What am I misunderstanding in the derivation of skin depth?

Question

Suppose we have an electric field $$E_x(z) = E^+e^{-\gamma z} + E^-e^{\gamma z}$$ where $$\gamma=\alpha+j\beta$$ is the complex propagation propagation constant, as defined in David Pozar's Microwave Engineering Section 1.4. The propagation factor then has form (in the time domain): $$e^{-\alpha z} \cos(\omega t - \beta z)$$ which represents a wave traveling in the +z direction.

From this, the skin depth is defined as $$\delta_s = \frac{1}{\alpha}$$ which is of course the distance that the wave travels in the +z direction over which it decays by 1/e.

This makes sense to me if we have an external wave incident upon a metal plate. Clearly, its depth of penetration is limited by the skin depth.

However, suppose instead that I have a wire (part of a simple voltage source + resistor circuit) carrying a sinusoidal current. Say this particular wire is oriented in the +z direction. Then the electric field wave is also oriented in the +z direction (J=σE). Would this not imply that after travelling one skin depth, my current is attenuated by 1/e? That obviously makes no sense:

1. It would imply that transmitting a non-dc current is effectively impossible as it just gets attenuated right away.
2. Typically, we see skin depth in a round conductor as creating a sort of 'donut' shaped conductor at high frequencies, which clearly does not align with whatever I'm picturing.

I suppose I must have a fundamental misunderstanding in the direction of the various components, and hope somebody can clear it up for me.

Answer 1

Amazingly, different boundary conditions lead to different field configurations. :)

Ok, not to get too flippant or anything... but, to say: the missing insight is about what fields and geometry are involved.

For the plane wave and plate (or more particularly, half-plane volume) configuration, there is a surface current induced by the incident (plus reflected) wave: eddy currents. But these flow in loops complementary to magnetic field lines in the incident wave; not continuously across the surface, like a DC current in a wire does.

For the wire configuration, we have different waves in question.

For a DC current, the field is longitudinal, and very small magnitude (~mV/m?). For AC, at frequencies where skin effect and transmission line behavior apply, we have a significantly greater electric field, radial to the wire, due to its instantaneous charge above the surroundings (for a lone wire in space, just that; for the transmission line, the excess charge with respect to the companion conductor). And correspondingly we have a skin effect depth into the wire surface.

Note we might still have that longitudinal component, but because the resistance of the conductor(s) is so much smaller than the impedance of the space around them, we can approximate it as ideal and ignore that component. (That is, the field might not be perfectly radial, but slightly inclined longitudinally; but we can ignore that for subsequent analysis.) (We might then add losses back in later, as an averaged effect or empirical parameter, so that we don't have to integrate over a nasty differential equation to solve for it by first principles; but, yes, that has also of course been solved.)

The key insight for high frequencies is that, power does not flow (or at least very much) inside the wire itself, but resides predominantly in the space between conductors -- the wires are wave guides, and the dielectric carries the bulk of the energy.

The most immediate consequence to field geometry, is the solutions involved in the skin effect calculation.

At mid frequencies, curvature of a, for example cylindrical wire, cannot be ignored. We must take account of how the field diffuses* into the conductor itself. Intuitively, the wave does not attenuate completely by the time it reaches the center, but some residual fully crosses the wire, and this causes wave interference with the field entering from the same side, and thus peaks and valleys (or outright reversal!) appear in the current profile within the wire. Solutions involve Bessel functions.

For curvature approaching zero, i.e. \$f \rightarrow \infty\$ or \$r \gg \delta\$, solutions look very decaying-exponential, as we should expect; but it's that mid-frequency case where \$r \sim \delta\$ where things get interesting (complicated).

*Skin effect is a diffusion transport mechanism. So you tend to get lots of square roots in results: the skin depth formula, impedance vs. frequency, etc.