On Skin Effect and Microstrip Lines

[chr0mium]

In the field of microwave engineering, skin effect is referred to when people talk about the fact that current, that is, electrons only flows on the surface of transmission lines, e.g. microstrip lines.

I've checked many EM books--everywhere skin effect is demonstrated with both E and H
parallel to the surface of a conductor, with the wave propagating into the conductor.

Now here is a gap between these two. For transmission lines, waves actually propagate in the direction parallel to the line instead of perpendicular to conductors. And if we draw the field lines, most of them lie in the surrounding dielectric materials instead of in the conductor.

Can anyone help me bring these two together?

 

[Delta2]

You make the assumption that the wave outside the transmission line propagates parallel to it but inside the line vertical to it. Like there is a 90 degrees refraction in the propagation of wave when it meets the transmission line conducting medium.

Of course this assumption is just to simplify the theoretical calculations, we can never have 90 degrees refraction angle.

 

[Bob S]

This is a very good question. In an ideal transmission line with zero resistance conductors, all the energy flows between the conductors as a transverse electric magnetic (TEM) wave. The currents in (ideally on) the conductors support the TEM wave (similar to the currents in the walls of a waveguide). In essence a TEM wave is the equivalent of a TE or TM wave below waveguide cutoff. There is a small electric field parallel to the surface of the conductor only if the conductor has a finite resistance. Otherwise, the boundary condition requires E_parallel = 0. This electric field arises not from the TEM wave but from the voltage drop of the currents in the conductors.

The Poynting vector P = E_T x H_T of the TEM wave points along the direction of the wave propagation. The Poynting vector of E_parallel crossed into the transverse H_T points into the conductor, meaning that power is flowing from the TEM wave into the resistive conductor.

Resistive conductor (skin effect) losses in coax cables (e.g., RG-8) are the dominant power losses at RF frequencies. The only other losses are dielectric and radiative, usually only at microwave frequencies.

A very thorough discussion of eddy (skin effect) currents and losses can be found in chapter X of Static and Dynamic Electricity (third edition) by Smythe.

Bob S

 

[chr0mium]

You make the assumption that the wave outside the transmission line propagates parallel to it but inside the line vertical to it. Like there is a 90 degrees refraction in the propagation of wave when it meets the transmission line conducting medium.

Of course this assumption is just to simplify the theoretical calculations, we can never have 90 degrees refraction angle.

Well this is refreshing. So in reality there is a vertical component close to the conducting medium? Is there any source that elaborates on this?

 

[chr0mium]

This is a very good question. In an ideal transmission line with zero resistance conductors, all the energy flows between the conductors as a transverse electric magnetic (TEM) wave. The currents in (ideally on) the conductors support the TEM wave (similar to the currents in the walls of a waveguide). In essence a TEM wave is the equivalent of a TE or TM wave below waveguide cutoff. There is a small electric field parallel to the surface of the conductor only if the conductor has a finite resistance. Otherwise, the boundary condition requires E_parallel = 0. This electric field arises not from the TEM wave but from the voltage drop of the currents in the conductors.

The Poynting vector P = E_T x H_T of the TEM wave points along the direction of the wave propagation. The Poynting vector of E_parallel crossed into the transverse H_T points into the conductor, meaning that power is flowing from the TEM wave into the resistive conductor.

Resistive conductor (skin effect) losses in coax cables (e.g., RG-8) are the dominant power losses at RF frequencies. The only other losses are dielectric and radiative, usually only at microwave frequencies.

A very thorough discussion of eddy (skin effect) currents and losses can be found in chapter X of Static and Dynamic Electricity (third edition) by Smythe.

Bob S

Thanks very much!

 

[Delta2]

Well this is refreshing. So in reality there is a vertical component close to the conducting medium? Is there any source that elaborates on this?

You mean vertical component of the poynting vector outside but close to the conductor? I guess there must be one which accounts for the radiative losses.

 

[chr0mium]

You mean vertical component of the poynting vector outside but close to the conductor? I guess there must be one which accounts for the radiative losses.

I was talking about a Poynting vector that is perpendicular the the conductor. It has been well answered by Bob.

The only doubt left is, given the conductors are ideal, will an AC current be on the surface, instead of distributed evenly? If so, then we have two separate phenomena both of which make the current flow on the surface, one is lossy, the other lossless. I'll try to find an answer from the Smythe book.

 

[Delta2]

I was talking about a Poynting vector that is perpendicular the the conductor. It has been well answered by Bob.

The only doubt left is, given the conductors are ideal, will an AC current be on the surface, instead of distributed evenly? If so, then we have two separate phenomena both of which make the current flow on the surface, one is lossy, the other lossless. I'll try to find an answer from the Smythe book.

Er what do you mean by ideal conductor? If u mean infinite conductivity then from the skin depth formula one gets that skin depth is zero, that is the current is flowing only on surface.

 

[chr0mium]

Er what do you mean by ideal conductor? If u mean infinite conductivity then from the skin depth formula one gets that skin depth is zero, that is the current is flowing only on surface.

If conductor is ideal, there is no $E_{parallel}$, thus there is no resistive conductor loss (or the so called skin effect)

 

[Bob S]

The only doubt left is, given the conductors are ideal, will an AC current be on the surface, instead of distributed evenly? If so, then we have two separate phenomena both of which make the current flow on the surface, one is lossy, the other lossless. I'll try to find an answer from the Smythe book.

There is only one current in or on the conductors; it is required by the boundary conditions for the TEM wave flowing between the microstrips. If there is a voltage drop (parallel to the current) in the conductor, then there are eddy current (skin effect) losses in the conductor, but there are no additional currents. So there is an additional electric field from the IR drop, but no additional current.

If your pulse signals are very high frequencies, the losses scale as the square root of frequency (creating both frequency-dependent attenuation and velocity dispersion), so the rise and fall times of pulses will round off. The losses are dependent on the impedance (Z₀) of the transmission line. Usually higher Z₀ is better.

Bob S

 

[Delta2]

In reality there are no ideal conductors so there is E_parallel in and on the conductor. Does this imply that there is E_parallel too in the dielectric space between the transmission line conductors?

 

[chr0mium]

Oh my, I think I have pinned down the source of my doubt. It all happened when I first learned about skin depth, where there was an equation of electric field in a good conductor

E_x = E_0 * e^{-\alpha * z} * cos(\omega * t - \beta * z)
\alpha = \sqrt{\omega \mu \sigma}

I mistakenly thought that if conductivity \sigma were to be infinite, e^{\alpha * z} would be 1 thus the magnitude E_x would be constant inside the conductor, and it is because the conductivity is not infinite that we have reduced E_x which is called skin effect. It's a simple mathematical mistake coupled with a plausible physical interpretation.

But the truth is, given an infinite \sigma, \alpha would be infinite, and e^{-\alpha * z} would be 0, meaning there wouldn't be any E_x inside a perfect conductor. It is only because in reality \sigma is not infinite that E_x could penetrate a good conductor somewhat, but only up to a "skin depth" beyond which E_x becomes trivial.

 

[Bob S]

In reality there are no ideal conductors so there is E_parallel in and on the conductor. Does this imply that there is E_parallel too in the dielectric space between the transmission line conductors?

Yes, you are absolutely correct. Consider the interesting example of a balanced line, like the 300-ohm TV lead-in line. The skin effect losses are equal in the two conductors, but the currents are in the opposite direction. So the two E_parallel's are equal in magnitude but opposite in direction. So the E_parallel changes polarity midway between the two conductors. In a microstrip, most of the E_parallel is in the strip, and very little in the ground plane.

Bob S