Derivations
You can skip this section if you'd like. It simply explains the math behind deriving the resistance from the ADC value.
ADC to Voltage
The formula to convert an ADC value to voltage is:
\[ V_{\text{out}} = {{V_{in}}} \times \frac{\text{adc_value}}{\text{adc_max}} \]
Where:
- adc_value: The value read from the ADC.
- v_in: The reference input voltage (3.3V for the Pico).
- adc_max: The maximum ADC value is 4095 (\( 2^{12}\) -1 ) for a 12-bit ADC.
Deriving Resistance from ADC Value
We combine the voltage divider formula with ADC Resolution formula to find the Resistance(R2).
Note: It is assumed here that one side of the thermistor is connected to Ground (GND). I noticed that some online articles do the opposite, connecting one side of the thermistor to the power supply instead, which initially caused me some confusion.
Votlage Divider Formula \[ V_{out} = V_{in} \times \frac{R_2}{R_1 + R_2} \]
Step 1:
We can substitue the Vout and make derive it like this
\[ {V_{in}} \times \frac{\text{adc_value}}{\text{adc_max}} = V_{in} \times \frac{R_2}{R_1 + R_2} \]
\[ \require{cancel} \cancel{V_{in}} \times \frac{\text{adc_value}}{\text{adc_max}} = \cancel {V_{in}} \times \frac{R_2}{R_1 + R_2} \]
Step 2:
Lets temperoarily assign the adc_value/adc_max to x for ease of derivation and finally subsitue
\[ x = \frac{\text{adc_value}}{\text{adc_max}} \]
Substituting x into the equation:
\[ x = \frac{R_2}{R_1 + R_2} \]
Rearrange to Solve \( R_2 \)
\[ R_2 = x \times (R_1 + R_2) \]
Expand the right-hand side:
\[ R_2 = x \times R_1 + x \times R_2 \]
Rearrange to isolate \( R_2 \) terms:
\[ R_2 - x \times R_2 = R_1 \times x \]
\[ R_2 \times (1 - x) = R_1 \times x \]
\[ R_2 = R_1 \times \frac{{x}}{{1-x}} \]
\[ R_2 = R_1 \times \frac{1}{\left( \frac{1}{x} - 1 \right)} \]
Step 3
Let's subsitute the x value back. We need 1/x, lets convert it. \[ \frac{1}{x} = \frac{\text{adc_max}}{\text{adc_value}} \]
Final Formula
\[ R_2 = R_1 \times \frac{1}{\left( \frac{\text{adc_max}}{\text{adc_value}} - 1 \right)} \]