Math Bridge: The I2C Pull-Up Window

← Back to Wired Communication Fundamentals
Math BridgeCore NetworkingStruggle-friendly runway

Why is an I2C pull-up a window, not a favourite value?

One thread from an open-drain line to the current floor and RC timing ceiling.

Pete, the packet guidePete guides
The one targetFind when no pull-up resistor can satisfy both limits.
The chapter case3.3 V, 0.4 V, 3 mA, 400 pF; 1,000 ns and 300 ns.
What it buys youReject a copied 4.7 kΩ value when the bus changes.

A field team has a real problem to settle: Why is an I2C pull-up a window, not a favourite value? They must decide what happens before they change i2c bus capacitance on the device. Predict the direction first.

See the relationship first

The figure reads from left to right. The blue card is i2c bus capacitance. The middle card uses this page's rule. The green card is standard-mode ceiling. Follow the arrows: set the input, use the rule, then read the result and its unit.

The audit later on checks more than one number. Here, the added model uses the baseline named below and holds every other chapter value fixed. That sentence bridges the fixtures, so the numbers do not change without a reason.

I2C bus capacitance changes standard-mode ceiling An input card leads through the page rule to the standard-mode ceiling result. SET INPUT ONE CONTROL USE RULE predict calculate check units READ RESULT
Follow the arrows. Capacitance changes timing, not the sink-current guarantee. That is why adding cable or devices can invalidate the same resistor.

Derive the baseline in four moves

  1. 1

    Name the input. The chapter baseline for i2c bus capacitance is 400.

  2. 2

    Name the rule. Rmin=(3.3-0.4)/0.003; Rmax=tr/(Cb ln9); tr,4.7k=4.7k·Cb·ln9

  3. 3

    Put in the chapter value. Set i2c bus capacitance to 400. The page rule gives standard-mode ceiling as 1138 ohm.

  4. 4

    Read the result. Keep ohm next to the value. Use it only within the limits on this page.

Predict, then change i2c bus capacitance

Try Predict what happens to standard-mode ceiling. Move one control, calculate, then check your idea.

400
Chapter baseline
Standard-mode ceiling

Observe Capacitance changes timing, not the sink-current guarantee. That is why adding cable or devices can invalidate the same resistor. Reset to 400 and compare standard-mode ceiling.

Explain Only i2c bus capacitance moves here. The other chapter values stay fixed.

Check yourself

What should you do before you trust the result?
Answer: Predict its direction, use the shown rule, keep the units, and reset to the worked baseline.
What does this small model leave out?
Answer: Only i2c bus capacitance moves. Field effects named in the page limits stay fixed.

Honesty note. This teaching model changes only i2c bus capacitance. It replays the chapter baseline; it is not a field prediction.

1. Devices pull low; the resistor pulls high

An open-drain output actively sinks current to make zero. It releases the line to make one, and the pull-up resistor then charges the bus capacitance toward the supply.

Pete: Smaller resistance rises faster but demands more sink current. Larger resistance saves current but rises more slowly.

2. Derive the two bounds

1

Current floorRmin=(VDD−VOL)/IOL keeps the low-state sink within its guarantee.

2

RC edgeV(t)=VDD(1−e^(−t/RC)).

3

Rise ceilingThe 10–90% rise time is RC ln9, so Rmax=tr,max/(C ln9).

3. A valid bus needs an overlap

Rmin=(VDD−VOL)/IOL; Rmax=tr,max/(Cb ln9); valid only if Rmin≤Rmax

If the timing ceiling falls below the current floor, changing to another ordinary resistor cannot fix the bus. Reduce capacitance, slow the mode, or use a stronger compliant driver or active pull-up design.

4. Try the bus capacitance

Rmin=(3.3−0.4)/0.003; Rmax=tr/(Cb ln9); tr,4.7k=4.7k·Cb·ln9

TryMove the bus capacitance. The current floor stays fixed while both timing ceilings shrink.

Current-safe floor
Standard-mode ceiling
Fast-mode ceiling
4.7 kΩ time constant
4.7 kΩ rise time
Fast-mode overlap?

ObserveAt 400 pF, Rmin≈967 Ω. Standard-mode allows only about 1.14 kΩ, while Fast-mode caps resistance near 341 Ω, below the current floor. A 4.7 kΩ pull-up rises in about 4.13 µs.

ExplainCapacitance changes timing, not the sink-current guarantee. That is why adding cable or devices can invalidate the same resistor.

Technical boundaries.

The values are the chapter's explicitly catalog-typical I2C example.

Real compliance uses device-specific VOL/IOL curves, input thresholds, distributed capacitance, leakage, level shifters, clock stretching, temperature, tolerances, probe loading, and the exact bus specification
Needs separate evidence

Use field evidence or a deeper model before release.

5. Reproduce the chapter values

(3.3−0.4)/0.003=967 Ω. At 400 pF, 1,000 ns/(400 pF·ln9)≈1,140 Ω and 300 ns/(400 pF·ln9)≈341 Ω. For 4.7 kΩ, τ=1.88 µs and tr=τln9≈4.13 µs.

6. Prove the installed edge

Record actual pull-ups, device sink guarantees, bus voltage, estimated and measured capacitance, cable and connector, level shifters, target mode, scope rise time at the worst node, low-level voltage, retries, and temperature range.

7. Check yourself

Why is Rmin a lower bound?
Answer: A still smaller resistor would demand more low-state sink current than the stated 3 mA guarantee.
Why does more capacitance lower Rmax?
Answer: Rise time is proportional to RC, so a larger C needs a smaller R to meet the same time budget.
Can continuity testing prove this bus?
Answer: No. A connected wire can still have an invalid low voltage or a rise time too slow for the selected mode.
Honesty boundary.

These are the chapter inputs, worked results, and named teaching assumptions.

3.3 V
Voltage or voltage-step value
0.4 V
Voltage or voltage-step value
3 mA
Current or responsivity value
400 pF
Capacitance value
100 kHz/1,000 ns
Frequency, sample rate, or event rate
400 kHz/300 ns
Frequency, sample rate, or event rate
967 Ω
Resistance or impedance value
1,140 Ω
Resistance or impedance value
341 Ω
Resistance or impedance value
4.7 kΩ
Resistance or impedance value
1.88 µs
Time, interval, or service-life value
4.13 µs
Time, interval, or service-life value

They are not universal component limits.

Return to wired fundamentals with the maths unpacked →