Three Formulas, Three Design Questions

Three Formulas, Three Design Questions

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

foundations
math-foundations
calculation-audit
electronics
Ada 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.

See the relationship before changing it

The figure reads from left to right. The blue card is led resistor. The middle card applies the page rule. The green card is led current. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

LED resistor changes led current An input card leads through the rule current = 1.3 V / resistor x 1000 to the led current result. INPUT PAGE INPUT APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrows. A larger resistor lowers current and heat, but also lowers light output.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 150 ohm.

  2. 2

    Name the relationship. current = 1.3 V / resistor x 1000

  3. 3

    Substitute with units. 1.3 / 150 x 1000 = 8.67 mA

  4. 4

    Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.

Predict, then change led resistor

Try Predict the direction of current = 1.3 V / resistor x 1000. Test another led resistor, then compare led current.

150 ohm
Chapter baseline
LED current

Observe A larger resistor lowers current and heat, but also lowers light output. Reset led resistor to 150 and compare led current.

Explain A larger resistor lowers current and heat, but also lowers light output.

Check yourself

What should you do before trusting a moved-control result?
Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.
What does this small model leave out?
Answer: Only led resistor moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only led resistor. It is not a field prediction.

TryThe chapter states three headline results — a 1 ms debounce time constant, a 13.5 pF antenna capacitor, and a 130-ohm LED resistor. Use Check derivation.
ObserveThe displayed ledger resolves 1 ms, 13.5 pF, 130-ohm at full precision. This audit re-derives all three, showing three formulas that answer three design questions. Check derivation shows this.
ExplainThe three readouts follow different physical laws: RC sets debounce time, inverse-square-root LC sets resonance, and voltage divided by current sets the LED resistor; standard component values then move each ideal result. Check derivation confirms it.

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.

Technical boundaries
The three formula checks deliberately do not simulate component tolerances, parasitic capacitance and inductance, diode nonlinearity, source impedance, or temperature; the RC, LC, and LED examples use nominal lumped values.

Work the audit first, then check the displayed derivation.

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