The ratio of inductance to capacitance is all it's about. Zo = sqrt(L/C). This works for lumped filters (where the product L*C defines the time constant, and therefore cutoff frequency), transmission lines, whatever.
Just as distributed filters use different lengths and widths of transmission lines (relative to the system impedance Zo), lumped filters use different values of L and C to implement interesting frequency-dependent reflections.
In a distributed line, the L and C are either imagined (infinitesimal) quantities, or an average, low frequency equivalent.
When any engineer speaks of "inductance" or "capacitance", that's really what they mean (whether they realize it or not): the low frequency approximation, of an impedance which exhibits similar behavior (Z ~= j*2*pi*F*L or -j / (2*pi*F*C)), over some finite range of frequencies.
Ratios are very powerful. Every time you see an impedance, that is derived from reactances rather than resistances (i.e., transmission through space, rather than absorption in a load, or emission from a source), then you can make some observation about its magnitude relative to the impedance of free space (377 ohms), or its length or delay, or both.
Because of this, we can derive the LF equivalents, absolutely independent of geometry, just from knowing a few bulk properties of transmission line (and assuming that it's a simple geometry -- namely, a minimum phase network). Knowing that Zo is 377 ohms, and that a piece of coax is 50 ohms, and that its velocity factor is 0.67, and that it has no magnetic loading, we know that:
- The velocity c_0 of an unloaded line, of any geometry, must be c. If c_0 < c, then it's lower by sqrt(mu_r * e_r).
- The line isn't magnetically loaded, so mu_r = 1 and e_r = 1/0.67^2 = 2.23. (The published dielectric constant for most polyethylenes is 2.25. c_0 is higher, and e_r lower, for teflon or foamed dielectrics!)
- Z_0 is lower than 377 ohms by the same factor, i.e., due to dielectric loading alone, the impedance of "unfree" space, in the cable, must be 252 ohms.
- So the rest of the reduction in impedance (50 versus 250 ohms, a factor of 5) is due to increased capacitance and reduced inductance (split evenly, or a factor of 5 in each), due to geometry.
- Finally, the permeability of free space is 1.257 uH/m, and the permittivity is 8.84 pF/m (note that sqrt(mu_0 / e_0) = 377 ohms). We have all the factors where the transmission line differs from this, so now we know:
- Inductivity is 5 times lower, or 0.25 uH/m.
- Capacitivity is 5 and 1/0.67^2 times higher, or 98 pF/m.
If you look up the datasheet for a 50 ohm coax like RG-58, you'll find very close to these numbers (one reference for RG-58 gives c_0 = 65.9% of c, and 101.05 pF/m; but no inductance figure, but we know that must be L = Z_0^2 * C = 0.253 uH/m).
You can repeat this process for anywhere there are two pieces of conductor nearby, and construct transmission line equivalents all over the place. Most of the time, you'll only be using this for the low frequency model anyway (e.g., transformer leakage inductance), but keeping in mind the deeper transmission line basis is very valuable when you need to consider frequencies above the low frequency limit.
Tim
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