What is the reflection coefficient (Gamma)?

Gamma (Γ) is a complex number describing how much of a signal is reflected at an impedance discontinuity. Its magnitude ranges from 0 (perfect match) to 1 (total reflection). It relates to normalized load impedance z by: Γ = (z − 1) / (z + 1).

What is normalized impedance?

Normalized impedance z = Z_load / Z0, where Z0 is the reference impedance (typically 50 Ω). Normalizing removes the system impedance so the Smith chart works for any reference.

How is VSWR calculated from Gamma?

VSWR = (1 + |Γ|) / (1 − |Γ|). A perfect match gives VSWR = 1; total reflection gives VSWR = ∞.

Return loss = −20 log10(|Γ|) in dB. Higher return loss means less reflected power. A value of 20 dB means only 1% of power is reflected.

What reference impedance does this tool use?

The tool works with normalized impedance only (Z0 = 1). To get actual impedance, multiply the normalized values by your system Z0 (e.g. 50 Ω): Z = z × 50.

What does mismatch loss mean?

Mismatch loss = −10 log10(1 − |Γ|²) in dB. It is the fraction of available power lost due to the impedance mismatch, separate from any dissipative loss.

Smith Chart Helper

Convert between reflection coefficient (Gamma) and normalized impedance for RF impedance matching on a Smith chart.

Input

Magnitude |Γ| (0 to 1)

Angle ∠Γ (degrees, −180 to 180)

Key Formulas

Gamma:Γ = (z − 1) / (z + 1)

Impedance:z = (1 + Γ) / (1 − Γ)

VSWR:(1 + |Γ|) / (1 − |Γ|)

Return loss:−20 log₁₀(|Γ|) dB

Mismatch:−10 log₁₀(1 − |Γ|²) dB

Enter values and click Calculate

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Summary

The Smith Chart Helper converts between reflection coefficient (Gamma) and normalized impedance using the standard bilinear transformation. Enter Gamma in polar form or supply normalized R and jX values to get VSWR, return loss, and mismatch loss instantly. All computations run client-side — no data leaves your browser.

How it works

Choose your input mode: Gamma (polar) or Normalized Impedance (R + jX).
Enter the reflection coefficient as magnitude (0–1) and angle (degrees), or enter normalized resistance and reactance.
The tool converts instantly between Gamma and normalized impedance using standard Smith chart formulas.
Key parameters — VSWR, return loss, mismatch loss, and reflection phase — are displayed automatically.
Use the results to plan matching networks or to read coordinates on a physical Smith chart.

Use cases

Verify impedance matching network designs for RF amplifiers and antennas.
Convert VNA (vector network analyzer) Gamma readings to impedance for circuit design.
Quickly determine VSWR and return loss from a measured reflection coefficient.
Cross-check Smith chart coordinate readings against calculated values.
Teach or study Smith chart concepts with instant numerical feedback.
Check whether a load is capacitive or inductive before selecting a matching topology.

Frequently Asked Questions

Last updated: 2026-06-11 · Reviewed by Nham Vu

Smith Chart Helper

Input

Key Formulas

Results

Reflection Coefficient (Gamma)

Normalized Impedance z = r + jx

Derived Metrics

Summary

How it works

Use cases

Frequently Asked Questions

Smith Chart Helper

Use the Smith Chart Helper

Input

Key Formulas

Results

Reflection Coefficient (Gamma)

Normalized Impedance z = r + jx

Derived Metrics

Summary

How it works

Use cases

Frequently Asked Questions