Bode plot: 10(S+10)/(S(S²+5S+4)(S+40))

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

N(s) = 10(s + 10) / [ s (s² + 5s + 4)(s + 40) ]
     = 10(s + 10) / [ s (s + 1)(s + 4)(s + 40) ]   since s² + 5s + 4 = (s + 1)(s + 4)

H(s) = 0.625 × (1 + s/10)
         / [ s (1 + s/1) (1 + s/4) (1 + s/40) ]

Constant term K₀ = 0.625, so 20 log K₀ = −4.08 dB.

Step 2: Corner frequencies

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 −4.1 dB at ω = 1 rad/s. The slope then changes at each corner as below.

Asymptotic magnitude plot (each * is the asymptote; 1 row = 10 dB):

  20 |*                                    
     | ******                              
   0 |       ******                        
     |             ***                     
 −20 |                ***                  
     |                   **                
 −40 |                     **              
     |                       **            
 −60 |                         ***         
     |                            ***      
 −80 |                               ***   
     |                                  ** 
−100 |                                    *
     ++-----------+-----------+-----------+
      0.1         1           10          100  ω (rad/s)

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 −7.3 dB, the asymptote gives −4.1 dB.