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Gain and Phase Margins Model Perturbation Flipbook PDF
1 Phase Margin & Gain Margin M. Sami Fadali Professor of Electrical Engineering University of Nevada 2 Stability Mar
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Stability Margins • What is the worst perturbation of the transfer function that will make the system marginally stable? • Marginal stability for open loop stable systems is when the contour goes through the point (1,0).
Phase Margin & Gain Margin M. Sami Fadali Professor of Electrical Engineering University of Nevada 1
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Gain and Phase Margins
Model Perturbation
Gain Margin: gain perturbation that makes the system marginally stable. Phase Margin: negative phase perturbation that makes the system marginally stable.
R(s) +
G(s)
C(s) G(s)
• G(s) = model perturbation • Gain Margin: G(s) = K (gain perturbation) • Phase Margin: G(s) = ej (phase lag perturbation) 3
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Margins on Polar Plot
Definitions of Margins
/
=gain crossover
/phase crossover
Gain Margin: additional gain that makes the system on the verge of instability. Phase Margin: additional phase lag that makes the system on the verge of instability.
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Polar plot of Unstable System
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Point on Bode Plot • Negative real axis for Nyquist plot . corresponds to an angle of • Magnitude of unity corresponds to zero dB. • For PM we need to find the phase angle at magnitude unity (0dB). • For GM we need to find the magnitude at an angle of .
Negative GM (dBs) and PM. 7
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Bode Plot of Stable System
MATLAB Margin >> [Gm,Pm,Wcg,Wcp] = margin(g) Gm = 2.5000 Pm = 22.5359 Wcg = 10.0000 (phase crossover freq.) Wcp = 6.0783 (gain crossover freq.)
>> margin(g) 9
Gain Margin Calculation
Example: Gain Margin
• Multiply numerator and denominator by the complex conjugate of the denominator. ∗
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• Solve for phase crossover (imaginary part zero)
∗
∗
• Equate the imaginary part of the numerator to zero and solve for the phase crossover ∗ frequency: • Calculate the gain margin 11
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Phase Margin Calculation
Calculate Gain Margin • Evaluate the magnitude at the phase crossover frequency
• Solve for gain crossover frequency (unity magnitude)
• Calculate the phase margin
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Phase Margin
Phase Margin Calculation • Solve for gain crossover (unity magnitude)
∠
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Delay Margin
Stability Margins >> g=zpk([],[-1,-5],10) Nyquist Diagram 1.5
Imaginary Axis
1 0.5 0
System: g Phase Margin (deg): 104 Delay Margin (sec): 1.12 At frequency (rad/s): 1.62 Closed Loop Stable? Yes
-0.5
= delay margin = time delay for the system to be on the verge of instability.
Zero/pole/gain: 10 ----------(s+1) (s+5)
• Transfer function with time delay
>> nyquist(g)
• System on verge of instability
-1 -1.5 -1
-0.5
0
0.5 Real Axis
1
1.5
2
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Delay Margin Calculation
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Example
• Equate angles:
• Solve for the delay margin Same as MATLAB answer. 19
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