Strain Gage Techniques: Wheatstone Bridge Circuit Analysis and Applications, Esquemas de Medição Eletrônica e Instrumentação

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Applying the

Wheatstone

Bridge Circuit

by Karl Hoffmann

W1569-1.0 en

Contents:

• 1 Introduction ...................................................................................................................................
• 2 Elementary circuits with strain gages..........................................................................................
  • 2.1 Measurements on a tension bar..............................................................................................
  • 2.2 Measurements on a bending beam .........................................................................................
  • 2.3 Measurements on a twisted shaft ...........................................................................................
  • moment ......................................................................................................................................... 3 Analysis and compensation of superimposed stresses, e.g. tension bar with bending
• 4 Compensation of interference effects, especially temperature effects. ...................................
• 5 Compensation of the influence of lead resistances. ..................................................................
• 6 Table on different circuit configurations ..................................................................................
• 7 Some remarks on the limits of the compensation of interference effects ...............................
• 8 The linearity error of the Wheatstone bridge circuit...............................................................
  • 8.1 Quarter bridge ......................................................................................................................
  • 8.2 Half bridge ...........................................................................................................................
  • 8.3 Full bridge ............................................................................................................................
• 9 Bibliography.................................................................................................................................

1 Introduction

In 1843 the English physicist, Sir Charles Wheatstone (1802-1875), found a bridge circuit for measuring electrical resistances. In this bridge circuit, known today as the Wheatstone bridge circuit, unknown resistances are compared with well-defined resistances [1]. The Wheatstone bridge is also well suited for the measurement of small changes of a resistance and is, therefore, also suitable for measuring the resistance change in a strain gage (SG). It is commonly known that the strain gage transforms strain applied into a proportional change of resistance. The relationship between the applied strain H (H = ¨L/Lo ) and the relative change of the resistance of a strain gage is described by the equation

' (^) k ˜ H R

R 0

The factor k, also known as the gage factor, is a characteristic of the strain gage and has been checked experimentally [2]. The exact value is specified on each strain gage package. In general, the gage factor for metal strain gages is about 2.

Below, the Wheatstone bridge circuit will only be considered with respect to its application in strain gage technique. Two different presentations are given in fig. 1: a) is based on the original notation of Wheatstone, and b) is another notation that is usually easier to understand by a person without a background in electrical or electronic engineering. Both versions are, in fact, identical in their electrical function.

Fig. 1: The Wheatstone bridge circuit

The four arms (or branches) of the bridge are formed by the resistors R 1 to R 4. The corner points (or nodes) are numbered and color-coded according to the HBM standard for designating the connections of transducers and instruments. If nodes 2 (black) and 3 (blue) - the so-called excitation diagonal - are connected to a known voltage UE (bridge input or excitation voltage) then a voltage U (^) A (bridge output voltage) appears between nodes 1 (white) and 4 (red), the so-called measurement diagonal. The value of the output voltage depends on the ratio of the resistors R 1 : R 2 and R 4 : R 3.

Note 1-3: For practical strain gage applications, the pairs R 1 , R 2 and R 3 , R 4 or all four resistors R 1 to R 4 should have the same nominal value to ensure that the relative changes of the individual bridge arms are proportional to the relative variation of the output voltage. It is not significant whether R 1 and R 4 (or R 2 and R 3 ) have the same or a different nominal resistance. This is the reason for always assuming R 1 = R 2 = R 3 = R 4.

Substituting equation (1) (¨R/R 0 = k · İ) in (4) we find

4 ˜H^1 H^2 H^3 ^ H^4

k U E

U A (5)

The signs of the terms are defined as follows (see also fig. 2):

With the given polarity of the excitation voltage UE: node (2) = negative, (3) = positive, we will have

positive potential at point (1), negative potential at point (4), if R 1 > R 2 and/or R 3 > R 4 ;

negative potential at (1), positive potential at (4), if R 1 < R 2 and/or R 3 < R 4.

For a.c. supply, the above conditions are true for the phase relations of UE and UA.

Please note: The changes of the absolute values of neighboring strain gages are subtracted if they have like signs. They are added if they have unlike signs. This fact can be used for some combinations or compensation methods, which will be discussed in detail separately.

Fig. 2: Illustration for the rules of the signs in the Wheatstone bridge

Note 1-4: With reference to the strain gages themselves, we have:

point (1) positive, point (4) negative if H 1 >H 2 and/or H 3 >H 4.

point (1) negative, point (4) positive if H 1 <H 2 and/or H 3 <H 4.

Consequently, the size of H 1 should be considered with respect to its effect on the resistance R. “Greater than” or “smaller than” must be used in an algebraic sense and not only for the absolute values, for example: + Pm/m > +5 Pm/m; +2 Pm/m >-20 Pm/m; -5 Pm/m >-50 Pm/m.

