Three Formulas, Three Design Questions

Ada re-derives this chapter’s own numbers step by step, at full precision

ADA · CALCULATION AUDIT

Three Formulas, Three Design Questions

The chapter states three headline results — a 1 ms debounce time constant, a 13.5 pF antenna capacitor, and a 130-ohm LED resistor. Each comes from a different formula: RC sets timing, LC sets frequency, and Ohm’s law sets current. This audit re-derives all three, showing three formulas that answer three design questions.

Companion to the chapter Conductors, Insulators, Semiconductors — every number here comes from that chapter.

Ada: This chapter states three headline results — a 1 ms debounce time constant, a 13.5 pF antenna capacitor, and a 130-ohm LED resistor. Each comes from a different formula, so let me re-derive all three from scratch.

• RC time constant. tau = R x C = 10,000 ohms x 0.0000001 F = 0.001 s = 1 ms. A capacitor reaches 1 - e^-1 = 0.6321 = 63.2% of final voltage in one tau and 1 - e^-5 = 0.99326 = 99.3% after 5 tau = 5 ms, so a 1 ms filter rides comfortably over sub-10 ms switch bounce.
• LC resonance. Inverting f = 1 / (2 pi x sqrt(LC)) gives LC = 1 / (2 pi x 433,000,000)^2 = 1.35103e-19. With L = 10 nH, C = 1.35103e-19 / 10e-9 = 1.35103e-11 F = 13.51 pF. Feeding 10 nH and 13.5 pF back into the forward formula returns 433.2 MHz, closing the loop.
• Ohm’s law LED. R = (3.3 V - 2.0 V) / 0.010 A = 130 ohms. Rounding up to a 150-ohm standard part drops the current to 1.3 V / 150 ohms = 8.67 mA and dissipates (0.010 A)^2 x 150 ohms = 0.015 W — safe for a 1/8 W (0.125 W) resistor.

Each formula answers a different design question: RC sets timing, LC sets frequency, and Ohm’s law sets current. Notice too that snapping each result to the nearest standard component nudges the real value — 130 ohms becomes 150, 13.51 pF becomes 13.5 — which is why the calculation is where the check starts, not where it ends.

Every number above is taken from the chapter’s own material and re-derived step by step.