Bounce Partial-Sum Cross-Scan - Magnetism, Induction, and Transmission Lines Step by Step

Each round trip bounces off both boundaries, and the load voltage is the running sum of only the load-arrival deltas while source-side arrivals do not add to load voltage. With load Gamma 1/2 and source Gamma -1/2 on a 4 ohm line, the load deltas of 12, -3, and 3/4 V sum to a partial load voltage of 39/4 V exactly for the ideal lossless-line 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.

Load-voltage arrivals are scanned through coefficient and partial sums. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Load voltage is a partial-arrival sum

Each load arrival contributes its incoming wave plus the load reflection. Source-side rows do not directly add to load voltage.

V_{\text{shown}}=\frac{39}{4}\ \text{V}

The load coefficient changes each load delta

Incoming voltage is fixed in these rows. The load delta changes with the boundary coefficient.

\begin{array}{c|c|c|c}V_{\text{in}}&\Gamma&V_{\text{out}}&\Delta V_{\text{load}}\\8\ \text{V}&0&0\ \text{V}&8\ \text{V}\\8\ \text{V}&\frac{1}{2}&4\ \text{V}&12\ \text{V}\\8\ \text{V}&\frac{-1}{2}&-4\ \text{V}&4\ \text{V}\\\end{array}

Partial sums keep the visible load history honest

The shown load voltage is a running sum of load-arrival deltas, not a caption attached to the final arrow.

\begin{array}{c|c}\Delta V_{\text{load}}&V_{\text{partial}}\\12\ \text{V}&12\ \text{V}\\-3\ \text{V}&9\ \text{V}\\\frac{3}{4}\ \text{V}&\frac{39}{4}\ \text{V}\\\end{array}