Wheeler Incremental Inductance Rule
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The incremental inductance rule, attributed to
Harold Alden Wheeler Harold Alden Wheeler (May 10, 1903 - April 25, 1996) was a noted American electrical engineer. Biography Wheeler was born in Saint Paul, Minnesota, to William Archibald Wheeler and Harriet Marie Alden Wheeler (a descendant of John and Prisc ...
by Gupta and others is a formula used to compute
skin effect Skin effect is the tendency of an alternating electric current (AC) to become distributed within a conductor such that the current density is largest near the surface of the conductor and decreases exponentially with greater depths in the co ...
resistance and internal
inductance Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The flow of electric current creates a magnetic field around the conductor. The field strength depends on the magnitude of the ...
in parallel
transmission line In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmis ...
s 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. Linternal = Lexternal(conductors receded) - Lexternal(conductors not receded). Skin effect resistance is assumed to be equal to the reactance of the internal inductance. Rskin = ωLinternal. Gupta gives a general equation with partial derivatives replacing the difference of inductance. : L_= \sum_ \ \frac \frac \frac : R_ = \sum_ \ \frac \frac = \omega L_ ::where ::: \frac is taken to mean the differential change in inductance as surface m is receded in the nm direction. ::: R_ = \frac is the surface resistivity of surface m. ::: \mu_m = magnetic permeability of conductive material at surface m. ::: \delta_m = skin depth of conductive material at surface m. ::: n_m = unit normal vector at surface m. Wadell and Gupta state that the thickness and corner radius of the conductors should be large with respect to the skin depth. Garg further states that the thickness of the conductors must be at least four times the skin depth. Garg states that the calculation is unchanged if the dielectric is taken to be air and that L = / where Z_c = the characteristic impedance and V_p = velocity of propagation = the speed of light. Paul, 2007, disputes the accuracy of R_ = \omega L_ at very high frequency for rectangular conductors such as
stripline Stripline is a transverse electromagnetic (TEM) transmission line medium invented by Robert M. Barrett of the Air Force Cambridge Research Centre in the 1950s. Stripline is the earliest form of planar transmission line. Description A strip ...
and
microstrip Microstrip is a type of electrical transmission line which can be fabricated with any technology where a conductor is separated from a ground plane by a dielectric layer known as the substrate. Microstrip lines are used to convey microwave-frequ ...
due to a non-uniform distribution of current on the conductor. At very high frequency, the current crowds into the corners of the conductor.


Example

In the top figure, if : L_0 is the inductance and Z_0 is the characteristic impedance using the dimensions \mathrm_0, \mathrm_0, and \mathrm_0, and : L_1 is the inductance and Z_1 is the characteristic impedance using the dimensions \mathrm_1, \mathrm_1, and \mathrm_1 then the internal inductance is : L_ = ( L_1 - L_0 ) = ( Z_1 - Z_0 ) / V_p where V_p is the velocity of propagation in the dielectric. and the skin effect resistance is : R_ = \omega ( L_1 - L_0 )


Notes


References

{{DEFAULTSORT:Wheeler Incremental Inductance Rule Signal cables Telecommunications engineering Transmission lines Distributed element circuits