Reflection coefficient from the load boundary

A uniform line permits a +z voltage wave and a −z voltage wave. Before a termination is specified, their amplitudes are independent constants fixed by boundary conditions. The load supplies one such boundary relation: total voltage at its terminals must equal Z_L times the total current entering it under the declared reference.

Place the load at z=0 and the source side at z<0. This coordinate choice is convenient but does not redefine +z. The incident wave still travels toward +z, into the load; the reflected wave travels toward −z.

If the load relation happens to be identical to the modal relation of a single forward wave, no backward wave is required. Otherwise a second traveling solution is needed so that BOTH load voltage and load current can satisfy the terminal relation simultaneously.

At z=0 the RF-00 solution gives v̲_L=V⁺+V⁻. Because line current is STILL referenced toward +z, i̲_L=(V⁺−V⁻)/Z₀: the backward voltage contribution carries the minus sign in current.

Define the load voltage-wave reflection coefficient as Γ_L=V⁻/V⁺, assuming V⁺ is nonzero. Then Z_L=v̲_L/i̲_L=Z₀(1+Γ_L)/(1−Γ_L). Solving this one algebraic equation produces the standard result.

The formula is therefore a boundary-condition result, not an independent law. Its signs are tied directly to the fixed current reference and to the definitions of forward and backward waves.

The symbol Γ_L above refers to the VOLTAGE-wave amplitudes. With the fixed +z current reference, I⁺=V⁺/Z₀ while I⁻=−V⁻/Z₀. Therefore the current-wave amplitude ratio is Γ_I=I⁻/I⁺=−Γ_L.

This does not mean the reflected electromagnetic wave has a mysterious extra sign. It means the same backward-traveling mode is being described using a current variable whose positive reference remains +z.

Stating which wave variable is being reflected prevents later errors at shorts, opens and ports. AoN uses Γ without a subscript for voltage-wave reflection only after this convention has been declared.

For Z_L=Z₀, the numerator of Γ_L vanishes: Γ_L=0. The forward wave alone already satisfies the load relation, so no reflected voltage wave is needed.

For an ideal open circuit, i̲_L=0. The only way (V⁺−V⁻)/Z₀ can vanish is V⁻=V⁺, hence Γ_L=+1. Voltage doubles at the boundary while current cancels.

For an ideal short circuit, v̲_L=0. Thus V⁻=−V⁺ and Γ_L=−1. Voltage cancels at the boundary while current contributions add under the fixed current reference.

In general Γ_L=|Γ_L|exp(jφ_Γ). Its magnitude is the reflected-to-incident VOLTAGE amplitude ratio at the load. Its phase is the phase shift of the reflected voltage amplitude relative to the incident voltage amplitude at the same reference plane.

For real positive Z₀ and a passive load Z_L=R_L+jX_L with R_L≥0, Γ_L lies inside or on the unit circle. The proof is direct: |Z_L+Z₀|²−|Z_L−Z₀|²=4Z₀R_L≥0.

Equality |Γ_L|=1 occurs for lossless purely reactive terminations, including ideal open and short limits. An active negative-resistance load can produce |Γ|>1 because the reflected wave can contain power supplied by the load; passive bounds must not be imposed on an active device.

On a lossless line with real positive Z₀, RF-00 gives P_inc=|V⁺|²/(2Z₀) and the magnitude of reflected power P_ref=|V⁻|²/(2Z₀). Their ratio is therefore |Γ_L|².

The net power delivered through the load plane is P_inc−P_ref=P_inc(1−|Γ_L|²). For a passive load this is nonnegative; for a purely reactive load it is zero even though nonzero voltage and current can exist at the boundary.

Return loss is a logarithmic amplitude-reflection metric: RL=−20log10|Γ| dB, so a smaller |Γ| gives a larger positive return loss. By contrast, a plotted voltage-reflection quantity such as 20log10|Γ| is negative for |Γ|<1. Do not confuse these opposite-sign dB conventions.

For complex reference impedance or general lossy-line power waves, squared voltage-wave amplitudes are not automatically power ratios. RF-04 will introduce the required power-wave definitions explicitly.

At a point z<0, the local reflected-to-incident voltage ratio contains the different propagation factors of the two waves: Γ(z)=Γ_L exp(2γz). Thus Γ is attached to a reference plane, not only to a physical load.

It is often clearer to measure distance d=−z≥0 from the load TOWARD the generator. Then Γ(d)=Γ_L exp(−2γd). On a lossless line γ=jβ, the magnitude is unchanged and the phase rotates by −2βd.

On a lossy line α>0, |Γ(d)|=|Γ_L|exp(−2αd): the reflected wave has made an extra round trip relative to the incident wave at the more distant reference plane. Chapter 03 will use the same transformation to derive input impedance along the line.

First terminate a lossless 50 Ω line in 75 Ω. Then Γ_L=(75−50)/(75+50)=0.200: the reflected voltage amplitude is 20% of the incident amplitude and is in phase at the load. Only 4.00% of incident power is reflected, so 96.0% is delivered to the passive load. The return loss is about 13.98 dB.

With a 25 Ω load on the same line, Γ_L=−1/3. Its magnitude is 0.333 but its phase is 180°: the sign cannot be captured by a magnitude-only statement.

For Z_L=j50 Ω on a lossless 50 Ω line, Γ_L=+j. Its magnitude is one and its phase is +90°. The load absorbs zero average power; the boundary stores and returns energy each cycle.

1. Starting only from v̲_L=V⁺+V⁻, i̲_L=(V⁺−V⁻)/Z₀ and Z_L=v̲_L/i̲_L, derive Γ_L.
2. Why does the backward CURRENT-wave reflection coefficient equal −Γ_L under AoN's fixed +z current reference?
3. For a 50 Ω line terminated in 100 Ω, calculate Γ_L, reflected-power fraction and delivered-power fraction.
4. For a 50 Ω line terminated in 25 Ω, calculate Γ_L and explain the physical meaning of its phase.
5. Use the boundary conditions directly to recover Γ=+1 for an open circuit and Γ=−1 for a short circuit.
6. For real Z₀>0 and passive Z_L=R+jX, prove |Γ|≤1 by comparing |Z_L+Z₀|² and |Z_L−Z₀|².
7. A lossless 50 Ω line is terminated by j50 Ω. Compute Γ and the average power absorbed by the load.
8. A load has |Γ_L|=0.1. Compute reflected-power fraction and return loss.
9. On a lossless line, Γ_L=0.5∠60° and βd=30°. Find Γ at a reference plane a distance d toward the generator.
10. A lossy line has α=0.08 Np/m and |Γ_L|=0.6. What is |Γ| 2 m toward the generator, and why is it smaller than |Γ_L|?

Chapter 02 will use the same Γ to build the standing-wave envelope rigorously: voltage maxima/minima, VSWR, phase spacing, open/short patterns and the information that magnitude-only standing-wave measurements can and cannot recover.