The engineer’s frequency ruler. Feed a system one pure tone at a time and it answers with a gain and a phase lag; sweep the tone across decades and you read the whole character of the loop — where it rolls off, and how much margin stands between it and howling instability. Magnitude in decibels, phase in degrees, both against a logarithmic frequency axis. Rendered, not quoted.
source Bode, H. W., Network Analysis and Feedback Amplifier Design, D. Van Nostrand, New York, 1945 — the book that introduced log-magnitude / log-frequency asymptotes. No stable DOI; scan at archive.org/details/dli.ernet.15701 (AMBER: uncanonical scan).
Take a first-order low-pass plant with time constant τ:
Evaluate on the imaginary axis, s = jω. The response is a complex number: its length is the gain, its angle is the phase.
At the corner ω = 1/τ the gain is 1/√2 → exactly −3.01 dB, phase exactly −45°. Past the corner each pole subtracts 20 dB/decade and asymptotically 90° of lag. Margins are read on a loop L(s): the gain margin lives at the −180° phase crossover, the phase margin at the |L|=1 gain crossover.
the-transfer-function → here. The transfer function H(s) is the algebra; the Bode plot is that same H, walked along s = jω and drawn — magnitude and phase by eye, decade by decade.
the-nyquist criterion, unrolled. Nyquist wraps the response around the origin in the complex plane; Bode cuts that contour and lays it flat on a log-frequency axis, which is how margins are actually measured on a bench.
Live re-check of the planted invariant: at the corner the gain must read −3.01 dB and the high-frequency roll-off must be −20 dB/decade. Trip the tamper in window 6 and this badge flips.
Plant H(s)=1/(τs+1), τ set by the slider. Loop for margins is fixed:
Frequency grid ω ∈ [10⁻², 10²] rad/s, log-spaced. Auxiliary LTI check block: A = [[0,1],[−2,−3]], B=[0;1], C=[1,0] — eigenvalues −1,−2 by construction.
Proven, closed-form, asserted at boot to 1e−6 (LTI block to 1e−8):
The magnitude alone does not tell you the phase — except for minimum-phase systems, where Bode’s own gain-phase theorem ties them together.
“−3 dB is where the system stops responding.” → it is the half-power point; output is still 70.7% of input and keeps falling smoothly.
“A healthy gain margin proves the loop is robust.” → GM and PM are one-axis-at-a-time cuts; a loop can pass both yet sit near instability under simultaneous gain-and-phase perturbation (the disk-margin gap).
“Phase always lags with frequency.” → a left-half-plane zero adds phase lead; only poles and RHP zeros push it down.
Planted void (disclosed): compute magnitude as |H|² instead of |H| — forgetting the square root. The −3 dB corner shifts and the roll-off doubles to −40 dB/dec. The Witness in window 7 catches it live.