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Gain Margin and Phase Margin

Gain Margin It is necessary to ascertain to what degree the system is stable. Intuitively, if A is only slightly less than unity the system is “almost unstable” and may be expected to behave in a highly oscillatory manner even though it is theoretically stable? To assign some quantitative measure to these considerations, the concept of gain margin is introduced. Using the nomenclature of Figure, Gain margin = 1/A Where A= (AR)ωco A gain margin of unity or less indicates an unstable system. We can make the following observations on the practical significance of gain margin: 1. It constitutes a measure of how far the system is from the brink of instability. 2. The higher the gain margin is above the value of 1 the more robust the closed-loop behavior will be: and thus the safer the operation of the controlled process. In other words, the higher the gain margin the higher the safety factor we use for controller tuning. 3. Typical specifications for design are that the gain margin should be greater than 1.7. This means that the AR can increase 1.7 times above the design value before the system becomes unstable. 4. If the various parameters are known very well, only small safety factors are needed (i.e., gain margins in the range 1.4 to 1.7). For systems with parameters poorly known, the safety factor must increase and the recommended values for gain margins are in the range 1.7 to 3. Phase Margin: Besides the gain margin there is another safety factor which is frequently used for the design of a feedback control system is the phase margin. It is the difference between 1800 and the phase lag at the frequency for which the gain is unity. Consider again Figure. Let φAR=1. be the phase lag at the frequency for which AR=1. Then phase margin is defined as: Phase margin=180 deg-φ_(AR=1) The phase margin therefore represents the additional amount of phase lag required to destabilize the system, just as the gain margin represents the additional gain for destabilization. Typical design specifications are that the phase margin must be greater than 30 deg. A negative phase margin indicates an unstable system.