text: To calculate the breakthrough voltage, a homogeneous electrical field is assumed. This is the case in a parallel-plate capacitor setup. The electrodes may have the distance . The cathode is located at the point .
  To get impact ionization, the electron energy  must become greater than the ionization energy  of the gas atoms between the plates. Per length of path  a number of  ionizations will occur.  is known as the first Townsend coefficient as it was introduced by Townsend.
  The increase of the electron current , can be described for the assumed setup as
  (So the number of free electrons at the anode is equal to the number of free electrons at the cathode that were multiplied by impact ionization. The larger  and/or , the more free electrons are created.)
  The number of created electrons is
  Neglecting possible multiple ionizations of the same atom, the number of created ions is the same as the number of created electrons:
   is the ion current. To keep the discharge going on, free electrons must be created at the cathode surface. This is possible because the ions hitting the cathode release secondary electrons at the impact. (For very large applied voltages also field electron emission can occur.) Without field emission, we can write
  where  is the mean number of generated secondary electrons per ion. This is also known as the second Townsend coefficient. Assuming that , one gets the relation between the Townsend coefficients by putting () into () and transforming: