Pickett N-515-T Electronic (CIE) — Volume 3 — The Back Face: The Decimal Point Locator and the Formula Panels

Figure 1 — The back of my N-515-T. Left half: the “Reactance and Resonance Decimal Point Locator,” with “REACTANCE PROBLEMS” and “RESONANCE PROBLEMS” sections on the slide. Right half: formula panels for temperature, impedance, transformers, Ohm’s law (DC and AC), frequency and wavelength, reactance, resonance, decibels, power factor, parallel resistors, coupled inductance and efficiency; a table of constants; the CIE seal; and “CLEVELAND INSTITUTE OF ELECTRONICS, CLEVELAND, OHIO 44114, PATENT NO. 3,120,342, MADE IN U.S.A.” Photo: my own.

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3.1 About This Volume

Turn the N-515-T over and it stops looking like a slide rule. There are no logarithmic cycles marked 1 to 10, no trigonometric scales — only rows of decimal numbers such as 0.001, 0.01, 0.1, 1, 10, 100, small arrows, unit symbols, and two boxed legends, Reactance Problems and Resonance Problems. This is the Reactance and Resonance Decimal Point Locator of US Patent 3,120,342, the reason the rule exists. The rest of the back is a printed reference card of the formulas a radio technician needed most. This volume explains the locator — what each row means, why the scales are spaced the way they are, and how to use it — and transcribes the formula panels.

3.2 The Problem the Locator Solves

Every reactance and resonance problem has the same awkward shape. The three formulas the patent singles out are

X_L = 2πfL X_C = 1 / (2πfC) f = 1 / (2π√(LC))

each with three quantities, so nine possible problems — find X_L given f and L, find L given X_L and f, and so on. In practice the quantities come in mixed units: capacitance in microfarads or picofarads, inductance in henrys, millihenrys or microhenrys, frequency in hertz, kilohertz or megahertz, reactance in ohms or megohms. To use the formulas as written, everything has to be converted to farads, henrys and hertz first — numbers like 3.5 × 10⁻⁴ and 2 × 10⁵ — and converted back afterwards.

The front face of the rule gives the digits of the answer to three figures, but not the decimal point. Geiger’s patent sets two objects for the back face: that problems “can be readily solved without need for conversion from one unit of measure to another,” and that “an accurate determination of the correct decimal point location is obtainable immediately and directly.” The answer is a second, deliberately crude slide rule whose scales are marked in practical units and whose only job is the order of magnitude.

3.3 How the Locator Is Built

Figure 2 — The locator on my rule. Upper body: three rows labelled C_µF or L_H, L_mH, and C_pF or L_µH. Slide, “Reactance Problems” section: rows f_Hz, f_kHz, f_MHz, and four arrow marks along the slide’s lower edge — X_C MΩ, X_L Ω, X_C Ω, X_L MΩ. Lower body: rows X_C, X_L and f. The slide continues to the right into the “Resonance Problems” section (Figure 1). Photo: my own.

3.3.1 Coarse logarithmic scales

Each of the locator’s scales is a row of equally spaced main marks, one decade apart: 0.001, 0.01, 0.1, 1, 10, 100 and so on. Between each pair of main marks are two intermediate marks labelled 2 and 5. The patent specifies their spacing precisely: they divide each decade into three spaces “whose lengths relative to each other are represented by the integers 3, 4 and 3.”

That ratio is not arbitrary. On a logarithmic scale the distance from 1 to 2 is log 2 = 0.301 of a decade, from 2 to 5 is log 2.5 = 0.398, and from 5 to 10 is log 2 = 0.301 — almost exactly 3 : 4 : 3. The locator is therefore a real logarithmic slide rule, but with one decade per division and only two subdivision marks: just enough resolution to tell 2 from 5, and nothing finer. A whole decade of the front face’s C scale — ten inches — is compressed into a little under half an inch of locator scale.

Because one length of scale stands for every unit, the same graduations can carry several rows of numbers for different units, each row offset by the right power of ten. That is how unit conversion is abolished: 1 µF and 1,000,000 pF are the same point on the scale, so they appear in the same column, in different rows.

3.3.2 The rows and their arrows

Table 1 — The rows and their arrows

Where	Row label	Meaning	Direction of increase
Upper body	CµF or LH	capacitance in microfarads, or inductance in henrys	← leftward
Upper body	LmH	inductance in millihenrys	← leftward
Upper body	CpF or LµH	capacitance in picofarads, or inductance in microhenrys	← leftward
Slide (reactance)	fHz, fkHz, fMHz	frequency	→ rightward
Slide (resonance)	CµF or LH, LmH, CpF or LµH	the second of L or C	→ rightward
Lower body	XC	capacitive reactance, ohms (or megohms, by arrow)	→ rightward
Lower body	XL	inductive reactance	← leftward
Lower body	f	resonant frequency	→ rightward, one decade per two divisions

Small arrows at the ends of each row point in the direction in which its values grow. The course explains why they are there: “Knowing the direction of progression of the scales makes it easier to set and read values.” The directions are chosen so that the right answer falls out of the geometry:

• On the upper body, values grow to the left, on the slide to the right. Setting a slide value opposite a body value therefore adds their logarithms in the slide’s displacement — the slide’s position encodes the product fC, fL or LC.
• X_C = 1/(2πfC) falls as fC rises, X_L = 2πfL rises as fL rises, so the X_C and X_L rows on the lower body run in opposite directions.
• Resonant frequency depends on the square root of LC, so a decade change in LC moves f by only half a decade. The f row on the lower body therefore has two divisions per decade, In the patent, the scale behind this row has a single intermediate mark midway between main marks, labelled 3 — standing for √10 ≈ 3.16, the halfway point of a decade on a logarithmic scale — and the frequency row carries small secondary figures 2, 3 and 6 within each decade.

3.3.3 The arrow marks

The answer is not read at a hairline but opposite an arrow mark printed on the slide’s lower edge, pointing down at the lower body. There are two groups:

• Reactance Problems: four arrows, labelled X_C Ω, X_C MΩ, X_L Ω and X_L MΩ, for reading a reactance in ohms or megohms.
• Resonance Problems: arrows labelled f_Hz, f_kHz and f_MHz, for reading a resonant frequency in the unit wanted.

Each arrow is displaced from the decade marks by the amount needed to absorb the constant factor in its formula — the 2π in the reactance formulas, the 2π and the square root in the resonance formula — together with the unit change from ohms to megohms or hertz to megahertz. The patent states the positions: the reactance arrows stand “at 0.159 of the length of the main scale spaces,” the resonance arrows at 0.318. (Those numbers are the values of 1/2π and 1/π.) On the patent drawing the lowest-unit resonance arrow falls beyond the end of the printed f row, and the patent says so: the row’s numbers were simply not extended that far.

3.4 Using the Locator

The course gives one rule for each type of problem (Part III, Topics 26 and 27).

Resonance: “Set the value of either C or L (it makes no difference which one) on the Resonance Problems Scale of the slide opposite the other given value on the upper body scale. Then read the approximate frequency on the bottom body scale opposite the appropriate arrow on the slide.”

Reactance: “The frequency on the part of the slide marked ‘Reactance Problems’ is placed opposite the given value of inductance or capacity on the upper body of the rule. The reactance is then read on the proper scale on the lower body opposite the appropriate arrow of the slide.”

Run backwards, the same settings find a missing L, C or f. The course is equally clear about the precision required: “A high degree of accuracy is never necessary in setting values on the Decimal Point Locator scales.”

Figure 3 — Part III, page 55: “To find resonant frequency of 40 mh with 0.03 µf: set 0.03 on scale Cµf opposite 40 on scale Lmh; then opposite arrow f_kc read 5 kc on scale f, or opposite arrow f_cps read 5000 cps.” Image: Cleveland Institute of Electronics, Electronics and Your Slide Rule, Part III, scanned by Joe Hernandez (KN2A), via RF Cafe, https://www.rfcafe.com/references/electrical/Cleveland-Institute-515-T-Slide-Rule-Manuals.htm.

3.4.1 Worked examples from the course, checked

Each of the following is a worked example in Part III. The “exact” figure was computed directly from the formula for this unit; the locator figure is the course’s own.

Resonance (Example in Topic 26) — 40 mH in parallel with 0.03 µF. On my rule’s labelling: set the hairline over 0.03 on the C_µF row of the upper body (between the 0.01 and 0.1 main marks, nearer the 2 than the 5); move the slide until 40 on its L_mH resonance row (between 10 and 100, nearer the 5) is under the hairline; read the f row opposite the f_kHz arrow. The arrow falls between the 3 and 6 marks of the decade 1–10, nearer 6: about 5 kHz. Exact: f = 1 / (2π√(0.040 × 0.03 × 10⁻⁶)) = 4,594 Hz. The locator’s job is done: the answer is a few kilohertz, not a few hundred hertz or tens of kilohertz.

Capacitance for resonance (Example 31) — What capacitance resonates with 350 µH at 200 kHz? Set the f_kHz arrow opposite 200 on the f row; set the hairline over 350 on the L_µH row of the upper body; read under the hairline on the slide’s C_µF row: about 0.003 µF. Exact: C = 1 / ((2π × 200,000)² × 350 × 10⁻⁶) = 1.809 × 10⁻⁹ F = 0.00181 µF.

Capacitive reactance (Example 32) — Reactance of 0.005 µF at 3 MHz. Set the hairline over 0.005 on the C_µF row; bring 3 on the slide’s f_MHz row under the hairline; opposite the X_C Ω arrow read about 10 Ω. Exact: 1 / (2π × 3 × 10⁶ × 5 × 10⁻⁹) = 10.61 Ω.

Inductance from reactance (Example 33) — What inductance has 4 MΩ of reactance at 3,000 kHz? Set the X_L MΩ arrow opposite 4 on the X_L row; set the hairline over 3,000 on f_kHz; read about 0.3 H on L_H. Exact: L = 4 × 10⁶ / (2π × 3 × 10⁶) = 0.212 H.

Frequency from reactance (Example 34) — At what frequency does 200 pF have a reactance of 25 Ω? Set the X_C Ω arrow opposite 25 on the X_C row; set the hairline over 200 on C_pF; read about 30 MHz on f_MHz. Exact: f = 1 / (2π × 25 × 200 × 10⁻¹²) = 31.8 MHz.

The locator answers are off by as much as a factor of about 1.6 (0.003 against 0.00181) — and that is by design. In the course’s practice problems the same pattern holds: 3 mH with 175 pF is given as “250 kc” (exactly 220 kHz) and 340 µH with 0.0034 µF as “200 kc” (exactly 148 kHz). The locator only has to be right to well within a factor of ten, and it always is. The front face then supplies the three significant figures: 459 for the first example, 181 for the second, 1061 for the third, 212 for the fourth, 318 for the fifth — exactly the “accurate” answers the course gives in its next topics (Examples 37–43). Volume 4 shows those front-face settings.

3.5 The Formula Panels

Figure 4 — The formula panels and constants on my rule, ending at the right in the CIE seal and the maker’s legend. Photo: my own.

The rest of the back face is a printed reference card. As printed on my rule, panel by panel:

Temperature

• C = 0.556 F − 17.8
• F = 1.8 C + 32
• Kelvin = C + 273

Impedance (beside the temperature panel)

• Z (series) = √(R² + (X_L − X_C)²) = √(R² + X²)
• Z (parallel) = RX / √(R² + X²)

Transformer

• N_P/N_S = E_P/E_S = I_S/I_P = √(Z_P/Z_S)

Ohm’s Law Formulas, DC Circuits

• I = E/R = √(P/R) = P/E
• R = E/I = P/I² = E²/P
• E = IR = P/I = √(PR)
• P = I²R = EI = E²/R

Ohm’s Law Formulas, AC Circuits

• I = E/Z = √(P / (Z cos θ)) = P / (E cos θ)
• Z = E/I = P / (I² cos θ) = E² cos θ / P
• E = IZ = P / (I cos θ) = √(PZ / cos θ)
• P = I²Z cos θ = IE cos θ = E² cos θ / Z

Frequency and Wavelength

• f_MHz = 3 × 10⁴ / λ_cm λ_cm = 3 × 10⁴ / f_MHz
• f_MHz = 984 / λ_ft λ_ft = 984 / f_MHz

Reactance

• X_L = 2πfL
• X_C = 1 / (2πfC)

Resonance

• f = 1 / (2π√(LC))

Decibels

• dB = 10 log (P₁/P₂)
• dB = 20 log (E₁/E₂)

Power Factor

• pf = P / EI
• pf = R / Z

Two Resistors Parallel

• R_t = R₁R₂ / (R₁ + R₂)
• R₁ = R_tR₂ / (R₂ − R_t)

Coupled Inductance

• L_t = L₁ + L₂ ± 2M
• K = M / √(L₁ × L₂)

