What is the best phase margin for a loop?

[anhnha]

What is the best phase margin for a loop?
I read somewhere that 90 degree is the ideal phase margin and 60 degree is practical. Could you explain why?
Can phase margin be larger than 90 degrees?

 

[schmitt trigger]

Complexity for one. You would have to create a very complex compensation to null out the system's poles and zeros. With discrete components very quickly one finds not only a lot of components, but components with non-standard values.

However, there are trends to start compensating in the digital domain, there are several power supply companies that are doing that, and claim either excellent phase margins or extremely fast transient responses.

Which brings me to the second point; transient response. Compensation slows the loop down. The struggle always is, besides a stable loop, a quick transient response. So once that you have achieved unconditional stability with enough margin to account for component tolerances, there is no value in increasing that margin if it will slow the transient response.

 

[LvW]

anhnha, I suppose your question is related to opamp circuits with feedback, OK?

1.) Let´s assume for simplicity that the open-loop gain of an opamp falls with a slope of -20 dB/dec.
The corresponding phase shift is -90 deg. Together with the phase inversion at the neg. input (assuming negative feedback, of course) we have a total phase shift of -270 deg.
As a consequence, the remaining phase margin is only 90 deg.

2.) However, this margin is the absolute maximum (which cannot be achieved) because each opamp has more poles at higher frequencies causing more phase shift.
Thus, in reality the phase margin always will be smaller than 90 deg.

3.) In the time domain, the stability margin (to be determined in the frequency domain) determines the dynamic properties of a circuit with feedback. This can be measured using the principle of step response.
It can be shown by calculation, measurement or simulation that a phase margin of 60 deg causes only a very small overshoot in the step response.
In contrast, a margin of only 20 deg will be connected with a step response that exhibits ringing and a heavy overshoot of app. 20%.

4.) That is the reason, we require - if possible - a phase margin of at least 60deg. for good stability and sufficient time behaviour.

 

[anhnha]

Well, thanks a lot, schmitt trigger and LvW.

LvW: you are right. I am talking about opamp circuits.
For my LDO, the phase margin at no load condition starts at 156 degree not 180 degree. That is strange to me.
Could you explain what determine the starting point of phase in bode plot?

 

[LvW]

Could you explain what determine the starting point of phase in bode plot?

For phase margin determination we need the loop gain. This is the gain (as a function of frequency) of all loop components (loop open) - including the sign inversion at the inv. opamp input.
Because DC stability requires negative feedback at DC the loop gain phase must start at 0 Hz with -180deg. As a consequence, the phase margin PM is determined at -360deg.
However, in many cases, the phase response is not measured or simulated but derived from the magnitude response (-20dB/dec correspond with -90 deg,....).
In this case, the sign inversion very often is NOT included in the phase function and the phase does start at 0 deg. (PM determination at -180deg).
In the latter case, we have NOT the loop gain phase PH_loop but (PH_loop+180) deg.

 

[anhnha]

Thank you.
I don't quite understand what you said. If possible could you explain more?

Because DC stability requires negative feedback at DC the loop gain phase must start at 0 Hz with -180deg.

IF so, my LDO regulator that has 156deg at 0Hz is not stable?
Could you explain why it is 156deg not 180 there?
What factors contribute to the phase at 0Hz?

 

[FvM]

Your LDO loop can't have 156° phase at 0 Hz, either 0 (non-inverting) or 180° (inverting loop gain). For stable DC operation, it must be inverting.

But how do you say the phase at 0 Hz is 156°? 0 Hz can't be measured in AC analysis, only an arbitrary small frequency. Apparently the frequency isn't low enough to see the approximate 0 Hz value.

 

[LvW]

Thank you.
I don't quite understand what you said. If possible could you explain more?

Because DC stability requires negative feedback at DC the loop gain phase must start at 0 Hz with -180deg

Each feedback stabilized circuit needs negative feedback for DC. Otherwise, there is no stable operating point. This applies to simple transistor stages as well as opamp circuits.
With other words: Within the whole loop must be one (or three) phase inversions (negative sign).
As a logical consequence, the phase of the loop gain function must start at -180 deg.
As mentioned by FvM: At 0 Hz (DC) a phase of other than 180 deg is impossible.

