Impedance of Free Space

1 Impedance of Free Space

1.1 Free Space as a Transmission Line

You may recall from our discussion of transmission lines that the (real-valued) impedance of a transmission line ${\displaystyle Z_{0}}$ was given by:

${\displaystyle Z_{0}={\sqrt {\frac {L}{C}}},\,\!}$

and that this impedance was independent of length. One way of understanding this, which made sense on a wire, was in terms of per-length inductance and capacitance, and using the respective impedances of inductors and capacitors to derive the relationship between voltage and current.

In free space, we obviously don’t have inductors and capacitors, nor do we even have current. However, there is something more fundamental that we can look at: for waves travelling in free space and waves travelling on transmission lines, the wave energy is sloshing between electric and magnetic forms. A capacitor stores energy as an electric potential, with energy ${\displaystyle U_{E}={\frac {1}{2}}CV^{2}}$. An inductor stores energy as a magnetic field, with energy ${\displaystyle U_{B}={\frac {1}{2}}LI^{2}}$. On the other hand, it follows from Maxwell’s equations in free space that the energy associated with an electrostatic field is ${\displaystyle U_{E}={\frac {1}{2}}\epsilon _{0}\left|E\right|^{2}}$, and ${\displaystyle U_{B}={\frac {1}{2}}\mu _{0}\left|B\right|^{2}}$ for a magnetic field.

By analogy, it should be apparent that ${\displaystyle \epsilon _{0}}$,${\displaystyle \mu _{0}}$ play the roles of ${\displaystyle C}$,${\displaystyle L}$ for quantifying the energy associated with electric and magnetic fields in free space, respectively. It may not be surprising, then, that the impedance of free space is simply:

${\displaystyle Z_{0}={\sqrt {\frac {\mu _{0}}{\epsilon _{0}}}}\approx 120\pi \ {\rm {Ohms}}.\,\!}$

1.2 Impedance Matching Free Space

One of the important consequences of acknowledging the impedance of free space is recognizing that, in order to get maximum power transfer into your antenna, and to avoid reflecting the incoming wave back out into space, you need to match the impedances of the signal path inside your antenna to 376.73 ${\displaystyle \Omega }$. In practice, this is very hard to achieve over a range of frequencies, and, in cases where amplifier noise dominate your system temperature, you may even have to explicitly violate good termination practices in order to integrate a low-noise amplifier as close to the incoming signal as possible.