Echo Polarity Reveals the Boundary - Magnetism, Induction, and Transmission Lines Step by Step

An echo-polarity-reveals-the-boundary ledger uses the returned sign to distinguish higher-load and lower-load mismatches at the line end. The +6 V and -4 V echoes are exact only for the stated reflection coefficients, incident step, and ideal round-trip timing row. Real time-domain measurements add line loss, dispersion, connector discontinuities, source impedance, finite rise time, trigger calibration, finite precision, and measurement uncertainty.

The returned sign distinguishes higher and lower load mismatches. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

A positive echo points to a higher load

The higher-load case returns 6 volts with a positive sign.

V_{-,\text{high}}=6\ \text{V}

A negative echo points to a lower load

The lower-load case returns -4 volts, so polarity distinguishes the boundary.

V_{-,\text{low}}=-4\ \text{V}

The polarity arrives only after the round trip

The source sees that sign at 8 seconds, not at launch.

t_{\text{echo}}=8\ \text{s}