Step 1 Find the corner frequencies 1τ and ωn, and mark them. The horizontal axis for ω is divided into several sections by these corner frequencies.
Step 2 (Low freq. section) Using the factor g sn, draw asymptotes at low frequencies where ω is less than the smallest corner frequency.
g sn
Magnitude plot
a line with slope n, passing through (ω = 1, |G(iω)| = g)
Phase plot
90°n
How to Sketch Bode Plot
Step 3 (Repeat Step 3 and 4 for each section) Draw asymptotes until the next corner frequency using the following table, and repeat this for each corner frequency.
τs + 1
Magnitude plot
increase the slope by 1 at the corner frequency 1τ
Phase plot
increase the phase by 90° at the corner frequency 1τ
1τs + 1
Magnitude plot
decrease the slope by 1 at the corner frequency 1τ
Phase plot
decrease the phase by 90° at the corner frequency 1τ
s2 + 2ζωns + ωn2ωn2
Magnitude plot
increase the slope by 2 at the corner frequency ωn
Phase plot
increase the phase by 180° at the corner frequency ωn
ωn2s2 + 2ζωns + ωn2
Magnitude plot
decrease the slope by 2 at the corner frequency ωn
Phase plot
decrease the phase by 180° at the corner frequency ωn
How to Sketch Bode Plot
Step 4 Sketch approximate magnitude / phase curves by the following guide.
τs + 1
Magnitude plot
increase the asymptote value by a factor of 1.4 at the corner frequency 1τ
Phase plot
increase (decrease) the asymptote value by 5.7° at ω = 110τ (at ω = 10τ)
1τs + 1
Magnitude plot
decrease the asymptote value by a factor of 0.7 at the corner frequency 1τ
Phase plot
decrease (increase) the asymptote value by 5.7° at ω = 110τ (at ω = 10τ)
s2 + 2ζωns + ωn2ωn2
Magnitude plot
multiply the asymptote value by a factor of 2ζ at the corner frequency ωn
Phase plot
the curve is close to the asymptote when ζ is small
ωn2s2 + 2ζωns + ωn2
Magnitude plot
multiply the asymptote value by a factor of 12ζ at the corner frequency ωn
Phase plot
the curve is close to the asymptote when ζ is small
Together · put G(s) in standard form
Rewrite the system using the standard factors in the table.
G(s) = 2000(s + 0.5)s(s + 10)(s + 50)
1 · Take out the constants
s + 0.5 = 0.5(1 + 2s)
s + 10 = 10(1 + 0.1s)
s + 50 = 50(1 + 0.02s)
2 · Combine the constants
2000 × 0.510 × 50 = 2
3 · Keep each standard factor visible
G(s) = 2s−1 (1 + 2s) 11 + 0.1s11 + 0.02s
Together · use the table
Match each factor. Complete one row at a time.
gsn
Start with slope n and phase 90°n.
1 + τs
At 1/τ: slope +1, phase +90°.
11 + τs
At 1/τ: slope −1, phase −90°.
Our factor
Match to table
Corner (rad/s)
Slope
Phase
2s−1
g = 2, n = −1
No corner
Start: −1
Start: −90°
1 + 2s
τ = 2 s
1/2 = 0.5
Add +1
Add +90°
11 + 0.1s
τ = 0.1 s
1/0.1 = 10
Add −1
Add −90°
11 + 0.02s
τ = 0.02 s
1/0.02 = 50
Add −1
Add −90°
Together · draw the magnitude
Start with 2/s. At each corner, change the current slope.
Frequency
Current slope
Magnitude sketch
0 < ω < 0.5
−1
2/ω
0.5 < ω < 10
−1 + 1 = 0
4
10 < ω < 50
0 − 1 = −1
40/ω
ω > 50
−1 − 1 = −2
2000/ω2
Find the starting slope before drawing.
Together · draw the phase asymptotes
Start at −90°. Add the phase change from each factor.
Frequency
Table rule
Phase asymptote
0 < ω < 0.5
−90°
−90°
0.5 < ω < 10
+90°
−90° + 90° = 0°
10 < ω < 50
−90°
0° − 90° = −90°
ω > 50
−90°
−90° − 90° = −180°
Use the Phase plot row in the reference table.
Frequency-Domain Specifications
Bode Plot
r(t) = R sin ωt ⇒ yss(t) = R|G(iω)| sin(ωt + ∠G(iω))
Substitute s = iω and complete the table. Keep ζ symbolic.
Sketch magnitude (log–log) and phase asymptotes. Mark the values at ω = ωn.
ωωn
ω
G(iω)
|G(iω)|
∠G(iω)
≪ 1
= 1
≫ 1
iPad setup
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