Propagation, attenuation and phase constants

Propagation, attenuation and phase constants

The propagation constant is separated into two components that have very different effects on signals:

The real part of the propagation constant is the attenuation constant and is denoted by Greek lowercase letter (alpha). It causes a signal amplitude to decrease along a transmission line. The natural units of the attenuation constant are Nepers/meter, but we often convert to dB/meter in microwave engineering. To get loss in dB/length, multiply Nepers/length by 8.686. Note that attenuation constant is always a positive number, if it was negative you'd violate the First Law of Thermodymamics (you never get something for nothing!)

The phase constant is denoted by Greek lowercase letter (beta) adds the imaginary component to the propagation constant. It determines the sinusoidal amplitude/phase of the signal along a transmission line, at a constant time. The phase constant's "natural" units are radians/meter, but we often convert to degrees/meter. A transmission line of length "l" will have an electrical phase of l, in radians or degrees. To convert from radians to degrees, multiply by 180/.
The two parts of the propagation constant have radically different effects on a wave. The amplitude of a wave (frozen in time) goes as cosine(l). In a lossless transmission line, the wave would propagate as a perfect sine wave. In real life there is some loss to the transmission line, and that is where the attenuation constant comes in. The amplitude of the signal decays as Exp(-l). The composite behavior of the propagation constant is observed when you multiply the effects of and .
The graph below represents wave propagation in a fairly lossy line, we made it lossy so you could observe the familiar exponential curve of amplitude decay. In this graph, =1 and =0.05. In practice we usually want to minimize loss, but this example is a very lossy line!

Phase constant versus wavenumber

We have a separate page on wavenumber. Phase constant and wavenumber are often treated as the same thing. Indeed, for TEM transmission lines (coax and stripline), the phase constant and wavenumber are equal. Wavequide is one case where you need to understand the difference between the two.

Wavenumber is denoted by lower case "k", and is a measure of how many cycles a wave has in a given length, for a traveling wave that is frozen in time.

Phase constant, phase velocity, frequency and wavelength

Let's examine the relationships between phase constant, frequency, phase velocity and wavelength, Recall that there are 2 radians in a wavelength, therefore the relationship between phase constant and wavelength is simply: