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.

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

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

Transforming these equations into frequency domain leads to:

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

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

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

$$\dfrac{\partial^2\underline{U}}{\partial z^2} = \underline{\gamma}^2 \cdot \underline{U}$$ (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:

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

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

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

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