RF Current Density Simulator

Explore Skin Effect in a Rectangular Conductor

Relative Current Density Map $|J_z|$

Click on the heatmap to select a cross-section

At radio frequencies (RF), alternating current (AC) no longer flows uniformly through a conductor. It tends to concentrate near the surface, a phenomenon known as the skin effect. This becomes critical in the "intermediate frequency regime," where the skin depth—the effective depth of current flow—is comparable to the conductor's dimensions.

This simulator visualizes an analytical model of this effect for a rectangular trace, like those found on a printed circuit board (PCB). Use the controls to see how changing the frequency, material, and trace size impacts the current density profile. Understanding this distribution is vital for predicting RF circuit performance, including losses, impedance, and thermal behavior.

Conductor
Width (w)
Height (h)
↑ y
→ x

This simulator now uses a two-term analytical model based on the formulation discussed in papers such as G. W. Pan's work. This model is a well-established approximation for the current distribution:

$$ J_z(x,y) = K \left[ \frac{\cosh(k_c x)}{\sinh(k_c a)} + \frac{\cosh(k_c y)}{\sinh(k_c b)} \right] $$

Where:

  • $K$ is a scaling constant calculated to ensure the total current $I = \iint J_z \,dx\,dy$ is normalized to 1A for the simulation.
  • The complex wave number $k_c$ is: $$k_c = \frac{1+j}{\delta}$$
  • The skin depth $\delta$ is: $$\delta = \frac{1}{\sqrt{\pi f \mu \sigma}}$$
  • $a$ is half the width ($w/2$) and $b$ is half the height ($h/2$).

The visualizations reveal key behaviors of RF currents. The 3D plot and cross-sections typically show a "U" shape, indicating that the current density is highest at the conductor's outer edges and corners, decaying towards the center. This is current crowding.

Implications for RF Design:

  • Increased Resistance: Because the current is squeezed into a smaller area, the effective AC resistance is higher than the DC resistance, leading to greater power loss (heat).
  • Changed Inductance: The non-uniform current alters the internal magnetic field, affecting the trace's inductance, which can change the characteristic impedance of transmission lines.
  • Thermal Hotspots: The corners and edges, with higher current density, will get hotter than the center of the trace.

Based on the current settings, observe how the current is distributed.

This simulator implements an analytical model based on the formulations and principles discussed in the following key academic papers. These are highly relevant for the calculation of current distribution and frequency-dependent resistance in rectangular conductors.

Primary & Related Model References

  • Pan, G. W., et al. (2009). "Current Distribution in a Rectangular Conductor." IEEE Transactions on Magnetics, 45(10), 4480-4483.

    Relevance: Provides a direct and clear derivation of the two-term analytical model for current density implemented in this simulator.

    DOI: 10.1109/TMAG.2009.2021673

  • Cao, M., et al. (2022). "Quasi-Analytical Calculation of Frequency-Dependent Resistance of Rectangular Conductors Considering the Edge Effect." Energies, 15(2), 503.

    Relevance: This recent paper discusses and applies the same two-term model to accurately calculate AC resistance, confirming its modern relevance.

    DOI: 10.3390/en15020503

  • Zhang, Y., et al. (2024). "An Improved Analytical Model for the AC Resistance of Rectangular Conductors at High Frequencies." Energies, 17(12), 2828.

    Relevance: Demonstrates the ongoing research and refinement of these models, proposing improvements for even greater accuracy.

    DOI: 10.3390/en17122828

Foundational Text

  • Stoll, R. L. (1974). The Analysis of Eddy Currents. Oxford University Press.

    Relevance: A classic textbook that provides the fundamental theory of eddy currents and diffusion of electromagnetic fields in conductors, which is the physical basis for the skin effect.