Circuit Theory

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.

We have seen how Maxwell's equations can be reduced to explain electro- and magnetostatic phenomena. In the employed descriptions, electric and magnetic field are independent of each other. In electrostatics we assumed a curl-free electric field, \(\nabla\times\mathbf{E}\left(\mathbf{r}\right)=0\). This equation is now replaced by the so-called Maxwell-Faraday equation \[\nabla\times\mathbf{E}\left(\mathbf{r},t\right)   =   -\partial_{t}\mathbf{B}\left(\mathbf{r},t\right)\ \]whereas the other equations, \(\nabla\cdot\mathbf{E}\left(\mathbf{r},t\right)=\rho\left(\mathbf{r},t\right)/\varepsilon_{0}\), \(\nabla\times\mathbf{B}\left(\mathbf{r},t\right)=\mu_{0}\mathbf{j}\left(\mathbf{r},t\right)\) and \(\nabla\cdot\mathbf{B}\left(\mathbf{r},t\right)=0\) remain unchanged with respect to an explicit time dependency. Please note that we do not work with the full Maxwell eqquations yet. This version is often termed the slowly varying fields approximation of Maxwell's equations and neglects radiation contributions. This formulation is nevertheless extremely useful. Employing the modularization of electrical engineering boiling down electrodynamics to the interaction of the three basic passive elements resistor, capacitor, we will be able to understand a lot of basic phenomena. Thus, we will have a good starting point for more involved circuit theory aspects and a decent starting point for full electrodynamics.