Question

The relation between resistance \(R\) and temperature \(T\) for a thermistor closely follows $$ R=R_{0} \exp \left[\beta\left(\frac{1}{T}-\frac{1}{T_{0}}\right)\right] $$ where \(R_{0}\) is the resistance, in ohms \((\Omega)\), measured at temperature \(T_{0}(\mathrm{~K})\) and \(\beta\) is a material constant with units of \(\mathrm{K}\). For a particular thermistor \(R_{0}=2.2 \Omega\) at \(T_{0}=310 \mathrm{~K}\). From a calibration test, it is found that \(R=0.31 \Omega\) at \(T=422 \mathrm{~K} .\) Determine the value of \(\beta\) for the thermistor and make a plot of resistance versus temperature.

Short answer

\( \beta \approx 2276 \, K \). Use the formula to plot resistance versus temperature.

Step-by-step solution

Write down the given information

The given information includes: \(R = 0.31 \,\Omega\) at \(T = 422 \, K\), \(R_0 = 2.2 \,\Omega\) at \(T_0 = 310 \, K\). The relation between resistance and temperature is given by the equation\[ R = R_0 \exp \left[ \beta \left( \frac{1}{T} - \frac{1}{T_0} \right) \right] \]

Substitute the known values into the equation

Substitute \(R = 0.31 \,\Omega\), \(R_0 = 2.2 \,\Omega\), \(T = 422 \, K\), and \(T_0 = 310 \, K\) into the equation:\[ 0.31 = 2.2 \exp \left[ \beta \left( \frac{1}{422} - \frac{1}{310} \right) \right] \]

Simplify the expression inside the exponent

Calculate \( \frac{1}{422} - \frac{1}{310} \):\[ \frac{1}{422} \approx 2.37 \times 10^{-3} \]\[ \frac{1}{310} \approx 3.23 \times 10^{-3} \]Subtract the smaller value from the larger value:\[ 2.37 \times 10^{-3} - 3.23 \times 10^{-3} = -0.86 \times 10^{-3} \]

Isolate \( \beta \) and solve

Using the simplified exponent, rewrite the equation:\[ 0.31 = 2.2 \exp \left( -0.86 \times 10^{-3} \beta \right) \]Divide both sides by 2.2:\[ \frac{0.31}{2.2} = \exp \left( -0.86 \times 10^{-3} \beta \right) \]\[ 0.1409 \approx \exp \left( -0.86 \times 10^{-3} \beta \right) \]Take the natural logarithm on both sides:\[ \ln(0.1409) = -0.86 \times 10^{-3} \beta \]\[ -1.958 = -0.86 \times 10^{-3} \beta \]Solve for \( \beta \):\[ \beta = \frac{1.958}{0.86 \times 10^{-3}} \approx 2276 \, K \]

Plot the resistance versus temperature

For plotting, the formula \( R = 2.2 \exp \left[ 2276 \left( \frac{1}{T} - \frac{1}{310} \right) \right] \) is used. Create a table of values for \(T\) ranging from some minimum to maximum temperatures. Calculate corresponding \(R\) values, plot these on a graph, and draw the curve.

Key concepts

thermistor

A thermistor is a type of resistor whose resistance changes significantly with temperature. Unlike standard resistors, which have very predictable and stable resistance over a range of temperatures, thermistors are designed to be sensitive to temperature changes. There are two main types of thermistors:

• NTC (Negative Temperature Coefficient) Thermistors: Their resistance decreases as temperature increases.
• PTC (Positive Temperature Coefficient) Thermistors: Their resistance increases as temperature increases.

The relationship between resistance and temperature in a thermistor is more complex than in standard resistors and often follows an exponential relationship, which we will discuss further.

temperature dependence

The resistance of a thermistor depends on its temperature. This temperature dependence is typically modeled using an empirical formula that shows how resistance varies with temperature. For many thermistors, the resistance as a function of temperature can be expressed in an exponential form. This allows for precise temperature measurements based on the resistance value. Understanding this temperature dependence is crucial for applications in temperature sensing and circuit protection, where thermistors are commonly used.

The exponential function captures the rapid change in resistance with small changes in temperature, which makes thermistors extremely sensitive and useful in many practical applications.

resistance calculation

Calculating the resistance of a thermistor at a specific temperature involves using the exponential relationship between resistance and temperature. Given the equation:
\( R = R_0 \exp\left[ \beta \left( \frac{1}{T} - \frac{1}{T_0} \right) \right] \)
where:

• \(R_0\) is the resistance at a reference temperature \(T_0\)
• \(T\) is the temperature at which we want to find the resistance
• \(\beta\) is a material-specific constant

To find \( \beta\), we can rearrange and simplify the equation by substituting known resistance and temperature values.
By applying logarithmic functions and algebraic manipulation, we isolate and solve for the variable of interest, in this case, \( \beta\). This calculated value then allows us to predict the thermistor's behavior at other temperatures.

exponential relationship

The exponential relationship in a thermistor means that resistance changes exponentially with temperature. This is represented by the equation:
\( R = R_0 \exp\left[ \beta \left( \frac{1}{T} - \frac{1}{T_0} \right) \right] \)
Breaking down the equation:

• \( R_0 \) is a baseline resistance at a specific temperature \( T_0 \).
• \( \beta \) is a material constant that determines how much the resistance changes with temperature.
• The term \( \frac{1}{T} - \frac{1}{T_0} \) inside the exponent captures the inverse relationship between temperature and resistance.

Because of the exponent, even small changes in temperature can lead to large changes in resistance. This makes thermistors extremely sensitive and useful for precise temperature measurements. Understanding and using the exponential relationship is key to effectively employing thermistors in real-world applications.