Slide Rules For Fun Part 2
As I was surfing the Internet yesterday, I ran across a question someone had asked about the Pickett N4-ES slide rule. I forgot to notice the date. Anyway the question was, “Is there some way to determine resonant frequency elegantly on the N4-ES rule versus the N3-ES rule?” That set me to thinking, “Is there?” Well, the formula to determine resonant frequency is as follows,
\[f_{r} = \frac{1}{2 \pi \sqrt{L C}}\]
On the slide rule, it is easier to calculate if we simplify the formula as such,
\[f_{r} = \frac{1}{2 \pi} = 0.159 \quad \Rightarrow \quad \frac{0.159}{\sqrt{L C}}\]
Suppose we have values where inductance is 8 uH, capacitance is 10 pF. We determine that 10 x 8 = 80, and uH is 10-6, pF is 10-12 for 10-18. On the N4-ES the square root scale is at the bottom of the front side stock. We slide the hairline (HL) to 8 on the D scale, and read 8.94 on the \(\sqrt{}\) scale.
Unfortunately, the non-elegant aspect comes where we have to transfer that value to the C scale. For division, the C and D scales are used, where the bottom number (8.94) on the C scale is set over the top number (0.159) of the D scale. The frequency is then read at the end of the C scale (index) on the D scale as 1.78.
For the decimal placement where \(\sqrt{80 ^{-18}} = 8.94^{-9}\), we examine \(\frac{15.9^{-2}}{8.94^{-9}}\), where (-2) - (-9) = 7 places. So applying that (107) to 1.78 gives the final answer of 17.8 MHz. By the way, the only thing slide rules don’t do is directly add and subtract, as it uses logarithms for calculations.
And to answer the original question, both the N3-ES and N4-ES require the same actions. And by converting \(\frac{1}{2 \pi}\) to 0.159, the process is vastly simplified. A good rule of thumb to assist in decimal placement1 is to recall that 10 uH and 10 pF gives about 16 MHz resonant frequency.
Another thing the slide rule does really simply, is determine areas and circumferences of circles, especially if the rule has a DF or CF scale. The area of a circle, \(\pi r^{2}\), and its derivative, circumference, \(\pi d\), is found easily. By setting the circle’s radius on the dual \(\sqrt{}\) scale, read the area on the DF scale. For circumference, set the HL over the circle’s diameter on the D scale, read the circumference under the HL on the DF scale. For example for a circle of 30’, the area is ~707 ft2, and the circumference is ~ 94 ft.
If the slide rule doesn’t have CF/DF scales, for area, set the B index to 0.7854 on the A scale (most rules have a mark there). Move the HL over the diameter on the C scale, and read the area under the HL on the A scale. For circumference, simply multiply \(\pi \times diameter\) normally.
And, just for fun, \(\pi = 3.141592653589793238462\) (21 decimal digits). Have a great day, and may God Bless you and yours.