# High-Speed Skin Depth: Calculating Material Impact

### Key Takeaways

• The skin depth is the concentration of the current density toward the surface of a conductor at higher frequencies.

• Conductive materials experience more significant losses at higher frequencies, especially magnetic materials.

• Calculating the skin depth requires only a few simple material values.

High-speed skin-depth interactions greatly influence the performance of coaxial cables.

As circuit frequencies increase, behavior departs from DC or low-speed characteristics. This statement is a cornerstone of high-speed design, but understanding the reason and the resultant best practices are entirely different scopes. While the result of high-speed skin depths is a greater loss of the circuit as the conductive surface used to transport current shrinks, the underlying electromagnetic behavior tells a more complicated story.

Skin Depth by Conductor and Frequency (in mm)

 60 Hz 1x103 Hz 1x106 Hz 1x109 Hz 1x1012 Hz Copper 8.417 2.062 0.065 0.002 6.52x10-5 Aluminum 10.59 2.593 0.082 0.003 8.2x10-5 Silver 8.184 2.005 0.064 0.002 6.34x10-5 Iron 0.291 0.072 0.002 7.118x10-5 2.251x10-6

## The Effects of High-Speed Skin Depth

For direct current, the current density is uniform across the volume of the conductor; however, this is not the case at significantly high speeds. The skin effect is a phenomenon regarding a conductor’s current density where the frequency of a current increases, and the depth to which the current penetrates the conductor becomes increasingly concentrated towards the conductor’s surface. The definition of the skin depth is the distance where the current is e-1 (~37%) of the surface value. As the skin depth shrinks, the area of the conductor reduces significantly in size as the conductor forms the “border” of the concentric polygon (e.g., a ring from a cylinder), meaning the effective resistance increases proportionally. Therefore, for the same current and conductor parameters (materials, shape, etc.), high frequencies inherently incur more energy loss and generate more heat than lower frequencies.

The mechanism for the skin depth arises from changes to the back electromotive force. Conductors produce a magnetic field around itself once current flows through the conductor, and any change to the current likewise induces a change in the magnetic field. The collapse or change in the magnetic field also creates an electric field that opposes the change, known as the counter-electromotive force (EMF). This counter EMF originates at the center of the conductor and pushes electrons away depending on the strength of the electric field. Simultaneously, an increase in the magnetic field (caused by a sharp change in the current, i.e., a high frequency) creates eddy currents around the magnetic field, which cancel toward the center of the conductor and sum toward the surface of the conductor. These various factors all contribute to the same result: a concentration of the free electrons toward the conductor's surface.

The skin effect process mediates two intrinsic material properties: resistivity, which measures the ratio of the electric field to the current density, and permeability, which influences the magnetic flux reduction response that occurs during a change in the overall magnetic field of the conductor. Practically, electronics manufacturers can only control the material selection of the product and design conductors to minimize the impact of the skin effect without undercutting total performance. The former consideration can change depending on the design intent: for example, while conductivity is an essential quality of material selection (more conductive materials experience less loss due to a corresponding reduction in resistance), high-frequency applications can be undone by overly conductive materials. Similarly, the permeability of a material is the deciding factor for the strength of the eddy currents formed in response to the induced magnetic field. The confluence of effects from resistivity and permeability go as far as to shape cable manufacturing in high-frequency power transfer:

• Silver plating can cost-effectively encapsulate nearly all of the current density and is an excellent conductive medium, although a protective surface finish is necessary to prevent oxidation.

• It’s more beneficial to bundle multiple small cables rather than a single large cable of equivalent volume because the combined surface area of the cable bundle facilitates greater current-carrying capacity and lower resistance.

• High-current power lines can use a steel core surrounded by an aluminum cable bundle. Aluminum is one of the better conductors, while the cable benefits from the mechanical properties of the steel with negligible impact on the electrical performance.

## Calculating Current Density and Skin Depth

Analytically, calculating the skin depth and other interrelated values is relatively straightforward. It’s easiest to begin with the current density for AC frequencies:

where J is current density, Js is surface current density, j is the imaginary number, d is the depth, and δ is the skin depth. The imaginary number indicates a signal's radian delay per skin depth penetration (i.e., greater skin depth penetration changes the phase and amplitude). Alternatively, one full wavelength penetration requires 2π skin depths. For the skin depth, a lossless form is available:

for frequencies less than 1/(ρε), where ρ is the material resistivity, ⍵ is the frequency, and μ is the permeability. This simplified form holds for materials that are good conductors, but materials that are not instead experience a skin depth that approaches an asymptotic limit rather than continually decreasing:

which effectively states that the skin depth is unaffected by poor conductive materials and should be negligible in most applications.

## Cadence Has Solutions for High-Speed Design

The effects of high-speed skin depth on performance are understandable through the lens of material selection and manufacturing solutions that minimize the losses or leverage the near-surface current density. As frequency increases, conductive materials experience greater losses due to the confinement of the current density to a small area of the conductor surface. Notably, this loss is magnified due to materials with considerable magnetic permeability, increasing losses due to the eddy currents formed. Modeling and simulation are critical for predicting electronic behavior, and Cadence’s PCB Design and Analysis Software suite provides comprehensive tools to determine signal parameters at varying frequencies. Alongside OrCAD PCB Designer, design teams can accelerate ECAD development without sacrificing quality.

Leading electronics providers rely on Cadence products to optimize power, space, and energy needs for a wide variety of market applications. To learn more about our innovative solutions, talk to our team of experts or subscribe to our YouTube channel.

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