Frequency Response

Wikipedia · Bode plot · CC BY-SA 4.0

Bode plots

Two plots stacked vertically. The top plot shows magnitude in decibels (20*log10 of the gain) versus log frequency. The bottom plot shows phase in degrees versus log frequency. Straight-line asymptotes approximate the curves. Each pole contributes -20 dB/decade of slope and -90 degrees of phase shift. Each zero contributes +20 dB/decade and +90 degrees.

Gain and phase margin

Gain margin: how much you can increase the gain before the system goes unstable. Measured at the frequency where the phase is -180 degrees. Phase margin: how much additional phase lag the system can tolerate. Measured at the frequency where the gain is 0 dB (the gain crossover frequency). Positive margins mean stable. Typical design targets: gain margin of at least 6 dB, phase margin of at least 45 degrees.

; Bode magnitude for a first-order system: H(s) = 1/(1 + s/wc)
; |H(jw)| = 1 / sqrt(1 + (w/wc)^2)
; In dB: 20*log10(|H|)

(define wc 10.0)  ; corner frequency

(define (log10 x) (/ (log x) (log 10)))

(define (magnitude-db w)
  (let ((ratio (/ w wc)))
    (* 20.0 (log10 (/ 1.0 (sqrt (+ 1.0 (* ratio ratio))))))))

(define (phase-deg w)
  ; phase = -arctan(w/wc), converted to degrees
  (let ((ratio (/ w wc)))
    (* (- (atan ratio)) (/ 180.0 3.14159265))))

(display "Bode plot data (wc=10 rad/s):") (newline)
(display "  w(rad/s)  |H|(dB)   phase(deg)") (newline)
(for-each
  (lambda (w)
    (display "  ")
    (display w)
    (display "       ")
    (display (magnitude-db w))
    (display "  ")
    (display (phase-deg w))
    (newline))
  (list 0.1 1.0 5.0 10.0 20.0 50.0 100.0))

import math

# Bode plot for H(s) = 1/(1 + s/wc)
wc = 10.0

print("Bode plot data (wc=10 rad/s):")
print(f"  {'w':>8}  {'|H| (dB)':>10}  {'phase (deg)':>12}")
for w in [0.1, 1.0, 5.0, 10.0, 20.0, 50.0, 100.0]:
    mag_db = 20 * math.log10(1.0 / math.sqrt(1 + (w/wc)**2))
    phase = -math.degrees(math.atan(w / wc))
    print(f"  {w:8.1f}  {mag_db:10.2f}  {phase:12.2f}")