Math Bridge: Antenna gain and ADC noise

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Math BridgePrototypingStruggle-friendly runway

What does a higher-gain gateway antenna trade away?

Compare a chip antenna and panel antenna at the same transmit power, then keep the ADC's separate noise floor in its own ledger.

Voltage Vera, the prototyping guideVoltage Vera guides
The one targetTurn antenna gain into EIRP, ideal range, and coverage share.
The chapter caseA 2 dBi onboard antenna and 9 dBi panel.
What it buys youA placement record that names directionality and ADC noise separately.

A field team faces an unresolved physical question: What does a higher-gain gateway antenna trade away? They must answer it before changing onboard gain ratio 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 onboard gain ratio. The middle card applies this page's relationship. The green card is panel gain ratio. 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.

Onboard gain ratio changes panel gain ratio An input card leads through the page relationship to the panel gain ratio result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. The antenna ledger describes where radio power goes. The ADC ledger describes how finely a voltage input is represented.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for onboard gain ratio is 9.

  2. 2

    Name the relationship. 2 dBi=1.58x and 2 dBm=1.58 mW 9 dBi=7.94x and 9 dBm=7.94 mW ideal range ratio=√(7.94/1.58)=2.24x coverage shares=63.1% and 12.6% ADC q=3.3/4096=0.806 mV; e RMS =0.233 mV

  3. 3

    Substitute the chapter fixture. Set onboard gain ratio to 9. The page ledger gives panel gain ratio as 7.94 times.

  4. 4

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

Predict, then change onboard gain ratio

Try Predict the direction of panel gain ratio. Move one control, calculate, then check your prediction.

9
Chapter baseline
Panel gain ratio

Observe The antenna ledger describes where radio power goes. The ADC ledger describes how finely a voltage input is represented. Reset the control to 9 and compare panel gain ratio.

Explain Only onboard gain ratio 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 onboard gain ratio moves. Field effects named in the page's technical boundary stay fixed.

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

1. Start with an imaginary equal radiator

An isotropic antenna is a reference that spreads power equally over a sphere. A real antenna reshapes that pattern. More gain in one direction means less coverage elsewhere. EIRP adds transmit power and antenna gain in dB; it does not say that every direction received the gain.

Voltage Vera: Higher gain can extend one planned path while making aim and placement more important.

2. Name every algebra move

1

Turn dBi into a power ratioGlinear=10GdBi/10.

2

Add EIRP in dBEIRP=Pt+G.

3

Compare ideal rangeAt fixed sensitivity, range scales with √G.

4

Compare coverage shareThe ideal solid-angle share scales with 1/G.

5

Keep ADC noise separateq=Vref/2N and eRMS=q/√12.

3. Reproduce the chapter case

2 dBi=1.58× and 2 dBm=1.58 mW
9 dBi=7.94× and 9 dBm=7.94 mW
ideal range ratio=√(7.94/1.58)=2.24×
coverage shares=63.1% and 12.6%
ADC q=3.3/4096=0.806 mV; eRMS=0.233 mV

The panel concentrates the same conducted power more strongly in its main direction. The ADC result belongs to a different measurement chain and must not be treated as radio noise.

4. Try the panel gain

TryMove the panel gain while the 2 dBi onboard antenna, transmit power, and ADC stay fixed.

Onboard gain ratio
Panel gain ratio
Onboard EIRP
Panel EIRP
Ideal range ratio
Onboard coverage share
Panel coverage share
ADC voltage step
ADC quantisation noise

ObserveMore panel gain raises directional EIRP and ideal range while shrinking the ideal covered share. The ADC step does not move because it is independent.

ExplainThe antenna ledger describes where radio power goes. The ADC ledger describes how finely a voltage input is represented.

Technical boundaries.

These are ideal gain-scaling and ADC-quantisation ledgers, not a coverage prediction or analogue qualification.

Range
Path loss, fading, interference, polarization, height, cable loss, and sensitivity set real range
Pattern
100/G is a teaching solid-angle share, not a real antenna radiation pattern
ADC
Noise, effective bits, input drive, and reference accuracy add to quantisation

Measure the installed radiation path and ADC input separately under representative conditions.

5. Turn the trade into platform evidence

For the radio, record the aim, mounting height, enclosure, cable, channel, transmit setting, packet result, and off-axis coverage. For the ADC, record the reference, mode, source impedance, settling, noise, and calibration. A platform comparison should keep those evidence paths distinct.

6. Record the evidence state

Store the platform, antenna model and gain, cable and connector loss, frequency, conducted power, orientation, installation geometry, receiver sensitivity, packet logs, ADC reference, bit depth, sample settings, raw codes, and firmware revision.

7. Check yourself

Why is 9 dBi about 7.94 times on a power scale?
Answer: 109/10=7.94.
Why is the ideal range ratio 2.24 rather than 5.01?
Answer: The gain ratio is 7.94/1.58=5.01, while range scales with its square root: √5.01=2.24.
Does the 0.233 mV ADC noise belong in the radio link budget?
Answer: No. It is the separate converter quantisation floor for the platform's analogue input.
Honesty boundary.

The arithmetic reproduces the chapter's catalog-typical 2 dBi, 9 dBi, and 12-bit ADC example.

2.24×
An ideal gain-only range ratio, not a site result
12.6%
A teaching solid-angle share, not a measured beam pattern
0.233 mV
Quantisation noise only, not total ADC uncertainty

Correct, not complete: this ledger does not qualify a gateway antenna, predict coverage, or certify ADC performance.

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