The Wheatstone Bridge

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Introduction In 1843 the English physicist Sir Charles Wheatstone (1802-1875) found a bridge circuit for the measurement of electrical resistances. In this bridge circuit, known today as the Wheatstone bridge, unknown resistances are compared with well defined resistances (1). The Wheatstone bridge is well suited also for the measurement of small changes of a resistance and, therefore, is also suitable to measure the resistance change in a strain gauge. It is commonly known that the strain gauge transforms strain applied to it into a proportional change of resistance. The relation between the applied strain ( = L/Lo) and the relative change of the resistance of a strain gauge is described by the equation (Equation 1)The factor k, also known as the gauge factor, is a characteristic of the gauge used for strain measurement and is to be checked experimentally (2). Below, the Wheatstone bridge circuit will only be considered with respect to its application in strain gauge techniques. Two different presentations are given in fig. 1: a) is based on the original notation of Wheatstone, and b) is another notation which 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.

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(Figure 1)The four arms (or branches) of the bridge are formed by the resistors R1 to R4. Input and output (the four nodes) are numbered and color coded according to HBM standards which is used throughout in all types of transducers and instruments. If the nodes 2 (black) and 3 (blue) - the energizing diagonal - are connected to a known voltage UE (bridge input or energizing voltage) then a voltage UA (bridge output voltage) appears across the so-called output diagonal, the nodes 1 (white) and 4 (red). The value of the output voltage depends on the ratio of the resistors R1 : R2 and R4 : R3. Note 1- 1: For the sake of clarity it is more convenient to consider the ratio of the output and the input voltage UA/UE. Therefore, the following equations are based on this ratio. Generally the equation (Equation 2)is valid, and for the case of the balanced bridge we have (Equation 3)Note 1-2: In all practical cases where strain gauges are used, the state of the balanced bridge is reached with more or less deviation. Moreover, all instruments designed for strain gouge measurements are equipped with balancing elements which allow to set the indication to zero for the initial state. This allows to use the equation (3) for all further considerations. Another assumption is that the bridge output remains unloaded electrically. The internal resistance of the instrument connected to this output must be large enough to prevent noticeable errors. This is assumed for all further considerations. If the resistors Rl to R4 vary, the bridge will be detuned and an output voltage UA Will appear. With the additional assumption that the resistance variation Ri is much smaller than the resistance Ri itself (which is always true for metallic strain gauges) second order factors can be disregarded. We now have the following relation (Equation 4)Note 1-3: For practical strain gauge applications the pairs Rl, R2 and R4, R3 should be equal or all four resistors Rl to R4 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 Rl and R4 (or R2 and R3) have the same or a different nominal value. This is the reason for assuming always R1 = R2 = R3 = R4 = R. The latter is valid when constant voltage is used for bridge excitation. Substituting equation (1) (LR/R = k x c) in (4) we find (Equation 5)The signs of the terms are defined as follows (see also fig. 2): With the given polarity of the energizing voltage UE: node (2) = negative, (3) = positive, we will have positive potential at point (1), negative potential at point (4), if Rl > R2 and/or R3 > R4. We will have negative potential at (1), positive potential at (4), if Rl Please note: The changes of neighboring strain gauges are subtracted if the sign is equal, they are added if they are of opposite sign. This fact can be used for some combinations or compensation methods which will be discussed in detail separately. Consequently, the amount of the strain 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 amounts, for example: +10 m/m > +5 m/m; +2 m/m > -20m/m; -5 m/m > -50 m/m. Note 1-5: Special features of strain measuring instruments. Here we would like to review the instruments commonly used in strain gauge measurement techniques. 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 UA/UE. 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 m/m = 1 x 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 * is equivalent to the actual strain present, if only one active gauge is in one arm of the bridge (quarter bridge configuration, see section 2) and if the gauge factor k of the gauge used corresponds with the calibration value of the instrument. The bridge arms 2, 3 and 4 are formed by resistors or by passive gauges. This means in fact, that 2 = 3 = 4 of equation (5) are zero and can be omitted. Some instruments are calibrated with a fixed gauge factor of k = 2, others have a "gauge factor selector" which can be set to the actual gauge factor of the gauge. With the condition k(gauge) = k(instr), we will have the indicated strain value equal to the measured strain * = 1. (Equation 6) If an instrument with a fixed calibration k = 2 is used, measured values must be corrected, because gauge factors may vary with the grid material and grid configuration. The correction formula is 1 = *(2/k). (Equation 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 operation manuals of the instruments. Please note that this sample paper on The Wheatstone Bridge is for your review only. 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