Phase margin

In electronic amplifiers, phase margin is the difference, measured in degrees, between the phase of the amplifier's output signal and -360°. In feedback amplifiers, the phase margin is measured at the frequency at which the open loop voltage gain of the amplifier and the closed loop voltage gain of the amplifier are equal.cite book
author=Paul Horowitz & Hill W
title=The art of electronics
year= 1989
page=§ 4.33 pp. 242-249
edition=Second Edition
publisher=Cambridge University Press
location=Cambridge MA
isbn=0521370957
url=http://books.google.com/books?id=bkOMDgwFA28C&printsec=frontcover&dq=intitle:Art+intitle:of+intitle:electronics&lr=&as_brr=0#PPA245,M1]

Phase margin and its important companion concept, gain margin, are measures of stability in closed loop dynamic [control] systems. Phase margin indicates relative stability, the tendency to oscillate during its damped response to an input change such as a step function. Gain margin indicates absolute stability and the degree to which the system will oscillate without limit given any disturbance.

The output signals of all amplifiers exhibit a time delay when compared to their input signals. For the most part, the delay is caused by internal resistances, current limits and capacitances within the amplifier. This delay causes a phase difference between the amplifier's input and output signals. If there are enough stages in the amplifier, at some frequency, the output signal will lag behind the input signal by one wavelength. In this situation, the amplifier's output signal will be in phase with its input signal though lagging behind it by 360°, i.e., the output will have a phase angle of –360°. This lag is of great consequence in amplifiers that use feedback. The reason: the amplifier will oscillate if the fed-back output signal is in phase with the input signal at the frequency at which its open loop voltage gain equals its closed loop voltage gain and the open loop voltage gain is one or greater. The oscillation will occur because the fed-back output signal will then reinforce the input signal at that frequency. ["Ibid", p. 245.] In conventional operational amplifiers, the critical output phase angle is –180° because the output is fed back to the input through an inverting input which adds an additional –180°.

In practice, feedback amplifiers must be designed with phase margins substantially in excess of 0°, even though amplifiers with phase margins of, say, 1° are theoretically stable. The reason is that many practical factors can reduce the phase margin below the theoretical minimum. A prime example is when the amplifier's output is connected to a capacitive load. Therefore, operational amplifiers are usually compensated to achieve a minimum phase margin of 45° or so. This means that at the frequency at which the open and closed loop gains meet, the phase angle is -135°. The calculation is: -135° - (-180°) = 45°. See Warwickcite book
author=K Warwick
title=An introduction to control systems
year= 1996
edition=Second Edition
page=Chapter 5, pp. 137-196
publisher=World Scientific
location=Singapore
isbn= 9810225970 (pb), or 9810215630 (hc)
url=http://books.google.com/books?id=EvgjDFf84l8C&pg=PA172&dq=compensation+%22phase+margin%22&lr=&as_brr=0&sig=cM4TxzoOeDhYDYD7I2ny7c1QT3Y#PPA170,M1] or Stoutcite book
author=David F Stout & Kaufman M
title=Handbook of operational amplifier circuit design
year= 1976
page=Sec. 3-4
publisher=McGraw-Hill
location=NY
isbn= 007061797X
url=http://books.google.com/books?id=PlWyAQAACAAJ&printsec=frontcover&dq=editions:ISBN007061797X] for a detailed analysis of the techniques and results of compensation to insure adequate phase margins. See also the article on pole splitting. Often amplifiers are designed to achieve a typical phase margin of 60 degrees. If the typical phase margin is around 60 degrees then the minimum phase margin will typically be greater than 45 degrees. A phase margin of 60 degrees is also a magic number because it allows for the fastest settling time when attempting to follow a voltage step input (a Butterworth design). An amplifier with lower phase margin will ring [Ringing is the displaying of a decaying oscillation for a portion of the output signal's cycle.] for longer and an amplifier with more phase margin will take a longer time to rise to the voltage step's final level.

A related measure is gain margin. While phase margin comes from the phase where the loop gain equals one, the gain margin is based upon the gain where the phase equals -180 degrees.

References

ee also

* BIBO stability
*Nyquist stability criterion
*Routh-Hurwitz stability criterion
*Root locus method
*Bode plots & phase margin
*Step response & phase margin