Note 1-5: Special features of strain measuring instruments

Here we would like to review the instruments commonly used in strain gage measurement technique. Equations (2) to (4) assume that a resistance variation in one or more arms of the bridge circuit produces a variation of the relative output voltage U (^) A/U (^) E. As a final result of the measurement this is only of limited importance. Since the actual strain value is more interesting, most of the special instruments have an indicator scale calibrated in “strain values”. The strain value 1 Pm/m 1 · 10-6^ m/m is used as the “Unit”, on older designs you may also find the designation “microstrain”.

All these special instruments are calibrated in such a manner that the indicated value H* is equivalent to the actual strain present, if there is only one active strain gage in bridge arm 1 (quarter bridge configuration, see Section 2) and if the gage factor k of the strain gage used corresponds to the calibration value of the instrument. The bridge arms 2, 3 and 4 are formed by resistors or by passive strain gages. This means, in fact, that H 2 = H 3 = H 4 of equation (5) are zero and can be omitted. Some instruments are calibrated using a fixed gage factor of k = 2, others have a “gage factor selector” which can be set to the actual gage factor of the strain gage. Thus, if

kSG = k (^) instr. then İ* = İ (6)

If an instrument with a fixed calibration value k = 2 is used, the measured values H* must be corrected because the gage factors of the strain gages vary according to the grid material and grid configuration. The correction formula is

k

2 H 1 H*˜ (7)

For all further considerations, these special features will be neglected since they are not necessary for a basic understanding of the bridge configuration. Further details may also be taken from the operating manuals of the instruments.

When strain measurements are taken, it is not the strain on the surface of the specimen that is of interest in most cases but instead the material stress.

Normal stress results in changes of length that can be measured with strain gages. The ratio of stress ı and strain H in the elastic deformation range is defined by Hooke’s Law [3]:

ı =H*E

E expresses Young’s modulus (modulus of elasticity) which is a material parameter.

clearly distinguished, such as compressive or tensile stress, as well as bending, shear or torsional forces. (The compensation of interferences is covered in detail in Section 4.)

The mechanical interrelations are only treated as far as this is absolutely necessary. A detailed presentation can be found in [3].

2.1 Measurements on a tension bar

With a tension bar, a strain İ 1 = ı/E in the direction of the force applied will occur. In the transverse direction, a transverse contraction H 2 = -ȣ · H 1 will occur. From this, a change of resistance 'R 1 = H 1 · k · R 1 will be found in strain gage no. 1. For strain gage no. 2, this difference will be 'R 2 = -ȣ · H 1 · k · R 2. Accordingly, this is also true for strain gages no. 3 and no. 4 respectively.

Fig. 4: The tension bar

Note 2-1: The factor ȣ, also called Poisson’s Ratio, depends on the material and is valid in the material’s elastic deformation range only. It has a value of around 0.3 for metals.

If all four strain gages are connected in the sequence of their indices to form a full bridge (see fig. 3a) an output signal as described by equation (5) is produced

1 2 3 4 >^1133 @

4 4 H  HHH HH H  QH k k U

U E

A .

Introducing ȣ § 0.3 and with İ 1 = İ 3 = İ and İ 2 = İ 4 § -0.3 · İ, the net signal will be

| 4 ˜ 2. 6 ˜ H 1 k U

U E

A (8)

Note 2-2: A bridge circuit with four active strain gages gives a signal 2.6 times the value of the strain İ 1 in the tension bar’s main direction of stress. Sometimes this factor is also called the bridge factor “B”. Therefore, a general form of equation (8) is also

4 ˜^ B ˜^ H^1

k U

U E

A (9)

If compressive forces are to be measured, the opposite signs of the strains H 1 to H 4 are valid.

If the strain H 1 in the main stress direction is unknown, this can be derived by transformation from equation (8):

E

A U

U

k

H 1 (10)

or, in the more general form of equation (9), transformed into

E

A U

U B k

˜ ˜

4 H 1 (11)

A half bridge circuit with only strain gages no. 1 and no. 2 as active strain gages gives only half the signal, because the factor B is only 1.3.

A combination of R 1 and R 3 formed by active strain gages and R 2 and R 4 with fixed resistors is also possible and gives B = 2. But this is without automatic compensation of the thermal expansion and other influences (see also Section 4). However, bending forces that are superimposed will be compensated.

2.2 Measurements on a bending beam

Fig. 5: The bending beam

The conditions in the case of the bending beam are a bit simpler and more favorable, see fig. 5. Here strain values are of the same absolute value but with the opposite sign. This means that strain

˜ B ˜H k ˜ H k U

U E

A 4

From this we can derive the strain value as k

U (^) A / UE H

If the sign of the torque is unknown, it can be found using the rules described in Section 1.

In the case of the twisted shaft, we can also use a half bridge with the active strain gages no. 1 and no. 2. The bridge factor will then be B = 2.

Note 2-3: If the measured signal must be transmitted from a rotating shaft via slip rings or similar devices (as is often required in torque measurements), the full bridge circuit should preferably be used. The advantage here lies in the fact that errors from the transmission path are considerably smaller than for a half or quarter bridge circuit (see Section 5).

