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Temperature Coefficient of Resistance
10 questions By Tony R. Kuphaldt
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Question 4 of 10
The electrical resistance of a conductor at any temperature may be calculated by the following equation:
$$R_T=R_r+R_r\alpha T-R_r\alpha T_r$$
Where,
RT = Resistance of conductor at temperature T
Rr = Resistance of conductor at reference temperature Tr
α = Temperature coefficient of resistance at reference temperature Tr
Simplify this equation by means of factoring.
Reveal answer$$R_T=R_r [1+\alpha (T-T_r)]$$
Follow-up question: when plotted on a graph with temperature (T) as the independent variable and resistance (RT) as the dependent variable (i.e. a two-axis graph with T on the horizontal and R on the vertical), is the resulting plot linear? Why or why not? How is it possible to tell just by looking at the equation, prior to actually plotting on a graph?
Notes:Just an exercise in algebra here!
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Question 5 of 10
Write an equation solving for the temperature of a conductor (T), given its resistance at that temperature (RT), its resistance at a standard reference temperature (Rr @ Tr), and its temperature coefficient of resistance at that same reference temperature (α @ Tr).
Reveal answer$$T= \frac{\frac{R_T}{R_r}-1}{\alpha}+T_r$$
Notes:Students may be able to find this equation in a textbook somewhere, but the point of this question is really to have them perform algebraic manipulation to derive this equation from another.
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Question 6 of 10
Precision wire-wound resistors are often made of a special metal alloy called manganin. What is it about this alloy that makes it preferable for use in precision resistor construction?
Reveal answerThe α value of manganin alloy is nearly zero.
Notes:Ask your students what a wire-wound resistor made of copper or iron wire might do, if subjected to changes in temperature.
An historical side-note: during World War II, allied forces made extensive use of analog computers for directing the firing of projectiles and the dropping of bombs. Unlike digital computers, which perform mathematical operations using on/off signals and are thus immune to errors caused by slight changes in component value, electronic analog computers represent physical variables in the form of continuous voltages and currents, and depend on the precision of its constituent resistors to produce precise results. I remember reading one of the pioneering engineers in that field describe great gains in accuracy being due mostly to improvements in resistor construction. Without some crucial improvements in resistor accuracy and stability, analog computers of the war-time era would have suffered from substantial inaccuracies. Of all things, the lowly resistor was an influential piece of the allied war effort!
