Near field magnetic & electric wave impedance formulas

Question

I am looking for the exact formulas for the impedance of an electric and magnetic wave in the near field region.

We have some approximates:

Z_{w_e}≈ 1 / (ω * ε₀ *r)

Z_{w_m}≈ (ω * µ₀ *r)

where r is the distance, ω is the angular frequency, ε₀ vacuum permittivity, µ₀ vacuum permeability

However these formulas are only approximations, and I believe the accurate formula involves complex numbers.

I could not find the full formula for this, I was hoping somebody here would know it.

@analogsystemsrf if you are talking about "Electromagnetic Fields and Waves - Lorrain and Corson" , I have quickly read through the book, and there are a lot of complex equations there, but most of them are talking about electric charges and electromagnetism is a conductor. I haven't seen anything about waves and wave propagation. I've went to page 263, where it detailed the Characteristic Impedance (ω * µ/ k ) but this is essentially the far field vacuum impedance equation, 376.73 Ohm. I could not find an equation for the near field. — user138887, Apr 27, 2017 at 17:39
I have also looked here dannex.se/theory/3.html , equation 46 and 47, but they are just alternate forms of the equations I provided above. So they are still approximates, and not the precise formula. I am looking for the accurate formula though. — user138887, Apr 27, 2017 at 18:13

Andy aka · Accepted Answer · 2017-04-27 11:21:15Z

1

I haven't got formulae but I have this picture that may help a little bit: -

answered Apr 27, 2017 at 11:21

Andy aka

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\$\begingroup\$ Yes I know about that relationship that is why I asked specifically about the near field. In the far field the equation becomes that of the free space (µ0*c). Also it looks like the relationship is inverted between the magnetic and electric field in the near field. However I think a true formula has to include in some way the wave propagation constant which has an imaginary element in it. \$\endgroup\$
– user138887
Apr 27, 2017 at 14:02

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user138887 · Accepted Answer · 2017-04-28 20:43:17Z

I have found a potential good answer:

A general formula for wave impedance, where we know that the magnetic impedance is essentially the inverse of the electric wave impedance.

To calculate the near field impedance, just use a common definition of the near field like:

r < λ / ( 2* π)

Which was defined by Ott & White, or other definitions depending on the situation: http://m.eet.com/media/1140939/19185-table_1.pdf

2 Answers 2