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Analysis of the effect of skin depth on different RF PCB structures

Analysis of the effect of skin depth on different RF PCB structures

Skin depth is usually used to describe the behavior of current flowing through circuit conductors, especially PCB circuits at RF/microwave frequencies. When direct current (DC) passes through the PCB circuit conductor, the current density in the conductor is evenly distributed. However, when a high-frequency sinusoidal current flows through a PCB conductor, the current density distribution inside the conductor will change. Compared to the surface of the conductor, the internal current density will become smaller and smaller. The skin depth of all conductors indicates the depth at which the current density on the surface of the conductor drops to 1/e. Skin depth is an important circuit board parameter that needs to be considered when designing high-frequency transmission lines or RF circuits, and it is also a factor that all circuit simulation modeling software needs to consider when modeling circuits in the RF/microwave frequency band.

The mathematical expression of skin depth ð is: ð = (1/πfµσ)0.5

Where f is the frequency, μ is the magnetic permeability, and σ is the electrical conductivity. By observing this formula, we can see that the skin depth of the conductor is inversely proportional to the frequency. So the skin depth in high-frequency circuit boards is actually a very small thickness.

Current density

In the case of DC direct current, 100% of the conductors are used to transmit current. All conductors connected to direct current have evenly distributed current density across their cross-sections. However, for the current changing at a sinusoidal frequency, the current density distribution inside the conductor is different, and the outer surface of the conductor has a larger current density than the inner and middle parts of the conductor. As the mathematical expression of skin depth indicates: as the frequency increases, the current density on the outer surface of the conductor will become larger and larger. At very high frequencies, the current density inside the conductor is very small, or even no current density. Most of the current density is concentrated on the outer surface of the conductor. In fact, the higher the frequency, the smaller the skin depth of the conductor.

So what is the actual skin depth of common conductors? Let's take copper as an example. The approximate value of μ is approximately equal to 1, and σ is approximately 5.8 × 107 S/m. The skin depth will decrease as the frequency increases. The skin depth of copper is 2.95μm (0.116mils) at 500MHz, 2.09μm (0.082mils) at 1GHz, 0.66μm (0.026mils) at 10GHz, 0.30μm (0.012mils) at 50GHz, and 0.23 at 80GHz μm (0.009mils). Obviously, most of the current density at millimeter wave frequencies is located near the surface of the copper conductor.

Analysis of the effect of skin depth on different RF PCB structures

So, when might the copper on the circuit be too thin to be a good conductor? Since the size of the circuit geometry is a function of wavelength, it decreases as the frequency increases. For high-frequency RF/microwave circuits, especially at millimeter wave frequencies, strict etching control of PCB copper wires is required. Some applications require very thin circuit board materials with copper foil, because thinner copper foil can better control the etching of PCB circuit features, such as strip lines and microstrip transmission lines. One-quarter ounce (0.25oz) copper foil is very thin copper, with a standard thickness of 8.89 microns (0.35mils). Compared with other thicknesses, this thin copper conductor provides sufficient skin depth for frequencies well below 500 MHz and well above 500 MHz.

Considering that the surface of the conductor has a higher current density at higher frequencies, the factor that may affect the skin depth of the conductor in the PCB is the surface roughness of the copper foil at the substrate-conductor interface. Because at higher frequencies, the current density increases toward the outer surface of the conductor, the rougher copper conductor surface, especially the roughness of the copper foil at the substrate-conductor interface, will increase the conductor circuit loss.

 In addition to increasing the conductor loss, the rough surface of the copper foil conductor will also reduce the phase response and phase speed of the circuit, making the performance of the circuit appear as if it is on a substrate with a higher dielectric constant (Dk). Therefore, for the substrate material of the same dielectric constant, a circuit with a smooth copper foil conductor surface has a lower effective Dk than a circuit with a rough copper conductor surface. The frequency that should be considered for the surface roughness of copper conductors is related to the skin depth of the conductor. When the skin depth is the same as the conductor surface roughness or thinner, the conductor surface roughness will have an impact on the performance of the RF/microwave circuit. For example, electrolytic (ED) copper usually has a surface roughness of about 2 μm RMS and will affect the RF performance of the circuit at a frequency of about 1 GHz. Rolled copper has a smoother surface roughness, about 0.35μmRMS, which will not affect the performance of RF/microwave circuits when it is less than 40GHz.

In-depth analysis

When designing and modeling high-frequency circuits, the actual skin depth is usually determined several times the theoretically calculated skin depth (up to 5 times the value of D) in order to carry out meaningful simulations. According to the calculation formula of D, the skin depth is related to the conductivity of the conductor. However, we should not only consider the electrical conductivity of copper, but also the electrical conductivity of any surface treatment that protects the copper wires of the PCB. The conductivity of most PCB surface treatments is lower than that of copper, resulting in a decrease in composite conductivity and an increase in skin depth. For example, for electroless nickel immersion gold (ENIG) surface treatment, the conductivity is a composite of nickel, gold and copper conductors. At lower frequencies, the current density is distributed across all three metal conductors. But at higher frequencies, the skin depth decreases, and only nickel and gold play the role of conductors. At very high frequencies, only gold is the conductor.

For the ENIG surface treatment method, since nickel is magnetic, its μ value is increased compared to copper (the μ value is higher than 1), resulting in a decrease in the skin depth during ENIG surface treatment. The use of this surface treatment will result in the effect of two factors. The magnetic permeability of nickel reduces the skin depth, while its lower conductivity increases the skin depth. In contrast, immersion silver is also used as the final surface treatment of copper wires on the PCB. Silver has higher conductivity than copper and is not magnetic. Therefore, the skin depth of copper conductors will be slightly reduced when using the immersion silver surface. However, very thin immersion silver surfaces are usually used, so unless at higher millimeter wave frequencies, such as 100 GHz and higher millimeter wave frequencies, the effect of this surface treatment may not be obvious.

Skin depth is a circuit characteristic that needs to be considered, especially at higher millimeter wave frequencies. Although the final surface treatment will also affect the performance of the PCB, the weight/thickness and type of the copper conductor will affect the performance of the RF/microwave circuit as well as the quality of the dielectric material and the quality of the substrate. Smooth and thin copper foil, such as rolled copper, can provide the skin depth and low conductor loss required for good high-frequency performance, thereby reducing the overall PCB circuit loss.