Efficiency

• Eff = output / input
• output = input × Eff
• e = 2.718 (2π)² = 39.5

Constants and conversions

Table 2 — Constants and conversions

√10 = 3.1623	1 HP = 746 watts	Q = XL/R = XC/R	√2 = 1.4142
1 in = 2.54 cm	1 HP = 42.4 BTU/min	2π = 6.2832	Eeff = 0.707 Emax
1 ft = 0.3048 m	1 liter = 1.06 qts	1/2π = 0.1592	Eave = 0.637 Emax
1 m = 39.37 in	1 kg = 2.20 lbs		Eave = 0.902 Eeff
1 mi = 1.609 km	1 oz = 28.35 g		1 radian = 57.3°

Every constant on the card checks. A few are worth a note:

• 3 × 10⁴ and 984. The speed of light is 3 × 10¹⁰ cm/s, so f in MHz times λ in cm is 3 × 10⁴; in feet, 2.998 × 10⁸ m/s ÷ 0.3048 m/ft = 9.84 × 10⁸ ft/s, giving 984. A 7.1 MHz amateur signal therefore has λ = 984 / 7.1 = 138.6 ft, or 3 × 10⁴ / 7.1 = 4,225 cm = 42.3 m.
• The sine-wave averages. E_eff (RMS) = E_max/√2 = 0.707 E_max; the full-wave average is 2/π = 0.637 E_max; and 0.637 / 0.707 = 0.901, printed as 0.902.
• 1 HP = 42.4 BTU/min. 746 W × 60 s/min ÷ 1,055 J/BTU = 42.4.
• (2π)² = 39.5. 39.478, the constant in the resonance formula squared — useful when working f = 1/(2π√LC) the long way round.
• Two resistors in parallel. The second form, R₁ = R_tR₂ / (R₂ − R_t), answers the bench technician’s question “what do I put across this resistor to get that value?” — see Volume 4.
• Coupled inductance. L_t = L₁ + L₂ ± 2M is the series total of two coils with mutual inductance M, + for series-aiding and − for series-opposing; K is the coefficient of coupling.

3.5.1 The seal and the legend

At the right-hand end of the slide is the circular CIE seal; at the right of the lower body the legend CLEVELAND INSTITUTE OF ELECTRONICS / CLEVELAND, OHIO 44114 / PATENT NO. 3,120,342 / MADE IN U.S.A. — the details used in Volume 1 to date my rule to no earlier than 1964.

The panel’s use of Hz and pF on the locator, where the 1958 patent and the 1966 manuals print cps, kc, mc and µµf, means that an owner working from the scanned manuals has to translate as he goes: cps is Hz, kc is kHz, mc is MHz, µµf (micromicrofarad) is pF. The scales themselves are unchanged.

Sources

• D. L. Geiger, Slide Rule, US Patent 3,120,342 (filed 16 October 1958, granted 4 February 1964) — the objects of the invention, the 3-4-3 subdivision, the rows of indicia and directional arrows, the 2-3-6 frequency subdivisions, the indicator marks at 0.159 and 0.318 (https://patents.google.com/patent/US3120342A/en). In this site’s Reference Library.
• D. L. Geiger, Electronics and Your Slide Rule, Part III, Cleveland Institute of Electronics, October 1966 printing — Topics 25–27 and Examples 31–34, with answers (https://www.rfcafe.com/references/electrical/Cleveland-Institute-515-T-Slide-Rule-Manuals.htm). In this site’s Reference Library: Parts I, II, III, IV.
• The formula panel and constants transcribed from my own rule (Figure 4) and cross-checked against the 1967 advertisement drawing and the RF Cafe photographs of another example.
• Every “exact” value was computed directly: f(40 mH, 0.03 µF) = 4,594 Hz; C(350 µH, 200 kHz) = 1.809 nF; X_C(0.005 µF, 3 MHz) = 10.61 Ω; L(4 MΩ, 3 MHz) = 0.2122 H; f(200 pF, 25 Ω) = 31.83 MHz; f(3 mH, 175 pF) = 219.7 kHz; f(340 µH, 0.0034 µF) = 148.0 kHz; 984/7.1 = 138.6; 3 × 10⁴/7.1 = 4,225; 746 × 60/1,055 = 42.4; (2π)² = 39.48; log 2 = 0.301, log 2.5 = 0.398.