Math Bridge: Moving Tracker Antenna Heading

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What happens when the tracker turns away from its patch antenna?

Turn antenna gain into a beamwidth estimate, then test how heading changes the link budget.

Radio Remi, the guideRadio Remi guides
The one targetCompare a broad omni with a narrow patch on a moving chassis.
The chapter case23 dBm LTE-M tracker; 2 dBi omni or 8 dBi patch.
What it buys youDesign a route test that includes turns, not one favourable heading.

A technician must decide whether patch eirp is safe before changing off-axis antenna loss 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 off-axis antenna loss. The middle card applies this page's rule. The green card is patch eirp. 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 off-axis antenna loss, so the numeric fixture does not switch without explanation.

Off-axis antenna loss changes patch eirp An input card leads through the rule patch EIRP = 23 dBm + 8 dBi - off-axis loss to the patch eirp result. INPUT PAGE INPUT APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrows. Turning away spends directional gain and lowers patch EIRP.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 15 dB.

  2. 2

    Name the relationship. patch EIRP = 23 dBm + 8 dBi - off-axis loss

  3. 3

    Substitute with units. 31 - 15 = 16.0 dBm

  4. 4

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

Predict, then change off-axis antenna loss

Try Predict the direction of patch EIRP = 23 dBm + 8 dBi - off-axis loss. Test another off-axis antenna loss, then compare patch eirp.

15 dB
Chapter baseline
Patch EIRP

Observe Turning away spends directional gain and lowers patch EIRP. Reset off-axis antenna loss to 15 and compare patch eirp.

Explain Turning away spends directional gain and lowers patch EIRP.

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 off-axis antenna loss moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only off-axis antenna loss. It is not a field prediction.

1. Start with direction

Antenna gain does not create transmit power. It redirects power. A patch can be strong when aimed at a tower and weak after the vehicle turns. An omni gives up peak gain to cover more headings.

Radio Remi: Ask “where does the power go?” before treating gain as a free link-budget number.

2. Name the algebra moves

1

Add gainEIRP = Pt + G.

2

Convert dBi to a ratioGlin = 10^(GdBi/10).

3

Estimate beamwidthFor a symmetric beam, θ = √(41253/Glin).

4

Apply the pattern lossEIRPoff = Pt + G − Loff.

3. Reproduce the two headings

Omni: 23+2=25 dBm; patch on-axis: 23+8=31 dBm; patch off-axis: 23+8−15=16 dBm

An 8 dBi symmetric Kraus estimate gives θ=80.9°. Its half-beam is about 40.4°. A 60° heading error sits outside that teaching beam.

4. Try one controlled change

θ=√(41253/10^(G/10)); EIRPpatch=Pt+G−Loff

TryChange only the heading error. The 23 dBm transmitter, antenna gains, and illustrative 15 dB off-axis loss stay fixed.

Symmetric beamwidth
Heading coverage proxy
Omni EIRP
Patch on-axis EIRP
Patch at this heading
Patch versus omni
Power-density ratio

ObserveAt 60°, the model applies the off-axis loss. The patch falls to 16.0 dBm, 9.00 dB below the 25.0 dBm omni. That is 0.126× the omni power density, not 0.355×; 0.355 is the corresponding field-amplitude ratio.

ExplainThe on-axis patch advantage exists only inside its assumed beam. A moving tracker cannot promise that heading, so the lab must measure the route in both travel directions and through turns.

Technical boundaries.

The Kraus beamwidth and step-like 15 dB loss are teaching approximations, not a real antenna pattern.

Pattern
Real antennas have smooth roll-off, sidelobes, nulls, polarization effects, and enclosure coupling
Network
Serving-cell direction, handover, clutter, interference, and power control also change
Compliance
Approved antennas, cable loss, and regional limits still govern the hardware

Use the approved pattern and route measurements for the installed chassis.

5. Read the chapter values carefully

The omni result is 25 dBm. The patch is 31 dBm on-axis and 16 dBm in the illustrative off-axis case. Its 80.9° beam covers 22.5% of a 360° heading circle. Those numbers justify a route-test question; they do not choose the antenna.

6. Carry the evidence forward

Record antenna model, approved bands, cable and connector loss, mounting plane, chassis orientation, route heading, serving cell, RSRP, SINR, transmit power, retries, handovers, weather, and repeated runs.

7. Check yourself

Does 8 dBi mean the patch creates more power?
Answer: No. It redirects available power into a narrower pattern.
Why is 9 dB below equal to 0.126× power density?
Answer: Power ratios use 10^(ΔdB/10). The /20 form applies to a field or voltage amplitude under matching assumptions.
Can this page certify the route?
Answer: No. It frames a mutation-sensitive test using illustrative pattern values.
Honesty boundary.

The mathematics is correct for its stated simplified pattern, but it is not a complete mobile-link model.

80.9°
Kraus symmetric-beam estimate
15 dB
Illustrative off-axis loss
0.126×
Power-density ratio for the stated 9 dB difference

A decision needs measured installed patterns, route evidence, and modem/network traces.

Return to LTE-M Lab with the maths unpacked →