Transmission line

See the impedance at the input of a line from its load, electrical length and attenuation.

Transform a load through a line

Load termination

Load resistance R (Ω)

Load reactance X (Ω)

Line impedance Z₀ (Ω)

Velocity factor

Phase velocity / c.

Frequency

Frequency unit

Length entered as

Physical length

Physical length unit

Physical length stays fixed as frequency changes.

Propagation model

Impedance at the input plane

25 Ω

Input VSWR: 2 : 1
Load VSWR: 2 : 1
Input return loss: 9.54 dB

◆ InputLoad ●

Line Z₀ 50 ΩLoad 100 + j0 Ω

Length 74.95 mmElectrical length 90 °

Smith chartReference 50 Ω

Load → Input at one frequency. Rotation keeps |Γ| constant.

More measurements

Reflection at the input & load planes

Both planes use the real characteristic impedance Z₀ as their reference. With the positive-time phasor convention, reflection rotates by twice the one-way electrical angle in the negative direction toward the input. Line loss reduces its magnitude over the outward and return paths.

Input plane

Complex reflection Γ: -0.33 + j0
Magnitude |Γ|: 0.33
Reflection phase: 180 °
Return loss: 9.54 dB
VSWR: 2 : 1

Load plane

Complex reflection Γ: 0.33 + j0
Magnitude |Γ|: 0.33
Reflection phase: 0 °
Return loss: 9.54 dB
VSWR: 2 : 1

Length, wavelength & phase delay

Physical length: 74.95 mm
Length in guided wavelengths: 0.25 λg
Guided wavelength: 299.79 mm
One-way phase delay: 250 ps

The delay uses phase velocity. It equals group delay within this constant-VF, nondispersive model, not for every real line. On a lossless line, half waves repeat the load and quarter waves invert its normalized impedance.

Attenuation & propagation constant

Matched one-way attenuation: 0 dB
Attenuation constant α: 0 Np/m
Phase constant β: 20.96 rad/m

The input return loss equals load return loss plus twice the matched one-way attenuation. A finite dB result may remain available when a very small linear reflection is outside the numerical range.

Generated frequency and length values display at most two decimals. Calculations and copied values retain full precision until you edit the field.

Model & assumptions

A line transforms load impedance $Z_L$ into the impedance $Z_{\mathrm{in}}$ seen at its input. For line impedance $Z_0$ , length $\ell$ and propagation constant $\gamma$ :

Z_{\mathrm{in}}=Z_0\frac{Z_L+Z_0\tanh(\gamma\ell)}{Z_0+Z_L\tanh(\gamma\ell)}

Impedances are in ohms and length is in metres. The propagation constant is $\gamma=\alpha+j\beta$ : $\alpha$ is amplitude attenuation in Np/m, $\beta$ is phase change in rad/m, and $j$ is the imaginary unit. Convert attenuation from dB/m to Np/m by multiplying by $\ln(10)/20$ .

The reflected wave makes a round trip. Its reflection coefficient at the input is therefore related to the load reflection by twice the one-way attenuation and phase:

\Gamma_{\mathrm{in}}=\Gamma_L e^{-2\gamma\ell}

Both reflection coefficients use the entered real $Z_0$ reference. The negative exponent uses the positive-time phasor convention. If total one-way matched loss is $A$ dB, input return loss is load return loss plus $2A$ dB. A higher input return loss caused by line loss does not mean more power reaches the load.

The line is uniform and single-mode, with real positive characteristic impedance, constant velocity factor and the entered point-frequency attenuation. A general lossy line can require complex characteristic impedance. Connectors and discontinuities are excluded. Phase delay equals group delay only in this constant-velocity model; quarter-wave inversion and half-wave repetition are exact only for a lossless line.

The Smith-chart path moves the reference plane from the load toward the input at the selected frequency. A lossless line rotates clockwise at constant reflection magnitude; attenuation moves the path inward. The circular marker is the load and the diamond is the input. This is not a frequency sweep or a wave animation. Above 16 complete chart rotations, only representable endpoints are shown. An unavailable tiny reflection is not replaced with an exact match.

Generated frequency and length fields display at most two decimals while preserving the full value for calculations and copying until edited. The line impedance is a positive, real characteristic impedance and the reference for both planes; it is not a target input impedance.

For example, a lossless 50 Ω quarter-wave line transforms a 100 Ω load to 25 Ω. A half-wave line repeats the 100 Ω load.

SourcesMIT, Electromagnetic Fields and Energy — Sinusoidal Steady State (opens in a new tab)University of Wisconsin, Physics 623 — Transmission Lines (opens in a new tab)Keysight, S-Parameter Design Techniques (opens in a new tab)

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