Incident and reflected waves on a transmission line
Derive forward and reverse voltage/current solutions, load reflection, open/short conditions, standing waves, power and reference-plane rotation without changing the current sign convention.
- Derive the two traveling-line solutions from telegrapher equations.
- Explain the minus sign of reflected current under a fixed +z reference.
- Decompose measured voltage/current into V+ and V−.
- Derive the reflection coefficient from load boundary conditions.
- Test matched, open and short-circuit limits.
- Derive the standing-wave envelope and VSWR.
- Derive the reference-plane phase rotation with d=−z.
- Compute incident, reflected and accepted average powers.
- Distinguish physical line impedance from VNA reference normalization.
- State lossless, single-mode and real-impedance assumptions.
This opening RF-04 lesson builds incident and reflected transmission-line waves before introducing scattering matrices. We preserve AoN's exp(+jωt) convention, the load at z=0 and generator at z<0, with current referenced along +z. First derive voltage and current waves, then introduce reflection coefficient, signed boundary conditions, power, standing waves and reference-plane motion.
0104.01.1 — Forward and reverse transmission-line modes
For a uniform two-conductor transmission line, the phasor telegrapher equations admit two independent solutions propagating in opposite spatial directions. With exp(+jωt) and +z pointing toward the load, the forward wave varies as exp(−γz) and the reverse wave as exp(+γz), where γ=α+jβ.
The forward current wave follows the forward voltage divided by the line characteristic impedance Z_c. The reflected current has the opposite sign because its direction of travel opposes the fixed positive current reference. This sign will control the short/open boundary conditions and scattering definitions.
0204.01.2 — Solve for incident and reflected voltage amplitudes
At one reference plane z=0, we have V=V⁺+V⁻ and Z_cI=V⁺−V⁻. Adding and subtracting gives V⁺=(V+Z_cI)/2 and V⁻=(V−Z_cI)/2; these are local traveling voltage amplitudes of the line.
No term labeled forward voltage may be read as the full measured port voltage when a reflection is present. The decomposition requires a specified Z_c and modal line model. In an ideal lossless 50 Ω coax or microstrip approximation Z_c is real, but lossy or multimode networks demand additional care.
0304.01.3 — Reflection from a load boundary
At the load plane, the reflection coefficient is defined as Γ_L=V⁻/V⁺. Substituting the traveling-wave decomposition into Z_L=V/I produces Γ_L=(Z_L−Z_c)/(Z_L+Z_c). The result follows from the boundary condition, not from a statistical power definition.
For passive purely resistive load and real positive Z_c, Γ_L lies between −1 and +1. A conjugate-matched reactive case requires the full complex expression, and the simple range statement cannot be applied blindly to arbitrary complex impedance.
Figure 1.1 — The load constrains the total field
One question: how must V⁺ and V⁻ combine so the total load voltage and current satisfy ZL at z=0?
The line occupies z<0 and the load is placed at z=0. The iL arrow keeps RF-00's fixed +z current reference; vL runs from − to +. The V⁺ and V⁻ arrows show voltage-wave propagation directions, not two current references.
0404.01.4 — Match, open and short terminations
At z=0 the boundary values are V(0)=V⁺(1+Γ_L) and I(0)=V⁺(1−Γ_L)/Z_c. A matched load Z_L=Z_c yields Γ_L=0, so there is no reflected wave.
An ideal open has I(0)=0 and hence Γ_L=+1; the load-end voltage doubles. An ideal short has V(0)=0 and Γ_L=−1, while the current doubles relative to the incident current. The signs are a direct test of the current convention.
0504.01.5 — Standing waves from interference
For a lossless line, set γ=jβ and write Γ_L=|Γ_L|exp(jφ). From V(z)=V⁺exp(−jβz)[1+Γ_L exp(+2jβz)], taking the modulus squared gives the standing-wave envelope with cosine phase φ+2βz.
Thus neighboring voltage maxima repeat every λ_g/2 and a maximum-to-minimum displacement is λ_g/4. The voltage standing-wave ratio is (1+|Γ|)/(1−|Γ|). It measures reflection magnitude but cannot by itself determine the absolute reflection phase.
0604.01.6 — Moving the reference plane
The local reflection coefficient compares the two traveling amplitudes at an observation plane. Since their longitudinal phases advance in opposite directions, Γ(z)=Γ_L exp(+2jβz) for z<0 measured toward the generator from the load plane at z=0.
Define d=−z≥0 as physical distance toward the generator: Γ(d)=Γ_L exp(−2jβd). This clockwise complex-plane rotation under increasing d is the same reference-plane convention used in RF-01 Smith-chart lessons. Magnitude stays unchanged only for an ideal lossless uniform line.
