Alpha Beta Estimator Calculator

Calculator Inputs

Example Data Table

Example measurements at f = 1.0×10⁹ Hz. Values are illustrative for a weakly attenuating guided wave.

Formula Used

For a monochromatic field propagating along z, a common model is:

E(z,t) = E₀ e^−αz cos(ωt − βz + φ₀)

• α is the attenuation constant (Np/m).
• β is the phase constant (rad/m).
• ω = 2πf is angular frequency (rad/s).

Two-point estimate

α = \ln(A₁/A₂) / Δz

β = \pm (Δφ / Δz)   with  Δφ in radians

Attenuation conversion: α(dB/m) = 8.686·α(Np/m).

Multi-point regression

Linearize both relationships and fit slopes using least squares:

\ln A(z) = \ln A₀ − αz

φ(z) = φ₀ ∓ βz

Phase unwrapping removes 360° jumps before fitting.

Derived values

λ = 2π/|β|,   v_p = ω/|β|,   γ = α + jβ

Professional Article

1) Meaning of α and β in propagation

Many experiments reduce wave travel to a complex propagation constant, γ = α + jβ. Here α controls exponential amplitude decay with distance, and β controls phase accumulation. Estimating both turns measurements into compact, comparable propagation parameters.

2) Measurement data that drives the estimate

The estimator uses distance z, a positive amplitude-like magnitude A(z), and phase φ(z) in degrees. Choose a z-range wide enough to reveal clear trends above noise. Enter frequency f so the tool can compute ω = 2πf and derived phase-velocity quantities.

3) Two-point method for quick laboratory checks

With two stations separated by Δz, attenuation follows α = ln(A₁/A₂)/Δz from A(z)=A₀e^−αz. Phase uses a single difference, |β| ≈ |Δφ|/Δz, after converting degrees to radians and applying your chosen sign convention.

4) Multi-point regression for higher confidence

Regression mode fits slopes to linearized relations: ln A(z) = ln A₀ − αz and φ(z) = φ₀ ∓ βz. Using more rows averages random errors and reduces sensitivity to one bad point. Reported R² values help validate whether the assumed models match your data.

5) Phase unwrapping and sign conventions

Phase instruments often wrap values modulo 360°. Unwrapping removes artificial jumps by adding or subtracting 360° so adjacent points remain continuous, which stabilizes the fitted slope. The sign selector aligns the estimate with your notation, preventing a common reporting error when using φ(z)=φ₀−βz versus a plus sign.

6) Derived wavelength and phase velocity

After estimating β, the tool computes λ = 2π/|β| and v_p = ω/|β|. Using the built-in example at f = 1.0×10⁹ Hz, phase shifts about −45° over 0.15 m, giving |β| ≈ 5.24 rad/m, λ ≈ 1.20 m, and v_p ≈ 1.2×10⁹ m/s.

7) Interpreting attenuation with real numbers

α is reported in Np/m and converted to dB/m using 1 Np = 8.686 dB. In the example, amplitude decreases from 1.000 to 0.860 across 0.15 m, implying α ≈ 1.005 Np/m (about 8.73 dB/m). Use these values to compare materials, estimate link margins, or evaluate loss reduction after design changes.

8) Practical use cases and reporting workflow

This workflow supports coax and waveguide characterization, acoustic ducts, optical guides, and controlled test fixtures where amplitude and phase are recorded versus distance. Export CSV for traceable analysis and PDF for lab notebooks. If R² is low, widen the z-span, reduce reflections, and improve phase referencing before drawing physical conclusions.

FAQs

1) What should I use as the amplitude input?

Use any positive quantity proportional to wave magnitude, such as envelope voltage, RMS field magnitude, or calibrated power-amplitude equivalent. Keep units consistent across rows; only ratios matter for α.

2) Why does the tool require frequency?

Frequency is needed for ω = 2πf and to compute phase velocity v_p from β. The estimated α and β themselves come from spatial trends.

3) When should I choose regression instead of two-point?

Use regression when you have more than two measurements or when noise and drift are present. The slope fit averages errors and provides R² values that help you judge data quality.

4) What does phase unwrapping change?

It removes artificial 360° jumps so the phase-vs-distance trend remains continuous. This prevents incorrect slopes when phases cross the ±180° or 0/360° boundary.

5) My α is negative. Is that possible?

A negative α usually means amplitude increased with distance due to gain, reflections, or measurement normalization. Check calibration, move the reference plane, or use regression to reduce point-to-point artifacts.

6) How do I interpret very large v_p values?

Large phase velocities can occur in dispersive structures and waveguides; they do not necessarily represent energy transport speed. For energy flow, group velocity is more relevant than v_p.

7) Can I use degrees or radians for phase input?

Enter phases in degrees, as shown in the labels and example. The calculator converts to radians internally before estimating β and computing derived quantities.