How Can Engineers Design a Temperature-Independent Resistor?

[ktobrien]

Homework Statement

An engineer needs a resistor with zero overall temperature coefficient of resistance at 20°C. She designs a pair of circular cylinders, one of carbon and one of Nichrome.

The device must have an overall resistance of R₁ + R₂ = 15.0 Ω independent of temperature and a uniform radius of r = 1.40 mm. Determine the lengths l₁ and l₂ of the carbon and Nichrome segments respectively. You may ignore thermal expansion of the cylinders and assume both are always at the same temperature.

Homework Equations

R₁+R₂=15
alpha₁+alpha₂=0
R=pl/A
p_carbon = 3.5e-5
p_nichrome = 1.5e-6

The Attempt at a Solution

I don't really know where to begin. The problem states that the 15ohms is independent of temperature. I know what that means but does it have any relevance to the problem? Any help getting started would be appreciated. Thanks

 

[berkeman]

Homework Statement

An engineer needs a resistor with zero overall temperature coefficient of resistance at 20°C. She designs a pair of circular cylinders, one of carbon and one of Nichrome.

The device must have an overall resistance of R₁ + R₂ = 15.0 Ω independent of temperature and a uniform radius of r = 1.40 mm. Determine the lengths l₁ and l₂ of the carbon and Nichrome segments respectively. You may ignore thermal expansion of the cylinders and assume both are always at the same temperature.

Homework Equations

R₁+R₂=15
alpha₁+alpha₂=0
R=pl/A
p_carbon = 3.5e-5
p_nichrome = 1.5e-6

The Attempt at a Solution

I don't really know where to begin. The problem states that the 15ohms is independent of temperature. I know what that means but does it have any relevance to the problem? Any help getting started would be appreciated. Thanks

One of those materials would have to have a negative temperature coefficient of resistance. Double check the rho numbers -- is one negative?

 

[ktobrien]

Rho is the resistivity. Alpha is the temperature coefficient. Both values of resistivity are positive as written. Carbons temperature coefficient is positive and nichromes is negative.

 

[berkeman]

Rho is the resistivity. Alpha is the temperature coefficient. Both values of resistivity are positive as written. Carbons temperature coefficient is positive and nichromes is negative.

Ah, thanks for the catch. My bad.

 

[ktobrien]

Could still use some help. Thanks.

 

[ideasrule]

The two alpha's don't need to add up to zero; R1 and R2 just need to sum to 15--not 15+0.2T or anything, just 15. All the other relevant equations are correct. Why not plug them all into R1+R2=15 and see what you get?

 

[ktobrien]

Plugging into the two formulas I got.

3.5e-5(l₁) + 1.5e-6(l₂) = 9.2363e-5

but this gives me 1 equation and 2 unknowns

 

[ktobrien]

Can someone please help me out?