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What is the primary purpose of a transfer function in control systems?
What is the primary purpose of a transfer function in control systems?
- To describe the system's response to sinusoidal inputs at different frequencies (correct)
- To analyze the stability of a system
- To design a control system's feedback loop
- To measure the system's impulse response
What is the gain margin of a system?
What is the gain margin of a system?
- The reciprocal of the gain at the frequency where the phase angle is -180° (correct)
- The ratio of the system's gain to its maximum allowable gain
- The difference between the system's gain and its maximum allowable gain
- The frequency at which the system's phase angle is -90°
What is the primary criterion for determining the stability of a control system using the transfer function?
What is the primary criterion for determining the stability of a control system using the transfer function?
- Bode plot criterion
- Nyquist criterion
- Nichols chart criterion
- Routh-Hurwitz criterion (correct)
What type of plot shows the magnitude and phase angle of a transfer function versus frequency?
What type of plot shows the magnitude and phase angle of a transfer function versus frequency?
What is the purpose of the Nyquist criterion in stability analysis?
What is the purpose of the Nyquist criterion in stability analysis?
What is the relationship between the gain margin and the stability of a control system?
What is the relationship between the gain margin and the stability of a control system?
What is the phase margin of a control system?
What is the phase margin of a control system?
What is the primary application of frequency response analysis in control systems?
What is the primary application of frequency response analysis in control systems?
What type of plot shows the real and imaginary parts of a transfer function?
What type of plot shows the real and imaginary parts of a transfer function?
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Study Notes
Control System Frequency Response
Transfer Function
- A mathematical representation of a control system's behavior in the frequency domain
- Describes the system's response to sinusoidal inputs at different frequencies
- Typically represented as a ratio of polynomials in the complex variable
s = σ + jω - Can be obtained through:
- Laplace transform of the system's differential equation
- Fourier transform of the system's impulse response
Gain Margin (GM)
- A measure of how much the gain of a system can be increased before it becomes unstable
- Defined as the reciprocal of the gain at the frequency where the phase angle is -180°
- Gain Margin (GM) = 1 / |G(jω)| at ω = ω180°
- A higher GM indicates a more stable system
Stability Analysis
- Frequency response analysis is used to determine the stability of a control system
- Stability criteria:
- Routh-Hurwitz criterion: all poles of the transfer function must lie in the left half of the s-plane
- Nyquist criterion: the Nyquist plot must not encircle the point -1 + j0
- Frequency response plots:
- Bode plot: magnitude and phase angle vs. frequency
- Nyquist plot: real vs. imaginary parts of the transfer function
- Nichols chart: magnitude and phase angle vs. frequency on a polar plot
Phase Margin (PM)
- A measure of how much the phase of a system can be shifted before it becomes unstable
- Defined as the angle between the phase plot and the -180° line at the gain crossover frequency
- Phase Margin (PM) = φ - (-180°) at ω = ωgc
- A higher PM indicates a more stable system
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