Math Bridge: Directional Deployment Link

← Back to WSN Production Deployment
Math BridgeWSNStruggle-friendly runway

What does the directional antenna buy?

Turn dBi into EIRP, ideal range, and collecting aperture without hiding the coverage trade.

Packet Pete, the guidePacket Pete guides
The one targetTranslate deployment antenna gain into three connected outputs.
The chapter case14 dBm radio; 2.15 dBi omni versus 8 dBi directional.
What it buys youAn auditable fixed-path trade, not “more range” alone.

A field team has a real problem to settle: What does the directional antenna buy? They must decide what happens before they change directional gain on the device. Predict the direction first.

See the relationship first

The figure reads from left to right. The blue card is directional gain. The middle card uses this page's rule. The green card is omni eirp. Follow the arrows: 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.

Directional gain changes omni eirp An input card leads through the page rule to the omni eirp result. SET INPUT ONE CONTROL USE RULE predict calculate check units READ RESULT
Follow the arrows. The /20 exponent appears because received power falls with distance squared. Aperture grows with linear gain and wavelength squared.

Derive the baseline in four moves

  1. 1

    Name the input. The chapter baseline for directional gain is 8.

  2. 2

    Name the rule. EIRPomni=14+2.15=16.15 dBm EIRPdir=14+8=22.0 dBm Δ=5.85 dB; range ratio=10^(5.85/20)=1.96x λ=0.125 m; Ae=0.00785 m²

  3. 3

    Put in the chapter value. Set directional gain to 8. The page rule gives omni eirp as 16.1 dBm.

  4. 4

    Read the result. Keep dBm next to the value. Use it only within the limits on this page.

Predict, then change directional gain

Try Predict what happens to omni eirp. Move one control, calculate, then check your idea.

8
Chapter baseline
Omni EIRP

Observe The /20 exponent appears because received power falls with distance squared. Aperture grows with linear gain and wavelength squared. Reset to 8 and compare omni eirp.

Explain Only directional gain moves here. The other chapter values stay fixed.

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 directional gain moves. Field effects named in the page limits stay fixed.

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

1. dBi describes direction

A directional antenna redistributes the radio's power. More energy reaches the intended direction, while less reaches other directions.

Packet Pete: A stronger fixed path and narrower useful coverage are the same physical decision.

2. Name every algebra move

1

Add in decibelsEIRP=Ptx+G.

2

Subtract the two EIRPsΔEIRP=EIRPdir−EIRPomni.

3

Undo the range logarithmd2/d1=10^(ΔEIRP/20).

4

Find wavelengthλ=c/f.

5

Convert dBi and find apertureGlinear=10^(GdBi/10); Ae=Glinear λ²/(4π).

3. Reproduce the deployment example

EIRPomni=14+2.15=16.15 dBm
EIRPdir=14+8=22.0 dBm
Δ=5.85 dB; range ratio=10^(5.85/20)=1.96×
λ=0.125 m; Ae=0.00785 m²

The range ratio is ideal free-space geometry. The aperture is an equivalent receiving area, not the panel's outline.

4. Try the directional gain

TryChange directional gain while the radio and omni reference stay fixed.

Directional gain
Omni EIRP
Directional EIRP
Gain difference
Ideal range ratio
Wavelength
Effective aperture
Linear gain

ObserveAt 8 dBi the directional path has 5.85 dB more EIRP than the omni and an ideal 1.96× range ratio.

ExplainThe /20 exponent appears because received power falls with distance squared. Aperture grows with linear gain and wavelength squared.

Technical boundaries.

The widget holds frequency and conducted power fixed.

Range
Free-space ratio omits terrain, clutter, Fresnel clearance, interference, and receiver mode
Pattern
Real antennas have lobes, nulls, efficiency loss, polarization, and mounting effects
Rules
EIRP limits and permitted bands depend on the deployment jurisdiction

Measure the installed pattern and link margin along required paths.

5. Match antenna shape to architecture

Use directionality when the gateway path is fixed and known. A changing or multi-direction topology may value broad coverage more than the ideal range multiplier.

6. Build the deployment record

Record conducted power, antenna model and gain, frequency, orientation, mounting, feedline, EIRP limit, required coverage, measured margin, failure exercise, owner, and retest trigger.

7. Check yourself

Why is directional EIRP 22.0 dBm?
Answer: Decibel terms add: 14 dBm conducted power plus 8 dBi gain.
Why is the range ratio not 5.85×?
Answer: dB is logarithmic and free-space power falls as distance squared, giving 10^(5.85/20)=1.96.
Does 0.00785 m² equal the physical panel face?
Answer: No. It is effective aperture under the ideal gain-wavelength relation.
Honesty boundary.

The radio and antenna figures are the chapter's explicit catalog-typical teaching case.

1.96×
Ideal fixed-path range multiplier
0.00785 m²
Effective aperture, not physical size
22.0 dBm
Before feedline and regulatory review

Correct, not complete: site evidence decides whether the antenna belongs.

Return to WSN Production Deployment with the maths unpacked →