Math Bridge: Fixed radios and current loops

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Math BridgePrototypingStruggle-friendly runway

How do fixed radios and long current loops survive the real site?

Trace one wireless budget through EIRP and one wired budget through cable resistance, burden voltage, and supply headroom.

Voltage Vera, the prototyping guideVoltage Vera guides
The one targetKeep two installation budgets honest.
The chapter caseA 20 dBm fixed gateway and a 500 m, 20 mA field loop.
What it buys youA handoff that separates directional-radio assumptions from cable headroom.

A field team faces an unresolved physical question: How do fixed radios and long current loops survive the real site? They must answer it before changing cable length 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 cable length is 500.

  2. 2

    Name the relationship. Gateway EIRP=14+6=20 dBm Wearable EIRP=0+0=0 dBm Ideal range ratio=10^(20/20)=10.0x Rcable=0.0842x500x2=84.2 ohm Vneeded=0.020x(84.2+250)=6.68 V

  3. 3

    Substitute the chapter fixture. Set cable length to 500. The page ledger gives cable resistance as 84.2 ohm.

  4. 4

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

Predict, then change cable length

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

500
Chapter baseline
Cable resistance

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

Explain Only cable length moves here. The other chapter fixtures remain fixed.

Source: content/prototyping/prototyping-kits-industrial-wearable.qmd — Phoebe's Field Notes: Why the Antenna and the 4-20 mA Loop Follow the Same Logic. 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 cable length moves. Field effects named in the page's technical boundary stay fixed.

Honesty note. This teaching model changes only cable length. It replays the chapter fixture; it is not a field prediction.

1. Begin with what stays fixed

A bolted gateway can keep its antenna pointed; a moving wearable cannot. A series current loop keeps the same current through each element, but its supply must still provide enough voltage for the cable and burden.

Voltage Vera: Gain redirects RF power; loop headroom pays for resistance. Neither creates energy.

2. Name every algebra move

1

Combine radio termsEIRP=Pt+G.

2

Compare ideal ranged2/d1=10^(ΔEIRP/20) when all other link terms match.

3

Count both conductorsRcable=rL×2.

4

Pay for cable and burdenVneeded=I(Rcable+Rburden).

5

Check headroomVheadroom=Vsupply−Vneeded.

3. Reproduce the 500 m case

Gateway EIRP=14+6=20 dBm
Wearable EIRP=0+0=0 dBm
Ideal range ratio=10^(20/20)=10.0×
Rcable=0.0842×500×2=84.2 Ω
Vneeded=0.020×(84.2+250)=6.68 V

The 24 V loop has 17.32 V of teaching headroom. A plain voltage signal carrying the same 20 mA would lose 1.68 V in that cable; the current loop keeps the measured current but still needs compliance voltage.

4. Try the cable length

TryExtend the one-way cable while wire gauge, loop current, burden, and supply stay fixed.

Cable length
Gateway EIRP
Wearable EIRP
EIRP difference
Ideal range ratio
Cable resistance
Loop voltage needed
Supply headroom
Plain-voltage loss
0-10 V error

ObserveCable length changes loop resistance and voltage headroom, but it does not alter the radio ledger.

ExplainBoth designs survive the site by budgeting a physical carrier, but RF orientation and wired compliance are different mechanisms.

Technical boundaries.

This combines two ideal ledgers for comparison; it does not model one shared signal path.

RF range
The 10× value holds only if receiver, frequency, propagation, loss, and orientation assumptions match
Loop
The transmitter also needs its own compliance voltage and protection allowance
Cable
Temperature, joins, barriers, shielding, capacitance, and installation rules matter

Measure the installed RF link and verify the exact transmitter's compliance budget over worst-case cable resistance.

5. Test the installation assumptions

Rotate the wearable and walk the body-shadow cases. Aim and detune-check the fixed gateway. For the loop, test full-scale current at minimum supply and maximum expected cable resistance.

6. Record the evidence state

Store radio power, antenna gain and pattern, orientation, frequency, receiver, site points, wire gauge, one-way length, loop current, burden, transmitter compliance, supply tolerance, and measured headroom.

7. Check yourself

Why does a 20 dB link difference give a 10× ideal range ratio?
Answer: Free-space amplitude falls with distance, so distance ratio uses 10^(ΔdB/20).
Why is the 500 m cable counted twice?
Answer: Current travels out and back, so the loop contains 1,000 m of conductor.
Does a current loop ignore cable resistance completely?
Answer: No. The current reading is preserved only while the supply and transmitter have enough compliance voltage.
Honesty boundary.

The arithmetic uses the chapter's catalog-typical radio, antenna, 24 AWG cable, 250 Ω burden, and 24 V supply examples.

10.0×
An ideal matched-link ratio, not a site range promise
6.68 V
Cable plus burden only, not the transmitter's full requirement
17.32 V
Teaching headroom before other loop drops and tolerances

Correct, not complete: these budgets do not qualify an industrial radio or field loop installation.

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