# Electromagnetic near field and the far field, Chapter two

**Editor’s note: **This book is useful for all designers who need to test or have their designs tested for emissions, radiation and susceptibility for EMC. It can also help a wide range of specialists in biology, medicine, labor safety, environmental protection, metrologists, EMF meter designers, testers and users, and even for those who must make legal decisions on the grounds of measurement results interpretation.

I will publish this chapter two which follows and another chapter 7, “Directional Pattern Synthesis” in the near future. Dust off your Maxwell’s equations and dig in!

Electromagnetic Measurements in the Near Field, 2nd Edition *by Pawel Bienkowski, Hubert Trzaska* SciTech Publishing © 2012

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**Chapter 2 **

**The Near Field and the Far Field **

The essential information for practical metrology is presented in this chapter, including a brief summary of the near-field properties as well as the basic equations and formulas related to fields generated by simple radiation sources.

**2.1 AN EMF GENERATED BY A SYSTEM OF CURRENTS **

Consider the calculation of the EMF at an arbitrary point, P, external to a volume containing arbitrary currents. Figure 2.1 shows a volume V, of arbitrary cross section, containing a system of arbitrarily oriented electric and magnetic currents, J and *J, respectively. The volume V is surrounded by an infinitely large, homogeneous, isotropic, linear, lossless medium of permeability e and permittivity µ. Electrical parameters are continuous on the boundary surface. The maximal linear size of the volume, V, is D (Fig. 2.1).

The E and H fields can be determined at an arbitrary point situated outside the volume, V, by solving Maxwell’s equations for the angular frequency, ω [1]:

Where Π and *Π are electric and magnetic Hertzian vectors:

**Figure 2.1 EMF in point P generated by currents in volume V. **

Where k = the propagation constant

r = the distance from the observation point P to a general point in the volume Q (R’, Ө', Ҩ') so

**r = R – R’**

with a resultant magnitude of:

where:

ß = an angle between R and RR',

R = the distance from the observation point to the center of the coordinate system,

R' = the distance from the general point to the center of the coordinate system.

Under the condition R' < R, the distance r may be presented with the use of a series expansion [Eq. (2.7)]:

If R >> D (where D is the maximal size of an arbitrary cross section of the volume V), it is possible to assume that r is parallel to R, so r ≈ R - R' cos ß. Then:

The index ∞ in the formulas indicates that they are valid for R >> D.

In this case the spatial components of E and H are given by:

where:

Π_{Ө∞}, Π_{Ҩ∞}, * Π_{Ө∞}, *Π_{Ҩ∞} = the spatial components of vector Π_{∞} and * Π_{∞}.

Z = wave impedance of the medium:

Z0 = intrinsic impedance of free space:

Equations (2.10–2.14) allow us to find the far-field EMF components of an arbitrary system of currents in volume V. The field may be characterized as follows:

• The EMF in the far-field is a transverse field [Eq. (2.12)].

• At an arbitrary point the EMF has a shape of the TEM wave [Eqs. (2.13) and (2.14)].

• Vectors E and H can have two spatial components that are shifted in phase; as a result, the field is elliptically polarized.

• The dependence of the E and H fields on . and . is described by the normalized directional pattern that is independent of R.

• The E and H components are mutually perpendicular and related by the wave impedance of a medium.

• The Poynting vector S = E × H is real and oriented radially.

To characterize the EMF properties in the far field, we have presented a straightforward solution of Maxwell’s equations. To get a fully generalized solution of the equations, it would be necessary to take into account a number of additional factors including: the diffraction of a wave caused by irregularities in a nonhomogeneous medium, the dispersion and nonlinear properties of the medium, the anisotropy of the material, and the superposition of waves when nonmonochromatic fields are being considered. Such a solution has not been fully formulated. However, a fully general solution is not crucial for metrology and the following sections are based on the previous analysis.

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