6.2 Temperature Coefficient of Resistance

Temperature Coefficient of Resistance

In electrotechnical systems, electrical resistance is never strictly constant across operating environments. As an electrical machine, luminaire, or distribution cable carries current, it undergoes thermal changes due to internal $I^2R$ power dissipation and fluctuations in ambient air temperature. The degree to which a material's electrical resistance changes in response to temperature variations depends directly on its atomic structure and is characterized by its temperature coefficient of resistance (symbolized by the Greek letter alpha, $\alpha$).

Understanding how temperature alters resistance is fundamental for practicing electricians. Cable current-carrying capacities, voltage drop thresholds, motor winding diagnostics, and statutory earth fault loop impedance ($Z_s$) verification all depend on precise thermal adjustments.

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Material Classifications: PTC, NTC, and Zero-Coefficient Alloys

Materials fall into three broad categories based on how their electrical resistance responds to temperature:

1. Positive Temperature Coefficient (PTC) — Pure Metals

All pure metals used in electrical installation work—such as copper, aluminium, silver, gold, and tungsten—possess a Positive Temperature Coefficient (PTC). For these materials, an increase in temperature produces an increase in electrical resistance.

• Microscopic Mechanism: In a metallic conductor, valence electrons are delocalized, forming an "electron gas" that drifts through a rigid crystalline lattice of positive atomic ions. At low temperatures, these lattice ions vibrate with relatively small amplitudes around their fixed equilibrium positions, allowing conduction electrons to drift with infrequent collisions (a longer "mean free path"). As temperature rises, thermal energy causes the positive lattice ions to vibrate with greater velocity and larger amplitudes. These intensified vibrations increase the cross-sectional collision frequency between drifting electrons and the lattice, impeding charge drift velocity and raising bulk electrical resistance.

2. Negative Temperature Coefficient (NTC) — Carbon and Semiconductors

Materials with a Negative Temperature Coefficient (NTC) exhibit an inverse response: as temperature increases, electrical resistance decreases.

• Microscopic Mechanism: In materials such as carbon (graphite), semiconductors (silicon and germanium), metallic oxide thermistors, and liquid electrolytes, valence electrons are tightly bound within covalent bonds at room temperature, leaving few charge carriers available in the conduction band. As thermal energy is supplied, covalent bonds are broken, releasing exponential cascades of free electrons and positive holes into the conduction band. This massive increase in the density of available charge carriers far outweighs the minor increase in lattice collision scattering, resulting in a dramatic net decrease in electrical resistance.
• Practical Applications: NTC thermistors are widely utilized as temperature sensors in electronic thermostats, boiler water temperature monitors, digital thermometers, and inrush-current limiters that safely suppress initial current spikes in power supplies.

3. Zero or Negligible Temperature Coefficient Alloys

Specialized metallic alloys have been engineered to exhibit virtually zero change in electrical resistance across wide working temperature ranges:

• Constantan (Eureka): Composed of 55% copper and 45% nickel, Constantan has a temperature coefficient of approximately $\alpha \approx \pm 0.000008\text{ per }^\circ\text{C}$ (essentially zero).
• Manganin: Composed of 84% copper, 12% manganese, and 4% nickel, Manganin exhibits a temperature coefficient of roughly $\alpha \approx \pm 0.00001\text{ per }^\circ\text{C}$.
• Engineering Importance: When constructing precision measurement instruments—such as ammeter shunts, voltmeter multipliers, Wheatstone bridges, and calibrated resistance decade boxes—resistor values must remain invariant regardless of whether ambient room temperature fluctuates or internal current flow generates heat. Using pure copper for an ammeter shunt would cause the meter calibration to drift by roughly 4% for every $10^\circ\text{C}$ rise; zero-coefficient alloys eliminate this thermal error.

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Mathematical Calculation of Temperature Variations

The temperature coefficient of resistance ($\alpha$) is defined as the fractional change in electrical resistance per degree Celsius ($^\circ\text{C}$) change in temperature from a given baseline temperature.

The Reference Temperature Formula

When calculating final resistance from a baseline reference temperature of $0^\circ\text{C}$:

$R_\theta = R_0 (1 + \alpha_0 \theta)$

Where:

• $R_\theta$ = Conductor resistance at operating temperature $\theta$ ($\Omega$)
• $R_0$ = Conductor resistance at $0^\circ\text{C}$ ($\Omega$)
• $\alpha_0$ = Temperature coefficient of the material at $0^\circ\text{C}$ ($\text{per }^\circ\text{C}$)
• $\theta$ = Conductor temperature in degrees Celsius ($^\circ\text{C}$)

The Practical Working Temperature Formula

In electrical contracting, baseline tests are rarely conducted at $0^\circ\text{C}$. Instead, measurements are typically taken at ambient workshop temperature (standardized as $\theta_1 = 20^\circ\text{C}$). To calculate the resistance at a second operating temperature $\theta_2$:

$R_2 = R_1 [1 + \alpha_1 (\theta_2 - \theta_1)]$

Where:

• $R_1$ = Initial resistance at temperature $\theta_1$ ($\Omega$)
• $R_2$ = Final resistance at temperature $\theta_2$ ($\Omega$)
• $\alpha_1$ = Temperature coefficient at initial temperature $\theta_1$ (for copper at $20^\circ\text{C}$, $\alpha_{20} \approx 0.00393\text{ /}^\circ\text{C}$, often rounded to $0.004\text{ /}^\circ\text{C}$)
• $(\theta_2 - \theta_1) = \Delta\theta$ = Temperature rise in degrees Celsius ($^\circ\text{C}$)

Worked Example 1: Contactor Coil Hot Resistance

Problem: The copper field coil of an AC contactor has a resistance of $24.0\ \Omega$ when measured in an unenergized panel at $20^\circ\text{C}$. After six hours of continuous energization under heavy factory load, the coil temperature stabilizes at $70^\circ\text{C}$. Calculate the operating resistance of the coil, taking $\alpha_{20} = 0.00393\text{ /}^\circ\text{C}$.

