The Divider and ADC Numbers

Ada re-derives this chapter’s own numbers step by step, at full precision

ADA · CALCULATION AUDIT

The Divider and ADC Numbers

Voltage, code, settling, and current, ~4 minutes

A divider reading is trustworthy only when the ratio, ADC code, source impedance, and battery current all tell the same story.

A divider built from two 220 kΩ legs looks like 110 kΩ at the ADC pin, and against a 5 pF sample capacitor that gives a 0.55 us time constant — needing roughly 5 us to settle, yet the chapter warns that an ADC sampling for only about 1 us leaves the capacitor short of the true divider voltage. Swapping to 10 kΩ legs shrinks that same settling need to 225 ns. This audit asks the question that trade-off invites: exactly how much power does the low-impedance divider cost against the high-impedance one, once the settling and current numbers are both carried through?

Companion to the chapter Resistive Dividers and ADC Loading — every number here comes from that chapter.

1. The 10 kΩ thermistor example is a ratio before it is a voltage

With a 10 kΩ fixed top resistor, a 10 kΩ bottom thermistor, and a 3.3 V rail, the midpoint sits halfway up the supply:

Vout = 3.3 × 10k / (10k + 10k) = 1.65 V

12-bit code = 1.65 / 3.3 × 4095 = 2047.5, about 2048 counts

When the bottom thermistor is 4.0 kΩ, the same equation gives the lower warm reading used above:

Vout = 3.3 × 4.0k / (10k + 4.0k) = 0.943 V

12-bit code = 0.943 / 3.3 × 4095, about 1170 counts; separation = 2048 − 1170 = 878 counts

2. The first warm-threshold example keeps its decision margin visible

For the 5 kΩ warm point and a threshold near 1500 counts:

Check	Arithmetic shown	Audit result
Warm voltage	3.3 × 5k / (10k + 5k)	1.10 V
Warm code	1.10 / 3.3 × 4095	About 1365 counts
Cool-side margin	2048 − 1500	548 counts
Warm-side margin	1500 − 1365	135 counts

3. The ADC loading example is a time-constant audit

The high-value divider saves current but gives the ADC a high source resistance:

Rsource = 220k || 220k = 110kΩ; tau = 110,000 × 5 pF = 0.55 us; 9 tau = 4.95 us, about 5 us

Rsource = 10k || 10k = 5kΩ; tau = 5,000 × 5 pF = 25 ns; 9 tau = 225 ns

The battery trade-off is the other side of the same choice: 3.3 V / (10k + 10k) = 165 uA, while 3.3 V / (220k + 220k) = 7.5 uA. The 220 kΩ pair draws about 165 / 7.5 = 22 times less current, but it needs a longer sample window, a buffer capacitor, or a buffer amplifier.

Every number above is taken from the chapter’s own divider and ADC examples and re-derived step by step.