Why is the horizontal axis “0 at the load” on the right?

The formulas use **z** as distance **from the load toward the generator**, which is a standard convention in many texts (load reference plane). The sketch places the **load** marker on the **right** and the **generator** on the **left** so the wave appears to propagate toward the load.

How does this differ from the “Transmission Line & Γ” simulator?

That page uses a **purely real** load **Z_L** as a quick sinusoid sketch. Here **X_L** is included, Γ is fully **complex**, the **|V|** envelope matches **|1+Γe^{j2βz}|**, and **VSWR / return loss** are emphasized for teaching.

What happens when Z_L = Z₀?

Then **Γ = 0**, **VSWR = 1**, and the envelope is **flat**: there is no reflected wave in steady state and **|V|** does not vary with position (for this ideal lossless model).

Does the plot show actual volts?

No. Amplitudes are **normalized** by **|V⁺|** and scaled for display. The shape of the standing pattern and the numerical **Γ**, **VSWR**, and **return loss** are the quantitative teaching targets.

VSWR & Standing Wave (Transmission Line)

This simulator ties together load impedance, reflection, and standing waves on a lossless transmission line at a single frequency. The voltage reflection coefficient at the load is Γ = (Z_L − Z₀)/(Z_L + Z₀) for complex Z_L = R_L + jX_L and real characteristic impedance Z₀. Its magnitude controls the voltage standing wave ratio VSWR = (1+|Γ|)/(1−|Γ|) (for passive loads with |Γ|<1). Along the line, measuring distance z from the load toward the generator, the total voltage phasor V(z) = V⁺(e^{-jβz} + Γ e^{jβz}) yields a position-dependent magnitude |V(z)|/|V⁺| = |1 + Γ e^{j2βz}|, which oscillates between 1−|Γ| and 1+|Γ|. The animation plots the normalized envelope (cyan fill / dashed bounds) and the instantaneous real part Re{V e^{jωt}} (magenta) after normalizing by 1+|Γ| so matched loads produce a traveling wave of constant amplitude. A small Γ phasor inset shows the complex reflection coefficient on the unit disk. Return loss −20 log₁₀|Γ| is shown in decibels. The model assumes no line loss, no dispersion, and a lumped load at z = 0; it complements the simpler real-valued transmission-line-reflection sketch and the Smith chart normalised to z = Z/Z₀.

Who it's for: Students learning transmission-line theory, VSWR meters, antenna feedpoints, and RF mismatches—after complex numbers but before full S-parameter cascades.

Key terms

Characteristic impedance Z₀
Load impedance Z_L
Reflection coefficient Γ
VSWR / SWR
Standing wave
Return loss
Phase constant β

How it works

Complex load Z_L on a lossless line: reflection coefficient Γ, VSWR, and the voltage standing-wave pattern along the line.

Frequently asked questions

Why is the horizontal axis “0 at the load” on the right?: The formulas use z as distance from the load toward the generator, which is a standard convention in many texts (load reference plane). The sketch places the load marker on the right and the generator on the left so the wave appears to propagate toward the load.
How does this differ from the “Transmission Line & Γ” simulator?: That page uses a purely real load Z_L as a quick sinusoid sketch. Here X_L is included, Γ is fully complex, the |V| envelope matches |1+Γe^{j2βz}|, and VSWR / return loss are emphasized for teaching.
What happens when Z_L = Z₀?: Then Γ = 0, VSWR = 1, and the envelope is flat: there is no reflected wave in steady state and |V| does not vary with position (for this ideal lossless model).
Does the plot show actual volts?: No. Amplitudes are normalized by |V⁺| and scaled for display. The shape of the standing pattern and the numerical Γ, VSWR, and return loss are the quantitative teaching targets.

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