How Can I Accurately Model And Measure The Skin Effect In A Single-layer Rectangular PCB Trace With A Thickness Of 35 Microns, Width Of 0.5 Mm, And A Frequency Range Of 10 MHz To 100 MHz, Considering The Non-uniform Current Distribution And The Impact Of Copper Roughness On The Effective Resistance And Inductance?

To accurately model and measure the skin effect in a single-layer rectangular PCB trace with a thickness of 35 microns, width of 0.5 mm, and a frequency range of 10 MHz to 100 MHz, follow these steps:

Modeling the Skin Effect

1. Calculate Skin Depth (δ):

   • Use the formula: ${ \delta = \sqrt{\frac{\rho}{\pi \mu f}} }$
   • Where ${ \rho = 1.68 \times 10^{-8} \, \Omega\text{-m} }$ (copper resistivity), ${ \mu = 4\pi \times 10^{-7} \, \text{H/m} }$, and ${ f }$ is the frequency.
   • At 10 MHz: ${ \delta \approx 0.0652 \, \text{mm} }$
   • At 100 MHz: ${ \delta \approx 0.0206 \, \text{mm} }$

2. Determine Effective Resistance (R_ac):

   • Calculate DC resistance: ${ R_{\text{dc}} = \frac{\rho l}{w t} }$
   • Use the approximation: ${ R_{\text{ac}} = R_{\text{dc}} \sqrt{1 + \frac{f}{f_c}} }$
     • Critical frequency ${ f_c }$ where ${ \delta = t }$: ${ f_c \approx 3.47 \, \text{MHz} }$
   • At 10 MHz: ${ R_{\text{ac}} \approx 1.89 \, \Omega/\text{m} }$
   • At 100 MHz: ${ R_{\text{ac}} \approx 5.27 \, \Omega/\text{m} }$

3. Account for Copper Roughness:

   • Effective resistivity ${ \rho_{\text{eff}} = \rho \left(1 + \left(\frac{2 A_r}{\delta}\right)^2\right) }$
   • For roughness ${ A_r = 1 \, \mu\text{m} }$, ${ \rho_{\text{eff}} }$ increases slightly, especially at higher frequencies.

4. Model Inductance (L):

   • Use the formula for inductance per unit length over a ground plane: ${ L \approx \frac{\mu_0}{2\pi} \ln\left(\frac{h + \frac{w}{2}}{\frac{w}{2}}\right) }$
   • Assume height ${ h = 1.6 \, \text{mm} }$: ${ L \approx 4 \times 10^{-7} \, \text{H/m} }$
   • Inductance remains relatively constant with frequency.

Measuring the Skin Effect

1. Fabricate the PCB Trace:

   • Create the trace with specified dimensions on a single-layer PCB.

2. Set Up Measurement:

   • Use a Vector Network Analyzer (VNA) with a coaxial cable (e.g., 50 Ω) connected to the trace.

3. Measure S-Parameters:

   • Measure ${ S_{11} }$ and ${ S_{21} }$ over the frequency range (10 MHz to 100 MHz).

4. Convert to Impedance:

   • Calculate impedance ${ Z }$ using ${ Z = Z_0 \frac{1 + S_{11}}{1 - S_{11}} }$, where ${ Z_0 = 50 \, \Omega }$.

5. Extract Resistance and Inductance:

   • From the impedance, derive resistance and inductance.
   • Resistance increases with frequency due to skin effect.

6. Compare and Refine Model:

   • Validate modeled values against measurements.
   • Adjust parameters like roughness for better accuracy.

Summary

By calculating the skin depth, determining the effective resistance, accounting for copper roughness, and modeling inductance, you can accurately model the skin effect. Measuring with a VNA provides experimental data to validate and refine the model, ensuring accurate characterization of the PCB trace's behavior.