Is it possible to simulate reflections by designing an RLC transfer function?

Question

Assume that you're knowing the PCB stack and the geometry of the conductor. You're computing the capacitance per mm and inductance per mm and also the total resistance of the conductor.

That means you can shape the transfer function from the eletrical RLC circuit:

To this transfer function.

So that means you got the transfer function of the conductor. Then you need the transfer function from the driver and the load.

Assume that the driver and the load is a logic IC. How can I find the resistance, capacitace and the inductace from them?

Answer 1

You are trying to model a transmission line as a lumped model when it's more complicated than that. The way you have to do it is you have to break your lumped model model up into discrete RLC "rungs" and chain a ladder out of that:

^{simulate this circuit – Schematic created using CircuitLab}

Using calculus and FEA, the idea is to keep breaking the R, L, and C's down to the limiting case (dR, dL, dC) such that when integrated, will amount to your lumped model. You can model and simulate several stages to get a "feel" for the reflection by varying the RLC (which defines the characteristic impedance) and the termination resistance. The lower net would represent the ground plane. And the LR links represent the copper trace. The C's are the capacitance between the trace and the groundplane of the PCB pre-preg. Units will nominally be "per unit length". But you really need to use principles from transmission line theory to do this right.

Answer 2

Is it possible to simulate reflections by designing an RLC transfer function?

It's possible if you are content with the simplistic results obtained from a single LC low-pass filter.

However, it's better to cascade several LC low-pass filters because you obtain a progressively more accurate representation of what a transmission line is really like as per my answer to another question: -

Converting each stage to an individual transfer function and trying to mathematically cascade them is missing the point. In other words, if you make the assumption that input and output impedances are identical, then it's possible to derive the transmission line's characteristic impedance and, this is much more relevant when analysing what your delayed digital signal looks like when it gets to the load.

Assume that the driver and the load is a logic IC. How can I find the resistance, capacitance and the inductance from them?

The driver and load have nothing to do with the transmission line's characteristic impedance; it's cable or PCB tracks and, the physical dimensions dictate the characteristic impedance and therefore what the R, L, C and G values are.

Consider an ideal 50 Ω cable (no R and G), it's characteristic impedance is \$\sqrt{\frac{L}{C}}\$ so, you might have a distributed inductance of 250 nH per metre and, a distributed capacitance of 100 pF per metre. Take the square root of their ratio and you get 50 Ω.

The take-away summary is that to accurately represent a transmission line you need to use many cascaded LC circuits. And, you need to use many LC circuits if you are to ensure that the full bandwidth of your input signal is at least ten times smaller than the resonant frequency of a single LC stage. If you don't do this you may get unsatisfactory results.

Answer 3

Sort of. But understand the limited scope in which it applies.

I won't go into quantitative analysis here, also because I haven't used it myself, but it's probably not too material anyway, but more just to point out the equivalence. In any case:

There exists an equivalence, between a lumped-element equivalent network, and a network constructed of transmission line segments.

The equivalence holds, as long as the TL impedances hold -- that is, within range of one resonance -- in other words, maybe an octave, give or take.

When we construct planar filter networks of TL segments, they are typically only good to as much range. Outside that range, there are higher-order passbands -- unintended and maybe undesired, and may be poorly controlled (bandwidth, flatness, impedance, etc. all over the place). Further refinement of the trace geometry (little nicks and steps here and there) may be employed to address this, or additional (higher frequency) filters can be cascaded together so that their stopbands overlap to give strong attenuation overall.

We could model the 1/4 or 1/2 wave stub resonance of a TL segment, as a single (read: first-order) RLC equivalent, such as shown (give or take perhaps dividing the inductor into parts before and after the capacitor for a tee equivalent, or using two capacitors for a pi section, or tee with tapped inductor, etc.), and the mapping gives the damping factor equivalent to SWR or what have you.

This is only valid for the resonance modeled, so it's not very useful for purposes where we normally need to consider transmission lines: that is, where multiple harmonics of the signal, interact with multiple resonances of the network; broadband or time-domain. Particularly when few lines are used (such as between logic devices in point-to-point digital links), the TL analysis is the easiest route.