The concept of a geometric form factor is very useful when thinking about transmission lines. It bundles up all the messy 2D geometry of cross-sections and field line patterns into one scalar quantity that is fixed for a particular configuration. Then you can think about the general concepts without getting bogged down in field integrals and field solvers.
To explore this idea, let’s start with a uniform relative dielectric constant εr like you’d have with stripline. (Later we’ll generalize to the case where the field lines “sees” a mixture of materials, for example dielectric and air for a microstrip.)
First define a unitless geometric form factor, F, such that capacitance per unit length, C, is:
…where ε0 is the vacuum permittivity, 8.9pF/m (or 0.22pF/inch) and εr is the relative permittivity (relative dielectric constant), ~4.2 to 4.6 in FR4 glass/resin.
In general, we need a 2D electrostatic solver to determine F but let’s first look at a geometry for which there’s an exact, textbook, analytical solution: a rod of radius a, with its center a distance b above a ground plane like this:
The form factor is:
Couple of thing to note about F:
Unitless: F is a function of a geometric ratio, b/a.
Logarithm of a smallish ratio: In practical cases, the numerator and divisor are not of wildly different orders of magnitude. You wouldn’t create a transmission line with a = 1 mm and b = 1 km! No, for the realistic range of values of b/a, the natural logarithm of the function here is of order 1. For example if b=2a, then F is 0.21.
Let’s plot it and two other configurations (illustrated below) and you’ll see that (apart from the “short circuit” corner case at a = b) the blue curve varies slowly with b/a:
The neat thing about the form factor is that when you have it for capacitance, you get inductance and impedance “for free.” To see the first of these, consider the propagation velocity, v, which is both:
…where c = vacuum speed of light = 0.3 m / ns or 12 inches / ns
(Strictly speaking, we can only use “pure” insulator εr if the metal is a perfect conductor, the skin depth is zero (surface currents). If the conductivity and skin depth are finite, we would need a “blend” of the dielectric and metal εrs.)
With a little algebra you can see that the inductance per unit length, L, is given by:
L = Fμ0
…where μ0 is the vacuum permeability, 1.3 μH/m or 32 nH/inch.
Thus, you can think of F as “how inductive the geometry is versus the free space value of 1.” If F < 1 the geometry is “more capacitive” than free space, if F > 1 it’s “more inductive.” The second benefit is that once you have L and C, you also have impedance. Again a little algebra shows that:
…where Z0 is the vacuum impedance, 377 Ω. The form factor and the relative dielectric constant determine how far the line impedance is from the “natural” value of 377 Ω. Lowering the form factor (making a “more capacitive” geometry) lowers the impedance, as does a higher relative dielectric constant. If εr = 4.2, and you want Z = 50 Ω, then pick your geometry such that F is 0.27.
If you take the ground plane and wrap it around the rod, you get co-ax:
The form factor (magenta line in the plot above) is a bit lower (more capacitive) as you’d expect from bring the ground plane closer.
If instead you squash the rod into a ribbon and put a second ground plane on top, you get stripline:
The result is a bit less capacitive (red line in the plot above), because the squashing increases the separation from ground.
A power and ground plane pair forms another, simple, special case that’s useful. In a power distribution network (PDN) the source (VRM) and load (the chip) impedances are low, so the design goal is ultralow impedance. We use wide, closely spaced planes. The form factor of such a geometry is easily derived from the parallel plate capacitor equation recast into “per unit length” format: C = ε0εra/b. If a >> b, then F is simply b/a, where a is the copper width and b is the height of the dielectric between the planes. I didn’t plot it above, but it would be a straight line through the origin of slope 1. The inductance per square is simply μ0b or 1.3 pH per square per μm of height or 32 pH per square per mil of height. For a typical 4 mil (~100μm) height that’s ~130pH per square.
You can generalize this idea to non-uniform dielectrics by using a blended or “effective” relative dielectric constant. An example is the IPC recommended approximation for microstrip. In a future posting, I’ll also show how it’s useful for lumped elements like inductors in the form of flat coils, and short and long solenoids.