Transmission line in the time domain

Slower than light

In a cable a signal does not travel at the speed of light but slower: the insulation between inner conductor and shield slows it down. By how much is given by the velocity factor VF. For RG58 and RG213 with solid polyethylene it is 0.66, with foam dielectric around 0.8 to 0.85, with ladder line, which is almost all air, around 0.9.

With VF 0.66 the signal needs 5.05 ns per metre. 15 m of RG58 there and back therefore take 151.6 ns. That is exactly what a TDR measures: it sends a steep step in and watches when and how something comes back.

Electrical length

Because the signal travels slower, the wavelength in the cable is shorter too. A λ/4 piece of RG58 for 14.2 MHz is not 5.28 m long but 0.66 times that, so 3.48 m. If you build a stub or a transformer from cable, you need to know the VF. If you don't, measure it: with a known length the time to the echo gives you the VF, with a known VF it gives you the length.

Measure cable length

The example “Measure cable length” sends a short pulse into 30 m of RG213 with an open end. The echo arrives after 303.2 ns. Times 0.66 times the speed of light and divided by two (there and back) that gives 30.0 m again. This is also how you find a crushed spot in the cable up the mast without climbing it. In the TDR plot you can switch the axis to metres: the page then uses the VF of each piece.

Loss

Every cable swallows part of the power, high frequencies more than low ones. A steep step contains lots of high frequencies. After a long, thin cable it therefore arrives flatter: the echo first rises quickly, then creeps slowly to its final value. The simulator treats the loss like skin effect, it grows with the square root of the frequency. That matches datasheets well on HF and the lower VHF range. Above that, dielectric losses add up, which are missing here: RG174 has 33 dB per 100 m at 100 MHz according to its datasheet, here it is 31.4.

Where Z changes, something comes back

A wave on a 50 Ω cable carries voltage and current in a ratio of 50 Ω. When it meets something with a different impedance, the ratio no longer fits. Part of it carries on, the rest comes back. How much is given by the reflection coefficient Γ = (Z − Z0) / (Z + Z0).

• Equal (50 Ω to 50 Ω): Γ = 0, nothing comes back.
• Higher (75 Ω after 50 Ω): Γ = 0.2, the echo has the same sign, the voltage at the start rises.
• Lower (50 Ω after 75 Ω): Γ = −0.2, the voltage at the start falls.
• Open: Γ = 1, everything comes back, the voltage doubles, the current drops to zero.
• Short: Γ = −1, the voltage collapses, the current doubles.

Reading the TDR

After the step, the start of the line first sees the voltage of the forward wave, 1 V by default. Every echo changes it. If it rises to 1.2 V, Γ = 0.2 and there is 50 · 1.2 / 0.8 = 75 Ω at that point. So a TDR shows you not only where something is but also what it is: a hill is more Ω, a dip less, a spike upwards a coil, one downwards a capacitor. Move the pointer over the TDR plot and you'll see below which Z shows up at that point.

This only holds cleanly for the first echo of each point. Later, waves bounce back and forth between the discontinuities and arrive several times. If the source itself is not 50 Ω, it reflects as well.

Coil and capacitor

A coil does not let current flow straight away. For the steep edge it is open at first, the echo jumps up and then sinks to the value of the resistor. A capacitor starts as a short circuit and then charges up. How fast that happens depends on L and C. The example “Antenna with coil” shows a load like an antenna that is too long: 35 Ω with a series coil.

Without formulas, with two ropes and a pulse, the tab What is Γ? in the Smith chart shows it.

A carrier is one step after another

When you key a continuous carrier, the first cycle runs along the cable like a step and comes back from the load. After that, forward and reflected waves run on the cable at the same time and add up. At some points the peaks always meet, so the voltage is high there (antinode). At others a peak always meets a trough, so it is low there (node). The pattern stays put, hence the name standing wave. In the plot “On the line” you see both: the instantaneous voltage and the range it swings in.

Antinodes and nodes

Two nodes are half a wavelength apart, in the cable that is the electrical length: at 14.2 MHz in RG58, 6.97 m. At an open end there is always a voltage antinode and a current node, λ/4 before it exactly the other way round. With the “Current” view you can see that the current antinodes sit at the voltage nodes.

SWR

The SWR is the ratio of antinode to node, (1 + |Γ|) / (1 − |Γ|): at Γ = 0.2 it is 1.5, with 100 Ω on 50 Ω it is 2. On a lossless cable it is the same everywhere. A lossy cable flatters the SWR at the start: what comes back has been attenuated twice. 30 m of RG58 with an open end lose 1.67 dB each way on 14.2 MHz, so at the start you only measure SWR 5.3 instead of infinity.

Measuring

What it all looks like in the frequency domain, with circles around the centre instead of waves on the cable, is shown by the Smith chart. The button under “Values” takes cable and load there, and a link from there brings both back. You can measure your real antenna and cable with a VNA under Measure antenna.

With a probe moving along the cable, and the point running round its circle in the Smith chart, see Cable and Smith.