Skin-Effect Inductance

Whenever you alter the path of current, you alter the inductance. Because the skin effect modifies the distribution of current within the conductor, it must also change the inductance of that conductor. You can observe this in very careful measurements of transmission-line inductance at high and low frequencies.

At frequencies well above the skin-effect onset frequency w d the current in a transmission line distributes itself in whatever way minimizes the overall inductance of the circuit (current follows the path of least inductance). The least-inductive distribution for a transmission line concentrates current around the periphery of the conductors, with little or no flux interior to the conductors. The value of inductance so obtained is called the external inductance of the transmission line L e .

The external inductance is defined at a frequency sufficiently high that the skin depth shrinks to much less than the wire thickness , but also sufficiently low that the wire still operates in a TEM mode (or for microstrips, a quasi-TEM mode ”see "Non-TEM Modes" in Section 5.1.5). The name external inductance applies because the definition assumes the skin effect has expunged all magnetic flux from the interior of each conductor; thus the circuit responds only to magnetic flux appearing external to the conductors themselves . The external inductance is the value of series inductance, in Henries per meter, computed by a 2-D field solver under the assumption that current rides on the surface of each conductor without penetrating deeply into the body of any of the conductors.

At frequencies well below w d , current in a transmission line redistributes itself to minimize the resistance of the circuit (current follows the path of least resistance). Because this distribution is not the same as the least-inductive distribution, the value of low-frequency inductance must by definition be higher than the minimum inductance L e . The difference in inductance between low and high frequency values is called the internal inductance of the transmission line, L i . It carries this name to remind you that the shift in inductance has to do with the penetration of flux internal to the conducting elements of the line.

The redistribution of current within the conductors at low frequencies affects their inductance, but not their capacitance. The creation of an analogous "internal capacitance " would require that electric fields penetrate the body of the conductors, something that does not happen for good metallic conductors at any reasonable operating frequency. When working with metallic conductors you may assume that charge remains bound to the surfaces at all frequencies of interest to digital designers. Around the skin-effect onset frequency there is no change in capacitance.

If a material of very low conductivity violates the assumption s >> w e , then electric fields will penetrate the conductor. Above a certain frequency w c = s / e , there will develop an electric-field skin effect accompanied by changes in capacitance somewhat analogous to the changes in internal inductance caused by the magnetic skin effect. The partial penetration of electric fields into a lightly doped silicon substrate is the root cause of the slow-wave effect (see Section 2.14, " Slow-Wave Mode On-Chip").

Models for predicting the series resistance and inductance of conductors are presented in the next section.

POINTS TO REMEMBER

• The distribution of current at high frequencies minimizes inductance.
• At DC, the path of least DC resistance creates a slightly higher inductance.
• Good models for skin effect take into account changes in both resistance and inductance with frequency.