## Basics of bridge circuit

The bridge circuit allows the determination of very small changes in resistance.
At the same temperature-induced changes in resistance and other interference are compensated for.
However, a prerequisite for a good compensation of interferences that the disorder equally affecting two adjacent resistors of the bridge circuit. The two connections Us+ and Us- are used to power the bridge circuit. The voltage between Us+ and Us- is also referred to as excitation voltage Us or bridge supply or sensor input. Us- is connected to the mass GND in the many measuring amplifiers.

The voltage change is measured via the connections Ud+ and Ud- if the bridge circuit is disexcited due to strain at one or more strain gauges.
The voltage between Ud+ and Ud- is referred to as "differential voltage Ud" or "bridge output" or "sensor output".

The bridge circuit is adjusted (the voltage between Ud+ and Ud- is 0 V) if the following condition is met:

R1/R2 = R4/R3

In most cases, the bridge circuit is supplied with DC voltage of 5V. Also common are 2.5V bridge power or 1.0V bridge power, e.g. when a low-impedance strain gauge with 120 ohms is used, or if the strain gauge is glued to a non-conductive material, so that a strong self-heating of the strain measuring strip must be prevented.

#### Bridge equation

The relationship between the bridge output Ud and the resistance change ΔR/R is for the full-bridge:

Ud/Us = R1/(R1 + R2)  -  R4/(R3 + R4) (Equation1a)

where:
ΔR/R: relative change in resistance
Ud / Us: relative bridge output
Derivation of the equation for the bridge circuit: Wheatstone bridge.pdf

#### Linearized form of bridge equation

For small changes in the difference voltage Ud, the linearized form of the bridge equation may be applied:

ΔUd/Us = 1/4 (ΔR1/R1 - ΔR2/R2 + ΔR3/R3 - ΔR4/R4) (Equation 1b)

#### Relationship between resistance change and strain

The relationship between resistance change ΔR/R and strain ε is defined by the k-factor of the strain gauge.

ΔR/R = k · ε (Equation 2)

If the elongation of an electrical conductor, a change in cross section of the electrical conductor to the sequence. The change in cross section of the electrical conductor is in turn connected to a change in electrical resistance.

From the condition that an "elongation" (positive elongation) of a conductor in volume still remains constant, resulting in a "necking" of the conductor. Conversely, results from an "upsetting" (negative strain) a "thickening" of the conductor.

This, purely geometrical effect results under the condition constant conductor volume to a linear relationship between resistance change and expansion:

The Propotionalitätsfaktor is referred to as k-factor of the strain gauge. For alloys whose volume remains constant while stretching (and this applies to all electrical conductor) results in a k-factor of 2.

1 ‰ elongation corresponds to 2 ‰ change in resistance, or
1000 .mu.m / m elongation corresponding 2 ‰ change in resistance, or
1000 e 6 strain corresponding 2000 E-6 resistance change

#### Relationship between strain and bridge output

The relationship between bridge output ΔUd/Us , supply voltage and strain follows from equations (1) and (2):

ΔUd/Us = 1/4 k (ε1 - ε2 + ε3 - ε4) (Equation 3)

For one quarter bridge is due ε2 = ε3 = ε4 = 0:

ΔUd/Us = 1/4 k ε1 (Equation 4a)

From equation (4) gives the relationship between display Ud / Us and strain ε for a quarter bridge:

ε1 = ΔUd/Us 4/k (Equation 5a)

In a k-factor of 2.0 an indication Ud/Us of 2.0 mV/V (= 0.002 V/V = 2.0E-3) means:

0.002 x 4/2.0 = ε1 = 0.004 = 4 ‰ = 4000 µm/m

2 mV/V therefore correspond to 4000 µm/m strain at a quarter bridge with k-factor. 2

The output signals of the bridge circuit with quarter bridge, half bridge and full bridge are summarized on this page:

straingauge-bridge-circuit

For a full bridge with 4 active strain gauges (with ε1=ε3=-ε2=-ε4 = ε)is applied:

ΔUd/Us = k ε (Equation 4b)

From equation (4b) one gets the relationship between the indication "ΔUd/Us" and the strain for a full bridge:

ε = ΔUd/Us / k (Equation 5b)

#### Task of a measuring amplifier

The relative bridge output Ud/Us is usually less than 0.1%. At 5V bridge excitation voltages Ud therefore must be measured from -0.5 to +0.5 mV. Therefore, a sense amplifier is used.
The amplifier has several functions:

• it provides a bridge supply voltage Us with highest stability available,
• it amplifies the difference voltage Ud and transforms them into an appropriate display value.

A measuring amplifier shows the basic setting usually the relative bridge output Ud/Us at.
The unit of the display is then mV/V.

By standardizing the bridge output Ud on the supply voltage Us, the display values ​​on measuring amplifiers with different supply voltage Us directly comparable. The default setting of the unit on the display of many measuring amplifier is therefore mV/V.

#### Advantages of the bridge circuit

With adjusted bridge circuit (R1/R2 = R4/R3) is the differential voltage between + Ud and -Ud equal to 0 volts. Since only resistance changes are recognized starting from 0 volts, the measuring range can be adapted to the requirements. "The gain can be as high as desired selected"

The different signs in equation (3) allow the compensation of disturbance:

1. the thermal expansion can be compensated: ε1 - ε2 + ε3 - ε4 = 0
2. mechanical strains that are not desired in the measuring direction are, can be compensated. This can be used, by detecting specific strains of opposite signs with strain gauges.
3. For each load case (bending, torsion, compression, shear) there are wiring diagrams that capture only the strain in a certain measuring direction.

#### Benefits of the standardized display in mV/V

• when displaying the absolute bridge output always takes an additional indication of the supply voltage used to be made,
• for calculating the expansion only the knowledge of the relative bridge output is required (see equation 5)
• through the calibration of measuring instruments for strain gages in mV / V, the measuring amplifiers from different manufacturers with different bridge supply voltages are interchangeable.
• The bridge equation also contains relative resistance changes ΔR1/R1.

The strain is also a relative size Δl/l0