Math Bridge: Stuck-Bus Fault Resistance

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Math BridgeSensorsStruggle-friendly runway

Turn a stuck-bus voltage into a fault resistance

One thread from a loaded divider to the chapter's half-rail clue and logic-HIGH boundary.

Phoebe, the physics guidePhoebe guides
The one targetEstimate leakage resistance from idle bus voltage.
The chapter case3.3 V, 4.7 kΩ, and a 1.65 V reading.
What it buys youDistinguish leakage from a hard short.

A field team faces an unresolved physical question: Turn a stuck-bus voltage into a fault resistance They must answer it before changing measured i2c idle voltage on the real device. Predict the direction first.

See the relationship before changing it

Set the input, apply the rule, then read the result with its unit.

The retained audit below checks several chapter fixtures. This added model holds every other chapter fixture fixed, so the numeric fixture does not switch without explanation.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for measured i2c idle voltage is 1.65.

  2. 2

    Name the relationship. Rf=RpuVbus/(VDD-Vbus); VIH=0.7VDD

  3. 3

    Substitute the chapter fixture. Set measured i2c idle voltage to 1.65. The page ledger gives estimated fault resistance as 4.70 kohm.

  4. 4

    Read the result. Keep kohm beside the value. Use it only inside the technical boundary on this page.

Predict, then change measured i2c idle voltage

Try Predict the direction of estimated fault resistance. Move one control, calculate, then check your prediction.

1.65
Chapter baseline
Estimated fault resistance

Observe Predict an increase in estimated fault resistance as the input increases over this range. Reset to the chapter baseline and compare.

Explain Only measured i2c idle voltage moves here. The other chapter fixtures remain fixed.

Source: content/sensors/sensor-interfacing-protocol-diagnostics.qmd — Start With the Measurement Story. Use the chapter equation and fixed ledger above; only the named input varies.

Check yourself

What should you do before trusting a moved-control result?
Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.
What does this small model leave out?
Answer: Only measured i2c idle voltage moves. Field effects named in the page's technical boundary stay fixed.

Honesty note. This teaching model changes only measured i2c idle voltage. It replays the chapter fixture; it is not a field prediction.

1. Replace “stuck” with a circuit

Moisture, damaged insulation, or a marginal input can behave like a resistance from SDA or SCL to ground. Together with the pull-up, it forms a voltage divider.

Phoebe: A multimeter gives more than HIGH or LOW. A middle voltage is a clue about the unintended path.

2. Write the loaded divider

Vbus=VDD Rf/(Rpu+Rf)

Rf is the unwanted path to ground. Rpu is the known pull-up to VDD.

1

Cross-multiplyVbus(Rpu+Rf)=VDD Rf.

2

Collect RfVbusRpu=Rf(VDD−Vbus).

3. Solve in the diagnostic direction

Rf=Rpu Vbus/(VDD−Vbus)

When Vbus is half of VDD, numerator and remaining-voltage denominator contain the same voltage, so Rf=Rpu. Near zero volts, Rf approaches a hard short; near VDD, it approaches an open circuit.

4. Try the measured voltage

Rf=RpuVbus/(VDD−Vbus); VIH=0.7VDD

TryMove the idle voltage from near ground toward the 3.3 V rail.

Estimated fault resistance
Logic-HIGH threshold
Fault at HIGH boundary

ObserveAt 1.65 V, the inferred fault is 4.70 kΩ—the same as the pull-up.

ExplainHalf the rail means equal divider resistances. The 2.31 V HIGH threshold corresponds to about 10.97 kΩ.

Technical boundaries.

The fault estimate fixes the 3.3 V rail and 4.7 kohm pull-up and treats the idle bus as one static resistance, with no device deliberately pulling low. Meter loading, intermittent contamination or contamination that does not behave like a fixed resistor, protection clamps, address conflicts, and edge timing require separate diagnosis.

Real diagnosis must isolate powered devices, confirm the pull-up and rail, consider meter loading, intermittent/non-ohmic contamination, ESD clamps, address conflicts, and edge-time faults with a scope
Needs separate evidence

Use field evidence or a deeper model before release.

5. Work the half-rail clue

Rf=4,700×1.65/(3.3−1.65)=4,700 Ω

That is not a hard short. It is a leakage path comparable with the pull-up.

6. Find the invalid-HIGH boundary

1

Threshold voltageVIH=0.7×3.3=2.31 V.

2

Back-solveRf=4,700×2.31/(3.3−2.31)=10,967 Ω≈11.0 kΩ.

A leakage path below roughly 11 kΩ can prevent a valid HIGH even though it is thousands of ohms away from a dead short.

7. Check yourself

What does half-rail voltage imply?
Answer: In this two-resistor model, the fault resistance equals the pull-up resistance.
What is 0.7×3.3 V?
Answer: The chapter's HIGH threshold is 2.31 V.
Does this model diagnose a missing pull-up?
Answer: No. It assumes the pull-up exists and models an added resistive path to ground.
Honesty boundary.

The fault estimate fixes the 3.3 V rail and 4.7 kohm pull-up and treats the idle bus as one static resistance, with no device deliberately pulling low. Meter loading, intermittent contamination or contamination that does not behave like a fixed resistor, protection clamps, address conflicts, and edge timing require separate diagnosis.

3.3 V rail
Bus supply voltage
4.7 kΩ pull-up
Bus pull-up resistance
1.65 V reading
Measured idle bus voltage
4.70 kΩ inference
Inferred resistance pulling bus toward ground
0.7VDD HIGH threshold
Minimum logic-high voltage as a supply multiple
2.31 V
Minimum accepted logic-high voltage
10,967 Ω
Fault resistance at logic-high threshold
11.0 kΩ boundary come from the chapter
Rounded fault resistance at logic-high threshold

The divider is a first diagnostic model, not proof of which physical component failed.

Return to the chapter with the maths unpacked →