**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.

using:

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.

*Tankit*i

_{1max}/ I

_{bmax}= ( θ – sin(θ)) / ( 2 sin(θ/2) – θ cos(θ/2) ) [Eq. 1]

Ib / Ibo = *Tankit* I_{bmax} / I_{bo} = RSGB's Ia / Ia(0) =
(θ/2 – tan(θ/2)) / π [Eq. 2]

where:

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.