Reflections at Boundaries

Echo Polarity Reveals the Boundary

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−,high=6 VV_{-,\text{high}}=6\ \text{V}
Positive echo polarityA higher load reflects a same-sign step.incidentreflectedline 4 ohmload 12 ohmGamma 1/2in 12 Vback 6 V

A negative echo points to a lower load

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

V−,low=−4 VV_{-,\text{low}}=-4\ \text{V}
Negative echo polarityA lower load reflects an inverted step.incidentreflectedline 4 ohmload 2 ohmGamma -1/3in 12 Vback -4 V

The polarity arrives only after the round trip

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

techo=8 st_{\text{echo}}=8\ \text{s}
Echo timingPolarity information returns after two trips.sourceloadlineoutecholength 12 mspeed 3 m/sdelay 4 slaunch 0 sone way 4 sround trip 8 secho 8 s