Standing waves and VSWR

Keep the load at z=0 and define d=−z≥0 as distance measured from the load toward the generator. For a lossless line γ=jβ. The incident voltage acquires exp(+jβd) while the reflected voltage acquires exp(−jβd).

Using V⁻=Γ_L V⁺, factor out the incident phase. The remaining bracket 1+Γ_L exp(−j2βd) controls the local voltage magnitude. Current has the same structure with a minus sign because the backward current contribution is reversed under the fixed +z reference.

Only the bracket affects the envelope. The common factor exp(+jβd) has unit magnitude and therefore changes phase but not |V|.

Write Γ_L=ρ exp(jφ), with ρ=|Γ_L|. Squaring the magnitude removes the common phase and gives one cosine interference term. Voltage and current carry opposite signs in front of that cosine.

Thus voltage maxima occur where current minima occur, and vice versa. The line is not alternating between 'voltage mode' and 'current mode'; both fields exist everywhere, but their local amplitudes are complementary because of the fixed current sign convention.

The spatial phase is φ−2βd. Moving a distance d changes the incident/reflected relative phase by twice the one-way phase accumulation.

For voltage, the largest magnitude occurs when the two voltage waves are locally in phase: cos(φ−2βd)=+1. The smallest occurs when they are locally opposed: cos(φ−2βd)=−1.

Therefore V_max=|V⁺|(1+ρ) and V_min=|V⁺|(1−ρ). Current swaps these locations, with I_max=(|V⁺|/Z₀)(1+ρ) and I_min=(|V⁺|/Z₀)(1−ρ).

For ρ=1 the ideal minimum becomes zero and the maximum doubles the incident amplitude. For ρ=0, maxima and minima collapse to the same value: there is no standing-wave contrast.

Two successive voltage maxima correspond to an increase of 2π in the argument φ−2βd. Hence 2βΔd=2π, so Δd=π/β=λ_g/2 because β=2π/λ_g.

A voltage maximum and its neighboring voltage minimum differ by π in the interference argument. Therefore 2βΔd=π and Δd=λ_g/4.

These are ENVELOPE spacings. The traveling phasor itself accumulates one-way phase βd; the factor of two appears because the standing pattern compares waves propagating in opposite directions.

Define the voltage standing-wave ratio S=VSWR=V_max/V_min for a lossless line. Substituting the extrema gives S=(1+ρ)/(1−ρ) for 0≤ρ<1.

The relation can be inverted: ρ=(S−1)/(S+1). Thus a magnitude-only standing-wave measurement recovers |Γ| without needing an absolute calibration of |V⁺|.

Matched load: ρ=0 gives S=1. As ρ approaches one, V_min approaches zero and S diverges. Reporting 'infinite VSWR' describes the ideal complete-reflection limit, not an experimentally infinite measured voltage.

Magnitude ρ controls contrast; phase φ controls translation of the pattern relative to the load plane. Voltage maxima satisfy φ−2βd=2πn. Once the coordinate convention is declared, one measured extremum position can therefore constrain φ modulo 2π.

An open circuit has Γ_L=+1, so φ=0 and the load itself is a voltage maximum/current minimum. A short circuit has Γ_L=−1, so φ=π and the load is a voltage minimum/current maximum.

For a general reactive load with |Γ|=1, minima still reach zero, but the entire node/antinode pattern shifts according to φ.

When 0<ρ<1, incident and reflected waves interfere spatially but their powers are unequal. RF-00 gave P_net=P_inc−|P_ref|=P_inc(1−ρ²) on a real-Z₀ lossless line.

Therefore a partially standing pattern can carry positive net power toward the load. Only the complete-reflection limit ρ=1 gives equal counter-propagating powers and zero net average power.

Amplitude probes alone reveal the envelope contrast and extrema positions, but they do not directly separate absolute incident/reflected phases without a declared spatial reference. A phase-sensitive VNA measurement contains more information than a scalar detector moved along the line.

For a lossy line, Γ(d)=Γ_L exp(−2γd), so its magnitude decreases as exp(−2αd) toward the generator. The local interference contrast therefore changes with position.

The incident and reflected amplitudes also vary differently with d, so successive maxima are not all equal and successive minima are not all equal. A single global VSWR number no longer captures the complete spatial pattern exactly.

Numerical audit: let |Γ_L|=0.5 on a lossless line. Then VSWR=3, V_max=1.5|V⁺| and V_min=0.5|V⁺|. If λ_g=0.20 m, adjacent maxima are 0.10 m apart and a max/min pair is 0.05 m apart. On a line with α=0.10 Np/m, the local reflection magnitude 1 m toward the generator falls to 0.5e^(−0.2)≈0.409.

1. Starting from the two-wave voltage solution in d, derive |v̲|²=|V⁺|²[1+ρ²+2ρcos(φ−2βd)].
2. Derive the corresponding current-magnitude expression and explain why voltage maxima coincide with current minima.
3. For |Γ|=0.4 and |V⁺|=2.0 V peak, calculate V_max, V_min and VSWR.
4. Invert VSWR=5 to obtain |Γ| and reflected-power fraction on a real-Z₀ lossless line.
5. A line has λ_g=12 cm. What are the distances between adjacent voltage maxima and between a maximum and the neighboring minimum?
6. For Γ_L=0.6∠120° on a lossless line, give the condition for voltage maxima and find the nearest nonnegative maximum distance if λ_g=18 cm.
7. Explain the voltage/current pattern at an ideal open circuit and at an ideal short circuit at the load plane.
8. A lossless line has |Γ|=0.5 and incident power 100 mW. Compute net load-directed power. Does the presence of a standing-wave pattern contradict that result?
9. A probe measures V_max/V_min=2.0 but has no phase reference. What property of Γ is determined directly, and what phase information remains ambiguous?
10. A lossy line has |Γ_L|=0.7 and α=0.05 Np/m. Find |Γ| at d=3 m toward the generator and explain why one global VSWR is not exact.

Chapter 03 will convert the same local reflection coefficient into input impedance: Z_in=Z₀(1+Γ)/(1−Γ), derive the tangent-form lossless line expression, audit open/short transformations and connect reference-plane motion to impedance transformation.