# What are the rationales behind real wire (transmission line model), real capacitor and real inductor model?

I have been pondering about how the real model of circuit elements (wire, capacitors and inductors) and transmission model are derived through intuition (based on physics). Here are my references from Practical Electronics for Inventors 4th Edition:

Wire (page 263):

Transmission line model (Heaviside model from https://en.wikipedia.org/wiki/Heaviside_condition):

Capacitor (page 96):

Inductor (page 140):

What I want to know is how these configurations are derived (practically or preferably intuitively through physics) and why are they true instead of others (for instance why in the real capacitor model the inductive element is not parallel instead of in series with the capacitive element).

My best guess is that just like how a RC/RL parallel circuits can be converted into their equivalent series configurations, somehow these configurations might be converted into other configurations. However, if this is the case, why in the "simplified real model", the leakage resistance of the capacitor is in parallel configuration while the leakage resistance of the inductor is in series configuration?

What I mean by intuition is how the configurations can be built upon reasoning in physics before experimenting to figure out the model. For instance, as I understand it, in the transmission line model, the capacitors terminals should be connected to both side instead of one side only is because the capacitive charges are charged up on two wire, not one.

• You get one of the things, measure how it's different from ideal, come up with a hypothesis equation to describe the inaccuracy, test the equation... pretty much the scientific method. We didn't just invent these models out of thin air. The left side of the "real inductor model" shows an intuition for why this model is worth trying, but it still had to be tested to match reality. Commented Jul 1, 2022 at 8:21
• No model is correct for how things actually work. They are just models and depending on situation you need to pick a model that is accurate enough for your needs. Commented Jul 1, 2022 at 9:03

The simple physical intuition behind these models is:

1. Any conductor is an inductor.
2. Any conductor is a resistor.
3. Any insulating space between conductors adds capacitance between them.
4. Geometric feature sizes scale with the resulting time constants.
5. Different feature sizes will have their own sub-models.

Recall that RL, RC, C-L, C||L sub circuits all have associated time constants.

That’s most of the intuition you need. Then there are orders of magnitude, eg. in free space, 1cm of wire has 10nH of inductance, and two parallel wires have about 1pF/cm divided by the logarithm of the ratio of their distance to their diameter, etc. Those are order of magnitude approximations only.

Since most parts we can make have features at different size scales, their RLC models will become more complex as we go up in frequency the model should work at, since the size of the features captured by the model gets smaller – and the smaller the feature size, the more of those features there will be (eg. surface roughness distributed over the entire area!).

The models that you have called real are not real, they are just more detailed than the simpler models.

We only ever model the world. The best model to use in any circumstance is the simplest one that will meet our requirements.

If we have simple requirements, say we are teaching batteries, bulbs and switches to a class of 8 year olds, then a wire that carries a current, and has the same voltage at all points on it, is perfectly adequate. That model holds up fairly well for the interconnections for low speed logic, DC and audio amplifiers as well.

But it's not good enough for a car wiring diagram. We will have to add resistance to the model, to handle the wire area that we need to specify to keep the voltage drops under control.

If we are working on a radio system, or high speed logic, then we need to treat wires as transmission lines.

How do we know when our model is detailed enough?

Try working with a simple model. If it's too simple, then it won't be able to make predictions over the range that we want.

How do we improve the model?

We characterise the differences between behaviour we observe, and what the simple model predicts. Then we hypothesise what might be accounting for the differences. For instance we might observe that an inductor becomes very high impedance at a particular frequency. This could be modeled by a capacitance across the inductor that resonates with it, so we try adding that to the model, and adjusting the value until it fits our observations. This feels like a quite intuitive step, as you might expect the turns close to one another would have some capacitance between them.

That level is far better than a pure inductor model, but it's probably not adequate for the inductance of parts of some antenna structures, for which you might have to use a full 3D electromagnetic simulation.

• I totally agree with your point that models are created through experimental observations. But what I want to know is why the models are presented in such configuration instead of others? Does your answer implies that this is also derived through experiments? Commented Jul 1, 2022 at 13:34
• A theory is only valid if it agrees with experiment, with observation. We ask the universe how it behaves, and it tells us through experiment. So yes, experiment determines what we accept in a model. Given two configurations to elaborate a model, we usually use Occam's Razor, we pick the simpler one ... as long it as it describes what we want to know. Einstein said 'Everything should be made as simple as possible, but not simpler'. Use a 'wire', until we have to use a 'transmission line'. Anybody that says their model is 'correct' is making some assumptions. Check their assumptions explicitly. Commented Jul 1, 2022 at 13:42
• I understand your point that theories must agree with experimental observations in order to be accepted. However, theories themselves can also be built upon intuition before we need to experiment to figure out that they coincides. For instance, as I understand it, in the transmission line model, the capacitors terminals should be connected to both side instead of one side only is because the capacitive charges are charged up on two wire, not one. Commented Jul 1, 2022 at 13:53
• why in capacitor the capacitive elements are in series with inductive ones while in inductor the capacitive elements are parallel with inductive ones All components will have some parallel capacitance, and some series inductance. If the component is a capacitor though, the stray capacitance will tend to be absorbed into the nominal component capacitance as a correction, and not shown separately as a stray, and similarly for the inductor. Unless of course the level of detail required is very high, and then they will be shown separately. Commented Jul 1, 2022 at 14:22
• @DivineMK Stray capaciitance is a property of the space between two conductors, so by its very nature occurs in parallel to each pair of wires to the device - for a transistor we distinguish between Cbe, Cce and Cbc. Stray resisitance and inductance are the property of a wire. Wires are in series with the device terminals, so the strays end up in series. Commented Jul 3, 2022 at 8:55

Most generally: a real component has an impedance that varies up and down, dependent on frequency.

