# What exactly are the field equations that hold in the "distributed-circuit" and the "lumped circuit"-models?

As far as I understood, there are 2 Models for electric circuits that aim to simplify Maxwells Equations (by reducing the number of degrees of freedom from infinite field-values to 2 variables (voltage and current) per circuit element).

I'm however confused about how these models are obtained from Maxwells Equations:

The lumped-circuit model is obtained by assuming the following approximations:

• No change of magnetic flux in every loop of the circuit
• No change of charge in every element of the circuit
• Only Wavelengths that larger then the whole circuits dimensions

I'm especially worried about the 4th point: What does it mean to ignore radiation? Obviously we don't ignore induction effects in coils.

The 2nd point doesn't seem as understandable as well: Imagine a capacitor, and a voltage U that drops across it's plates. The conductors that connect the capacitors plates have the same potential as the respective plates. I now have a hard time imagining that the charge density inside the conductors doesn't change at all when the voltage drop across the capacitor changes. Of course the change density doesn't change enough to affect any electric fields in the lumped circuit model, but it still changes.

What's even more confusing is that the transmission-line model is not deduced from maxwells equations, but instead built up from the lumped-circuit model. I'd like to know what the limitations of this model are, but it seems very complicated to estimate this.

I think all in all my questions can be boiled down to "What are the field equations that hold in the lumped circuit model"? Are there terms originating in Maxwells Equations that are completely ignored? Are there some that are approximated just to quadratic order? Or to linear order? Does the constant c play a role here?

PS: This question is not a duplicate, the following two questions:question 1 und question 2 don't deal with the question on radiation.

• I don't have time for a full answer, but take into account that the derivation of the lumped-circuit model presented in most physics and circuit theory textbooks is way too simplified. If you want a full derivation, have a look at this paper or at R. M. Fano, L. J. Chu, and R. B. Adler, Electromagnetic fields, energy, and forces, MIT Press, 1968. Finally, the transmission line model can be actually derived directly from Maxwell's equations, see e.g. L. B. Felsen et al, Electromagnetic Field Computation by Network Methods, Springer, 2009. Jul 5, 2020 at 15:34
• The linked paper and Fano's book use two completely different approaches, and the latter is probably easier. Jul 5, 2020 at 15:41
• More generally, Felsen, Marcuvitz and Schwinger (yes, the Nobel prize) are those credited for the development of network-theoretical methods for the solution of Maxwell's equations, so you may want to have a look into their works. A derivation of the transmission-line model from Maxwell's equation can for instance be found also in this chapter ny Milton and Schwinger. Jul 5, 2020 at 15:50

What does it mean to ignore radiation?

It means we can assume that all power generated by the power generators in our model is consumed by the power consumers in our model.

It is a necessary approximation to have the potential at each circuit node be uniquely defined.

Obviously we don't ignore induction effects in coils.

The induction effect in a coil is not a result of radiation. It's the result of energy storage in a confined (not radiating) magnetic field.

Imagine a capacitor, and a voltage U that drops across it's plates. The conductors that connect the capacitors plates have the same potential as the respective plates. I now have a hard time imagining that the charge density inside the conductors doesn't change at all when the voltage drop across the capacitor changes.

The lumped circuit approximation is an approximation. In this case we're approximating the change in charge density in the conductors as zero. In many circuits this approximation produces adequately accurate results.

In other circuits, we might add a parasitic capacitance to our model to include this effect, while retaining the lumped circuit approximation otherwise.

In others we might have to forgo the lumped circuit approximation and solve the circuit as a distributed structure.

What's even more confusing is that the transmission-line model is not deduced from maxwells equations, but instead built up from the lumped-circuit model.

As far as I know, you can develop the transmission line model either way. Either start with a finite-element model of inductors and capacitors and show that they support a travelling wave. Or start with the field equations and show that they can be approximated by potentials and currents at the ends of the transmission line.

I'd like to know what the limitations of this model are, but it seems very complicated to estimate this.

Generally, the simple transmission line model fails when the line is not uniform, or when there are multiple modes present on the transmission line, and when coupling between modes needs to be considered. This could include situations like circuit trace bends, transitions between layers on a PCB, transitions between geometries that nominally have the same characteristic impedance but have different mode structures, etc.

Of course we can simply elaborate our quasi-lumped transmission line model to include additional variables and equations to account for these effects, but it's quite easy to find geometries that haven't been studied before and where it's easier to simply use a field solution method rather than try to generate a new simplified model.

What are the field equations that hold in the lumped circuit model"

Possibly this question has a better answer that's just beyond my engineering knowledge of physics, but generally we don't talk about field equations in lumped circuit models.

We use Kirchhoff's Laws (which amount to conservation of charge and uniqueness of the electrostatic potential) and the constitutive relations of the individual circuit elements to solve circuits in the lumped approximation.