# The Wheatstone bridge circuit

The Wheatstone bridge can be used in various ways to measure electrical resistances:

• for the determination of the absolute value of a resistance by comparison with a known resistance
• for the determination of relative changes in resistance

The latter method is used in strain gauge techniques. It enables the relative changes of resistance in the strain gauge, which are usually around the order of 10-4 to 10-2 Ω/Ω to be measured with great accuracy.

The picture below shows two different illustrations of the Wheatstone bridge which are however electrically identical: a) shows the usual rhombus type of representation which Wheatstone used;
b) is a representation of the same circuit which is more clear for the electrically untrained person.

The four arms or branches of the bridge circuit are formed by the resistances R1 to R4. The corner points 2 and 3 of the bridge designate the connections for the bridge excitation voltage Vs; the bridge output voltage VO, the measurement signal, is available on the corner points 1 and 4.

The bridge excitation is usually an applied, stabilized direct or alternating voltage Vs.

Note:
There is no generally accepted rule for the designation of the bridge components and connections. In the literature there are all kinds of designations and this is reflected in the bridge equations. Therefore, it is essential that the designations and indices used in the equations are considered together with their positions in the bridge networks in order to avoid misinterpretation.

If a supply voltage Vs is applied to the two bridge supply points 2 and 3 then this is divided up in the two halves of the bridge R1, R2 and R4, R3 as a ratio of the corresponding bridge resistances, i.e. each half of the bridge forms a voltage divider.

If the bridge is not balanced, this is caused by the difference in the voltages from the electrical resistances on R1, R2 and R3, R4. This calculated as:

If the bridge is balanced

the bridge output voltage VO is zero.

With a preset strain, the resistance of the strain gage changes by the amount ΔR. This gives:

For strain measurements, the resistances R1 and R2 must be equal in the Wheatstone bridge.
The same applies to R3 and R4.

With a few assumptions and simplifications the following equation can be found (further explanations are given in the HBM book “An Introduction to Measurements using Strain Gauges”):

In the last calculation step, the term ΔR/R must be replaced by the following:

Here k is the k-factor of the strain gauge, ε is the strain. This gives:

The equations assume that all the resistances in the bridge change. This situation occurs for example in transducers or with test objects performing a similar function. In experimental tests this is hardly ever the case and usually only some of the arms of the bridge contain active strain gauges, the remainder being made up of bridge completion resistors. Designations for the various forms such as quarter bridge, half bridge, double quarter or diagonal bridge and full bridge are commonplace.

Depending on the measurement task one or more strain gauges are used at the measuring point. Designations such as full bridge, half bridge or quarter bridge indicate such arrangements, although actually these are not correct. In fact, the circuit used for the measurement is always complete and is either fully or partially formed by the strain gauges and the specimen. It is then completed by fixed resistors, which are incorporated in the instruments.

Transducers generally have to comply with more stringent accuracy requirements than measurements for experimental tests. Therefore, transducers should always have a full bridge circuit with active strain gauges in all four arms.

Full bridge or half bridge circuits should also be used in stress analysis if different kinds of interferences need to be eliminated. An important condition is that cases of different stresses are clearly distinguished, such as compressive or tensile stress, as well as bending, shear or torsional forces.

The table below shows the dependence of the geometrical position of the strain gauges, the type of bridge circuit used and the resulting bridge factor B for normal forces, bending moments, torque and temperatures. The small tables given for each example specify the bridge factor B for each type of influencing quantity. The equations are used to calculate the effective strain from the bridge output signal VO/VS.

Note:
A cylindrical shaft is assumed for torque measurement in example 13, 14 and 15. For reasons of symmetry, bending in X and Y direction is allowed. The same conditions also apply for the bar with square or rectangular cross section.

Explanations of the symbols: