Temperature-Independent Resistor Combo

[Angie K.]

Homework Statement

For some applications, it is important that the value of a resistance not change with temperature. For example, suppose you made a resistor from a carbon resistor and a Nichrome wire-wound resistor connected together so the total resistance RT is the sum of their separate resistances. What value should each of these resistors have (at 20 °C) so that the combination is temperature independent? Express your answer in terms of RT and the temperature coefficients of resistance of carbon αC and Nichrome αN (specified at 20 °C). Note that αC < 0 but αN > 0.

Hints Given: The total resistance of the series combination is simply the sum of the separate resistances. Use this fact to write an expression for R(T), the resistance of the combination as a function of temperature T. How can you express the condition that R is independent of T at T = 20°C? (Hint: The function R(T) must have zero slope at T = 0). Remember that αC < 0; then check that your resistances RC and RN are positive.

The answer requires two separate equations, one for Resistivity of Carbon and another Resistivity of Nichrome.

Homework Equations

R = pl/A where p = resistivity coefficient and l is the length

R(T) = R0(1+a(T-T0)
a = alpha (temperature coefficient)

The Attempt at a Solution

Using the equation R(T) = R0(1+a(T-T0) I tried to plug in the value for resistivity of Carbon (3-60)*10^-5 and same for the Nichrome but I'm not sure whether to add them into one equation or use separate equations?

 

[Merlin3189]

... Express your answer in terms of RT and the temperature coefficients of resistance of carbon αC and Nichrome αN ...

Homework Equations

[/B]R = pl/A where p = resistivity coefficient and l is the length

I tried to plug in the value for resistivity of Carbon (3-60)*10^-5 and same for the Nichrome ...

Notice: this is not about resistivity, but the temperature coefficient of resistance.
I would leave these as simply αC and αN until the maths is done. Then you can look up the proper values at hyperphysics here, looking in the correct column.

Your second relevant equation is correct and that is what you use.
Write the expression for the carbon resistor R_C(T) and write the expression for the Nichrome resistor R_N(T) - they will look very similar.

Now the hint,

Hints Given: The total resistance of the series combination is simply the sum of the separate resistances.

So now write the expression for the total resistance R(T).
Because the two parts are so alike, you can easily rearrange it into a form suggested by the real big hint,

(Hint: The function R(T) must have zero slope at T = 0).

My hint at this point is, y = mx + c (but whether this means anything to you depends on who taught you maths and what bit of jargon they chose to use!)

See where you get to now.

 

[Angie K.]

I just don't understand why it is relevant that the total resistance is the sum of the separate resistances. The answer asks for Rc and Rn. What does the total resistance have to do with the equation?

 

[Merlin3189]

I just don't understand why it is relevant that the total resistance is the sum of the separate resistances. The answer asks for Rc and Rn. What does the total resistance have to do with the equation?

You are required to ensure the total resistance does not vary with temperature.
So you need an equation for the total resistance, which does not depend on temperature.

The given formula you are using, R(T) = R₀(1 + α(T - T₀)) does depend on temperature T
UNLESS the constant α happens to be zero, in which case the equation reduces to
R(T) = R₀(1 + 0x(T - T₀)) or ust R(T) = R₀(1 ) = R₀

Now we know α is not zero, but when we add two equations like this,
ie we add the formula for the carbon resistor and the formula for the nichrome resistor to get an equation for the total resistance,
then the T is multiplied not just by α, but by a more complex term involving α_C, α_N, R_C and R_N
That complex term CAN be equal to zero, so that the total resistance equation no longer depends on T

So you want an equation for the total resistance which looks like,
R(T) = (any old bunch of constant terms) + T*(a mixture of α_C, α_N, R_C and R_N terms)
then you set (a mixture of α_C, α_N, R_C and R_N terms) = 0, so that the total resistance does not depend on T

 

[Merlin3189]

Just to add: you won't get an actual value for R_C and R_N, because you don't know the total resistance required.
What you need is the ratio of $R_C : R_N$ (or as a fraction $\frac {R_C}{ R_N}$ )
Say this came to $\frac {R_C}{ R_N} = 0.7$ (or $\frac{0.7}{1}$ ) or as a ratio $R_C : R_N = 0.7 : 1$
and you want to make a resistance of 500Ω
then $R_C = 500 * \frac{0.7}{1.7} = 206Ω$ and $R_N = 500 * \frac{1}{1.7} = 294Ω$