Math Bridge: Smart-Parking Radio Ledger

← Back to Smart Cities
Math BridgeApplicationsStruggle-friendly runway

Why does one tiny parking report take about a second?

Carry distance and spreading factor into airtime, charge, link margin, and cell sag.

Radio Remi, the radio guideRadio Remi guides
The one targetTurn LoRa symbol timing into battery charge.
The chapter case868 MHz, 2 km, 12 bytes, SF12, 125 kHz, and 40 mA.
What it buys youAn auditable explanation for the one-second budget line.

A technician must decide whether free-space path loss at 868 mhz is safe before changing gateway distance on the real device. The result is unresolved until the rule and units are checked. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is gateway distance. The middle card applies this page's rule. The green card is free-space path loss at 868 mhz. 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 model keeps those stated values fixed and changes only gateway distance, so the numeric fixture does not switch without explanation.

Gateway distance changes free-space path loss at 868 mhz An input card leads through the rule loss = 20 log10(distance in m) + 31.22 dB to the free-space path loss at 868 mhz result. INPUT PAGE INPUT APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrows. Longer distance increases spreading loss at the fixed carrier frequency.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 2000 m.

  2. 2

    Name the relationship. loss = 20 log10(distance in m) + 31.22 dB

  3. 3

    Substitute with units. 20 log10(2,000) + 31.22 = 97.2 dB

  4. 4

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

Predict, then change gateway distance

Try Predict the direction of loss = 20 log10(distance in m) + 31.22 dB. Test another gateway distance, then compare free-space path loss at 868 mhz.

2000 m
Chapter baseline
Free-space path loss at 868 MHz

Observe Longer distance increases spreading loss at the fixed carrier frequency. Reset gateway distance to 2000 and compare free-space path loss at 868 mhz.

Explain Longer distance increases spreading loss at the fixed carrier frequency.

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 gateway distance moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only gateway distance. It is not a field prediction.

1. Start with the physical story

The gateway does not demand one second directly. Distance consumes link margin; a higher spreading factor makes each symbol longer so the receiver can recover a weaker packet. Longer symbols keep the transmitter on and move more charge.

Radio Remi: Link budget upstream becomes battery charge downstream.

2. Name every algebra move

1

Find wavelengthDivide light speed by 868 MHz.

2

Price distanceUse 20 log10(4πd/λ) for the free-space floor.

3

Time one symbolDivide 2 to the SF power by 125 kHz.

4

Count symbolsAdd preamble and payload symbols, including coding and low-rate optimisation.

5

Convert time to chargeMultiply seconds on air by 40 mA, then by 2,880 reports/day.

6

Check the cellMultiply pulse current by internal resistance.

3. Reproduce the chapter case

λ=3×10⁸/868×10⁶=0.346 m
FSPL=20 log10(4π×2000/0.346)=97.2 dB
Tsym=2¹²/125000=32.768 ms
ToA=(8+4.25+23)Tsym=1.155 s
Qtx/day=2880×40×1.155/3600=36.96 mAh/day
ΔV=0.040×15=0.600 V

The chapter rounds the radio state to one second; the explicit LoRa timing reproduces that order without pretending retries or the urban channel are known.

4. Try one real input

TryMove spreading factor from 7 to 12. Predict which grows faster: margin or airtime.

Spreading factor
Wavelength
Free-space loss
Ideal margin
Packet airtime
Charge per report
Daily TX charge
Pulse sag
Loaded voltage

ObserveAt SF12, ideal free-space margin is large, but the packet lasts about 1.16 seconds and the pulse drops 0.60 V across 15 ohms.

ExplainEach extra SF step lengthens symbols while improving sensitivity. Airtime is the battery price of that processing gain.

Technical boundaries.

This is a single ideal uplink with fixed radio current.

Channel
Buildings, vehicles, antenna placement, interference, and weather add losses beyond free space.
LoRaWAN
Join, headers, regional duty cycle, retries, acknowledgements, and downlinks change airtime and energy.
Cell
Resistance and open-circuit voltage change with chemistry, temperature, state of charge, and ageing.

Correct, not complete: this ledger does not prove urban coverage or ten-year field life.

5. Use the result in the design

Measure the installed link, select the lowest robust spreading factor, report on events where possible, and verify the supply during the actual current pulse.

6. Record the evidence state

Keep frequency plan, distance, payload, SF, bandwidth, coding rate, antenna, RSSI/SNR, retry count, current trace, cell temperature, resistance, and firmware.

7. Check yourself

Why does SF12 consume more charge than SF7?
Answer: Its symbols are much longer, so the same radio current flows for more time.
Does 53 dB of ideal margin prove a city link?
Answer: No. Free space excludes installation and urban losses, interference, antenna defects, and protocol behavior.
Why check voltage as well as mAh?
Answer: A cell can retain charge yet brown out when pulse current crosses its internal resistance.
Honesty boundary.

The arithmetic reproduces the chapter's 868 MHz, 2 km, 12-byte, 40 mA, and one-second-scale screening case.

Channel
Buildings, vehicles, antenna placement, interference, and weather add losses beyond free space.
LoRaWAN
Join, headers, regional duty cycle, retries, acknowledgements, and downlinks change airtime and energy.
Cell
Resistance and open-circuit voltage change with chemistry, temperature, state of charge, and ageing.

Correct, not complete: this ledger does not prove urban coverage or ten-year field life.

Return to Smart Cities with the maths unpacked →