Utilization of Skin Effect for Intra-plane Z-axis Partitioning in Circuit Boards

Skin effect is a term that describes the tendency of the current density of high radio frequency currents to become “crowded” toward the surface skins (surface boundaries) of a conducting material. The extent to which the current density is formed toward the surface relates to the depth of the primary radio frequency current below the surface, hence the term skin depth.

For direct current (DC) and lower frequency alternating current (AC) applications (e.g. below one megahertz, where the skin depths are typically larger dimensional values), the cross section of most conductors will be fully involved in the current transfer. This full involvement causes a uniform density of current distribution throughout the conductor cross-section.

Figure 1

Increasingly at higher frequencies, the presence (and location) of magnetic flux impacts the distribution of current density through the cross-section of the conductor (Figure 2).

Figure 2

For practical descriptions, the combination of the material characteristics, frequency (which in combination determines the skin depth), and distribution of flux pattern will determine the redistribution of current density throughout the cross-section. The flux pattern distribution circumscribing the conductor is the motivating influence to direct the alterations as applied to current density.

Figure 3

Since the motivating factor for current density is the shape and formative presence of flux at high frequency, the direction of the current density distribution in the cross-section of a conductor will be altered by the location of the flux. In Figure 4, the flux density is formed intensely in the boundary between two conductors that carry the current (and form the flux) in an opposing phase relationship. Due to the interrelationship of flux to the distribution of current density at high frequencies, this suggests that the current densities in the two conductors will be “crowded” into the two mutually opposing specific surfaces that correspond to the distribution of the flux density.

Figure 4

Examination of the effect and influence illustrated above suggests that in multilayer circuit boards, the arrangement of the plane layers that propagate high frequency currents in opposing directions may be effective in establishing partitioning of common-mode coupling effects within the Z-Axis. This projection assumes that there exists a sufficient number of skin depths that are actively available within each plane layer in the frequency spectra of interest.

To emphasize the formation of skin effect, the higher the frequency, the smaller the skin depth – and the more conductive and/or the more permeable the material (at higher frequency), the smaller the skin depth. Given this observation, in terms of skin effect, the smallest skin depths occur with most conductive materials at higher permeability (assuming that the permeability is evident as a characteristic of the material at high frequency) and at the highest frequencies.

Figure 5

To illustrate the extent of skin effect, Figure 6 describes the percentage of current capture toward the surface, which is expressed as a percentage of current density as related to skin depths. The conclusion yielded is that 5 skin depths are required to capture approximately 99 percent of the current density.

Figure 6

Figure 7

The significance of skin effect as a characteristic benefit at higher frequencies toward intra-plane partitions (within the Z-axis of a circuit board) may be recognized by reviewing the skin depth of annealed copper in the tabulation shown in Figure 8.

Figure 8

With the understanding that a copper weight in the plane of a circuit board of one ounce represents a thickness of approximately 1.4 mils, it is observed that at higher spectral frequency distributions, planes within circuit boards may be utilized for signals, signal categories, and power partitioning within circuit boards.

The parameters within a conductor that function to influence skin depth includes the values of relative conductivity and relative magnetic permeability. These are also the parameters that are significant to consider in terms of shielding performance. In practical terms, most approaches toward shielding relate the conductivity of the material as a value relative to copper. The value of magnetic permeability is relative to the permeability of free space.

For reference, the conductivity of annealed copper is given as the symbol σ, where

σ = 5.82 × 10⁷ mhos/meters for copper

with the relative values for other metals assigned the symbol σ_r.

σ_r is a numerical value that results by applying the factor indicated by that designated for σr to the value of the reference, σ.

The permeability of free space is given the symbol μ where

μ = 4π × 10^-7 Henrys/meter for free space

with the relative values of other materials assigned the symbol μ_r.

μ_r is a numerical value that results by applying the factor indicated by that designated for μr to the value of the reference, μ.

Copper and many other reasonably conductive materials, such as aluminum, beryllium, brass, bronze, gold, platinum, silver, tin, or zinc, vary in conductivity relative to each other. However, these listed materials are understood to mutually possess a magnetic permeability that is equal to the permeability exhibited in space [μr = μ]. Consequently, for these materials, the skin depth will vary as a function only of conductivity with the materials of higher conductivity, yielding the smaller skin depths. For materials that exhibit permeability values that are greater than free space, the skin depths will also be influenced. Since a characteristic of permeability includes increased efficiency in propagating flux density and since flux density is involved in gathering current density toward the surface of conductors, materials with greater permeability and reasonable conductivity will exhibit smaller dimensions of skin depths compared to those where μr = μ. This observation, however, is not uniformly correct! At higher frequencies, above for example 5 GHz, many ferrous metals exhibit a relative permeability of only 1 even though at lower frequency the value may be 1000! Under this example, the prevalent parameter controlling skin depth would be that of conductivity. If the relative conductivity characteristic of the material displays greater resistance than that of the reference value of annealed copper, then at microwave bandwidth the skin depth of the ferrous metal could be greater. These delineations are specific to each alloy configuration.

For example, typical steels (assuming μ_r = μ × 1000) that may be utilized for system-product packaging can exhibit skin depths that are a dimensional factor of 5 to 10 less than those of annealed copper, but only at lower frequencies. Because, however, the conductivity of various compositions of “steels” can vary with carbon content and general molecular densities, the skin depths from one category of steel to another would not be anticipated to be equal among the alloys. Stainless steels, in particular those of Austenitic types, can be alloyed to the level of being essentially non-magnetic, despite the name steel. Austenitic stainless steels typically have high contents of chromium and nickel within the alloy. These components both exhibit a value of μ_r = μ and can comprise approximately 30% of the steel alloy, suggesting that the total value of relative permeability may be half the value of cold-rolled steels. Since the base of stainless steels is often formed with low carbon content steel, the conductivity is also compromised. Generally the reported relative conductivity value for stainless steels is noted to be (σr) of only 0.02 × σ! [1] Consequently, the values suggest that due to the combination of low relative permeability and low relative conductivity, stainless steel alloys yield comparatively inefficient (e.g. deep) skin depths as shields. The low values of conductivity provided by stainless steel alloys also imply that the shielding performance for reflection losses are comparatively limited as well.

The value of skin depth is yielded by

(in meters since the values of r and r are expressed with relationship to meters)

where, ω = 2π f where f is in Hertz. [2]

References

1. Military Handbook, MIL-HB-419A.
2. International Telephone and Telegraph Corporation, Reference Data for Radio Engineers, 4th Edition, American Book, Stratford Press, New York, New York, 1956.

W. Michael King is a systems design advisor who has been active in the development of over 1,000 system-product designs in a 50 year career. He serves an international client base as an independent design advisor. Many terms used for PC Board Layout, such as the “3-W Rule”, the “V-plane Undercut Rule”, and “ground stitching nulls”, were all originated by himself. His full biography may be seen through his web site: www.SystemsEMC.com. Significantly, he is the author of EMCT: High Speed Design Tutorial (ISBN 0-7381-3340-X) which is the source of some of the graphics used in this presentation. EMCT is available through Elliott Laboratories/NTS, co-branded with the IEEE Standards Information Network.