Wave propagation in a transmission line is the movement of voltage and current along the conductors as a travelling wave, with energy carried in the electric and magnetic fields around them. The wave moves at the phase velocity, which is set mainly by the dielectric, and reflects wherever the impedance changes. Forward and reflected waves combine to give the voltage actually measured at any point.
This article looks at propagation in both the frequency and time domains. The background is in the introduction to transmission line theory, and the quantities used here are defined under transmission line parameters. All of it sits within the RF fundamentals section.
The wave equation on a line
Combining the telegrapher’s equations gives a wave equation for voltage. On a lossless line:
∂2V/∂z2 = LC ∂2V/∂t2
Its general solution is any function of z − vt plus any function of z + vt, with v = 1/√(LC). One moves forward and the other moves backward, and neither changes shape on a lossless, dispersion-free line. This is the point often missed in a first course: a line does not only carry sine waves, and pulses travel along it intact.
For a sine wave at angular frequency ω the forward component is:
v(z, t) = V+ e−αz cos(ωt − βz)
The term ωt − βz shows the wave moving. A point of constant phase moves at dz/dt = ω/β. The factor e−αz shows the decay caused by loss, which is the subject of transmission line losses and power handling capacity.
How energy moves along the line
Current in the conductors and voltage between them are two views of the same field. In a TEM line the electric and magnetic fields are perpendicular to each other and to the direction of travel, and the Poynting vector gives the power flow through the space around the conductors. The conductors guide the wave, but most of the power travels in the dielectric, which is why dielectric material sets the speed. A line with a mixed dielectric, as in microstrip, gives a speed between those of air and the substrate.
Phase velocity and group velocity
Phase velocity vp = ω/β is the speed of a point of constant phase. Group velocity vg = dω/dβ is the speed of the envelope, and so of the information and energy. On an ideal TEM line, ω and β are proportional, so vp = vg. Where they differ, the line is dispersive.
Dispersion in waveguide
A hollow waveguide is strongly dispersive near cutoff. For the TE10 mode, with fc = c/2a:
vp = c / √(1 − (fc/f)2) and vg = c √(1 − (fc/f)2)
In WR-90, with a = 22.86 mm, fc = 6.557 GHz. At 10 GHz, vp = 1.32c and vg = 0.755c, and the guide wavelength is 39.7 mm. The phase velocity exceeds c, but no information travels at that speed. The product vpvg equals c2.
Microstrip also shows mild dispersion, since the field moves further into the substrate at higher frequency and εeff rises. Wideband signals on long lines can suffer from the resulting group delay variation.
Step response of a line
The time domain gives the clearest view of propagation. When a source drives a line, it first sees only Z0, since the wave has no knowledge of the load. The first wave launched is set by a voltage divider between the source resistance Rs and Z0:
V1 = Vs × Z0 / (Z0 + Rs)
That wave travels to the far end, arrives one delay later, and reflects with ΓL. The reflected wave returns and reflects again at the source with ΓS. The voltage at the load is the running sum of all the arrivals. This bookkeeping is called a lattice or bounce diagram, and it is the basis of time-domain reflectometry.
Worked example: bounce sequence on a mismatched line
A 1 V step drives a 2 m, 50 Ω coaxial line with velocity factor 0.66. The source resistance is 25 Ω and the load is 100 Ω.
- One-way delay: 2 / (0.66c) = 10.1 ns.
- Launched wave: 1 × 50 / 75 = 0.667 V.
- Load reflection: ΓL = (100 − 50)/(100 + 50) = +1/3.
- Source reflection: ΓS = (25 − 50)/(25 + 50) = −1/3.
Each arrival at the load adds the incoming wave multiplied by 1 + ΓL, and the incoming wave falls by a factor of ΓLΓS = −1/9 per round trip.
| Time (ns) | Change at load (V) | Load voltage (V) |
|---|---|---|
| 10.1 | +0.889 | 0.889 |
| 30.3 | −0.099 | 0.790 |
| 50.5 | +0.011 | 0.801 |
| 70.8 | −0.001 | 0.800 |
The result settles to the DC value 1 × 100/125 = 0.80 V, as it must. Figure 1 plots the sequence. Note the overshoot on the first arrival, which is a real voltage on the load, and comes entirely from the line.
The transmission line terminations article explains how to remove such ringing by matching the source or load.
Standing waves
With a continuous sine wave the forward and reflected waves overlap everywhere. Their sum at distance d from the load has the magnitude:
|V(d)| = |V+| × |1 + Γ e−j2βd|
The voltage is no longer equal along the line. It has fixed maxima and minima, spaced a quarter wavelength apart, as Figure 2 shows. The envelope repeats every half wavelength, not every wavelength, because the reflected wave accumulates phase on both the outward and return journeys.
The ratio of maximum to minimum is the voltage standing wave ratio, or VSWR. It equals (1 + |Γ|)/(1 − |Γ|), and it is the subject of the VSWR and return loss page. The standing wave is why impedance varies along a line: voltage and current change with position, so their ratio does too. The Smith chart records this as a rotation around the chart centre.
Effect of loss on a travelling wave
Loss changes the picture in two ways. The wave decays as it moves, so forward and reflected waves are unequal at most points and the standing wave is strongest near the mismatch. Loss also depends on frequency, so a sharp step loses its fast edges first. This is why a pulse arriving at the end of a long cable has a rounded rise, even when the line is well matched.
Frequently asked questions
How fast does a signal travel in a transmission line?
It travels at c/√εeff. That is about 0.70c in solid PTFE coax, typically around 0.85c in foam cable, and near 0.55c in FR-4 microstrip. The types of transmission lines article gives more detail on velocity factor.
What is the difference between a travelling wave and a standing wave?
A travelling wave moves along the line and has the same amplitude at every point, apart from loss. A standing wave is the fixed pattern produced by forward and reflected waves together. Energy still travels on a partly reflected line, but the amplitude varies with position.
Why can phase velocity exceed the speed of light in a waveguide?
Phase velocity describes the movement of a phase pattern, not of energy or information. The group velocity, which carries the energy, stays below c. The wave can be pictured as bouncing between the guide walls at an angle, with the pattern on the wall moving faster than the wave itself.
Does the wave know about the load when it first starts?
No. The source sees Z0 until a reflection arrives, one round trip later. For this reason, the launched voltage depends on Z0 and not on the load. With a continuous wave this start-up has long since finished, and the steady-state input impedance applies.
Related topics
The size of each reflection is covered in reflection and transmission coefficients, and measured wave ratios are expressed through S-parameters.