Bode plot: 20(s+5)/(S(S+20)(s²+80s+200))

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) = 20(s + 5) / [ s (s + 20)(s² + 80s + 200) ]
     = 20(s + 5) / [ s (s + 20)(s + 2.583)(s + 77.42) ]

H(s) = 0.025 × (1 + s/5)
         / [ s (1 + s/20) (1 + s/2.583) (1 + s/77.417) ]

Constant term K₀ = 0.025, so 20 log K₀ = −32.04 dB.

Step 2: Corner frequencies

s² + 80s + 200 has real roots −40 ± √1400 = −2.583 and −77.42 (ζ = 2.83 > 1), so it splits into two simple factors.

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 −32 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 |  ******                             
     |        ******                       
 −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 = 2.583 rad/s the exact gain is −42.3 dB, the asymptote gives −40.3 dB.