Signals, Filters & Control — Cheat Sheet
Impedance, resonance, filter response, Bode margins and sampling in one place.
Reactance & impedance
| Inductive | X_L = 2πfL |
| Capacitive | X_C = 1/(2πfC) |
| Series RLC | Z = R + j(X_L − X_C) |
| Magnitude / phase | |Z| = √(R² + X²), θ = atan(X/R) |
| Sign of θ | + inductive (I lags), − capacitive (I leads) |
Resonance
| Frequency | f₀ = 1/(2π√(LC)) |
| Series Q | Q = (1/R)√(L/C) — lower R, higher Q |
| Parallel Q | Q = R√(C/L) — higher R, higher Q |
| Bandwidth | BW = f₀ / Q |
| Series at f₀ | Minimum |Z| = R — a short |
| Parallel at f₀ | Maximum |Z| = R — an open |
First-order filters
| RC corner | f_c = 1/(2πRC) |
| RL corner | f_c = R/(2πL) |
| At the corner | −3 dB (×0.707) and 45° phase |
| Roll-off | −20 dB/decade = −6 dB/octave |
| Time constant | τ = RC = L/R; f_c = 1/(2πτ) |
| Step response | 63% in 1τ, 95% in 3τ, 99% in 5τ |
Higher order
| Roll-off | −20n dB/decade for n poles |
| LC corner | f_c = 1/(2π√(LC)) |
| Butterworth Q | Q = 0.707 (maximally flat) |
| Peaking | Q > 0.707 peaks at f_c — amplifies the ringing |
| Order needed | n ≥ attenuation(dB) / (20 × decades) |
Sampling
| Nyquist | f_s > 2 f_max |
| Practical rate | 5–10× the signal bandwidth |
| Alias frequency | |f_signal − n·f_s| |
| Anti-alias | Filter BEFORE the ADC — after is too late |
| Corner placement | Well below f_s/2, not at it |
Loop stability
| Open loop | L(s) = plant × controller |
| Gain margin | 1/|L| at ∠L = −180°; target ≥ 6 dB |
| Phase margin | 180° + ∠L at |L| = 1; target 45–60° |
| Delay margin | PM(rad) / ω_gc — extra dead time tolerable |
| Dead time | Costs phase, not gain — the invisible margin killer |
| Bandwidth | ≈ gain crossover frequency |
Decibels
| Voltage ratio | dB = 20 log₁₀(V₂/V₁) |
| Power ratio | dB = 10 log₁₀(P₂/P₁) |
| −3 dB | ×0.707 voltage, ×0.5 power |
| −20 dB | ×0.1 |
| −60 dB | ×0.001 |
Formula reference — verify against the governing standard. Not a substitute for engineering judgment.