Calculating Total I2R Losses in a Conductor with Both AC and DC Currents

[Phrak]

References are available to calculate skin effect, R_AC/R_DC given wire gauge and frequency. But my problem is a complication of this simple calculation.

Say I have a 1 Amp RMS AC current component and a selected wire size that gives me a skin effect of 7X the DC resistance. But I also have a hefty 10 Amp DC current component.

I'm interested in calculating the total I²R losses due to both the AC and DC currents.

How are these two combined?? For the sake or argument, the DC resistance of the wire is 0.01 Ohms.

1) I can treat them as separate and independent currents and get a small power loss:

P = 0.01 Ohms * 10² Amps² + 0.07 Ohms * 1² Amps² = 1.07 Watts.

Or 2) I can imagine that the 7X skin effect also impedes the DC current flow.

P = 0.07 Ohms * 10² Amps² + 0.07 Ohms * 1² Amps² = 7.07 Watts.

The results are wildly different between the two.

Which method--or a third method, is correct to first order? I mean, never mind the finer details. I just want to know what my power loss is to +/-20%.

 

[berkeman]

I believe they are independent effects. Except for the fact that each effect heats up the wire, which increases its resistance some. That is the only interaction that I see.

 

[Bob S]

The 1.07 watts is correct. Only the ac current flows in the "skin depth" on the outer surface of the conductor. If you go to stranded wire of the same gauge, the total skin depth resistance is less.

Bob S

 

[Phrak]

This http://www.dartmouth.edu/~sullivan/litzwire/skin.html" is the only semi-useful link I've found on skin effect.

Its intent is to examine skin effect in transformers where wires are subjected to the magnetic influence of their neighbors, it’s pretty thick reading.

Given, without proof or motivation, is the resistive power loss due to non-sinusoidal waveforms as the Fourier sum of the power losses due to each sine wave independently.

P = R_DC Σ_j I²_j F_r(ω_j)

Supposedly, this includes DC.

I could over-analyze this problem to death and include the influences of other neighboring wires 'n everything, or get back to finishing the project. But I do like overanalyzing. :)

Is it too difficult to derive the equation for skin effect of an arbitrary waveform, where at any given moment, the radial force due to a radial distribution of charge cancels the radial force due to the time-rate-change of magnetic field due to longitudinal current?

 

[Mike_In_Plano]

Hey Phrak,

I had to do this one in college. For the simple wire, a breakdown of the currents by frequency is sufficient.
When designing transformers / inductors, flux at gaps can increase the loss by cutting across the copper, but this is usually neglected unless you have a very large gap, and when this is a problem, I've never seen a definitive way to compute the loss.

- Mike

 

[Phrak]

Hey Phrak,

I had to do this one in college. For the simple wire, a breakdown of the currents by frequency is sufficient.
When designing transformers / inductors, flux at gaps can increase the loss by cutting across the copper, but this is usually neglected unless you have a very large gap, and when this is a problem, I've never seen a definitive way to compute the loss.

- Mike

Thanks Mike. Do you have a reference source for the frequency independence that specifically includes f=0?

Secondly, after taking a short look at the simpler problem for a single frequency and single wire, I have to solve some sort of Bernoulli equation using about 3 or 4 out of 4 of Maxwell's equations that are both time and radially variant: J_z(r,t), E_r(r,t), E_z(r,t) and B_phi(r,t). Does this sound familiar?

 

[Bob S]

Thanks Mike. Do you have a reference source for the frequency independence that specifically includes f=0?

Secondly, after taking a short look at the simpler problem for a single frequency and single wire, I have to solve some sort of Bernoulli equation using about 3 or 4 out of 4 of Maxwell's equations that are both time and radially variant: J_z(r,t), E_r(r,t), E_z(r,t) and B_phi(r,t). Does this sound familiar?

In Smythe "Static and Dynamic Electricity" third edition on pages 372-4 the equations for the ac current density vs. radius in a solid cylindrical conductor are written out using modified Bessel functions of order zero with complex arguments. He gives the effective ac skin effect resistance vs. frequency on page 373.

Bob S.

 

[Studiot]

Electrical Engineer rule of thumb Bessel function for copper (wire)

skindepth in mm = 260 / $$\sqrt{f}$$ (in cycles per second)

 

[Phrak]

In Smythe "Static and Dynamic Electricity" third edition on pages 372-4 the equations for the ac current density vs. radius in a solid cylindrical conductor are written out using modified Bessel functions of order zero with complex arguments. He gives the effective ac skin effect resistance vs. frequency on page 373.

Bob S.

Doh! Of course I meant Bessel rather than Bernolli.


...I finally found some cheats here http://home.swipnet.se/swi/bessel/bessel.html#Basic_theory" but he does kind of jump into the middle of it, unexpectedly beginning with the electric field wave equation in a conductive medium, that I'm not familiar with.