Bode plot: 40(s+1)/((2s²+10s)(s²+2s+10))

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

H(s) = 40(s + 1) / [ (2s² + 10s)(s² + 2s + 10) ]
     = 20(s + 1) / [ s (s + 5)(s² + 2s + 10) ]   since 2s² + 10s = 2s(s + 5)

H(s) = 0.4 × (1 + s/1)
         / [ s (1 + s/5) (1 + 0.2 s + s²/10) ]

Constant term K₀ = 0.4, so 20 log K₀ = −7.96 dB.

Step 2: Corner frequencies

The quadratic s² + 2s + 10 has ωn = √10 = 3.162 rad/s and 2ζωn = 2, so ζ = 0.316 (underdamped).

Step 3: Magnitude plot

The net number of poles at the origin is 1, so the low-frequency asymptote has slope −20 dB/decade and passes through −8 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 |            ********     
     |                    *    
 −20 |                     *   
     |                      *  
 −30 |                       * 
     |                        *
     ++-----------+-----------+
      0.1         1           10  ω (rad/s)

At ωn = 3.162 the quadratic pole has ζ = 0.316, so the true curve rises about 4 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) = −270°. Exact phase values at the corners are in the table above; join them with a smooth curve starting at −90° and ending at −270°.

Accuracy check: at the first corner ω = 1 rad/s the exact gain is −4.4 dB, the asymptote gives −8 dB.