VSWR, Return Loss & Mismatch Loss Calculator

Method, formulas and references

Reflection coefficient and standing waves

A load Z_L on a line of characteristic impedance Z₀ reflects part of the incident wave. The reflection coefficient and the voltage standing-wave ratio are (Pozar §2.3):

The voltage on the line swings between (1 + |Γ|) and (1 − |Γ|) times the incident amplitude |V⁺|; VSWR is the ratio of the two.

Return loss, mismatch loss and power

|Γ|² of the incident power is reflected and 1 − |Γ|² reaches the load. Return loss says how far below the incident wave the reflection is; mismatch loss says how much power the load does not receive because of the reflection, with a matched (Z₀) source. A load with R = 0 (short, open or pure reactance) reflects everything: |Γ| = 1, VSWR and ML are infinite and RL is 0 dB. A perfect match gives |Γ| = 0, VSWR = 1, ML = 0 dB and an infinite return loss.

Sign convention. Return loss is a positive number of decibels. A network analyser displays the same quantity as S11 = 20 log₁₀|Γ|, which is negative. A negative value typed as return loss is therefore read as S11 and its sign dropped; the result table shows both.

Each quantity is converted with 1 − |Γ|² formed directly from the input (for example 4·VSWR/(VSWR + 1)² or 4RZ₀/|Z_L + Z₀|²), so very good and very poor matches keep full precision.

Mismatch between a source and a load

When the source is also mismatched (reflection coefficient Γ_s), the re-reflected wave adds to the incident one with a phase that is usually unknown. The power the load receives from a source with available power P_avs is the transducer gain of a perfect through connection (Pozar §12.1 with S₂₁ = 1, S₁₁ = S₂₂ = 0):

Over every phase, |1 − Γ_sΓ_l| lies between 1 − |Γ_s||Γ_l| and 1 + |Γ_s||Γ_l|. Relative to the same load fed from a matched source, the delivered power therefore lies between

This is the mismatch uncertainty of a power measurement (Fluke states it as a power error of 1 − 1/(1 ± |Γ_s||Γ_l|)²). The total mismatch loss of (4) is the sum of both single-port mismatch losses plus 20 log₁₀ of the same two limits; it reaches 0 dB when the two reflections cancel (Γ_s = Γ_l*). Measuring the complex Γ of both ports, for example with a vector network analyser, replaces the range by a single value.

Cascaded VSWR

A lossless two-port, such as an impedance step or a connector, maps the reflection coefficient behind it onto its input by a bilinear transformation that keeps the unit circle (Pozar §2.4, §4.3). With the phase between the two mismatches unknown, the input reflection coefficient lies between

which in VSWR is exactly VSWR_high / VSWR_low to VSWR₁ · VSWR₂. The tests check this against a phase sweep of a load behind an impedance step. Loss between the two discontinuities narrows the range; further discontinuities widen it, by the same product rule.

References

1. D. M. Pozar, Microwave Engineering, 4th ed., Wiley, 2012: §2.3 (terminated lossless line, Γ, SWR, return loss), §2.4 (Smith chart), §2.6 (generator and load mismatches), §4.3 (scattering matrix) and §12.1 (two-port power gains).

Data sources

1. Fluke Calibration, "Minimizing RF mismatch errors and uncertainties", application note 6005964a-en, 2015
   Power error = 1 − 1/(1 ± |Γs||Γl|)², |Γ| = (VSWR − 1)/(VSWR + 1), return loss = 20 log|Γ|⁻¹. Checked against the mismatch limits here.
2. NAWCWD TP 8347, Electronic Warfare and Radar Systems Engineering Handbook, section 6-2, VSWR, reflection coefficient and mismatch loss (reproduced by RF Cafe)
   Mismatch loss = −10 log(1 − ρ²); worked example VSWR 2:1 → ρ 0.333, ML 0.51 dB, RL 9.54 dB, 11 % reflected, used as a test case.