Q5Electromagnetics Waves

Relationship between VSWR and impedance at standing-wave extrema: at the point along a transmission line where the voltage standing-wave pattern is maximum, the line's input impedance is purely real and equal to Z0×VSWR; at the point where the voltage is minimum, the impedance is likewise purely real, equal to Z0/VSWR. Correspondingly, the normalized impedance zn = Z/Z0 takes the extreme real values:

Derivation: at a voltage maximum, the incident and reflected waves add in phase, so Γ is real and positive there (Γ = |Γ|), giving zn = (1+Γ)/(1−Γ) = (1+|Γ|)/(1−|Γ|) = VSWR directly (comparing with the definition of VSWR itself). At a voltage minimum, a half-wavelength (or quarter-wavelength net rotation on the Smith chart) away, Γ has rotated to become real and negative (Γ = −|Γ|), giving zn = (1−|Γ|)/(1+|Γ|) = 1/VSWR. For VSWR = 2, this gives zn,max = 2 and zn,min = 0.5 — the two extreme (and only purely real) normalized-impedance values seen anywhere along this particular mismatched line, with the actual normalized load impedance zn = Zn itself (i.e. the normalized impedance quoted at the load position, not merely at the pattern extrema) generally taking some complex value between these two real bounds depending on the load's own reflection-coefficient phase angle, but constrained overall so that its magnitude lies between 1/VSWR and VSWR — i.e., 0.5 ≤ |zn| ≤ 2 for this VSWR = 2 case, with the exact value requiring either the actual complex ZL or the load's distance from the nearest voltage minimum to pin down precisely.

Geometric interpretation on the Smith chart: these two bounds, zn,max = VSWR and zn,min = 1/VSWR, are exactly the two points where the constant-VSWR circle (of radius |Γ| = (VSWR−1)/(VSWR+1)) crosses the horizontal (real) axis of the Smith chart — one crossing on the right side (at z = VSWR, the voltage-maximum point) and one on the left side (at z = 1/VSWR, the voltage-minimum point). Every other point on that same constant-VSWR circle corresponds to some complex normalized impedance at some other position along the line, all sharing the same VSWR but differing in resistance and reactance; this is precisely why, on a Smith chart, VSWR is read off as the single number where a constant-|Γ| circle crosses the right-hand real axis, without needing to separately compute the magnitude of a generally complex zn at an arbitrary point along the line.