# Nyquist Diagramm, Open loop transfer function and stability

This is a general question about how to determine if an electronic system is stable. A system is unstable when the closed loop transfer function goes to infinity, ie when the denominator of the transfer function goes to 0.

If I consider a linear regulator, the system can be represented as follow :

where the closed loop transfer function is :

It is possible to get the open loop transfer function by opening the closed loop of the system as follows :

And we get from Va and Vb the open loop transfer function :

So we can write the closep loop transfer function as follows :

In order to get the closep loop transfer function to go to the infinity, we need to have :

Then in a nyquist diagram the point of instability in this case is not the "-1 point" but the "+1 point" ???? What do you think about it ?

Thank you very much and have a nice day !

• "A system is unstable when the closed loop transfer function goes to infinity, ie when the denominator of the transfer function goes to 0". This statement is not true. Consider the closed loop TF $\frac{1}{s+1}$ and $\frac{1}{s-1}$. The denominator goes to zero at s=-1 and s=1 respectively. One is stable and the other unstable. – AJN Aug 21 '20 at 10:07
• Why are you asking us "what do you think about it"? Do you want us to confirm Nyquist? Of course, the critical point depends on our choice: what is loop gain? With ot without the minus sign at the summing junction.... – LvW Aug 21 '20 at 10:11
• Which is weird is that it is generally said that the closed loop transfer function is unstable when the gain of the open loof transfer function is equl to 1 and its argument is equal to -180 ° which correspond to "-1 + 0j" in the the Nyquist diagram. So if i find via my measurement a transfer function equal to minus - the open loop tranfer function. I want to move the "-1 + 0j" in the the Nyquist diagram at "+1 + 0j" ? – Jess Aug 21 '20 at 10:15
• As I have mentioned: It depends how you define the "loop gain" function - with or without the minus sign. For my opinion, the minus sign must be included because that is the way we measure/simulate the loop gain (in this case: +1). Remember, in some cases, the sign inversion takes place NOT at the summing junction but anywhere within the feedback loop. In such a case, it would be crazy and confusing to suppress this minus sign in the loop gain anylysis. – LvW Aug 21 '20 at 11:07
• Ok thank you very much ! I have to find more informations about this subject ! I happy to know that it was not wrong what I did ;) – Jess Aug 21 '20 at 11:27

Apologies for chiming in, I'm a lay man, not an EE graduate, not friends with Laplace and Euler.

The one thing I know: for a feedback loop to diverge, at some point along the loop topology, at a frequency where the phase shift is an integer multiple of 360 degrees (including 0 degrees at DC), the gain must be > 1. If this is the case of 0 phase shift and plain DC operation, the system just drifts away immediately into its physical limits. No oscillation, just a one-way road. If this happens at some non-zero frequency, the result is a divergent oscillation.

You need not use the Nyquist plot, you can also imagine this in a bode plot, where the horizontal axis is frequency and the vertical axis has two plots (curves): gain and phase.

In the classic general schematic of a feedback loop (that you have drawn), where the summing point has a + and - input (like an op-amp has), the input where the signal gets "fed back" into the summing device (say an op-amp) is typically inverting. That's already 180 degrees phase shift. So in order to oscillate, the rest of the system need not cumulate a phase shift of 360*, because 180* is enough. 180* inversion + 180* phase shift and you're touching the first sweet spot of 360*, where if the gain is greater than 1, the oscillating system keeps picking up energy (amplitude). Note that further sweet spots are spaced by 360*, and therefore DC is NOT one of them (by definition of negative feedback?). This is why at DC, a simple "follower" op-amp topology with a 1:1 negative feedback is inherently stable (one way to put it).

I understand that negative feedback is a key intermediate goal or key tool of any control system or amplifier design, aiming for overall stability. That's why the comparator or "summing element" input for feedback is typically inverting. But considering the general universe of unstable feedback loops, some/many of which are inadvertent, strictly speaking the loop need not contain an explicit summing point or op-amp of this kind (inverting), or can contain several of them - and the "reference input" (with a plus sign) is perhaps also just a circumstance, or there can be several of them, the reference can be somehow "inherent" etc. So... when you see the stability criterion written as "gain at 360*", you know that this refers to the general principle - whereas if the criterion speaks about "gain at 180*, must be negative and lower than -1", I'd say that the author merely refers to the typical practical scenario with an op-amp like summing device (or comparator) having an inverting feedback input. It is the most practical topology to implement a system of "automatic control based on deviation from a reference". I mean to say that the latter (negative feedback) is merely a specific case of the former (general loop stability theory), and the author has just taken a small specific shortcut in the math.