 

[anhnha]

FvW:

Yeah, I think you are right.
I just measured phase at 1Hz. I tried a much smaller frequency and it is -179.8 degree now.
However, the result is a bit strange to me.
When I run from 1u Hz to 20Hz, it starts at -180 degree.
When I run from 1Hz to 20Hz, it starts at 156 degree.
Is this normal?

- - - Updated - - -
LvW:

Thanks!

At 0 Hz (DC) a phase of other than 180 deg is impossible.

I'd like to know why?
Are there any references?

 

[LvW]

I'd like to know why?
Are there any references?

Can you think of any passive element that can cause a phase shift at DC?
Active units (amplifier stages) can have only positive or negative gain values.

 

[anhnha]

Thank you.

Can you think of any passive element that can cause a phase shift at DC?

Nope!

Active units (amplifier stages) can have only positive or negative gain values.

I think you are right but I actually can't explain it.
For opamp, I usually see that there are two phase shift 0 (non-inverting) and 180 (inverting).
But why it is impossible for opamp to have other phase shift at 0Hz?
I know almost next to nothing about this.

 

[LvW]

But why it is impossible for opamp to have other phase shift at 0Hz?
I know almost next to nothing about this.

An amplifier consusts of several parts - active and passive.
Which parts can cause phase shifts? L and C.
Also at 0 Hz? No. There is no "phase" and no definition for "phase shift" at zero Hz.

 

[anhnha]

Thank you.
Could you help me with this question?
My LDO loop gain (I broke the loop and inserted inductor and capacitor for loop gain simulation) has phase starting at 180 degree, and at 0dB gain, the phase is 87 degree. So, is the phase margin = 180 -87 = 93 degree?

 

[LvW]

anhnha, may I ask you to read again my post#5?
I have explained that the correct loop gain simulation must show a phase shift of -180deg at 0 Hz. This is what you have measured - and this is correct, fine!
Then, the phase margin PM is the difference between the actual phase at unity gain and the stability limit at -360 deg (identical to 0 deg). Thus your margin is PM=87 deg.

 

[SIDDHARTHA HAZRA]

No its 87 .... PM is how much more phase shift is required to make the total phase shift of the system 180. For your case the system phase has already shifted by 93 so to make it unstable it will need 87deg more shift. So the margin left in your system is 87deg .... that is why we call it margin

 

[anhnha]

Thank you.
I am a bit confused.
With 360 degree phase shift for oscillation, are you talking about positive feedback?
**broken link removed**
According to this link, with positive feedback, the circuit will oscillate if 1- Aβ = 0. This condition means that | Aβ | = 1 and the phase angle of Aβ is zero or an integral multiple of 360.
But for negative feedback, the circuit will oscillate if 1+ Aβ = 0. This condition means that | Aβ | = 1 and the phase angle of Aβ is 180.

 

[LvW]

Yes - of course. When the phase shift of the loop gain is -360 deg we have pos. feedback and, therefore, this is the stability limit for an amplifier.

- - - Updated - - -

Perhaps it is helpful to add the following (for clarification):
Speaking about "negative" feedback, we mean: feedback which is intentionally negative (for low frequencies).
However, for higher frequencies each negative feedback will turn into positive feedback . This cannot be avoided because of the phase shifts from the amplifier.
But he circuit will remain stable if the gain is already below unity if the phase has reached the 360 deg. limit.

 

[anhnha]

Thank you, SIDDHARTHA HAZRA and LvW.

I just measured phase margin of LDO for a different case and now phase margin is 93 degree. But according to post #3, the maximum phase margin is 90 degree.
Is there something wrong here?

 

[Dominik Przyborowski]

the PM could be higher than 90 degrees, i.e. when transfer function contains low freq. dominant pole, LHP zero and nondominant pole after a next decade.

 

[anhnha]

Thanks, Dominik Przyborowski.
Could you explain a bit more?
My DC gain is 80dB. The dominant pole will decide where the unity gain is.
Because gain decreases 20dB/dec this means unity gain frequency is at four decades away from origin.
Why not there is a zero some where about four decade?