Practical applications
Chapter 7 Variable Resistance Sensors
Chapter Objectives
1. To review the relationship between voltage and resistance in order to understand the characteristics of the potentiometer.
2. To understand the behavior of the linear and rotary displacement potentiometers.
3. To study the strain gage and its characteristics.
7-1 - Introduction
Of all the sensing devices available for transducing, the potentiometer device is perhaps the easiest to understand, the least expensive to construct, and the simplest to install. Its basis of operation lies within the conversion of a resistance change to a change in voltage or current: namely, the application of Ohm's law. However, another type of variable-resistance device, the strain gage, is perhaps not quite as straightforward as the potentiometer in operation, as we will find out in the discussions below. However, to understand either device, an understanding of Ohm's law and the concept of resistivity is certainly helpful. A review of how voltage divider networks operate will also contribute to an understanding of these devices.
7-2 - Ohm's Law, Resistivity, and the Voltage Divider Rule
7-2.1 - Ohm's Law
The quantities of resistance, voltage, and current are all related by the expression I = E
This relationship is called Ohm's law. The relationship holds regardless of whether you are dealing with direct current or voltage or with alternating current or voltage.
Furthermore, if you are dealing with alternating current or voltage, it makes no difference if you are working with their RMS values, their peak-to-peak values, or simply with their peak values. Ohm's law works equally well within any of these concepts.
Another important idea that is a key to the understanding of how strain gages operate is the concept of resistivity. Resistivity is the electrical resistance measured for any material having a uniform cross-sectional area and is usually stated in terms of the material's length and/or cross-sectional area. In other words, resistivity is resistance that has been stated in terms of a unit length or area. The unit of resistivity in the English system of measurements is the Ω-ft. or the CM-Ω/ft, where the abbreviation CM represents the circular mil. In the SI system of measurements the unit of resistivity is the ohm. The circular mil unit of area is often associated with wire because of its circular cross-sectional area. This unit is discussed in more detail below.
7-2.2 - Resistivity
Resistivity is determined by the following four quantities:
1. The type of material (cork, iron, glass, water, etc.) 2. The length of the material
3. The material's temperature
4. The material's cross-sectional area
In our discussion here we are specifically interested in the resistivity of wire. A wire's cross-sectional area is usually circular in shape. Its composition is usually steel, copper, or some other similar highly conductive material. The electrical resistance of wire can be varied either by changing its length, changing its temperature, or changing its cross-sectional area. If the wire's length is changed, its resistance will vary directly as the length is changed; if the cross-sectional area changes, the resistance will vary inversely as the area changes. Mathematically,
R A
α
l (7-2)We can eliminate the proportional symbol in eq. (7-2) by replacing it with a proportionality constant, ρ (rho), and inserting an equals sign. If we do this, eq. (7-2) now becomes
R = A
ρ
l (7-3)where
R = resistance (Ω)
ρ = proportionality constant, called resistivity (Ω-CM/ft, Ω-m) l = length (ft, m)
A = area [usually expressed in circular mils (CM; see the discussion below); also, m2)
The unit circular mil is often used in stating wire dimensions since wire diameters are often expressed in mils rather than inches. Therefore, since 1 inch = 1000 mils , and
d2
A= 4
π
(7-4)a wire having a diameter of DM mils will have for its cross-sectional area,
2
Therefore, by dividing eq. (7-5) by π/4, we can determine the number of circular mils contained within the square-mil results that were calculated with eq. (7-5). In other words,
Find the resistance of a 500-ft length of copper wire 0.013 in. in diameter. Assume a temperature of 68°F and ρ = 10.37 ohm-CM/ft.
Solution: Converting the diameter to mils, we get 12 mils. Note that this conversion . is done simply by moving the decimal point to the right three places. Then, using eq.
(7-7), we obtain Placing this result into eq. (7-3) to find R yields
(
10.37 ohm-cm/ft 500 ft)( )
R = 169 CM
= 36.68Ω
7-2.3 - Voltage Divider Rule
Figure 7-1 shows a circuit comprised of three resistors, R1, R2, and R3, wired as shown to a voltage source ES. We would like to find the value of Eout. To analyse this circuit we first notice that ES also occurs across the two resistors, R1 and R2. We can neglect R3 since it does not affect the value of ES. We must now decide on how ES is going to divide across R1 and R2. The voltage across R2 is the same voltage, E2, which is the voltage we are interested in finding.
To solve this problem, we first determine the total current flowing through both R1 and R2. Using Ohm's law [eq. (7-1)] we find this total current, IT, to be
Figure 7.1 Voltage divider rule.
Equation (7-9) is the basic form of the voltage divider rule. The only change in the equation is to transpose the ES and the R2 to make the resultant equation a little easier to memorize. In other words,
out S 2
1 2
E = E R
R + R (7-10)
To find the voltage drop across R1 in Figure 7-1, eq. (7-10) would be rewritten as
1 S 1
1 2
E = E R
R + R (7-11)
Example 7-2
Calculate the voltage drops across both R1 and R2 in Figure 7-1.
Solution: To find the voltage drop across R1, we use eq. (7-11):
1
7-3 - Potentiometer
As stated earlier, the potentiometer is a relatively easy device to understand once the concept of the voltage divider is understood. We now investigate this device to learn how it operates and why it is such a popular and versatile sensing device in many industrial applications.