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### Lossless transmission line

In many practical applications, conductor loss is low $R\to 0$, and dielectric leakage is low $G \to 0$. 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 $\gamma = j\beta$ 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 $e^{-\alpha x}$.

 medium attenuation constant $\alpha$ [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 $e^{-2\alpha x}$. The power radiated by an antenna falls off as $1/r^{2}$. 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. $R<< \omega L$ 1and $G << \omega C$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. $R<< \omega L$ and $G << \omega C$, then in Equation lossytl2, $RG<< \omega ^2 LC$.

Taylor’s series for function $\sqrt {1+x}= \sqrt {1\,-\,j\,\left ( \frac {R}{\omega L}+\frac {G}{\omega C} \right )}$ in Equation lowtleq1 is shown in Equations taylorser1-taylorser2.

The real and imaginary part of the propagation constant $\gamma$ are:

We see that the phase constant $\beta$ is the same as in the lossless case, and the attenuation constant $\alpha$ 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, $\beta =const \, \omega$, then the phase velocity is a constant $v_p=\frac {\omega }{\beta }=\frac {1}{const}$, 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 $\gamma$, 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 $\frac {R_{sd}}{2 \pi } \left (\frac {1}{a} + \frac {1}{b} \right )$ $\frac {2 \pi \sigma }{\ln b/a}$ $\frac {2 \pi \epsilon }{\ln b/a}$ $\frac {\mu }{2 \pi } \ln b/a$

Where $R_{sd} = \sqrt {\pi f \mu _m/\sigma _m}$ is the resistance associated with skin-depth. $f$ is the frequency of the signal, $\mu _m$ is the magentic permeability of conductors, $\sigma _c$ is the conductivity of conductors.