Phase margin
Encyclopedia
In electronic amplifier
s, phase margin (PM) is the difference between the phase
, measured in degrees, of an amplifier's output signal (relative to its input) and 180°, as a function of frequency. The PM is taken as positive at frequencies below where the open-loop phase first crosses 180°, i.e. the signal becomes inverted, or antiphase; it is negative beyond. In negative feedback
, a negative PM at a frequency where the loop gain
exceeds unity guarantees instability, thus positive PM is a "safety margin" that ensures proper operation of an amplifier and, more generally, active filters, under various loads. In its simplest form, involving ideal negative feedback
voltage amplifiers with non-reactive feedback, the phase margin is measured at the frequency where the open loop voltage gain
of the amplifier equals the desired closed loop DC
voltage gain.
More generally, PM is defined as that of the amplifier and its feedback network combined (the "loop", normally opened at the amplifier input), measured at a frequency where the loop gain
is unity, and prior to the closing of the loop through tying the output of the open loop to the input source in such a way as to subtract from it.
In the above loop-gain definition, it is assumed that the amplifier input presents zero load. To make this work for non-zero-load input, the output of the feedback network needs to be loaded with an equivalent load for the purpose of determining the frequency response
of the loop gain.
It is also assumed that the gain vs. frequency crosses unity gain with a negative slope and does so only once. This consideration matters only with reactive and active
feedback networks, as may be the case with active filters.
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. 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. In conventional operational amplifier
s, 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 Warwick
or Stout
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; see ringing artifacts
. 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.
Electronic amplifier
An electronic amplifier is a device for increasing the power of a signal.It does this by taking energy from a power supply and controlling the output to match the input signal shape but with a larger amplitude...
s, phase margin (PM) is the difference between the phase
Phase (waves)
Phase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...
, measured in degrees, of an amplifier's output signal (relative to its input) and 180°, as a function of frequency. The PM is taken as positive at frequencies below where the open-loop phase first crosses 180°, i.e. the signal becomes inverted, or antiphase; it is negative beyond. In negative feedback
Negative feedback
Negative feedback occurs when the output of a system acts to oppose changes to the input of the system, with the result that the changes are attenuated. If the overall feedback of the system is negative, then the system will tend to be stable.- Overview :...
, a negative PM at a frequency where the loop gain
Loop gain
Loop gain is an engineering term used to quantify the gain of a system controlled by feedback loops. As such, the concept of loop gain is useful in a variety of disciplines. Traditionally, most of those have been in the field of electronics, telecommunications, or control systems...
exceeds unity guarantees instability, thus positive PM is a "safety margin" that ensures proper operation of an amplifier and, more generally, active filters, under various loads. In its simplest form, involving ideal negative feedback
Negative feedback
Negative feedback occurs when the output of a system acts to oppose changes to the input of the system, with the result that the changes are attenuated. If the overall feedback of the system is negative, then the system will tend to be stable.- Overview :...
voltage amplifiers with non-reactive feedback, the phase margin is measured at the frequency where the open loop voltage gain
Gain
In electronics, gain is a measure of the ability of a circuit to increase the power or amplitude of a signal from the input to the output. It is usually defined as the mean ratio of the signal output of a system to the signal input of the same system. It may also be defined on a logarithmic scale,...
of the amplifier equals the desired closed loop DC
Electricity
Electricity is a general term encompassing a variety of phenomena resulting from the presence and flow of electric charge. These include many easily recognizable phenomena, such as lightning, static electricity, and the flow of electrical current in an electrical wire...
voltage gain.
More generally, PM is defined as that of the amplifier and its feedback network combined (the "loop", normally opened at the amplifier input), measured at a frequency where the loop gain
Loop gain
Loop gain is an engineering term used to quantify the gain of a system controlled by feedback loops. As such, the concept of loop gain is useful in a variety of disciplines. Traditionally, most of those have been in the field of electronics, telecommunications, or control systems...
is unity, and prior to the closing of the loop through tying the output of the open loop to the input source in such a way as to subtract from it.
