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Tool 10 · Available

Transmission Line Calculator

Evaluate input impedance, phase, delay, and standing-wave behavior for lossless or lossy lines.

Typical inputs
Lossless/lossy mode, complex/open/short load, Z0, frequency, length, velocity factor, and attenuation.
Calculated output
Complex input impedance, input/load Γ, phase, delay, wavelength, VSWR, and attenuation.
Model status
Release C · documented and numerically tested
Engineering question

What impedance appears at the input after this line transforms the load?

Uniform-line input impedance

The general tanh(γℓ) relation handles loss. Open and short are analytic terminations; physical and electrical length inputs remain explicitly distinct.

Propagation model
Input impedance25.000000 − j0.000000 Ω25 Ω ∠ -0.0000°
Electrical length
90.000000° · 0.250000 λ
Physical length
74.9481 mm
Guided wavelength
299.792 mm
One-way delay
250 ps
Load Γ
0.333333 ∠ 0.0000°
Input Γ
0.333333 ∠ -180.0000°
Input VSWR
2.000000
Matched line attenuation
0.000000 dB one way
Propagation constant
0.00000000 + j20.958450 m⁻¹

Calculation path

  1. γ=α+jββ=2πλg\gamma=\alpha+j\beta\qquad\beta=\frac{2\pi}{\lambda_g}
  2. Zin=Z0ZL+Z0tanh⁡(γℓ)Z0+ZLtanh⁡(γℓ)Z_{\mathrm{in}}=Z_0\frac{Z_{\mathrm L}+Z_0\tanh(\gamma\ell)}{Z_0+Z_{\mathrm L}\tanh(\gamma\ell)}
  3. For the e^(jωt) convention used here, a lossless line rotates Γ by −2βℓ toward the generator.
  4. Matched loads repeat at every length; half waves repeat finite loads; quarter waves invert impedance as Z₀²/ZL.
Model

What this calculator is doing

The general tanh(γℓ) input-impedance equation uses γ = α + jβ. Open and short are analytic limits rather than huge or tiny substitute values.

Decision

How to read the result

Compare Zin at the actual source plane. Moving the plane rotates Γ on a lossless line; loss also reduces its magnitude toward the generator.

Equations & conventions

  • Zin=Z0ZL+Z0tanh⁡(γℓ)Z0+ZLtanh⁡(γℓ)Z_{\mathrm{in}}=Z_0\frac{Z_{\mathrm L}+Z_0\tanh(\gamma\ell)}{Z_0+Z_{\mathrm L}\tanh(\gamma\ell)}
  • γ=α+jβ\gamma = \alpha + j\beta
  • γ = α + jβ is propagation constant per metre.
  • α converts entered dB/m to nepers/m; β = 2π/λ.
  • ℓ is one-way line length.

Independently checked example

Checked example: a lossless 50 Ω quarter-wave line terminated in 100 Ω transforms to 25 Ω; a half wave repeats 100 Ω.

Common mistakes

  • Using physical length without velocity factor.
  • Substituting an arbitrary huge resistance for a true open limit.

Assumptions, validity & omissions

  • Uniform single-mode line with real positive Z₀.
  • Velocity factor and attenuation are constant over this point calculation.
  • Open and short limits are evaluated analytically.

Sources & model provenance

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