Reflection Coefficient Calculator
Enter the characteristic impedance Z0 and a complex load ZL = R + jX. Get Γ in polar and rectangular form, VSWR, return loss, mismatch loss, power reflected and a Smith chart plot.
Γ = (ZL − Z0) / (ZL + Z0)Derived quantities
Step-by-step solution
Smith chart
The amber point is Γ. The dashed circle is the constant-|Γ| (constant-VSWR) locus. Grid circles are normalized resistance r = R/Z0, arcs are normalized reactance x = X/Z0.
Common loads on your Z0
| Load ZL | Γ | |Γ| | VSWR | Return loss | Reflected |
|---|
FAQ
What do the extreme values mean?
A matched load (ZL = Z0) gives Γ = 0, so nothing is reflected. A short circuit gives Γ = −1 (∠180°), an open circuit gives Γ = +1 (∠0°), and a pure reactance gives |Γ| = 1: all power is reflected and none is absorbed.
How are VSWR and return loss defined?
VSWR = (1 + |Γ|) / (1 − |Γ|), from 1 (matched) to infinity (total reflection). Return loss = −20·log10|Γ| dB, so a higher value means a better match. Mismatch loss = −10·log10(1 − |Γ|²) dB is the power lost to reflection.
Does it work with a complex Z0?
No. This tool assumes a real Z0, which is the usual case for low-loss coax, microstrip and waveguide near their design frequency.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| Γ | Reflection coefficient (complex, at the load) | dimensionless |
| Z_L | Load impedance (complex) | Ω |
| Z₀ | Characteristic impedance of the transmission line | Ω |
| |Γ| | Magnitude of the reflection coefficient | dimensionless |
| VSWR | Voltage Standing Wave Ratio (derived from |Γ|) | dimensionless |
What it means
The reflection coefficient Γ at the load of a transmission line is given by Γ = (Z_L − Z₀) / (Z_L + Z₀), where Z_L is the load impedance and Z₀ is the characteristic impedance. Γ is a complex number that describes how much of the incident wave is reflected. If Z_L = Z₀, then Γ=0, meaning no reflection (perfect match). If Z_L is an open or short circuit, |Γ|=1, meaning total reflection. The reflection coefficient is used to calculate the voltage standing wave ratio (VSWR) and return loss. It is fundamental in RF and microwave engineering. Example: For a 75Ω load on a 50Ω line, Γ = (75−50)/(75+50) = 25/125 = 0.2. This means 20% of the incident voltage is reflected, causing a small standing wave.
Worked example
Reflection Coefficient – Practical Example
Real‑World| Parameter | Value |
|---|---|
| Z₀ | 50 Ω |
| ZL | 75 Ω |
| Formula | Γ = (ZL − Z₀) / (ZL + Z₀) |
Common mistakes
Watch unit consistency and the assumptions behind the formula; misapplying it outside its valid conditions is the most frequent error.Applications
Reflection coefficient Γ = (Z_L − Z₀)/(Z_L + Z₀) describes how much of a wave is reflected at a load. It is used to quantify mismatch and to design impedance matching networks. Engineers use it to analyse return loss, to evaluate antenna performance, and to design filters. By minimising |Γ|, they achieve better power transfer. This parameter is fundamental in RF and microwave engineering.
- Impedance matching and matching network design
- Return loss and VSWR measurement
- Antenna tuning and efficiency analysis
- Transmission line fault detection
- Educational understanding of wave reflection
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