1st order factors: the constant term. I.E. (S+ωc) or ωc(1+ωcS)
2nd order factors: the square root of the constant term I.E. (S2+2ζωcS+ωc2) or ωc2(1+ωc2ζS+(ωcS)2)
factor out these constant terms then find 20log∣K∣
make a table of ω and corresponding exact magnitude in dB (for guidance), slope of the line leaving ω and phase angle in degrees (you can use calculator's complex mode to find the exact numbers)
connect the dots 🤣
Find some analytical values
Gain Crossover Frequency ωG : the intersection of magnitude line with x-axis. I.E. 20log∣GH∣=0
Phase Crossover Frequency ωp : the intersection of phase curve with -180 degrees line. I.E ∠GH=−180∘
Gain Margin: the distance from x-axis (0) - the magnitude at ωp I.E. Gm=−20log∣GH(ωp)∣
Phase Margin: the distance from the phase at ωG - the -180 degrees line I.E. Pm=180+∠GH(ωg)
Stability analysis and gain tweaks
System is stable if ωg<ωp , unstable if ωg>ωp and critical stable when they're equal
to change gain margin multiply old gain by K:20logK=Gmold−Gmnew I.E. Knew=Kold×1020Gmold−Gmnew
to change phase margin, multiply the gain by a K that would produce the needed shift that makes the corresponding magnitude intersect with x-axis.
to make critically stable system change gain margin to 0.
Ex) Consider a closed loop system whose open loop transfer function is GH=s(s+1)2(s+10)K(s+5)
Draw a Bode diagram of the open-loop transfer function for K=50 then
I. Determine the gain crossover frequency, phase crossover frequency, gain margin, and phase margin.
II. Solve again for k=1.
III. Determine the value of gain K such that the gain margin is 40dB.
IV Determine the value of gain K such that the phase margin is 70 deg.
V. Determine the value of gain K for oscillation system.
Find corner frequencies: 1,5,10
Make a table of corresponding magnitude and phase angle
ω
0.01
0.1
1
5
10
100
1000
20log∣GH∣
67.96
47.87
22.07
-12.28
-28.15
-86.05
-146.02
Distance (mm)
44
31
14
-8
-18
-56
-95
Slope
-20dB/d
-20dB/d
27.95 -60dB/d
-40dB/d
-60dB/d
-60dB/d
-60dB/d
∠GH
-91.09
-100.85
-174.4
-228.95
-240.14
-266
-269.60
Distance (mm)
-13
-14
-25
-33
-34
-38
-39
Draw the plot
I. Exact values
gain cross over frequency = 2.917 phase margin = 180 + arg(GH(2.917)) = -38.16
phase crossover frequency =1.114 gain margin = -20log|GH(1.114)| = -20.17
II. Exact values
gain cross over frequency = 0.42 phase margin = 180 + arg(GH(0.42)) = 46.83
phase crossover frequency =1.1 gain margin = -20log|GH(1.1)| = 13.5
III. K=50×1020−20.17−40=0.049 => omega p = 1.11 => gain margin = 39.94
IV. angle + 180 = 70 => angle = -110 => omega = 0.19 = magnitude = 42.08 gain difference = -42.08 => new gain 50×1020−42.08=0.393 #
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