Get our free email newsletter

Far-Field Criterion for Wire‑Type Antennas

This article presents the derivation of the-far field criterion for two fundamental wire-type antennas: electric (or Hertzian) dipole and magnetic dipole.

1. Electric (Hertzian) Dipole

Hertzian dipole, shown in Figure 1, consists of a short thin wire of length l, carrying a phasor current 0 = I0, positioned symmetrically at the origin of the coordinate system and oriented along the z-axis. 

Figure 1: Hertzian dipole

The complete fields of the Hertzian dipole, at a distance r from the origin, can be obtained from the vector magnetic potential A shown in Figure 1, (see [1] for the derivations), and can be expressed as, [2],

- Partner Content -

A Dash of Maxwell’s: A Maxwell’s Equations Primer – Part One

Solving Maxwell’s Equations for real-life situations, like predicting the RF emissions from a cell tower, requires more mathematical horsepower than any individual mind can muster. These equations don’t give the scientist or engineer just insight, they are literally the answer to everything RF.

  (1.1)

  (1.2)

  (1.3)

where η0 is the intrinsic impedance of free space, and β0 is the phase constant. 

With this electromagnetic wave, we associate wave impedance, defined as

- From Our Sponsors -

  (1.4)

A far-field criterion (distance r from the antenna) for Hertzian dipole (and other wire-type antennas) is derived from the requirement that the magnitude of the wave impedance in far field is equal to the intrinsic impedance of free space.

  (1.5)

Using Eqns. (1.2) and (1.3) in Eq. (1.4) we get

  (1.6)

or

  (1.7)

Multiplying the numerator and denominator by j(β0r)3 we obtain

  (1.8)

or

  (1.9)

Letting,

  (1.10)

we obtain

  (1.11)

We can evaluate this expression at different distances (in terms of the wavelength) from the antenna, for instance, (r = λ0/2π, r = λ0, r = 3λ0). When evaluated at r = 3λ0, the wave impedance becomes

  (1.12)

The result in Eq. (12) leads to the far-field criterion for the Hertzian dipole (and other wire‑type antennas):

  (1.13)

2. Magnetic Dipole

Magnetic dipole, shown in Figure 2, consists of a small thin circular wire loop of radius a, carrying a current 0 = I0, positioned in the xy plane, with the center of the loop at z = 0.

Figure 2: Magnetic Dipole

The complete fields of the magnetic dipole, at a distance r can be expressed [2],

  (2.1)

  (2.2) 

  (2.3)

The wave impedance for magnetic dipole is defined as

  (2.4)

Using Eqns. (2.1) and (2.3) in Eq. (2.4) we get

  (2.5)

or

  (2.6)

Multiplying the numerator and denominator by j(β0r)3 we obtain

  (2.7)

or

  (2.8)

Letting,

  (2.9)

we obtain

  (2.10)

Evaluating Eq. (2.10) at r = 3λ0 we get

  (2.11)

Thus, the magnitude of the wave impedance 

  (2.12)

We have arrived again at the far-field criterion as

  (2.13)

References

  1. Bogdan Adamczyk, Foundations of Electromagnetic Compatibility with Practical Applications, Wiley, 2017.
  2. Clayton R. Paul, Introduction to Electromagnetic Compatibility, Wiley, 2006.

Related Articles

Digital Sponsors

Become a Sponsor

Discover new products, review technical whitepapers, read the latest compliance news, and check out trending engineering news.

Get our email updates

What's New

- From Our Sponsors -

Sign up for the In Compliance Email Newsletter

Discover new products, review technical whitepapers, read the latest compliance news, and trending engineering news.