### Lossless transmission line

In many practical applications, conductor loss is low , and dielectric leakage is low . These two conditions describe a lossless transmission line.

In this case, the transmission line parameters are

- Propagation constant
- Transmission line impedance will be defined in the next section, but it is also
here for completeness.
- Wave velocity
- Wavelength

### Voltage and current on lossless transmission line

On a lossless transmission line, where current and voltage simplify to

### What does it mean when we say a medium is lossy or lossless?

In a lossless medium, electromagnetic energy is not turning into heat; there is no amplitude loss. An electromagnetic wave is heating a lossy material; therefore, the wave’s amplitude decreases as .

medium | attenuation constant [dB/km] |

coax | 60 |

waveguide | 2 |

fiber-optic | 0.5 |

In guided wave systems such as transmission lines and waveguides, the attenuation of power with distance follows approximately . The power radiated by an antenna falls off as . As the distance between the source and load increases, there is a specific distance at which the cable transmission is lossier than antenna transmission.

### Low-Loss Transmission Line

This section is optional.

In some practical applications, losses are small, but not negligible.
^{1}and
^{2}.

In this case, the transmission line parameters are

- Propagation constant
We can re-write the propagation constant as shown below. In somel applications, losses are small, but not negligible. and , then in Equation lossytl2, .

Taylor’s series for function in Equation lowtleq1 is shown in Equations taylorser1-taylorser2.

The real and imaginary part of the propagation constant are:

We see that the phase constant is the same as in the lossless case, and the attenuation constant is frequency independent. All frequencies of a modulated signal are attenuated the same amount, and there is no dispersion on the line. When the phase constant is a linear function of frequency, , then the phase velocity is a constant , and the group velocity is also a constant, and equal to the phase velocity. In this case, all frequencies of the modulated signal propagate at the same speed, and there is no distortion of the signal.

### Transmission-line parameters R, G, C, and L

To find the complex propagation constant , we need the transmission-line parameters R, G, C, and L. Equations for R, G, C, and L for a coaxial cable are given in the table below.

Transmission-line | R | G | C | L |

Coaxial Cable | ||||

Where is the resistance associated with skin-depth. is the frequency of the signal, is the magentic permeability of conductors, is the conductivity of conductors.