#### Skin Effect Background

It is well known that the characteristic impedance of a transmission line will vary depending on the bandwidth of the signals that are expected to pass through it, according to the following formulas:

The vital parameter to consider in the context of the Skin Effect is the resistance, R, which starts to become dominant and affects the impedance at frequencies above 1 GHz.

The is because the signal propagates along the outer surface of the trace instead of propagating uniformly along the entire cross-section. As the frequency increases, this phenomenon will become worse. It can be inferred from this that the resistance of the conductor will not be constant but a function of the frequency. This will naturally affect the characteristic impedance of the line.

#### A Bit of Physics

According to Faraday's Law, an alternating (AC) current flowing through a conductor will create a magnetic field and flux. The change in flux will create induced electric currents in the conductor. These currents are called "Eddy Currents", also known as "Foucault currents".

*Figure 2 - Development of Eddy Currents (also known as Foucault Currents)*

The figure depicts the Eddy Currents generated on a cross-section of any conductor (in this case a conductor in a PCB).

It is important to note that Eddy Currents are generated around magnetic field lines. It can be seen that these current loops are divided into i1 and i2. The current i1 flows in the envelope of the current I that created it, and the current i2 flows inside the conductor in the opposite direction to the current that generated it. As a result, the total current inside the conductor is small, and most of the flow occurs on the envelope of the conductor.

(Note – the loop currents are drawn as separate loops, but in practice, they merge into a single loop).

#### Skin Depth

The skin depth is the parameter that determines how deep into the material the current will penetrate. For FR4 materials, the calculation will be according to the following formula:

This parameter indicates the depth of the "ring" or "strip" of the perimeter of the conductor where the current flows.

#### From which frequency is the skin effect observed?

Placing a frequency of 10 MHz in the skin depth formula above, the skin depth in an FR4 conductor is 21 um, while a 1 oz copper layer is 34 um thick. This means that even at low frequencies, the Skin Effect starts to come into play.

At a frequency of 1 GHz, the skin depth will be 2 um, so the attenuation will be significantly higher.

**Calculation of Conductor Resistance Depending on Frequency:**

At frequencies below 10 MHz, we will use the following formula:

At frequencies above 10 MHz, we will use the following formula:

RDC is the resistance per length for DC currents.

ρ is the bulk resistivity of copper.

w is the line width of the signal trace.

t is the geometrical thickness of the signal trace.

The Total Resistance is Composed of a DC Component and an AC Component. The total resistance will be the sum of the DC and AC components as described in the following formula:

#### What is the Proximity Effect?

We already know that Eddy Currents cause the Skin Effect in a conductor through which a current flows. But what happens when (as in any PCB) there is a conductor for the signal, and below or above it, there is another conductor layer, as in the following example:

*Figure 3 - The Proximity Effect*

Due to the Skin Effect, the current density is concentrated on the outer surface of the conductor. Still, not in a homogeneous manner, and this is because there is a return conductor layer below the signal layer. This phenomenon is called the Proximity Effect, and it also results from Eddy Currents, but this time, this phenomenon occurs due to the Eddy Currents that are generated not in the signal conductor itself, but in the return layer.

#### 3D Simulations

Figure 4 shows the results of the simulations that were performed using the Hyperlynx 3D EM (courtesy of Mentor Graphics). It can be seen that as the signal

frequencies increase, the current density on the surface of the conductor decreases.

*Figure 4 - 3D Simulations of the Skin Effect*

#### What is Surface Roughness?

It is known that the surface of conductors in the PCB manufacturing process is not uniform – ("Surface Roughness"). This phenomenon must also be taken into account in addition to the Skin effect and the Proximity effect as an additional loss mechanism in High-Speed channels and PCB traces.

#### Conclusions:

Several parasitic effects occur when high-speed signals are involved.

Both vortex currents and PCB manufacturing processes cause these effects.

It is essential to understand and evaluate the magnitude of parasitic effects to be able to transmit high-speed signals on low-cost materials such as FR4 and to adopt design and layout techniques that will reduce the impact on signal attenuation and distortion.

As always, it is essential to perform simulations, especially if the signals are above 1 GHz.

It is important to remember that a digital signal is composed of the sum of the odd-integer harmonic frequencies of the signal's fundamental frequency. Therefore, the parasitic effects' severity will differ per each harmonic integer frequency.

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