Field Guide · term

Also known as: Γ, gamma, rho, voltage reflection coefficient

The reflection coefficient (Γ, sometimes ρ) is the complex ratio of the wave reflected from an impedance boundary to the wave incident on it.1 For a load ZL terminating a line of characteristic impedance Z0, it is Γ = (ZL − Z0) / (ZL + Z0). Its magnitude runs from 0 for a perfect match to 1 for total reflection, and its phase records where along the wave cycle the reflection occurs. Γ is the single quantity from which return loss and standing-wave ratio are derived.

ZL incident reflected = Γ · incident Γ = (ZL − Z0) / (ZL + Z0) Z0
At a mismatched boundary part of the incident wave reflects; Γ is the complex ratio of that reflection to the incident wave, set entirely by ZL and Z0.

How it works

When a travelling wave reaches a boundary where the impedance changes, it cannot simply continue unchanged — the boundary conditions on voltage and current must be satisfied on both sides. The only way to satisfy them is for part of the wave to reflect. Solving those conditions gives the defining formula Γ = (ZL − Z0)/(ZL + Z0), where both impedances may be complex, so Γ itself is generally complex.

Three special cases build intuition:

  • Matched load, ZL = Z0. The numerator is zero, so Γ = 0. Nothing reflects; all power enters the load. This is the design goal.
  • Open circuit, ZL → ∞. Γ → +1. The wave reflects fully and in phase; voltage doubles at the open end.
  • Short circuit, ZL = 0. Γ = −1. The wave reflects fully but inverted; voltage is forced to zero at the short.

For a passive load |Γ| never exceeds 1, because a passive termination cannot reflect more power than it received. The fraction of power reflected is |Γ|², so a Γ magnitude of 0.1 sends back only 1 % of the incident power even though it is a tenth of the incident voltage.

In practice

Γ is not just a load property; it varies with position along a lossless line, rotating in phase by 720° per wavelength while keeping constant magnitude. That rotation is exactly what a Smith chart plots, which is why impedance matching can be read off as movement around the chart. On a two-port device the input reflection coefficient looking into port 1 is the S-parameter S11, and a vector network analyzer measures Γ directly in both magnitude and phase across frequency.

Two derived numbers repackage |Γ| for convenience. Return loss is −20·log₁₀|Γ| in decibels, and the standing-wave ratio is (1 + |Γ|)/(1 − |Γ|). All three describe the same mismatch.

Relevance to SDR

The reflection coefficient at an SDR’s antenna port determines how much of the captured signal actually crosses into the receiver rather than bouncing back up the feedline. A feedpoint impedance that drifts from 50 Ω as you tune across a band raises |Γ| and quietly costs signal, which for a weak trunking control channel can be the difference between a lock and dropped frames. On transmit-capable radios a high |Γ| also means power reflected back toward the power amplifier, a stress worth avoiding.

GopherTrunk works on the IQ samples produced after the front end, so it never sees Γ directly — the reflection has already happened in the antenna and cabling by the time samples reach the decoder. The concept still matters to operators because minimising |Γ| with a resonant antenna and a clean feedline is the cheapest way to raise the signal-to-noise ratio the software depends on.

Sources

  1. Reflection coefficient — Wikipedia, the Γ = (ZL − Z0)/(ZL + Z0) definition and its relation to VSWR and return loss. ↩

See also