {1} / R_ {2}}$${\ displaystyle R_ {x}}$

${\ displaystyle R_ {x} = {\ frac {a} {b}} \ cdot R_ {v}}$

The accuracy essentially depends on the mechanical ratio and the comparison resistance. In the historical application a galvanometer was used to display the detuning. In order to carry out the zero adjustment more precisely, there is a button in series with the indicator, since a movement of the pointer is easier to recognize than a position. ${\ displaystyle a / b}$${\ displaystyle R_ {v}}$

Wheatstone bridge with sliding wire potentiometer

The equivalent resistance should be of the order of magnitude because the accuracy decreases towards the ends of the sliding wire. ${\ displaystyle R_ {v}}$${\ displaystyle R_ {x}}$

### Further development

The Wheatstone measuring bridge is only used today for precision measurements, see also calibration . Due to the high accuracy of the digital multimeters and the availability of precision operational amplifiers , measurement methods with direct display can be used almost everywhere.

Wheatstone measuring bridges as laboratory measuring devices like the one shown are therefore no longer in trade and professional use, whereas the modification to the deflection resistance measuring bridge is.

The Wheatstone bridge is not suitable for measuring small resistances (guide value <1 Ω), as the lines and terminals that connect the resistance to be measured to the measuring device falsify the measurement. The Thomson Bridge was created from the Wheatstone Bridge . This is also no longer in trade and professional use. For an alternative, see resistance meter . ${\ displaystyle R_ {x}}$

Instead of ohmic resistors with a DC voltage for the supply, impedances can also generally be measured with an AC voltage supply , see AC voltage bridge .

## Knockout procedure

In the measurement technology of non-electrical quantities, the Wheatstone bridge is of considerable importance for absorbing small changes in resistance from the balanced state. Then it works as a transmitter , e.g. B. in context

### invoice

In these cases a voltage arises as a measure of a change in resistance ; the bridge works according to the deflection method . Specifically: If the balanced state changes, → then arises according to the equation set out at the beginning ${\ displaystyle U_ {5}}$${\ displaystyle \ Delta R}$${\ displaystyle R_ {1}}$${\ displaystyle R_ {1}}$${\ displaystyle R_ {1} + \ Delta R_ {1}}$

${\ displaystyle {\ frac {U_ {5}} {U_ {0}}} = {\ frac {R_ {1} + \ Delta R_ {1}} {R_ {1} + \ Delta R_ {1} + R_ {2}}} - {\ frac {R_ {3}} {R_ {3} + R_ {4}}}}$

With the detuning and the bridge ratio becomes ${\ displaystyle v = {\ frac {\ Delta R_ {1}} {R_ {1}}}}$${\ displaystyle k = {\ frac {R_ {2}} {R_ {1}}} = {\ frac {R_ {4}} {R_ {3}}}}$

${\ displaystyle {\ frac {U_ {5}} {U_ {0}}} = {\ frac {1 + v} {1 + v + k}} - {\ frac {1} {1 + k}} = v {\ frac {k} {(1 + v + k) (1 + k)}}}$

As long as            or            the approximation applies ${\ displaystyle | v | \ ll 1 + k}$${\ displaystyle | \ Delta R_ {1} | \ ll R_ {1} + R_ {2}}$

${\ displaystyle {\ frac {U_ {5}} {U_ {0}}} \ approx v {\ frac {k} {(1 + k) ^ {2}}} \ quad;}$       then is proportional to !${\ displaystyle U_ {5}}$${\ displaystyle \ Delta R_ {1}}$

The function has a maximum at and has the value there . This means that the bridge has a maximum of sensitivity when it is symmetrical (all resistors are equal when they are adjusted = ). Then ${\ displaystyle y = f (k) = {\ frac {k} {(1 + k) ^ {2}}}}$${\ displaystyle k = 1}$${\ displaystyle f (1) = {\ tfrac {1} {4}}}$${\ displaystyle R}$

${\ displaystyle {\ frac {U_ {5}} {U_ {0}}} = {\ frac {1} {4}} \ {\ frac {\ Delta R_ {1}} {R}}}$

Example: Relative change in resistance . Then . That is 25 digits (number steps) if the voltmeter resolves the measuring range of 200 mV into 2000 digits. ${\ displaystyle \ Delta R_ {1} / R = 10 ^ {- 3}; \ U_ {0} = 10 \; \ mathrm {V}}$${\ displaystyle U_ {5} = 2 {,} 5 \; \ mathrm {mV}}$

That means: Without knowing the resistance exactly, small changes can be determined with the quality that can be determined. While the subtraction of two measured values ​​of almost the same size always leads to very unreliable results, here the difference is formed in the circuit and as such can be measured directly and reliably! ${\ displaystyle U_ {5}}$

If all four resistors are allowed a small change out of the adjustment, then one obtains in the above arrangement with a symmetrical bridge

${\ displaystyle {\ frac {U_ {5}} {U_ {0}}} = {\ frac {1} {4}} \ left ({\ frac {\ Delta R_ {1}} {R}} - { \ frac {\ Delta R_ {2}} {R}} - {\ frac {\ Delta R_ {3}} {R}} + {\ frac {\ Delta R_ {4}} {R}} \ right)}$

Note rule for the signs: Based on the influence of the change in any resistance on , the change in an adjacent resistor in the bridge is entered with the opposite sign and the change in the diagonally opposite resistor with the same sign. ${\ displaystyle U_ {5}}$

Example: If two neighboring resistances change by +2 ‰ each, then their influence is canceled out. ${\ displaystyle U_ {5}}$

### Applications in electronics

Silicon pressure sensor with diffused resistors

This equation is built on to a considerable extent in microelectronics and in sensor technology . Resistors sensitive to expansion can react to deformation with a positive or negative change in resistance, depending on the application of the resistors, and complement each other in the equation, while temperature influences, which have the same effect on all, cancel each other out. Resistances that are placed on an elastic base record forces, pressures, torques, etc. Small relative changes in length below 10 −4 can still be recorded. The picture shows a pressure measuring device using this technology: a membrane made of silicon, which has high-quality elastic properties, is deformed by pressure; Resistances have diffused in at places with particularly strong bends; with three bond wires each half of a Wheatstone bridge is created.

When measuring the temperature using a resistance thermometer , only one of the bridge resistors is made changeable, in this case changeable by the temperature. The measuring effect is quite comfortable: the resistance of a standardized platinum resistance thermometer doubles in the range of 0 ... 266 ° C. It is therefore possible to work with an asymmetrical bridge,, which reduces the sensitivity but increases the range in which the linear approximation applies. In addition, when connected in a three-wire circuit, the bridge circuit eliminates the effects of temperature on the resistance of the supply lines. ${\ displaystyle k \ gg 1}$

## literature

• Elmar Schrüfer: Electrical measurement technology. Measurement of electrical and non-electrical quantities. 5th revised edition. Hanser, Munich a. a. 1992, ISBN 3-446-17128-2 , pp. 226-228.
• Siemens Aktiengesellschaft (Hrsg.): Electrical measuring technology. 5th edition. Siemens, Berlin a. a. 1968, pp. 114-123.
• Wilhelm H. Westphal : Physics. 22.-24. Edition. Springer, Berlin a. a. 1963, p. 301.