# How do you calculate the impedance of a parallel-wire balanced feedline (ladder line)?

I've looked around and maybe I'm just searching for the right terms to find the answer. I know that the impedance of ladder line has to do with the distance between wires:

What is the generalized form to calculate feedline impedance between two parallel wires separated by some distance?

Does velocity factor or wire radius affect that impedance?

• There is an entire chapter in The ARRL Handbook on this. Parameters are the diameter of each wire, the distance between them, and the dielectric constant of the material between them. I think the dielectric parameters control the velocity factor. The equation works for both parallel wire transmission line and coax (with one diameter negative). Nov 25, 2021 at 1:27
• I remember seeing the transmission line equation for characteristic impedance, but I'm not finding a direct reference for it. Nov 25, 2021 at 2:19

Yes, wire diameter matters. Assuming the two wires are round and of equal diameter, the impedance is $$\frac{120}{\sqrt{\epsilon_r}} \mathrm{acosh}\,\frac{D}{d}$$, where $$\epsilon_r$$ is the relative permittivity (dielectric constant) of the material between the wires, $$D$$ is the distance between wire centers, and $$d$$ is the diameter of the wires.

In cases where the spacing is significantly greater than the diameter, you can use the simpler formula $$\frac{120}{\sqrt{\epsilon_r}} \ln \frac{2D}{d}$$. This has less than 1% error when $$\frac{D}{d}$$ is more than 3.6.

The velocity factor is simply $$\frac{1}{\sqrt{\epsilon_r}}$$, as user10489 points out.

Estimating $$\epsilon_r$$ can be tricky; for ladder line you need to figure something based on the material and the "fill factor" (how much of the space between the wires is occupied by plastic and how much of it is air). The goal of "open wire" feedline is to reduce this fill factor to nearly 0, which makes $$\epsilon_r$$ decrease to nearly 1, and VF increase to nearly 1.

The dielectric constant of polyethylene is around 2.3; the $$\epsilon_r$$ of ladder lines made with PE webbing is between around 1.2 and 1.55, based on published velocity factors between 0.8 and 0.91.

• 120 is actually an approximation for the impedance of free space over pi... but the impedance of free space happens to be really close to 120 pi ohms, with less than 0.1% error. Nov 25, 2021 at 4:25
• I seem to remember a more generalized form of this equation that also handled different wire diameters. Also, it it correct to assume that the dielectric of webing would be the weighted (by linear space) average of the material and air? Nov 26, 2021 at 12:45
• @user10489 I think it's closer to volume (roughly of the rectangle between the two wires) than area, but it's probably more complicated than that. I don't think anyone really computes it on paper, you either do a numerical simulation with something fancy enough to model the dielectric, or you just build it and measure :) Nov 28, 2021 at 3:40
• @user10489 and yes, there is a formula for differing wire diameters, I don't remember anymore where I saw it. And lossy feedline is a whole other ball game. Nov 28, 2021 at 3:58

The ARRL Handbook and the ARRL Antenna Book both have an entire chapter on transmission lines. However, the chapter in the Antenna book is more detailed.

Velocity factor is $$VF = \frac{1}{\sqrt\epsilon}$$ where $$\epsilon$$ is the dielectric constant of the material between the wires.

• While this is true, I'm not sure it answers the question since the question was asking what the formula for impedance is. Nov 25, 2021 at 14:29
• I had intended to add the other equation, but I can't find a generalized form of it. I keep finding forms that leave out terms by making specific assumptions, like lossless, air dielectric, or coax. Still looking. Nov 25, 2021 at 16:51