In control systems or linear amplifiers with a global feedback, phase shift along the frequency domain is really just another way to speak about propagation delay (a time domain concept). Elements of the signal chain along the loop each have a non-zero propagation delay. And, as backward time travel does not exist, the propagation delay cannot be negative. Always non-zero and positive. Imagine an element, like an amplifier stage, or some actuator, having a fixed response time = a constant in the time domain. In the frequency domain, this translates into a phase shift that grows with frequency. Because you compare constant time to a gradually shorter period of some oscillation. And, you typically have several such "constant time delays" along the loop. This is how most feedback-heavy amplifiers (I've cut my teeth on audio stuff) and control systems obtain their typical phase response in the frequency domain bode plot, where the phase shift only ever grows with frequency. And the most popular way (or the only way?) to escape the grim reaper's axe (stability criterion) is to deliberately suppress loop gain at higher frequencies - just enough (or safely enough by a big margin) to miss the criterion from below, already at the first 360* multiple. This is typically implemented using a local negative feedback (capacitor or work-alike) across the fastest individual inverting gain stage along the loop, lending a notable "low-pass filter" characteristic to the individual stage (and adding 90 degrees of phase shift if memory serves). This also means that the "control based on deviations" gets gradually less acurate at higher frequencies. That's the downside/cost of having a stable control system...

In well behaved "feedback-heavy" amps and control systems, the gain only ever drops with frequency, and the phase shift only ever grows.

Should you dare to ask for examples of more interesting phase responses, take a look at resonant cells. Either based on lumped elements (LC circuits) or unterminated transmission lines tuned for some popular wavelength fraction (lambda half or quarter, and their integer multiples). These turn the phase right or left, depending on deviation from the resonant frequency. These are hardly ever included in feedback-heavy circuits (at least not deliberately) as the dancing phase shift can be difficult to handle. In radio-frequency stuff, they are either used deliberately to filter the spectrum, or they also happen inadvertently - and can even provoke parasitic oscillation, loved by RF designers. The one circuit class where this is eagerly used is, you guessed it, the oscillator...

Note that if your circuit has multiple violations of the stability criterion across the spectrum, the strongest one will take over. This is probably because, as the divergent oscillation runs into the physical limits of the system (such as the power supply rails in an amplifier), the gain stages lose some gain. To the extent, that the overall gain stabilizes at just enough to sustain a constant oscillation at the most prominent "stability violating pole". Thus, at the lesser stability violations, the gain typically drops below the stability criterion. In a spectrum plot, apart from harmonics caused by the limitation, you can possibly also see minor poles, corresponding to the other open-loop gain/phase violations.

There are neighboring topics such as the "damping factor" or "resonator quality factor" (possibly different names for essentially the same phenomenon). In feedback-heavy systems, damping is related to the gain margin at integer multiples of 360*, even if the stability criterion is not violated...

• Thank you for your detailed answers :) "The one thing I know: for a feedback loop to diverge, at some point along the loop topology, at a frequency where the phase shift is an integer multiple of 360 degrees (including 0 degrees at DC), the gain must be > 1" Are you talking about the gain of the open loop system ? – Jess Aug 22 '20 at 9:20
• @Jess: heh this is where you have caught me with my pants down. You are reminding me to do my homework. Let me say this: closed loop gain is a ratio between output taken at some arbitrary point in the loop, vs. a "reference" inserted at some other arbitrary point. I.e. viewed from outside. Whereas: I'm speaking about gain >1 at some arbitrarily chosen point where you cut the loop (at least as a mental experiment, if this is difficult in the real world). Gain >1 between signal that departs into the loop and signal that arrives from the loop. So I'm speaking open loop gain here, am I ? – frr Aug 22 '20 at 19:53
• Thank you for your answer :) You probably talking about open loop gain ! – Jess Sep 7 '20 at 6:38