Calculating the Gauge of a Copper Wire for a Thermometer Bridge Circuit

[PaxFinnica96]

Homework Statement

a. The resistance thermometer bridge circuit shown in FIGURE 1 has a designed maximum temperature of 200°C, ignoring the effects of connecting wire resistance. If the connecting loop is 250 m determine the smallest gauge (swg) of copper wire which must be used if the indicated maximum temperature is to be less than 202°C.
The temperature coefficient of resistance of the thermometer is 0.0042 °C–1 and the resistance of the thermometer is 112 Ω at 20°C. Assume the connecting leads are at 20°C.
b. Using the gauge of copper wire calculated in part (a), calculate the maximum indicated temperature using a 3-wire system (as shown in FIGURE 2) over a distance of 125 m.

Relevant Equations

Rt = R0(1 + at)

R1RT = R2RS

R = pL/A = Resistance = (Resistivity) x Length of wire / cross-sectional area

A = (pi x d2) / 4

d = squareroot(4A /pi)

Hi All,

Really struggling to get my head around what I'm doing wrong - every time my answer of wire diameter comes out too small for what's included in the table of SWG. I must be doing something wrong somewhere - my attempt is below.

Any help is very much appreciated.

 

[kuruman]

1 m = 10³ mm
1 m² = 10⁶ mm²
4.22 x 10^-8 m² = ? mm²

 

[PaxFinnica96]

1 m = 10³ mm
1 m² = 10⁶ mm²
4.22 x 10^-8 m² = ? mm²

Ah ok, but 4.22 x10^-8 m^2 = 0.0422 mm^2 however I cannot see this value of diameter in the SWG table provided - do you think it would be easier to calculate the ohm/m for copper instead or is there another error in my method?

 

[kuruman]

Ah ok, but 4.22 x10^-8 m^2 = 0.0422 mm^2 however I cannot see this value of diameter in the SWG table provided - do you think it would be easier to calculate the ohm/m for copper instead or is there another error in my method?

1. You don't need to calculate ohm/m; it is given in the table you posted.
2. Can you explain how you think this bridge circuit is supposed to work? Specifically, how does this "fundamental interval" enter the picture?

 

[PaxFinnica96]

Ok, well I believe the bridge is designed to be balanced so that the value of resistance, R_S, can be calculated and thus related to the resistance (ant therefore temperature) of the thermometer. It is this value I have used to incorporate into my rearranged resistivity equation to determine the wire diameter. Am I right to use the fundamental interval of the thermometer to calculate the wire size of the connecting loop?

 

[kuruman]

The question is more complex than simply calculating a diameter and looking it up.
First, you misunderstood the table. The numbers have units of Ohm/m. These are not resistivities but resistances per unit length for a given gauge. The dimensions of resistivity are Resistance times Length.
Secondly, the resistance of the thermometer is given as $R_0=112~\Omega$ at 20 ^oC. You should not have to recalculate it but should use it in the equation for the temperature dependence of the thermometer.
Thirdly, the question asks you to "determine the smallest gauge (swg) of copper wire which must be used if the indicated maximum temperature is to be less than 202°C." Why would a thermometer that has a maximum designed temperature of 200 ^oC give a reading of 202 ^oC? Stated differently, suppose you determined independently that the thermometer is at 200 ^oC, yet the bridge balances at a thermometer resistance appropriate to 202 ^oC, where did the extra resistance come from?