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I'm aware that the impedance of free space (377 ohms in SI units) doesn't have a whole lot of physical significance, because in natural EM units impedance is dimensioness and (e.g. in Lorentz-Heaviside units) the impedance of free space just equals 1.

But I occasionally see some argument for why it's conceptually useful to think of free space as actually having an impedance, rather than thinking of "the impedance of free space" as solely being a fancy way of saying "1". E.g. this answer compares the propagation of an EM wave through free space to a long LC circuit, and argues that the electric and magnetic fields of the EM wave behave mathematically similarly to the voltage and current of the analogous circuit. This analogy allegedly gives some people better intuition for the behavior of EM waves.

But an LC circuit doesn't have any resistance, but only reactance. It seems to me that in this analogy, the (metaphorical) "impedance" would be pure imaginary, i.e. solely a reactance.

Is there any conceptual utility in thinking of the impedance of free space as a complex impedance, which could in principle have a generic complex phase, as opposed to just a (conceptually simpler) reactance?

tparker
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2 Answers2

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An LC circuit is indeed reactive and its impedance is complex imaginary i.e., a reactance, but a transmission line is not an LC circuit. A transmission line is an infinite periodic LC ladder and its impedance is real because it is infinite long and is matched at every section for the rest.

I disagree with the claim that the impedance of free space is fictitious. It is true that the $377\Omega$ figure is SI dependent, but so is the resistance of the $47k\Omega$ resistor, no more and no less.

In fact, when you drive an antenna with a coaxial cable and match the antenna to the cable what you are doing is having a two-step transformer, first step is the $377\Omega$ being matched to the input of the antenna and we call the result the radiation impedance, and the 2nd step is to match that to the coax while in between you place various reactive elements to tune out the near field, ie., reactive field effects of the antenna that would make its impedance not real. Viewed this way an antenna is nothing but an impedance transformer between $E/H$ of the wave and $V/I$ of the terminals.

While vacuum is lossless so its impedance is a real number, the same is not true for a lossy medium such as propagation in salt water (sea), or in the earth. There indeed you have a complex "E/H" impedance and as you can imagine there is an enormous literature written on it. If interested, see the works of Ronold King on submerged antennas and some such.

hyportnex
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It seems that the answer is very simple: I had forgotten that the word "impedance" is used to refer to two related but distinct concepts.

The impedance of a single circuit element in a finite A/C circuit is defined as the ratio of the phasor representations of the voltage and current through that circuit element. For a simple LC circuit with an inductor and a capacitor wired in series, the equivalent impedance of the circuit is $Z = -i(\omega L + 1/(\omega C))$ and depends on the driving frequency.

The characteristic impedance of a transmission line is defined as the ratio of the phasor representations of the voltage and current for the incoming (one-directional) wave along an infinite series of circuits connected in parallel. For an ideal lossless transmission line consisting of an infinite series of LC circuits with no resistance or dielectric conductance, the characteristic impedance of the entire line equals the very different expression $\sqrt{L/C}$, independent of the driving frequency. This quantity is real, even through the circuit element impedance of each circuit element in the line is pure imaginary.

The physical interpretation of these quantities is different. A transmission line can have a real characteristic impedance, but this is not entirely analogous to a resistance, i.e. a real circuit element impedance. For example, in the lossless case described above, there is no energy dissipation even though the characteristic impedance is real. So the characteristic impedance of a transmission line isn't referred to as a "resistance" even if it's real. (In some sense, the physical intuition is reversed for characteristic impedance - a purely real characteristic impedance is the least lossy situation.)

The "impedance" in "impedance of free space" refers to characteristic impedance, not circuit element impedance. So it's purely real, but it would be misleading to describe it as a resistance.

tparker
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