Math Bridge: LoRa Spreading Factor and Link Budget

A field team faces an unresolved physical question: How does a higher spreading factor buy budget without adding power? They must answer it before changing spreading factor on the real device. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is spreading factor. The middle card applies this page's relationship. The green card is chips per symbol. Walk the arrows once: 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.

Walk the arrows. Processing gain lowers the detectable signal floor; it does not create RF power, and the longer symbol raises airtime and capacity costs.

Derive the baseline in four named moves

1
Name the input. The chapter baseline for spreading factor is 12.
2
Name the relationship. At SF10: 2¹⁰ = 1024 chips and Ts = 1024/125000 = 8.192 ms raw budget = 14 + 2 + 6 - 3 - (-130) = 149 dB allowable loss = 149 - (10 + 12 + 4) = 123 dB At SF12: relative ideal gain = 10 log10(4096/1024) = 6.02 dB
3
Substitute the chapter fixture. Set spreading factor to 12. The page ledger gives chips per symbol as 4096.
4
Read the result. Keep the stated output unit beside the value. Use it only inside the technical boundary on this page.

Predict, then change spreading factor

Try Predict the direction of chips per symbol. Move one control, calculate, then check your prediction.

Observe Processing gain lowers the detectable signal floor; it does not create RF power, and the longer symbol raises airtime and capacity costs. Reset the control to 12 and compare chips per symbol.

Explain Only spreading factor moves here. The other chapter fixtures remain fixed.

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 spreading factor moves. Field effects named in the page's technical boundary stay fixed.

1. Start with the physical story

A higher spreading factor uses more chips for each symbol. The receiver combines more observations, improving its decision against noise, while the transmitter power stays unchanged and the symbol lasts longer.

Eddie: Name the physical limit first; the algebra then has one honest job.

2. Name every algebra move

Count chipsRaise two to the spreading factor.

Find relative gainTake 10 log10 of the chip-count ratio to SF10.

Move the receiver floorSubtract that gain from the SF10 sensitivity.

Close the ledgerAdd gains, subtract losses and sensitivity, then subtract all reserved margins.

3. Reproduce the chapter case

At SF10: 2¹⁰ = 1024 chips and Ts = 1024/125000 = 8.192 ms
raw budget = 14 + 2 + 6 − 3 − (−130) = 149 dB
allowable loss = 149 − (10 + 12 + 4) = 123 dB
At SF12: relative ideal gain = 10 log10(4096/1024) = 6.02 dB

The ideal SF12 screen becomes 129.02 dB while symbol time becomes 32.768 ms; real data sheets may show slightly different sensitivity steps.

4. Try one real input

TryMove spreading factor while transmit power, antenna gains, fixed loss, bandwidth, and the three reserved margins stay fixed.

ObserveAt SF12, the ideal gain over SF10 is 6.02 dB, allowable loss is 129.02 dB, and symbol time is four times the SF10 value.

ExplainProcessing gain lowers the detectable signal floor; it does not create RF power, and the longer symbol raises airtime and capacity costs.

Technical boundaries.

This is a bounded formula screen, not a deployment approval.

Sensitivity: Use the selected radio's data sheet and settings; ideal 3.01 dB steps are a teaching screen.
Range ratio: The distance ratio assumes a free-space exponent and identical losses.
Capacity: Higher SF affects airtime, duty cycle, collisions, and network capacity.

Correct, not complete: use the measured state named above before release.

5. Use the result in the lab

Choose the lowest spreading factor that closes measured margin with reserve, then verify delivery, airtime, duty cycle, and collision load.

6. Record the evidence state

Keep radio and firmware, bandwidth, coding and spreading settings, conducted power, antenna gains, losses, reserved margins, RSSI, SNR, airtime, and delivery.

7. Check yourself

Does SF12 transmit more RF power than SF10?

Answer: No. It spreads each symbol across more chips and lets the receiver integrate longer.

Why does symbol time become four times longer from SF10 to SF12?

Answer: Chip count doubles at each step, so two steps multiply 1024 chips by four.

Does 129.02 dB prove deployment range?

Answer: No. Terrain, clutter, interference, regulatory limits, antenna installation, and traffic still need evidence.

Honesty boundary.

The bridge keeps calculation, chosen inputs, and field evidence separate.

Computed: Chip count, symbol time, relative processing gain, ideal sensitivity, raw budget, reserve, and allowable loss.
Specified: Radio floor, bandwidth, transmit power, gains, fixed loss, and reserve policy.
Observed: RSSI, SNR, delivery, collisions, airtime, duty cycle, and installed reach.

Correct, not complete: this page does not certify hardware, coverage, safety, capacity, or compliance.