Timing, filters, and analog design

Time constants — τ = RC or L/R; a first-order step reaches 63.2% after τ and is within 1% after about 5τ; the time to reach a voltage is t = −τ·ln(1 − V/Vs). LC gives the resonant frequency 1/(2π√LC) and √(L/C).

Filters & Bode — first-order RC and RL low- and high-pass (fc = 1/(2πRC) or R/(2πL)), a series RLC band-pass (output across R; Q = √(L/C)/R, bandwidth f₀/Q), and a second-order RLC low-pass (output across C). The plots show magnitude in dB and phase from three decades below to three above, and the design rows give the R, C, or L for a target frequency, rounded to standard values.

Reactance & resonance — Xc = 1/(2πfC), XL = 2πfL, and the L or C that resonates with the other at f.

555 timer — astable high time ln2·(R1 + R2)·C (or ln2·R1·C with a diode across R2), low time ln2·R2·C; monostable pulse ln3·R·C ≈ 1.1RC. The design panel solves R1 and R2 for a frequency and duty cycle; duties at or below 50% need the diode. The waveform shows the output and the capacitor swinging between ⅓ and ⅔ of the supply.

Op-amp — inverting −Rf/Rin, non-inverting 1 + Rf/Rin, differential Rf/Rin·(V2 − V1) with matched resistors, summing −Rf·ΣVi/Ri, and a follower. Outputs are clipped to the rails you enter; bandwidth is estimated as GBW divided by the noise gain. The table lists standard-value pairs for a target gain.

Zener — the largest series resistor that keeps the Zener in regulation at minimum input and maximum load, rounded down to E12, and the worst-case Zener and resistor power.

Linear regulator — dissipation (Vin − Vout)·I + Vin·Iq, junction temperature without a heatsink, and the heatsink rating needed (θSA = (Tj,max − Ta)/P − θJC − θCS). Presets are typical datasheet figures; θJA in particular depends on your PCB copper.