# Equation that expresses a negative feedback phase shift

#### jaydnul

Joined Apr 2, 2015
175
Hi,

Curious if there is an expression for a negative feedback system that experiences phase shift. For example, at 0 degree phase shift, the additional gain is 0 because it is fully negative feedback. At 180 phase shift the additional gain is infinite because you have full positive feedback. Is there an equation that expresses this and all the intermediate phase shifts?

Thanks

#### Audioguru again

Joined Oct 21, 2019
6,441
Negative feedback has near 180 degrees of phase shift. Then 100% of negative feedback produces a gain of 1.
But the phase shift changes with frequency because of stray or added capacitances.

An opamp has an added internal capacitor to make the negative feedback 100% at a frequency below where stray capacitance creates enough positive phase shift to cause oscillation.

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#### BobTPH

Joined Jun 5, 2013
8,105
0 degree phase shift, the additional gain is 0 because it is fully negative feedback. At 180 phase shift the additional gain is infinite
No.

#### jaydnul

Joined Apr 2, 2015
175
Lol cool

#### MrAl

Joined Jun 17, 2014
10,906
Hi,

Curious if there is an expression for a negative feedback system that experiences phase shift. For example, at 0 degree phase shift, the additional gain is 0 because it is fully negative feedback. At 180 phase shift the additional gain is infinite because you have full positive feedback. Is there an equation that expresses this and all the intermediate phase shifts?

Thanks
Hi,

Yes, and it's called the Transfer Function (of the system).

For the most general expression, we have:
T(s)=G(s)/(1+G(s)*H(s))

G(s) is the feed forward function, H(s) is the feedback function. The transfer function T(s) is as above.
Both G(s) and H(s) can have any phase shift, but of course some will cause instability which results in either oscillation or the output ramps up or down to one of the power supply rails.
In that expression H(s) is usually negative feedback. We call it negative feedback because an increase in feedback results in a decrease of the output. It's not that it has to be perfectly negative like -K or something like that (K a constant here).

I got this drawing from Wikipedia but I modified it to be more concise and more general, and of course used circles to show the input and output as that is more normal for these kinds of drawings.

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#### Ya’akov

Joined Jan 27, 2019
8,522
Are you asking about taking a negative feedback signal, which will be 180° out of phase by definition, and applying a phase offset to that (already 180° phase shifted) signal?

#### MrAl

Joined Jun 17, 2014
10,906
Are you asking about taking a negative feedback signal, which will be 180° out of phase by definition, and applying a phase offset to that (already 180° phase shifted) signal?
Hi there Ya'akov,

I am not sure you can say that because that would only be the case in some situations.
I think a better way to state it, if we have to, is to say that negative feedback is a signal that when its amplitude increases the output decreases, and if the amplitude decreases the output increases.

Maybe another way to say that is to say that negative feedback is feedback that subtracts from some feed forward signal. The reason I say this is because the feedback phase shift can be several values and still cause a decrease in the output with an increase in the feedback signal.

An example...

G=(s+1)/((s+2)*(s+3))
H=1/(s+1)

Now assuming the summing junction subtracts from the input signal as is the usual case for negative feedback, we have:
Ts=Gs/(1+Gs*Hs)

and this system is stable for an input step change.

The feedback signal phase alone is:
ph=-atan(w)
and that means that the phase shift can vary widely over frequency, which of course means it will not always be 180 degrees. Since this feedback subtracts from the input, that would make it not always -180 degrees.

There are times when it will always be -180 degrees, but that would be for only very specific cases like H being a constant. That is very possible, but far from the general case. An example would be an inverting (or even non-inverting) op amp amplifier where the feedback is a constant value which never changes in phase.

There's also a related measure we call phase margin. That relates the phase to the ability of the system to remain stable.

#### Ya’akov

Joined Jan 27, 2019
8,522
Hi there Ya'akov,

I am not sure you can say that because that would only be the case in some situations.
I think a better way to state it, if we have to, is to say that negative feedback is a signal that when its amplitude increases the output decreases, and if the amplitude decreases the output increases.

Maybe another way to say that is to say that negative feedback is feedback that subtracts from some feed forward signal. The reason I say this is because the feedback phase shift can be several values and still cause a decrease in the output with an increase in the feedback signal.

An example...

G=(s+1)/((s+2)*(s+3))
H=1/(s+1)

Now assuming the summing junction subtracts from the input signal as is the usual case for negative feedback, we have:
Ts=Gs/(1+Gs*Hs)

and this system is stable for an input step change.

The feedback signal phase alone is:
ph=-atan(w)
and that means that the phase shift can vary widely over frequency, which of course means it will not always be 180 degrees. Since this feedback subtracts from the input, that would make it not always -180 degrees.

There are times when it will always be -180 degrees, but that would be for only very specific cases like H being a constant. That is very possible, but far from the general case. An example would be an inverting (or even non-inverting) op amp amplifier where the feedback is a constant value which never changes in phase.

There's also a related measure we call phase margin. That relates the phase to the ability of the system to remain stable.
It appeared the TS was referring to negative feedback as applied to OPAs. I was trying to figure out the reference was for 0°.

#### jaydnul

Joined Apr 2, 2015
175
Thanks for the comments! Let me try to restate my question better:

i run a stability analysis and i get loop gain and loop phase. Whats the equation that combines these two plots as if it was still closed loop. Basically how do i get the LG/1+LG plot that includes the peaking that a low phase margin would cause in that plot. For example is it LG/1+LG+tan(phase)? (Thats not it, just an example of the form im looking for)

adding this plot to make sense of it. How do i combine black and blue plots into an equation that gives me red plot:

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