B Equation

The B equation, also called the Beta parameter method, is the simplest way to convert thermistor resistance into temperature. It uses a single material constant provided by the manufacturer to approximate the thermistor’s behavior.

The B equation formula

\[ \frac{1}{T} = \frac{1}{T_0} + \frac{1}{B} \ln \left( \frac{R}{R_0} \right) \]

In this equation, T is the temperature in Kelvin that we want to find. R is the measured resistance at the unknown temperature T.

\( T_0 \) is the reference temperature, usually 298.15K (25°C), where the thermistor’s resistance is known. \( R_0 \) is the resistance at the reference temperature \( T_0 \), often 10kΩ for common thermistors.

B is the B-value of the thermistor, a material constant. And, the ln represents the natural logarithm function.

Understanding the parameters

R₀ (Reference resistance):

This is the thermistor’s resistance at a known temperature, usually 25°C. For example, a “10kΩ thermistor” has R₀ = 10,000 ohms at 25°C. This value should be specified in the datasheet.

B (Beta value):

The B value is a constant usually provided by the manufacturers. The value describes how quickly resistance changes with temperature. Common values range from 3000 to 4000 K.

The datasheet often specifies B over a temperature range, such as B₂₅/₈₅ = 3950, meaning the Beta value is 3950 between 25°C and 85°C.

Temperature in Kelvin:

The temperature in the B equation must be in Kelvin (Kelvin = Celsius + 273.15), not Celsius. Convert to Kelvin before using the equation, and subtract 273.15 from the result to get Celsius again.

Example Calculation

Let us say the measured resistance of the thermistor is 10,475 Ω, which corresponds to R in the equation. To calculate the temperature, we substitute this value along with the reference temperature \( T_0 = 298.15K \) (25°C), the reference resistance \( R_0 = 10k\Omega \), and the thermistor B-value of 3950 into the B equation.

Step 1: Calculate the resistance ratio \[ \frac{R}{R_0} = \frac{10475}{10000} = 1.0475 \]

Step 2: Calculate the natural logarithm of the ratio \[ \ln(1.0475) = 0.04641 \]

Step 3: Apply the B equation \[ \frac{1}{T} = \frac{1}{298.15} + \frac{1}{3950} \times 0.04641 \]

\[ \frac{1}{T} = 0.003354 + 0.00001175 = 0.003366 \]

Step 4: Calculate T by taking the reciprocal \[ T = \frac{1}{0.003366} = 297.1K \]

Step 5: Convert to Celsius \[ T = 297.1 - 273.15 = 23.95°C \]

The measured resistance of 10,475Ω corresponds to a temperature of approximately 24°C.

Rust function

fn calculate_temperature(current_res: f64, ref_res: f64, ref_temp: f64, b_val: f64) -> f64 {
    let ln_value = (current_res/ref_res).ln();
    // let ln_value = libm::log(current_res / ref_res); // use this crate for no_std
    let inv_t = (1.0 / ref_temp) + ((1.0 / b_val) * ln_value);
    1.0 / inv_t
}

fn kelvin_to_celsius(kelvin: f64) -> f64 {
    kelvin -  273.15
}

fn celsius_to_kelvin(celsius: f64) -> f64 {
    celsius + 273.15
}

const B_VALUE: f64 = 3950.0;
const V_IN: f64 = 3.3; // Input voltage
const REF_RES: f64 = 10_000.0; // Reference resistance in ohms (10kΩ)
const REF_TEMP: f64 = 25.0;  // Reference temperature 25°C

fn main() {
    let t0 = celsius_to_kelvin(REF_TEMP);
    let r = 10475.0; // Measured resistance in ohms
    
    let temperature_kelvin = calculate_temperature(r, REF_RES, t0, B_VALUE);
    let temperature_celsius = kelvin_to_celsius(temperature_kelvin);
    println!("Temperature: {:.2} °C", temperature_celsius);
}