Waves on Transmission Lines

This section should derive the existence of the voltage and current waves on a transmission line. This way, it also proofs that the definitions from the last section make sense.

Figure 1.1: Infinite short piece of transmission line

Figure 1.1 shows the equivalent circuit of an infinite short piece of an arbitrary transmission line. The names of the components all carry a single quotation mark which indicates a per-length quantity. Thus, the units are ohms/m for $ R'$, henry/m for $ L'$, siemens/m for $ G'$ and farad/m for $ C'$. Writing down the change of voltage and current across a piece with length $ \partial z$ results in the transmission line equations.

$\displaystyle \dfrac{\partial u}{\partial z} = - R' \cdot i(z) - L' \cdot \dfrac{\partial i}{\partial t}$ (1.5)

$\displaystyle \dfrac{\partial i}{\partial z} = - G' \cdot u(z) - C' \cdot \dfrac{\partial u}{\partial t}$ (1.6)

Transforming these equations into frequency domain leads to:

$\displaystyle \dfrac{\partial\underline{U}}{\partial z} = -\underline{I}(z)\cdot (R' + j\omega L')$ (1.7)

$\displaystyle \dfrac{\partial\underline{I}}{\partial z} = -\underline{U}(z)\cdot (G' + j\omega C')$ (1.8)

Taking equation 1.8 and setting it into the first derivative of equation 1.7 creates the wave equation:

$\displaystyle \dfrac{\partial^2\underline{U}}{\partial z^2} = \underline{\gamma}^2 \cdot \underline{U}$ (1.9)

with $ \underline{\gamma}^2 = (\alpha+j\beta)^2 = (R'+j\omega L')\cdot(G'+j\omega C')$. The complete solution of the wave equation is:

$\displaystyle \underline{U}(z) = \underbrace{\underline{U}_1 \cdot \exp(-\under...
...ace{\underline{U}_2 \cdot \exp(\underline{\gamma}\cdot z)}_{\underline{U}_b(z)}$ (1.10)

As can be seen, there is a voltage wave $ \underline{U}_f(z)$ traveling forward (in positive $ z$ direction) and there is a voltage wave $ \underline{U}_b(z)$ traveling backwards (in negative $ z$ direction). By setting equation 1.10 into equation 1.7, it becomes clear that the current behaves in the same way:

$\displaystyle \underline{I}(z) = \underbrace{\dfrac{\underline{\gamma}}{R'+j\om...
...U}_f(z) - \underline{U}_b(z) \right) =: \underline{I}_f(z) + \underline{I}_b(z)$ (1.11)

Note that both current waves are counted positive in positive $ z$ direction. In literature, the backward flowing current wave $ \underline{I}_b(z)$ is sometime counted the otherway around which would avoid the negative sign within some of the following equations.
Equation 1.11 introduces the characteristic admittance $ \underline{Y}_L$. The propagation constant $ \underline{\gamma}$ and the characteristic impedance $ \underline{Z}_L$ are the two fundamental properties describing a transmission line.

$\displaystyle \underline{Z}_L = \dfrac{1}{\underline{Y}_L} = \dfrac{\underline{...
..._b} = \sqrt{\dfrac{R'+j\omega L'}{G'+j\omega C'}} \approx \sqrt{\dfrac{L'}{C'}}$ (1.12)

Note that $ \underline{Z}_L$ is a real value if the line loss (due to $ R'$ and $ G'$) is small. This is often the case in reality. A further very important quantity is the reflexion coefficient $ \underline{r}$ which is defined as follows:

$\displaystyle \underline{r} = \dfrac{\underline{U}_b}{\underline{U}_f} = -\dfra...
... = \dfrac{\underline{Z}_e - \underline{Z}_L}{\underline{Z}_e + \underline{Z}_L}$ (1.13)

The equation shows that a part of the voltage and current wave is reflected back if the end of a transmission line is not terminated by an impedance that equals $ \underline{Z}_L$. The same effect occurs in the middle of a transmission line, if its characteristic impedance changes.

\fbox{$\underline{U} = \underline{U}_f + \underline{U}_b$} \fbox{$\underline{I} = \underline{I}_f + \underline{I}_b$}
\fbox{$\underline{U}_f = \frac{1}{2}\cdot (\underline{U} + \underline{I}\cdot\underline{Z}_L)$} \fbox{$\underline{I}_f = \frac{1}{2}\cdot (\underline{U}/\underline{Z}_L + \underline{I})$}
\fbox{$\underline{U}_b = \frac{1}{2}\cdot (\underline{U} - \underline{I}\cdot\underline{Z}_L)$} \fbox{$\underline{I}_b = \frac{1}{2}\cdot (\underline{I} - \underline{U}/\underline{Z}_L)$}

This document was generated by Stefan Jahn on 2007-12-30 using latex2html.