Math Bridge: Antenna Mismatch from VSWR

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Math BridgeCellular IoTStruggle-friendly runway

How much forward power comes back from a mismatched antenna?

Turn VSWR into reflection coefficient, reflected fraction, mismatch loss, and delivered RF power.

Radio Remi, the guideRadio Remi guides
The one targetCalculate what VSWR says—and does not say—about RF power.
The chapter case23 dBm forward power at VSWR 1.5:1 and 3:1.
What it buys youMake antenna attachment and matching a measurable bring-up gate.

A technician must decide whether delivered rf level is safe before changing antenna mismatch 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 antenna mismatch loss. The middle card applies this page's rule. The green card is delivered rf level. 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 antenna mismatch loss, so the numeric fixture does not switch without explanation.

Antenna mismatch loss changes delivered rf level An input card leads through the rule delivered level = 23 dBm forward power - mismatch loss to the delivered rf level result. INPUT PAGE INPUT APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrows. Greater mismatch loss leaves less forward power at the antenna.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 1.25 dB.

  2. 2

    Name the relationship. delivered level = 23 dBm forward power - mismatch loss

  3. 3

    Substitute with units. 23 - 1.25 = 21.75 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 antenna mismatch loss

Try Predict the direction of delivered level = 23 dBm forward power - mismatch loss. Test another antenna mismatch loss, then compare delivered rf level.

1.25 dB
Chapter baseline
Delivered RF level

Observe Greater mismatch loss leaves less forward power at the antenna. Reset antenna mismatch loss to 1.25 and compare delivered rf level.

Explain Greater mismatch loss leaves less forward power at the antenna.

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 antenna mismatch loss moves here. Field effects named in the technical boundary stay fixed.

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

1. Follow the wave to the load

A feedline expects a matching load. A mismatch sends part of the forward wave back toward the transmitter. VSWR summarizes the standing-wave pattern created by the forward and reflected waves.

Radio Remi: First find the amplitude reflection coefficient. Then square it to get a power fraction.

2. Name the algebra moves

1

Recover reflectionΓ=(VSWR−1)/(VSWR+1).

2

Square for powerPref/Pfwd=Γ².

3

Keep delivered fractionPdel/Pfwd=1−Γ².

4

Convert to lossLmismatch=−10log10(1−Γ²).

3. Reproduce the two matches

1.5:1 ⇒ Γ=0.200 ⇒ 4.00% reflected ⇒ 0.177 dB loss

At 3:1, Γ=0.500, so 25.0% reflects and mismatch loss is 1.25 dB. With 23 dBm forward power, the matched-load delivery estimate is 21.8 dBm.

4. Try one controlled change

Γ=(VSWR−1)/(VSWR+1); L=−10log10(1−Γ²)

TryChange only VSWR. The forward power stays at 23 dBm.

Reflection coefficient
Reflected power
Delivered power
Mismatch loss
Delivered RF level
Delivered RF power

ObserveAt VSWR 3.00:1, Γ is 0.500, reflected power is 25.00%, delivered power is 75.00%, mismatch loss is 1.25 dB, and the delivered level is 21.8 dBm or about 150 mW.

ExplainΓ is an amplitude ratio, so it must be squared for power. As VSWR grows, Γ approaches one and the delivered fraction approaches zero. The open-circuit limit is total reflection.

Technical boundaries.

Mismatch loss is not the same as radiated-power or amplifier-safety certification.

Antenna
Efficiency, cable loss, connector loss, pattern, polarization, and enclosure absorption remain
Radio
Protection, foldback, calibration, harmonics, temperature, and duty cycle depend on hardware
Measurement
VSWR must cover the installed bands and physical states, with suitable calibration

Follow the module vendor's approved load, antenna, and bring-up procedure.

5. Take the open limit honestly

As VSWR tends to infinity, Γ tends to 1, so reflected power tends to 100%. The finite slider stops at 10:1; it does not simulate amplifier damage or a protection circuit.

6. Carry the evidence forward

Record approved antenna, cable and connector, calibration plane, S11 or VSWR across every band, enclosure state, ground plane, temperature, forward and reflected power, radio protection behaviour, current, and the vendor limit.

7. Check yourself

Why is 0.500 reflection coefficient equal to 25% reflected power?
Answer: Reflection coefficient is an amplitude ratio, and power follows its square: 0.5²=0.25.
Does 21.8 dBm mean 21.8% of power?
Answer: No. dBm is logarithmic power referenced to 1 mW.
Can this calculation prove the amplifier is safe?
Answer: No. It does not model the transmitter's protection, temperature, or vendor limits.
Honesty boundary.

The equations correctly translate VSWR under the stated transmission-line model, but they do not describe the whole RF chain.

1.25 dB
Mismatch-only loss at 3:1
150 mW
Delivered RF estimate from 23 dBm forward
Open load
Limiting reflection case, not a damage prediction

Hardware acceptance requires calibrated RF measurements and vendor-defined safety gates.

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