Power Tip 26: Current distribution in high-frequency conductors

(Additional editor's note: We've updated this Power Tips entry to include a short, informative, no-nonsense video by the author. The video player is at the bottom—so check it out.)

In this Power Tip, we will look at the effective resistance of conductors in free space and wound structures. Figure 1 shows the first example. It is the cross-section of a single wire in free space carrying a high-frequency current. If the current were DC, the current density, displayed as varying colors, would be the same throughout.

However, as the frequency increases; the current moves toward the outside of the conductor as indicated by the red and orange colors. This “crowding” is called the skin effect. Depth of penetration is defined as the distance from the outer surface to the point where the current density has fallen to 1/e of the outer surface current density. For copper, this depth is:  

where f is in megahertz and depth is in cm.

Figure 1: Current crowds the outer surface at high frequency.
(Click on image to enlarge)

Figure 2 shows the current distribution in a flat conductor in free space. Rather than being equal around the conductor’s surface, it tends to flow in the narrow edges.  However, it still has the same depth of penetration. This greatly increases the resistance as most of the conductor has a very low current density.

Figure 2: Current concentrates near ends of conductor at depth of penetration.
(Click on image to enlarge)

To circumvent the current distribution issue in a flat conductor, it is usually placed directly over either a second conductor or a ground plane with an equal current flowing in the opposing direction.

Figure 3 shows an example of this where the opposing currents draw each other to the adjacent surfaces of the two conductors. The depth of penetration remains the same. The current is mostly contained in an area bounded by the depth of penetration and the width of the conductor, rather than the depth of penetration by the thickness of the conductor as in Figure 2. Therefore, the AC resistance of these conductors is significantly lower than the free air case. 

Figure 3: Opposing currents pull to adjacent surfaces.
(Click on image to enlarge)

Figure 4 shows the cross section of a layer-wound structure. Here, the top two conductors (3 and 4) carry the same current in the same direction, while the bottom two (1 and 2) carry equal currents in a direction opposite of the top layers. This might represent layers in a transformer with a two-to-two-turns ratio. As in the previous example, current on the windings are drawn to opposing faces.

However, an interesting phenomenon occurs. In windings 1 and 4, current is drawn to the inner surface and it induces current flow on the surfaces of windings 2 and 3 in the opposite direction. The overall current in windings 2 and 3 is flowing in the opposite direction, so the current density on the inner surface is much larger. This is called proximity effect and makes layered structures operating at high frequency problematic.

One way around this issue is to rearrange the conductor stack up. Rather than using two adjacent layers of windings with current flowing in the same direction, interleave the windings so that current flows in the proper direction on both sides.

Figure 4: Opposing current flow on adjacent windings greatly increases losses.
(Click on image to enlarge)

Dowell¹ developed an analytical model to calculate the increases in the AC resistance of conductors with various thicknesses and layer configurations (Reference 1). Figure 5 presents his results. The x-axis of the graph normalizes the layer thickness to the depth of penetration, and the y-axis presents the AC resistance normalized to the DC resistance. A family of curves is presented depending on the number of layers in the windings.

Once the conductor thickness approaches the skin depth, the number of layers for a reasonable AC/DC ratio becomes small. Also, note the lower curve for half a layer. In this case, the windings are interleaved and the resistance increase is significantly smaller than the single layer case.

Figure 5: Dowell shows how lossy layer-wound structures are.
(Click on image to enlarge)

To summarize, as frequency is increased, current distribution in a conductor will drastically change. In free air, a circular conductor will be lower resistance at high frequency than a flat one. However, the flat conductor is much better when used with a ground plane, or if it is located near a conductor carrying the return current. Please join us next month when we will discuss paralleling power supplies using the droop method.

References
1. P.L. Dowell, “Effects of eddy currents in transformer windings,” Proceedings of the IEEE, vol 113, no 8, pp. 1387-1384, Aug. 1966.
2. Lloyd Dixon, “Coupled Filer Inductors in Multi-Output Buck Regulators,” Texas Instrument, section 3, page 4.

Robert Kollman is a Senior Applications Manager and Distinguished Member of Technical Staff at Texas Instruments. He has more than 30 years of experience in the power electronics business and has designed magnetics for power electronics ranging from sub-watt to sub-megawatt with operating frequencies into the megahertz range. Robert earned a BSEE from Texas A&M University, and a MSEE from Southern Methodist University.