0704.01.7 — Peak-phasor wave power balance
For a lossless line with real Z_c and peak phasors, the forward and backward average powers are |V⁺|²/(2Z_c) and |V⁻|²/(2Z_c). Interference does not create net power: its cross terms cancel in the real part of VI* when Z_c is real.
Therefore P_net=P⁺−P⁻ and reflected power fraction equals |Γ_L|². At Z_c=50 Ω, V⁺=1 V peak and a resistive 75 Ω load, Γ_L=0.2, incident power 10 mW, reflection 0.4 mW and load acceptance 9.6 mW. These values describe power at the specified plane, not instrument calibration.
0804.01.8 — Line Zc versus measurement reference impedance
A modal characteristic impedance Z_c is a property of the line and propagation mode; a VNA reference impedance R_ref is a selected normalization in the data format. They coincide in many 50 Ω systems but are not synonyms, and arbitrary complex reference impedances require carefully distinguished wave definitions.
The following RF-04 lessons will formalize voltage versus power wave normalization, the S matrix, reciprocity and passivity. Here we use a real positive reference equal to Z_c only when converting our voltage amplitudes to waves, so that the factor one half in peak-phasor power remains explicit.
Worked practice — traveling waves and reflection
- Derive voltage and current forward/backward solutions from the telegrapher equations and state the sign convention.
- Recover V⁺ and V⁻ from V and I measured at one reference plane.
- Calculate the reflection coefficient for Z_c=50 Ω and Z_L=75 Ω.
- Derive Γ_L=+1 for an ideal open and Γ_L=−1 for a short from the boundary conditions.
- For |Γ|=0.2 calculate VSWR and the maximum/minimum voltage ratio.
- Derive the spatial cosine factor in |V(z)|² and give the distance between neighboring extrema.
- For Γ_L=0.5∠30° and d=λ_g/8, calculate Γ(d) on a lossless line.
- At V⁺=1 V peak, Z_c=50 Ω and Γ_L=0.2, compute incident, reflected and accepted powers.
- Explain the difference between line characteristic impedance and chosen VNA reference impedance.
- Describe why simple positive/negative traveling-wave powers require qualification for a lossy or multimode guide.
Worked solutions
1. With +jωt, uniform line solutions have exp(−γz) and exp(+γz). Because the reference current points toward +z, forward current is +V⁺/Z_c while backward current is −V⁻/Z_c. Adding the two yields the equations for V(z) and I(z).
2. At z=0, V=V⁺+V⁻ and Z_cI=V⁺−V⁻. Add to obtain 2V⁺=V+Z_cI, and subtract to obtain 2V⁻=V−Z_cI. Divide both equations by two; the decomposition presumes an identified line characteristic impedance.
3. Γ_L=(75−50)/(75+50)=25/125=0.2. Thus 20% of the incident voltage amplitude is reflected with the same phase at the load, but the reflected power fraction is 0.2²=0.04.
4. For an open, I(0)=V⁺(1−Γ_L)/Z_c=0, so Γ_L=1 and V(0)=2V⁺. For a short, V(0)=V⁺(1+Γ_L)=0, hence Γ_L=−1 and I(0)=2V⁺/Z_c.
5. VSWR=(1+0.2)/(1−0.2)=1.5. The peak voltage envelope is 1.2|V⁺|, the minimum is 0.8|V⁺|, and their ratio is 1.5.
6. V(z)=V⁺e^(−jβz)(1+Γ_L e^(+2jβz)). Square its modulus to obtain |V|²=|V⁺|²[1+|Γ|²+2|Γ|cos(φ+2βz)]. Period 2π in 2βz corresponds to λ_g/2; maxima/minima alternate after λ_g/4.
7. Move d=λ_g/8 toward the generator: 2βd=2(2π/λ_g)(λ_g/8)=π/2=90°. Thus Γ(d)=0.5∠(30°−90°)=0.5∠−60°.
8. Incident power is |V⁺|²/(2Z_c)=1/(100)=0.01 W=10 mW. Reflected power is Γ²×10=0.4 mW. Net accepted power at the lossless load is 10−0.4=9.6 mW.
9. Z_c depends on transmission-line fields, geometry and materials. The VNA reference R_ref is the numerical and measurement-port wave normalization; although both are often 50 Ω they can differ, requiring a deliberate conversion.
10. For a lossy line Z_c may be complex and each wave decays spatially. Then the simple real-power decomposition |V⁺|²/(2Z_c)−|V⁻|²/(2Z_c) is not a valid real-valued formula; specify physical power and modal normalization instead.
Next: reference impedances, voltage-wave and power-wave normalization, including unequal and complex references.