Question
The resistance of a thermistor is measured to be $2.25 k\Omega$ at $30 \degree C$ and $1.17 k\Omega$ at $60 \degree C$. Its material constant $\beta$ is _________ K (rounded off to two decimal places).
The resistance of a thermistor \( R \) at a temperature \( T \) is modeled by the equation:
\( R = R_0 e^{\frac{\beta}{T}} \)
where \( R_0 \) is the resistance at a reference temperature and \( \beta \) is the material constant. Assuming the base \( e \), the equation can be rearranged for two temperature points:
- \( R_1 = 2250 \, \Omega \) at \( T_1 = 303 \, \text{K} \) (converted from \( 30 \degree C \))
- \( R_2 = 1170 \, \Omega \) at \( T_2 = 333 \, \text{K} \) (converted from \( 60 \degree C \))
Using the equation for both temperatures, we get:
\( \frac{R_1}{R_2} = e^{\beta\left(\frac{1}{T_2} - \frac{1}{T_1}\right)} \)
By taking the natural logarithm on both sides:
\( \ln\left(\frac{R_1}{R_2}\right) = \beta\left(\frac{1}{T_2} - \frac{1}{T_1}\right) \)
Simplifying for \( \beta \):
\( \beta = \frac{\ln\left(\frac{R_1}{R_2}\right)}{\frac{1}{T_2} - \frac{1}{T_1}} \)
Substituting the values:
- \( \frac{R_1}{R_2} = \frac{2250}{1170} \approx 1.9231 \)
- \( \frac{1}{T_1} = \frac{1}{303} \approx 0.0033 \, \text{K}^{-1} \)
- \( \frac{1}{T_2} = \frac{1}{333} \approx 0.003003 \, \text{K}^{-1} \)
Then:
\( \ln(1.9231) \approx 0.654 \)
\( \frac{1}{303} - \frac{1}{333} = 0.0033 - 0.003003 \approx 0.000297 \, \text{K}^{-1} \)
\( \beta = \frac{0.654}{0.000297} \approx 2203.37 \, \text{K} \)
Final result: \( \beta \approx 2203.37 \, \text{K} \), within the expected range 2160-2210.
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Important Questions from Thermistor
Which one of the following transducers requires power supply for its operation?
- Which of the following statements is false?
Which of the following is NOT true about thermistor and resistance temperature detectors?
______ is a type of resistor whose electrical resistance varies with changes in temperature.
Which of the following is NOT an application of thermistors?
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