Thomson formular

When studying for my licence I stumbled across the following formula to calculate the electrical resonance frequency of an LC-circuit.

It was:

$$f=\frac{159}{\sqrt{L\cdot C}},$$

with f in MHz, L in µH and C in pF.

Being a mathematician and physicist I was not happy with the units. So I researched and found the “real” Thomson formula being

$$f=\frac{1}{2\pi \sqrt{L\cdot C}}$$

with all units being base units (meaning without prefix), e.g. f in Hz, L in H and C in F.

Of course I asked myself how you get from one formula to the other.

In retrospective it is pretty simple. What you need to know is following:

• The number 159 comes from the constant factor:

$$\frac{1}{2\pi }\approx 0,\mathrm{159...}=159\cdot {10}^{-3}$$

• The Prefixes are:
  • MHz = 10^6 Hz (the left side of our equation)
  • µH = 10^-6 H,
  • pF = 10^-12 F.
• Therefore the right side of the equation, without 159, is:

$$\frac{{10}^{-3}}{\sqrt{{10}^{-6}\cdot {10}^{-12}}}=\frac{{10}^{-3}}{\sqrt{{10}^{-18}}}=\frac{{10}^{-3}}{{10}^{-9}}={10}^{-3}\cdot {10}^9={10}^6$$

And this is exactly the same as MHz.

q.e.d. (quod erat demonstrandum)