understand that the resistance of a thermistor decreases as the temperature increases (it will be assumed that thermistors have a negative temperature coefficient)

Resistance and Resistivity

In this section we explore how the intrinsic property of a material – its resistivity – determines the resistance of a component, and how temperature influences resistance, particularly for thermistors with a negative temperature coefficient (NTC).

Key Definitions

• Resistance (\$R\$): The opposition to the flow of electric current, measured in ohms (Ω).
• Resistivity (\$\rho\$): An intrinsic property of a material that quantifies how strongly it resists current flow, measured in Ω·m.
• Conductivity (\$\sigma\$): The reciprocal of resistivity, \$\sigma = 1/\rho\$, measured in siemens per metre (S·m⁻¹).
• Temperature coefficient of resistance (\$\alpha\$): Describes how resistance changes with temperature. Positive for metals, negative for many semiconductors.

Relationship Between Resistance and Resistivity

The resistance of a uniform conductor of length \$L\$ and cross‑sectional area \$A\$ is given by

\$R = \rho \frac{L}{A}\$

From this equation we see that:

• Longer conductors have greater resistance.
• Wider conductors have lower resistance.
• Materials with higher resistivity produce higher resistance for the same geometry.

Typical Resistivity \cdot alues

Temperature Dependence of Resistance

For most conductors (metals) the resistance increases with temperature and can be approximated by

\$R = R0\,[1 + \alpha (T - T0)]\$

where \$R0\$ is the resistance at a reference temperature \$T0\$ and \$\alpha\$ is the positive temperature coefficient.

For many semiconductors, including NTC thermistors, the opposite occurs: resistance falls as temperature rises. The behaviour is exponential rather than linear.

Thermistors – Negative Temperature Coefficient (NTC)

A thermistor is a semiconductor resistor whose resistance varies strongly with temperature. For an NTC thermistor the relationship is commonly expressed as

\$R = R0 \, e^{\beta\left(\frac{1}{T} - \frac{1}{T0}\right)}\$

where:

• \$R0\$ – resistance at the reference temperature \$T0\$ (usually 25 °C or 298 K),
• \$\beta\$ – material‑specific constant (typically 3000–5000 K),
• \$T\$ – absolute temperature in kelvin (K).

Because \$1/T\$ decreases as \$T\$ increases, the exponent becomes more negative, causing \$R\$ to decrease.

Qualitative Behaviour

1. At low temperature the thermistor’s charge carriers are few; resistance is high.
2. Heating the thermistor provides energy to promote electrons across the band gap, increasing carrier concentration.
3. More carriers mean a larger current for a given voltage, so the measured resistance drops.

Practical Example

Suppose an NTC thermistor has \$R0 = 10\ \text{k}\Omega\$ at \$T0 = 298\ \text{K}\$ and \$\beta = 3500\ \text{K}\$. Find its resistance at \$T = 350\ \text{K}\$.

Solution:

\$R = 10\,000\ \Omega \times e^{3500\left(\frac{1}{350} - \frac{1}{298}\right)}\$

\$\frac{1}{350} - \frac{1}{298} = 0.002857 - 0.003356 = -0.000499\ \text{K}^{-1}\$

\$R = 10\,000\ \Omega \times e^{3500 \times (-0.000499)} = 10\,000\ \Omega \times e^{-1.7465}\$

\$R \approx 10\,000\ \Omega \times 0.174 \approx 1.74\ \text{k}\Omega\$

The resistance has fallen to roughly 17 % of its value at 25 °C, illustrating the strong negative temperature coefficient.

Key Take‑aways

• Resistance depends on both geometry (length, area) and material resistivity.
• Resistivity is temperature‑dependent; for metals it rises with temperature, for many semiconductors it falls.
• NTC thermistors exhibit an exponential decrease in resistance as temperature increases, described by the \$\beta\$‑parameter equation.
• Understanding this behaviour is essential for designing temperature‑sensing circuits and for compensating temperature effects in electronic devices.