1---
2app_version: 0.7.0
3precision: 3
4form_title: Transmission-Line Impedance Transformation
5form_description: What impedance does a length of feedline actually present at its input? Enter the line's Z0, the load impedance, its physical length and velocity factor, and the frequency.
6form_inputs: Line impedance Z0 (Ω)|z0, Load resistance (Ω)|r_load, Load reactance (Ω)|x_load, Physical length (m)|len_m, Velocity factor|vf, Frequency (MHz)|freq_mhz
7form_outputs: |Z_in| (Ω)|z_in_mag, Phase angle|z_in_phase, VSWR at the load|vswr
8---
9# Transmission-Line Impedance Transformation
10// "My antenna isn't 50 Ω, but a random length of feedline changes what my
11// rig actually sees — what IS it seeing?"
12//
13// The line-input impedance equation, exact for any lossless line length:
14// Zin = Z0 * (ZL + i*Z0*tan(βl)) / (Z0 + i*ZL*tan(βl))
15// β = 2π/λ, so βl is a pure ratio (radians) — `in deg` shows it in degrees.
16
17z0 = 50 // Ω, the feedline's characteristic impedance
18r_load = 120 // Ω, load resistance (e.g. an off-resonance dipole)
19x_load = 80 // Ω, load reactance (+inductive, −capacitive)
20len_m = 12.5 // m, physical length of the run
21vf = 0.66 // velocity factor (0.66 typical for solid-PE coax)
22freq_mhz = 14.2 // MHz
23
24z_load = r_load + x_load * 1i
25
26speed_of_light = 299792458 // m/s
27wavelength_m = speed_of_light / (freq_mhz * 1e6) * vf
28beta_l = 2pi * len_m / wavelength_m // electrical length, radians
29t = tan(beta_l)
30
31z_in = z0 * (z_load + z0 * t * 1i) / (z0 + z_load * t * 1i)
32z_in_mag = abs(z_in)
33z_in_phase = arg(z_in) in deg
34
35"Load impedance": z_load
36"Electrical length": beta_l in deg
37"Input impedance": z_in
38"|Z_in| (Ω)": z_in_mag
39"Phase angle": z_in_phase
40
41# How mismatched is this, really?
42gamma = abs((z_load - z0) / (z_load + z0))
43vswr = (1 + gamma) / (1 - gamma)
44
45"Reflection coefficient |Γ|": gamma
46"VSWR at the load": vswr
47
48// A special case worth knowing: an exact quarter-wave (βl = 90°) always
49// gives Zin = Z0² / ZL, independent of everything else about the line.
50qw_check = z0^2 / z_load
51"Zin an exact ¼λ of this Z0 would give": qw_check
52