Shielding Formulas for Near Fields
R.C. Hansen
Consulting Engineer
Tarzana, CA
This article reviews extremely low frequency (ELF) formulas for magnetic shielding effectiveness. In particular, the simple threeterm Schelkunoff formula is accurate at all frequencies if the proper nearfield, or farfield, wave impedance is used.
Electromagnetic shielding of rooms or equipment through the use of metal walls or sheets has been important for many decades, and is now even more necessary with the burgeoning number of RF transmitters of all types, both inside and outside of buildings. Many devices such as electronic typewriters and computers emit ELF radiation. For plane wave sources there are wellknown simple formulas for transmission through a metal sheet (or through a multiplelayer wall) using ABCD or equivalent chain matrices. Often the frequencies of interest are sufficiently low that the metal wall is in the near field of a source or test antenna. For these cases, the shielding effectiveness (SE) (the ratio of the incident field to the exit field) may be significantly different than for the plane wave case. This article reviews formulas for the nearfield case.
SCHELKUNOFF SHIELDING THEORY
Cylindrical and spherical shields were investigated by Schelkunoff^{1} many decades ago. He showed that SE can be written as three terms (in decibels) representing the primary reflection at the airmetal interface, the attenuation (diffusion loss) for a single pass through the metal wall and a correction term to account for the internal reflections and additional attenuation in the material. These formulas have been applied to plane sheets and used by many workers. The terms, of course, result from the exact formula for transmission of a plane wave through a sheet of arbitrary material.^{2,3} To recapitulate, the three terms are
? additional internal attenuation
and reflection loss
The SE, in decibels, is the sum
SE = SER + SEL + SEIR
The reflection coefficient G is
The sheet thickness is t and its impedance and propagation constant are Z_{sheet} and . The critical contributions of Schelkunoff were to formulate transmission through a wall  long before the Radiation Laboratory work  and to recognize that nearfield effects could be subsumed into the external wave impedance Z_{wave} .
At low frequencies most antennas (and inadvertent radiators) are loops. The small loop nearfield wave impedance is usually computed as the ratio of the wellknown nearfield components E_{f } /H_{ } . Whitehouse^{4} demonstrated that the correct wave impedance uses the ratio of transverse fields, where H_{trans } = H_{r} sin + H_{ } cos . (Transverse fields apply when the plane of the loop is parallel to the wall; other loop orientations will have a different wave impedance.) This relationship results in a wave impedance of
where
R  =  distance from the source loop 

 to the receiving loop 
k  =  2p / 
 =  120 p 
Whitehouse modified Equation 6 to include the effect of loop radius, replacing R with√R^{2 } + a^{2} where a is the loop radius. As will be shown, use of this modified nearfield wave impedance produces very good results. The wall must be infinitely large or, if part of a shielded room, sufficiently large to encompass most of the incident field.
LOOPTOLOOP CALCULATIONS
In a classic paper, Moser^{5} obtained the SE of a metal wall when the source antenna was an electrically small loop, and when the wall was in its near field. The loop magnetic field was expanded into an integral of plane waves; each plane wave component was propagated through the wall. This work was recently extended to include an electrically small receiving loop, again in the nearfield regime.^{6} This formulation matches the experimental setup,^{5} which was looptoloop, and is rigorous as long as the loops are sufficiently small to maintain constant current.
THEORIES vs. EXPERIMENTS
Carefully measured data on the SE of metal sheets are sparse; Moser's^{5} data are used here. He measured three materials  copper, aluminum and steel  each roughly 1/16 and 1/8inch thick. Both transmit and receive loops were 7 cm in diameter. The power source was a 200 W McIntosh amplifier with a low impedance, providing a roughly constant voltage on the source loop. The receive loop open circuit voltage was measured both with the sheet present and absent. For a fixed distance between loops, the SE was found to be roughly independent of the distance of the sheet from the transmit loop (as predicted by the plane wave spectrum theory). As long as the sheet is large enough to encompass most of the field, the nearfield effect is from one loop to the other; the looptoloop distance affects the wave impedance.
Using the correct wave impedance, the exact looptoloop calculations (and the Schelkunoff formulation results) are compared here with the Moser measured data. Figure 1 shows data for 1/16inch copper sheet with looptoloop spacing of 3 cm. It is apparent that the agreement between numerical results from the exact theory and measurements is very good. The Schelkunoff formulation results are very close to the exact results. Figure 2 shows data for a steel sheet with conductivity of 9.86E6 S/m and a permeability of 112. A looptoloop spacing of 3 cm was used. Again, agreement between theoretical results and the measurements is very good. For the copper sheet, the thickness equals one skin depth at 1.733 kHz; for the iron sheet this condition occurs just below 0.1 kHz. For larger looptoloop distances, the Schelkunoff results are an excellent match to the exact results. To illustrate the contribution of the individual terms in the Schelkunoff formula, the copper sheet at 2 kHz was used as an example. The face reflection is 15.6 dB, skin depth loss is 9.3 dB and internal reflection loss is 0.6 dB, for a total of 25.5 dB.
CONCLUSION
Schelkunoff's formulas are excellent for calculating nearfield SE of metal sheets, provided the transverse nearfield wave impedance is used. The formulas are also excellent for farfield (plane wave) calculations, where the free space wave impedance is used.
References
1.? S.A. Schelkunoff, Electromagnetic Waves , Van Nostrand, 1943, Section 8.18.
2.? W.M. Cady, M.B. Karelitz and L.A. Turner, Radar Scanners and Radomes , Vol. 26, Rad. Lab. Series, McGrawHill, 1948, Section 12.5.
3. T.E. Tice (editor), Techniques for Airborne Radome Design , Technical Report AFALTR66391, Vol. 1, December 1966, Chapter 2, AD811 355.
4. A.C.D. Whitehouse, "Screening: New Wave Impedance for the Transmissionline Analogy," Proc. IEE , Vol. 116, July 1969, pp. 11591164.
5. J.R. Moser, "Lowfrequency Shielding of a Circular Loop Electromagnetic Field Source," Trans. IEEE , Vol. EMC9, March 1967, pp. 618.
6. R.C. Hansen and J.R. Moser, "Loopshieldloop Shielding Effectiveness," Trans. IEEE , Vol. EMC41, May 1999, pp. 144146.