In the above loop-gain definition, it is assumed that the amplifier input presents zero load. To make this work for non-zero-load input, the output of the feedback network needs to be loaded with an equivalent load for the purpose of determining the frequency response
Frequency response
Frequency response is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterize the dynamics of the system. It is a measure of magnitude and phase of the output as a function of frequency, in comparison to the input...
of the loop gain.
It is also assumed that the gain vs. frequency crosses unity gain with a negative slope and does so only once. This consideration matters only with reactive and active
Passivity (engineering)
Passivity is a property of engineering systems, used in a variety of engineering disciplines, but most commonly found in analog electronics and control systems...
feedback networks, as may be the case with active filters.
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. 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
Feedback
Feedback describes the situation when output from an event or phenomenon in the past will influence an occurrence or occurrences of the same Feedback describes the situation when output from (or information about the result of) an event or phenomenon in the past will influence an occurrence or...
. The reason: the amplifier will oscillate
Electronic oscillation
Electronic oscillation is the continuous recurrence of the same electrical periodic waveform.The recurrence may be in the form of a varying voltage or a varying current. The waveform may be sinusoidal or some other shape when its magnitude is plotted against the duration of its cycle...
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. In conventional operational amplifier
Operational amplifier
An operational amplifier is a DC-coupled high-gain electronic voltage amplifier with a differential input and, usually, a single-ended output...
s, 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
Frequency compensation
In electrical engineering, frequency compensation is a technique used in amplifiers, and especially in amplifiers employing negative feedback. It usually has two primary goals: To avoid the unintentional creation of positive feedback, which will cause the amplifier to oscillate, and to control...
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 Warwick
or Stout
for a detailed analysis of the techniques and results of compensation to insure adequate phase margins. See also the article on pole splitting
Pole splitting
Pole splitting is a phenomenon exploited in some forms of frequency compensation used in an electronic amplifier. When a capacitor is introduced between the input and output sides of the amplifier with the intention of moving the pole lowest in frequency to lower frequencies, pole splitting causes...
. 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
Butterworth filter
The Butterworth filter is a type of signal processing filter designed to have as flat a frequency response as possible in the passband so that it is also termed a maximally flat magnitude filter...
design). An amplifier with lower phase margin will ring
Ringing artifacts
In signal processing, particularly digital image processing, ringing artifacts are artifacts that appear as spurious signals near sharp transitions in a signal. Visually, they appear as bands or "ghosts" near edges; audibly, they appear as "echos" near transients, particularly sounds from...
Ringing is the displaying of a decaying oscillation for a portion of the output signal's cycle; see ringing artifacts
Ringing artifacts
In signal processing, particularly digital image processing, ringing artifacts are artifacts that appear as spurious signals near sharp transitions in a signal. Visually, they appear as bands or "ghosts" near edges; audibly, they appear as "echos" near transients, particularly sounds from...
. 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.
See also
- BIBO stabilityBIBO stabilityIn electrical engineering, specifically signal processing and control theory, BIBO stability is a form of stability for linear signals and systems that take inputs. BIBO stands for Bounded-Input Bounded-Output...
- Nyquist stability criterionNyquist stability criterionWhen designing a feedback control system, it is generally necessary to determine whether the closed-loop system will be stable. An example of a destabilizing feedback control system would be a car steering system that overcompensates -- if the car drifts in one direction, the control system...
- Routh-Hurwitz stability criterionRouth-Hurwitz stability criterionThe Routh–Hurwitz stability criterion is a necessary and sufficient method to establish the stability of a single-input, single-output , linear time invariant control system. More generally, given a polynomial, some calculations using only the coefficients of that polynomial can lead to the...
- Root locus methodRoot locusRoot locus analysis is a graphical method for examining how the roots of a system change with variation of a certain system parameter, commonly the gain of a feedback system. This is a technique used in the field of control systems developed by Walter R...
- Bode plots & phase margin
- Step response & phase margin
- Ringing artifactsRinging artifactsIn signal processing, particularly digital image processing, ringing artifacts are artifacts that appear as spurious signals near sharp transitions in a signal. Visually, they appear as bands or "ghosts" near edges; audibly, they appear as "echos" near transients, particularly sounds from...