VI. PLAN ECONÓMICO FINANCIERO
6.12 Estado de Ganancias y Pérdidas
The resistance of most metals increases reasonably linearly with temperature in the range −100 to +800 °C. The general relationship between the resistance RTΩ of a metal element and temperature T °C is a power series of the form:
where R0Ω is the resistance at 0 °C and α, β, γ are temperature coefficients of resistance. The magnitude of the non-linear terms is usually small. Figure 8.2(a) shows the variation in the ratio RT/R0with temperature for the metals platinum, copper and nickel. Although relatively expensive, platinum is usually chosen for industrial resistance thermometers; cheaper metals, notably nickel and copper, are used for less demanding applications. Platinum is preferred because it is chemically inert, has linear and repeatable resistance/temperature characteristics, and can be used over a wide temperature range (−200 to +800 °C) and in many types of environments. It can be refined to a high degree of purity, which ensures that statistical variations in resistance, between similar elements at the same temperature (Section 2.3.2), are small. A typical platinum element has R0= 100.0 Ω, R100= 138.50 Ω, R200= 175.83 Ω, α = 3.91 × 10−3°C−1 and β = −5.85 × 10−7°C−2. The change in resistance between
8.1 RESISTIVE SENSING ELEMENTS 153
the ice point and the steam point, i.e. R100− R0, is called the fundamental interval; in the above element this is 38.5Ω. The maximum non-linearity as a percentage of f.s.d. (eqn [2.5]) between 0 and 200 °C is +0.76%.
The standard IEC 751: 1983 (BS EN 60751: 1996)[2]lays down tolerance limits on the maximum variation in resistance between platinum elements at a given temperature. For class A elements the tolerance limits are ±0.06Ω at 0 °C and ±0.20Ω at 200 °C; for class B elements the tolerance limits are ±0.12 Ω at 0 °C and ±0.48Ω at 200 °C. The amount of electrical power produced in the element should be limited in order to avoid self-heating effects (Chapter 14); in a typical element 10 mW of power causes a temperature rise of 0.3 °C.
One type of element is constructed using the partially supported arrangement shown in Figure 8.2(b).[1]Here fine platinum wire is wound into a very small spiral and is inserted into axial holes in a high purity alumina insulator. A small quantity of glass adhesive is introduced into the holes and the unit is fired, thus securely fixing a part of each turn onto the alumina; the remainder of the wire is free to move. The diagram also shows the element housed in a stainless steel protective sheath.
Resistive temperature elements made from semiconductor materials are often referred to as thermistors. The most commonly used type is prepared from oxides of the iron group of transition metal elements such as chromium, manganese, iron, cobalt and nickel. The resistance of these elements decreases with temperature – in other words there is a negative temperature coefficient (NTC) – in a highly non-linear way. Figure 8.3 shows typical thermistor resistance–temperature characteristics which can be described by the relationship:
[8.4] Rθ K β θ = exp Figure 8.2 Metal resistive temperature sensors: (a) Resistance/ temperature characteristics of commonly used metals (b) Typical construction of platinum element probe.[1]
where Rθis the resistance at temperature θ kelvin; K and β are constants for the thermistor. A commonly used alternative equation is:
[8.5]
where Rθ1Ω is the resistance at reference temperature θ1K, usually θ1= 25 °C = 298 K. Thermistors are usually in the form of either beads, rods or discs (Figure 8.3); bead thermistors are enclosed in glass envelopes. A typical NTC thermistor has a resistance of 12 kΩ at 25 °C (298 K), falling to 0.95 kΩ at 100 °C (373 K), and β = 3750 K.[3]The manufacturer’s tolerance limits on the above figures are ±7%, i.e. ±840Ω, at 25 °C, and ±5%, i.e. ±47.5 Ω, at 100 °C, which is far wider than for metal elements. The element time constant is 19 seconds in air and 3 seconds in oil, and the self-heating effect is 1 °C rise for every 7 mW of electrical power. Thermistors with positive temperature coefficients (P.T.C.) are also available; the resistance of a typical element increases from 100Ω at −55 °C to 10 kΩ at 120 °C.
Thick-film polymers can be used as resistive sensors for temperature and
humidity measurement.[4]These are pastes consisting of a polymer matrix (usually epoxy, silicone or phenolic resin) that binds together the filler particles. If the filler material is metallic, e.g. silver flakes or copper particles, then the paste has similar resistive properties to a metal. If carbon particles are used as the filler material, then the paste has the resistive properties of a semiconductor. Solvents are then added to give the paste the thixotropic properties of an ink so it can be screen printed on to a substrate (often alumina). Following the printing process, the solvents are dried off and the pastes are cured at temperatures that do not affect the physical and chemical stability of the substrates. The printed patterns used for carbon and silver paste are shown in Figure 8.4(a).
Figure 8.4(b) shows the variation in resistances with temperature for silver and carbon polymers. Silver polymers exhibit a linear increase in resistance with tem- perature. This is described by a linear approximation to eqn [8.3], i.e.
Rθ Rθ β θ θ = exp − 1 1 1 1 Figure 8.3 Thermistor resistance–temperature characteristics and types (graph (a) after Mullard Ltd[3]).
8.1 RESISTIVE SENSING ELEMENTS 155
An alternative linear approximation for T between 25 and 200 °C is:
where α = 3.732 × 10−3°C−1.[4]
In contrast the carbon paste cured at 220 °C exhibits semiconductor or thermistor- like characteristics, described by eqns [8.4] and [8.5]: here the constant β = 136 K.
Carbon polymers also show a variation in resistance with humidity. The polymer matrix absorbs water, causing it to swell. This causes the carbon particles embedded in the matrix to move further away from each other, thus reducing the number of conductive paths and consequently increasing the resistivity of the material. Figure 8.4(c) shows the variation in resistance with humidity in the range 30% to 90% for carbon polymers cured at 160 °C and 200 °C and operated at 25 °C and 60 °C. The resistance–humidity characteristics are non-linear; the higher the curing temperature the greater the non-linearity.[4]
RT= R25[1 +α(T − 25)] [8.7]
RT= R0(1 +αT ) [8.6]
Figure 8.4 Thick-film
polymer resistive sensors: (a) Printed patterns for carbon paste (left) and silver paste (right) (b) Changes in resistance of silver and carbon filled pastes with temperature (c) Changes in resistance of carbon polymer pastes with relative humidity (Figures 8.4(b) and (c) are reprinted by permissions of the Institute of Measurement and Control and the University of Southampton and N. M. White[4]).