Sensor Trigger

The divider. One loop, one current: I = E/(R₁+R₂+r), and the output is that current through the lower resistor, so Vout = E·R₂/(R₁+R₂+r). As the sensor drifts, the ratio drifts, and the dashed curve is that drift plotted in volts.

The sensors. The thermistor follows R = 10 kΩ·e^(−θ/22 °C) + 150 Ω and the LDR follows R = 1 MΩ·e^(−L/12 %) + 200 Ω. Both fall exponentially, so a steady ramp in the world becomes a curved ramp in volts.

The trip line. A comparator fires the moment Vout crosses the reference V_T, so sliding the line up or down slides the firing instant along the curve — that is exactly how a thermostat’s set screw places its switching point in time. The divider is most sensitive where the fixed resistor matches the sensor, because dVout/dR is largest there.

Lost volts. A tired cell keeps I·r for itself, so r belongs in the denominator. Leave it out and every threshold you compute sits above the voltage the divider can actually reach.

The next step. Hang a capacitor across the output and it charges through the divider’s Thévenin resistance, giving a hold-off time τ = C·R₂(R₁+r)/(R₁+R₂+r) that hides transients shorter than τ. That exponential is worked through in Circuit Calculus.