Gain & phase margins are key parameters utilized in control systems to determine a system’s stability and durability.
Understanding these concepts is important for engineers & designers working with feedback systems because they provide information about how close a system is to instability & how much variation it can withstand before becoming unstable.
Phase Margin
The phase margin (PM) of a control system is measured by how far the phase of the loopgain transfer function deviates from 180 degrees at the gain crossover frequency.
It is measured in degrees, and a positive phase margin indicates stability.
A larger positive phase margin represents better stability, whereas a negative phase margin suggests possible instability.
To calculate phase margin, use the formula:
Phase Margin (PM) = ∠ LG(_{jω GCF}) + 180 ^{∘}
The phase of the loopgain transfer function at the gain crossover frequency is represented by ∠LG(jω GCF).
Gain Margin
Gain margin (GM) indicates how far the loopgain transfer function may be increased before the system gets unstable.
It is commonly measured in decibels (dB), and a positive gain margin indicates stability.
A negative gain margin indicates that the system is inherently unstable.
To calculate gain margin, use the following formula:
Gain Margin (GM) = 1/∣ LG(_{ j ω P C F }) ∣.
The magnitude of the loopgain transfer function at the phase crossover frequency (PCF) is denoted as ∣LG(jω PCF)∣.
Importance in Control Systems

Both margins give a numerical estimate of how close a system is to instability. A system with broad margins can withstand more variations in system parameters without getting unstable.

Engineers employ gain and phase margins to develop control systems that meet specified performance requirements while being robust in the face of system dynamics uncertainties.

Engineers can use these margins to maximize system performance while maintaining stability, minimizing oscillations (or) unpredictable actions in feedback systems.

Bode plots are commonly used to depict gain and phase margins, which show how gain & phase shift change with frequency. This graphical depiction allows engineers to rapidly detect stability margins & make educated design decisions.
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