Math Bridge: The I2C LOW-Level Divider

A field team has a real problem to settle: Why does the I2C pull-up floor equal 967 ohm? They must decide what happens before they change pull-up resistance on the device. Predict the direction first.

See the relationship first

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.

Derive the baseline in four moves

1
Name the input. The chapter baseline for pull-up resistance is 4700.
2
Name the rule. Ron=0.4/0.003=133 ohm; Rpu,min=(3.3-0.4)/0.003=967 ohm
3
Put in the chapter value. Set pull-up resistance to 4700. The page ledger gives predicted low as 0.091 V.
4
Read the result. Keep V beside the value. Use it only within the limits on this page.

Predict, then change pull-up resistance

Try Predict what happens to predicted low. Move one control, calculate, then check your idea.

Observe Predict a decrease in predicted low as the input increases over this range. Reset to the chapter baseline and compare.

Explain Only pull-up resistance moves here. The other chapter values stay fixed.

Source: content/networking-core/wired-comm-i2c-transactions.qmd — Phoebe's Field Notes: The Voltage Divider Hiding Inside The Pull-Up Spec. Use the chapter equation and fixed ledger above; only the named input varies.

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 pull-up resistance moves. Field effects named in the page limits stay fixed.

1. Open drain means the resistor completes the circuit

The device pulls SDA or SCL LOW through a transistor; the external resistor pulls it HIGH when every transistor releases. During LOW, both components carry the same current in series.

Pete: Keep the units and the assumptions beside every number.

2. Name every algebra move

Infer transistor resistanceRon,max=VOL,max/IOL,max.

Write the dividerVOL=VDD·Ron/(Rpu+Ron).

Solve the floorRpu,min=(VDD−VOL,max)/IOL,max.

3. The Ohm-law floor and divider are identical

Ron=0.4/0.003=133 Ω; Rpu,min=(3.3−0.4)/0.003=967 Ω

At the worst-case values, 967 Ω plus 133 Ω carries 3 mA and leaves exactly 0.4 V across the transistor. Larger pull-ups reduce sink current and LOW voltage but make the RC rise slower.

4. Try one controlled change

Ron=0.4/0.003=133 Ω; Rpu,min=(3.3−0.4)/0.003=967 Ω

TryMove the pull-up from the 967 Ω floor toward the common 4.7 kΩ value.

ObserveAt 4.7 kΩ, the worst-case divider predicts about 0.091 V LOW, roughly 0.309 V below the 0.4 V limit.

ExplainIncreasing Rpu helps the LOW-level current ledger, but another formula—the RC rise-time ceiling—eventually limits how large it may be.

Technical boundaries.

Ron is an equivalent worst-case model, not a guaranteed constant.

device VOL/IOL curves: Sink-characteristic evidence
temperature, supply, and leakage: Operating-corner evidence
bus capacitance and pull-up tolerance: Rise-time evidence
level shifters and speed-mode timing: Topology and protocol evidence

All need datasheet and oscilloscope evidence.

5. Reproduce the chapter values

Ron,max=0.4/0.003=133 Ω. At Rpu=967 Ω, VOL=3.3×133/(967+133)=0.400 V. At 4.7 kΩ, the same model gives about 0.091 V, more than four times below the limit.

6. Carry the evidence forward

Record VDD, every device’s VOL/IOL rating, pull-up tolerance, measured bus capacitance, speed mode, rise/fall time, leakage, level shifters, temperature range, and oscilloscope captures at the furthest devices.

7. Check yourself

Why is 967 Ω a minimum?

Answer: A smaller resistor would demand more than 3 mA at the 0.4 V limit.

Does 4.7 kΩ always pass?

Answer: Not automatically; LOW level improves, but RC rise time can fail on a capacitive or fast bus.

Is Ron really fixed at 133 Ω?

Answer: No. It is the equivalent resistance implied by the worst guaranteed datasheet point.

Honesty boundary.

These are the worked values and named assumptions for this bridge.

3.3 V: Bus supply
0.4 V: LOW-voltage limit
3 mA: Sink-current limit
133 Ω: Equivalent worst-case Ron
967 Ω: Pull-up floor
4.7 kΩ: Worked pull-up
0.091 V: Predicted LOW
0.309 V: LOW margin

Ron is an equivalent worst-case model, not a guaranteed constant. Device VOL/IOL curves, temperature, supply, leakage, bus capacitance, pull-up tolerance, level shifters, and speed-mode timing all need datasheet and oscilloscope evidence.