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
  $$\begin{eqnarray} \gamma =\sqrt{(R+j\omega L)(G+ j\omega C)} \nonumber \\ \nonumber \gamma= \sqrt{j \omega L j \omega C} \\ \nonumber \gamma = j \omega \sqrt{L C} = j \beta \end{eqnarray}$$
• Transmission line impedance will be defined in the next section, but it is also here for completeness.
  $$\begin{eqnarray} Z_0=\sqrt{\frac{R+j\omega L}{G+ j\omega C}} \nonumber \\ \nonumber Z_0=\sqrt{\frac{j\omega L}{ j\omega C}} \\ \nonumber Z_0=\sqrt{\frac{L}{C}} \end{eqnarray}$$
• Wave velocity
  $$\begin{eqnarray} v=\frac{\omega}{\beta} \nonumber \\ \nonumber v=\frac{\omega}{\omega \sqrt{LC}} \\ \nonumber v=\frac{1}{\sqrt{LC}} \end{eqnarray}$$
• Wavelength
  $$\begin{eqnarray} \lambda = \frac{2 \pi}{\beta} \nonumber \\ \nonumber \lambda = \frac{2 \pi}{ \omega \sqrt{LC}} \\ \nonumber \lambda =\frac{2 \pi}{\sqrt{\epsilon_0 \mu_0 \epsilon_r} } \\ \nonumber \lambda = \frac{c}{f \sqrt{\epsilon_r}} \\ \nonumber \lambda = \frac{\lambda_0}{\sqrt{\epsilon_r}} \nonumber \end{eqnarray}$$

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 $e^{-\alpha x}$.

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, $RG<< \omega ^2 LC$.

  $$\begin{eqnarray} \gamma =\sqrt{(R+j\omega L)(G+ j\omega C)} \\ \gamma= j\omega \sqrt{ L\, C}\sqrt{1\,-\,j\,\left( \frac{R}{\omega L}+\frac{G}{\omega C} \right)-\frac{R G}{\omega^2 L C}} \label{lossytl2} \\ \gamma\approx j\omega \sqrt{ L\, C}\sqrt{1\,-\,j\,\left( \frac{R}{\omega L}+\frac{G}{\omega C} \right)}\label{lowtleq1} \end{eqnarray}$$

  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.

  $$\begin{eqnarray} \sqrt{1+x}=1+\frac{x}{2}-\frac{x^2}{8}+\frac{x^3}{16}-... \,for\, |x|<1 \label{taylorser1} \\ \gamma \approx j\omega \sqrt{ L\, C} \sqrt{1\,-\,j\,\left( \frac{R}{\omega L}+\frac{G}{\omega C} \right)}= j\omega \sqrt{ L\, C}\left(1-\frac{j}{2} \left( \frac{R}{\omega L}+\frac{G}{\omega C} \right)\right) \label{taylorser2} \end{eqnarray}$$

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

  $$\begin{eqnarray} \alpha= \frac{ \omega \sqrt{ L\, C} }{2} \left( \frac{R}{\omega L}+\frac{G}{\omega C} \right) \\ \beta= \omega \sqrt{ L\, C} \end{eqnarray}$$

  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.

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.