Fourier, Tankit, and the RSGB’s F8 Curve
Chuck Varney
10 December 2015
 Last edit: 10 March 2016

This page documents my identification of calculations that could have been used to create a plot in the Radio Society of Great Britain's (RSGB) Radio Communication Handbook as described by Keith Kunde in his Tankit documentation, page 2.  (Folks at the RSGB were unable to tell Keith how the plot originated.)

Given this plot, where curve F8 is the RSGB handbook plot in question:


and this plot from Kunde's Tankit documentation, derived by curve fitting data pairs that he extracted from the F8 curve above:



. . .as described here:



I constructed this plot comparing the Tankit and RSGB F8 plots to a plot I derived from Fourier equations:

   Figure 1.  RSGB F8, Tankit, and Fourier-derived results for Ib / Ibo between 1 and 25 and i1 / Ib from 1 to 1.6.


   1. The Tankit F8 curve fit polynomial

   2. Nine data pairs picked off a scan of the RSGB F8 plot

   3. Two equations based on Fourier analysis, in which θ is the plate current conduction
    angle in radians.

   i1 / Ib = Tankit i1max / Ibmax  =  ( θ – sin(θ)) / ( 2 sin(θ/2) – θ cos(θ/2) )           [Eq. 1]

   Ib / Ibo = Tankit Ibmax / Ibo     =  RSGB's Ia / Ia(0) = (θ/2 – tan(θ/2)) / π            [Eq. 2]


      i1 is the amplitude of the Fourier fundamental frequency component
      Ib is the DC plate current with signal present
      Ibo is the no-siganl DC plate current

      See the Equation Appendix below for how Eq. 1 and Eq. 2 were derived.

The Tankit and Fourier-derived curves in Figure 1 essentially lie atop one another. Over an Ib / Ibo range of 1 to 1000,
the absolute difference in their i1 / Ib values only exceeds 0.0014 for Ib / Ibo less than ~1.82. The difference peaks
at 0.092 at which point Ib / Ibo is ~1.06 and θ is ~275°. This is illustrated graphically in Figure 2.


  Figure 2.  Absolute difference in Tankit and Fourier-derived i1 / Ib over a 1 to 1000 range of Ib / Ibo.

The RSGB F8 data points in Figure 1 closely correspond to the Tankit and Fourier curves for Ib / Ibo from 5 to 20,
but depart for lesser values. The maximum absolute difference between F8 i1 / Ib values and the Tankit and
Fourier-derived values is 0.00064 over the Ib / Ibo range from 5 to 20, and is equal to or less than 0.0060
at Ib / Ibo of 2.5. This is illustrated in Figure 3.


   Figure 3.  Absolute differences between extracted F8 data points and the corresponding Tankit and Fourier-derived
values for i1 / Ib over the Ib / Ibo range of 2.5 to 20.



Equation Appendix

 Equations 1 and 2 given above were calculated from these equations:

     i1 =  (imax / 2 π) (θ – sin(θ)) / ( 1 – cos(θ/2) )                                          [Eq. 4]   

     Ib =  (imax / 2 π) ( 2 sin(θ/2) – θ cos(θ/2) ) /  ( 1 – cos(θ/2) )                  [Eq. 5]

     Ibo = - imax cos(θ/2) / ( 1 – cos(θ/2) )                                                       [Eq. 6] 

     Equation 1 is Eq. 4 divided by Eq. 5.

     Equation 2 is Eq. 5 divided by Eq. 6.

Eq. 4 and Eq. 5 are Fourier integral solutions for the Fourier DC and fundamental frequency components of a current pulse with peak value of imax, taken from page 16 of a thesis by Oliver Isler.

Eq. 6 for Ibo was pieced together, initially taking it as equal to – cos(θ/2), as given for Iidle on a webpage by J.R. Mathison.
The factor imax / ( 1 –  cos(θ/2) ) was introduced to accord with the scheme used for Eq. 4 and Eq. 5, which normalize results to 1 so that a factor imax can be used.

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