Modeling skin effect in inductors

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Modeling Skin Effect in Inductors
Ken Kundert Designer’s Guide Consulting, Inc.

Version 1a, October 2001

Presents a technique for modeling skin effect losses in inductors. Includes a model written in Verilog-A as well as a model based on a new type of basic element named a fracpole.

This manuscript wasoriginally written in October 2001. It was last updated on May 12, 2006. You can find the most recent version at www.designers-guide.org. Contact the author via e-mail at ken@designers-guide.com. Permission to make copies, either paper or electronic, of this work for personal or classroom use is granted without fee provided that the copies are not made or distributed for profit or commercial advantage andthat the copies are complete and unmodified. To distribute otherwise, to publish, to post on servers, or to distribute to lists, requires prior written permission.

Copyright © 2006, Kenneth S. Kundert – All Rights Reserved

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Background

1.0 Background
In the Coilcraft application note on modeling RF inductors [2], the inductor is modeled as shown in Figure 1. Component H in themodel represents the skin-effect loss. In the

FIGURE 1.

RF inductor model.

Cp Rs H

Rp

L

ap-note, it is modeled in two different ways. In the PSPICE version, it is written as 1 + jfZ( f ) = V ( f ) = ----------------- . ----------I( f ) H For other simulation software, the model is written as Z( f ) = k f . (2) (1)

These two models are not consistent in two ways. First, (1)has a corner at 1 Hz whereas (2) does not. Second, Z( f ) in (1) is complex while in (2) it is real. Once significantly above 1 Hz the real and imaginary are equal for the PSPICE model. Thus, for measurements made well above 1 Hz, k and H are related by 1 k = ----------2H (3)

Having a purely real impedance that varies with f is not physically realizable (it would be non-causal, as shown in theappendix) and so is clearly incorrect. However, the corner frequency at 1 Hz in the PSPICE model seems artificial. I expect that it was included to prevent the impedance from going to zero at DC, which causes problems for PSPICE. Instead of (1) and (2), skin effect is modeled here in a more traditional way, jfZ( f ) = -------- . H (4)

This model involves a non-integer power for f and so is adistributed model [1]. In other words, Z( f ) cannot be exactly represented using a finite number of poles and zeros and the model cannot be implemented exactly by combining a finite number of lumped components (resistors, capacitors, and inductors).

2.0 Modeling Skin Effect
A transfer function of jf is approximated over a finite range of frequencies with an equal number of real poles and zerosalternating and evenly spaced in a logarithmic

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Modeling Skin Effect

sense over that range, as shown in Figure 2. The range of the approximation is from f0 to

FIGURE 2.

Modeling the

jf nature of skin effect using a collection of real poles and zeros.

fz1 fp1 fz2 fp2 fz3 fp3 fz4 fp4 fz5 fp5

f1 log Z

mf0

log f

f1, with the impedance flattening out at frequencies outside of this range. The range of frequencies over which skin effect must be accurately modeled can be determined by examining its contribution to the overall impedance of the inductor. The impedance of a representative inductor is shown in Figure 3. Notice that the impedance is separated into

FIGURE 3.

Modeled impedance ofan RF inductor separated into real (R) and imaginary (X) parts.
Coilcraft A01T Mini Spring 1kΩ 100Ω 10Ω 1Ω 100mΩ 10mΩ

1/(2πfCp)

2πfL

Rp

X f --------2H 2 R
Rs
10kHz 100kHz 1MHz 10MHz 100MHz 1GHz 10GHz 100GHz

1mΩ

its real and imaginary parts (the resistance and the reactance). The resistance represents the loss. It is important to accurately model the resistive portion in the...