3 Analysis and compensation of superimposed stresses,

e.g. tension bar with bending moment

As discussed in Section 2, pure tensile or compressive forces as well as pure bending moments or torque cannot be assumed in all cases. In most practical cases, either two types or all three are combined. For example, this situation can be demonstrated for a beam under tensile force F and a bending moment MB.

Fig. 7: Tension bar with bending moment superimposed

Fig. 8: Tension/bending beam according to fig. 7 with strain gages applied

The following strain components can be detected on this beam:

a) The total strain HO on the upper side of the beam and the total strain HU underneath the beam.

b) The strain HN caused by the normal force (tensile or compressive).

c) The strain component HB caused by the bending moment.

The composite strain values, HO and HU, can be measured with one strain gage each on the upper side and underneath the beam, orientated along the beam axis, see fig. 8.

Each strain gage has to be inserted into a quarter bridge (see fig. 3c) separately and its measured value must be recorded. The strain gage on the upper side delivers the sum HO = HN + HB , the strain gage underneath the difference HU = HN - HB. The strain components can be computed from these two values with the help of the relationship

Fig. 10: Tension /bending beam according to fig. 7 fitted with longitudinal and transverse strain gages

When designing a force transducer, the aim is always to measure the forces in the direction of the transducer axis only instead of the bending strains resulting from eccentric or side loads, etc.

In this case, a combination of strain gages should be used in accordance with fig. 4. Strain gages must be symmetrical both with regard to their position and specification. All four strain gages form a complete Wheatstone bridge circuit, which is optimum for electrical symmetry.

The absolute strain values (or the proportional changes in strain gage resistance) are subtracted if they appear with opposite signs in opposite bridge arms. It is obvious that the signals resulting from the normal force are added (see also fig. 4) and signals from the bending force compensate one another. The components from the bending force are distributed as follows:

SG no. 1: + İB SG no. 2: - ȣ · İB SG no. 3: - İB SG no. 4: +ȣ · İB

The absolute values of the corresponding signals are equal, which leads to complete compensation (see also Section 4).

This principle can also be applied to compensate for axial forces in the bending beam or bending forces in the twisted shaft (see also Sections 2.2 and 2.3).

Note 3-1: Generally speaking, the four strain gages of such a transducer could also be connected in this way into a half bridge, by series connecting strain gages no. 1 and no. 3 and strain gages no. 2 and no. 4 into one bridge arm each, as illustrated in fig. 11.

Fig. 11: Half bridge with four strain gages

This circuit is symmetrical with respect to the position of the strain gages but not with respect to the connection leads within the overall bridge circuit, which must be completed with the complementary half bridge located inside the instrument. In addition, the sensitivity is only half the sensitivity of the full bridge. The measured values of series-connected strain gages will certainly not add up. The following relationship is true for the circuit shown in fig. 11:

¸¸ ¹

· ¨¨ ©

§ 

' ' 

' ' 2 4

2 4 1 3

1 3 4

1 R R

R R R R

R R U E

U A (14)

From this, it can easily be seen that compared to equation (4) we get only half the signal.

SG no. İ M İ W

1 Positive Positive

2 Negative Positive

3 Positive Positive

4 Negative Positive

We can see that the thermal component in the total strain İ W has the same sign for all strain gages, provided they are all subjected to the same change in temperature. Based on equation (5) we find the expression

(^) M W M W M W M W @ (^) M E

A k k U

U H H H H H H H H 4 H 4 1 2 3 4 4        ˜

The interference effect İ W has been compensated.

The same method of compensation can be used for a half bridge.

This method of compensation cannot be implemented with the quarter bridge , which is widely used in experimental stress analysis. The only way to overcome this is to use a “compensating strain gage” to expand the quarter bridge into a half bridge. It is very important to select and apply the compensating strain gage correctly in order to achieve the correct result. The compensating strain gage:

a) must have the same physical properties as the “active” strain gage

b) must be applied in a spot where it will be subjected to the same interference effect as the active strain gage.

c) must only be subjected to the interference effect and never to the quantity to be measured İM or its side effects.

If a suitable spot cannot be found on the specimen itself, an extra piece of material can be used. The compensating strain gage can be bonded to it and the arrangement can then be placed near the active strain gage. This will lead to a proper compensation of the interference effects (of temperature) without influences from the stress, since there is no mechanical coupling. The piece with the compensating strain gage should be made from the same material as the original specimen with the active strain gage, especially if the interference effect will lead to deformation of the material, such as thermal expansion, compression by hydrostatic pressure, magneto-striction, etc.

Note 4-2: The temperature response of a strain gage can be adjusted so that the temperature effect on the material can be compensated – within technical feasibility. Therefore, each HBM strain gage (exception: LD20 high strain gage) will be adapted to the thermal expansion coefficient of a certain material. This adaptation considerably reduces the error caused by temperature.

The remaining temperature response is specified on the strain gage package so that the temperature error can be corrected mathematically.