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# Understanding a Wheatstone Bridge Strain Gauge Circuit

29-04-2020

A strain gauge basically works on the principle of a simple metal conductor wire that tends to have an impact on its length, cross-sectional area and resistance due to applied stress. This is due to the fact that the length of the wire is directly proportional to the resistance of the wire and inversely proportional to its cross-sectional area, as per the following equation.

$$R= ρ \frac{L}{A}$$
Here, R=resistance of the wire,
ρ=resistivity
L=length of the wire
and A= area of the cross-section of the wire

The value of strain caused by the stress is determined by measuring a change in resistance of the gauge, as a result of the change in the dimensions of the object due to the applied force. This change in resistance is smaller when compared to the resistance of the strain gauge and hence needs to be measured accurately to determine the strain. To measure this small change in resistance, a Wheatstone Bridge Circuit is used. It enables relative changes of strain gauge in the order of 10-4 to 10-2 Ω/Ω to be measured with great accuracy.

A typical Wheatstone bridge circuit consists of a simple network of four resistors of equal resistances connected end to end to form a square as shown in the below figure. Across one pair of diagonal corners of the circuit, an excitation voltage is applied and across the other pair, the output of the bridge is measured. The output of the bridge, i.e the value of V0 depends on the ratio of resistances of the resistors, i.e. R1:R4 and R2:R3. When the bridge is balanced and no strain is induced upon the strain gauge, the relationship between the four resistances can be expressed as:

$$\frac{R1}{R4}=\frac{R2}{R3}$$

Now, as V0 = V+ – V , this implies V0 = 0

### Forms of Wheatstone Bridge Circuits

In some applications, depending on the measurement task, usually, only some of the bridge arms contain active strain gauges, the remainder consisting of bridge completion resistors. This includes bridge arrangements such as quarter bridge, half bridge, or diagonal bridge. For applications with very stringent accuracy requirements, a full bridge arrangement is preferred.

Quarter Wheatstone Bridge

In case of an increase in the resistance of one of the resistors in the bridge due to the applied force, the bridge no longer stays balanced. This configuration is known as a quarter bridge strain gauge. Here,  $$\frac{R1}{(R4+ΔR)}≠ \frac{R2}{R3} and V0≠0$$

Now, using Voltage divider formula, we get
$$VC= \frac{R1}{(R1+R2)}VEX$$
$$VD=\frac{(R4+ΔR)}{(R4+ΔR)+R3})VEX$$
Also, voltage drop V0 can be expressed as:
$$V0= \frac{(R+ΔR)}{(R+(R+ΔR))}VEX - \frac{R}{(R+R)}VEX [Since R1=R2=R3=R4=R]$$
On further simplification, we get
$$V0=\frac{(Vex ΔR)}{(4R+2ΔR)}$$
Now, since R >> ΔR and 4R >> 2ΔR, we get the following final equation:
$$V0=\frac{Vex}{4} \times \frac{ΔR}{R}$$

In the above equation, ΔR/R is the electrical strain and the gauge factor is termed as the ratio of the electrical strain and the mechanical strain. Therefore, the above equation can be re-written as:
$$V0=\frac{Vex}{4} \times k \times ϵ$$
Where k = gauge factor
and ϵ= Mechanical strain
In a quarter bridge circuit, as the distance between the strain gauge and the other three resistances is unknown, there may be a substantial amount of wire resistance that can impact the measurement. In this case, the strain gauge resistance will not be the only resistance being measured, but the wire resistance will also contribute to the output voltage measurement. This undesired effect can be minimised with the addition of a third wire connecting directly to the upper wire of the strain gauge.

Half Wheatstone Bridge
When we mount two active strain gauges on a bending beam, placed one at the front and one at the back, we get a half-bridge arrangement. This is because half of the four resistors in the circuit are now strain gauges. This arrangement will have both the strain gauges respond to the induced strain, thereby making the bridge more responsive to the applied force. As compared to the quarter bridge configuration, the half-bridge circuit yields twice the output voltage for a given strain, thereby improving the sensitivity of the circuit by a factor of two. Let’s take an example of a cantilever beam that is clamped to a lab bench, and a weight is applied at the end of the beam. The setup has a strain gauge attached on the top surface of the beam, and another is attached at the bottom surface (see case 1). As the beam gets strained due to the applied force, the top strain gauge is stretched, while the bottom strain gauge is compressed. Hence, the output voltage, in this case, can be expressed as:

$$V0=\frac{Vex}{4} \times \frac{(ΔR3 - ΔR4)}{R}$$
$$V0=\frac{Vex}{4} \times k \times (ϵ3-ϵ4) (with ϵ3=-ϵ4=ϵ)$$
$$V0=\frac{Vex}{2} \times k \times ϵ$$

If the strain gauges are connected diagonally opposite to each other as shown in the above figure (case 2), the output voltage will be expressed as:
$$V0=\frac{Vex}{4} \times \frac{(ΔR2 - ΔR4)}{R}$$
$$V0=\frac{Vex}{4} \times k \times (ϵ2+ϵ4) (with ϵ2=ϵ4=ϵ)$$
$$V0=\frac{Vex}{2} \times k \times ϵ$$

Full Wheatstone Bridge
On substituting all the resistors of the Wheatstone bridge circuit with four active strain gauges, we get a full-bridge arrangement. This configuration enables large outputs of strain-gage transducers, improves temperature compensation and offers even greater sensitivity. A full-bridge strain gauge Wheatstone bridge gives linear output than other configurations as the output voltage is directly proportional to an applied force, with no other approximation involved, unlike the quarter and half-bridge configurations. In this circuit, two of the diagonally attached strain gauges, which are mounted on the bottom of the cantilever beam are compressed, while the other two are stretched during the process. This induces both positive and negative changes in the resistances in the circuit. On applying force, the resistances of all the strain gauges change by the amount ΔR, thus giving the following equation:
$$V0=\frac{VEX}{4} \times (\frac{ΔR1}{R1} - \frac{ΔR2}{R2} + \frac{ΔR3}{R3} - \frac{ΔR4}{R4})$$
$$V0= \frac{VEX}{4}\times \frac{k}{4}(ϵ1-ϵ2+ϵ3-ϵ4)$$

Both half-bridge and full-bridge configuration are known to provide greater sensitivity and better accuracy. However, it is not always possible to bond complementary pairs of strain gauges to the test specimen. For these specific cases, the quarter bridge configuration is preferred. Therefore, depending on the size, mounting restrictions and the nature of the application, a suitable arrangement is used.

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