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Sensors and Signal Conditioning

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Resistive Sensors

传感器英文课件第2章

Sensors based on the variation of the electric resistance of a device are very common. That is because many physical quantities affect the electric resistance of a material. This chapter discusses sensors based on a variation in resistance. It describes their fundamentals (sensing principle, dynamic model, limitations, advantages), technology, equivalent electric circuit, and applications. The different resistive sensors are classified by the physical quantity being measured as mechanical, thermal, magnetic, optical, and chemical variables.

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2.1 PotentiometersA potentiometer is a resistive device with a linear or rotary sliding contact (Figure 2.1).

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Figure 2.1 (a) Ideal (linear) potentiometer connected as a voltage divider and (b) its symbol; the arrow indicates that the resistance variation responds to a mechanical action. (c) The wiper can also be connected to an end terminal to obtain a variable resistor (rheostat).

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The resistance between that contact and the bottom terminal is

where p is the resistivity, A is the cross section, l is the length, x is the distance traveled from the bottom terminal, and a is the corresponding length fraction.

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A potentiometer is a zero-order system. Equation (2.1) means that the resistance is proportional to the travel of the wiper. This implies the acceptance of several simplifications that may not necessarily be true. First, we assume that the resistance is uniform along the length l. Second, we assume that the sliding contact gives a smooth resistance variation, not a stepped one, and therefore that the resolution is infinite. Furthermore, the mechanical travel is usually larger than the electrical travel.

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For (2.1) to be valid, if the potentiometer is supplied by an alternating voltage, its inductance and capacitance should be insignificant. For low values of the total resistance RT, the inductance may be significant, particularly in models with wound resistive elements. For high values of RT, the parasitic capacitance may be important. Furthermore, resistors drift with temperature. Temperature changes can arise not only from fluctuations in ambient temperature but also from selfheating due to the finite power that the potentiometer dissipates. Friction and inertia of the wiper also limit the model validity. Finally, noise associated with the wiper contact limits resolution.

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Most of these limitations are outweighed by the advantages of this device. It is simple and robust and yields a high-level voltage with high accuracy relative to its cost. Potentiometers comprise a resistive element, a wiper, an actuating or driving rod, bearings, and housing. linear and rotary

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2.2 Strain Gages Fundamentals: Piezoresistive EffectStrain gages are based on the variation of resistance of a conductor or semiconductor when subjected to a mechanical stress. The electric resistance of a wire having length l, cross section A, and resistivi

ty p is

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When the wire is stressed longitudinally, each of the three quantities that affect R change and therefore R undergoes a change given by

The change in length that results when a force F is applied to a wire, within the elastic limit (Figure 2.6a), is given by Hooke's law,

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Consider a wire that in addition to a length l has a transverse dimension t (Figure 2.6b). A longitudinal stress changes both l and t. According to Poisson's law, we have

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For a wire of circular cross section of diameter D, we have

The change in resistivity as a result of a mechanical stress is called the piezoresistive effect. P. W. Bridgman showed that, in metals, the percent changes of resistivity and volume are proportional:

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where C is the Bridgman‘s constant. For the usual alloys from which strain gages are made, 1.13< C< 1.15. For platinum, C=4.4. By applying (2.10), the change in volume can be expressed as

Therefore if the material is isotropic, within the elastic limit, (2.7) leads to

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where G is the gage factor, defined as the factor inside the square brackets.

Therefore for small variations, the resistance of the metallic wire is

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When a semiconductor is stressed, in addition to its dimensional change, both the number of carriers and their average mobility change. Unlike metals,the resistivity change under stress dominates over the dimensional change.

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G is from about 40 to about 200.

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Semiconductors with a relatively low number of carriers yield large gage factors, but they are temperaturesensitive and depend on the stress; that is, they are nonlinear. Semiconductors with a relatively high number of carriers have smaller gage factors, but these are less temperature and stress-dependent.