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The 3 most common devices in electric circuits are inductors, resistors and capacitors. The relation between current and (terminal) voltage across these elements is often used in circuit analysis, namely: \begin{align} U = RI \\ U = L \frac{d}{dt} I \\ \frac{d}{dt} U = \frac{I}{C} \end{align}

When the assumptions of the lumped element model (no charges in inductors connecting the elements, no change of magnetic flux outside elements) don't hold anymore, people mostly worry about wether Kirchhoff's laws are still applicable. I'd like to ask another question, namely wether the relations written above can still hold, even with the most ideal resistors, inductors and capacitors possible.

My own answer would be that this is impossible, but I'm not sure wether I made a mistake:

Let's assume there is an electric field $\vec{E}_0$ throughout the whole circuit, created by charges inside and (possibly) outside the circuit. If I define voltage as the negative line integral of the overall electric field, then the voltage-drop across a capacitor is affected by $\vec{E}_0$, and doesn't depend solely on $Q$, the charge inside the capacitor, anymore. However, defining voltage as the negative line integral of the overall electric field is the only definition by which $U=RI$ holds, because this is just a reformulation of $$\vec{j} = \sigma \vec{E}$$

Is my reasoning valid, and this is another caveat when talking about "voltage" outside the realm of the lumped-circuit model?

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In microwave circuit analysis, it is standard practice to introduce voltages, currents, impedances, etc., such that Kirchhoff's laws (1&2) hold. This is based on the observation that waveguides sustain a well-defined discrete set of propagating modes. Take, for example, a junction to which one or several waveguides are connected and launch a wave in one of the guides far away from the junction so that all the evanescent modes between the launcher or loads and the junction can be ignored, and the frequency is such that only a single propagating mode may exist in each waveguide.

Let $V_k, I_k$ denote the amplitudes of the $E$ and $H$ propagating field, resp., in the $k^{th}$ guide. Then it is possible to show that there is a matrix $\bf{Z}$ or $\bf{Y}=\bf{Z}^{-1}$ that depends only on the geometry of the conductors and $\bf{V}=\bf{Z}\bf{I}$ or $\bf{I}=\bf{Y}\bf{V}$. (This is almost always true except for a handful of singular cases, such as open or short circuits, etc., but these can still be handled with scattering matrices that always exist.) For an excellent description see, [1]

[[1]: https://archive.org/details/in.ernet.dli.2015.16056/page/n145/mode/2up Montgomery, Dicke, Purcell: PRINCIPLES OF MICROWAVE CIRCUITS, chapter 5]

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  • $\begingroup$ This looks VERY promising! Just for my understanding (I'm not familiar with the terminology, especially with the term "junction": is junction essentially what I called "element" in my question above? $\endgroup$ Jun 19, 2020 at 21:55
  • $\begingroup$ Imagine a circular pipe and a rectangular pipe, and between them some kind of box (maybe filled with dielectric, ferrite, etc.). The pipes are connected to the box. The box is the junction and it can be represented as a circuit of a theoretically infinite number of $i=Cdv/dt$ capacitors and $v=Ldi/dt$ inductors. $\endgroup$
    – hyportnex
    Jun 19, 2020 at 22:00
  • $\begingroup$ it may take some time for me to appreciate your answer, because the introduction of "voltage" is totally different to anything I've ever seen before. I'm reading through chaper five at the moment. In the beginning, the author mentions Kirchhofs laws. Are they proven for "his" definition of voltage later on? $\endgroup$ Jun 19, 2020 at 22:40
  • $\begingroup$ @hypnortnex I'm reading through chapter 5 at the moment. What still is a missing puzzle piece kind of is that after the "new" definitions of voltage and current are introduced, the author neither reconciles them with the "standard" definition, nor does he give a proof for kirchhoffs laws. Do you know where to find those in the Book you mentioned? $\endgroup$ Jun 20, 2020 at 11:17
  • $\begingroup$ if you really want to get into this then read Collin:Field Theory of Guided Waves or Collin: FOUNDATIONS FOR MICROWAVE ENGINEERING. The "Theory" is more advanced and is only about "propagation" stuff, the "Foundations" is simpler but more also about other unrelated stuff, as well. The age-old standard is actually the Montgomery book, read the earlier chapters, or Marcuvitz: Waveguide Handbook archive.org/details/WaveguideHandbook/page/n9/mode/2up, first few chapters. Enjoy! (I did....) $\endgroup$
    – hyportnex
    Jun 20, 2020 at 13:38

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