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Transmission Lines

Transmission lines are very important **historically** since they enabled the **communication** over long distances. Other than that, they are very interesting from a didactical perspective:

- one-dimensional systems that can be characterized by very
**few parameters** (in the end, it's only the impedance \(Z\))
- very good example for
**slowly varying fields** approximation, where \(\nabla\times\mathbf{B}\left(\mathbf{r},t\right)\approx\mu_{0}\mathbf{j}\left(\mathbf{r},t\right)\) and thus the coupling of electric and magnetic field is solely given by Faraday's law \(\nabla\times\mathbf{E}\left(\mathbf{r},t\right)=-\dot{\mathbf{B}}\left(\mathbf{r},t\right)\)
- fundamental physical effects like
**wave propagation** and **reflection** can be explained which will be a good bridge to complicated relativistic phenomena

With these facts in mind, we hereby dedicate a whole section to transmission lines.

One of the most important concepts in **optics** are Fabry-Perot resonances of certain **cavities**. Nevertheless, we can understand this resonance phenomenon on the basis of wave propagation in **transmission lines**! You can also find out how this principle is used in **nanophotonics**.

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For technical applications it is extremely important to be able to **transfer** as much **energy** from a transmission line to some load. In this problem we will learn how this is achieved and understand the basic principles signal **reflection** and **impedance matching**.

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We consider a transmission line in a **lumped model**. This will allow you to find the so-called **telegrapher equation** which describes the **signal propagation** along transmission lines in general. You will also see how such devices can even also be used as **qu**antum **bits**.

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