What is the specific formula for the relationship between the resistivity of carbon brush leads and temperature?
Carbon brush leads are often made of metal materials (such as copper, copper alloys, etc.), and the relationship between their resistivity and temperature follows the temperature characteristics of metal resistance. The following empirical formula can usually be used to describe it:
ρₜ = ρ₀[1 + α(t - t₀)]
Among them:
ρₜ represents the resistivity at temperature t (unit: Ω· m);
ρ₀ represents the resistivity at the reference temperature t ₀ (usually taken as t ₀=20 ℃, unit: Ω· m);
α is the temperature coefficient of resistance (unit: ℃⁻¹), which reflects the rate of change of material resistivity with temperature. The α value of different metal materials is different. For example, the α value of pure copper at 20 ℃ is about 0.00393 ℃⁻¹, and the α value of copper alloys may vary slightly due to differences in composition;
T is the actual temperature (unit: ℃);
T ₀ is the reference temperature (usually 20 ℃).
This formula is applicable to the conventional operating temperature range of carbon brush leads (generally between -50 ℃ and 150 ℃), where the metal has not undergone significant phase transitions or structural changes, and the resistivity changes approximately linearly with temperature. When the temperature exceeds this range, the error of the formula may increase due to changes in the internal crystal structure of the metal, and it needs to be corrected based on experimental data of specific materials.
For example, if it is known that the resistivity of a copper carbon brush lead at 20 ℃ is 1.72 × 10 ⁻⁸Ω· m, when the temperature rises to 50 ℃, its resistivity can be calculated according to the formula as ρₜ=1.72 × 10 ⁻⁸× [1+0.00393 × (50-20)] ≈ 1.72 × 10 ⁻⁸× 1.1179 ≈ 1.92 × 10 ⁻⁸Ω· m, which has a small deviation from the actual measurement value and can meet the estimation requirements in engineering applications.
ρₜ = ρ₀[1 + α(t - t₀)]
Among them:
ρₜ represents the resistivity at temperature t (unit: Ω· m);
ρ₀ represents the resistivity at the reference temperature t ₀ (usually taken as t ₀=20 ℃, unit: Ω· m);
α is the temperature coefficient of resistance (unit: ℃⁻¹), which reflects the rate of change of material resistivity with temperature. The α value of different metal materials is different. For example, the α value of pure copper at 20 ℃ is about 0.00393 ℃⁻¹, and the α value of copper alloys may vary slightly due to differences in composition;
T is the actual temperature (unit: ℃);
T ₀ is the reference temperature (usually 20 ℃).
This formula is applicable to the conventional operating temperature range of carbon brush leads (generally between -50 ℃ and 150 ℃), where the metal has not undergone significant phase transitions or structural changes, and the resistivity changes approximately linearly with temperature. When the temperature exceeds this range, the error of the formula may increase due to changes in the internal crystal structure of the metal, and it needs to be corrected based on experimental data of specific materials.
For example, if it is known that the resistivity of a copper carbon brush lead at 20 ℃ is 1.72 × 10 ⁻⁸Ω· m, when the temperature rises to 50 ℃, its resistivity can be calculated according to the formula as ρₜ=1.72 × 10 ⁻⁸× [1+0.00393 × (50-20)] ≈ 1.72 × 10 ⁻⁸× 1.1179 ≈ 1.92 × 10 ⁻⁸Ω· m, which has a small deviation from the actual measurement value and can meet the estimation requirements in engineering applications.
