Bode plot: 10 S(S+3)/((S+1)(S²+2S+16))

A Bode plot has two curves against log ω: the gain 20 log|H(jω)| in dB and the phase ∠H(jω). The asymptotic (straight-line) magnitude plot is drawn by writing H(s) in time-constant form, finding the corner frequencies and changing the slope by ±20 dB/decade for each simple factor (±40 for each quadratic factor).

Step 1: Standard form

G(s) = 10 s (s + 3) / [ (s + 1)(s² + 2s + 16) ]

H(s) = 1.875 × s (1 + s/3)
         / [ (1 + s/1) (1 + 0.125 s + s²/16) ]

Constant term K₀ = 1.875, so 20 log K₀ = 5.46 dB.

Step 2: Corner frequencies

The quadratic s² + 2s + 16 has ωn = 4 rad/s and 2ζωn = 2, so ζ = 0.25. There is a zero at the origin, so the low-frequency slope is +20 dB/decade.

Step 3: Magnitude plot

The net number of zeros at the origin is 1, so the low-frequency asymptote has slope +20 dB/decade and passes through 5.5 dB at ω = 1 rad/s. The slope then changes at each corner as below.

Asymptotic magnitude plot (each * is the asymptote; 1 row = 5 dB):

  10 |                   *     
     |           ******** ***  
   0 |        ***            **
     |     ***                 
 −10 |  ***                    
     |**                       
     ++-----------+-----------+
      0.1         1           10  ω (rad/s)

At ωn = 4 the quadratic pole has ζ = 0.25, so the true curve rises about 6 dB above the asymptote there (resonant peak, −20 log 2ζ); the actual peak is near ωn.

Step 4: Phase plot

Low-frequency phase = 90° × (zeros at origin − poles at origin) = 90°. Each simple factor changes the phase by 90° (zero: +, pole: −), spread over the range 0.1ωc to 10ωc with 45° at ωc; each quadratic factor changes it by 180°, with ±90° at ωn. High-frequency phase = 90° × (m − n) = −90°. Exact phase values at the corners are in the table above; join them with a smooth curve starting at 90° and ending at −90°.

Accuracy check: at the first corner ω = 1 rad/s the exact gain is 3.4 dB, the asymptote gives 5.5 dB.