Solution:

1. Identify known values:
   • $R_1 = 24.0\ \Omega$
   • $\theta_1 = 20^\circ\text{C}$
   • $\theta_2 = 70^\circ\text{C}$
   • $\Delta\theta = 70 - 20 = 50^\circ\text{C}$
   • $\alpha_1 = 0.00393\text{ /}^\circ\text{C}$
2. Substitute into the formula: $R_2 = 24.0 \times [1 + 0.00393 \times 50]$
3. Evaluate bracket terms: $0.00393 \times 50 = 0.1965$ $1 + 0.1965 = 1.1965$
4. Calculate final resistance: $R_2 = 24.0 \times 1.1965 = 28.716\ \Omega \approx 28.72\ \Omega$

Worked Example 2: Temperature Rise Diagnostic in an Electric Motor

Problem: During machine maintenance, an induction motor stator winding has a cold resistance of $4.50\ \Omega$ at an ambient room temperature of $15^\circ\text{C}$. Following a 4-hour full-load thermal run, the hot winding resistance measures $5.58\ \Omega$. Determine the final operating temperature of the winding, taking $\alpha_{15} = 0.0040\text{ /}^\circ\text{C}$.

Solution:

1. Transpose the equation for temperature rise $\Delta\theta$: $R_2 = R_1 [1 + \alpha_1 \Delta\theta] \implies \frac{R_2}{R_1} = 1 + \alpha_1 \Delta\theta \implies \frac{R_2}{R_1} - 1 = \alpha_1 \Delta\theta$ $\Delta\theta = \frac{\frac{R_2}{R_1} - 1}{\alpha_1}$
2. Substitute measured values: $\frac{R_2}{R_1} = \frac{5.58}{4.50} = 1.24$ $\Delta\theta = \frac{1.24 - 1}{0.0040} = \frac{0.24}{0.0040} = 60^\circ\text{C}$
3. Calculate final temperature $\theta_2$: $\theta_2 = \theta_1 + \Delta\theta = 15^\circ\text{C} + 60^\circ\text{C} = 75^\circ\text{C}$

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The BS 7671 Rule of Thumb: The 20% Increase and the 0.8 Factor

In practical UK electrical installation inspection and testing under BS 7671 (IET Wiring Regulations), temperature variations have direct statutory safety implications. This is most critical when verifying earth fault loop impedance ($Z_s$).

The Operational Problem

• Initial Verification Testing: Carried out before an installation is energized and loaded, under cold ambient site conditions—typically assumed to be $20^\circ\text{C}$.
• Full-Load Operating State: When an installation carries design current ($I_b$), continuous $I^2R$ power dissipation heats the copper conductors to their maximum permissible continuous operating temperature:
  • $70^\circ\text{C}$ for standard thermoplastic (PVC) insulated cables.
  • $90^\circ\text{C}$ for thermosetting (XLPE / LSOH) insulated cables.

The 20% Resistance Increase Multiplier

To determine how much copper conductor resistance increases between ambient testing ($20^\circ\text{C}$) and maximum thermoplastic operating temperature ($70^\circ\text{C}$):

$\Delta\theta = 70^\circ\text{C} - 20^\circ\text{C} = 50^\circ\text{C}$ $R_{70} = R_{20} \times [1 + 0.00393 \times 50] = R_{20} \times [1 + 0.1965] = R_{20} \times 1.1965 \approx 1.20 \times R_{20}$

The 0.8 Rule-of-Thumb Factor for Maximum $Z_s$ Verification

To ensure that protective devices (such as BS EN 60898 circuit breakers or BS 88 fuses) disconnect automatically within the required statutory time limits ($0.4\text{ seconds}$ for $230\text{ V}$ final circuits up to $32\text{ A}$), the operating earth fault loop impedance must not exceed the maximum values published in BS 7671 and the IET On-Site Guide ($Z_s \le Z_{s(\text{tabulated})}$).

However, the maximum values published in BS 7671 are calculated for conductors operating at their maximum operating temperature of $70^\circ\text{C}$. If an electrician measures $Z_s$ at $20^\circ\text{C}$, the measured reading will naturally be lower than the impedance during a live fault under load.

To confirm compliance during initial testing at ambient temperatures, the electrician can:

1. Multiply the measured ambient loop impedance by $1.20$ to project its value at $70^\circ\text{C}$: $Z_{s(\text{operating})} = Z_{s(\text{measured})} \times 1.20 \le Z_{s(\text{tabulated})}$
2. Or, divide both sides by $1.20$ to compare the measured value directly against a reduced limit: $Z_{s(\text{measured})} \le \frac{Z_{s(\text{tabulated})}}{1.20} = Z_{s(\text{tabulated})} \times 0.833$

To provide an additional safety margin covering supply voltage variations and earthing conductor variations, BS 7671 and the IET On-Site Guide specify a conservative 0.80 multiplier (known as the "rule-of-thumb factor"):

$Z_{s(\text{measured})} \le 0.80 \times Z_{s(\text{tabulated})}$

If a 32 A Type B MCB has a tabulated maximum $Z_s$ of $1.37\ \Omega$ in BS 7671, the maximum permissible reading on an instrument at $20^\circ\text{C}$ is: $Z_{s(\text{test})} \le 1.37 \times 0.80 = 1.096\ \Omega \approx 1.10\ \Omega$