Reflection Boundary Cross-Scan - Magnetism, Induction, and Transmission Lines Step by Step

The reflection coefficient sign and size come from the load-to-line impedance mismatch, so this cross-scan varies one of load or incident step while holding the other fixed. With a 4 ohm line the 2, 4, and 8 ohm loads give Gamma of -1/3, 0, and 1/3, and at 6 V incident the 8 ohm load reflects 2 V exactly for the ideal sharp-boundary model; real lines add attenuation, dispersion, finite rise time, impedance tolerance, source and load mismatch, cable losses, connector parasitics, and timing and voltage measurement uncertainty.

Reflection sign and size are scanned across load and incident step. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

The boundary coefficient comes from load mismatch

A higher load sends back a same-sign step. The coefficient and reflected voltage are checked from the same boundary source.

\Gamma={Z_{\text{load}}-Z_{\text{line}}\over Z_{\text{load}}+Z_{\text{line}}}=\frac{1}{3}\qquad V_{\text{back}}=2\ \text{V}

Load impedance controls reflection sign

Line impedance and incident step are fixed. Lower, matched, and higher loads give negative, zero, and positive reflection rows.

\begin{array}{c|c|c|c|c}Z_{\text{line}}&Z_{\text{load}}&\Gamma&V_{\text{in}}&V_{\text{back}}\\4\ \Omega&2\ \Omega&\frac{-1}{3}&6\ \text{V}&-2\ \text{V}\\4\ \Omega&4\ \Omega&0&6\ \text{V}&0\ \text{V}\\4\ \Omega&8\ \Omega&\frac{1}{3}&6\ \text{V}&2\ \text{V}\\\end{array}

Incident step scales the reflected step

The coefficient is fixed here. The reflected voltage follows the incident voltage one for one.

\begin{array}{c|c|c|c|c}Z_{\text{line}}&Z_{\text{load}}&\Gamma&V_{\text{in}}&V_{\text{back}}\\4\ \Omega&8\ \Omega&\frac{1}{3}&3\ \text{V}&1\ \text{V}\\4\ \Omega&8\ \Omega&\frac{1}{3}&6\ \text{V}&2\ \text{V}\\4\ \Omega&8\ \Omega&\frac{1}{3}&12\ \text{V}&4\ \text{V}\\\end{array}