Wheeler incremental inductance rule

{{short description|Rule of thumb for estimating skin effect resistance of parallel transmission lines}}

File:Wheeler Incremental Inductance Rule.png

The incremental inductance rule, attributed to Harold Alden Wheeler{{cite journal |journal=Proc. IRE |title=Formulas for the Skin Effect |author-first=H. A. |author-last=Wheeler |volume=30 |issue=4 |pages=412–424 |date=September 1942 |doi=10.1109/JRPROC.1942.232015 |s2cid=51630416 }} by Gupta{{Citation |last1=Gupta |first1= K. C. | last2=Garg | first2=Ramesh | last3=Bahl | first3= I. J. |year= 1979 |title= Microstrip Lines and Slotlines |publisher= Artech House |isbn= 0-89006-074-6}}{{rp|101}} and others{{Citation |last1=Garg |first1=Ramesh | last2=Bahl | first2=Inder| last3=Bozzi| first3=Maurizio |year= 2013 |title= Microstrip Lines and Slotlines | edition=3 |publisher= Artech House |isbn= 978-1-60807-535-5}}{{rp|80}} is a formula used to compute skin effect resistance and internal inductance in parallel transmission lines when the frequency is high enough that the skin effect is fully developed. Wheeler's concept is that the internal inductance of a conductor is the difference between the computed external inductance and the external inductance computed with all the conductive surfaces receded by one half of the skin depth.

: L_internal = L_external(conductors receded) − L_external(conductors not receded).

Skin effect resistance is assumed to be equal to the reactance of the internal inductance.

: R_skin = ωL_internal.

Gupta{{rp|67}} gives a general equation with partial derivatives replacing the difference of inductance.

:$L\_\backslash mathrm\{int\}=\; \backslash sum\_\{m\}\; \backslash \; \backslash frac\; \{\backslash mu\_m\}\; \{\backslash mu\_0\}\; \backslash frac\; \{\backslash partial\; L\}\; \{\backslash partial\; n\_m\}\; \backslash frac\; \{\backslash delta\_m\}\; \{2\}$

:$R\_\backslash mathrm\{skin\}\; =\; \backslash sum\_\{m\}\; \backslash \; \backslash frac\; \{R\_\{\backslash mathrm\{s\}\; m\}\}\; \{\backslash mu\_0\}\; \backslash frac\; \{\backslash partial\; L\}\; \{\backslash partial\; n\_m\}\; =\; \backslash omega\; L\_\backslash mathrm\{int\}$

::where

:::$\backslash frac\; \{\backslash partial\; L\}\; \{\backslash partial\; n\_m\}$ is taken to mean the differential change in inductance as surface m is receded in the n_m direction.

:::$R\_\{\backslash mathrm\{s\}\; m\}\; =\; \backslash frac\; \{\backslash omega\; \backslash mu\_m\; \backslash delta\_m\}\; \{2\}$ is the surface resistivity of surface m.

:::$\backslash mu\_m\; =$ magnetic permeability of conductive material at surface m.

:::$\backslash delta\_m\; =$ skin depth of conductive material at surface m.

:::$n\_m\; =$ unit normal vector at surface m.

Wadell{{Citation |last=Wadell |first= Brian C. |year= 1991 |title= Transmission Line Design Handbook |publisher= Artech House |isbn= 0-89006-436-9}}{{rp|27}} and Gupta{{rp|67}} state that the thickness and corner radius of the conductors should be large with respect to the skin depth. Garg{{rp|80}} further states that the thickness of the conductors must be at least four times the skin depth. Garg{{rp|80}} states that the calculation is unchanged if the dielectric is taken to be air and that $L\; =\; Z\_\{\backslash mathrm\{c\}\}\; /\; V\_\{\backslash mathrm\{p\}\}$ where $Z\_\{\backslash mathrm\{c\}\}$ is the characteristic impedance and $V\_\{\backslash mathrm\{p\}\}$ the velocity of propagation, i.e. the speed of light. Paul, 2007,{{Citation |last=Paul |first=Clayton R. |year= 2007 |title= Analysis of Multiconductor Transmission Lines |publisher= Wiley |url=http://read.pudn.com/downloads740/ebook/2953214/TRANSMISSION_LINES.pdf}}

{{efn|Wheeler incremental inductance rule … should not be used for conductors of rectangular cross-section, because , … resistance and internal inductance reactance are not equal.}}{{rp|149}} disputes the accuracy of $R\_\{\backslash mathrm\{skin\}\}\; =\; \backslash omega\; L\_\{\backslash mathrm\{int\}\}$ at very high frequency for rectangular conductors such as stripline and microstrip due to a non-uniform distribution of current on the conductor. At very high frequency, the current crowds into the corners of the conductor.