# The Wheatstone Bridge Circuit

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

• 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 with regard to strain gauge techniques. It enables 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 image below shows two different illustrations of the Wheatstone bridge which are electrically identical: figure a) shows the usual rhombus representation in which the Wheatstone is used; and figure b) is a representation of the same circuit, which will be clearer for an 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 V0 , that is 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 existing 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 along with their positions in the bridge networks in order to avoid misinterpretation.

If a supply voltage Vs is applied to the bridge supply points 2 and 3, then the supply voltage 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.

The bridge can be imbalanced, owing to the difference in the voltages from the electrical resistances on R1, R2 and R3, R4. This can be calculated as follows:

if the bridge is balanced and

where the bridge output voltage V0 is zero.

With a preset strain, the resistance of the strain gauge changes by the amount ΔR. This gives us the following equation:

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 determined (further explanations are given in the HBM book “An Introduction to Measurements using Strain Gauges”):

In the last step of calculation, 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 us the following:

The equations assume that all the resistances in the bridge change. For instance, this situation occurs in transducers or with test objects performing similar functions. In experimental tests, this is hardly ever the case and usually only some of the bridge arms contain active strain gauges, the remainder consisting 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. Although designations such as full bridge, half bridge ,or quarter bridge are used to indicate such arrangements, actually they 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 within the instruments.

Transducers generally have to comply with more stringent accuracy requirements than measurements pertaining to 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 for 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 related to symmetry, bending in X and Y direction is allowed. The same conditions also apply for the bar with square or rectangular cross sections.

Explanations of the symbols:

 T Temperature Fn Longitudinal, normal force Mb Bending moment Mbx, Mby Bending moment for X and Y directions Md Torque εs Apparent strain εn Longitudinal, normal strain εb Bending strain εd Torque strain ε Effective strain at the point of measurement ν Poisson’s ratio Active strain gauge Strain gauge for temperature compensation Resistor or passive strain gauge