Current distribution in high-frequency conductors

where f is in megahertz and depth is in cm.

Figure 1 Current gathers on the outer surface at high frequency

Figure 2 shows the current distribution in a flat conductor in free space. It tends to flow in the narrow sides, rather than being equal across the conductor surface. However, it still has the same penetration depth. This greatly increases the resistance, since most of the conductor has very low current density.

Figure 2 Current concentration near the conductor end at penetration depth

To circumvent the current distribution issues of flat conductors, they are often placed directly on top of a second conductor or ground plane, where the currents are equal and opposite. Figure 3 shows an example where the opposing currents pull each other to the adjacent surfaces of the two conductors. The penetration depth remains the same. The current is primarily contained in a region bounded by the penetration depth and the conductor width (rather than the penetration depth and conductor thickness as shown in Figure 2). Therefore, the AC resistance of these conductors is much lower than that of free space.

Figure 3 Reverse current pulls to adjacent surfaces

Figure 4 shows a cross section of a layer winding 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 current in the opposite direction to the layers above. This can represent the layers of a 2-2 turns ratio transformer. As in the previous example, the current of the windings is drawn to opposite sides. However, an interesting phenomenon occurs. In windings 1 and 4, the current is drawn to the inner surface, which induces current in windings 2 and 3 in the opposite direction. The total current of windings 2 and 3 is flowing in opposite directions, so the current density on the inner surface is greater. This phenomenon is called the proximity effect, which can cause problems for layer structures operating at high frequencies. One way to solve this problem is to rearrange the conductor stack and stagger the windings so that the current flows in the correct direction at both ends, rather than using two adjacent layer windings with current in the same direction.

Figure 4 Reverse current in adjacent windings greatly increases losses

Dowell ¹ developed an analytical model to calculate the increase in AC resistance for conductors of varying thickness and layer structure (see Reference 1). The results are shown in Figure 5. The X-axis of the graph normalizes the layer thickness to the penetration depth, while the Y-axis shows the AC resistance normalized to the DC resistance. These curves are plotted as a function of the number of layers in the winding. Once the conductor thickness approaches the skin depth, the number of layers for a reasonable AC/DC ratio becomes smaller. Also note the low curve for 1/2 layer. In this case, the windings are interleaved and the resistance increase is much less than for the single layer case.

Figure 5 Dowell illustrates the case of a high loss layer winding structure

In summary, as frequency increases, the current distribution in a conductor changes dramatically. In free space, a round conductor has lower resistance at high frequencies than a flat conductor. However, when used with a ground plane, or when it is located near a conductor carrying a return current, a flat conductor is better. Next time, we will discuss how to connect power supplies in parallel using the droop method, so stay tuned.