When that impedance is flat for much of the range, we call it a resistor.

When that impedance is sloped upward for much of the range, we call it an inductor.

When that impedance is sloped downward for much of the range, we call it a capacitor.

Real components don't hold that slope forever, nor perfectly (i.e., slightly other than frequency raised to a power of -1, 0 or 1). To model these errors, we add other elements.

The elements can be solved in an algorithmic manner. Since RLC (lumped element) circuits have rational polynomial characteristics, we use Padé approximants to fit a circuit to the measured curve.

The degree of accuracy we wish to fit that model, determines the order of the approximation: how many components we use to fit it.

Which in turn is determined by our need for the model. If we're just sketching something out, the ideal element (pure R, L or C) may do. If we're budgeting power losses in a converter, say, we may need the RL or RC model; if switching harmonics are involved, we might need it valid over several decades of frequency, in which case still more elements may be required.

The models shown above are indeed sufficient for many practical purposes, so they're worth documenting as such. They are certainly not exhaustive models; indeed, the corresponding diagrams suggest you can subdivide these components ultimately into infinite elements to finally get exact accuracy. Somewhere between one and infinity, there is likely sufficient accuracy for a given application.

As for your particular question: why series or parallel? I'll answer this with my own question: so what if there was a parallel inductor [to the capacitor]? What would happen at DC, then? Would its impedance still approach infinity (or R_L)?

A more subtle question we might ask, is: how much (or little) inductance could be in parallel, that could still be within experimental error -- because, after all, this is not just an approximation exercise, but subject to noisy measurements as well?

Or more general still: what series RLC combination (or further combinations in turn, recursively for each element) could be in parallel with the capacitor, with what value inductor(s), and still result in the observed impedance curve?

A simple calculation shows that, for a capacitor to exhibit say 1GΩ leakage, as measured within 1 hour, and measuring the same value for three months: the minimum possible inductance in parallel is of the order $$\ L \ge \frac{(1\,\textrm{G}\Omega) (7.776 \,\textrm{Ms}) }{ 2 \pi}\$$. If it were less than this, the impedance would be decreasing noticeably over those months.

Likewise, for inductance in series with R_L, for it to stabilize within an hour, $$\ L \le \frac{(1\,\textrm{G}\Omega) (3600 \,\textrm{s}) }{ 2 \pi}\$$. Still an impossibly large value -- and at that, negligibly small in comparison with our 1GΩ resistor, so we have no reason to bother including it in our model, and scratch it out as irrelevant.

Physically realistic values aren't everything, but we do prefer them. There is no physical mechanism for parallel inductance in a capacitor, so we have no reason to try and put it in our model.

That's not to say we should never use them. Consider the quartz crystal resonator. This is a piezoelectric crystal between two metal plates, which therefore have some capacitance to each other (and their surroundings), but due to the piezoelectric effect of the quartz (or other) material, there is an exchange between electrical and mechanical energy. Voltages applied to the crystal are reflected as mechanical vibrations, and vice versa.

If nothing's [separately] coming in as mechanical vibration, then we're only concerned with the effect of those internal vibrations on, and by, the electric field: we can draw a 2 or 3-terminal component, with a dominant capacitance between terminals, and model the effect of the electric field on the mechanical vibration on the electric field. When we do this, we find the capacitor has a resonant network in parallel with it, having a rather high impedance: this reflects the mechanical resonance of the crystal itself, as sensed by electromechanical effect.

And as it turns out, these resonances can be extremely sharp -- meaning both that, the frequency range over which they act is very narrow, and that they couple to the electric circuit surprisingly well despite the otherwise fairly weak piezo effect (the crystal is only moving, perhaps some nanometers, at normal signal levels). So we end up with, say a 4MHz, ESR=100Ω crystal, having motional inductance and capacitance of ~kH and ~fF. These are clearly nonphysical values -- you can't actually wind a 1kH inductor, at least not for anywhere near 4MHz, and the thing doesn't seem to contain a coil structure at all. But the model is nonetheless a good fit for the measurements, and so we accept this model as electrically correct, and just smile and nod at the otherwise-ridiculous numbers within.

• I understand that your point is models are obtained through empirical data and I really appreciate the detailed examples you provided to prove this point. However, what I am looking for (if existed) is the intuitive explanation for these configurations (similar to the one I added at the end of my post). Commented Jul 2, 2022 at 13:05
• @DivineMK By this explanation, the configurations reflect how terms and factors are combined to give a best fit to the characteristic, for a given degree of approximation. The math is intuitive to me; granted, that's after many years studying these things, and I don't know your level. This is a dynamics question (electricity in motion, AC), so a calculus-level answer seems apt. ... More hand-wavingly, you can fit slopes to the Bode plot of a component, and the result reflects RLC values and their arrangement (which can be demonstrated easily e.g. in SPICE). Commented Jul 2, 2022 at 19:31